ion of chromatographic selectivity_1986

JOURNAL OF CHROMATOGRAPHY LIBRARY - volume 35

optimization of chromatographic selectivity a guide to method development

This Page Intentionally Left Blank

JOURNAL OF CHROMATOGRAPHY LIBRARY - volume 35

optimization of chromatographic selectivity a guide to method development

Peter J. Schoenmakers Philips Research Laboratories, P. 0. Box 80.000,5600 JA Eindhoven, The Netherlands

ELSEVl ER Amsterdam - Oxford - New York - Tokyo 1986

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgwhartstraat 25 P.O. Box 21 1 , l OOO AE Amsterdam, The Netherlands

Distributors for the United Statas and C . d a :

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ISBN 0-444-42681-7 (Vol. 35) ISBN 0444-41616-1 (Series)

0 Elsevier Science Publishers B.V.. 1986

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying. recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V./Science & Technology Division, P.O. Box 330, 1000 AH Amsterdam, The Netherlands.

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Printed in The Netherlands

CONTENTS

Journal of Chromatography Library (other volumes in the series)

PREFACE

ACKNOWLEDGEMENTS

CHAPTER I INTRODUCTION 1.1 CHROMATOGRAPHY 1.2 SEPARATION - THE COLUMN

1.2.1 Retention times and capacity factors 1.2.2 Distribution coefficients 1.2.3 Selectivity 1.2.4 The phase ratio

1.3 RESOLUTION 1.4 EFFICIENCY

1.5 OPTIMIZATION 1.6 PEAK CAPACITY 1.7 METHOD DEVELOPMENT

1.7.1 An organized approach 1.7.2 Method development in the laboratory

1.4.1 The plate number

REFERENCES

CHAPTER 2 SELECTION OF METHODS 2.1 CLASSIFICATION O F CHROMATOGRAPHIC TECHNIQUES 2.2 SELECTION O F CHROMATOGRAPHIC METHODS

2.3 CHARACTERIZATION A N D CLASSIFICATION METHODS 2.2:1 Expert systems

2.3.1 Polarity; Solubility parameters 2.3.2 The Rohrschneider characterization scheme 2.3.3 The Snyder solvent classification scheme 2.3.4 Summary

REFERENCES

CHAPTER 3 PARAMETERS AFFECTING SELECTIVITY 3.1 GAS CHROMATOGRAPHY

3.1.1 Gas-liquid chromatography (GLC) 3.1.2 Gas-solid chromatography (GSC) 3.1.3 The use of retention indices

3.2 LIQUID CHROMATOGRAPHY 3.2.1 Liquid-liquid chromatography (LLC) 3.2.2 Liquid bonded phase chromatography

IX

XI11

XV

1 1 2 2 4 5 5 7 8 9 9

14 15 15 18 19

20 20 21 23 24 24 27 31 35 36

37 37 37 43 45 47 52 56

V

3.2.2.1 Reversed phase chromatography (RPLC) 3.2.2.2 Polar bonded phases

3.2.3 Liquid-solid chromatography (LSC)

3.3.1 Ion-exchange chromatography (IEC) 3.3.2 Ion-pair chromatography (IPC)

3.4 SUPERCRITICAL FLUID CHROMATOGRAPHY (SFC) 3.5 CLASSIFICATION OF PARAMETERS

REFERENCES

3.3 SEPARATION OF IONS IN LC

3.5.1 Summary of parameters for selectivity optimization

CHAPTER 4 OPTIMIZATION CRITERIA 4.1 INTRODUCTION

4.1.1 Separation of two peaks 4.1.2 Separation in a chromatogram

4.2 ELEMENTAL CRITERIA 4.2.1 Peak-valley ratios 4.2.2 Fractional peak overlap 4.2.3 Separation factor 4.2.4 Discussion

4.3 CHROMATOGRAMS 4.3.1 Sum criteria 4.3.2 Product criteria 4.3.3 Minimum criteria 4.3.4 Other criteria 4.3.5 Summary

4.4 COMPOSITE CRITERIA 4.4.1 Number of peaks 4.4.2 Analysis time 4.4.3 Column independent time factors 4.4.4 Time corrected resolution products

4.5 RECOMMENDED CRITERIA FOR THE GENERAL CASE 4.6 SPECIFIC PROBLEMS

4.6.1 Limited number of peaks of interest 4.6.2 Programmed analysis 4.6.3 Dealing with solvent peaks

REFERENCES

CHAPTER 5 OPTIMIZATION PROCEDURES 5.1 INTRODUCTION

5.1.1 Univariate optimization 5.1.2 Local vs. global optima 5.1.3 Characteristics of optimization procedures 5.1.4 Definitions

5.2 SIMULTANEOUS METHODS WITHOUT SOLUTE RECOGNITION 5.3 THE SIMPLEX METHOD

56 74 76 82 82 93

101 105 108 113

116 116 116 117 119 119 123 125 127 131 131 134 140 144 145 146 146 148 151 153 158 158 158 165 167 169

170 170 173 176 177 179 179 183

VI

5.4 REDUCTION O F THE PARAMETER SPACE 5.4.1 Full factorial designs 5.4.2 Scouting techniques

5.5 INTERPRETIVE METHODS 5.5.1 Simultaneous interpretive methods 5.5.2 Iterative designs 5.5.3 Summary

5.6.1 Single channel detection 5.6.2 Dual-channel detection 5.6.3 Multichannel detection

5.6 PEAK ASSIGNMENT A N D RECOGNITION

5.7 SUMMARY .

REFERENCES

CHAPTER 6 PROGRAMMED ANALYSIS 6.1 THE APPLICATION O F PROGRAMMED ANALYSIS 6.2 PARAMETERS AFFECTING SELECTIVITY IN PROGRAMMED

ANALYSIS 6.2.1 Temperature programming in G C 6.2.2 Gradient elution in LC

6.3.1 Optimization of programmed temperature G C 6.3.1.1 Sequential methods 6.3.1.2 Interpretive methods 6.3.1.3 Discussion 6.3.1.4 Selectivity optimization 6.3.1.5 Summary

6.3.2 Optimization of programmed solvent LC 6.3.2.1 Simplex optimization 6.3.2.2 Systematic optimization of program parameters 6.3.2.3 Interpretive methods for selectivity optimization 6.3.2.4 Discussion 6.3.2.5 Summary

6.3 OPTIMIZATION O F PROGRAMMED ANALYSIS

REFERENCES

CHAPTER 7 SYSTEM OPTIMIZATION 7.1 INTRODUCTION 7.2 EFFICIENCY OPTIMIZATION

7.2.1 Open columns vs. packed columns 7.2.2 Gas chromatography (open columns) 7.2.3 Liquid chromatography (packed columns) 7.2.4 Summary

7.3 SENSITIVITY OPT1 M IZATION 7.4 INSTRUMENT OPTIMIZATION

7.4.1 Gas chromatography (open columns) 7.4.2 Liquid chromatography (packed columns)

188 188 191 199 200 220 233 233 236 239 24 1 245 250

253 253

257 258 260 266 269 269 273 275 276 276 276 277 279 284 290 292 294

296 296 299 299 300 302 305 305 310 314 316

VII

7.4.3 Summary REFERENCES

LIST OF SYMBOLS AND ABBREVIATIONS

AUTHOR INDEX

SUBJECT INDEX

318 318

321

329

333

VIII

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XI

The aim of this book is to be of help to those involved in the process of developing chromatographic methods. I have tried to write a text that is comprehensible and useful for both chromatographers with some experience, and for novices to the field with a background in science.

The fundamentals of chromatography are not covered in detail; the reader is referred to one of the introductory textbooks or courses on the subject.

Method development in chromatography today requires skills, knowledge and, above all, experience. Therefore, it is a particularly difficult field to enter for newcomers. I feel that an organized approach to method development, as presented in this book, may shift the emphasis from experience to knowledge. In this way, it may help newcomers to understand the process of method development. Also it may open the way for those already involved in method development to go beyond their personal experience and to apply different chromatographic techniques and optimization procedures.

The approach followed should be equally beneficial for chromatographers who do not develop their own methods but wish to improve (optimize) existing ones.

Procedures for developing and optimizing chromatographic separations have attracted increasing attention not only from researchers, but also from instrument manufacturers. Already, several of the procedures described are commercially available. The approach followed does not include describing existing methods. One reason for not doing this is that the elements that constitute a complete optimization package can be discussed and understood separately. Therefore, an existing method may be good in one respect, but poor in another. A second reason is that whereas complete optimization packages may be expected to change a great deal in the next few years, I expect this to be much less true for the underlying principles; so I would like to think that the material presented here will still be of value in the years to come.

This book is intended to be a critical assessment of procedures for method development and selectivity optimization. It is not intended to be a survey of available information, therefore references to the literature are included only when they are relevant to the text. Consequently, a number of references have been omitted. No doubt, some may also have been overlooked.

I am very grateful to a number of people who have reviewed the manuscript of this book (or parts of it) at various stages during the preparation. Together, they are responsible for an immense number of corrections, improvements and clarifications.

My gratitude to: Hugo Billiet (Delft, The Netherlands), Pieter de Bokx (Eindhoven, The Netherlands), Cherie Goewie (Utrecht, The Netherlands), Ernst Lankmayr (Graz, Austria), Pamela Naish (Cambridge, Great Britain), Charles Perkins (Cambridge, Great Britain), Frank Verhoeven (Eindhoven, The Netherlands),

XI11

and especially my teacher for many years

from whom I am still learning, and my wife,

for correcting the final manuscript and one or two other things.

Leo de Galan (Delft, The Netherlands),

Dana Conron,

Eindhoven, February 1986 Peter Schoenmakers

XIV

ACKNOWLEDGEMENTS

I acknowledge permission to reprint previously published material from the following publishers:

American Chemical Society, Washington D.C., U.S.A. Figures 3.3, 3.4, 3.20, 5.22, 5.36, 5.38.

Marcel Dekker Inc, New York, NY, U.S.A. Figure 3.18.

Elsevier Science Publishers, Amsterdam, The Nederlands. Figures3.2, 3.5, 3.6, 3.13, 3.14,3.15, 3.19, 3.22,3.25, 3.29, 3.30, 3.31, 3.33, 5.1, 5.2, 5.7,

5.8,5.9,5.11,5.13,5.14,5.15,5.16,5.17,5.20,5.21,5.23,5.24,5.25,5.26,5.27,5.37, 6.8, 6.10, 6.11, 6.12, 6.14, 6.15, 6.16.

Preston Publications Inc., Niles, IL, U.S.A. Figure 2.3.

Royal Chemical Society, London, Great Britain. Figure 3.1.

The following authors are acknowledged for permission to reprint material originally published in Chrornatographia by Friedr.Vieweg und Sohn Verlagsgesellschaft mbH, Wiesbaden, West Germany:

L.de Galan (Delft, The Netherlands)

CXiuiochon (Washington D.C., U.S.A.)

J.F.K.Huber (Vienna, Austria)

Figures 3.7, 3.8, 3.16, 3.26, 5.5, 5.29, 5.30, 5.31, 5.32, 5.33, 5.34.

Figures 5.18, 5.19.

Figure 3.10.

Figure 3.20, 3.23 and 3.24 were adapted from original drawings of J.C.Kraak (Amsterdam, The Netherlands).

Figure 3.12 was adapted from an original provided by H.M.van den Bogaert (Eindho- ven, The Netherlands).

Figures 4.4 and 4.5 were taken from unpublished work of A.C.J.H.Drouen (Delft, The Netherlands).

xv

CHAPTER I

- Mobile Phase ~ ~ l , ~ ~ ~ ~

INTRODUCTION

- -

Data

Handling i r q k Detection - Sample

Introduction - - --

In this chapter the concepts of chromatography, as far as they are relevant to the context of this book, will be outlined.

The chromatographic system, the column, and the basic fundamentals of chromatographic separations will be briefly discussed.

The extent of separation can be quantified in terms of the resolution obtained between two consecutive chromatographic peaks. This resolution can be expressed in terms of three elemental characteristics of chromatographic separation: retention, selectivity and efficiency. The influence of each of these three factors on resolution will be discussed.

1.1 CHROMATOGRAPHY

Chromatography can be defined as the separation of molecules by differential migration*, i.e. separation is achieved on the basis of different speeds of transportation for different molecules.

In this book column chromatography will be discussed almost exclusively, although occasional reference will be made to thin layer chromatography (TLC), the fundamentals of which are not different from those of column chromatography.

Furthermore, this treatment is limited to those forms of chromatography which involve two phases (a stationary and a mobile phase) and in which the necessary differences in speed of migration are caused by differences in chemical interactions between the molecules of the different sample components (“solutes”) and the two chromatographic phases, as well as between the solute molecules themselves. Interaction chromatography is sometimes used as a term to describe such systems.

Separations that are achieved on the basis of the size of the molecules (e.g. size exclusion chromatography) are not dealt with in this book. Such separations are not selective, and hence there is no selectivity to be optimized.

I

Figure 1 .l: Schematic representation of a chromatograph.

A schematic representation of a chromatograph is given in figure 1 . I . This figure applies to all kinds of column chromatography, but the various boxes will have different contents for different chromatographic techniques, notably for gas chromatograhy (GC) and for liquid chromatography (LC) (for definitions see section 2.1).

* In this broad definition some techniques which are not usually considered as chromatography are included, for example field flow fractionation (FFF) techniques and electrophoresis. However, isotachophoresis is not included.

I

For GC the mobile phase delivery box could consist of a gas cylinder, a reducing valve and a flow controller. For LC a high pressure pump will be required. In this book the instrumentation required for chromatography will not be discussed. Only where the equipment used is relevant to the cause of optimization of selectivity will it feature in the present text (e.g. sections 5.6 and 7.4).

The rest of this book will focus on the thick box in the centre of figure 1.1, identified as separation.

1.2 SEPARATION - THE COLUMN

The chromatograph is built around the column, in which the actual separation takes place. The column accommodates the two chromatographic phases: the stationary phase, which remains in the column, and the mobile phase, which is transported through it. Separation is achieved because different sample components (solutes) show different distributions over the two phases. A solute, having such a high affinity towards the stationary phase that it resides in this phase exclusively, will stay in the column indefinitely. A solute, that does not enter the stationary phase at all, will be transported through the column at the same speed at which the mobile phase is transported. In chromatographic terms, the latter is called an “unretained solute.

If a column is packed with porous particles, then an unretained solute is assumed to be swept through the entire volume of the column that is occupied by the mobile phase, either outside the particles or in the pores. A solute that does not enter any of the pores is called a (completely) “excluded” solute. Throughout the remainder of this book we will assume that the solutes will not be (partially or completely) excluded from the pores.

1.2.1 Retention times and capacity factors

The above discussion can be quantified as follows. A solute i distributes itself over the two phases, resulting in a total quantity qiem to be present in the mobile phase (m), and a quantity qi.s in the stationary phase (s). The solute molecules which find themselves in the mobile phase will be transported through the column at the same speed (u) as the molecules of the mobile phase. However, this is only a fraction of all the solute molecules, so the average speed for all solute molecules will be only a fraction of u given by

9i .m v. = u , ’ qi.m+ 4i.s

where vi is the migration speed, the average speed at which the solute band travels through the column. The time tR,ineeded for the solute band to elute from the column is determined by the column length and the average migration speed

tR,i = L / v i .

tR,i is called the retention time of the solute. Similarly, the time which a mobile phase molecule will spend in the column is

to = L / u

2

to (frequently also denoted by t,,,) is known under different names: the hold-up time, mobile phase time, or unretained time. The combination of eqns.( 1 .l), (I .2) and (1.3) yields

By definition, the capacity factor (k,) of the solute i is

and hence

c , ~ = (1 + k,) t o . (1.6)

Eqn.(l.6) is the fundamental equation for retention in chromatography. Throughout this book, extensive use will be made of the capacity factor as a convenient means to describe retention. A major advantage of the use of k for this purpose is the fact that it is a dimensionless quantity. It follows from eqn.( 1.6) that

I

0 t lmin I -

Pigure 1 .L: Schematic chromatogram illustrating the meaning ot various retention parameters

3

where fRVi is the net retention time of the solute, i.e. the (average) time which a solute molecule spends in the stationary phase.

Eqm(l.7) also shows that the capacity factor k can easily be determined from the chromatogram. This is illustrated in figure 1.2. If a signal at t = t , is obtained in the chromatogram, then the quantities to, t,, and tkcan all bemeasured directly. The capacity factor can either be calculated from eqn.(l.7), or determined from a calibration line as shown in figure 1.2. Two points can be used to construct the line, for instance k= 0 at the occurance of the unretained peak and (a fictive point) where k = - 1 at the time of injection ( t = 0). The capacity factor of any peak in the chromatogram can be determined very easily in this way. However, to avoid inaccuracies if high k values occur, the calibration line may be constructed by using eqn.(l.7) once for a point at a high value of k.

1.2.2 Distribution coefficients

The quantity q of the solute i in one of the phases is the product of the average concentration (7) of i in that phase (where the average is taken along the length of the column) and the volume of that phase. Hence, for the capacity factor (eqn.l.5) we find

The ratio Zi,s/Ci,m is a constant if the distribution isotherm*, i.e. a plot of q s vs. qm, is linear. This is usually the case at high dilutions. Preferably, all (analytical) chromatography is performed in this linear region. The distribution coefficient in terms of concentrations (KJ may be defined as

Kc,i = c i , J q m . (1.9)

Since K, may be independent of the solute concentration, but will always be a function of the temperature (and pressure), the term distribution coefficient is to be preferred to the alternatives: distribution constant and equilibrium constant. If the distribution isotherm is linear, K , will also equal the ratio of average concentrations in eqn.(l.8), and hence

ki = &. Vs/ V , (1.10)

Eqn.(l .lo) relates retention in chromatography (k) to a thermodynamic parameter (KJ. The so-called phase ratio Vs/ Vm is a characteristic of the column**.

* If the stationary phase is asolid surface, then the term adsorption isotherm is more commonly used. ** However, in some kinds of chromatography (e.g. reversed phase liquid chromatography, see section 3.2) the phase ratio may vary with variations in the mobile phase composition.

4

1.2.3 Selectivity

It was stated at the beginning of this section that solutes are separated in a chromatographic column on the basis of differences in their speed of migration through the column. We can define the relative retention (aj,) of two peaks as

a.. = t i J / (1.11) I'

In this equation i represents the first eluting peak of a peak pair and j the last eluting peak. Hence, by definition a is always larger than unity. Sometimes a is called the separation factor, which is somewhat unfortunate terminology because separation is influenced by other factors than just a (see section 1.3)*. a is the chromatographic parameter that is most directly related to the selectivity of the phase system. In this book, therefore, the word selectivity will often be associated with a. Using eqns.(l.7) and (1.10) we can write two other equations for aj,:

a.. = kj / ki (1.12) J'

and

a,. = K . / K,i J' C J

(1.13)

Eqn(l.12) is very useful in practice, because it expresses a directly in terms of the capacity factors. We will make frequent use of this equation throughout this book.

Eqn.(l.l3) relates a to the distribution coefficients. Since no phase ratio term appears in eqn.( 1.1 3), it is clear that the selectivity (a) of the chromatographic system is determined only by thermodynamic factors.

The relative retention will be affected only by those factors which affect the distribution coefficients, i.e. .- the solute .- the mobile and the stationary phase (together constituting the phase system) .- the temperature .- the pressure.

The effect of the pressure on a and on k is usually negligible. Only in some particular cases (e.g. in supercritical fluid chromatography, SFC; see section 3.4) will it be a relevant parameter.

1.2.4 The phase ratio

The phase ratio V,/ V,,, occurs in eqn.(l .lo) as one of the factors that determine retention ( :k) in chromatography. We can influence the phase ratio by varying one or more of several parameters:

* In chapter 4 we will define a separation factor S which provides a more realistic measure of the contribution of chromatographic retention to separation.

5

- The type of column In particular, we can choose between open (capillary) columns and packed columns*.

A wall coated open tubular (WCOT) column has a much smaller phase ratio than a packed column, due to the small surface area of the wall. - The column diameter

If open columns are used, then the phase ratio will vary with the column diameter (provided that the film thickness is kept constant). The cross-sectional area of the column (and hence the mobile phase volume) is proportional to the square of the column diameter, while the wall area is proportional to the diameter itself. Hence, the phase ratio is inversely proportional to the column diameter. - The sugace area

The area available for the stationary phase will directly affect the phase ratio. If a solid material is used as the stationary phase in a packed column, if a liquid phase is deposited on a solid adsorbent with a constant film thickness, or if chemically bonded phases are employed, the phase ratio (through V,) will be directly proportional to the available surface area. The surface area of an adsorbent is usually given per unit weight (i.e. the specific surface area in m2/g). However, it should be noted that the relevant quantity is the surface area per unit volume (m2/ml) in the packed column. - The column porosity

This is the fraction of the column volume that remains available for the mobile phase after packing. There are two contributions to the total column porosity. One part of the volume available to the mobile phase is in between the particles (interparticle space). For uniform, spherical particles this is about 40% of the column volume. The second contribution is due to the very porous structure of materials with large specific surface areas. This makes a significant part of the intraparticle volume available to the mobile phase (usually 20 to 30% of the column volume). - Thefilm thickness of a liquid stationary phase

Clearly, with all other factors constant, V, will increase linearly with the film thickness (this is also true for the phase ratio V,/ V,, as long as V, < V,,,). For solid adsorbents this effect does not occur. For chemically bonded phases the (mono-)layer thickness is not as well defined as the film thickness of a bulk liquid, and neither is the description of variation in the layer thickness as straightforward as it is for liquids (see section 3.2.2).

In general, the effective volume of a stationary phase (Vb can be increased in a predictable manner by increasing the surface area, but only for liquids can the same be said for increasing the film thickness.

Obviously, there are many ways to influence the capacity factors. However, the effects described above are predictable (see section 4.2.3) and in a sense trivial. It is worth noticing at this point.that certain parameters do not at all affect the capacity factor and therefore do not at all affect chromatographic selectivity. These parameters include column length, flow rate and the diameter of packed columns. This renders these parameters irrelevant to the selectivity optimization process. In some cases they may be considered as parameters

* For gas chromatography (and for supercritical fluid chromatography) there is a real choice. Open columns may theoretically be used in liquid chromatography as well, but their diameter should then be so small that they do not yet form a realistic alternative to packed columns in practice.

6

during the course of the optimization in conjunction with other parameters, which do affect the selectivity. For example, a decreased temperature in GC may lead to an increased selectivity and hence allow an increased flowrate to compensate for the increase in analysis time. In most cases, however, it is sensible to consider the parameters that do not affect the selectivity separately after completion of the optimization process. Some comments on how to choose the values of these parameters will be made at the end of this book (chapter 7).

1.3 RESOLUTION

We have seen that the selectivity in chromatography can be related to the relative retention of two solutes. However, this parameter does not describe the actual separation between two chromatographic peaks. There are two factors which determine whether or not two peaks are completely resolved, as is illustrated in figure 1.3. The relevant parameters are the distance between the peaks and their width. The distance can be expressed as the difference in retention times (AtR), while the peak width at the peak base (usually determined by drawing tangent lines along the slopes of the peaks) can be denoted by w.

Figure 1.3: Two chromatographic peaks illustrating the definition of resolution (eqn.l.14).

The resolution (R, ) between two peaks is now defined as

R, = 2AtR / (w, + w2) . (1.14)

Hence, the resolution is equal to unity if the distance between two peaks equals the average

7

peak width. It should be noted that R, is a dimensionless quantity and that At, and w should be expressed in the same units (e.g. seconds or mm on a recorder trace).

1.4 EFFICIENCY

Ideally, chromatographic peaks are of Gaussian shape. In practice, because of finite sample concentrations, inhomogeneities in the stationary phase, dead volumes in the system and various other factors, they usually are not. In LC peaks tend to be less symmetrical than in GC.

Nevertheless, to a first approximation the Gaussian peakshape can be assumed for a chromatographic peak. Iffit) is the signal (detector response) as a function of time and t, is the retention time of the peak, then a Gaussian peak can be described by

t - t t , 2 fit) = hexp -1/z (T)

(1.15)

In eqm(l.15) h is the height at the peak maximum, A is the peak area, and a i s the standard deviation of the peak (in time units), a measure of its width.

Some characteristics of a Gaussian peak are summarized in figure 1.4. It can be seen from this figure that the variance of a genuinely Gaussian peak can be determined by measuring the peak width at some fixed fraction of the peak height. The peak width at the

1

t h/hrnm

0.607

0.5

0.135

Figure 1 . 4 Some characteristics of a Gaussian peak.

8

base is usually very hard to measure accurately, so that a measurement of the peak width at half height ( w , / ~ ) is usually considered to be a more practical proposition. However, most measurements of peak width suffer from lack of precision and accuracy. It is beyond the scope of this book to discuss other ways for characterizing the peak shape, peak width and peak symmetry in chromatography.

1.4.1 The plate number

The efficiency of a chromatographic system (i.e. the column plus the instrument) is usually expressed in terms of the number of theoretical plates (4, which may be defined as follows:

N = (tR/o)2. (1.16)

Here the peak shape is assumed to be Gaussian, with 0 being the standard deviation. It follows from figure 1.4 that eqn.(l.l6) can also be written as

N = 5.54 ( t , /w,,2)2 (1.1 6a)

or

N z 16(t,/w)2 (1.16b)

where wl I2 and w are the peak width at half height and the peak width at the baseline, respectively. Another convenient equation can be derived for instruments that provide information on both the peak height (h) and the peak area (A):

N = 2 n ( t , h / A ) * . (1.17)

In applying eqn.( 1.1 7) one should be aware of the units involved. N is dimensionless, so that if t , is expressed in seconds and h in mV, A should be expressed in mV.s.

From the number of plates in the column, the “height equivalent of a theoretical plate” (HETP), usually abbreviated to “plate height” (H), can easily be calculated:

H = L / N (1.18)

where L is the length of the column. The plate height will vary with the flow rate (u) of the mobile phase through the column.

This variation can be characterized by an H vs. u curve. Such a curve shows a minimum plate height at some optimum value of u. Again, this will not be discussed any further in this book and the reader is referred to one of many general textbooks on chromatography.

1.5 OPTIMIZATION

Eqns.( 1.6), (1.12), (1.14) and (1.1 6) can now be combined to yield the key equation for

9

the optimization of resolution in chromatography. If we combine eqm(l.14) and (1.1 6) we find

- tR,2- ‘R,I . - vxi R, = - (Ol + ‘R.1 + ‘R.2 2

or, in terms of net retention times (eqn.l.7) fk.2- tk.1 . - fi

R, = ‘R.1 + tk,2 + 2t0

which upon division of all retention times by to turns into

R, = k2-kl -5 k , + k 2 + 2 2

(1.19)

(1.19a)

(1.20)

Eqn.( 1.20) is in itself useful and it will be used later in the book (section 4.2.3). However, a more generally useful equation can be found by some manipulation:

h - k , k,+k2 fi k,+k2 k , + k , + 2 2

R , = - . .- (1.21)

and using eqn.( 1.12) to define aand introducing the average capacity factor E = (k, + k2)/2 for the two peaks we find

a-I E vxi a+l l + E 2 .

R, = -.-.-

’ r

O 1 2 3 L 5 a-

’ ’ 0 10,000

0’ N-

5,000

(1.22)

Figure 1.5: Influence of (a) the relative retention (a), (b) the (average) capacity factor (x) and (c) the number of theoretical plates (N) on t_he resolution ( R J according to eqn.(1.22). In each case the two other parameters are kept constant. k and N are assumed not to equal zero, and a not to equal one.

10

Eqn.(1.22) is one of several forms of the same general equation. It is preferred here because it is symmetrical (towards the two peaks) and exact.

Eqn.( 1.22) shows that there are three factors which together determine the chromatographic resolution. The influence of each of these factors can be discussed independently of the two others, as is shown in figure 1.5. In this figure R, is plotted against a, x and N (from top to bottom). In each of the three plots the two other parameters are kept constant. Figure 1.5a shows the variation of the factor (a- l ) / (a+ 1) with a. Note that by definition a> 1. It is seen from the figure that this factor, and therefore R , increases more or less regularly when a increases. If a= 1, then R, equals zero. The value of R, will slowly increase with increasing a. Even when a= 2 only one third of the maximum value for the a factor of 1 has been reached. In thermodynamic terms this is already a very large selectivity, since the two capacity factors (i.e. the distribution coefficients) should differ by a factor of two (eqns.l.12 and 1.13). Hence, to increase the resolution a is of extreme importance when its value is close to one, but even at higher values resolution will benifit substantially from an increase in the thermodynamic selectivity.

From figure 1.5b it appears that the variation of R, with x is not dissimilar to the dependence on a. If z= 0, then R, will always equal zero (no matter how high the value of a may be). The factor z/(l +x) will increase regularly with increasing z. However, at k = 1 already half of the limiting value of 1 has been reached. When %=9 this is 90%, so that very little can be gained in terms of resolution by increasing the k value further. Moreover, higher values imply longer analysis times (eqn.l.6) and are therefore less attractive.

The optimum z value can be found if we combine eqa(1.22) with eqn.(l.l8) and eqn.( 1.3):

-

so that with eqn.(l.6) b e average retention time for a pair of peaks ( T R ) becomes

t R = to( l+k) = R:

(1.23)

(1.24)

a can be treated as independent from the average z value and if we assume as a first approximation that H / u is also independent of x, then the optimum z value would be such as to minimize the factor.(l +z)3/p. This function is plotted in figure 1.6 and it can be seen from this figure that a minimum value occurs at %=2*. Since eqn(1.24) shows a shallow minimum with respect to z (see figure 1.6), a range of 1 < k < 5 is usually considered as the optimum for the separation of mixtures of two components [103]. This range is indicated in the figure. However, when more than two peaks are present, the range of capacity factors will also be determined by the peak capacity of the chromatogram (section 1.6).

* In a more detailed analysis Snyder [loll found a? optimum around 3 <z< 5 for packed columns. Guiochon [lo21 found an optimum value around k = 2 in a detailed study of capillary columns,

11

Figure 1 .5c illustrates the variation of the resolution with the plate number N. The square root function initially rises steeply from a value of zero at N = 0. The rate of increase in I6 becomes quickly less with increasing N. At N= 2500 I6 equals 50. A value of 100 is reached for N= lO,OOO, and 200 for N=40,000. These three values of N may be seen as typical for three different kinds of chromatography: packed column GC, packed column LC and capillary column GC. Within the constraints of any of these techniques, another factor of 2 in resolution by increasing N by a factor of 4 will be hard to achieve.

Improving resolution by increasing the plate number will lead to longer analysis times. If all other conditions (column or particle diameter, flowrate etc.) remain constant, then the plate count will be proportional to the column length (eqn.l.18 with H constant), and so will be the analysis time (eqn. 1.2). Hence, R , can be increased by a factor of two at the expense of a factor of four in analysis time. Usually, the situation is even worse. Merely increasing the column length to increase N will lead to an increased pressure drop over the column. Once the pressure drop becomes too high, other parameters will have to be varied besides the column length, for example the particle diameter in packed columns may be increased. This results in a decrease in the pressure drop, but also in an increase in H, leading to a further increase in the required column length and hence to a further increase in analysis time (eqn.l.2). It can be shown that in such a pressure-limited situation a twofold increase in R, can be achieved from a fourfold increase in N at the cost of a sixteenfold increase in analysis time (see also sections 4.4.3 and 7.2.3).

We conclude from the above that the efficiency (flm in eqm(l.22) is the least rewarding factor for increasing the resolution.

First, the (average) capacity factor (z) should be brought into the optimum range, for instance by varying the temperature (GC) or the mobile phase composition (LC).

Second, The selectivity (a) may be improved. This is extremely rewarding when ais close to unity, but even at much higher values considerable improvement may be obtained. a

I

5 - 10 k-

Figure 1.6: Variation of the factcr (1 +E)3/p as a function of z. The minimum observed at E = 2 represents theoptimum value for &in terms ofminimum analysis time(eqn.l.24). Thearrow indicates the optimum working range (1 < k < 5) for the separation of two peaks.

12

a =11

N = 1600 ._ k=O

Figure 1.7: Illustration of the effects of the capacity factor, the selectivity and the plate number on the appearance of chromatographic peaks. For explanation see text. - Middlechromatogram: 5 pairs of peaks, all with a= 1.1 and N = 1600. k values increase from left to right (k=0,1, 3, 5 and 8). Top left: k = 3 , a= l.O_S, N = 1600. Bottom left: k = 3 , a= 1.25, N = 1600. Top right: k = 8 , a = l . l , N=500. Bottom right: k = 8 , a = l . l , N=5000.

may be varied by changing the nature or the composition of the stationary phase (GC or L,C) or the mobile phase (LC)*.

Third, the efficiency (N) may be adapted to meet the requirements set by the values of k and a. The value of N is determined by the column characteristics and the flow rate. While increasing the plate count may require great sacrifices in terms of analysis time, the reverse is also true. If the values of k and a allow the use of a column with a low N value to achieve the separation, then the analysis time may be reduced dramatically.

and N using some actual chromatographic peaks. The (horizontal) chromatogram in the middle contains a series

-

Figure 1.7 illustrates the dependence of R, on a,

* Increasing a is generally advantageous for the separation of two peaks. If a series of solutes needs to be separated, then an increase in one of the a values may have an adverse effect on the required analysis time. The problem of how to distribute the peaks over the chromatogram will be discussed extensively in chapter 4.

13

of peak pairs, for which a (1.1) and N (1,600) are constant. It is seen how the resolution increases with increasing capacity factors. When z = O the two peaks show complete overlap. The resolution is then found to increase quickly from the first peak (actually a pair of overlapping peaks) at x= 0 to the third pair around z= 3. R, may still be increased from there by increasing z further. However, the improvement in going from x= 3 to z= 8 is not dramatic.

The peak pair at z= 3 is also shown in figure 1.7 for a= 1.05 (top) and a= 1.25 (bottom). When a= 1.05 only one peak can be observed (R, = 0.37), while a= 1.25 yields an excellent separation ( R , = 1.67).

The peak pair at I= 8 is shown in figure 1.7 for three different values of N. N= 500 (top right), N = 1600 (middle) and N = 5000 (bottom right). The improvement in R, for each factor of about 3 in N is clear from the figure.

1.6 PEAK CAPACITY

The optimal working range in terms of capacity factors will not only be determined by the considerations of analysis time given in the previous section, but also by the number of peaks present in the chromatogram. The theoretical peak capacity (n,,) of a chromatogram can be found from [lo41

(1.25)

where k, and k, are the capacity factors of the first and the last peak, respectively, and R, is the required resolution between each pair of successive peaks.

0 5 log N -

Figure 1.8: Variation of the theoretical peak capacity (n& of a chromatogram withihe plate count (N). for three ranges of capacity factors: (a) 0.2< k < 2 , (b) 1 > k< 10 and (c) 0.5< k<20.

14

In figure 1.8 the theoretical peak capacities are given for some typical ranges of capacity factors as a function of the number of plates. The resolution values were taken to be equal to one. In the logarithmic plot of figure 1.8 a straight line is obtained with a slope of 1/2. A typical packed column with 5000 plates turns out to yield a peak capacity between 17 (case a) and about 50 (case c). An open column with 200,000 plates may accommodate 100 peaks with capacity factors between 0.2 and 2 (case a).

Davis and Giddings [lo51 have argued that the theoretical peak capacity is usually not even approached. Instead, they conclude that in order to provide a 90% probability for a compound of interest to appear as a pure peak in the chromatogram, the available peak capacity should exceed the theoretically required value (eqn.1.25) by a factor of 20. If we consider the same range of capacity factors this results in an excess plate count of about a factor of 400.

In the treatment of Davis and Giddings the peaks are supposed to be randomly distributed over the chromatogram. Optimization of selectivity can be seen as the process to fight statistics and to approach the theoretical peak capacity as closely as possible.

1.7 METHOD DEVELOPMENT

In this section we will discuss a general approach to method development in chromatography. Developing chromatographic methods is a difficult task. Unlike the situation for most other methods of instrumental analysis, chemistry is at the heart of successful method development in chromatography. Good, reliable instrumentation is a prerequisite for performing sound chromatographic separations. However, the best possible instrumentation will not provide the required separation without a sensible choice of the phase system (stationary and mobile phase) and the operating conditions. Almost this entire book (chapters 2 to 6) is intended to assist in making that sensible choice.

Usually, those who develop chromatographic methods rely on knowledge, experience and skill. This makes the field an especially hard one for newcomers to enter. The overview in this section provides an organized approach to method development, which is intended to introduce (relative) newcomers to the remainder of the book. Therefore, this section will contain many references to subsequent chapters and sections.

An organized approach to method development may also be beneficial to the more experienced chromatographer. It may provide an incentive to go beyond the limits of personal experience and knowledge. Subsequent chapters in this book are intended to provide sufficient information to allow the development and optimization of separation methods in a variety of different chromatographic techniques.

1.7.1 An organized approach

An organized approach to method development may be based on a series of actions as described below.

1. Getting to know the sample

It will seldom be necessary to analyse samples about which nothing is known, except

15

that they are green, liquid, and smelling. Usually, much more is known about the sample and it is important to acquire as much information as possible about the sample on beforehand. This includes obvious questions, such as which are the (expected) components in the sample, which components are of interest, as well as less direct questions, involving such characteristics as: - volatility of the sample (presence of a non-volatile fraction) - possible solvents - presence of acids and/or bases - history of the sample (e.g. details about the synthesis). Answers to such questions are essential in order to choose the most appropriate chromatographic technique.

There is a second kind of sample, which is hardly ever found in a real-life method development laboratory. That is the sample one knows everything about. Even if this might seem to be the case, it is good to be aware of the possibility that unexpected components may be present in the sample. These may include contaminants, side products, degradation products, metabolites, etc..

2. Reading up

It is always advisable to search the literature for possible solutions to a paPticular problem. The task of re-inventing the wheel should not be thought of lightly. Even a brief scan of the literature may reveal much about the problem. In most cases this will provide clues as to which chromatographic technique might be used, as well as a starting point with regard to a possible phase system. Because separations from the literature can often not be reproduced exactly and because of differences between different samples, this starting point is usually not the final stage in a method development process and a further optimization is usually required.

3. Method selection

If the literature does not suggest a particular chromatographic technique to be applied, then a method should be selected on the basis of the information we have about the sample and some rules of thumb. In chapter 2 we will discuss methods of going about the selection of a chromatographic technique. This chapter may help us to select a particular chromatographic technique (for instance gas-liquid chromatography or reversed phase chromatography). Chapter 2 also describes some general methods for the characterization and classification of phase systems and solvents, but it does not tell us how to realize a separation once an appropriate technique has been selected.

4. Recording a chromatogram

After a particular chromatographic technique has been selected a chromatogram should be recorded. This is not always straightforward, since recording a chromatogram requires that the components in the sample show reasonable retention times (capacity factors). Often, therefore, we have to adjust the chromatographic system in order to get all sample components to appear as a peak (but not necessarily a well-resolved one) in the

16

chromatogram. This can be done by adapting what we may call the primary parameters of the separation. The primary parameters for different chromatographic techniques will be discussed in chapter 3. That chapter is divided into different sections for different techniques, so that once a particular technique has been selected (using information from the literature or from chapter 2) the required knowledge about the effects of different parameters on retention and selectivity is readily available. The most relevant parameters for each technique are summarized at the end of the chapter (table 3.10) for quick reference.

In some cases we may speed up the selection of appropriate primary parameters with the help of programmed analysis, i.e. temperature programming in GC or solvent programming in LC. Another useful scouting technique may be thin layer chromatography (TLC). Possibilities for establishing the appropriate values of the primary parameters will be discussed in section 5.4.

Ideally, the eventual chromatogram is arrived at under constant conditions (i.e. isothermal in GC, isocratic in LC). Nevertheless, it may be impossibble to achieve a signal (peak) for each component in the sample under constant conditions. In that case it may be necessary to use programmed analysis methods, in which one of the (primary) parameters is varied (“programmed) during the separation. Programmed analysis will be discussed in chapter 6.

5 . Optimization of the separation

Once we have recorded a chromatogram, the separation may not be satisfactory. Indeed, the chances are that not all peaks will be well-resolved. In that case, we will have to optimize the separation. Eqm(1.22) is the key to this optimization step. The three factors which together determine the value of R , may to some extent be considered independently. In fact, in recording the chromatogram we assumed that the retention (capacity factor) may be optimized independently of the two other factors in the resolution equation. Recording the chromatogram is equivalent to adapting the capacity factors such that they are neither too high nor too low. The range of 1 < x < 10 may be considered as optimal (see section lS), but when the sample contains very few components or when very many plates are available, a somewhat lower range (e.g. 1 to 5 or 0.5 to 2) may be preferred (see section 1.6).

Another factor that contributes to R, is the plate count N. However, we have seen in section 1.5 that optimization through an increase in N is expensive, not only in terms of equipment and columns, but also in terms of analysis time. Therefore, as long as the shape of the peaks and the plate height (length of the column divided by N) are satisfactory, we should not rely on the number of plates for optimization, unless as a last resort. Methods which may be used to optimize the chromatographic system with respect to the required number of plates will be described in chapter 7.

The third factor in the resolution equation is the most vital one for the optimization of the separation. Since this factor involves the selectivity (a) we may talk about “Selectivity optimization”. We have seen in section 1.5 that R, is very sensitive even to small changes in a if the components are difficult to separate (i.e. a close to 1).

Although the primary parameters will usuallly affect the selectivity to some extent, we may not use them to vary a, because they will affect the capacity factors to a much greater

17

extent and push them out of the optimum range. Therefore, we usually need to look at different parameters for selectivity optimization.

The parameters that may be used for optimizing the selectivity in various chromatographic techniques may be referred to as secondary parameters. They will be discussed in chapter 3 and summarized in table 3.10.

Method Development Advanced. flexible instrumentation Various injectors, detectors Sophisticated data handling Multichannel detectors

6. Optimization of the chromatographic system

Level 1

Once we have realized optimum capacity factors and optimized the selectivity, we can use the resolution equation to calculate the number of plates that is required to achieve baseline resolution (R,=1.5). The required number of plates will to a large extent determine the kind of column and instrumentation needed to perform the separation. This will be briefly discussed in chapter 7.

1.7.2 Method development in the laboratory

After we have discussed the development of chromatographic methods, this final section of chapter 1 will discuss the role of method development in the modem laboratory. We have seen that in developing methods we should aim at simple, rapid analyses. Program- med analysis in a routine situation should be avoided whenever possible and a high degree of automation should be feasible.

To achieve this goal, method developers should ideally find themselves in the opposite situation, being equipped with flexible, advanced instrumentation, including a variety of possible injectors and detectors, facilities for temperature or solvent programming, etc.. Multichannel detectors are very useful, as they may be of assistance in recognizing the different sample components when they move about in the chromatogram during the selectivity optimization process (section 5.6).

Figure 1.9: 1llustration.of three instrument levels in a modern laboratory handling a large number of chromatographic analyses.

18

Figure 1.9 shows a scheme of a large laboratory in which three levels of instrumentation may be distinguished. The first level is the method development level, where the instruments are equipped with many features and options. This provides the method developer with a high degree of flexibility. At level two analyses are performed which require the use of special instrumentation, such as programming or column-switching options, sophisticated detection systems, etc.. The third level is that of simple dedicated equipment, which is ideally used for all routine analysis.

It should be the aim of the method developer to try and develop methods which allow the use of level 3 instruments instead of level 2 ones as much as possible. If this can be achieved, the money spent on sophisticated instrumentation for method development will not be wasted.

REFERENCES

101. L.R.Snyder, J.Chromatogr.Sci. 10 (1972) 369. 102. G.Guiochon, Adv.Chromatogr. 8 (1969) 179. 103. L.R.Snyder and J.J.Kirkland, Introduction to Modern Liquid Chromatography, 2nd

Edition, Wiley, New York, 1979, p.54. 104. J.C.Giddings, AnaLChern. 39 (3967) 1027. 105. J.M.Davis and J.C.Giddings, AnaLChern. 55 (1983) 418.

19

CHAPTER 2

SELECTION OF METHODS In this chapter we will first classify the different chromatographic techniques on the

basis of the mobile and stationary phases used (section 2.1). This section introduces some of the terminology and abbreviations used in the remainder of the book.

Section 2.2 provides a rough scheme for the selection of an appropriate chromatographic technique. This scheme can be used if only a few physical characteristics of the sample are available and if the literature has not provided a starting point for further optimization.

Section 2.2.1 briefly addresses the possibility to incorporate a scheme for the selection of chromatographic methods in a computer program, a so-called expert system. This is a relatively recent proposition, and progress may be expected in this area.

Section 2.3 describes various methods for the characterization and classification of mobile and stationary phases for chromatography. In section 2.3.1 the solubility parameter is introduced as a quantitative definition of the word "polarity". Section 2.3.2 describes the characterization of GC stationary phases according to Rohrschneider and section 2.3.3 the classification of LC solvents according to Snyder. The applicability of the different methods is summarized in section 2.3.4.

2.1 CLASSIFICATION OF CHROMATOGRAPHIC TECHNIQUES

The most common way to classify the different chromatographic techniques is by the nature of the two phases involved. The mobile phase can be a gas (gas chromatography, GC), a liquid (liquid chromatography, LC) or a supercritical fluid (SFC)*.

The nature of the stationary phase can be incorporated in this nomenclature. The convention hereby is to denote the character of the stationary phase by inserting one or two letters in the middle. Hence, LSC and LLC are both forms of liquid chromatography (LC; mobile phase is a liquid), while the stationary phase is a solid (S) and a liquid (L), respectively.

Chemically bonded phases (CBP's) are very commonly used in LC, and occasionally also in GC. Such phases cannot be seen as either a solid or a liquid. The common term I2011 used for LC involving such phases is bonded phase chromatography (BPC)**. To be consistent, the stationary phase identification should follow that of the mobile phase in defining the chromatographic system. Hence, LBPC should be used for liquid chromatography using chemically bonded stationary phases.

A summary of the different chromatographic techniques is given in table 2.1.

* A supercritical fluid is a substance above its critical pressure and temperature. ** (L)BPC is not thesame as reversed phase chromatography (RPLC). RPLC can bedefined as a form of liquid chromatography in which the mobile phase is more polar than the stationary phase, a situation that does not occur in GC. RPLC will be discussed extensively in section 3.2.2.1.

20

Table 2.1: Classification of chromatographic techniques

Stationary Mobile Phase phase

Gas Liquid Supercritical (G) ( L) Fluid (SF) -

Undefined G C LC SFC

Chemically bonded (BP) GBPC LBPC (S)FBPC (1)

(1) Supercritical fluid solid chromatography (SFSC) may be abbreviated to fluid-solid chromatography (FSC), etc.

In this book the attention will mainly be focussed on the most popular chromatographic techniques, i.e. G C and LC. Some comments will be made regarding SFC in section 3.4.

2.2 SELECTION OF CHROMATOGRAPHIC METHODS

Before we can inject a sample into a chromatograph, we should be able to decide upon which of the above chromatographic techniques is suitable for the separation problem at hand. Clearly, this requires some information about the sample. Completely unknown samples may require a combination of different techniques. For example, an unknown liquid may contain a volatile fraction that can be anaiysed by G C and a non-volatile fraction, for which LC is required. In many cases, we know something about the sample, enough to decide which of the many possible analytical techniques can be applied. A scheme to decide on the appropriate chromatographic technique based on the nature of the sample is shown in figure 2.1.

The first question to be answered is whether or not the sample is volatile enough to be analyzed by GC. G C columns are currently available with upper temperature limits of around 350 "C. Hence, compounds should be sufficiently volatile at this temperature to be analyzed by GC. A second requirement for the sample, which becomes the more relevant the higher the temperatures used, is the thermal stability of the sample, both in the column, as well as in the injector, which in conventional GC is operated at a temperature slightly above that of the column*. Because of the limited stability of organic substances at higher temperatures, extremely high temperatures do not seem to be very

* Cold (on-column) injectors can be used in capillary GC for the analysis of relatively high boiling (low volatile) solutes, so that only the upper temperature limit of the column itself remains relevant.

21

Y P / I

*tile? 9 n t

U U

Figure 2.1: Scheme for the selection of chromatographic techniques. For explanation see text.

useful in GC. The solid adsorbents that may be used up to such very high temperatures aggravate the problem of thermal lability by their catalytic activity. Therefore, the majority of all organic substances is not compatible to GC.

For the volatile samples we then have a choice between GSC and GLC. For all but the permanent gases, which possess a very high intrinsic volatility and need strong specific adsorption sites to be sufficiently retained, GLC is usually the preferred method.

For those samples that are not compatible with GC, the first question to ask involves the size (molecular weight) of the solute molecules. Their size should be compared to the pores of the packing materials that can be used in LC. If the size of the molecules is not negligible relative to the (average) pore size, then part of the pores and hence part of the stationary phase present in the column will not be accessible to the solute molecules. Hence, the simple relationship between chromatographic retention and thermodynamic distribution (eqn.l.6) loses its significance. To avoid that, wide pore materials can be used for the separation of large molecules (e.g., proteins) based on their distribution over the two phases [202].

The effect of limited penetration of the pores by the largest molecules may also be applied beneficially for the separation of very large molecules. Depending on the size of the molecules (in solution), they will be more ore less excluded from the pores, and hence the retention times will be affected. This effect is used in size exclusion chromatography (SEC) or gel permeation chromatography (GPC). In this technique, any interactions between the solute molecules and the stationary phase are purposefully avoided. The solute molecules remain exclusively in the mobile phase, but the accessible mobile phase volume, and hence the retention volume, may vary between the total volume of the mobile phase and the so-called exclusion volume, which is the total volume of mobile phase outside the pores. The latter elution volume applies to very large solute molecules (excluded solutes),

22

which will therefore appear first in the chromatogram. SEC (GPC) is the preferred method if the solute molecular weight exceeds about 3000 (see for instance ref. [201], p.270). Because SEC (GPC) is a separation technique based on the size of the molecules, rather than on (thermodynamic) selectivity, it will not receive further attention in this book.

For the samples that will be subjected to other (so-called interactive) LC techniques, the next question involves the nature of the solvent in which the sample has been or can be dissolved. If this is a non-polar solvent, such as n-hexane, then the sample solution is compatible with Normal Phase LC (NPLC), in which mobile phases with a relatively low polarity are used in combination with more polar stationary phases (see section 3.2.3). In this form of chromatography solid adsorbents (such as silica or alumina) may be used as stationary phases (LSC). Alternatively, polar chemically bonded stationary phases may be used (see section 3.2.2).

If the sample solvent is polar, then the question needs to be answered whether or not the sample is ionic. If this is not the case, then Reversed Phase LC (RPLC) should be applied. In this technique polar mobile phases are combined with less polar stationary phases, typically chemically bonded hydrocarbon chains (section 3.2.2). If the sample is ionic, then it matters whether the sample consists of strong or of weak ions. Strong ions will have to be separated on the basis of their behaviour as ions, either by an ion-exchange mechanism (using a stationary phase that contains ionic groups) or by ion pairing LC (in which an ionic reagent is added to the mobile phase). These techniques are described in section 3.3. For weak ions we have a choice. They may be separated as non-charged molecules by RPLC, if the ionization can be supressed by a suitable choice of the pH. Alternatively, they may be separated as charged ions, using either of the two methods described above.

If there is no information on the ionic nature of the sample, it may be subjected to RPLC first. If the sample consists of ionic solutes under the conditions of the separation, which is usually manifested by low retention times and therefore insufficient separation, then positive or negative ion-pairing reagents (counter ions) may be added to enhance the retention. The advantage of this strategy is that the same column may be used for RPLC as for ion-pairing chromatography.

The merits of all the different methods involving the distribution of solutes over a selective system of a mobile and a stationary phase are discussed in the next chapter.

2.2.1 Expert systems

A scheme for the selection of phase systems such as figure 2.1 is based on logical reasonings and on experience. In order to obtain guidance on how to go about the chromatographic separation of a new kind of sample, the alternative for going through such a scheme is to study the literature or to ask an expert. One may try to formalize the knowledge and experience of the expert and combine it with information from the literature into a computer program called an “Expert System” [203].

Some steps towards the creation of such an expert system for the selection of phase systems in chromatography have recently been taken. Karnicky et al. have reported on attempts to build such a system for LC [204]. The selection of the most appropriate phase system for a chromatographic separation is a complicated matter. The choice will be largely determined by the characteristics of the sample (see figure 2.1) and by the analytical

23

demands. However, secondary factors, such as the availability of columns and hardware, as well as the background, experience and personal preferences of the user may also play an important role. Because of this complex, partly subjective style of decision making, the development of an expert system for the purpose of phase selection in chromatography is a difficult task.

According to Klaessens et al. [203] a program must be able to “solve substantial problems” in order to deserve the qualification “expert system”. Although the substantial problems are definitely there to be solved, and hence expert systems may be of use in chromatography, there is no program good enough yet to deserve the name.

Even if they prove successful, expert systems will not take away the need for optimization of selectivity. Rather, they may be complementary in that they may provide the platform for appropriate initial experiments, from which the optimization procedure may take off.

2.3 CHARACTERIZATION AND CLASSIFICATION METHODS

2.3.1 Polarity; Solubility parameters

Polarity is a key word in many discussions about chromatography, chromatographic selectivity and chromatographic separations. Nevertheless, it is often unclear what exactly is meant by the word. According to Rohrschneider, the polarity of a stationary phase in GC may be measured from its ability to retain polar compounds [205]. Although this appears to be a good description from the point of view of the practical (gas-)chromatographer, it is not a sound definition in the sense that it leaves us with the question of defining what is a polar solute.

Clearly, an objective and quantitative measure for the polarity of compounds of chromatographic interest is needed. Such a quantitative measure may be found in the solubility parameter introduced by Hildebrand [206]. The solubility parameter is defined as the square root of the cohesive energy density (c):

S 2 = C = - E / v (2.1)

where Sis the solubility parameter, E the cohesive energy and v the molar volume. All the interactive forces between the different molecules of a species contribute to the cohesive energy. Hence, molecules which exhibit strong interactions give rise to high values of E. Because of the definition of E(relative to the ideal gas phase) it is a negative quantity. Table 2.2 shows some examples of solubility parameters for compounds of chromatographic interest .

The solubility parameter is commonly expressed in units of ca11/2cm-3’2. One such unit corresponds to 2.05 lo3 Pa”’. In the remainder of this book, the units of the solubility parameter will usually be omitted for reasons of convenience.

In table 2.2 values are given for a variety of materials, including both typical solvents and typical stationary phases. The inclusion of the latter involves some rigorous assumptions, because the simple definition above (eqn.2.1) bears no relevance for solid adsorbents. Nevertheless, by looking at table 2.2 the usefulness of the solubility parameter as a quantity to describe polarity in quantitative terms becomes instantly apparent. The

24

Table 2.2: Some examples of solubility parameters for compounds of chromatographic interest

Compound 6 ref. (cal'%m - 3'2)

Water Alumina Methanol Pyrocarbon Acetonitrile Methylene chloride 1 ,CDioxane Tetrahydrofuran Toluene Ethyl acetate Alkanes Perfluorinated alkanes

25.52

15.85

13.14 10.68 10.65 9.88 9.57 9.53

- 16

N 14

- 7 - 5

207 208 207 208 207 207 207 207 207 207 207 206

table reflects the chemist's intuition , with water being the most polar solvent at the top, and alkanes as common non-polar solvents at the bottom. Only perfluorinated alkanes are less polar than alkanes according to the solubility parameter model.

An important aspect that deserves attention is the extreme polarity of water. Its solubility parameter is around 25, while all other compounds in table 2.2, polar or non-polar, are found between about 5 and 16.

Besides showing the usefulness of the solubility parameter for the quantification of polarity, table 2.2 also illustrates the shortcomings of the model. On the basis of its solubility parameter alone, methylene chloride will be expected to behave quite similar to dioxane, and toluene similar to ethyl acetate. However, in both cases there are considerable differences between the solvents in practice. For example, dioxane is miscible with water in all proportions, while methylene chloride is virtually insoluble in water. Clearly, to account for differences in behaviour between compounds of similar polarity a refinement of the model is needed.

At the basis of such a refinement should be the different interactions which together constitute the total cohesive energy. These can be classified as follows: - Dispersion interaction arises from the mutual interaction of the electron clouds of interacting molecules. Since every molecule has such a cloud of electrons, all compounds, polar or non-polar, exhibit dispersion interaction. - Dipol orientation reflects the energy gain when the negative end of an electric dipole

moment is required, not all molecules can participate in this kind of interaction. For instance, the non-polar alkanes do not possess a permanent dipole moment. - Dipole induction interaction occurs when a permanent dipole induces a temporary dipole in a neighbouring molecule that does not necessarily possess a dipole moment of its own. - Acid-base interactionscan be seen as a combination of all the processes in which electrons

is surroun 4h ed by positive poles of similar or other molecules. Since a permanent dipole

25

or protons (hydrogen ions) are transferred between molecules. Such interactions can only occur if both functions are present, i.e. one molecule (or functional group) must act as an acid, another as a base. Again, completely non-polar molecules cannot take part in any such interactions.

The different types of interactions that may occur in three different kinds of mixtures are indicated in table 2.3. Non-polar (or apolar) compounds are those in which dispersion interaction forms the only contribution to the cohesive energy. Clearly, the interactions that occur in apure non-polar compound are equivalent to those in a mixture of two such compounds.

Table 2.3: Summary of specific interactions occurring in different kinds of mixtures.

Non-polar/non- Polar/ Polar/polar polar non-polar (polar compounds) (non-polar compounds)

Dispersion + Orientation -

Induction -

Acid-Base -

We may now write the total cohesive energy density as the sum of a series of contributions, each due to one of the interactions described above:

where the subscripts refer to total (7), dispersion (4, orientation (o), induction (ind), acid (a) and base (b). Two different kinds of terms appear in eqn.(2.2). Quadratic terms represent dispersion and orientation interaction. For induction and acid-base interaction double product terms occur. The first two types of interactions are called symmetrical, because the two participating molecules are equivalent. In the latter two types of interactions, the different molecules play diffeFent roles. Either one molecule actively induces a dipole in another passive molecule, or one acts as an acid and the other as a base.

Table 2.4 lists the individual contributions or partial polarities for the solutes that appear in table 2.2. From this table, it is clear that a distinction can now be made between molecules of similar overall polarity. Much of the cohesive energy of toluene is due to dispersion interaction, whereas dipole orientation is more important in ethyl acetate. Orientation interaction is of more relevance in methylene chloride than it is in dioxane,’ which shows a considerable contribution from induction interaction.

26

Table 2.4 Some examples of partial solubility parameters (call%m -3'2) for compounds of chromatographic interest [209].

Compound Partial polarities (cal'%m -3/2)

Water Alumina Methanol Pyrocarbon Acetonitrile Methylene chloride 1 ,4-Dioxane Tetrahydrofuran Toluene Ethyl acetate Alkanes Perfluoroalkanes

25.52

15.85 - 14 13.14 10.68 10.65 9.88 9.57 9.53 - 7 - 5

N 16 7.2 10.8 7.2

7.3 8.0 8.10 8.0 8.5 7.6

14

N 7 N 5

? 9.8 3.9 small 5.8 4.38 1.2 3.3 0.8 3.6 0 0

? ? 0.1 small 0.2 1 .o 3.4 0.3 0.7 1.4 0 0

21.7 14.2 11.4 2.5 17.1 5.4 small small 10 4 4 1 small 2.0 6.2 1.5 8 0.4 3.5 1 0 0 0 0

2.3.2 The Rohrschneider characterization scheme

Rohrschneider [205,210] has developed a scheme for the characterization of stationary phases for gas chromatography. The scheme is based on the retention index (I). The retention index is a dimensionless retention parameter, designed to be independent of flow rate, column dimensions and phase ratio. The retention index of a solute is defined as 100 times the number of carbon atoms in a hypothetical n-alkane, which shows the same net retention time as that solute. This definition is illustrated in figure 2.2. By plotting the logarithm of the net retention time against the number of carbon atoms in n-alkanes, a straight line is obtained. The net retention time for a solute may then be located on the vertical axis, and the retention index found on a horizontal scale, which represents 100 times the scale for nc

The definition of the retention index does not necessarily require that the calibration line in figure 2.2 is linear. However, the observed linear relationship allows us to express the retention index in a simple mathematical formula:

(2.3)

where ?;,,is the net retention time of the solute x, t i s n the net retention time of the n-alkane that elutes just before the solute, and t;,,+, the net retention time of the n-alkane that elutes just after the solute.

The retention index can be obtained experimentally in a straightforward manner, can be measured accurately and is relatively robust with regards to small deviations in the temperature [210]. In the Rohrschneider scheme, Z values are measured at 100 "C.

27

nc - I 1

500 600 700

I -

/

I

8

I I

800

Figure 2.2 Illustration of the definition of the retention index in GC. ncis the number of carbon atoms in n-alkanes, 1 the retention index and t i the net retention time.

Rohrschneider assumed that the polar interaction of stationary phases can be characterized by substracting the retention indices observed on a completely non-polar phase from those obtained on a polar phase. As a completely non-polar stationary phase, squalane was selected. For the polar interactions of a solute on a polar stationary phase we may now write:

(2.4)

Because squalane is not entirely non-polar, (small) negative values for AZ are sometimes obtained for very non-polar stationary phases [211].

Rohrschneider further assumed that the polar contribution to the retention index can be described by a simple linear equation, involving a series of constants which are solely dependent on the solute (a to e) and a series of constants which depend solely on the stationary phase (x. y, z, u and s)*. The following equation may now be applied:

A Z = a . x + b - y + c . z + d . u + e - s (2.5)

* The lack of alphabetical order in the stationary phase parameters is due to the fact that three parameters (x,y and z) were initially thought to be sufficient [205]. At a later stage, Rohrschneider found the need for two additional parameters [210].

28

The Rohrschneider scheme has no significance, untii the parameters for five different solutes (or five different stationary phases) are known. With fewer parameters, the system is undefined, while more than 25 known parameters create a problem of consistency. Realizing that the characterization scheme is completely empirical, Rohrschneider made a very convenient choice for the characterization of stationary phases. The probe solutes and their parameters are listed in table 2.5.

The stationary phase parameters may now readily be obtained from the retention indices of the five test probes, using the following equations:

= "methyl ethyl ketone /loo

' = "nitromethane /loo

s = AZpyridine 1100 (2.10)

Hence, stationary phases can be characterized very quickly by measuring the retention indices of the five probe solutes. On the other hand, the characterization of solutes is not so easy, for a combination of reasons. In the first place, a set of five equations with five unknowns has to be solved. In the second place, the retention indices of the solute need to be obtained on five different stationary phases with known Rohrschneider constants, as well as on squalane. Hence, six different columns are needed. Moreover, in order to obtain reproducible data, very careful experimentation is required. It is especially difficult to maintain a squalane column. In this light, the choice of squalane as a reference phase has been unfortunate. Therefore, the Rohschneider scheme has become extremely popular for the characterization of stationary phases, and not for the characterization of both phases and solutes, allowing the prediction of retention indices through equation 2.5.

Some representative examples of GC stationary phases with their Rohschneider constants are shown in table 2.6.

Table 2.5: Rohrschneider probes and their parameters. Retention indices on squalane taken from ref. [212]. Temperature: 100 O C .

Benzene 100 0 0 0 0 649 Ethanol 0 100 0 0 0 3 84 Methyl ethyl ketone 0 0 100 0 0 53 I Nitromethane 0 0 0 100 0 457 Py r i d i n e 0 0 0 0 100 695

Table 2.6: Rohrschneider constants for some typical stationary phases for GC. Data taken from ref. [212].

Stationary phase %Ph (1) x Y Z U S

Silicone SE-30 Silicone OV-1 Silicone OV-101 Apiezon L Silicone OV-3 Silicone OV-7 Dinonylphthalate Silicone OV-1 1 Silicone OV-17 Silicone OV-22 Silicone OV-25 Carbowax 4000 DEGS(2) ~

ODPN(3) TCEP (4)

- 0.16 0.20 0.50 0 0.16 0.20 0.50 0 0.16 0.20 0.50 - 0.32 0.39 0.25 10 0.42 0.81 0.85 20 0.70 1.12 1.19 - 0.84 1.76 1.48 35 1.13 1.57 1.69 50 1.30 1.66 1.79 65 1.58 1.80 2.04 75 1.76 2.00 2.15 - 3.22 5.46 3.86

4.93 7.58 6.14 - 5.88 8.48 8.14 - 6.00 8.71 7.94

-

0.85 0.85 0.85 0.48 1.52 1.98 2.70 2.66 2.83 3.27 3.34 7.15 9.50

12.58 11.53

0.48 0.48 0.48 0.55 0.89 1.34 1.53 1.95 2.47 2.59 2.81 5.17 8.37 9.19 9.40

(1) (2) Diethyleneglycol succinate (3) /?, /3’ -0xydipropionitrile . (4) 1,2,3-Tris(2-cyanoethoxypropane)

Percentage of phenyl groups in methylsilicone polymer in OV-series

Table 2.6 shows that some stationary phases show exactly the same Rohschneider constants. The fact that these phases also show identical chromatographic selectivity in practice forms an indication for the validity of the Rohrschneider approach. Any of these phases can be selected, but it would be a waste of time to investigate the selective effect of more than one of them. The choice will now be based on secondary considerations, such as stability, temperature range, availability and cost. Indeed, one of the first consequences of the Rohrschneider characterization scheme was for some manufacturers to reconsider the program of available stationary phases and to remove obsolete ones (see e.g. ref. [212], p.62). It is seen that not all non-polar stationary phases are identical, and that minor differences in selectivity may be anticipated from the use of Apiezon L instead of one of the silicone polymers.

Table 2.6 shows a series of silicone polymer stationary phases from the OV series. These are dimethylsiloxane polymers, with a varying percentage of the methyl groups replaced by phenyl groups. It turns out that the Rohrschneider constants closely follow this increase in phenyl group percentage. The constants therefore appear to be a reliable indication of the polarity of the stationary phase.

Finally, some truly polar low molecular weight liquid stationary phases are listed in table 2.6. Initially, it was difficult to synthesize stationary phases of high polarity but yet high molecular weight (implying low volatility and hence a high temperature limit). In recent

30

years, an increasing number of polar stationary phases with a good temperature stability has become available.

The Rohrschneider constants obtained for a large number of stationary phases may serve as the basis for a statistical analysis, from which a limited number of stationary phases which are most likely to yield different selectivity effects can be selected.

1 . Classify stationary phases for GC according to their polarity and speczjic interactions. 2.. Compare diyerent stationaryphases, in order to allow a more systematic search for the

3.. Select a limited number of relevant stationary phases for general applications.

We may conclude that the Rohschneider system allows us to:

optimum stationary phase.

The Rohrschneider scheme has been modified by McReynolds [213]. The modifications include the use of some more convenient test probes (for instance, ethanol often yields very low retention indices, nitromethane often yields poorly shaped peaks), the use of a somewhat higher temperature and a factor of 100 to avoid decimal points in the stationary phase parameters. The McReynolds probes are shown in table 2.7. Considering these modifications, I feel that no justice is done when “McReynolds constants” are tabulated. It appears to be more appropriate to refer to these constants as “Modified Rohrschneider constants”, or “Rohrschneider constants, modified according to McReynolds”. An additional set of five probes may be used according to McReynolds, but the five extra parameters are not very helpful for characterization purposes.

2.3.3 The Snyder solvent classification scheme

A convenient way to classify solvents of chromatographic interest in terms of their polarity and the specific chemical interactions is the empirical scheme proposed by Snyder [214,215]. This scheme is based on experimental (gas chromatographic) distribution coefficients for three test solutes (“probes”) on a large number of stationary phases, which were published by Rohrschneider [216]. The probe compounds are ethanol (e), 1 ,Cdioxane (6) and nitromethane (n ) . The experimental values for the distribution coefficients undergo several empirical modifications: 1. The distribution coefficients are corrected for the solvent molecular weight through multiplication by the molar volume of the solvent ( V J :

Table 2.7: McReynolds probes and their parameters. Retention indices on squalane taken from ref. [211]; temperature: 120 “C.

Solute a b C d e L a l a n e

Benzene n-Butanol 2-Pentanone Nitropropane Pyridine

1 0 0 0 0 0 I 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

653 590 627 652 699

31

K b = K , . v, (2.1 1)

K , is the experimental distribution coefficient and K b the corrected value. This correction is required, because any measure for the interactions that occur in certain solvents should be more related to the ratio of mole fractions than to the ratio of concentrations of the solute in the liquid phase and in the gas phase. We may assume the molar volume of the gas phase to be constant and hence irrelevant if our purpose is a classification of solvents. However, the molar volumes of solvents vary a great deal. The K , values for n-octane in various hydrocarbon solvents vary up to a factor of 3.9 between cyclohexane and squalane [216]. The Kb values vary by a more realistic factor of 1.5 [214]. 2. A correction is made for the molar volume of the solute:

log KL = log Kh - log K, (2.12)

where K, is the distribution coefficient of a hypothetical n-alkane with the same molar volume as the solute. K, is approximated by

log K, = (vJ163) log K O (2.13)

where K O is the K b value for n-octane, vx the molar volume of the solute and the constant factor 163 represents the molar volume of n-octane. Eq.(2.13) implies that a straight line through the origin is assumed for a plot of log Kb vs. vx for n-alkanes. This is a reasonable approximation, but not exactly true. Therefore, the K;Z values obtained for n-alkanes other than n-octane will not be equal to zero. However, the compilation of Rohrschneider [216] does not allow a more precise estimate of K, to be made.

After the corrections described above to transfer the K , values into KZ values, Snyder suggests the following definitions for four classification parameters:

Polarity

P' = log K;,, + log KEd + log KZn

where e, d and n denote ethanol, dioxane and nitromethane, respectively.

Proton acceptor parameter

x, = log K&jP'

Proton donor parameter

xd = log K:,d/P'

Strong dipole parameter

x, = log K%,/P'

32

(2.14)

(2.15)

(2.16)

(2.17)

Table 2.8: Solvent classification parameters according to Snyder. Data taken from ref. [215].

P' xf? xd X" Group

Ethanol 4.3 0.52 0.19 0.29 I1

Nitromethane 6.0 0.28 0.3 1 0.40 VI I D i o x a n e 4.8 0.36 0.24 0.40 VIa

Methanol n-Propanol i-Propanol n-Butanol t-Butanol i-Pentanol n-Octanol

Benzene Toluene Chlorobenzene

n-Hexane Isopropyl ether Dichloromethane THF Chloroform Acetonitrile Water

5.1 4.0 3.9 3.9 4.1 3.7 3.4

2.7 2.4 2.7

0.1 2.4 3.1 4.0 4.1 5.8 10.2

0.48 0.54 0.55 0.59 0.56 0.56 0.56

0.23 0.25 0.23

(1) 0.48 0.29 0.38 0.25 0.3 1 0.37

0.22 0.19 0.19 0.19 0.20 0.19 0.18

0.32 0.28 0.33

(1) 0.14 0.18 0.20 0.41 0.27 0.37

0.3 1 0.27 0.27 0.25 0.24 0.26 0.25

0.45 0.47 0.44

(1) 0.38 0.53 0.42 0.33 0.42 0.25

VII VII VI I

(I) (2) Close to group VIII.

Irrelevant, because of low P' value.

Table 2.8 shows the P' and x values for a number of solvents of chromatographic interest. The table first shows the values for the three probe solutes. It is clear that the definitions applied for the x values do not imply that the probes show a value of unity (or 100) for one of the parameters, as was true in the scheme of Rohrschneider (see section 2.3.2). It is therefore wrong to conclude that a compound with a high value for xd closely resembles dioxane, because in that case dioxane would not resemble itself more closely than 24%!

It is then clearly demonstrated in the table that the scheme succeeds in classifying solvents that are chemically similar. Seven additional aliphatic alcohols (besides ethanol) are shown in the table, and while their P' values vary from 3.4 to 5.1, the values for the selectivity parameters are virtually identical for all alcohols. Similarly, benzene, chlorobenzene and toluene appear to show identical selectivity, and hence appear as one class.

The bottom part of table 2.8 shows some solvents of particular interest to LC. It is again seen that water has a very high polarity. However, the polarity parameter P' , because it

33

is based on an empirical interpretation of GC data, is less useful as a quantitative measure for polarity in LC than the solubility parameter S(section 2.3.1). For example, acetonitrile appears to be more polar than methanol in table 2.8. The reverse order is observed from the solubility parameters in table 2.2. In the practice of LC, methanol turns out to be more polar than acetonitrile, and indeed the polarity (solvent strength) of mixed solvents can be predicted quantitatively from solubility parameters (see section 3.2).

The main strength of the Snyder scheme is for the classification of solvent selectivity. We have seen from table 2.8 that solvents that are chemically similar yield similar selectivity parameters. This type of classification can be made on the basis of structural information alone. However, the Snyder scheme goes one step further, in that it classifies different chemical classes into a single selectivity group. From the definition equations (2.14 through 2.17) we see that the three selectivity parameters are correlated by the equation

x, + X d + x, = 1 (2.18)

and hence, that the polarity (P' )plus two interaction parameters characterize a compound completely. If we disregard the polarity, the selectivity can be adequately depicted in a two-dimensional figure. This is commonly done in a triangular diagram, as shown in figure 2.3. Figure 2.3a in the top left corner illustrates the interpretation of the figure. For an arbitrary point in the triangle, it is shown how the values for x, x d and x, can be located on the axes.

n

Figure 2.3: Solvent classification according to Snyder. The insert (figure 2.3a) shows how the location of a solvent in the triangle is related to the valuk for the selectivity parameters x, (eqn.2.15), xd (eqn.2.16) and x, (eqn.2.17). For identification of classes see table 2.9. Figure taken from ref. [215]. Reprinted with permission.

34

All the common solvents are confined to a limited part of the triangle, as was already suggested by table 2.8. Figure 2.3 shows eight (partially overlapping) circles, which represent a classification of 81 studied solvents into eight classes. The classes are numbered by Snyder with Roman numerals, and the classification of the solvents is shown in the last column of table 2.8. Class VI may be subdivided into two subclasses (a and b), but this subdivision serves no practical purposes [215]. Chloroform is represented by the little circle in the bottom left of the triangle. Clearly, it is outside the circle for group VIII. Similarly, the little circle at the top right (representing triethylamine), which is closest to group I, falls outside the classification. However, apart from these two eccentric ones, all solvents can readily be classified, and a summary of the resulting classification is given in table 2.9. A more complete listing of individual solvents into classes is given in ref. [215].

Table 2.9: Solvent classification according to Snyder [215]. Classes are illustrated in figure 2.3.

Group Solvents

I I1 111 IV V VI

VII

VIII

Aliphatic ethers Aliphatic alcohols F'yridine derivatives, THF, sulfoxides Glycols, acetic acid Dichloromethane, 1,2-dichloroethane a. Aliphatic ketones and esters, dioxane b. Sulfones, nitriles Aromatic hydrocarbons, halo substituted aromatic hydrocarbons, nitro compounds, aromatic ethers Fluoroalkanols, water

It can be seen from table 2.9 that the Snyder classification scheme enables us to classify a large number of solvents into a limited number of classes. Chemically similar compounds (e.g., homologues) are found in a single class, but also some very different chemical functionalities are grouped together. Group VII forms a good example. The important practical consequence of the classification is that if a certain solvent (e.g., toluene) does not provide sufficient chromatographic selectivity, it is unlikely that any other solvent in the same group (VII) will do so. Hence, nitromethane does not need to be tested experimentally. Instead of choosing a solvent from all 81 entries in Rohschneider's compilation, the choice is now limited to 8 solvent classes. Hence, what the Rohrschneider characterization scheme does for the selection of stationary phases for GC is achieved with the Snyder classification scheme for the selection of solvents for LC.

2.3.4 Summary

We may summarize the conclusions obtained in sections 2.3.1 to 2.3.3 in terms of the following recommendations:

35

1. The solubility parameter may be used to characterize the overall polarity of compounds (solutes, mobile and stationaryphases) in chromatography. Zt may also be used to predict the polarity (solvent strength) of mixtures (see section 3.2).

2. The Rohrschneider scheme may be used to characterize stationary phases for gas chromatography, to compare direrent stationary phases and to select a limited number of phases out of a large number.

3. The Snyder scheme may best be used to classifi the many solvents that may potentially be used in chromatography into a limited number of categories. This classification may then be used to aid in the selection of solvents which show dizerent selectivity.

REFERENCES

201. L.R.Snyder and J.J.Kirkland, An Zntroduction to Modern Liquid Chromatography,

202. F.V.Warren and B.A.Bidlingmeyer, J.Liq.Chromatogr. 8 (1985) 619. 203. J.W.A.Klaessens, G.Kateman and B.G.M.Vandeginste, Trends Anal. Chem. 4 (1985)

204. J.Karnicky, T.Schlabach, Sj.van der Wal and S.Abbott, 8th Intern. Symp. Column

205. L.Rohrschneider, J.Chromatogr. 17 (1965) 1. 206. J.H.Hildebrand, J.M.Prausnitz and R.L.Scott, Regular and Related Solutions, Van

207. R.Tijssen, H.A.H.Billiet and P.J.Schoenmakers, J.Chromatogr. 122 (1976) 185. 208. B.L.Karger, L.R.Snyder and C.Eon, AnaLChem. 50 (1978) 2126. 209. P.J.Schoenmakers, H.A.H.Billiet and L.de Galan, Chromatographia 15 (1982) 205. 210. L-Rohrschneider, J.Chromatogr. 22 (1966) 6. 21 I . The Chrompack Guide to Chromatography; Theory and Applications, Chrompack,

212. W.R.Supina, The Packed Column in Gas Chromatography, Supelco Inc., Bellefonte,

213. W.O.McReynolds, J.Chromatogr.Sci. 8 (1970) 685. 214. L.R.Snyder, J.Chromatogr. 92 (1974) 223. 215. L.R.Snyder, J.Chromatogr.Sci. 16 (1978) 223. 216. L.Rohrschneider, Anal.Chem. 45 (1973) 1241.

Second edition, Wiley, New York 1979.

114.

Liq. Chromatogr., New York, 1984.

Nostrand Reinhold, New York 1970.

Middelburg, NL, 1981.

PA, 1974.

36

CHAPTER 3

PARAMETERS AFFECTING SELECTIVITY Before starting any optimization process, we should ask ourselves what exactly it is we

want to optimize. In general terms our goal is to obtain the best possible chromatogram for a particular purpose. In the next chapter we will discuss criteria by which to judge the quality of a chromatogram. In the present chapter, we will describe the parameters that influence retention and selectivity, and see which parameters we might consider (or exploit) during the optimization process.

Where possible, we will derive simple relationships between retention and the relevant parameters. For reasons of clarity, we will express all equations in terms of the capacity factor (k). Obviously, the simplest possible equations will be most useful for optimization purposes. Ideally, we will be looking for linear relationships, since straight lines allow straightforward interpolation.

At the end of the chapter (section 3.5) we will summarize the relationships that are recommended for the various parameters in different kinds of chromatography.

3.1 GAS CHROMATOGRAPHY

3.1.1 Gas-liquid chromatography (GLC)

In this form of chromatography retention can readily be expressed in thermodynamic terms. The definition equation for the capacity factor (k ) is

where q i is the total quantity of the solute in either phase, Ci the average solute concentration and Y the total volume of the indicated phase in the column.

If we assume very dilute solutions (as is usually the case in chromatography), we can write for the concentration in the stationary phase

where xi,s is the mole fraction of the solute in the stationary phase, and p, and M, are the density and the molecular weight of this phase, respectively.

For the mobile (gaseous) phase we can write

Pi = l / v i , m = ~

~ R T ’ (3.3)

where vi , , , is the partial molar volume of the solute in the gaseous phase, p i its partial pressure, and (= p i vi , , , / R T ) the compressibility coefficient. Since we are dealing with gases at low pressures, we may usually (and definitely for the present purpose) assume the ideal gas law to be valid. Hence, we may assume to be equal to one.

37

For dilute solutions, we may use Henry’s law to describe the partial pressure of the solute:

where cs is the activity coefficient (at infinite dilution) of the solute in the stationary phase and py is the vapour pressure of the pure solute. Hence, for the ratio of concentrations we find from eqn~(3.2) to (3.4):

Because this ratio is independent of the concentration of the solute, it equals the ratio of the average concentrations, and hence

where ns is the total number of moles of the stationary phase present in the column. From eqn.(3.6) we conclude that there are two solute-dependent factors that affect

retention. In the first place, this is the vapour pressure of the pure solute. p? is a strong function of the temperature (see below) and therefore, temperature may be used as a parameter to influence retention. However, the vapour pressure is a pure component property and it cannot be changed at will. Differences in the vapour pressure of two solutes (or differences in variation of vapour pressure with temperature) may or may not provide us with a means to achieve separation. When the vapour pressure is not sufficiently different, we need to create differences in the second solute-dependent factor.

This is the activity coefficient of the solute in the stationary phase (rf“,). The value of 7 is determined by molecular interactions between the solute and the stationary phase. Therefore, (chemically) different stationary phases will lead to different values for 7. This explains the availability of many different stationary phases for GC, many of which show different selectivity (see section 2.3.2).

Temperature

Eqn.(3.6) provides a good insight into the variation of retention with temperature in GLC. Both the activity coefficient and the vapour pressure of the solute vary with temperature in an exponential way. For the activity coefficient we can write

In es = h d R T - s d R , (3.7)

where h, and sE are the (partial molar) “excess” enthalpy and entropy respectively*.

equation: The vapour pressure can be approximated with the common Clausius-Clapeyron

* The excess quantities measure the deviation of the solution of i in s from the “perfect” solution, for which Raoult’s law is obeyed, i.e. es = 1.

38

Inpp = h , / R T - s , / R ,

where h, and s, are the molar heat and entropy of vaporization. If we now combine eqns.(3.6), (3.7) and (3.8) we find

h,+ h" S E + % + I n s , Rn Ink = In T - - + - R T R Vrn

(3.9)

i.e. an equation of the form

I n k = l n T + A / T + B (3.10)

where A and B are constants. According to this equation, we can expect to obtain a straight line if we plot In (k/ T)

versus 1/ T. This somewhat odd-looking relationship corresponds to the more common procedure of plotting the logarithm of the specific retention volume ( Vg) against the reciprocal temperature. Vg is defined as the net retention volume at standard temperature (0 "C) and per unit weight of stationary phase:

(3.1 1)

Figure 3.1 provides an illustration of the relationship between the specific retention

A great disadvantage of using V, is that ws is not usually known. Alternatively, we may write eqk(3.10) as

volume and the temperature.

Tln(k/T) = A + B T

1.01 L 1 I I I I

26 27 28 29 30 31 32 10'IT -

(3. I Oa)

3

Figure 3.1: Example of the linear dependence of the logarithm of the specific retention volume (eqn.3.11) on the reciprocal temperature. Stationary phase : silicone oil 702. Solutes: n-alcohols (number of carbon atoms as indicated in the figure). Figure taken from ref. [301]. Reprinted with permission.

39

(a)

Figure 3.2 Fundamental linear relationship between retention and temperature in GLC according to eqn.(3.l0b). Stationary phases: (a) SE-30 (silicone oil) and (b) Carbowax 20M (polyethylene-gly- col). Solutes: n-alkanes (number of carbon atoms as indicated in the figure). Figure taken from ref. [302]. Reprinted with permission.

or, because k is proportional to the net retention volume Vx,

Tln(VR/T) = A‘ + B ’ T . (3.10b)

where A’ and B’ are constants.

plots is clear. Figure 3.2 shows an example of a plot according to eqn. (3.10b). The linearity of these

The stationary phase

The properties of the stationary phase manifest themselves in the activity coefficient in eqn.(3.6). A very simple expression for the activity coefficient can be obtained from the concept of solubility parameters (see section 2.3.1). This expression can be seen as a special form of Hildebrand’s regular mixing rule, and it reads [303].

In cs = ( V / R T ) (6i - 6b2 + In (v i /vs ) + (1 - v / v > , (3.12)

where the subscript i denotes the solute, and s denotes the stationary phase. v is the molar volume. The first term on the right hand side of eqn.(3.12) is the enthalpic (“regular”) contribution to the activity coefficient. The other two terms together form the entropic (“athermic”) part. Unfortunately, this very simple equation does not usually yield quantitatively reliable results, unless, perhaps, for the interaction of solutes of low-polarity with phases of low polarity. However, eqn.(3.12) does provide us with a simple means to explain and understand the role of the “polarity” of solutes and stationary phases in GLC.

Almost always, the enthalpic contribution dominates the right hand side of eqn.(3.12), in which case we can use the approximate expression

40

In = (v i /R7‘) (Si - SJ’. (3.13)

Clearly, the right-hand side of eqn.(3.13) is always positive. Consequently, it leads to activity coefficients which are above unity, and hence (since the capacity factor is inversely proportional to y) towards decreasing retention. When the polarity of the solute and the stationary phase are similar, then only the (small and negative) entropic contribution to the activity coefficient will remain, y will be close to unity, and retention will mainly be determined by the pure solute’s vapour pressure.

The vapour pressure, in its turn, is closely related to the boiling point. Indeed, this case is often encountered in the GLC of low polarity solutes on low polarity stationary phases (“boiling point separation”).

Because of the occurrence of the excess quantities h , and sE in eqn.(3.9), the coefficients in eqn.O.10) for the temperature dependence of the retention are a function of the stationary phase. Hence, every stationary phase may be expected to yield a different optimum temperature, at which the capacity factors of all sample components fall in the optimum range. Therefore, to make a fair comparison between two different stationary phases for a given separation problem*, the (potentially different) optimum temperature should be established for each of them and the resulting chromatograms should be compared. The common practice of characterizing (and consequently comparing) stationary phases at a standard temperature is a very convenient one. Nevertheless, it may give rise to erroneous conclusions in some cases.

Mixed stationary phases

The choice of a stationary phase is no longer a discrete variable once mixtures of stationary phases are considered. For binary mixtures the following relationship is usually observed**

(3.14)

where cp denotes the volume fraction, and the subscripts A and B represent two different stationary phases. According to eqn.(3.14) a mixture of two stationary phases behaves as the sum of two individual contributions. The same result may be obtained using a 50150 mixture of the two phases A and B, as with two similar columns (of half the original length) coated with pure A and pure B, respectively, coupled in series. Indeed, in practice both methods may have certain advantages.

Figure 3.3 shows an example of a plot of the retention in GLC versus the composition of the stationary phase. In this and many other cases straight lines are observed. Figure 3.4 shows an exceptional case, in which no straight lines were observed.

Eqn.(3.14) is usually, but not always obeyed in practice. Moreover, it is hard to predict when deviations will occur. Apparently [306], not only the two stationary liquids involved,

* For some suggestions on how to select different stationary phases see section 2.3.2. ** It should be noted that eqn.(3.14) applies only when the total volume of the stationary phase is kept constant.

41

'PDNP - Figure 3.3: Variation of retention (distribution coefficient) with the composition of the stationary phase in GLC at three different temperatures (indicated in the figure in "C). Stationary phase: mixtures of squalane and dinonylphthalate (DNP). Solutes: (a) n-octane, (b) cyclohexane, (c) methylcyclohexane and (d) tetrahydrofuran. Straight lines observe eqn.(3.14). Figure.taken from ref. [304]. Reprinted with permission.

500 '\ 'I

\'\\ '\f\

L 20

630

580 K

/f ,'

0.5 'PDNP - 130

0 0.5 PDNP-

Figure 3.4 : Variation of retention (distribution coefficient) with the composition of the stationary phase in GLC. Stationary phase: mixtures of squalane and dinonylphthalate (DNP). Solutes: (a) cyclohexane, (b) methylcyclohexane, (c) benzene, and (d) toluene. Temperature: 30 OC. Curved lines are obtained, which do not obey eqn.(3.14). Figure taken from ref. [305]. Reprinted with permission.

42

but also the procedure that is used to form the mixture and to coat the mixture on the column is relevant in this respect. Nevertheless, eqn.(3.14) may be assumed valid, until there are indications to the contrary.

After eqk(3.14) turned out to be obeyed by many systems in practice, a model was developed that could provide a physical picture. This so-called diachoric model [306] explains the fact that the two components of the mixed phase behave independently by demixing on a microscopic scale. Hence, the stationary phase is assumed to consist of little patches or droplets of either pure A or pure B. Obviously, such a model does explain obeyance of eqn.(3.14), while it also gives a handle to explain deviations from linearity in terms of complete mixing of the two phases.

It is useful at this point to realize that with the composition of the stationary phase being a continuous variable and with retention and selectivity being strong functions of temperature, the optimum composition may also be expected to vary with temperature. Ideally therefore, temperature and stationary phase composition should be optimized simultaneously (see section 5.1.1). Moreover, once different lengths of columns with the individual stationary phases are applied instead of real mixtures, it is in theory feasible to optimize the temperature of each of the columns, as well as the ratio of column lengths simultaneously.

3.1.2 Gas-solid chromatography

We can describe the retention behaviour in GSC in similar terms as we did for GLC in the previous section. However, we have to reconsider our definitions of the concentration in the stationary phase and of the distribution coefficient. It is common practice to express the concentration of the solute adsorbed onto the stationary phase in terms of moles per gram of adsorbent, i.e.

c: = 4JWS 7 (3.1 5)

where w, is the weight of the sorbent present in the column. Therefore, the partition coefficient

C (3.16) K = - - ? - c: - Vrn

Cm 4rn Ws

is now expressed in units of ml/g. An expression for Henry's adsorption law can be formulated that is analogous to eqn.(3.4) [307]:

4, = Hi A , Pi 9 (3.17)

where Hi(units: mol.atm-'.m-*) is Henry's adsorption coefficient for the solute ion the solid adsorbent. Hi is determined mainly by the interactions between the stationary phase and the solute, and hence it takes over the role of the activity coefficient in GLC. Note that the vapour pressure of the pure solute does not appear as an independent entry in eqn.(3.17). A, is the surface area of the adsorbent. Using eqn.(3.3) with = 1 for the mobile phase, we find from a combination of eqns.(3.15), (3.16) and (3.17) that

K , = Hi A, R T / w, = His R T , (3.18)

43

where s is the surface area per unit weight of the adsorbent (m2/g), usually called the specific surface area. The capacity factor now becomes

and finally for the relative retention:

a , . = k / k , = HfH,. I.'

(3.19)

(3.20)

Hence, the selectivity in GSC is determined by the values of the Henry adsorption coefficients of the two solutes.

Eqn.(3.19) describes the ideal case in which the adsorption isotherm of the solute is linear and the carrier gas does not adsorb onto the stationary phase. This simple situation is not always encountered, but analytical equations can be derived for many other cases [308]. In fact, the practical conditions in GSC are more often non-ideal than is the case in GLC. The adsorption isotherm can only be approximated as linear at very low concentrations. In other words, solute capacities are usually lower in GSC. Surface heterogeneities play a role, especially on inorganic adsorbents such as silica and alumina. These stationary phases are also sensitive to contaminations. Consequently, the observed peak shapes and retention times may be affected by the history of the column ("conditioning") and by the water content of the carrier gas.

Because the pure component vapour pressure is not one of the determining factors for retention in GSC (eqn.3.19), GSC is especially useful for the separation of permanent gases and other solutes of low molecular weight. Even at room temperature the interaction with the stationary phase may be high enough to retain permanent gases on GSC columns. Under GLC conditions a reasonable retention for compounds with a very high vapour pressure is only possible if activity coefficients can be achieved that are very much smaller than one, and this is not a common situation.

Because of the specific application area of GSC in the world of small molecules, the number of components to be separated is usually small. For most practical problems, therefore, specific stationary phases are readily available. Hence, GSC is not the most fertile soil for selectivity optimization.

Temperature

The temperature of the column is the most important parameter in GSC. Its effect on retention can be described by the same equations as were used in GLC, as can be seen from a comparison of eqm(3.6) and (3.19). According to eqn.(3.10) a straight line will be obtained by plotting In (k/ 7) vs. (1 / 7). However, a plot of In k(or In VR) vs. the reciprocal temperature also yields approximately straight lines, as is illustrated in figure 3.5.

Other parameters

Apart from the temperature, the only other factor that may affect the retention and selectivity in GSC is the nature and, theoretically, the composition of the stationary phase.

44

Figure 3.5 : Variation of retention (net retention volume per unit surface area of stationary phase) with the temperature in GSC. Stationary phase: GTCB (carbon). Solutes: o/p-xylene (l), m-xylene (2), n-octane (3), ethylbenzene (4), toluene (5), n-heptane (6), methylcyclohexane (7), exo-5-methyl- norbornene (8), endo-5-methylnorborner1e (9), norbornane (10) and norbornene (1 1). Figure taken from ref. [309]. Reprinted with permission.

The materials that can be used vary from inorganic oxides such as silica and alumina to organic polymers such as styrene-divinylbenzene copolymers.

The inorganic materials are more stable at higher temperatures. Both silica and alumina are strongly affected by the water content of the surface. Therefore, the reproducibility and the repeatability of the separation largely depend on the “conditioning” (equilibration) of the column. Alumina shows a remarkable affinity to hydrocarbons. For this reason it is eminently suitable for the separation of volatile hydrocarbon fractions, including alkane isomers up to about 5 carbon atoms.

Due to both the magnitude of the interactions with the surface, as well as to the surface heterogeneity, both alumina and silica are less useful for the separation of polar compounds.

This latter effect is not encountered to the same extent when organic (solid) polymers are used as the stationary phase. Hence, more or less polar substances with a high volatility may be eluted from such columns as symmetrical peaks.

Mixed beds of stationary phases in a single column have not found much application in GSC. However, for the separation of complex gas mixtures, more than a single column is often used in a configuration that involves switching valves, so that only part of the sample is subjected to a second column (see also section 6.1).

3.1.3 The use of retention indices

‘The retention index ( I ) was introduced in section 2.3.2 as a reproducible means for reporting CC retention data. The retention index was therefore found to be highly useful as the basis for a characterization scheme for stationary phases in GLC. In this chapter, however, the capacity factor and not the retention index has been used to find expressions, which describe the influence of the various relevant parameters on the retention. Besides

45

the fact that the use of capacity factors is consistent with other sections in this chapter, there is also a more fundamental reason behind this choice.

The definition of the retention index in terms of capacity factors is

I - In k,-In k,

In k,+ -In k, I = 100 [ n + (3.21)

If we use eqn.O.10) for the variation of the capacity factor with the temperature we find

(3.22)

In eqns(3.21) and (3.22) i denotes the solute and n and n+ 1 the preceding and the following n-alkane, respectively (see figure 2.2). It is seen that there is a hyperbolic relationship between the retention index and the temperature. Although over small sections of the hyperbola a linear approximation is often used, this is not a sound basis for temperature optimization, especially not since a straight line can easily be obtained by plotting In ( k /

An even more complicated result is obtained if we express the retention index as a function of the stationary phase composition. A combination of eqm(3.14) and (3.21)

vs. 11 T (eqn.3.10).

(a 1

12-

10 -

20 LO 60 80 100 - % Polybutadiene 0 20 LO 60 80 100 - % Polybutadiene

(3.23)

Figure 3.6 : Variation of retention with the composition of the stationary phase in GLC. Stationary phase: styrene-butadiene polymer blends and copolymers, the butadiene fraction is plotted on the horizontal axis. (a) Specific retention volumes for three n-alkanes and benzene. V is proportional to the capacity factor. (b) the retention index for benzene. The solid line is calculated from the straight lines in figure 3.6a. The circles (polymer blends) and triangles (copolymers) represent experimental data. Figure taken from ref. [310]. Reprinted with permission.

46

Figure 3.6 illustrates the variation of (a) the specific retention volume ( Vg which is proportional to the capacity factor) with the composition of a mixed stationary phase, and (b) the variation of the retention index for benzene with the composition. It is clear from these figures that, whereas straight lines are observed for the variation of the capacity factor with the composition, the retention index varies in a highly non-linear manner.

Clearly, for the purpose of selectivity optimization, the capacity factor ( k ) is greatly to be preferred to the retention index (I).

3.2 LIQUID CHROMATOGRAPHY

We will first approach liquid chromatography by assuming that both phases are bulk fluids (i.e. LLC), and generalize our approach later. For LLC we can define a thermodynamic equilibrium constant (Kth) as

(3.24)

i.e. the ratio of the solute activities in the two phases. A suitable standard state (at which by definition a = 1) should now be defined for each phase. As long as we strictly observe this definition, we are free to choose a convenient one. Hence, we opt for the pure solute i as standard state for both phases*, since in that case the activities are equal at unit concentration (1 00% i), and therefore at all concentrations:

and, using the definition for the activity coefficient

For the high dilutions encountered in LC this can be transformed into

(3.25)

(3.26)

(3.27)

where q is the amount of solute i (in number of moles) present in either phase. Finally, by definition

(3.28)

* 'This definition is equivalent to the one implicitly assumed in Henry's law (section 3.1.1). An alternative way to arrive at eqm(3.25) is to consider the thermodynamic potential (p). A condition for equilibrium is that p should be equal in the two phases of a chromatographic system for each compound. Therefore, we can write for the solute

pi,s = p!s + R T In aLs = pi,m = p!m + R T In ai,m

Eqn.(3.25) follows immediately from this equation, if we again choose for the same standard state (the pure solute) in both phases (i.e. pys=py,,).

47

The last factor on the right-hand side of eqn.(3.28) is clearly a phase ratio, independent of the solute. Hence, the activity coefficients determine retention in liquid chromatography. In other words, retention is determined by the molecular interactions of the solute with the stationary and with the mobile phase.

This situation is fundamentally different from the one in GC, where interactions with the stationary phase and the vapour pressure of the pure solute were the relevant factors (see section 3.1.1). In LC, both the interactions in the mobile phase and in the stationary phase can be influenced in order to optimize the selectivity of the system, and neither is beyond control in the sense that vapour pressure is in GC.

The activity coefficient can be expressed in terms of solubility parameters (eqn.3.12). Neglecting the (small) entropy correction terms we find

In ki = ( v / R T ) { (ai - 6,J2-(Si - 6J2} + In(n,/n,J = ( v / R T ) {(6,+6, - 26,)(6,-6,)} + ln(n/n,J. (3.29)

The above equation is very approximate. It involves many assumptions and approximations [311], and it is not adequate for a quantitative description or prediction of retention in LC. However, because of its simplicity, it provides us with a very elegant means to explain many of the features of modem LC in qualitative terms.

Since eqn.(3.29) will not be used quantitatively, we may take some additional liberties by assuming that it not only provides us with some insight into LLC systems, but may also be used for the qualitative interpretation of other forms of LC, notably LSC and LBPC as well. Indeed, this is not a new appproach, and a value for the solubility parameters of some solid surfaces has been suggested in the literature [303,312].

Retention

According to eqn.(3.29) retention (k) varies exponentially with the polarity of the solute (ai). It can easily be seen from eqn.(3.29) that as soon as the solubility parameter product (6, + 6, - 26,)(6, - 6,) becomes significantly positive or negative, the capacity factor will either be impractically high or uselessly low. Hence, to a first approximation reasonable values for k will be obtained when the above product is about equal to zero. This can be achieved in two ways: 1. Choosing the mobile and the stationary phase of roughly the same polarity (i.e. 6, M 6,).

While this has the desired effect on retention, the remedy is equally effective for all solutes, independent of the value of 6, and therefore it creates a very non-selective phase system (see the discussion on selectivity below).

2. Alternatively, the first factor can be minimized, by taking the solute polarity to be roughly intermediate between the polarities of the two phases:

Si M (6, + &,)I2 . (3.30)

This very simple rule of thumb for the selection of LC phase systems (a phase system is a combination of a mobile and a stationary phase) is illustrated in figure 3.7. In figure 3.7 three horizontal axes have been drawn, representing (from top to bottom)

the solubility parameter of the stationary phase, the solute and the mobile phase. Each

48

n-alkylphases silica

25

calv? cm-3’2

fluoroalkcines 1

reversed phase straight phase

T metianol 1 alkanes THF acetonitrile water

Figure 3.7: Illustration of the selection of phase systems for LC according to eqn.(3.30). A solute with a polarity of 12.5 (middle scale) can be eluted from silica (S,= 16; top scale) with a non-polar mobile phase (Sm=9; bottom scale) or with a polar solvent in a reversed phase system. The shaded areas indicate the latitude with respect to the selection of the mobile phase. Figure taken from ref. [311]. Reprinted with permission.

(non-horizontal) line in figure 3.7 represents a phase system for which eqm(3.30) is obeyed. Clearly, a vertical line again denotes a system for which the polarities of the two phases are equal. According tp the above discussion this is a completely non-selective phase system.

In figure 3.7 two lines have been drawn, which represent two examples of possible phase systems for the elution of a solute with Si= 12.5 (cal”*.~m-~’~). The line with a positive slope connects a moderately polar mobile phase (6, = 9) with a polar stationary phase (a,= 16). Because the polarity of the stationary phase exceeds that of the mobile phase (a,:> S,), this is by definition a normal (or straight) phase system.

The reverse is true for the line drawn in figure 3.7 with a negative slope (a,< a,), and for this reason this system is called a “reversed phase” (RP) system. The particular line in figure 3.7 connects a typical non-polar (alkane-like) phase with 6,= 7 to a polar mobile phase with 6,= 18. Such a mobile phase could for instance be created by mixing methanol ( 6 ~ 16) with water ( 6 ~ 2 5 ) in the correct proportions. Since a very wide range of mobile phase polarities can be covered with mixtures of methanol and water, or even tetrahydrofuran (THE 6s 10) and water, the reversed phase system is a very flexible one. Without changing the column (stationary phase), it can be applied for the elution of a wide variety of solutes.

Alongside the two lines drawn in figure 3.7 as examples for phase systems, dotted areas are indicated towards the mobile phase axis. This is done to indicate that, when eqn.(3.30) is strictly obeyed, a small capacity factor is expected to result. By choosing the mobile phase polarity slightly further away from that of the solute, the capacity factor can be moved into the optimum range. However, the margin thus created is usually very small and

49

eqn.(3.30) may serve as a good (albeit qualitative) indication of the behaviour of LC systems.

Selectivity

From eqn.(3.29) a very simple expression for the selectivity of a phase system may be derived. For the relative retention (aj.> of two solutes with similar molar volumes ( v i = vj) we find

In aj.i = In kj - In ki = (2v i /R7' ) (Si-Sj) (6,-Ss). (3.31)

Hence, if we want to separate a given pair of solutes (i.e. Si and are fixed), the general approach involves maximizing the only variable left to control in eqn.(3.31), the factor

By definition, the relative retention is larger than (or equal to) 1. Thus, for a normal phase system, where Ss> S,, it follows from eqm(3.3 1) that Sj> 6 , and hence themore polar solute will elute last. Again, the reverse is true for a reversed phase system. Because the signs of the two factors in eqn.(3.31) which involve solubility parameters will always be the same, we may state that it is the absolute difference between the polarities of the two phases that should be maximized. Therefore, the selectivity of a phase system ( V ) may be defined as

(6, - as)*.

(3.32)

Unfortunately, we cannot just use any combination of phases that would constitute a very selective phase system. For example, we might want to opt for the combination of an RPLC column with typically 6,yz 7, with pure water (a,= 25.5) as.the mobile phase, which would result in a selectivity ( V ) of about 18. However, in this particular phase system only a very polar solute with 6,s 16 would satisfy eqm(3.30). It will come as no surprise that in this particular (highly selective) phase system all but the very polar solutes will have extremely high capacity factors. In fact, for a given solute, once a given column (stationary phase) has been selected, the appropriate mobile phase can readily be obtained from eq~(3.30) (or graphically from figure 3.7). From a substitution of eqm(3.30) in eqn.(3.32) we find

v = 2 (6, - 6.J . (3.33)

This equation shows that we should ideally select a stationary phase with a polarity that is very different from that of the solute. Indeed, the recommendation to use normal phase chromatography (high S,J for non-polar solutes (low Si) and reversed phase chromatography (low 6.J for the separation of polar solutes (high Si) is not new. However, this rule of thumb is much too simple. A complication is caused by the availability of appropriate mobile phases. For instance, to satisfy eqn.(3.30) for the elution of non-polar solutes ( 4 ~ 7 ) from a silica column (6,yz 16), a mobile phase with 6,- - 2 would be required.

* This factor can easily be shown to be of equally great importance for the separation of two molecules with similar polarities but with different molar volumes.

50

Clearly, this is an impossible proposition. Practical mobile phase polarities will be restricted to the range between 6,w7 (for alkanes) and 6, = 25.5 (for water).

For some selected stationary phases, the selectivity that can be achieved as a function of the solute polarity is shown in figure 3.8. Eqn.(3.33) forms the basis for this figure, but the practical limits for the mobile phase polarity are respected. For example for a reversed phase column (represented by the line AT in figure 3.8 and denoted by the letters RP) the maximum solute polarity is just over 16 when pure water is used as the mobile phase. Phases with intermediate polarity are represented in figure 3.8 by a set of two lines in a V-shape. The set of lines denoted by LSC represents a typical normal phase adsorption material with a polarity of around 16 [312]. This stationary phase can be combined with a less polar mobile phase down to 6, = 7, yielding the line with a negative slope in figure 3.8. This branch represents the common application of normal phase adsorption chromatography.

However, the polar adsorbent may also be combined with an even more polar mobile phase up to a mobile phase polarity of 25.5 and a solute polarity of about 21. Hence, this line with positive slope in figure 3.8 represents the use of polar adsorbents in the reversed phase mode for the separation of very polar solutes.

Interestingly enough, according to figure 3.8, the non-polar stationary phase (reversed phase column) will always lead to a higher selectivity than the more polar stationary phase (normal phase column), apart from the range for very polar solutes, where the polar stationary phase is used in the reversed phase mode. Indeed, there has been a recent interest in separations of this kind on silica, and some polar chemically bonded phases (see section 3.2.2) are especially useful in this respect. The separation of sugars on an amino-type column is a good example.

The line TW in figure 3.8 is dashed. This is the virtually non-existent situation where water is being used as the stationary phase. Although this suggestion is not at all practical, it is clear from figure 3.8 that a very high selectivity could be obtained for polar solutes.

Figure 3.8 : Calculated selectivities according to eqn. (3.33) for various stationary phases as a function of solute polarity. RPF = perfluorinated reversed phase; RP = reversed phase; PC = pyrocarbon; LSC = alumina, silica. Figure taken from ref. [31 I]. Reprinted with permission.

51

Therefore, it seems an interesting challenge to try and create stationary phases of a polarity much higher than that of silica.

There are two other phases indicated in figure 3.8. The first is a so-called pyrocarbon material. Such a stationary phase is formed by pyrolizing an organic layer on a silica substrate. The idea is to combine the mechanical strength of silica with the chemical inertness of carbon. The value of 14 used here can be thought of as typical for carbonaceous materials. These materials do not seem to behave like non-polar phases in the tradition of chemically bonded phases for RPLC, but rather like phases of intermediate polarity. Hence, as for silica, they may be most useful in the reversed phase mode for the separation of very polar molecules using aqueous mobile phases.

The thin line in the top left of figure 3.8 denoted by RPF represents a very non-polar perfluorinated (chemically bonded) stationary phase. Perfluorinated alkanes are known to behave like even less polar materials than the alkanes themselves 1313). Although it is theoretically possible to use such materials as a mobile phase (for instance for the separation of low polarity solutes on a silica column), figure 3.8 suggests that it will be more rewarding to use perfluorinated materials as the stationary phase. Of course, this proposition would also be more cost effective. Indeed, such materials have been studied by several researchers. The general conclusion of these studies turns out to be that there might be an overall increase in selectivity relative to conventional RPLC systems, but that this effect is overshadowed by very large specific effects (i.e. selectivity towards specific solutes)[314]. Therefore, perfluorinated materials should be seen as alternative rather than as superior stationary phases for RPLC.

The behaviour of the perfluorinated phases as discussed above illustrates the fact that the solubility parameter model, despite its charms, may only be used as a crude approximation. The occurrence of specific deviations from the general rule may at least be made plausible by differentiating between different kinds of molecular interactions, and by introducing so-called partial solubility parameters or partial polarities [303,3 121 (see also section 2.3.1). However, such an extension greatly increases the complexity of the model, without increasing its predictive value correspondingly.

3.2.1 Liquid-liquid chromatography

A liquid-liquid system can be created by coating a particulate matter with a thin layer of a liquid phase, similar to the way packed columns are used in GLC. To maintain such an LLC column, the stationary phase should be insoluble in the mobile phase, just as GLC phases need to be involatile at the temperature of operation. Unfortunately, “in- solubility” is an absolute demand that can at best be approximated in practice. The solubility of the stationary phase in the mobile phase becomes even more critical once some flexibility is desired with regard to the choice of the mobile phase. For example, mixtures of several pure solvents are usually required in order to adapt the eluotropic strength (polarity) of the mobile phase such that the capacity factors fall in the optimum range.

Because complete immiscibility of the two phases cannot usually be accomplished, practical measures will have to be taken to avoid bleeding of the stationary phase from the column, for example pre-saturation of the eluent with the stationary phase or the inclusion of a small coated column (coated with the same stationary phase as the analytical

52

column) in the flow stream before the injector. Basically, both the above remedies are very similar.

Despite such measures, LLC columns with “insoluble” stationary phases are not very stable, for instance due to a disturbance of the system every time a sample is injected. The reproducibility of retention data on these LLC columns is generally unsatisfactory, and ‘‘aging” of the column tends to occur rapidly. Apparently, the stationary phase is not merely dissolved from the column because of its “solubility” in the mobile phase, but it may also be eroded from the column because of mechanical processes (shear forces) or by a solution-precipitation mechanism, which causes the stationary phase to be redistribu- ted within the column. These effects may be enhanced by temperature changes within the column, due to viscous ‘heat dissipation and inadequate temperature control. Indeed, thermostatting of the column (and the eluent reservoir) is vital for the proper operation of LLC systems.

Another problem associated with LLC is that of mixed retention mechanisms. Ideally, the solid support in LLC binds the molecules of the stationary phase with strong adsorptive forces, but it does not exert these forces on solute molecules. Clearly, this ideal situation can never be realized completely [3 151.

For all these reasons, it will be understandable that LLC systems have been virtually replaced by chemically bonded phases (section 3.2.2) in current LC practice. Consequent- ly, the various parameters of interest for the optimization of these systems will not be discussed extensively. With regard to the influence of temperature and mobile phase composition on retention and selectivity, it is suggested that the same relationships may be used for “insoluble” LLC stationary phases as are used for LBPC. LLC systems have been used extensively for the separation of ionic compounds by means of ion-pairing techniques. Such systems will be discussed in section 3.3.2.

The main parameters in LLC are the polarities of the mobile and the stationary phase. Increasing the polarity difference between the phases enhances both the selectivity of the system (figure 3.7) and the stability, due to a reduced mutual solubility of the phases.

In LLC systems there is not a substantial difference between the selectivity characteristics in the normal phase and the reversed phase mode. The choice of either will mainly be determined by the sample. Polar samples (in polar solvents) will preferably be injected in a reversed phase system and non-polar samples in a normal phase system.

Within the framework of the given polarity of a phase, its composition may still be varied for optimization purposes (see the discussion about iso-eluotropic mixtures in section 3.2.2). However, the mutual solubility of the two phases is not only determined by their polarity, so that changes need to be considered carefully. In conventional LLC systems, changes to the stationary phase are hard to make, because they may require a lengthy “re-coating’’ procedure.

Dynamic LLC systems

A promising way to create LLC systems with sufficient stability is the use of immiscible ternary mixtures to create what is called a “dynamic” (or “solvent-generated”) LLC system. The principle of such a phase system is illustrated in figure 3.9. This figure shows an example of a thermodynamic phase diagram of a mixture of three components (A, B and C). Both the binary mixtures A + B and A + C are miscible in all proportions.

53

However, this is not the case for solvents B and C and therefore there is a range of binary and ternary compositions at the bottom of figure 3.9 where two liquid phases are formed. If the three solvents are brought together in proportions that correspond to a composition that is situated inside this area, such as the point indicated by a dot in figure 3.9, then two liquid phases will be formed according to the “nodal” line through this point. The position of the point on the nodal line will determine the ratio of the amounts of the two immiscible phases formed.

The two liquids thus formed are immiscible, but in thermodynamic equilibrium. Therefore, we may speak of a dynamic system of two immiscible phases. Figure 3.10 shows an example of a practical system applied to create a dynamic LLC phase system. A practical phase system can be created by pumping a mobile phase through a column, the composition of which corrresponds to a ternary mixture that is in dynamic equilibrium with another mixture (the two mixtures can be connected by a nodal line). If the mobile phase is the more polar one of the two ternary mixtures in equilibrium, then a non-polar (hydrophobic) solid support must be used and a reversed phase system can be generated. If the mobile phase is the less polar of the two mixtures in equilibrium, a polar support is required.

The phase ratio of the system will largely be determined by the specific surface area of the solid support.

Because of the equilibrium between the two phases, dynamic LLC systems are considerably more stable than the conventional LLC systems. If the equilibrium is disturbed by the injection of a sample, then it will soon be restored once the sample starts to move along the column. LLC systems offer a great flexibility with regards to the choice of phasesystems. We have

seen above (figure 3.8) that the choice of available mobile and stationary phases

A

Figure 3.9 : Schematic phase diagram for a ternary system of three liquids, two of which are not miscible in all proportions. A mixture that corresponds to the composition M in the figure will “demix” according to the nodal line LN. Two liquid phases are formed that correspond to the compositions of L and N in the ratio kn.

54

Figure 3 .10 Example of a phase diagram for a ternary system used to create a dynamic LLC system. Components: Ethanol (EtOH), Acetonitrile (ACN) and Iso-octane (2,2,4-trimethylpentane; TMP). I - V nodal lines. Circles: compositiods determined experimentally by titration (full circles) and GC (open circles). Figure taken from ref. [315]. Reprinted with permission.

determines the possible (general) selectivity that can be achieved in an LC system. Potentially, LLC systems allow us to use the entire range of the triangle ATW in figure 3.8. This is neither true for RPLC systems (section 3.2.2), nor for LSC systems (section 3.2.3). The possibility to form a truly homogeneous, highly polar stationary phase is a real advantage of LLC systems.

Other advantages of LLC systems include the possibility to form reproducible, homogeneous stationary phases, a large sample capacity and a large “contamination capacity” (i.e. LLC columns are not easily polluted by contaminants in the mobile phase or the samples) [315]. Because the LLC system is generally well-defined, it allows a more rigorous theoretical treatment than other forms of LC. In particular, LLC retention data correlate well with liquid-liquid partition coefficients obtained from independent (“sta- tic”) experiments.

A disadvantage of LLC systems relative to other forms of liquid chromatography (LSC, LBPC) is the long time it takes to create a phase. For dynamic LLC systems, every new mobile phase necessarily requires the creation of a different stationary phase. This may require 50 to 170 (depending on the pore size of the support) times the volume of the mobile phase in the column to be pumped through the system [315].

The phase diagrams of figures 3.9 and 3.10 will be affected considerably by a change in temperature. Therefore, the temperature should be controlled very carefully, as indeed is necessary for all LLC systems.

Summary

1. LLC systems are generally not very stable and not very easy to use in practice. 2. The use of dynamic LLC systems may help to overcome some of these problems. 3. LLCsystems ofler a great degree offrexibility with regard to the possible choice of mobile

and stationary phases. 4. Well-defined LLC phase systems can be made reproducibly.

55

5. LLC systems offer high sample capacities and contamination capacities. 6. Temperature is a critical factor for both the stability and the selectivity of LLC systems. 7. Every change in the mobile or stationary phase requires a long equilibration time.

3.2.2 Liquid bonded phase chromatography

3.2.2.1 Reversed phase chromatography ( R PLC)

RPLC is currently by far the most popular of all LC techniques [316]. Two reasons for that have already been identified when we discussed LC in terms of solubility parameters. First, a single RPLC column offers great flexibility for the chromatography of a wide variety of solutes by using mixtures of water and an organic solvent as the mobile phase (figure 3.7). Second, the overall selectivity of the RPLC system is almost always superior to that of other LC systems (figure 3.8). Also, in the previous section, some practical disadvantages were described for LLC systems, which have resulted in the almost exclusive use of bonded phases for RPLC. The advantages and disadvantages of RPLC will be summarized at the end of this section. It suffices here to point out that the emphasis put on RPLC in this long section is amply warranted, from a theoretical as well as from a practical point of view.

Even more than in other LC techniques, the exact mechanism of retentionin RPLC is unclear. Certainly, a simple picture that would enable us to derive unambiguous equations for the variation of retention with the various parameters of interest cannot yet be drawn. Unfortunately, there has been too much speculation in the literature throughout the last decade, often accompanied with insufficient experimental data to justify the conlusions drawn. Therefore, it is not surprising that there are many different propositions for expressions to describe the retention behaviour in RPLC.

In the following pages we will discuss the parameters which are relevant for selectivity optimization and some possible quantitative relationships.


Almost exclusively, chemically bonded phases (CBPs) are now being used in RPLC, the vast majority of the applications being achieved on silica-based phases, modified with long

CH3 r-----1 I I L - - - - - J I

i - O t H + C l t S i - R

C Y

CH3 r------ l I

-0fH + R ’ O t S i - R L - - - - - - J I

CH3

Figure 3.1 1: Schematic illustration of the reaction of a silica surface with a monofunctional reagent (dimethyl-alkyl-ethoxysilane). Figure taken from ref. 131 7). Reprinted with permission.

56

R I

EtO -Si - OEt I 0 I

Si / I \

B HO-& --OH

I 0 I a

/I\

OH\ /R

7 9 I \

OEt R

Si / \ 0 0

\ / Si / \

/I\ /I\ Si Si

R I I 0

Si /

Figure 3.12: Schematic illustration of the possible products formed by the reaction of a silica surface with a trifunctional reagent (alkyltriethoxysilane).

alkyl chains [316]. Typically, the silica surface, featuring reactive silanol (-SOH) groups is brought to react with reactive chloro- or alkoxysilanes according to the reaction shown in figure 3.1 1.

In figure 3.1 1 a mono-ethoxysilane is used as an example. Alternatively, a trifunctional reagent such as a triethoxy-alkylsilane may be used, to yield what is commonly referred to as a “polymeric” material. The various possible products from the reaction of trifunctional silane molecules with the silica surface are shown in figure 3.12.

The term polymeric phases arises from the fact that trifunctional reagents may just as well react with each other as with the silica surface under the influence of (inevitably present) traces of water. Hence, the resulting material is not necessarily a well-defined monomolecular layer. Moreover, for every silanol group that disappears during the reaction, two new ones are potentially formed once the product is brought in contact with water. Many of these newly formed silanol groups can subsequently be removed by reaction with a small monofunctional silane (e.g. trimethylchlorosilane, TMCS)*. Also,

* This so-called end-capping process is common practice in the synthesis of bonded phases for RPLC, whether mono- or trifunctional reagents are used. This is done in order to keep the number of remaining silanols to a minimum.

57

many hydroxyl groups can be removed by a heat treatment of the product, because different ligands attached to the silica surface will be “cross-linked” at elevated temperatures. However, a number of the silanol groups will definitely remain present. The presence of these remaining (“residual”) silanols is unwanted, because they may contribute to the retention process, yielding mixed retention mechanisms and increased band-broadening.

Because the silica surface is effectively shielded by the hydrophobic layer of long chain silanes, the silanol groups will only exert their influence on solute molecules by long range (electrostatic) interactions. Hence, the presence of silanol groups will be felt more easily at higher pH values (pH > 5 ) where the silanol groups become increasingly negatively charged, and for basic solutes, which may be positively charged at these pH values. Polyelectrolyte molecules (see below) also tend to be affected by the charge of the surface, as they are large enough to experience the electrostatic forces of a number of ionized silanol groups.

The resulting CBPs are usually identified by the length of the alkyl chain. For example, when the number of carbon atoms in the alkyl chain (nJ is equal to 18, by far the most popular chain length [318], we speak of an octadecylsilica (ODS) or of an RP-18 phase. The second most popular chain length [318] is the octylsilica or RP-8 (nc= 8). Apart from the characteristics of the starting material (specific surface area, pore size distribution) and the reagent used, the alkyl chain length is the only variable to be considered.

Upon increasing the alkyl chain length, the retention (k) will initially increase exponentially, i.e. In k increases linearly with nc *. However, when the chain length is increased further, the increase in retention diminishes and the capacity factor becomes roughly independent of the chain length. This is illustrated in figure 3.13. The “breaking point” in the In k vs. nc curves was defined by Berendsen and de Galan [319] as the critical chain length (nz). nr appears to vary with the solute. Tentatively, it increases with the size of the solute molecules.

The initial increase of In k with nc is usually larger for solutes for which the absolute retention is larger. Hence, in a plot of In k vs. nc the lines tend to diverge towards larger nr As a rule, therefore, the relative retention (a) increases with nc until the critical chain length is reached (for both solutes). Above this point a will become roughly constant. Although there are exceptions to this rule [319], it does imply that almost always the best selectivity is obtained with n,values above the critical chain length. This is usually realized with ODS phases, which is the largest chain length that is commercially available**.

As a conclusion, ODS materials are understandably the most widely applied RPLC stationary phases, and the stationary phase chain length is a variable that will usually not be of interest as a single variable for the optimization of selectivity.

* Note that this observation is in conflict with both a liquid-liquid-like behaviour of the hydrocarbo- naceous layer (in which case k is expected to increase linearly with nJ, and with adsorption of the solute on top of this layer (in which case k would be virtually independent of nJ. ** Even larger alkyl chain lengths have been used to synthesize CBPs, but they would definitely be more expensive if turned into commercial products. Since the critical chain length is usually well below 18 [319], such expensive materials would not usually have major advantages over the existing ODS materials.

58

t jot I

rn-cresol --.---- -- .--____ -.

, I I I I I

nc - 18 2-2 01-' '

1 3 6 10 14

Figure 3.13: Variation of the capacity factor with the length of chemically bonded alkyl chains of the stationary phase using monomeric phases (nJ. Mobile phase: methanol-water (8020). Solutes: n-alcohols and aromatic solutes as indicated in the figure. Asterisks indicate the critical chain length. Figure taken from ref. [319]. Reprinted with permission.

The mobile phase

In the previous discussion the possibility to use mixtures as the mobile phase has already been mentioned. It was tacitly assumed that a mixture of (for example) methanol and water has a polarity in between those of the two pure constituents. This will generally be the case*.

The extent to which retention in RPLC can be made to vary with the composition of the mobile phase is enormous. For almost all solutes retention will be impractically low in some pure organic solvent (methanol, THF) and impractically high in pure water. Hence, to achieve reasonable retention times, a mixture of water and an organic solvent (a so-called modifier) is usually required.

We have seen before (section 3.2) how this can be explained in terms of solubility parameters, and it was also concluded that RPLC offers superior selectivity for a great variety of samples.

First of all, let us recall the basic equation for LC retention in terms of solubility parameters:

In ki = ( v / R T ) { (6, + 6, - 26,)(S, - 6,)} + In (n,/nJ. (3.29)

* Exceptions to this rule will only be observed when two compounds are mixed that exhibit very strong mutual interactions, for example two compounds that give a chemical reaction, or an acid and a base.

59

To a first approximation, we can write for the solubility parameter of a mixed mobile phase 13201

6, = I: (Pi 4. i

(3.34)

where (P is the volume fraction and the subscript j denotes the individual components of the mixture. For a binary mixture of water (W) and an organic modifier (a ) eqn.(3.34) reads

and because the sum of the volume fractions must equal one

where cp= cp,, the volume fraction of the organic modifier. If we combine eqns.(3.29) and (3.36) we find

In ki = (vi/RT) { 6, + (~(6 , -6~) + 6, - 2Si} { 6, + cp(6,-6,) - a,} + ln(n,/n,)

(3.37) = (vi/RTr) { (p2(6, - 6,)' + 2(~(6, - 6,) (6, - Si) + (6, + 6, - 26,)(6, - as)} + ln(n,/n,).

Clearly, eqm(3.37) is a quadratic equation of the form

l n k = A q ? + B c p + C (3.38)

in which the coefficient A is expected to be positive (see eqn.3.37), B large and negative (because 6, > 6, and 6, B SJ, and Cis the natural logarithm of the capacity factor in pure water. Eqn.(3.38), as well as the expectations for the sign and magnitudes of the coefficients as expressed above, is obeyed very well in practice [321,322]. Only for mobile phase compositions close to (P=O (mobile phases with 10% or less organic modifier in water) may considerable deviations be observed [323]. Hence, eqm(3.38) can be used for the description of the retention as a function of the (binary) mobile phase composition, but the coefficient Cdoes not necessarily give an accurate estimate for the retention in pure water [323].

A similar quadratic equation can be used to describe retention in a ternary eluent, where water is mixed with two organic modifiers, the volume fractions of which are denoted by (P, and (p2 [324]:

In k = A, (P; + A2 4 + Bl (P1 + B2 (P2 + c + D (Pl(P2 (3.39)

and for quarternary mixtures the obvious expansion would be:

In k = A, ( P , ~ + A, (P: + A, ( P ~ ~

+ B , (P1 + B, (P2 + B3 (P3 + c + D12 (P1 (Pz

+ D13 (P1 (P3 + D23 (P2 9 3 ' (3.40)

The same quadratic equations for retention as a function of composition have been

60

derived by Jandera et al. [325] in other terms (“interaction indices” rather than solubility parameters) but in a very similar way. Melander and Horvath [326] arrive at an equation which is quadratic, but they try to describe their results by a linear relationship between In k and p. Weijland et al. [327] use equivalent quadratic expressions without an underlying model. Their equations are less practical, because the necessary condition that the sum of all volume fractions equals one is not implicitly contained in the expressions, so that a set of two equations remains. They also allow a third order term to allow deviations from the quadratic model.

Although the quadratic equations for retention (In k) as a function of mobile phase composition (rp) provide a good description of experimental data, they are inadequate to describe retention within experimental error. For binary mixtures the standard’deviation between the quadratic equation (eqn.3.38) and experimental data is typically between 5 and 10% (depending on the solute) [322]. For ternary systems average deviations of 10 to 20% are typical [324]. However, the inclusion of additional (higher order) terms at will is not an attractive way to improve the description of the experimental data. We will discuss this more fully in chapter 5 (section 5.5).

Alternative expressions

Several other equations have been suggested for RPLC. Purnell et al. [328] suggested plotting 1 / k vs. rp. A straight line in such a plot would correspond to the same assumptions as were used to explain the validity of eqn.(3.14) in GLC. However, straight lines are not observed, but instead plots of l /k vs. pshow a pronounced curvature towards the (paxis. Purnell et al. [328] proceeded to describe l / k as a function of p using the following four-parameter hyperbolic equation:

(3.41)

This equation may be used not only for RPLC, but also for other forms of liquid chromatography.

Melander and Horvath [329] have suggested a five-parameter hyperbolic equation to describe k itself as a function of rp:

A ~ + ~ r p + c k =

A’ cp+ B‘ (3-42)

Eqn. (3.42) is an example of a so-called “rational function”. Functions of this kind are renowned for their flexibility in describing curves without physical modelling.

Lu Peichang and Lu Xiaoming derived a three-parameter parameter equation, which reads [3301

I n k = a + B p + c l n q . (3.43)

Obviously, eqn.(3.43) will not be able to yield a successful description of the retention behaviour in the range of low rp values. Therefore, it was subsequently modified to

Ink = a + B + c l n ( 1 + d q ) . (3.44)

61

None of the equations (3.41) to (3.44) appears to offer a better compromise between the accuracy of the description and the complexity of the model than does the quadratic equation (3.38).

Linear approximation for binary mixtures

The quadratic expressions facilitate the description of retention over large ranges of composition. Such large ranges are usually of limited interest. Both very small (insufficient resolution) and very large (excessive retention times) capacity factors are unattractive. The most useful range of capacity factors is between 1 and 10 (cf. sections 1.5 and 1.6). Over this limited range eqn. (3.38) for a binary mixture can usually be approximated quite adequately by a straight line [322,331]:

Ink = In ko - S q , (3.45)

where k, is an imaginary (extrapolated) capacity factor in pure water*. Although a part of the curve for binary mixtures can be approximated by a straight line, this does not imply that a part of the retention surface in ternary mixtures can be approximated by a plane. Straight line approximations can only be used for quasi-binary mixtures, i.e. ternary mixtures in which the ratio of the volume fractions of the two organic modifiers is constant. Two of such straight lines (for different ratios) do not usually define a plane.

Initially, Snyder [331] suggested that the slope S in eqn.(3.45) would be independent of the solute, i.e. S would be a constant for a given stationary phase and two given mobile phase constituents. For example, for methanol-water mixtures on an ODS column S was claimed to be about 7 [331]**. Hence, eqn.(3.45) would be approximately valid over a composition span of 30%( = 2.3 x l O O / S ; there is a span of 2.3 units in In k between k = 1 and k = 10).

Since then, however, it has been shown that the value of S does vary systematically with the retention behaviour of the solute [322,333]. If binary mixtures of water and methanol are used as the mobile phase, S tends to increase with an increase in the absolute retention. This is illustrated by the diverging set of lines in figure 3.14***.

In the methanol-water system, a linear correlation between the coefficients S and In ko has been observed by several researchers [322,333], and the coefficients p and q describing this linear relationship as

S = p + qlnk , (3.46)

* What is true for the coefficient C in eqm(3.38) is certainly not less true for In k, in eq~(3.45). Both coefficients cannot be relied upon to provide accurate estimates of the (logarithmic) capacity factor in pure water [323,332]. ** Snyder used decimal logarithms instead of natural ones, causing a difference of a factor of 2.3 between the value given here and the literature data [331]. *** Note that the simple solubility parameter model (eqn. 3.37) predicts the coefficient B to be dependent on the (polarity of the) solute and, moreover, predicts the magnitude of B to increase if the solute polarity decreases. For RPLC this implies an increase of the slope with an increase of the retention.

62

water methanol

'T

Figure 3.14: Variation of retention with the binary mobile phase composition for methanol-water mixtures on an ODS column. Solutes: naphthalene ( ), anisole (0) and phenol (x). Thin lines: eqn. (3.38) for k < 50; thick lines: eqn.(3.45) for 1 < k < 10. The diverging straight lines suggest an increase of the slope parameter S (eqn.3.45) with increasing capacity factors Figure taken from ref. [322]. Reprinted with permission.

have been estimated with remarkable consistency, despite the use of different solutes and different columns for the evaluation.

The parameters found for the coefficients in eqn.(3.46) are summarized in table 3.1. An example showing the validity of eqn(3.46) for 32 aromatic solutes is given in figure 3.15. Although some justifiable comments can be made regarding the use of linear regression techniques on logarithmic equations [334], the correlation described by eqm(3.46) certainly appears to be significant, and it may be used in an elegant way to reduce the parameter space for the optimization of RPLC separations (section 5.4.2).

Eqm(3.46) thus appears to hold reasonably well for the methanol-water system. However, it is obeyed much less strictly in the system tetrahydrofuran-water and not at all in the system ACN-water. There appears to be no sensible explanation yet for these anomalies. Data observed on the correlation between In k, and S for these two binary mixtures are also included in table 3.1.

The concept of iso-eluotropic mobile phases

As we saw above (eqn.3.34), the solubility parameter concept provides a very simple rule for approximating the polarity of a mixture. For a binary mobile phase containing water (W) and methanol (Me) the sum of the two volume fractions should equal 1, hence

(3.47)

is a very simple equation for the polarity of such a mixture. According to eqn.(3.47), a given mixture of methanol and water will have a given polarity somewhere in between the polarity of pure methanol and that of pure water. Of course, the same solubility parameter

63

may be obtained with several other mixtures. In general, a mixture of a given polarity may be formed by mixing two solvents, one with a higher polarity than the required value and the other with a lower polarity.

. I 10

5 -

-

.

/*

5=2.27+079 Ink, corr coeff. :0.98

C ~

5 Ink, - 10

Figure 3.15: Variation of the slope with the intercept for linear retention vs. composition curves in RPLC for 32 aromatic solutes on an ODS column using methanol water mixtures as the mobile phase. Parameters S and In k, correspond to eqn. (3.45). Figure taken from ref. [322]. Reprinted with permission.

Table 3.1: Parameters describing the linear relationship between the slope and the intercept of linear retention vs. composition curves in RPLC (eqn.3.46). Data taken from refs. [322] and [333] (1).

Stationary phase Modifier p 9 r (2)

Lichrosorb RP-18 Hypersil ODS Hypersil ODS Nucleosil 10-RPI 8 Nucleosil 10-RP18 Lichrosorb RP-18

MeOH 2.27 0.79 0.98 MeOH 3.73 0.74 0.96 MeOH 3.62 0.79 0.96 MeOH 3.55 0.69 0.94 MeOH 2.97 0.75 0.93 MeOH 3.50 0.73 0.98

Lichrosorb RP-18 ACN 5.87 (3) - - 0.06

Lichrosorb RP-18 THF 4.33 0.78 0.76

(1) Data in ref. [333] are acompilation from other sources. The values for the intercept were multiplied

(2) Correlation coefficient. (3) Average S value, since no correlation was observed.

by a factor of 2.3 (In 10) to account for the use of natural logarithms in the present text.

64

To a first approximation (eqn.3.29) we may expect mixtures of the same polarity to yield the same capacity factors. In other words, mixtures with the same solubility parameters are expected to have the same eluotropic strength, and therefore they might be called iso-eluotropic mixtures. If we use T H F (T) instead of methanol in a binary mixture with water, the following equation relates two iso-eluotropic mixtures

where cpr is the volume fraction of THF. From this equation it follows that

'Me - 'W c p r = 'PMe '

ST-',

(3.48)

(3.49)

If we use 15.85 for the solubility parameter of methanol, 25.52 for water and 9.88 for T H F (see table 2.2), we find that

Hence, a mixture of 50% methanol in water is expected to yield roughly the same capacity factors as a mixture of 31% T H F in water. Similarly, for acetonitrile with a solubility parameter of 13.14, we find that

(3.51)

These very simple relationships can be verified experimentally as is shown in figure 3.1 6. The iso-eluotropic compositions of binary mixtures of T H F and acetonitrile with water have been plotted against the binary methanol-water composition. The thin straight lines indicate the theoretical relationships from solubility parameter theory (eqns. 3.50 and 3.51). The thick lines correspond to average experimental data over large numbers of solutes [335]. An (average) experimental data point can be found as follows.

For a particular solute, a capacity factor of 3 may be found in a 50150 mixture of methanol and water. For the same solute, a mixture of 34 %THF in water may also yield a capacity factor of 3. For a different solute, the capacity factor in a 50150 methanol/water mixture may be 30, and the same capacity factor may be observed with a 28/72 THF/water mixture. The average composition for many solutes that yield the same capacity factor as the 50/50 methanoVwater mixture yields the (average) experimental point on the solid line for THF at cp = 0.5 in figure 3.16.

Due to specific effects, the corresponding compositions of methanol and THF will not be exactly the same for all solutes. Conversely, when the iso-eluotropic composition is taken as the average of that observed for many solutes (or from solubility parameter theory), some solutes will be eluted later than with the original methanol/water mixture, and some will be eluted earlier. The relative differences may amount to a factor of two for certain solutes. This should not be seen as an error in establishing iso-eluotropic mixtures. Rather, it enables us to exploit iso-eluotropic mixtures to enhance selectivity, whilst keeping retention roughly constant. This principle is widely used for the optimization of selectivity in LC.

Figure 3.1 6 shows that there is good agreement between (solubility parameter) theory

65

and experiment for the composition of iso-eluotropic mixtures in RPLC. The great advantage of this is that the composition of iso-eluotropic mixtures may be predicted for other organic modifiers than THF and acetonitrile. In table 3.2 a selection of practically feasible RPLC modifiers is given [336]. The table lists their solubility parameters and their selectivity group classification according to Snyder (see section 2.3.3). Solvents within a given group show very similar selectivities in gas chromatography (see table 2.8). Therefore, it may be expected that the specific effects observed in LC will also be similar for modifiers in the same group. For each modifier, the percentage that is equivalent to one percent of methanol in binary mixtures with water (A,) is listed in the table.

Extension to multicomponent mixtures

We can easily extend the above treatment to iso-eluotropic mixtures that contain more than one modifier. Let us rewrite eqn.(3.51) in a simplified form

(3.52)

which relates the volume fraction of a modifier j in a binary mixture with water to the volume fraction of methanol in a binary reference mixture ( ( P ~ ~ , ~ ~ ~ ) . A, denotes the ratio of solubility parameter differences:

(3.53)

According to the simple solubility parameter model any mixture of two iso-eluotropic mixtures (same value for s) would yield a mixture that is iso-eluotropic to the original two (eqn.3.34). It then follows from eqa(3.52) that for any ternary mixture of two iso-eluotropic binaries the following equation holds

I' I

Figure 3.16: Iso-eluotropic compositions for binary mixtures of THF and acetonitrile in water, relative to methanoVwater mixtures. The solid lines represent the average experimental compositions for a large number of solutes. The thin lines represent calculated compositions from solubility parameter theory (eqns.3.50 and 3.51). Figure taken from ref. [311]. Reprinted with permission.

66

(3.54) pMe,ref = P M e -k Pj’’j 9

which can be expanded to multicomponent mixtures formed by blending a series of iso-eluotropic binary mixtures

(3.55)

It follows from eqn.(3.54) that iso-eluotropic ternary mixtures fall on a straight line in a figure where the two variables pMe and pj constitute the axes. Iso-eluotropic quarternary mixtures constitute a. plane in a three-dimensional space, and so on.

Eqm(3.54) and (3.55) are very convenient for the definition of iso-eluotropic mixtures and for the calculation of the eluotropic strength of multicomponent mobile phases, in terms of a corresponding binary methanol/water mixture.

The solubility parameter model appears to work very well for the prediction of iso-eluotropic mixtures in LLC and RPLC. However, in LSC the retention mechanism is very different from the one that was assumed at the outset of this section, and hence a different model should be applied to allow the description and possibly prediction of the eluotropic strength in LSC. This model will be described in section 3.2.3.

Temperature

Unlike the relationship between retention and composition, the temperature dependence of retention in RPLC is beyond dispute. A typical “van ’t Hoff-type” equation may be used:

Table 3.2: Iso-eluotropic mixtures for RPLC.

Modifier (in water)

6 (1) Aj (2) Selectivity

to MeOH (~aI’’*.cm-~’*) relative group (3)

Methanol Ethanol n-Propanol i-Propanol

Acetonitrile

Acetone

THF 1,4-Dioxane DMSO

15.85 13.65 12.27 12.37

13.14

10.51

9.88 10.65 13.45

1 0.81 0.73 0.74

0.78 0.64

0.62 0.65 0.80

I1 I1 I1 I1

VIb VIa

111 111 111

(1 ) ref. [303] (2) Eqn.(3.53) with Sw=25.52. (3) See section 2.3.3.

67

In k = A h / R T - A s / R + ln(ns/n,,,), (3.56)

where Ah and As are the (partial molar) enthalpy and entropy effects for the partition of the solute over the two phases, R is the gas constant and Tthe absolute temperature. n,/ nm is a phase ratio term (cf. eqn.3.29). Schematically, the temperature effect may be described by

I n k = A / T + B , (3.57)

where the coefficient A is usually positive, so that retention will decrease when the temperature is increased.

The effect of changes in temperature on retention and selectivity is not very large. Certainly, the mobile phase composition (water content) has a much more drastic effect on the retention. However, what was true for GC (cfsection 3.1) is also true for LC. Temperature and composition cannot be seen as independent variables, and a different optimum (mobile phase) composition is likely to be observed at a different temperature (see section 5.1.1).

Snyder et al. [337] have demonstrated the combined effects of the composition of a binary mixture and temperature on the retention and selectivity. An increase of the temperature has the predictable effect of a decrease in retention, with little effect on the selectivity. Since there is usually only a small margin for which the retention of all the solutes in a given sample is neither too high nor too low, drastic changes in the temperature in order to enhance the selectivity cannot be applied. However, a decrease in retention due to an increase in the temperature can be compensated by increasing the water content of the mobile phase. In the case in which methanol-water mixtures are used as the mobile phase, this is likely to result in an increase in the selectivity, because of the regular pattern of In k vs. cp lines diverging towards cp=O (cf.figure 3.14). Hence, an increase in the temperature combined with an increase in the water content of the mobile phase will usually result in an increased selectivity, while retention may be kept constant.

A disadvantage of the operation of RPLC columns at elevated temperatures may be a more rapid detoriation of the column because the silica is more rapidly dissolved in the mobile phase. This effect may also lead to a reduced reproducibility of the system (peakwiths and capacity factors).

A simple but useful equation to express the mutual effects of temperature and mobile phase composition on retention has been described by Melander et al. [338]:

Ink = A, cp(1- T J T ) + A , / T + A , , (3.58)

where A,, A,, and A, are constants for a given solute using a given stationary phase and two given mobile phase components. T, is the so-called compensation temperature*, at which the retention is independent of the mobile phase composition. For all practical

* The term compensation temperature arises from the compensation between enthalpy and entropy at the temperature T, This temperature turns out to be virtually independent of the solute in an RPLC system [339]. The magnitude of T, (200 - 300 "C) usually implies that it is a hypothetical rather than a practical temperature.

68

purposes T, can be considered as an arbitrary coefficient, the value of which may be determined from experimental retention data.

A minimum of four experimental data points is required to estimate the parameters in eqn.(3.58), similar to the experimental design employed by Snyder et ul. [337]. Of course, eqn.(3.58) can only be applied over a limited range of compositions, for example the range over which 1 < k < 10. To describe retention as a function of both temperature and composition over wider ranges of the latter, more complicated equations need to be used. A quadratic equation for the relationship between retention and composition' (eqn.3.38) can be combined with eq~(3.57) to yield

I n k = A ' ( l - u / T ) q 2 + B ' ( l - b / T ) q + C'(1-c/T) , (3.59)

where A', B' , C' , u, b and c are all constants.

tion it turns into eq~(3.57). At constant temperature this equation reduces to eqn.(3.38), while at constant composi-

The pH of the mobile phase will affect retention in liquid chromatography if the structure of the solute molecules in solution is affected by the pH. This will clearly be the case if the solute species may occur in a protonated or a non-protonated form, dependent on the pH. The pH may also affect the capacity factors of ions and neutral molecules for which this is not the case, but in this case the effects are usually small.

1. different forms of the solute (e.g. protonated vs. non-protonated) may exist in the

2. the (relative) occurrence of the different forms of the solute changes upon variation of

If two different forms of the solute exist, then the first requirement will usually be met, especially for simple monofunctional solutes. The most obvious example is the dissociation of a weak acid (HA) in an aqueous environment:

Retention in RPLC may be expected to be a strong function of the pH if

mobile phase, which show different retention times, and

the pH.

HA + H,O 7t A - + H 3 0 + . (3.60)

Because HA is a neutral (uncharged) molecule and A - is a negatively charged ion, the

The second condition depends on the range of pH variation with respect to the retention between the two species of A may be expected to be very different.

equilibrium constant of the solute. The dissociation constant of HA is defined as

K, = [A-1 [H,O+l [HA] (3.61)

so that for the ratio between the two different species of A ( r A )

rA = [A-]/[HA] = K,/[H,O+] (3.62)

or

69

Eqn.(3.63) shows that the greatest variation of rA can be expected around pH = pK,, where rA = 1. When the pH exceeds the p K , value by two units, then rA = 100, so that more than 99% of the solute is dissociated. When the pH is two units below pK,, then rA = 0.01 and less than 1% dissociation occurs. Hence, the second condition will be met if the pH is varied in the region of the pK, value of the solute. Since for silica-based RPLC columns the working range is limited to 2 < pH < 7, solutes with 1 <pK,< 8 may be expected to show variations in retention upon changing the pH over the entire range allowed. In other words, for weak acids with a pK, value in the above range the pH is a parameter of interest.

A similar argument can be set up for basic solutes. The following schematic reaction may serve as an example:

X + H,O e XH+ + OH- (3.64)

and a dissociation constant may be defined as

so that

rx = [XH+] / [A = K, / [OH-] (3.66)

and

log rx = log K, + log [OH-] = 14-pH-pKb. (3.67)

Hence, the greatest variation in rx is observed for pH values around pH = 14 - pK, *. On silica-based RPLC columns the range of 2 < p H < 7 will correspond to bases with 7<pK,< 12. For these very weak bases pH is a relevant parameter in RPLC. Much stronger bases @K,< 6) will be protonated completely, or almost completely in RPLC using silica-based CBPs. On these columns, their ionization cannot be suppressed, and they should be chromatographed as ions, either on ion-exchange or on ion-pairing systems (see section 3.3 below).

There has been much recent interest in the development of column packings for RPLC that can be used over a wider pH range. Potential materials include organic polymers, carbon packings and (modified) alumina. Clearly, such stationary phases will be most relevant for the separation of basic solutes.

If a buffer is present in the mobile phase, then the dissociation will be controlled by the pK, value of the buffer. Because of the pH limitations of silica-based stationary phases, only weak acids can be used as buffer compounds. As an example, we will consider the dissociation of a weak acid (HA) in the presence of a buffer acid (HB).

~~

* In an aqueous environment the p K , and p K , values, which correspond to the same reaction, may be related by pK,+pK,=14.

70

The following equation can readily be derived [340]:

[A -1 rA =-

HA1

[A-I[H,O+l [HBI [B-I [HA1 [B-I[H,O+l [HBI

.- - -

(3.68)

where Ka.A is the dissociation constant of the solute, K a , B the dissociation constant of the buffer, rB the dissociation ratio of the latter and K , , is a constant. Hence, the dissociation ratio of A is proportional to the dissociation ratio of the buffer. The ratio rB is a more practical quantity than the pH in mobile phases other than pure water [341]. The pH is ill-defined in such mobile phases, but the ratio of the salt and the acid that constitute the buffer in pure water a known quantity.

An equation for the observed capacity factor (kobs) can be derived if it is assumed that the acid-base equilibrium kinetics are fast enough to be considered as instantaneous. This corresponds to the observation of a single sharp peak for the ionizable species in the chromatogram. In that case it may be assumed [340,342,343] that for the weak acid HA

[ HA1 [A-I [HA1 + [A -1 + k A [HA]+[A-] kobs = k H A

(3.69) - - HA + k ~ r ~ 1 + rA

or with eqn(3.63)

(3.70)

Eqm(3.70) is a three-parameter equation which describes the relationship between retention and pH for weak acids. If the dissociation ratio of a buffer ( r B = [B-]/[HB]) is used as a variable instead of pH, then a combination of eqm(3.68) and (3.69) leads to

(3.70b)

which again is a three-parameter equation. Similar equations can also be derived for weak bases.

Figure 3.17 shows the observed capacity factor for a weak acid as a function of the pH. Arbitrary values of k, = 1 (for the solute ion) and k H A = 9 have been assumed. It is clear from the figure that only in the vicinity of the p K , value can a large effect of the pH on retention be observed. If the pH range studied ranges from pH = p K a + 1 upward, then the variation in kobs is less than 1O0/oof the total variation between k , and kHA. A similar minor effect is observed at pH values below p K a - 1. Figure 3.17 shows that complicated sigmoidal curves are obtained for retention as a function of the pH over a range encompassing the pK value of the solute. Simple approximations are not possible in this

71

range. Plotting different transformations of the variables will not simplify the picture*. A practical example of the variation of retention with pH in RPLC can be found in

fig.5.17. Three experiments are in principle sufficient to establish the three coefficients in

eqn(3.70) for a given solute. In practice this is only true if the three experiments are taken at such values of the pH (relative to p K J that a sensible estimate of all three coefficients can be made. This implies one experiment within one pH unit of the p K , value, one experiment at a higher and one at a lower pH. If the p K , value of a solute is known, then the retention behaviour can be estimated from a minimum of two experiments. Another way to reduce the minimum number of required experiments is to assume a negligible capacity factor for the charged species. Of course, once more experimental data points become available initial assumptions about the value of any of the coefficients in eqn.(3.70) can be abandoned.

Multivalent ions

So far we have discussed the behaviour of monoprotic acids and monofunctional bases. The behaviour of bifunctional or trifunctional acids and bases is not completely different.

I , L

PKo+3 PH - pKa -3 PKO

0 L, '

Figure 3.17: Schematic illustration of the variation of the observed capacity factor (kobs) with the pH for a weak acidic solute in RPLC according to eqn.(3.70). The retention line is calculated from eqn.(3.70) assuming k, = 1 and k H A = 9.

* This may only be done if both k , and k,,, are known, for instance form measuments performed at pH values well above and well below the p K , value. In that case the following transformation of eqn.(3.70) will result in a straight line:

72

The most important equilibria are those in which one of the two species is ionized, while the other is neutral. In many situations in RPLC both the monovalent and di- (tri-, etc.) valent ions have negligible capacity factors in comparison with the neutral molecule. Therefore, often all but one of the dissociation constants of an oligovalent species can be neglected. Equations for the influence of the pH on the retention of diprotic and oligoprotic substances can be found in ref. [316], p.239 et seq..

More value should be assigned to the possible formation of multivalent ions if separations based on the ionic character of the solute are considered (section 3.3).

A special place is taken by polyelectrolytes. These are molecules containing a large number of ionizable groups, each of which can be assumed to behave in a manner similar to a single weak acid or base. However, the combination of a large number of these functionalities gives rise to a total charge of the molecule that varies continuously from a large positive number at a low pH to a large negative number at a high pH. Proteins are good examples of such polyelectrolytes. At some pH, the overall charge of the molecule will be zero. This does not imply that there is no charge on the molecules, but only that the total number of positive charges equals the number of negative charges. This point of zero electric charge is called the isoelectric point, and the pH value at which it occurs is denoted as PI. Polyelectrolytes show a variation of retention with pH over the entire range. However, it should be noted here that their chromatographic behaviour is complicated. It is not usually possible to isolate different retention mechanisms. In the chromatography of proteins ionic mechanisms, physico-chemical interactions and size exclusion may all play a role [344].

Ionic strength

The ionic strength of the eluent will affect the retention of both neutral and ionized species. For non-charged molecules the effects of increasing the ionic strength of the eluent can be understood as an increase in the mobile phase polarity, leading to an increase in retention (“salting-out effect”).

For charged species it was shown by van der Venne et al. [345] and by Otto and Wegscheider [346] that the so-called Davies equation can be used to describe the effect of the ionic strength on solute retention quantitatively. This equation reads

(3.71)

where k, is the capacity factor at zero ionic strength, A a temperature-dependent constant (equal to 0.512 at 25 “C [345]), z the charge of the solute ions and 1 the ionic strength (M). It can be seen from the above equation that the ionic strength will have a similar effect on solutes of similar charge. Only if the charges of the components to be separated are different can the ionic strength be used to vary the chromatographic selectivity.

Summary

At the end of this section we may summarize the advantages and disadvanatages of RPLC. The parameters that may be used for the optimization of the selectivity will be summarized in section 3.5.1 (table 3.10~). The major advantages of RPLC are:

73

1. RPLC is a veryflexible chromatographic method. It can be applied to a great variety of samples (see figure 3.7).

2. Theoretically, RPLC ofiers a superior selectivity for all but the very polar samples (see figure 3.8).

3. RPLCcolumns with chemically bonded (alkyl) stationaryphasesprovide rapid equilibration, high eflciency and symmetrical peaks.

4. RPLC is compatible with aqueous samples, but also with a number of organic sample solvents.

5. The aqueous mobile phases used in RPLC allow the use of buffers in the mobile phase. This may lead to improved selectivity and eficiency. Secundary (ionic) equilibria other than acid-base dissociation may also be used (see section 3.3.2).

Some disadvantages of RPLC remain. These may be summarized as follows: 1. The current alkyl modified silica stationary phases are not truly non-polar. This leads

to a reduction of the overall selectivity (see figures 3.7 and 3.8). 2. Residual silanol groups are present on the surface, which may have a negative effect on

the peak shape of basic solutes and poly-electrolytes in particular. 3. Current RPLC stationary phases are only stable over a limited range of pH. A reliable

working range is 2<pH< 7. 4. The mechanism of RPLC is still poorly understood and reliable quantitative theories are

not yet available. It appears from this summary, that most of the disadvantages of RPLC are not fundamental, but connected to the use of a particular kind of stationary phases (alkyl modified silicas). It may therefore be expected that advances in this area will further enhance the potential of RPLC.

3.2.2.2 Polar bonded phases

Besides the typical RPLC stationary phases with n-alkyl groups, various other functional groups may be chemically bonded to the surface in a similar way as described in section 3.2.2.1. A selection of possible bonded bases, arranged roughly in order of increasing polarity, is given in table 3.3.

Apart from the perfluorinated phases, the polarities of the CBPs in table 3.3 are typically intermediate between those of the non-polar alkyl phases and polar adsorbents such as silica. As we saw in figure 3.8, such phases may be operated with a polar eluent in the Reversed Phase mode, or with a non-polar (or weakly polar) eluent in the Normal Phase mode. The elution order of the sample components will be reversed in these two cases. A clear example of this phenomenon has been described by Kirkland [347] for the separation of some urea herbicides using a CBP with aliphatic ether groups.

We also saw in figure 3.8 that moderately polar stationary phases may be most useful in the reversed phase mode for the separation of very polar solutes, which cannot be sufficiently retained on reversed phase (alkyl) materials. A good example of this is the separation of carbohydrates on amino bonded phases.

Chemically bonded phases may also be tailored to a specific separation problem. A case in point is the synthesis of chiral stationary phases for the separation of optical isomers.

Another application of polar bonded phases, which is beyond the scope of this book,

74

Table 3.3: A selection of chemically bonded phases featuring different functional groups.

Type Functional group

RP-F (Perfluorinated) - CnF*n+l

RP-n (n-alkyl) - CnH2n + 1

Cyclohexyl TP Phenyl -0 Cyano - C = N

Diol - c& - CEL

Amino - NH2

is to be found in size exclusion chromatography, where the chemical modification is aimed at minimizing rather than enhancing specific interaction between the solutes and the stationary phase.

In principle, the composition of the stationary phase may be varied by using “mixed phases”. Phases which incorporate different functional groups in a given ratio have been synthesized (see for example ref. [348]). However, retention may not be expected to vary linearly with the composition of such mixed phases in a manner similar to what is observed with mixed liquid phases in GLC (section 3.1.1). This, combined with the complications involved in preparing mixed phases and the irreversibility of bonding reactions, excludes the composition of mixed CBPs as a practical parameter for optimization purposes.

The stability of polar bonded phases is generally considered to be less than that of n-alkyl phases, because the Si-0-Si-C bonds are less effectively shielded against nucleophi- lic attacks (ref. [316], p.125).

The mechanism of separation on polar bonded phases is not clear. Due to their limited proliferation, no theories have been developed solely to descibe this particular form of liquid chromatography. Instead, descriptions from other fields may be applied. If polar bonded phases are used in combination with more polar mobile phases in the reversed phase mode, then the same rules may be applied as in RPLC (section 3.2.2.1) to describe the effects of the mobile phase, the temperature, the pH, etc..

When used with less polar mobile phases in the normal phase mode, LBPC may be treated similarly to LSC (section 3.2.3). The same solvents may be recommended for use in these two forms of LC (ref. [349], p.284). However, the use of particular bonded phases may impose some constraints on the choice of solvents. For instance, amino type columns would be modified (reversibly) by mobile phases containing acetic acid and (irreversibly) by acetone.

75

3.2.3 Liquid-solid chromatography (LSC)

In liquid-solid chromatography (LSC) the solute is distributed between a liquid mobile phase and a solid surface. A distribution coefficient may be defined in an analogous way as in GSC (eqn.3.16): ‘a,, = <,s/Ci,m 2 (3.16)

where Ka,, is the adsorption coefficient (ml/g), c:,~ the concentration of the solute on the stationary phase (mole/g) and c,,, as usual, the concentration in the mobile phase (mole/ml).

In principle, the adsorption isotherm is non-linear, i.e. K,., varies with varying c , , ~ . Only at very low concentrations may an approximately linear distribution isotherm be assumed. Therefore, LSC techniques suffer from a low sample capacity.

Snyder [350] has given an early description and interpretation of the behaviour of LSC systems. He explained retention on the basis of the so-called “competition model”. In this model it is assumed that the solid surface is covered with mobile phase molecules and that solute molecules will have to compete with the molecules in this adsorbed layer to (temporarily) occupy an adsorption site. Solvents which show a strong adsorption to the surface are hard to displace and hence are “strong solvents”, which give rise to low retention times. On the other hand, solvents that show weak interactions with the stationary surface can easily be replaced and act as “weak solvents”. Clearly, it is the difference between the affinity of the mobile phase and that of the solute for the stationary phase that determines retention in LSC according to the competition model.

Snyder [350] formulated the following equation to describe the above effect quantitatively:

log K,., = log V, + a (Sp - A , 8) . (3.72)

In this equation is the adsorption energy of the solute on a standard adsorbent. A, is the adsorption area of the solute molecule. 8 is the adsorption energy of the solvent per unit area on the same standard adsorbent*, usually referred to as the solvent strength or eluotropic strength. a is the activity of the adsorbent and V, is the volume of the adsorbed solvent per gram of stationary phase. Hence, V, can be seen as a compensation factor for the dimensions of K,.,. K a / V, is a dimensionless quantity.

The choice of units and standards for the remaining variables is arbitrary. The following conventions were followed by Snyder [350]: standard adsorbent (a= 1) : dry alumina standard solvent (8 = 0) : n-pentane standard solute area (A,=6) : benzene**

* 8 is the solvent adsorption energy (S$ equivalent to area of the solvent (As). Hence, the competition factor can be rewritten as

for the solute) divided by the adsorption

sp - A i 8 = / t i ( - g --) - q Ai As

This expression shows more clearly that the competition is based on the adsorption energy per unit area, i.e. e / A , vs. e / A r ** This convention for A, implies that one unit in A, corresponds to approximately 8.5 A*.

76

Using these conventions, values for a (different adsorbents), 8 (different solvents), 9 and A i (different solutes) can be established from chromatographic experiments (for procedures see ref. [350]).

Eqn. (3.72) describes retention in LSC in terms of separate parameters for the adsorbent (V,, a), the solute (9, A i ) and the solvent (8). As such, it has proved invaluable for the interpretation of retention and selectivity phenomena in LSC. For example, the effect of a change in the solvent using the same stationary phase and the same solute can easily be understood in terms of a variation in EO.

Unfortunately, this is no longer true once several parameters are changed at the same time. For instance, the value of 8 appears to depend not only on the solvent, but also on the adsorbent. Hence, different 8 values have been tabulated for different adsorbents. Some examples of values for common solvents are collected in table 3.4.

Clearly, the values for 8 are different on silica and alumina. They are very different from those observed on carbon, which is partly, but not entirely due to the different conventions used in this case by Colin et al. 1351). Likewise, the solute parameters Ai and

will be different for different adsorbents. Hence, eqn.(3.72) gives a consistent description for one particular stationary phase. A new set of parameters will have to be established for each new adsorbent.

As a consequence, the influence of the stationary phase on retention and selectivity cannot be explained on the basis of eqn.(3.72).

Influence of the mobile phase

The influence of the solvent in LSC is described by the solvent strength parameter 8.

Table 3.4 Eluotropicstrength (E’, eqn.3.72) of some common solvents for LSC. Data taken from refs. [349] en [351].

&O 6 Select. Solvent (1) group

silica alumina carbon (2)

Alkanes 0.01 0.01 0.10-0.17 7 1 Chlorobutane 0.20 0.26 0.13 8.42 V Chloroform 0.26 0.40 0.18 9.87 Vll l Methylene chloride 0.32 0.42 0.13 10.68 V Isopropylether 0.34 0.28 - 7.60 I

THF 0.44 0.57 0.14 9.88 I11 Acetonitrile 0.50 0.65 0.04 13.14 VI Methanol 0.7 0.95 0 (3) 15.85 I 1

Ethyl acetate 0.38 0.58 0.13 9.57 VI

(1) Solubility parameter in ca1”2.cm-3’2 [303]. (2) See section 2.3.3. (3) By definition, see ref. [351].

77

According to eqn.(3.72) a higher value of 8 will result in a lower value for the capacity factor. It can be concluded from table 3.4 that the solvent strength on silica and alumina stationary phases roughly increases with increasing polarity (6) of the solvent, but that there is no quantitative correlation between these two solvent properties. For example, ethers are much stronger solvents (especially on silica) than can be anticipated on the basis of their solubility parameters.

Although the number of pure solvents that is compatible with the LSC system is much larger than it is for RPLC (section 3.2.2.1), mixtures are frequently employed as mobile phases. Snyder [350] has developed the following equation for the dependence of the solvent strength (8) on the composition (N, the mole fraction of the stronger solvent B) in a binary mixture

G B = + (l /an& log { N b 10 anb(G - 4) + 1 - Nb} (3.73)

where nb is the molecular size of the solvent B and a is again the activity of the stationary phase. Figure 3.18 shows the variation of GB with the composition for a series of binary mixtures in which n-pentane is the weaker of the two solvents. The experimental points in the figure show that at least for these mixtures eqn.(3.73) provides a good quantitative description. According to eqn.(3.72) log (1 / K,,J is proportional to G B and the same is true for log (l /k). Therefore, the variation of retention with composition in LSC appears to be rather complicated.

I , I I I I I t I I , I I I I I t I J 20 LO 60 80 100

% Blvlv) - Figure 3.18: Variation of the eluotropic strength (GB) with the composition of the mobile phase in LSC (eqn.3.73). Weak solvent is n-pentane; Strong solvents (from bottom): carbon tetrachloride, n-propylchloride, methylene chloride, acetone, pyridine. Figure taken from ref. [349]. Reprinted with permission.

78

I 2 I

Figure 3.19: Variation of retention with composition in LSC according to the simplified linear relationship of eqn.(3.74). Stationary phase: Lichrosorb ALOX T (Alumina). Mobile phase: n-propanol (rp is volume fraction) in n-heptane. Solutes: lumisterol (l), tachysterol (2), calciferol (3) and ergosterol (4). Figure taken from ref. [357]. Reprinted with permission.

However, a much simpler relationship between retention and composition may often be observed in practice. It was suggested by Soczewinski [352,353] that a plot of In k vs. In X, (where X , is the mole fraction of the stronger solvent B) would yield a straight line, according to

I n k = c - n I n X , , (3.74)

where c and n are both constants. In fact, as pointed out by Jandera and ChuraEek [354], Soczewinski’s eqn.(3.74) is a simplified version of the treatment of Snyder.

Figure 3.19 shows an example of the linear variation of retention with composition according to eqm(3.74). In this figure the logarithm of the volume fraction (q) of the stronger solvent is plotted on the horizontal axis. Plotting In X , will lead to a similar linear plot. The simple equation of Soczewinski (eqn.3.74) often yields a very good description of experimental data in LSC [353,355,356].

Small percentages of strong solvents (so-called “modulators”) are seen to have a drastic effect on the value of .z0 (see figure 3.18). This is especially true for the most polar solvent of all, water. Therefore, the water content of the mobile phase and the extent to which the

79

stationary surface is covered with water are critical parameters in LSC. Besides a drastic reduction in the retention time, an improved peak shape may be the result of the addition of water to the mobile phase. Both the column efficiency and the reproducibility of the analysis may improve as a result.

Isosluotropic mixtures

Figure 3.20 is a graphical representation of the eluotropic strength in LSC. This nomogram was originally published by Saunders [358]. The solvent strength parameter 8' increases from left to right in the figure as indicated on the top axis. Every other horizontal line represents a particular binary combination of two solvents with the compositions indicated. It is clear that the scale division on these lines, which correspond to eqn.(3.73), is highly non-linear.

The vertical dashed line illustrates how a series of iso-eluotropic mixtures can be located. Mixtures of about 75% methylene chloride in n-hexane, 49% diethyl ether in n-hexane, 50% methylene chloride in 2-chloropropane, 46% diethyl ether in n-hexane, 1.5% acetonitrile in 2-chloropropane and 0.1% methanol in 2-chloropropane all show similar solvent strengths (8 = 0.30).

As in RPLC, these iso-eluotropic mixtures are expected to yield similar capacity factors, but may give rise to certain specific effects towards certain (types of) solutes, which may be exploited to enhance separation.

Of the many different iso-eluotropic solvent mixtures, not all are equally attractive from

solvent strength E O (silica) 0 .05 .10 .15 .20 .25 .30 .35 .LO .L5 .SO .55 .60 .65 .70 .75

0510

0 1 3 5 10 50 :1w i in H~

l ' f i ' 1 I I 1 ' 1 1

0 1 3 5 10 p 1w : I Il%Et20inHx

O N ) 3odm

7 1 I , * , I ,

1-p yz in i Pr CI

I I I I % EtzO in iPrCl 0 .5 1 i2 '3 5 I0 30 50100

I I I % A C N ~ ~ ~ P ~ C I -5 1 2 3 5 10 zp :050 100 , I I I IOhMeOHiniPrCI

0 ; : ' 0 60100

%EtzO in MC 01235 10 lU)

Hx , hexane iPrCI.isopropyl chloride 7 ? 5 2; 3p loo MC,methylene chloride Et20,ethyl ether l w ] % A C N in EtzO

MeOH. methanol ! I %MeOH in EtzO

I ! I I %ACN in MC

:IH%MeOHinMC 0 510

ACN. acetonitrile 0 .5 1 2 3 5 10 30

1 0 , 'P % MeOH in ACN 01 35 'p ,

Figure 3.20: Nomogram illustrating the solvent strength of various binary solvent mixtures for LSC. Vertical (dashed) line illustrates a series of iso-eluotropic solvents (see text). Figure taken from ref. [358]. Reprinted with permission.

80

a practical point of view. Moreover, to allow a reasonably efficient search for optimum conditions, it is necessary to select a few solvents.

Glajch et al. [359] have suggested the use of methylene chloride, methyl t-butyl ether and acetonitrile as modifiers in n-hexane (see section 5.5.1).

Eqn.(3.73) suggests that any mixture of two solvents with the same 8 value (iso-eluotropic solvents) will also have the same eluotropic strength. This would allow the application of a similar strategy for the definition of iso-eluotropic multicomponent mobile phase mixtures as was used for RPLC in section 3.2.2.1. In practice, the situation in LSC has proved to be more complicated, because an effect described as “solvent localization” limits the validity of eqm(3.72) and (3.73) if polar components (such as acetonitrile or methyl t-butyl ether) are present in the mobile phase. This makes it difficult to calculate the composition of iso-eluotropic mixtures for LSC with sufficient accuracy for optimization purposes [360-3631.

In practice, as a first approximation, it may be assumed that mixtures of iso-eluotropic solvents can be used. If the resulting solvent strength is either too high or two low, it may be corrected by the addition of more or less n-hexane.


For many years silica has been the dominant adsorbent used in LSC. Silica has the advantage of abundant availability. It can be obtained commercially in different particle sizes (spherical or aspheric materials), different specific surface areas and different pore size distributions. Because the specific surface area is usually large (up to about 400 m2/g), the sample capacity of silica is relatively high.

A disadvantage of the availability of many different silicas is the limited reproducibility of the packing materials. Apart from the factors described above, the chromatographic behaviour of the silica can be affected by chemical factors such as the structure of the surface (affected by heat treatments and by washing the column with acidic or basic solutions), the history of the material (previous usage) and the presence of contaminants (e.g. metal ions). The water content is another major factor. Physically adsorbed water can be removed from or added to the surface, but water bound to the surface as silanol groups ‘(chemisorption) cannot be introduced or removed once the silica is packed into the column.

In general, different silicas from different manufacturers may show large differences in retention and selectivity and even between different batches of the same (nominal) product from the same manufacturer the differences may be considerable. For these reasons, it is often necessary to re-optimize the separation (mobile phase) if a new column is installed for an existing separation method.

Historically, alumina used to be one of the standard adsorbents for LSC. Snyder (ref. [350], chapter 11) has compared the chromatographic selectivity of silica and alumina surfaces extensively. Alumina may offer some advantages over silica, especially for separations that can be enhanced at high pH values. In recent years therefore, there has been a revival of interest in alumina and its applications in LC [364].

Another material that has recieved considerable attention in recent years is carbon [365]. Carbon can be used as an adsorbent for LC in one of several forms, such as pyrolytic carbon (either as such or as a thin layer covering silica particles), glassy carbon and

81

graphitized carbon. An especially interesting carbon material for LC appears to be the porous graphitized carbon (PGC) described by Gilbert et al. 13661.

The advantages of carbon surfaces are their chemical inertness and stability. However, it is difficult to prepare carbon packings with the same specific surface area, pore size (distribution) and pore volume as typical silicas. Moreover, although purely carbonaceous materials will in theory be highly non-polar reversed phase materials, all materials prepared so far are found to behave as fairly polar surfaces, requiring fairly non-polar (high methanol content) mobile phases for the elution of the solutes. Figure 3.8 suggests that the present carbon-based materials are most useful for the RPLC separation of polar compounds.

Temperature

The effects of temperature in LSC are similar to those observed in RPLC. Eqn.(3.57) may be used for a quantitative description. The temperature is usually not considered to be a relevant parameter. A typical change of 2% in k for 1 "C variation in the temperature (ref. 13491, p.390) may, however, warrant a careful control of the column temperature.

3.3 SEPARATION OF IONS IN LC

The LC methods discussed before were based mainly on physico-chemical interactions between the solute on the one hand and the two chromatographic phases on the other. Although we have seen that in RPLC the degree of ionization of weakly acidic or basic solutes may be a major factor in the control of retention and selectivity, the ionic species themselves were not exploited purposefully to realize or enhance the separation. In fact, in a typical RPLC system all fully ionized solutes will show little retention and therefore little resolution can be achieved between different ions. The methods described in this section make positive use of the ionic character of solutes to create a chromatographically selective system.

3.3.1 Ion-exchange chromatography (IEC)

In ion-exchange chromatography (IEC) ionic or, rather, ionizable groups (R) are permanently present on the surface of the stationary phase. In the absence of the solute, these groups are all masked by a counterion (0, which is present in the mobile phase in a constant concentration. Retention is based on an opposite charge between the solute ion (denoted below as the anion X- or the cation YH+) and the ionic groups on the stationary phase. The counterion has a charge similar to that of the solute ions. Typically, the following schematic reactions can be used to describe the IEC process:

E R + U - + X- 3 R+X- + u- (3.75)

or 3 R-U+ + YH+ e = R-YH+ + U + . (3.75a)

In the first case we speak of anion-exchange, since the exchanged ions U- and X - are negatively charged. The second reaction illustrates a cation-exchange process. It is clear that different stationary phases will typically be used for the two types of IEC. Thus, there

82

cahon exchanger fraction

strong weak

0 2 L 6 8 1 0 1 2

weak

0 2 L 6 8 10 12 - PH

Figure 3.21: Variation of the capacity (fraction of charged functional groups) of typical ion-exchange materials with the pH of the mobile phase. Top: cation-exchangers; bottom: anion exchangers. Ionized fraction is fraction associated with counterions. Remaining fraction is associated with H30+ or OH- ions.

are anion-exchange columns, with positive groups on the surface, and cation-exchange columns with negative groups.

A further differentiation can be made from a classification in strong and weak ion-exchange materials. A strong ion-exchanger possesses functional groups which are always ionized. This implies that over the practical range of pH values the capacity of the stationary phase remains unaltered. Weak ion-exchangers are affected by the pH. Weak cation-exchangers gradually lose their exchange capacity if the pH is decreased. With weak anion-exchangers this occurs when the pH is increased. The variation of the degree of ionization of the functional groups on the surface, a quantity that is directly proportional to the ion-exchange capacity, of some typical stationary phases is illustrated in figure 3.21. Examples of the different kinds of exchangers are given in table 3.5.

The different stationary phases can also be classified on the basis of their physical structure. Pellicular materials, the particles of which consist of a hard (glass) core covered by a thin layer of an ion-exchange resin, may be used if a moderate efficiency and a small ion-exchange capacity are acceptable, but not if the column is required to have a high

Table 3.5: Examples of different types of ion-exchange materials.

Cation

~~

Anion

Strong Sulfonate - so, Quaternary amines - NR:

Weak Phosphonate - Po;- Tertiary amines - NRZ Carboxylate -coo- Secondary amines - NR+

83

stability. Glass beads covered with a styrene-divinylbenzene cross-linked copolymer backbone to which the functional (ion-exchange) groups are chemically bonded may be employed over the entire range 1 < pH < 13 [3671.

As an alternative to pellicular materials, microparticulate stationary phases may be used. These are either based on organic resins or on inorganic oxides. The latter class contains bare oxides, as well as chemically bonded phases, which may be synthesized in a way similar to that described in section 3.2.2.1, but the functional end group is now an ionic one.

Resin based materials offer a greater chemical stability (large pH range), whereas silica based materials are mechanically more stable and allow a wider range of (organic) solvents to be used. All microparticulate phases offer a high column efficiency and a large ion-exchange capacity. Therefore, in modem HPLC they are usually preferred to pellicular packings.

The retention and selectivity in IEC are influenced by a number of parameters, which we will discuss below.

Counterion concentration

The concentration of the counterion can be used to control the retention in IEC. It plays a role similar to that of the eluotropic strength of the eluent in RPLC or LSC, in that it affects retention much more than it does selectivity. The capacity factor can be related to the distribution coefficient of the solute ( D J

k = D , ( V , / V , J . (3.76)

D, is the ratio of the total concentrations of solute ions in the two phases. The total concentration may involve “free” ions, protonated ions, absorbed molecules and all other forms of the sofute ion X:

(3.77) D, = I: [“A?’], / I: rlr.],,, .

For the exchange of strong ions on strong exchangers very simple equations result:

D, = [RxJ / [ X - 1 (3.78)

for (monovalent) anions or

D, = [RYH] / [YH+] (3.78a)

for cations. For a weak monovalent anion the situation is slightly more complex and the distribution coefficient becomes

A similar equation can be written for a weak cation:

(3.79a)

84

If we introduce the exchange equilibrium constant ( Km), which reads for monovalent anions (see eqn.3.75)

then we find with eqn.(3.78)

and with eqn.(3.79)

(3.80)

(3.81)

(3.82)

where K,,is the acid dissociation constant for solute X(eqn.3.61). A similar equation can readily be derived for cations. From eqm(3.81) and (3.82) it is clear that for monovalent counterions, D (and hence the capacity factor k) is inversely proportional to [ U-1, SO that

In k = In [U-] + In k, , (3.83)

where k, is the capacity factor observed at a (hypothetical) unit concentration of coun terions.

According to eqn.(3.83) a plot of In k vs. In [ U-] will yield a straight line, with a slope that is independent of both the kind of solute and the kind of counterion. The slope will only vary if the valence of the solute (a) or the valence of the counterion (b) changes. This can easily be accounted for in eqn.(3.83), which reads for the general case

In k = (a /b ) In [vl -t In k, . (3.84)

The validity of eqn.(3.84) is demonstrated in figures 3.22% b and c. In figures 3.22a and b the retention of some nucleotides in IEC is shown. The counterion is monovalent potassium dihydrogen phosphate. The figure shows a series of straight lines, the slopes of which are in good agreement with the predicted values from eqa(3.84): 0.96 for the monophosphates (solutes 1 to 5), 1.85 for the diphosphate (solute 6) and 3.03 for the triphosphate (solute 7).

Figure 3 .22~ illustrates the validity of eqn.(3.84) for the IEC separation of some inorganic anions. The counterion in this example is a mixture of monovalent and divalent phthalate ions. At the selected pH of 5.3 the average charge of the phthalate ions is 1.52. According to eqn.(3.84) this would result in slopes for monovalent solute ions of about 0.66 and about 1.3 for divalent ions. These figures correspond reasonably well with what is observed in figure 3.22b [3681.

From eqn.(3.84) and figures 3.22a, b and c we conclude that the concentration of counterions in IEC is a primary parameter which may be used to vary retention, i.e. to bring the capacity factor into the optimum range. Only the selectivity between solutes of different valencies will be affected considerably by changes in the concentration of the counterion.

85

lo1

J

‘.

Figure 3.22: (a) and (b) Examples of the variation of retention (log k) with counterion concentration (log c) for nucleotides according to eqn.(3.84). Mobile phase: potassium dihydrogen phosphate in water, pH=3.15. Stationary phase: Perisorb AN. Solutes: 1 = thymidine 5’-monophosphate, 2 = ribothymidine 5’-monophosphate, 3 = deoxyuridine 5’-monophosphate, 4 = deoxyguanosine 5’-monophosphate, 5 = guanosine 5’-monophosphate, 6 = guanosine 5’-diphosphate, I = guanosine 5’-triphosphate. Figures taken from ref. [357]. Reprinted with permission. (c) (see opposite page) Examples of the variation of retention (log k) with counterion concentration (log c) for some monovalent and divalent inorganic anions according to eqn.(3.84). Mobile phase: Phthalic acid in water (pH = 5.3). Stationary phase: Vydac 302 IC silica-based ion-exchanger. Solutes as illustrated in the figure. System peak corresponds to the retention time of the phthalate ion. Figure taken from ref. [368]. Reprinted with permission.

Mixed retention mechanisms

In the above discussion we have assumed that no other retention mechanisms play a role in IEC other than an ion-exchange process. For instance, we have assumed in eqm(3.78) to (3.82) that other forms of the solute, such as the protonated form of an anion ( H X ) , are not present in the stationary phase. In practice, this assumption is not always correct. For this reason, different’ packing materials with the same functional groups may show different selectivities. For example, Rabel[367] has illustrated the differences in selectivity for the separation of nucleotides on two different strong anion-exchangers. The selectivity on a pellicular material was markedly different from that on a silica-based stationary phase. The percentage of cross-linking in an organic resin may also affect the selectivity [369].

86

\ * H2 POL

I\'-..- - 0.2

-0.L -2.7 -26 -2.5 -21, -23 -2.2 -23 -2.0

log c-

Mixed retention mechanisms are most evident in the separation of polyelectrolytes. These are large, multivalent molecules, which possess polar and non-polar groups (or "sites") on the surface of the molecule in solution, that may interact physically with the backbone of the ion-exchanger. The most important examples of polyelectrolytes are proteins. IEC has long been the major tool for the separation of proteins by HPLC, but it is being replaced more and more by RPLC [344]. One of the reasons for this is that due to mixed retention mechanisms broad and non-symmetrical peaks are common for the IEC separation of proteins.

Type of counterion

The type of counterion used may affect the retention considerably. The eluotropic strength of the different counterions is usually expressed as an eluotropic series. An example of this is shown in table 3.6.

Also, the type of counterion may have an effect on selectivity. Especially for anions dramatic differetices are sometimes observed (see e.g. ref. [370]).

Similarly, the choice of the buffer may have a considerable effect on the selectivity in IEC. Phosphate has been recommended as the most useful general purpose buffering agent 13671. If other buffers are used, the selectivity needs to be re-optimized.

The pH will affect the capacities of weak ion-exchangers as well as the dissociation of weak anions and cations (e.g. eqn.3.82). In these cases, the pH may be the most relevant parameter in the IEC separation process.

Analytical equations can be derived for all combinations of solutes, stationary phases

87

and counterions. However, especially if we do not limit ourselves to monovalent ions, these equations may become quite complex algebraically. For the simple case of a monovalent anion on a strong anion exchanger using a strong counterion, the relationship is given by eqn.(3.82). We find that the capacity factor of the solute is proportional to the relative dissociation, since the last factor in eqn.(3.82) may be rewritten as

(3.85)

This factor will approach unity if [H,O+] 4 KQ,x, i.e. pH > pK,, , . Hence, at pH values well above its pK, . , value, the solute will be completely dissociated. At pH = p K , . , the dissociation will be SOo/o and it will further decrease when pH < pK, . , Some schematic examples of the effect of the pH on the retention of weak acids and bases on strong ion-exchangers are shown in figure 3.23.

On weak ion-exchangers the effective relative dissociation is the product of the dissociation of the solute and that of the stationary phase. The effective relative dissociation is then

(3.86)

where s denotes the stationary phase (ion-exchanger). This may result in a maximum retention at a particular pH value, as is illustrated in figure 3.24.

Table 3.6: Typical eluotropic series for anions and cations in IEC. Ions are listed in order of increasing elution strength.

Anions Cations

Fluoride Hydroxide (OH-) (1) Acetate Formiate Chloride Thiocyanate Bromide Chromate Nitrate Iodide Oxalate Sulfate Perchlorate Citrate Hydroxide (OH-) (2)

Lithium Hydronium (H,O+) (1) Natrium Ammonium Potassium Rubidium Cesium Silver Manganese Copper (11) Calcium Strontium Barium Trivalent anions Hydronium (H,O+) (2)

(1) On stong cation and anion exchange columns. (2) On weak exchangers.

88

"strong anion exchanger" I

.:: "strong anion exchanger"

0 2 L 6 8 1 0 1 2 1 L - PH

&ax \ m 1 .J exdanger"\ \ \

"strong cation i \ \

0 2 L 6 8 1 0 1 2 1 L - PH

Figure 3.23: Variation of the capacity factor (relative dissociation coefficient, see eqn.3.85) as a function of the pH for some weak acids and bases on strong ion-exchange materials. p K , values refer to different solutes.

0 2 L 6 8 1 0 1 2 1 L

"weak cation exchanger "

0 2 L 6 8 1 0 1 2 1 L - PH

Figure 3.24 Variation of the capacity factor (relative dissociation coefficient, see eqn.3.85) as a function of the pH for some weak acids and bases on weak anion-exchangers @ K b = 6 ; top) and cation-exchangers @ K , = 7; bottom). p K , values in the figure correspond to different solutes.

Figure 3.25 shows an example of the effects of pH in IEC in practice.

Temperature

Frequently, IEC separations are carried out at elevated temperatures. This is because the kinetics of the ion-exchange process may be improved dramatically. The efficiency of the column may be increased by a factor of three if the temperature is increased from 30 to 70 O C [3711. An additional advantage of an increased column temperature is a decrease in the eluent viscosity and hence a reduced pressure drop over the column.

89

Increasing the temperature leads to a decrease in retention. However, in some cases this is accompanied by marked changes in the selectivity (see e.g. ref. [371]).

Organic modifiers

Organic modifiers may be added to the mobile phase in IEC in order to optimize the

9 PH ___)

6 7 8

6 7 8 9 - PH

Figure3.25: Experimental variation of the retention with pH for somenucleobases (a)and nucleosides (b) in IEC. Stationary phase: Aminex A-28. Mobile phase: 5 mM citrate - 5 m M phosphate buffer in 50-50 ethanol-water. Temperature: 70 "C. Figure taken from ref. [371]. Reprinted with permission.

90

selectivity for a particular separation. According to Rabel [367], not more than 10% of modifier may be added, because otherwise the dominant retention mechanism may be partition or adsorption rather than ion-exchange. However, in practice it is not relevant what the mechanism behind a separation is. It is only relevant that optimum resolution is obtained for all solutes.

Therefore, the addition of large amounts of organic modifiers may be considered as a parameter for the optimization of IEC separations on inorganic stationary phases (with or without a chemically bonded exchange group). For organic based materials, the amounts of organic modifiers that can be added to the mobile phase may be limited by swelling of the polymer.

The addition of organic modifiers may lead to either an increase or a decrease in retention. Moreover, the effects can differ considerably for different solutes, as is illustrated in figure 3.26 for some basic alkaloid drugs using unmodified alumina as the ion-exchange material.

The organic modifier content of the mobile phase can be used to optimize the separation. This has for example been shown for alkaloids [373,374] and for nucleosides and nucleobases [372].

Ion chromatography

Ion chromatography is a fashionable phrase used for the IEC separation of inorganic ions. Initially [375], the term was used exclusively to describe an ion-exchange LC system, equipped with a special background suppression column and a conductivity detector. The suppression column is used to reduce the conductivity background of the mobile phase. If a weak counterion is used (such as the bicarbonate ion), then an exhange of sodium ions against protons in the suppressor column will lead to a greatly reduced background conductivity.

Nowadays, however, ion chromatography may describe any of various techniques for the liquid chromatographic separation of inorganic ions [376]. Therefore, the parameters that are relevant for IEC in general bear the same kind of relevance in ion chromatography. The most prolific application of ion chromatography is the simultaneous determination of a series of common inorganic anions. These include monovalent ions (fluoride, chloride, bromide, iodide, nitrate and nitrite) as well as multivalent ones (sulfate, sulfite, phosphate). Therefore, both the retention and the selectivity are strongly affected by the concentration of the counterion. This is illustrated in figure 3.22~.

Usually, the stationary phases for ion chromatography have a low capacity, in order to reduce the background signal in conductivity detection. Selectivity in ion chromatography can be optimized along the same lines as other chromatographic methods [368].

Gradient elution

Many separations in IEC require the use of gradients, i.e. not all components of the sample can be eluted at one given composition of the mobile phase, so that this composition has to be changed during the elution. Since for weak solute ions the pH has the largest effect on retention, pH gradients are the most effective. However, salt (ionic strength) gradients are both more general and more predictable in their effects [377].

91

I.,,,,,,,. 0 20 Lo 60 80

MeOH content/% -

t i,,.,,,,, 0 20 Lo 60 80

ACN content/% - Figure 3.26: Variation of the retention of alkaloid drugs in IEC with the concentration of various organic modifiers in the mobile phase. Conditions: citric acid and trimethylammonium hydroxide; Figures (a) and (b): 0.01 M, pH=6; Figure (c): (see opposite page) 0.002 M, pH=6. Column: Spherisorb A 10 Y (alumina). Solutes: cocaine ( 0 ), dihydromorphine (*), morphine (El), dihydroco- deine (A), ephedrine (0) and brucine (x). Figure taken from ref. (3731. Reprinted with permission.

Conclusion

A number of parameters may affect retention and selectivity in IEC. Because of the subtle differences in selectivity between different ion-exchangers, it is often necessary to optimize the separation in a specific situation. The common way to do this [367] is by varying a single parameter at a time, keeping all the others constant. As we will see in chapter 5 (section 5.1.1), this is not the most appropriate way to approach the optimization

92

I I I

0 20 Lo 60 80 THF content/%-

of chromatographic separations, and therefore IEC is one of the techniques that may benefit substantially from a systematic approach to method development and optimization.

3.3.2 Ion-pair chromatography (IPC)

Ion-pair chromatography (IPC) is based on the principles of ion-pair extraction [378]. The underlying idea is that in a two-phase (liquid-liquid) system of which one phase is aqueous and the other one organic, ions will predominantly be found in the aqueous layer, and neutral molecules in the organic one. This will be true for most organic molecules, and the more so the larger and the less polar the molecules become. It will also be true for most ions, the more so the smaller and more polar the separate ions (X and r) are. If X- is a solute anion and if Y+ is a so-called pairing ion present in the aqueous phase, then a simple equilibrium scheme for ion-pair extraction can be drawn, as is shown in figure 3.27.

In figure 3.27 K%qy is the equilibrium coefficient in the aqueous phase:

(3.87)

and K,, it the usual distribution coefficient. In a chromatographic system, the capacity factor will again depend on the distribution

coefficient for the solute X, which reads for the example in which the aqueous phase is the 1 mobile one:

(3.88)

organic

Figure 3.27: Simple mechanism for ion-pair extraction. Ion-pair formation occurs in the aqueous phase. The ion-pair XY is distributed over the two phases.

93

If the product K”Xq,[ Y+] < 2 , which is usually the case at common concentrations of the pairing ion Y, then eqn.(3.88) simplifies to:

D, = K , , KIqy [ Y+] (3.89)

and hence, with eqm(3.76)

k = K , , Kyy [ Y+] (VJ V,,,) (3.90)

which for a given solute Xand pairing ion Yon a given column yields a straight line defined by

In k = In [Y’] + In k, , (3.91)

where, as in eqm(3.83) k, is the capacity factor at a (hypothetical) unit concentration of the ion Y. Of course, the above treatment can readily be adapted to account for cationic solutes and to normal phase ion-pair chromatography (NP-IPC), in which the aqueous phase is the stationary one.

According to eqn.(3.90), k is proportional to the concentration of the pairing ion, with the proportionality constant being determined by the distribution coefficient for the neutral molecule ( K x y ) and by the dissociation constant for this molecule into the two separate ions X and Y The first factor is affected by the same parameters as retention in the LC of non-ionic solutes (section 3.2). The latter factor will be determined by the nature of the solute ion and the pairing ion and by the composition (ionic strength, pH, modifier content) of the aqueous phase.

Above we have described a very simple mechanism for ion-pair extraction. The mechanism in practical IPC experiments is usually not quite as simple. There are several complicating factors.

In the first place, we have not considered the pH influence on the dissociation of weak solute ions or pairing ions. The effect of the pH in IPC will be addressed below.

aqueous “2/ X Y J

Figure 3.28: Illustration of the mechanism of IPC. The solute ion X, the pairing ion Y and the ion-pair X Y are all distributed over the two phases. Ion-pair formation occurs in both phases (reactions along horizontal lines). “Ion-exchange’’ reactions may also occur. These reactions involve solute ions in one phase and pairing ions in the other. These reactions can be found along diagonal lines in the figure.

94

In the second place, we have assumed that both of the ions (X and Y) are found exclusively in the aqueous phase. This is a great simplification of the true mechanism of IPC. Figure 3.28 shows a more realistic version of figure 3.27.

According to figure 3.28 the solute X can be retained by at least three different mechanisms: - Partition of the solute ion between the two phases (characterized by K,) - Ion-pairing between the solute ion X- and the pairing ion Y+ (characterized by KYy),

followed by partition of the ion-pair XY over the two phases (characterized by KXY) . - Ion-exchange reactions, between solute ions in one phase and pairing ions in the other

(these reactions can be found along diagonal lines in figure 3.28 and can be characterized by the ion-exchange equilibrium coefficients K g g and Q).

Other reactions, such as the distribution of the pairing ion over the two phases and the two possible ion-exchange reactions in which the pairing ion and not the solute ion transfers from one phase into the other can also take place, but do not have a direct effect on the retention of the solute X.

Figure 3.28 reduces to the simple mechanism of figure 3.27 if both K, and K are very small. If K , is small (i.e. the solute molecule is mainly in the mobile phase) but K is large (the pairing ion is mainly absorbed into the stationary phase), then the mechanism of retention in IPC becomes similar to that of IEC. Typical ion-pairing as well as typical ion-exchange mechanisms may play a role in practical IPC systems.

Normal phase or reversed phase

Both normal and reversed phase separations are possible in IPC. Normal phase systems (NP-IPC) are usually LLC systems. The aqueous phase, containing the pairing ion, is coated onto a silica surface. In order to change the kind or often even the concentration of the pairing ion it is necessary to coat another column. In some cases the pairing ion may be dissolved in the (organic) mobile phase (e.g. long chain fatty acids or amines), but the equilibration of the system will still take a long time. As in all other LLC systems, adequate thermostatting is vital for the performance of the system.

A major advantage of the use of normal phase systems may be the possibility to use UV-absorbing (or even fluorescent) pairing ions for the separation of non-UV absorbing solutes. If a UV absorbing pairing ion is in the (aqueous) stationary phase and if this is subsequently eluted from the column as an ion-pair in the presence of sample ions, then a very sensitive detection may be possible (see e.g. ref. [379]).

The mechanism of NP-IPC will be very similar to that of ion-pair extraction, i.e. the simple mechanism of figure 3.27.

Reversed phase ion-pair systems (RP-IPC) could be of the LLC type, but the use of chemically bonded (alkyl) phases has become increasingly popular, because of the increased stability and flexibility of the system. Even if an LLC system is used for RP-IPC, then a chemical modification of the surface is still required to coat an organic liquid on the particles of (for example) silica.

The system is usually equilibrated by adding the pairing ion to the mobile phase and pumping this through until a stable baseline is obtained. Equilibration of reversed phase

95

columns with pairing ions may take up to several hours, depending on the hydrophobicity (chain length) of the pairing ion and on the flow rate. This is much longer than the equilibration times in regular RPLC (without the use of pairing ions), but compares favourably with the effort needed to change the stationary phase in NP-IPC.

The most common way to create an RP-IPC system is to use a genuine chemically bonded reversed phase column (e.g. C18; see section 3.2.2.1) and to use large pairing ions with a hydrophobic alkyl chain dissolved in the mobile phase. This technique was introduced by Knox and Laird, who named it soap chromatography [380]. Because of the usuatly long alkyl chains of the pairing ions, the use of C18 phases is to be recommended in order to avoid effects that are related to the critical chain length (see section 3.2.2.1).

Table 3.7 summarizes the advantages and disadvantages of NP-IPC and RP-IPC. Because of the ease of operation, the latter technique is currently by far the more popular one. Because of this, most of the following discussion will be focussed on RP-IPC.

Table 3.7: Comparison of NP-IPC and RP-IPC systems

NP-IPC RP-IPC

Primary parameters Organic phase pH; pairing ion concentration and chain length; counterion concentration

Organic phase Variable Fixed

Temperature control Critical Not critical

Change kind of pairing ion Difficult Easy (1) Change concentration of pairing ion Difficult Easy (1)

Prevailing mechanism IP-extraction Dynamic IEC

(1) Equilibration time increases with increasing hydrophobicity (chain length) of the pairing ion.

Effect of pairing ion concentration

Because of the complexity of the IPC mechanism, the effect of the counterion concentration is not usually as simple as was suggested by eqm(3.91). One reason for this is that the distribution isotherm for the pairing ion Y is not linear, i.e. K in figure 3.28 is not a constant. Some typical distribution isotherms are shown in figure 3.29.

The figure shows that the initial addition of smalt amounts of pairing ion to the mobile phase leads to a predominant absorption of these ions in the stationary phase. However, the stationary phase quickly becomes saturated, the distribution curves flatten and K decreases. Therefore, the effect of changes in the concentration of the pairing ion is bound to be very different at low Concentrations than it is at higher ones. For this reason,

96

1 10 pm 100

Figure 3.29 Examples of distribution isotherms of a pairing ion (tetrabutylammonium) in a reversed phase system. Left: logarithmic representation; right: conventional (Langmuir) representation. Stationary phase: Lichrosorb RP-18. Mobile phase: indicated percentages of methanol in aqueous phosphate buffer (25 mM H,PO, and 25 mM NaH2P0,, pH = 2.1-3.4, bromide concentration 200 mM. Temperature: 25 "C. Figure taken from ref. [381]. Reprinted with permission.

Crombeen et al. [382] and Bartha er al. [381] have suggested that the concentration of pairing ions in the stationary phase be used as the main descriptive parameter in IPC. However, this parameter is not very convenient in practice, because the distribution isotherms need to be known.

In practice, the retention (In k) is usually plotted against the logarithm of the concentration of the pairing ion in the mobile phase (In [ fl). In such plots straight lines (or slightly curved but monotonic lines) are usually observed over wide ranges in concentration. An example is shown in figure 3.30.

Kind of pairing ion

Some examples of typical pairing ions are given in tabel 3.8. An extensive list of pairing ions and their applications can be found in ref. [367].

The kind of pairing ion will mainly be determined by the nature of the sample ions. Clearly, for anionic samples a cationic pairing ion will be required and vice versa. Also, if the ions to be separated are small polar ions, then a pairing ion with a large, non-polar group should be used to enhance the extraction of the ion-pair in the organic phase. If the sample ions are large, then small pairing ions may be used.

A combination of two large ions is also sometimes used. The resulting ion-pair may have its charge deeply buried, so that it can be extracted with very non-polar solvents. This possibility is of more interest for NP-IPC than it is for RP-IPC, because of the flexibility to choose an appropriate organic solvent in the former technique.

The nature of the pairing ion greatly affects retention, but it will also affect theselectivity (e.g. ref. [384]). Each pairing ion will require a different concentration to yield capacity factors in the optimum range.

97

-----

I I I I I I

0 2 5 10 20 50 RnlmM -

Figure 3.30: Example of variation of retention in IPC with the concentration of the pairing ion (octanesulfonate) in the stationary phase (P; upper of the two horizontal axes) and in the mobile phase (P,,,; lower axis). The axis for P, is linear, while the one for P,,, is not. Stationary phase: Hypersil ODs. Mobile phase: lOoh methanol in water. Solutes: 1 = homo-vanillic acid, 2 = 5-hydroxyindol-3-acetic acid, 3 = 3,4-dihydroxyphenylacetic acid, 4 = tyrosine, 5 = L-DOPA, 6 = dopamine, 7 = octopamine, 8 = adrenaline, 9 = 3,4-dihydroxymandelic acid, 10 = noradrenaline. Figure taken from ref. [383]. Reprinted with permission.

Table 3.8: Examples of pairing ions for IPC.

Pairing ion Structure Type

Cations: Tetra-alkylammonium (1) Tri-alkylammonium A1 kylammonium

NR: Strong NHR: Weak NH,R: Weak

Anions: Chloride c1- Strong Bromide Br- Strong Iodide I - Strong Perchlorate ClO, Strong

RSO, Strong

RSO, Weak RHPO, Weak

Alkyl sulfonate Toluene sulfonate Naphthalene sulfonate Alkyl sulfates Alkyl phosphates

(1) Either symmetrical ions, such as tetrabutylammonium, or non-symmetrical ones, such as cetyltrimethylammonium, may be used.

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1.6- log k

I 0.8

0-

-0.8, 7 9 11 13

chain length -

Figure 3.31 :Example of the effect of the chain length of the pairing ion on retention in IPC. Pairing ions: alkyl-benzyl-dimethyl ammonium with different lengths of the alkyl chain. Organic phase: chloroform; Aqueous phase: 0.1 mM of pairing ion in water. Solutes: sodium cromoglycate (circles) and acid red dye I (triangles). Figure taken from ref. [385]. Reprinted with permission.

-

Influence of the chain length of the pairing ion

A way to influence retention without greatly altering the selectivity is to use different pairing ions from the same homologous series. This offers the possibility to vary the retention more or less continuously. An example of the effect of the chain length of the pairing ion is shown in figure 3.31. It is seen in this figure that approximately straight lines are obtained if the logarithm of the capacity factor is plotted against the chain length of the pairing ion.

The length of the pairing should be such as to create a stable system with a good capacity. This implies that one should work on the plateau of the distribution isotherm (figure 3.29). The required chain length will depend on a series of factors, including the type of the pairing ion and the modifier content of the mobile phase.

Organic modifiers

Organic modifiers may affect both the retention and the selectivity in RP-IPC. Most of all, the amount of modifier added determines the distribution isotherm of the pairing ion and therefore the retention. A secondary factor involves the solubility of the pairing ion in the mobile phase. Large pairing ions may require large amounts of organic modifier to allow them to be dissolved sufficiently in the mobile phase.

In principle, the type of organic modifier(s) may be used as a parameter to optimize the selectivity of the RP-IPC system. However, because of the wide choice of other parameters, this possibility has not yet been extensively investigated.

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Interdependent parameters

From the preceding brief discussion it is apparent that three parameters in RP-IPC are highly intertwined and cannot be chosen independently: 1. the type and concentration of the pairing ion 2. the chain length of the pairing ion 3. the organic modifier content of the mobile phase.

To find a compromise for a mobile phase with neither too large a chain length (because of slow equilibration) nor too high a modifier content (because of the suppression of ionization), but yet optimum capacity factors and stable operating conditions is an optimization problem on its own.

As soon as weak solute ions or weak pairing ions form part of the system, the pH is a vital parameter in the optimization of IPC separations. Often it is a very selective parameter, when the different solute ions have different p K , values. The use of silica based reversed phase packings limits the variations of the pH to the range between about 2 and 7. Because of this limited range of operation, basic solutes often require the use of ion pairing reagents in RPLC, because they are fully ionized in the practical pH range, and give rise to highly non-symmetrical peaks in conventional RPLC.

Kind of buffer

The buffer used in RP-IPC has a minor effect on the selectivity. Therefore, the choice of the buffer will be mainly determined by practical considerations, especially by the solubility in the mobile phase. Phosphate and citrate buffers allow a wide range of pH to be used. Acetate buffers are also frequently employed.

The buffer concentration should be sufficient to yield a stable system after injection of the solute, but has an otherwise negligible effect on retention and selectivity if the counterion concentration is kept constant (see below).

Counterion concentration

As in IEC, the counterion concentration has a considerable effect on the retention in IPC. In IPC the counterion is charged similar to the solute molecules, but opposite to the pairing ion. For example, for the separation of anionic solutes, the pairing agent may be a sodium sulfonate, in which the sulfonate is the pairing ion and sodium the counterion. The addition of a buffer salt (e.g. sodium phosphate) and a neutral salt (e.g. sodium bromide) may also contribute to the concentration of the counterion. Because of the similar retention mechanism, the counterion concentration has a similar effect on retention and selectivity in RP-IPC as in IEC.

The practice to keep the total concentration of the counterion constant by adding varying amount of the neutral salt leads to a more regular behaviour of the RP-IPC system and is therefore to be recommended [386].

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Temperature

The temperature wiil affect both retention and efficiency in IPC, but not to the same extent as it does in IEC. IPC is usually a much more efficient technique (in terms of plate counts) than is IEC and therefore ambient temperatures usually yield satisfactory results.

Summary

The preferrred form of ion-pairing chromatography is RP-IPC. This is a complex chromatographic method. Unlike in other kinds of chromatography, there is notone but a series of primary parameters (type, concentration and chain length of the pairing ion, pH, organic modifier content) which together determine the capacity factors, but also may give rise to large variations in the selectivity. As in RPLC, the type of organic modifier may be used to optimize the selectivity of the system.

3.4 SUPERCRITICAL FLUID CHROMATOGRAPHY (SFC)

In supercritical fluid chromatography (SFC), the mobile phase is neither a gas nor a liquid. The definition of a supercritical fluid can be illustrated in a classical p- T diagram such as figure 3.32. In this figure three phases are located in their specific domain: the solid phase (S), the gas phase (G) and the liquid phase (L). At the triple point (tp) all three phases may coexist. The line that separates the gas and the liquid phase is the vapour pressure curve. This line ends at the critical point cp. Above this point, no distinction can be made between the gaseous and the liquid state. A compound for which either the pressure or the temperature is above the critical value can therefore not be identified as a liquid or a gas. The term supercritical fluid is usually reserved for those conditions at which both the pressure and the temperature exceed the critical value. This area in figure 3.32 can be found in the top right comer and is bounded by the two dashed lines. These dashed lines do not correspond to a phase change between conditions on either side. As was shown by Lauer et al. [387], there is a gradual change in the solvent properties (e.g. density, viscosity) if we

T-

Figure 3 . 3 2 Phase diagram illustrating the domains of the solid (S), gaseous (G) and liquid (L) phases as a function of pressure (p) and temperature (7). tp is the triple point, at which three phases co-exist. cp is the critical point, which forms the end of the vapour pressure curve (between tp and cp). The area in the top right corner indicated by SF represents the domain of supercritical fluids.

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pass the dashed lines under conditions of constant pressure (moving horizontally in figure 3.32) or constant temperature (moving vertically).

Retention in supercritical chromatography is affected by the nature of both the mobile and the stationary phase. A variety of stationary phases, including high boiling liquids, polymer films, solid supports and chemically bonded monolayers, has been used.

The choice of possible mobile phases is more limited. The critical properties (critical pressure pc and temperature TJ should be within practical reach. Moreover, stable compounds are required, which do not show disintegration at elevated temperatures and pressures. Also, the mobile phase must not be too agressive towards the materials used in the column (usually silica-based phases) and the instrumentation (mainly stainless steel). Therefore, mobile phases that are extremely interesting from a chemical point of view, such as supercritical ammonia and, especially, supercritical water, have found little use so far. Table 3.9 lists some possible mobile phases for SFC together with their chemical properties.

There are several reasons why SFC may gain its place as a separation technique alongside GC and LC in the years ahead. From a fundamental point of view, the diffusion coefficients under typical SFC conditions are lower than those typically encountered in gases, but higher than those found in liquids. The viscosity of supercritical fluids is usually higher than that of typical gases, but much lower than that of common liquids. At the same time, supercritical fluids are good solvents for many low-volatile solutes, which are not compatible with GC. Therefore, SFC may offer the possibility to separate non-volatile

Table 3.9: Some suggested solvents for SFC. Data taken from refs. [388] and 13891. Asterisks indicate preferred solvents.

Carbon dioxide* Nitrous oxide

Ethane Ethene n-Butane &Butane n-Pentane* n-Hexane

Diethyl ether THF Ethyl acetate Acetonitrile Methanol 2-Propanol Ammonia Water

31.0 36.4

32.2 9.2 152.0 134.9 196.5 234.2

193.5 267.0 250.1 274.8 239.4 235.1 132.4 374.1

72.8 71.5

48.2 49.7 373 36.0 33.3 29.3

35.9 51.2 37.8 47.7 79.9 47.0 111.3 217.6

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samples much faster and/or more efficiently (higher numbers of theoretical plates) than with LC.

From a practical point of view, SFC may allow the use of many different detection principles, including both typical LC detectors (UV-absorbance, fluorescence) and typical GC detectors (flame ionization, mass spectrometry). Also, capillary SFC seems to be well within the posssibilities of current technology, while capillary LC is not.

Promising applications of SFC include group separations (paraffins, olefins and aromatics) in petrochemical samples, monitoring of supercritical extraction processes (caffeine from coffee, nicotine from tabacco) and oligomer separations. However, it is in the field of applications that SFC has yet to prove its value. Unique separations that can be accomplished with SFC, but not with either GC or LC, have yet to be demonstrated.

The mobile phases that have been used most extensively to date are n-pentane and carbon dioxide. Pentane has the advantage that it is a liquid under ambient conditions, so that it can be handled and pumped in the same way as mobile phases for liquid chromatography. On the other hand, its critical temperature is relatively high (almost 200 "C) and it is a highly inflammable compound.

Carbon dioxide has a vapour pressure of about 50 bar at room temperature. It is therefore more difficult to handle, and it can only be pumped as a liquid when it is cooled down to sub-ambient temperatures. However, carbon dioxide is non-flammable and non-toxic, which makes it very attractive from a practical point of view. Also, the critical properties of carbon dioxide are very mild.

Mobile phase density

The main factor that influences the retention in SFC is the density of the mobile phase. For a given eluent, the density is a function of the pressure and the temperature. At a given temperature, the retention varies with the pressure in a rather coomplicated way, as is illustrated for the retention (In k) of naphthalene using CO, as the mobile phase in figure 3.33a. Figure 3.33b represents the same data, but now retention is plotted against the density of the mobile phase. It is seen that smooth curves are obtained, which hardly vary with the temperature. Of course, to obtain the same mobile phase density, a much higher pressure is required when the temperature is increased from 35 to 50 OC.

Figure 3.33b is seen to be very similar to a typical plot of retention (In k) vs. composition (p) in RPLC (see e.g. figure 3.14). Hence, a quadratic equation may be used to describe the relationship between retention (In k) and density @) in SFC

(3.92)

Composition of the mobile phase

Both pentane and carbon dioxide are solvents of low polarity. The polarity may be increased by the addition of suitable modifiers to the mobile phase. Such modifiers have a pronounced effect on the retention. The decrease in retention upon the addition of modifiers seems to resemble what is observed in LSC (see section 3.2.3). Apart from the effect on retention, the addition of polar modifiers to the mobile phase also has a marked effect on the peak shape. Especially in the case of more polar solutes, the addition of

103

modifiers to the mobile phase has become increasingly popular. The nature and concentration of organic modifier is a parameter that may be used for the optimization of SFC separations. Some initial work in this direction has been reported by Randall [392].

3.0-

2.5

2.0

1 1.5- logk

1.0

0.5

0.0

0.5

-1.0

-1.5

1 -2.5 0 30 do 50 60 70 80 90 100 110 120

platm - -

-

- -

-

-

-

-

-

-2.0- -2.0 -1.6 -1.2 -0.8 -0.L 0

Figure 3.33: Retention (In k) as a function of (a) pressure, (b) mobile phase density and (c) the logarithm of the mobile phase density in SFC at three different temperatures. Mobile phase: carbon dioxide. Stationary phase: ODS. Solute: naphthalene. Figure taken from ref. [390]. Reprinted with permission. Experimental data from ref. 13911.

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Stationary phase effects

It is not yet clear what type of stationary phase will be most useful for SFC. Liquid stationary phases will almost inevitably be of insufficient stability. Polymeric films of various thickness and varying degree of cross-linking have been used. Preferably, such polymeric phases should be covalently bonded to the column wall (open columns) or a solid support (packed columns). Alternatively, solid adsorbents or chemically bonded monolayers may be applied.

To a first approximation [393] the selectivity (a) on a given stationary phase may be expected to be independent of the mobile phase density. Consequently, the problem of stationary phase selection is similar to that encountered in GC. In GC each stationary phase will require a given temperature at which the capacity factors are in the optimum range. In SFC, each stationary phase will require a given mobile phase density. Different phases may be compared at their individual optimum conditions.

3.5 CLASSIFICATION OF PARAMETERS

In this chapter we have discussed the parameters that affect the selectivity in various chromatographic methods. The parameters we have encountered can roughly be divided into three categories: 1. Thermodynamic parameters (T). These include temperature and pressure. 2. Stationary phase parameters (S), which include the nature and composition of the

stationary phase. 3. Mobile phase parameters (M) which include the nature and composition of the mobile

phase, pH, nature and concentration of additives such as buffers, salts, ion-pairing agents or complexing agents.

Two further categories of parameters may be defined, which do not affect the selectivity: 4. Capacity parameters (C) i.e. those parameters which affect the phase ratio: film

thickness, surface area, column diameter (open columns), porosity (packed columns). 5. Physical parameters (P) column length, flowrate, particle size, column diameter

(packed columns).

The capacity parameters do not affect the selectivity (a), but they do have an effect on the capacity factor ( k ) and hence on the resolution (Rs; see eqn.1.22). The physical parameters only affect the resolution through the efficiency (N) . They also have an effect on the retention time through the hold-up time to (see eqn.l.6).

Physical parameters may be used to trade off increased resolution against decreased analysis time. Ideally, this is done separately from the selectivity optimization process, because the effects are simple, predictable, and independent of the parameters that do affect the selectivity.

Independent parameters with a simple effect on the resolution may be optimized sequentially, i.e. one after the other (section 5.1.1). Hence, after the selectivity has been optimized, the shortest possible column length or the highest possible flowrate may be established that will provide sufficient resolution. Adapting the length of the column is the preferred strategy, because it will lead to both faster analysis and lower pressure drops.

105

Some further comments on optimization of the physical parameters will be made in chapter 7.

The capacity parameters allow a variation of the capacity factor (and hence the resolution) independent of the selectivity. However, all these parameters are difficult to vary, since they almost always require new columns to be used. Moreover, the range of variation offered by these parameters is too limited for them to be generally useful in optimization schemes (see also section 4.2.3).

Therefore, the thermodynamic parameters (T), the stationary phase parameters (S) and the mobile phase parameters (M) are the ones we should consider if we wish to select the most relevant parameters for the optimization of chromatographic selectivity.

Of these three categories the thermodynamic parameters can be varied most easily. However, temperature has a major effect on retention in GC, but only a minor effect on the selectivity. In LC its effect is never very large, except from some ionic separations. Pressure is only relevant as a parameter for SFC. We conclude that temperature should head the list of optimization parameters in GC, and pressure (possibly in combination with temperature) in SFC.

Stationary or mobile phase optimization

In GC we cannot use the nature and composition of the mobile phase to vary the selectivity. Hence, the nature and consequently the composition of the stationary phase should be used for optimization purposes. There are many more stationary phases

Table 3.10a: Summary of parameters for selectivity optimization. Asterisks indicate preferred parameters.

Method: Gas Liquid Chromatography (GLC)

Section: 3.1.1 and 3.1.2

Primary parameter(s) Suggested relationship

Type Parameter Plot Shape Eqn.no.

Gas Solid Chromatography (GSC)

T c Temperature * C c Film thickness C d Surface area

In(k/T) vs. 1/T linear 3.10 k vs. d,y linear k vs. s linear

Secondary parameter(s) Suggested relationship


S d Stationary phase S c Stationary phase composition k vs. cp linear 3.14

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Table 3.10b: Summary of parameters for selectivity optimization. Asterisks indicate preferred parameters.

Method: Liquid-Liquid Chromatography (LLC) Section: 3.2.1



M c S c

Polarity of mobile phase * Polarity of stationary phase

C c Phase ratio k vs. V J V , linear 1.10



M d S d T c Temperature In k vs. 1/T linear 3.57 M c PH (1) M c Ionic strength (1)

Nature of mobile phase * Nature of stationary phase

(1 ) May be used if the more polar phase is aqueous.

available than could possibly (or, indeed, sensibly) be tried. After some different stationary phases (see section 2.3.1) have been tried, the possibility of using a mixture of two stationary phases could be considered as a final step in the optimization.

In LC we have a choice between optimizing the stationary phase parameters or the mobile phase parameters. Obviously, the latter can be changed more readily. Advantages of mobile phase optimization are: 1. many parameters offer a great flexibility, 2. the mobile phase can easily be changed, 3. there are good possibilities for automation, 4. the investment required for columns is low.

On the other hand, if the stationary phase parameters are being optimized, other advantages may occur: 1. stable and reproducible operation with simple mobile phases is possible, 2. mobile phases may be selected with a low cost, low viscosity and low toxicity.

Hence, optimization of the mobile phase offers advantages mainly during the optimization process, while optimization of the stationary phase offers its main advantages after the optimization. So far, researchers have been more concerned with the optimization process

107

itself than with the usefulness of their results. Hence, the mobile phase parameters have been optimized almost exclusively. In the future we may see an increased use of the optimization of stationary phase parameters, especially on the part of column manufacturers. This will result in the availability of reproducible systems for optimized separations on dedicated stationary phases for GC as well as for LC.

35.1 Summary of parameters for selectivity Optimization

Table 3.10 summarizes the parameters of interest for the various chromatographic techniques described in this chapter. A distinction is made between primary and secondary parameters.

Table 3.10~: Summary of parameters for selectivity optimization. Asterisks indicate preferred parameters.

Method: Reversed Phase Liquid Chromatography (RPLC) Section: 3.2.2



M c Mobile phase polarity * (modifier content) In k vs. cp quadr. 3.38 or In k vs. cp (1) linear 3.45

M c PH (2) In k vs. pH curved 3.631 68/70



M d M c T c Temperature In k vs. 1/T linear 3.57 S d S d Nature of buffer S s Stationary phase chain length In k vs. ne linear (4) M c Ionic strength In kvs. I hyperboiic 3.71 M d Nature of buffer

( 1 ) Linear approximation for 1 < k < 10. (2) pH is a primary parameter for ionizable solutes. (3) Modifier coontent fa - multicomganent mobile phases can be estimated using table 3.1, (4) Approximately linear up to “critical chain length” (see section 3.2.2.1).

Nature of modifier(s) (3) * Ratio of modifier concentrations.*

Nature OF stationary phase

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Table 3.10d: Summary of parameters for selectivity optimization. Asterisks indicate preferred parameters.

Method Liquid-Solid Chromatography (LSC) Section: 3.2.3


T v ~ e Parameter Plot Shape Eqn.no.

M c Eluotropic strength In k vs. In X , linear 3.74

Secondary parameter@) Suggested relationship


M d Nature of modifier(s)* M d Modulators (1)

S d

(1) Modulators are compounds which can be added to the mobile phase in small quantities to affect

T c Temperature In kvs. 1 / T linear 3.57 Nature of stationary phase

peak shape and/or selectivity (e.g. water, tri-ethylamine).

Primary parameters are those which have a large effect on retention. Usually, these parameters d o not affect selectivity to the same extent. Therefore, these are the parameters that can be used to bring the capacity factors of the solute into the optimum range.

Secondary parameters may affect retention, but always affect selectivity. In fact, ideally the parameters should be selected such that the retention ( k ) is kept roughly constant (i.e. in the optimum range) while the selectivity (a) can be varied. If the secondary parameters do affect retention, then sometimes this ideal situation can be approached by the simultaneous variation of two (or more) parameters a t the same time. Examples of this may be found in chapter 5.

Capacity parameters are not often used as primary optimization parameters in chromatography. Therefore, they are only included in table 3.10 in those cases in which they are used with some frequency. It should be noted, however, that changing one of the capacity factors usually involves the use of a completely different column and is therefore unattractive. Although changing the capacity parameters affects retention in an essentially predictable way, changing the column (packing material, film thickness, etc.) may give rise to unexpected second order phenomena. This is a second reason for which capacity parameters should not be recommended as primary optimization parameters.

Parameters that affect neither the capacity factors nor the selectivity (such as column length or flow rate) will not be found in this table.

The parameters are classified by two different letters. The capital letters correspond to the classification given above, i.e. thermodynamic parameters (T), stationary phase

parameters (S), mobile phase parameters (M), capacity parameters (C) and physical parameters (P).

The lower case letters indicate a second classification of the parameters. Three different types of parameters are indicated: - continuous parameters (c), which can take on all values between given limits (e.g.

- discrete parameters (d), which can only take on certain values (e.g. the kind of

- stepwise parameters (s), which can take on a series of discrete values (e.g. the chain

temperature),

stationary phase),

length of alkylsulfonate ions in ion-pair chromatography).

Table 3.10a lists the parameters that may be used in the two modes of GC discussed in this chapter (GLC and GSC). Because of the similarity of these two techniques, they have been combined in one table.

To optimize the capacity factors, the temperature may be adapted. Temperature is the most commonly used primary parameter in GC. Alternatively, the film thickness (in GLC) or the surface area of the stationary phase (in GSC) may be used to change the capacity

Table 3.10e: Summary of parameters for selectivity optimization. Asterisks indicate preferred parameters.

Method Ion Exchange Chromatography (IEC) Section: 3.3.1



M c Counterion concentration * In k vs.ln[ v] linear 3.84 Mc p H ( l ) * In k vs. pH , sigm. 3.85/86 S d Exchange capacity k vs. capacity linear

Secondary parameter@) Suggested relationship

Type Parameter Plot Shape Eqn.no. ~

T c Temperature In k vs. l / T linear 3.57 M d Type of modifier(s) M c Concentration of modifier(s) M d Type of counterion M d Type of buffer S d Type of stationary phase

(1) pH is a primary parameter if weak solutes, counterions or ion-exchangers are involved in the separation process.

110

Table 3.10f: Summary of parameters for selectivity optimization. Asterisks indicate preferred parameters.

Method: Ion Pairing Chromatography (IPC) Section: 3.3.2



M c p H ( l ) * In k vs. pH sigm. M s Chain length of pairing ion * In kvs. n, linear M c Pairing ion concentration * In k vs.ln [ r] curved M c Modifier content * In k vs. cp quadr. 3.38



M d Type of pairing ion * M d Type of modifier * M d Type of counterion M c Conc.of counterion M d Type of buffer

S d T c Temperature In k vs. 1 / T linear 3.57

Type of statationary phase

factors. However, in order to change either of these parameters another column is required. Because of the availability of packing materials, the surface area is a discrete parameter.

The most common secondary parameter is the kind of stationary phase used, which is obviously a discrete parameter. Because the capacity factors will usually differ on different phases, several parameters will have to be varied at the same time. For example, if another stationary phase is chosen, the temperature may be adapted to bring the capacity factors back into the optimum range.

Table 3.10b lists the optimization parameters that may be used in LLC. Clearly, the polarities of the two phases largely determine retention and selectivity. The exact composition of (preferably) the mobile phase may be varied to optimize the separation (i.e. variations in the nature and the concentration of mobile phase components, without substantial variations in the polarity). Even if the temperature is not a major optimization parameter, adequate temperature control is required in all LLC experiments. Therefore it may be experimentally straightforward to exploit temperature as a secondary optimization parameter.

Table 3 .10~ lists the relevant parameters for RPLC. The polarity (modifier content) of the mobile phase is the main primary parameter for most samples, although for some

111

weakly acidic or basic solutes the pH may have an even larger effect on the capacity factors.

A series of secondary parameters may be exploited. Changing the nature of the organic modifier is the most common and probably the most rewarding parameter to use. If ternary and quaternary mobile phases are considered, then the ratio between the concentrations of different modifiers becomes a continuous parameter that may be optimized.

Most separations are optimized by considering solely the influence of the concentration of modifier(s) in the mobile phase and the pH. However, for particular (or particularly difficult) separation problems there is a series of additional parameters that might be considered.

Table 3.10d lists the parameters for LSC. Again, most separations may be optimized by optimizing the eluotropic strength (primary parameter) and the nature (secondary parameter) of the mobile phase. The latter parameter involves the preparation of different iso-eluotropic mixtures containing different solvents, or small quantities of very polar components ("modulators"). As in the case of RPLC, there are several additional parameters that are not frequently exploited.

Polar chemically bonded stationary phases (section 3.2.2.2) may be used as an alternative stationary phase for both RPLC and LSC, if variations in the mobile phase do not result in an adequate separation. If polar CBPs are used in combination with more polar mobile phases (reversed phase mode), then table 3.10~ may be used to find the most appropriate optimization parameters. If operated in the normal phase mode, table 3.10d

Table 3.10g: t

Summary of parameters for selectivity optimization. Asterisks indicate preferred parameters.

Method: Supercritical Fluid Chromatography (SFC) Section: 3.4



M c Density (I) * In k vs. p quadr. 3.92 M c Concentration of modifier In k vs. Q, curved



M d S d M d

Nature of mobile phase Nature of stationary phase Type of modifier *

(1) The mobile phase density is determined by the combination of the pressure and the temperature.

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may be used. However, it should be noted that strictly speaking we do not deal with liquid-solid chromatography. A notable difference is that the water content of the mobile phase has a much less dramatic influence in normal phase LBPC than it has in LSC.

The parameters that play a role in ion-exchange chromatography (IEC) are summarized in table 3.10e. A combined optimization of the two primary optimization parameters (pH and counterion concentration) may already be used to optimize the chromatographic selectivity of the system. However, different buffers or counterions are often investigated for their effect on the selectivity. As a rule, elevated temperatures are used to increase the efficiency rather than the selectivity of the system.

Table 3.10f lists the most relevant parameters for ion-pairing chromatography (IPC). Here there are four major primary parameters, which cannot be seen as independent. Hence (see section 5.1 .l), these four parameters should preferably be optimized simultaneously. Sensible upper and lower limits may be set for each of the parameters and an optimized separation may result from the process. If this is not the case, there are still many secondary parameters that could be exploited.

Tables 3.10e and 3.10f suggest that the ionic separation methods IEC and IPC are the most complicated ones. Both methods involve several mutually dependent primary parameters and a series of additional secondary optimization parameters. Therefore, these techniques are bound to be the subject of many optimization studies in the future.

Finally, table 3.10g shows the relevant parameters for SFC. The density of the mobile phase (determined by the combination of pressure and temperature) is the main parameter for this technique. Several possible secondary parameters are listed in the table. Because SFC is not yet a mature technique, the list of secondary parameters may still undergo some changes.

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301. A.B.Littlewood, G.S.G.Phillips and D.T.Price, J.Chem. SOC. (1955) 1480. 302. J.Takacs, P-Rajcsanyi, L.Kaplar and LOIacsi, J. Chromatogr. 41 (1969) 438. 303. R.Tijssen, H.A.H.Billiet and P.J.Schoenmakers, J.Chromatogr. 122 (1976) 185. 304. C.-F.Chien, M.M.Kopecni and R.J.Laub,. AnaLChem. 52 (1980) 1402. 305. M. W.P.Harbison, R.J.Laub, D.E.Martire, J.H.Purnel1 and P.S. Williams, J. Phys.

306. R.J.Laub in: Th.Kuwana (ed.), Physical Methods in Modern Chemical Analysis,

307. J.A.Barker and D.H.Everett, Trans. Faraday SOC. 58 (1962) 1608. 308. D.C.Locke, J.Phys.Chem. 69 (1965) 3768. 309. E.V.Kalashnikova, A.V.Kiselev, D.P.Poshkus and K.D. Shcherbakova, J.Chroma-

310. J.Klein and H.Widdecke, J.Chromatogr. 147 (1978) 384. 31 I . P.J.Schoenmakers, H.A.H.Billiet and L.de Galan Chromatographia 15 (1982) 205. 312. B.L.Karger, L.R.Snyder and C.Eon, AnaLChem. 50 (1978) 2126. 313. J.H.Hildebrand, J.M.Prausnitz and R.L.Scott, Regular and Related Solutions, Van

314. H.A.H.Billiet, P.J.Schoenmakers and L.de Galan, J.Chromatogr. 218 (1981) 443. 315. J.F.K.Huber, M.Pawlowska and P.Mark1, Chromatographia 17 (1983) 653.

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316. W.R.Melander and Cs.Horvath, in: Cs.Horvath (ed.), HPLC, Advances and Perspec-

317. G.E.Berendsen, Preparation and Characterization ofwell-defined Chemically Bonded

318. R.E.Majors, H.G.Barth and C.H.Lochmueller, Anal.Chem. 56 (1984) 300R. 319. G.E.Berendsen and L.d.Galan, J.Chromatogr. 1% (1980) 21. 320. A.F.M.Barton, Chem.Rev. 75 (1975) 731. 321. P.J.Schoenmakers, H.A.H.Billiet, R.Tijssen and L.de Galan, J.Chrornatogr. 149

322. P.J.Schoenmakers, H.A.H.Billiet and L.de Galan, J.Chromatogr. 185 (1979) 179. 323. P.J.Schoenmakers, H.A.H.Billiet and L.de Galan, J.Chrornatogr. 282 (1983) 107. 324. P.J.Schoenmakers, H.A.H.Billiet and L.de Galan, J.Chromatogr. 218 (1981) 261. 325. P.Jandera, H.Colin and G.Guiochon, Anal.Chem. 54 (1982) 435. 326. W.Melander and Cs.Horvath, Chromatographia 18 (1984) 353. 327. J.W.Weyland, C.H.P.Bruins and D.A.Doornbos, J.Chromatogr.Sci. 22 (1984) 31. 328. M.Mcann, J.H.hrnel1 and C.A.Wellington, Faraday SOC. Symp.Series 15 (1980) 82. 329. W.R.Melander and Cs.Horvath, Chromatographia 18 (1984) 353. 330. Lu Peichang and Lu Xiaoming, J.Chromatogr. 292 (1984) 169. 331. L.R.Snyder, J.W.Dolan and J.R.Gant, J.Chromatogr. 165 (1979) 3. 332. C.E.Goewie, U.A.Th.Brinkman and R.W.Frei, Anal.Chem. 3 (1981) 2072. 333. T.L.Hafkenscheid and E.Tomlinson, J.Chromatogr. 264 (1983) 47. 334. S.T.Balke, Quantitative Column Liquid Chromatography, a Survey of Chemometric

335. P.J.Schoenmakers, H.A.H.BiHiet and L.de Galan, LChromatogr. 205 (1981) 13. 336. P.J.Schoenmakers, A.C.J.H.Drouen, H.A.H.Billiet and L.de Galan, Chromatogra-

337. J.R.Gant, J.W.Dolan and L.R.Snyder, J.Chromatogr. 185 (1 979) 153. 338. W.R.Melander, B.-K.Chen and Cs.Horvath, J.Chromatogr. 185 (1979) 99. 339. W.R.Melander, D.E.Campbel1 and Cs.Horvath, J.Chromatogr. 158 (1978) 21 5. 340. S.N.Deming and M.L.Turoff, AnaLChem. 50 (1978) 546. 341. C.J.C.M.Laurent, H.A.H.Billiet and L.de Galan, Chromatographia 17 (1983) 394. 342. Cs.Horvath, W.Melander and I.MolnBr, Anal.Chem. 49 (1977) 142. 343. D.J.Pietrzyk and C.H.Chu, AnaLChem. 49 (1977) 860. 344. M.T.W.Hearn, Adv. Chrom. 18 (1980) 93. 345. J.L.M.van de Venne, J.L.H.M.Hendrikx and R.S.Deelder, J.Chromatogr. 167

346. M.Otto and W.Wegscheider, J.Chromatogr. 258 (1983) 11. 347. J.J.Kirkland, Anal.Chem. 43 (1971) 36A. 348. B.Feibush, M.J.Cohen and B.L.Karger, J.Chromatogr. 282 (1983) 3. 349. L.R.Snyder and J.J.Kirkland, Introduction to Modern Liquid Chromatography, 2nd.

350. L.R.Snyder, Principles of Adsorption Chromatography, Dekker, New York, 1976. 351. H.Colin, C.Eon and G.Guiochon, J.Chromatogr. 122 (1976) 223. 352. E.Soczewinski, AnaLChem. 41 (1969) 179. 353. E.Soczewihski and W.Golkiewicz, Chromatographia 4 (1971) 501. 354. P.Jandera and J.ChuraEek, J.Chromatogr. 91 (1974) 207.

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355. ESoczewinski, J.Chromatogr. 130 (1977) 23. 356. L.R.Syder and H.Poppe, J.Chromatogr. 184 (1980) 363. 357. P.Jandera, M.Janderova and J.ChuraEek, J.Chromatogr. 148 (1978) 79. 358. D.L.Saunders, AnaLChem. 46 (1974) 470. 359. J.L.Glajch, J.J.Kirkland and L.R.Snyder, J.Chromatogr. 238 (1982) 269. 360. L.R.Snyder and J.L.Glajch, J.Chromatogr. 214 (1981) 1. 361. J.L.Glajch and L.R.Snyder, J.Chromatogr. 214 (1981) 21. 362. L.R.Snyder, J.L.Glajch and J.J.Kirkland, J.Chromatogr. 218 (1981) 299. 363. L.R.Snyder and J.L.Glajch, J.Chromatogr. 248 (1982) 165. 364. C.J.C.M.Laurent, A Reappreciation of Alumina in Liquid Chromatography, Ph.D.

365. J.H.Knox, K.K.Unger and H.Mueller, J.Liq.Chromatogr. 6 (suppl.1) (1983) 1. 366. M.T.Gilbert, J.H.Knox and B.Kaur, Chromatographia 16 (1982) 138. 367. F.M.Rabel, Advan.Chromatogr. 17 (1979) 53. 368. P.R.Haddad and C.E.Cowie, J.Chromatogr. 303 (1984) 321. 369. P.B.Hamilton, AnaLChem. 35 (1963) 2055. 370. Sj. van der Wal and J.F.K.Huber, J.Chromatogr. 135 (1977) 305. 371. R.Eksteen, P.Linsen and J.C.Kraak, J.Chromatogr. 148 (1978) 413. 372. Sj. van der Wal and J.F.K.Huber, J.Chromatogr. 102 (1974) 353. 373. C.J.C.M.Laurent, H.A.H.Billiet and L.de Galan, Chromatographia 17 (1983) 394. 374. C.J.C.M.Laurent, H.A.H.Billiet and L.de Galan, J.Chromatogr. 285 (1983) 161. 375. E.Sawicki in: E.Sawicki, J.D.Mulik and E.Wittgenstein (eds.), Ion Chromatographic

Analysis of Environmental Pollutants, Ann Arbor Science, Michigan, 1978, p.1. 376. P.R.Haddad and A.L.Heckenberg, J.Chromatogr. 300 (1984) 357. 377. P.Jandera and J.ChuraEek, Gradient Elution in Column Liquid Chromatography,

378. G.Schill in: J.A.Marinsky and Y.Marcus (eds.), Zon Exchange and Solvent Extraction,

379. J.Crommen, B.Fransson and G.Schill, J.Chromatogr. 142 (1975) 107. 380. J.H.Knox and G.R.Laird, J.Chromatogr. 122 (1976) 17. 381. A.Bartha and Gy.Vigh, J.Chromatogr. 260 (1983) 337. 382. J.P.Crombeen, J.C.Kraak and H.Poppe, J.Chromatogr. 167 (1978) 219. 383. H.A.H.Billiet, A.C.J.H.Drouen and L.de Galan, J.Chromatogr. 316 (1984) 231. 384. R.Modin and G.Schill, Talanta 22 (1975) 1017. 385. E.Tomlinson, C.M.Riley and T.M.Jefferies, J.Chromatogr. 173 (1979) 89. 386. A.Bartha, H.A.H.Billiet, L.de Galan and Gy.Vigh, J.Chromatogr. 291 (1984) 91. 387. H.H.Lauer, D.McManigill and R.D.Board, AndChern. 55 (1983) 1370. 388. W.Asche, Chromatographia 11 (1971) 411. 389. R.C.Reid, J.M.Prausnitz and T.K.Sherwood, The Properties of Gases and Liquids,

Third edition, McGraw-Hill, New York, 1977. 390. P.J.Schoenmakers, J.Chromatogr. 315 (1984) 1. 391. U.van Wasen, I.Swaid and G.M.Schneider, Angew.Chemie 92 (1980) 585. 392. L.G.Randall, Hewlett-Packard Technical Paper no. 102, 1983.

thesis, Delft, 1983.

Elsevier, Amsterdam, 1985.

Vo1.6, 1974, Chapter 1.

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CHAPTER 4

OPTIMIZATION CRITERIA Before any optimization process can be started, the goals of the process should be

defined unambiguously. For chromatographic separations this is not always a straightforward matter. A chromatogram is more than a simple unique number in one dimension. Nevertheless, we want to reduce the information contained in the chromatogram to a single number during the course of the optimization process.

An additional complicating factor is that the goals of the chromatographic optimization may vary considerably from one case to another. For example, all peaks may need to be separated, or just some relevant peaks in a complex chromatogram; large series of presumably identical samples may have to be run in a quality control situation, or a screening method may need to be developed for a relatively large number of potentially present drugs or pollutants.

The aim of this chapter is to translate such different analytical goals into objective functions, i.e. into different criteria which can be the objective goals of an optimization process.

In the literature many different terms are used for such criteria: (chromatographic) response functions, objective functions or (chromatographic) optimization functions. Throughout the rest of this chapter, the neutral term optimization criteria will be used.

4.1 INTRODUCTION

In this introduction the possible goals of an optimization process will be investigated. In the following sections we will then try to translate these goals into simple mathematical algorithms. At the end of this chapter the different goals and the recommended optimization criteria will then be summarized.

4.1.1 Separation of two peaks

The resolution of two chromatographic peaks has been defined in terms of retention times and bandwidths in chapter 1:

At R , =

'/2( w, + w2) and it was shown that for symmetrical (Gaussian) peaks eqn.(4.l) can be transformed into a very useful equation (see section 1.5):

The fact that resolution as defined by eqm(4.1) can be related so elegantly to the fundamental parameters of the separation process (i.e. a, and N) is a great advantage for the use of resolution (R,) to quantify the extent of separation of a pair of chromatographic peaks. However, in opting for R, we need to accept all characteristics

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of this quantity, which may not always be advantageous. The following three characteristics appear to be relevant: 1. The resolution (R, ) is independent of the (relative) height(s) of chromatographic peaks.

This is a fair proposition only if we work in the linear range of chromatographic operation, i.e. if the peak height increases linearly with the injected quantity and the peak width remains constant. However, even then differences in the ratio of the two peak heights will lead to differences in the resolution (subjectively) observed by the chromatographer. A detailed description can be found in ref. [401], pp.34-48.

2. R , may be expressed in terms of fundamental chromatographic quantities (eqn.4.2) for symmetrical (Gaussian) peaks, but for such peaks only. Ideally, all chromatographic peaks fall within the above qualification. However, in the practice of GC and in particular LC, this is usually not true.

3. R, cannot easily be estimated from a chromatogram using eqn.(4.1), since this requires knowledge of both retention times and peak widths. Establishing the latter from the chromatogram requires tedious manual measurements or complex mathematical algorithms for integrators or computers. In either case, the resulting estimates for the peak widths are not usually very reliable. The use of eqn.(4.2) with a given number of theoretical plates yields more reproducible results, but again it assumes the peaks to be Gaussian. In this case only retention times need to be established from the chromatogram. N may be obtained from independent measurements or from a “test chromatogram”. It may be a fixed number, but also a function of the capacity factor (k) . A disadvantage of the use of eqn.(4.2) may be a variation of N with time, for instance due to a gradual deterioration of the column. Such a process is not accounted for if the resolution is characterized without obtaining an up-to-date measurement for the peak width.

In section 4.2. we will define other criteria that may be used to characterize the separation between a pair of adjacent peaks in a chromatogram (so-called elemental criteria).

From the above it is obvious that the following three questions need to be addressed in this chapter: 1 . Should the criteria used to characterize the extent ofseparation of apair of adjacentpeaks

in a chromatogram be affected by the relative peak heights ? 2. Should the shape of the peaks be reflected in the value of the criterion ? 3. Can a criterion be defined which bears relation to fundamental chromatographic

parameters, but is yet conveniently obtained from the chromatogram ?

4.1.2 Separation in a chromatogram

To some extent the quantification of the amount of separation in a chromatogram can be seen as an expansion of the characterization of the separation achieved for each pair of successive peaks. However, a straightforward expansion of a criterion for a pair of peaks, for instance a summation of individual R, values, may easily yield numbers that do not at all correspond to the chromatographer’s own (subjective) opinion of what constitutes a good chromatogram.

This is easily illustrated by the two chromatograms shown in figure 4.1. These two

117

(a1 1

( b l 1

0 1 5 k-

Figure 4.1: Two schematic chromatograms, constructed with N = 10,000. The capacity factors of the peaks in the chromatograms are listed in table 4.1.

Table 4.1: Resolution data for the chromatograms of figure 4.1

Chromatogram Peak no. k RS = R S

4.1.a

1 1

2 1 .I 2.9

3 1.25

1.22

1.72

4.1.b

1 1

2 1.1 25.3

3 5

1.22

24.07

chromatograms are identical, apart from the position of the last peak. Table 4.1 lists the k values and the resolution factors (calculated from eqn.4.2) which correspond to the two chromatograms of figure 4.1. Also given in the table is the sum of all resolution factors in each chromatogram. Due to the improved resolution of the last two peaks, the sum of the resolution values is much higher for the bottom chromatogram, suggesting this one to

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be vastly superior. However, as long as all peaks are of equal importance, every experienced chromatographer would prefer the top chromatogram to the bottom one, because it would yield a much shorter analysis time.

The condition that all peaks must be of equal importance is relevant in this context. For example, an analyst who is only interested in the quantization of the last peak in the chromatogram might well prefer the bottom one. On a very much shorter column this would yield a very fast resolution of the last peak from the (unresolved) rest of the chromatogram. In the case that all peaks are (or must be) considered to be of equal importance, we will speak of the general case. If only a few peaks are of interest, or if some peaks are of more importance than others, we will speak of specific cases.

The above discussion has led to the following three questions: 1 . How can we expand criteria that measure the resolution between apair of succesivepeaks

to criteria that measure the quality of separation achieved in an entire chromatogram? 2. How should analysis time be reflected in such criteria ? 3. May we use the same criteria for the general case and for specific cases, or how should

we adapt the criteria to serve these dzfferent purposes?

4.2 ELEMENTAL CRITERIA

The resolution between two peaks has been defined in chapter 1 and this definition has been reviewed in section 4.1.1. In this section we will define and investigate various other criteria that may be used to quantify the extent of separation between a pair of adjacent peaks in a chromatogram. We will refer to these criteria as “elemental criteria”. Later in this chapter the elemental criteria will serve as the basis of criteria for judging the extent of separation in entire chromatograms.

4.2.1 Peak-valley ratios

Three definitions of peak-valley ratios are illustrated in figure 4.2. All of them express the extent of separation as some measure of the depth of the valley between two peaks divided by some measure of the peak height. The first criterion (P) measures the depth of the valley relative to the interpolated peak height as shown in figure 4.2.a. The corresponding expression is:

P = f / g (4.3)

where g is the interpolated peak height, i.e. the height to the baseline of the line connecting two peak tops at the location of the valley, and f i s the depth of the valley relative to the interpolated baseline. This criterion was suggested by Kaiser [402].

The second peak-valley ratio was suggested by Schupp [403] and is illustrated in figure 4.2.b. In this criterion, which we will refer to as the median peak-valley ratio ( Pm), the depth of the valley at a point midway between two successive peaks urn) is measured relative to the average peak height (g,) which equals the interpolated peak height at that point. The corresponding equation is:

119

\ P=flg

Figure 4.2: Three definitions for peak-valley ratios as elemental criteria to quantify the extent of separation between a pair of adjacent peaks in a chromatogram. (a) Peak-valley ratio ( P ; eqn.4.3) according to Kaiser, (b) median peak-valley ratio (P,; eqn.4.4) according to Schupp and (c) (opposite page) the valley-to-top ratio ( P ; eqn.4.5) according to Christophe.

120

p , = f , / g , . (4.4)

The third criterion, the valley-to-top ratio ( Pv), was introduced by Christophe [404]. It measures the height of the valley relative to the height of either of the peak tops. Hence, for a cluster of two peaks as in figure 4.2, two values of P, can be obtained, one for each peak. If the ratio of the height of the valley (v) to the peak height (h) is subtracted from unity, the resulting definition is very similar to the previous two. It is illustrated in figure 4.2.c and the appropriate equation for P, is

P, = 1 - v / h (4.5)

The parameters used in eqns.(4.3), (4.4) and (4.5) are all illustrated in figure 4.2. All three definitions for peak-valley ratios are very similar. According to the last

definition, a value for the valley-to-top ratio (P,) can be assigned to each peak rather than to each pair of peaks. However, in the case of two Gaussian peaks of equal heights all three definitions yield exactly the same results. Even if the relative peak heights vary, the first two definitions will still yield comparable results. The definition for P, implies that the value will be higher for the larger peak and lower for the smaller peak (proportionally to the relative height).

The peak-valley ratios vary from zero for separations where no valley can be detected, to unity for complete separation. It ought to be noticed that a P value equal to zero does not necessarily imply that two solutes elute with exactly the same retention time (or k value). There is a threshold separation below which the presence of two individual bands in one peak only leads to peak broadening or deformation, without the occurrence of a valley. In these cases R , values are indeed not equal to zero, because by definition (eqn.l.14) R, is proportional to the difference in retention times.

121

Three characteristics of peak-valley ratios are: 1. P can readily be estimated from a chromatogram. 2. In theory P will vary with varying relative peak heights (or areas) of the two peaks

involved. A simulation for Gaussianpeaks reveals that this variation is small for

3. Because of its pragmatic definition P automatically reflects peak asymmetry and it ' both P and P,, but it is substantial for P, [405].

can be applied to peaks of all shapes, not exclusively to Gaussian peaks.

For Gaussian peaks of equal height the value of the peak-valley ratio (then the same according to all three definitions) can readily be expressed in terms of R,. This can be done by relating the parameters f, g and v (see figure 4.2) to the parameters that describe a Gaussian peak (IS and h). For the first of a pair of Gaussian peaks (peak A) we can write (eqn. 1.1 5):

A ( t ) = h, exp - ' I2 - (4.6)

while a similar expression holds for peak B. If we now substitute t = 1/2 ( t , + tB) and assume the values of IS for close peaks to

be approximately equal (cr, z 0, FS 5) we find for the combined signal v (see figure 4.2.c):

v =g-f=A(*)+B(.) t + t , ' A + ' B

2 2

= h, exp - ' /2 (t;i:) - + h,exp - ' /2

2

w (hA + h,) exp - ' I 2

= (h , + h,)exp -(2 R:)

and given that

g = (h , + h,) / 2

eqns.(4.7) and (4.8) can be combined to yield

(4.10)

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Eqn.(4.10) gives the relationship between P and R , for two Gaussian peaks, assuming that oA w a, and assuming that the position of the peak tops is not significantly altered because of peak overlap. For Gaussian peaks of equal height P= P,= P, and eqn.(dlO) applies to all three criteria. Calculations performed on simulated Gaussian peaks [405] confirm that both P and P, closely follow the theoretical curve described by eqn.(4.10), even when the relative peak heights vary. Clearly, for non-symmetrical peaks (e.g. typical solvent peaks) the value of P will be affected by the peak heights. However, if the first of a pair of peaks is a “solvent peak” it may be well-nigh impossible to use any of the definitions in figure 4.2 to establish either the peak-valley ratio (P) or the median peak-valley ratio (PA from the chromatogram. Only a pragmatic ratio between the top of the observed peak and the height of the valley on the solvent front preceding the peak may be established from the chromatogram.

Eqm(4.10) also provides insight into the threshold value for P for symmetrical peaks. According to eqn(4.10) P will be estimated as zero for

or R,y < 0.59 . (4.1 1)

This figure applies to Gaussian peaks, but clearly, for peaks of other shapes there will also be some threshold value below which changes in the extent of separation will not be reflected in P.

Wegscheider et al. [406] have modified P so that it will also reflect baseline noise:

P’ = f / ( g + 2 n ) (4.12)

where n is the (peak-to-peak) noise level on the baseline. According to eqn.(4.12), P‘ will decrease when the noise level increases, as well as when the absolute peak heights (reflected in f and g ) decrease. If noise is a significant factor, eqn.(4.12) may provide a more realistic evaluation of the merits of the actual separation than does eq~(4.3). Eqm(4.4) and (4.5) can be modified analogously to account for the influence of baseline noise on P, and P,.

Because of the great similarity between the definitions for P and P,, we will not try to establish superiority of one over the other. To a large extent, the choice for one of them will depend on the software that is available to obtain P values from a chromatogram.

The choice between P or P, on the one hand and P, on the other will be determined by whether or not an influence of the (relative) peak heights is wanted (see discussion below).

4.2.2 Fractional peak overlap

An obvious criterion by which to judge the extent of separation of chromatographic peaks, especially for the optimization of a quantitative analysis, is the fraction of the peak that is free of overlap from adjacent peaks. The definition for this so-called fractional overlap criterion is illustrated in figure 4.3. An equation to describe the fractional overlap is

123

(4.13)

where A, is the area of the nth peak and A,,, .- , and A,.,, , are the areas it shares with the preceding and the following peaks, respectively.

An -An,n-1 -An.n+l

An FO =

Figure4.3: Illustration of the definition of the fractional overlap (FO) as a criterion for the separation of a pair of adjacent peaks in a chromatogram.

Clearly, FO gives a good indication of the accuracy with which a peak can be quantitatively determined in a chromatogram. However, it is not the same as the error involved in quantitative analysis. The latter is affected not only by the extent of separation (reflected in FO), but also by the algorithms or programs used to establish the peak area.

If the peaks are assumed to be Gaussian and if the exact peak positions and peak widths are known (the latter are very difficult to obtain accurately from a chromatogram), then FO can be calculated. But even then, the calculation is fairly complicated and simple equations relating FO to the difference in retention times and the standard deviations of the two peaks cannot be derived. For non-Gaussian, non-symmetrical peaks FO can only be estimated if the profiles of each of the individual peaks in the chromatogram can be established. This can be done in a purely mathematical way by “deconvolution”. This requires some mathematical function that describes the shape of the real peak with some degree of accuracy, and preferably also knowledge of the number of peaks actually present in the part of the chromatogram. It also requires complex computer programs.

A more practical way to obtain the profiles of the individual peaks may be a sensible application of modem multichannel detection techniques (see section 5.6.3). It should be noted that neither mathematical deconvolution nor multichannel detection can be a

124

substitute for chromatographic separation. They only serve to illustrate that sophisticated techniques are required if FO is to be used as a criterion by which to judge the separation between adjacent peaks in a chromatogram. At present, therefore, the FO criterion seems to be a merely theoretical proposition.

Liquid

4.2.3 Separation factor

Solid; CBP

In chapter 1 (eqn.l.20) we have seen that the resolution (R,) can be described as the product of two factors, one covering the chemical and physical characteristics of the separation and one reflecting the column efficiency:

Open columns

k , - k , fi .- R s = k , + k , + 2 2

Column diameter ( k cc dc-’)

(4.14)

Packed columns

The first factor in eqn.(4.14) combines the effects of the capacity factor ( k ) and the selectivity (a) on the resolution. To some extent, k and a can be varied independently for the purpose of optimization. Notably, k varies with the phase ratio (eqn.l.10) while adoes not. Hence, if the largest value for a is observed in conditions where k values are either too high or too low, variations in the phase ratio may be used to realize an optimum separation.

The parameters that can be used for this purpose have been classified as “capacity parameters” in section 3.5. These parameters and the ways in which they affect the capacity factors are summarized in table 4.2.

Surface area ( k a S,)

Table 4.2: Summary of parameters which affect retention (k), but do not affect selectivity (a). The proportionalities given assume all other parameters to be constant.

Stationary phase

I ( k a E - ’ )

For various reasons the “capacity parameters” listed in table 4.2 will not often be used to optimize k values.

125

In the first place column characteristics will often be determined by practical conditions, such as availability of columns and materials and instrumental considerations.

In the second place, the parameters listed in table 4.2 cannot always be varied independently and, moreover, will have side effects on yet other parameters. All the capacity parameters affect the phase ratio (V,/V,,J. If all other parameters are kept constant, then the film thickness and the surface area will affect V,, the porosity will affect V,,, and the diameter of open columns will affect both V,,, and V,. However, it is often impossible to keep all other parameters constant. For instance, it would be very difficult to vary the porosity without changing the surface area. An example of the effect of variations in the capacity parameters on other parameters is the decrease in the number of theoretical plates in the column that usually accompanies an increase in the stationary phase film thickness in GLC.

Thirdly, the parameters in table 4.2 turn out to be proportional or inversely proportional to k, whereas other parameters which affect both k and a, such as temperature in GC and mobile phase composition in LC, have an exponential effect on k (see table 3.10). Hence, even if higher a values can be obtained at some temperature or composition outside the range where k is optimal, chances are that the parameters listed in table 4.2 do not offer sufficient flexibility to move k values back into the optimum range.

For all these reasons, it is usually realistic to treat R, as the product of only two independent factors according to eqn.(4.14). The first factor will depend on the retention ( k ) and the second factor will reflect the efficiency of the chromatographic system (N). Since we are most interested in small values for R, (i.e. k , M k J , the variation of N with k can be neglected as a first approximation*, and the two factors can be treated as independent. Therefore, we can define a separation factor independent of the column efficiency:

(4.15) h - k , S = k , + k , + 2

This separation factor was first suggested by Ober [407] an it has been used more recently by Jones and Wellington [408] and by Schoenmakers and Drouen [409,410]. For a given value of S, the number of plates required to realize a given value of R, ( Nne) can easily be obtained from

N,, = 4 ( R, / S ) 2 . (4.16)

S has the advantagethat it is obtained from the chromatogram much more readily than is the case for R,. To establish the value of S no estimate of the peak width is required. Moreover, if we substitute k = (t- @ / t o in eqn.(4.15), we find that

(4.17)

Hence, S can be obtained directly from the retention times of two successive peaks,

* While this may be true for a pair of adjacent peaks in the chromatogram, it may not be quite as valid an approximation if an entire chromatogram is considered (section 4.3).

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without the use of an estimate for the hold-up time (to). Conversely, if S is calculated from k values, the to value used to obtain the latter will not affect the value of S.

We may summarize the advantages of the separation factor S as follows: 1 . S is directly related to chromatographic theory (as is RJ. 2. S can readily be obtained from the chromatogram. 3. N o estimate for the hold-up time to is required to establish S . The disadvantages are: 1 . Use of S implies the assumption of Gaussian peaks. 2 . The plate count N is assumed to be constant throughout (parts ofl the chromato-

gram (i.e. N is independent of k), as well as constant in time.

The second disadvantage can largely be removed by expressing the plate count (N) as a function of the capacity factor (k). This has been demonstrated by Svoboda [41 I]. If N is not assumed to be a constant, but some function f(k) of k, then we assume that the peak-broadening process is determined by the properties of the column and the phase system and not by the properties of the solute (e.g. its diffusion coefficient). In other words, if two very different solutes elute with the same capacity factor, we would expect the widths of the two peaks to be the same. While this may not always be completely true, it appears that a useful refinement of the elemental criterion is possible in this way. The function f(k) will add an extra factor to eqn.(4.15):

* f(k) . S’ = k2 - k, k , + k , + 2

(4.1 5a)

However, the main reason for preferring S over R, (i.e. that no estimate is required for N) is now no longer relevant, and therefore, it is more appropriate to introduce the function f(k) into eqn.(4.14):

k, - k, f( k)1’2 .- R , = k,+k,+2 2

(4.14a)

In order to estimate k (and hence f( k)), an estimate for the hold-up time to is required. However, this can be avoided if N is expressed as a function f( V,) of the retention volume.

4.2.4 Discussion

A comparison of various elemental criteria has been reported by Knoll and Midgett [412] and by Debets et al. [413]. Figure 4.4 shows the variation of some of the criteria for the separation of pairs of chromatographic peaks as a function of the time difference between the peak tops (At = t2 - t , ) . By definition, R, (and hence S) varies linearly with At. The peak-valley ratios (P) and the fractional overlap both increase rapidly with increasing AZ at first, but level off towards At =: 4 (T to reach the limiting value of 1. At high values of At, R , and S will keep increasing, while the other criteria will not.

Figure 4.5 shows the variation of the fractional overlap criterion with At for three different values of the ratio of peak heights (A). These data were calculated for Gaussian peaks. It is clear that FO will be lower for peak ratios different from unity. Similar

127

calculations reveal that P and P, are virtually independent of A (for Gaussian peaks), but that P, varies with A as expected.

FO accurately describes the real extent of quantitative separation obtained in a chromatogram. If an analysis is optimized on the same column on which it will later be run as a routine separation, then this is a fair criterion. If however, the analysis will eventually be run on a different column of a potentially different length (or diameter, etc.), then it will often be hard to predict the value of FO on that other column. In the case where

2 L 6 8 AL .-*

0

Figure 4.4: Variation of some elemental criteria as a function of the difference in retention times between the two solutes. Data calculated for Gaussian peaks of equal height. Courtesy of Anton Drouen [405].

'I

-0 2 6 8

Figure 4.5: Variation of the fractional overlap criterion (FO) and the resolution ( R J as a function of the difference in retention times between the two solutes. FO data calculated for Gaussian peaks of varying peak height ratios ( A = h, /h , ) . Courtesy of Anton Drouen [405].

128

FO equals unity this becomes quite impossible. In other words, given the final column for routine analysis, very large values of At are unattractive, since they do not increase the value of FO, but do lead to an increase in analysis time. If, however, we can tailor our column to the result of the optimization procedure (i.e. to the number of plates required), then large values of At leading to very large values of R, are indeed significant. Hence, in the case where the column dimensions can be chosen after completion of the optimization of selectivity, the use of R , or S is preferred, because of the clear and simple relationship between these criteria and the required number of theoretical plates.

The same argument holds even more strongly with respect to peak-valley ratios. Not only is there a range of large At values for which P = 1, there is also a considerable threshold range in which At > 0, but there is no discernible valley between the peaks and hence P = 0. For Gaussian peaks of equal height this threshold range was shown to equal a resolution of 0.59 or less (eqn.4.11).

If the analysis to be optimized involves a sample in which the relative peak areas are expected to be constant (for instance in a quality control situation), then a criterion may be used that is affected by the relative peak height (A), i.e. FO or P , may be used. If this is not the case, then a criterion should be selected that does not vary with A (R,$ or S; P or P J . This will avoid the very unattractive situation in which the location of the optimum is a function of the (quantitative) composition of the sample, so that in theory there may be different optimum conditions for every single sample!

This effect will be most pronounced in the case where a solvent peak dominates the chromatogram and solutes need to be analyzed on the tail of this peak (see section 4.6.3).

A similar argument holds for the influence of the peak shape on the separation criterion. In the non-linear part of the distribution isotherm, the shape of the peak will be a function of the injected quantity. Hence, once again, the location of the optimum may be affected by the composition of the sample. Also, the effect of column dimensions on the peak shape may be hard to predict, and the peak shape may to a large extent be determined by the characteristics of the instrument, rather than of the column. Therefore, if the composition (or the concentration) of the sample can be expected to vary considerably, and if it is desirable that the result of an optimization process can be extrapolated to different columns (of the same type) and to different instruments, then it is advisable to use criteria that are not affected by the relative peak areas, nor by the shape of the peaks.

For practical evaluation FO is a very unattractive criterion. Its variation with At and with the peak area ratio A is similar to that of the peak-valley ratio P,. Pand P , are similar to each other in all respects. P , may be obtained from the chromatogram slightly more easily than P, because it only requires location of the peak tops, and not of the valleys. To calculate R , from the chromatogram an estimate of N is required. Scan be estimated very easily, using only the retention times of individual peaks.

Below a certain threshold resolution, no valley can be observed between two adjacent peaks in a chromatogram. In that case the value for any of the peak-valley ratios would equal zero. In theory, the value for R, and S would exceed zero for any two peaks that have different retention times (At > 0). In practice, this difference vanishes if the presence of two peaks cannot be discerned from the chromatogram. However, the occurrence of ill-resolved peaks in a chromatogram may be recognized visually at resolutions well below 0.6 (the threshold value below which P equals zero for Gaussian peaks of equal height) (see ref. [401], figure 2.1 1, p.38). Moreover, there are several techniques which may be of

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FO equals unity this becomes quite impossible. In other words, given the final column for routine analysis, very large values of At are unattractive, since they do not increase the value of FO, but do lead to an increase in analysis time. If, however, we can tailor our column to the result of the optimization procedure (i.e. to the number of plates required), then large values of At leading to very large values of R, are indeed significant. Hence, in the case where the column dimensions can be chosen after completion of the optimization of selectivity, the use of R , or S is preferred, because of the clear and simple relationship between these criteria and the required number of theoretical plates.

The same argument holds even more strongly with respect to peak-valley ratios. Not only is there a range of large At values for which P = 1, there is also a considerable threshold range in which At > 0, but there is no discernible valley between the peaks and hence P = 0. For Gaussian peaks of equal height this threshold range was shown to equal a resolution of 0.59 or less (eqn.4.11).

If the analysis to be optimized involves a sample in which the relative peak areas are expected to be constant (for instance in a quality control situation), then a criterion may be used that is affected by the relative peak height (A), i.e. FO or P , may be used. If this is not the case, then a criterion should be selected that does not vary with A (R,$ or S; P or P J . This will avoid the very unattractive situation in which the location of the optimum is a function of the (quantitative) composition of the sample, so that in theory there may be different optimum conditions for every single sample!

This effect will be most pronounced in the case where a solvent peak dominates the chromatogram and solutes need to be analyzed on the tail of this peak (see section 4.6.3).

A similar argument holds for the influence of the peak shape on the separation criterion. In the non-linear part of the distribution isotherm, the shape of the peak will be a function of the injected quantity. Hence, once again, the location of the optimum may be affected by the composition of the sample. Also, the effect of column dimensions on the peak shape may be hard to predict, and the peak shape may to a large extent be determined by the characteristics of the instrument, rather than of the column. Therefore, if the composition (or the concentration) of the sample can be expected to vary considerably, and if it is desirable that the result of an optimization process can be extrapolated to different columns (of the same type) and to different instruments, then it is advisable to use criteria that are not affected by the relative peak areas, nor by the shape of the peaks.

For practical evaluation FO is a very unattractive criterion. Its variation with At and with the peak area ratio A is similar to that of the peak-valley ratio P,. Pand P , are similar to each other in all respects. P , may be obtained from the chromatogram slightly more easily than P, because it only requires location of the peak tops, and not of the valleys. To calculate R , from the chromatogram an estimate of N is required. Scan be estimated very easily, using only the retention times of individual peaks.

Below a certain threshold resolution, no valley can be observed between two adjacent peaks in a chromatogram. In that case the value for any of the peak-valley ratios would equal zero. In theory, the value for R, and S would exceed zero for any two peaks that have different retention times (At > 0). In practice, this difference vanishes if the presence of two peaks cannot be discerned from the chromatogram. However, the occurrence of ill-resolved peaks in a chromatogram may be recognized visually at resolutions well below 0.6 (the threshold value below which P equals zero for Gaussian peaks of equal height) (see ref. [401], figure 2.1 1, p.38). Moreover, there are several techniques which may be of

129

help in confirming the purity of the peaks obtained in a chromatogram during an optimization process (see section 5.6).

In some optimization procedures the capacity factor is known as a function of the parameters. These so-called “interpretive optimization methods” will be described in section 4.5. From known capacity factors R, and S can be calculated much more easily than peak-valley ratios and, moreover, from known capacity factors the R, or S values can be calculated, no matter how small the difference between the two capacity factors is. In other words, the resolution of a pair of peaks can be calculated in a range in which it would be very difficult to obtain an estimate for the resolution from an actual chromatogram. Therefore, the use of R, or S as a criterion to judge the separation in combination with interpretive optimization methods enables us to appreciate variations in the resolution in the range of 0 < R,< 0.6. Such variations are very significant because (i) on a different (more efficient) column the separation with the highest value for R, is most easily realized and (ii) on the same column, improvements in resolution in the range 0 < R, < 0.6 will help to send the optimization process in the right direction.

Hence, in combination with interpretive methods the use of R, or S as the resolution criterion appears to be always advantageous.

A further refinement may be sought by incorporating a function f(k) to describe the dependence of the plate count on the capacity factor (see eqns. 4.14a and 4.15a).

The characteristics of the different criteria are summarized in table 4.3. Table 4.4 lists the recommendations formulated above for the use of different criteria.

Table 4.3: Characteristics of different elemental criteria for measuring the extent of separation of a pair of chromatographic peaks.

Criterion Affected by Reflects Transfer Ease of actual towards calcu-

Peak area Peak separation other lation ratio shape columns

- + / - + + + / - - + / - + + + +

P - + + + / - ( I ) + / - - + + + / - (1) + / - + + + - + / - pm

P“

- - - FO + + + + (1) Indirectly via eqn.(4.10), but only in the range where (approximately) 0.05 < P<0.95.

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Table 4.4 Recommendations for the use of different elemental criteria for measuring the extent of separation of a pair of chromatographic peaks. The preferred criteria are given, while possible alternatives appear in brackets.

Optimization on final analytical column

YES NO Interpretive method (1)

YES NO

(1) See section 5.5. (2) The noise level can be incorporated in analogy to eqn. (4.12).

4.3 CHROMATOGRAMS

We will base our discussion about criteria by which to judge the quality of an entire chromatogram on the elemental criteria for pairs of chromatographic peaks, which have been defined in chapter 1 (R, and a) and in the previous section . We will look at several ways of combining the numbers for all individual pairs of peaks into a single number. We will then discuss the influence of other parameters, such as the analysis time and the number of peaks on the proposed criteria.

Initially, the discussion will be focussed on the general case (see section 4.1), in which all peaks in the chromatogram are considered to be of equal importance and all peaks have to be separated. At the end of this chapter, we will discuss some specific cases, for which the requirements are different.

4.3.1 Sum criteria

Summation of resolution values has been used by Berridge [414] and summation of separation factors has been suggested by Jones and Wellington [408].

In the introductory section of this chapter it was shown that a straightforward summation of resolution values does not yield a satisfactory criterion for the evaluation of complete chromatograms (see figure 4.1 and table 4.1). A problem that can readily be appreciated from the example given there, is that the sum of R, values will be determined mainly by the largest values of R, that occur in the chromatogram. For example, in the

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chromatogram shown in figure 4.lb, the sum of resolutions turns out to be 25.3. Of this, 24.07 or 95% is due to the excessive separation of the last two peaks. However, in judging the separation, the chromatographer will immediately refer to the separation of the first two peaks, rather than to that of the last pair. This is correct, because the first two peaks determine the efficiency of the chromatographic system that is required to realize the separation of all three peaks (see also section 4.3.3 below).

Apparently, it is the occurrence of very large R, values that causes problems. Obviously, this problem can then be avoided by substituting for R, one of the criteria which level off for very large time differences between the two peaks (see figure 4.4). In this way, the contribution of the abundantly separated pairs of peaks in a chromatogram is limited. The resulting sum of FO or P values is to a much smaller extent determined by the largest values, although in some extreme cases large contributions may still obscure important changes in small ones. For example, if 20 pairs of peaks were to occur in a chromatogram and each of these pairs were almost separated to the baseline (P=O.9), then the sum ZP would equal 18. This is less than if 19 out of the 20 pairs of peaks were amply separated (P= l), but two peaks showed complete overlap (P= 0), giving rise to a value for ZP of 19. Assuming that all peaks are of equal importance, the latter chromatogram is obviously inferior. Of course, this is a hypothetical example, but it illustrates a potential limitation of the criterion ZP.

Figure 4.6 illustrates the dependence of ZP and ZR, on the number of plates in the

50 100 fi-

Figure 4.6: Variation of the sum of peak-valley ratios as a function of the number of plates for the two chromatograms ( a and b) shown in figure 4.1, and for a third chromatogram (c), shown in figure 4.8. P was calculated from eqn.(4.10). Negative values for P were set equal to zero. The sum of resolution values is shown as a dashed line for chromatogram a only.

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column. This figure has been calculated from the data in table 4.1, assuming that the chromatograms of figure 4.1 have been run on a series of columns with different Nvalues. Obviously (eqn.1.22), ZR, increases linearly with V N . ZS (not indicated in figure 4.6) is independent of N (see eqn.4.15) The behaviour of ZP is vastly different.

When there are no plates (N = 0), there is complete overlap of all peaks (ZP= 0). For chromatogram b this soon changes, since only a handful of plates is necessary to achieve baseline separation of the last two peaks. For a while, the ZP value for chromatogram b is then larger than that for chromatogram a, because two peaks are distinguishable instead of one. However, at still higher values of N the last two peaks in chromatogram a start to be resolved, soon followed by the first two peaks in both chromatograms. Eventually, abundant resolution will be achieved for all pairs of peaks, all values of P will equal unity and the ZP values for both chromatograms will be the same (ZP= 2).

From this point on the use of ZP no longer enables us to quantify the quality of the chromatogram, because it does not differentiate between chromatograms a and 6. Moreover, above a certain threshold number of plates (around 10,000 in the example of figure 4.6) the ZPcriterion becomes very insensitive to the number of plates and to changes in the relative peak positions, unless these changes have a significant effect on R , values between about 0.6 and 1.5.

From the above discussion the following five conclusions can be drawn: 1. ZR, is not a useful criterion for judging the quality of a chromatogram, since its value

is determined largely by the largest values of R, that occur in the chromatogram. i.e. by thepairs ofpeaks which are the least relevant for the realization ofa separation. Thesame is true for the sum of separation factors (ZS).

2. ZP gives a better representation of the actual separation achieved on a given column, since there is a limit to the contribution of amply resolvedpairs ofpeaks.

3. Above a certain threshold value for the number of plates, a pair of peaks will become irrelevant for the determination of ZP. When this threshold is reached for all pairs of peaks. ZP will have reached its limiting value. Changes in N will no longer be reflected in ZP, whereas changes in the (relative) retention times become increasingly irrelevant as N increases.

4. Below a certain threshold value for the number ofplates, all P values will be zero and again ZP becomes insensitive to changes and provides no information about the chromatogram.

5. The values of ZR, and ZS on another column with a different plate count can easily be predicted, since ZR, is proportional to V N and since ZS is independent of the plate count. For ZP, the situation is less straightforward. The value of ZP is relevant for the separation on a given column, but cannot be extrapolated from there.

As the main conclusion from this section it appears that ZR, (and hence ZS) is not a useful criterion, and that ZP may be used for a comparison of chromatograms on a single column. A problem that remains is the fact that ZP becomes a very insensitive criterion once the limiting value (equal to n - 1 if n is the number of peaks in the chromatogram) is approached, as well as in the range in which one or more of the P values become equal to zero. In section 4.3.4 we will investigate whether composite criteria (involving for instance the analysis time) can be used to avoid this problem.

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4.3.2 Product criteria

A second major category of proposed criteria to ress the extent of separation in an

of one of the elemental criteria defined before. Taking the products of these criteria is equivalent to taking the sum of the logarithms, for instance

entire chromatogram is that in which the product is taken \ f the values for all pairs of peaks

l I R , = e x p ( Z I n R , ) . (4.18)

The use of the sum of logarithms may have a slight disadvantage in the case where a value of zero occurs for one of the pairs of peaks. If any of the peak-valley ratios (P, P, or Pv) is used, then this problem is aggravated because these criteria take on a value of zero below a certain threshold resolution. The obvious way around this problem, however, is to set the sum of logarithms equal to minus infinity or to a large negative number once a value of zero occurs.

A summary of the use of product functions in the literature is given in table 4.5. In some cases, the products were part of composite criteria involving other factors or terms. We will come back to these criteria in section 4.3.4. It is clear from table 4.5 that product criteria have been used more extensively than have sum criteria.

Table 4.5: Summary of product criteria proposed in the literature to express the extent of separation achieved in a chromatogram.

Elemental Product criterion proposed by criterion

Ref.

RS Glajch et al., JC 199, (1980),57

S Schoenmakers et al., Chr 15, (1 983),688

FO Smits et al. ZAC 273, (1975),1

P Morgan and Deming, JC 112, (1975),267 Watson and Carr, AC51, (1979),1835 Spencer and Rogers, AC 52, (1980),950

P' Wegscheider et al., Chr 15, (1982),498

41 5

409

416

417 41 8 419

406

Explanation of abbreviations: A C = Analytical Chemistry Chr = Chromatographia JC = Journal of Chromatography ZAC = Fresenius Zeitschrift fur Analytische Chemie

One obvious advantage of product criteria is that the result will be mainly determined by the smallest values for the elemental criterion, i.e. by the least resolved pairs of peaks.

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All the criteria that have been discussed in section 4.2 will equal zero if one pair of peaks shows complete overlap and therefore once such a situation occurs the resulting product will be zero as well.

/ I /

5 10 ~ 1 1 0 ~ -

Figure 4.7: Variation of the product of peak-valley ratios as a function of the number of plates for the two chromatograms ( a and b) shown in figure 4.1, and for a third chromatogram (c), shown in figure 4.8. P was calculated from eqm(4.10). Negative values for P were set equal to zero. The product of resolution values is shown as a dashed line for chromatogram a only.

Figure 4.7 shows the variation of the products of R, and P values as a function of the number of theoretical plates for the chromatograms of figure 4.1 and an additional chromatogram shown in figure 4.8. Plotting the products against N yields a straight line for nR,. In general, l7R, will be proportional to N, where p is the number of pairs of peaks in the chromatogram. Therefore, the differences between l7R, and l7P will be more pronounced for chromatograms with large numbers of peaks.

As with the sum criteria, the use of P instead of Rs does not yield a simple relationship for the variation with N. However, it is very clear from figure 4.7 that the differences between the l7P values for the chromatograms a and b are much smaller than the differences in the ZP values (figure 4.6). This illustrates that the value of l 7 P is mainly determined by the least separated pair of peaks, i.e. the first two peaks, which are the same in both chromatograms.

All product criteria will be zero if any single pair of peaks is completely unresolved. For FO, R, and S this situation theoretically only occurs if the retention times of two peaks are equal. For peak-valley ratios a value of zero is estimated from the chromatogram below a certain threshold sewration, which for Gaussian peaks corresponds to R, < 0.59

135

(eqn.4.11). Thus, l7P equals zero once any pair of peaks shows no discernable valley. Of course, there is an important increase in separation if R, is increased from 0 to 0.6. This is a serious drawback to the use of l7P as an optimization criterion, since it does not acknowledge definite improvements below a certain threshold and it illustrates once more that the elemental criterion P may be used only if the optimization process is carried out on the final analytical column.

Even more than ZP, l7Pis a threshold criterion. Its value is zero or one, with only a small range over which intermediate values occur. Threshold criteria can be used to allocate areas in which a certain condition is fulfilled. They divide the parameter space (i.e. the space formed by all the parameters considered in the optimization process, see section 5.1.3) into areas where a certain condition is met and areas where this is not the case. In the case of l7P there is a diffuse boundary in between the different areas.

So far, all the criteria that have been discussed have suggested either that chromatogram b is superior to chromatogram a, or that both chromatograms have equal credentials. The

k- 0

Figure 4.8 Three schematic chromatograms. Constructed for N= 10,000. Capacity factors are listed in table 4.6.

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main reason for this is that the capacity factors of the last peaks in the chromatograms in figure 4.1 are vastly different, i.e. we are comparing two chromatograms which on the same column under identical conditions would show vastly different analysis times. Clearly, some correction is required once this is the case.

Figure 4.8 shows the two chromatograms of figure 4.1, together with a third chromatogram (c), in which the capacity factors of the first and the last peak equal those observed in chromatogram b, but the separation of the first two peaks has been improved dramatically. Table 4.6 lists the data for all three chromatograms. The capacity factors are

Table 4.6: Data for capacity factors, elemental criteria and for criteria judging the extent of separation in the entire chromatograms. Chromatograms are shown in figure 4.8. Criteria for pairs of peaks: separation factor (S, eqn.4.15), resolution ( R , eqn.4.14) and peak-valley ratio ( P , eqn.4.10). Criteria for entire chromatograms: sum criteria (section 4.3.1), product criteria (section 4.3.2), normalized resolution product (r, eqn.4.19), calibrated normalized resolution product (r*, eqn.4.21) and minimum resolution (section 4.3.3). For discussion see text.

Chroma- Peak k S R S P togram number

1 1 0.0244 1.22 0.898

0.0345 1.72 0,995 a 2 1.1

3 1.25 r = 0.97 2 0.059 2.9 1.893 r* = 0.13 n 8.4.10-4 2.1 0.893 Rs,min = 1.2

1 1 0.0244 1.22 0.898

0.481 24.1 1 3 5 r = 0.18

b 2 1.1

z: 0.51 25.3 1.898 r* = 0.18 n 1.2.10 29.4 0.898 Rs,min = 1.2

1 1 0.273 13.6 1

0.263 13.2 1 3 5 r = 1.00

C 2 2.5

z: 0.54 26.8 2 r* = 0.98 n 7.2.10 180 1 Rs,min = 13.2

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, used to calculate S, R, and P. The latter is estimated from eqn.(4.7). Also, the various sum and product criteria are shown in the table. Since there is a constant factor of 50 (vN12) between S and R, we will focus on R, and P only.

In all three chromatograms the threshold number of plates for separation appears to have been approached, so that CP and are only slightly affected by the differences between the chromatograms. No distinction can be made between the sum criterion CP and the product criterion nP, The difference between the two is 1 for all three chromatograms. Both criteria yield marginally higher values for chromatogram b in comparison to.chromatogram a. Chromatogram c yields the maximum values of 2 for ZP and 1 for nP. In fact, it is well into the region in which the peak-valley ratio is completely insensitive to variations in the capacity factors. If the resolution (R,) equals 2, then eqn.(AlO) yields a Pvalue of 0.999. In chromatogram c of figure 4.8 the resolution between each pair of peaks is about 13.

ZR, is much higher for chromatograms band c than it is for chromatogram a. However, it is about equal for the bottom two chromatograms. Hence, ZR, is more sensitive to changes in the capacity factor of the last peak than it is to changes in the extent of separation. n R , does yield a much higher value for chromatogram c than it does for chromatogram b. Hence, nR, can be used for a quantitative comparison of chromatograms of similar length (capacity factor of the last peak). When the length of the chromatogram changes (for instance in going from chromatogram a to chromatogram b), nR, is not a useful criterion.

Normalized resolution product

Drouen et al. [410] have recognized this problem and proposed a solution by using a product of normalized resolution values. They divide every value of R, by the average R, value (x,), where the average is taken over all the pairs of peaks in the chromatogram:

where n is the number of peaks and

(4.19)

(4.20)

The average S value (3 is defined analogously. Because both R,s and x,$ are proportional to V N , the normalized product of R, values

is equal to that of the S values, and both are independent of the number of plates. The normalized resolution product ( r ) will vary from zero, in the case where one or more pairs of peaks show no resolution, to one, if the resolution is equal for all the pairs of peaks in the chromatogram. Therefore, in choosing r as the optimization criterion the aim is to achieve an equal distribution of the peaks over the chromatogram.

This aim is realized much better in chromatogram c in figure 4.8 than it is in chromatogram b. The r values given in table 4.6 illustrate that fact. For chromatogram c the r value is (almost) equal to 1, while for chromatogram b it is not larger than 0.18.

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Chromatogram a is very different from chromatogram c, but it also shows good spacing of the peaks over the chromatogram and hence an r value of about 1.

Clearly, r values do reflect the distribution of the peaks over the chromatogram and are not seriously affected by the absolute k values.

Chromatogram c appears to show a much longer analysis time than does chromatogram a. However, if we are free to define the column dimensions after the selectivity optimization process, chromatogram c can be the basis for a very quick separation on a very short column.

Calibrated normalized resolution product

In theory, all peak pairs may show equal R , values, but the peaks may occur very late (with very high capacity factors). A bunch of peaks may move about through the chromatogram and will yield the same value for any of the criteria discussed so far as long as the mutual resolution factors between all the different pairs of peaks remain unaltered. To avoid bunching of the peaks at some high value of k, Drouen et al. have suggested the inclusion of a hypothetical peak at t = to in the calculation of r [410]*. This yields the calibrated normalized resolution product (r*), defined as

n - I n-1

i = O i = O r* = ll (R, i , i+, / R,) = ll (Si,i+l 1s)

where

(4.21)

(4.22)

The r* values for the three chromatograms in figure 4.8. are also shown in table 4.6. It appears from these values that r* is very high (close to the maximum value of 1) for chromatogram c, but that the bunching of peaks around k = l in chromatogram a is effectively penalized. The introduction of the hypothetical peak at t = to has the effect of “calibrating” or “anchoring” the real peaks in the chromatogram to a starting point.

The optimum value of r (or r*) does not correspond to a unique chromatogram, but rather to a series of chromatograms, each of which has the peaks spread out at constant resolution intervals in the chromatogram. In other words, the absolute value for R, or S may vary, but all the normalized values are equal to 1.

r* is higher for chromatogram b than it is for chromatogram a. If we include a peak at t = to, then the value for S between this peak and the first real peak in the chromatograms is 0.33 (R, = 16.7; P= 1). This means that for chromatogram a one high value for S occurs (0.33) in combination with two low values (0.02 and 0.03), while in chromatogram b two high values (0.33 and 0.48) are combined with one low value (0.02). If the goal is to make all S values equal, both situations are equally bad. Nevertheless, because of the much smaller k value for the last peak (and given the equal resolution for the first two real peaks) chromatogram a may well be preferred to chromatogram b.

* A hypothetical peak assumed at any time after t = to [410] will give rise to considerable problems, because peaks may be distributed over the chromatogram before as well as after this imaginary peak.

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Although r* appears to be very useful as a criterion that strives towards a clear and objective goal in selectivity optimization, it is still not perfect once two chromatograms are compared which show very different capacity factors for the last peak, especially when both r* values are low.

The use of P , in sum and product criteria

The use of P, in sum or product criteria creates a special problem, because two values for P, can be calculated for each peak in the chromatogram. This applies to isolated as well as to ill-resolved peaks and it also applies to the first and the last peaks observed in the chromatogram. The number of P, P,, R,, or S values in an entire chromatogram usually equals n-1 (where n is the number of peaks). If an imaginary peak is assumed to be present at t = to, the number of values for the elemental criteria becomes n. We can deal with this problem in three ways. 1. Use all P, values without correction.

This is a useful approach if no other criteria are to be considered, so that the criteria based on P, will not have to be compared with other criteria.

A considerable disadvantage of this approach is that large improvements in the resolution may go by unnoticed.

If two values for P, for each peak are obtained, then this strategy corresponds to the use of (112) ZP, or VnP, as the optimization criterion. This third approach appears to be the most correct one, since it creates a common basis for all sum and product criteria (i.e. those based on P, P , P, R , and S), which may allow a comparison between the different propositions.

Another peculiar aspect of the use of P, is that its value is by definition proportional to the height of the peak (see also section 4.2.4). Hence, for a pair of peaks with a certain valley between them, the P, value will be largest for the largest peak and proportionally smaller for the smallest one. Since sum criteria are mainly determined by the largest values for the elemental criteria (see section 4.3.1), we may expect that the value of ZP, will be affected most by the largest peaks (major components) in the sample. On the other hand, product criteria are affected most by the smallest values for the elementa1 criteria (see section 4.3.2) and hence n P , will be determined mainly by the minor components in the sample, i.e. the smallest peaks which are detected and considered relevant in the optimization process. Because one very small value (close to zero) for P, will largely determine the value for the entire product, this emphasis on small peaks is much greater than the emphasis put on the large peaks by the use of ZP,,.

We might say that if P, is used as the elemental criterion, a weighing factor is automatically built into the (sum or product) criterion for the entire chromatogram, which puts the emphasis either on the major or on the minor components in the sample.

2. Use of the lowest value for P, that occurs on either side of a peak.

3. Use of an average value for P,.

4.3.3 Minimum criteria

The lowest value for a which occurs in a chromatogram has been used extensively in GC [420] and LC [421-4231 (see also section 5.5) as a criterion to quantify the extent of

140

separation achieved in a chromatogram. This so-called minimum a (amin) criterion is set equal to the lowest value for (I that occurs for any pair of peaks in the chromatogram.

However, the value of a is not a good indication for the separation of a pair of peaks. For example, if an a value of 1.05 occurs somewhere early in the chromatogram, say around k = 0.5, then the corresponding S value is 0.008 and 60,000 plates are required to achieve adequate resolution (R,= 1) of the two peaks. If the same value for awere to occur around k = 3 , then S would equal 0.018 and to realize an R, value of 1 about 12,000 theoretical plates would be sufficient.

Clearly, it is advisable to substitute R, (a quantity which depends on the plate count) or S (independent of N) for a. In judging a chromatogram on the basis of the*minimum value for R, ( Rs.min) or S (Smin), it becomes very easy to estimate the number of plates that is required to realize the separation with sufficient but not excessive resolution. For instance, if the final result of a selectivity optimization process is a chromatogram with an Rs.min value of 0.5 on a column with 2,500 theoretical plates, then a column with 10,000 plates will yield an Rs,min value of 1 under identical conditions.

During a selectivity optimization process the Rs.min or Smin criterion can be used in two ways: 1. By setting a threshold value (e.g. Rs.min = I ) , above which the result is acceptable. If x

is the threshold value, we can describe this criterion as

(4.23)

This criterion may be used during a sequential optimization process (see chapter 5), leading to an acceptable result and to completion of the optimization process once the threshold value has been reached. Alternatively, it may be used to establish ranges of conditions in the parameter space for which the result will be acceptable. This latter approach has been followed by Glajch et al. 14151, by Haddad et al. [424] and by Weyland et al. [425] and was referred to as resolution mapping by the former. Within the permitted area(s) secondary criteria are then required to select the optimum conditions. For example, the conditions at which the k value of the last peak ( k J is minimal while the minimum value for R,sexceeds 1 may be chosen as the optimum. Such a composite criterion can be described as

min k , f l Rs,min > x . (4.24)

2. A second way to use the Rs,min criterion is to try and maximize the value for Rs.min. In other words, one may strive towards conditions at which the lowest value for R, observed in the chromatogram is as high as possible. We can describe this criterion as

(4.25)

This criterion aims at a chromatogram that can be realized with the lowest possible number of theoretical plates. Indeed, if the highest possible value of Rs.min has been reached, this automatically corresponds to the lowest number of ptates required (see section 4.4.3).

Although the goat of achieving the separation with a minimum number of plates appears

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to be clear and unambiguous, the resulting chromatogram is not well-defined. The fact that Rs,min has the highest possible value reveals nothing about the remainder of the chromatogram. A very simple example can be found from the two chromatograms in figure 4.1, which yield the same value for Rs,min. Moreover, merely because of the direct relationship between R, and k (see section 1 S), R, values will tend to be larger for larger values of k. Hence, in striving towards a maximum value for one may be striving implicitly towards very high values for k. Therefore, the criterion described by eqn.(4.25) should only beconsidered as a criterion for a selectivity optimization process if the overall capacity factors are not expected to change considerably.

In those cases in which the overall capacity factors do vary, it is more realistic to use the Rs,min criterion in the way as described under 1. above. Eqn.(4.23) can be used to define the boundaries within which acceptable resolution can be achieved. It has the advantage that no implicit aim towards high capacity factors is present in the criterion. A disadvantage of the use of eqn.(4.23) is that the boundaries defined by the threshold criterion will change when the acceptance level is changed, i.e. they will be different if x in eqn.(4.23) is set equal to 1 from when x= 1.5.

Along the same line, the boundaries will change when the number of plates is changed, i.e. when another column is used, or even when the flow rate is changed on a given column. If subsequently secondary criteria are used to define a unique set of optimum parameters rather than an acceptable range (eg. the hierarchic criterion of eqn.4.24), then the location of the optimum may very well depend on the threshold value (x) selected by the user, or on the column used during the process of selectivity optimization. For example, if the following criterion had been used

min k, n Rs,min > 1 (4.26)

and the resulting chromatogram had an Rs,min value of 1.3, then necessarily a different optimum would have been found than if the criterion

min k, n Rs,min > 1.5 (4.26a)

had been used, or equivalently if the original optimization process were repeated on a column having a factor of 2.25 (1.5*) fewer theoretical plates than the original one.

In theory, the problem of the result being dependent on the number of plates used during the selectivity optimization process can be circumvented by setting a limiting value for S, rather than for R,. Nevertheless, the problem that the resulting location of the optimum will depend on the arbitrarily selected value for the threshold x still remains. There seems to be no logical way to define a single unambiguous value for Smin which can be used in all cases.

It is interesting at this point to notice the similarity between the use of l 7P and the threshold criterion of eqn. (4.23). In figure 4.9 this is illustrated by plotting IlP, n R , and Rs,min > 1 as a function of N for the two chromatograms of figure 4.1. Clearly, both l7P and Rs,min serve as threshold criteria, with the boundaries of the acceptable areas being abrupt when the minimum resolution criterion is used and diffuse for n P . This invites the use of IlP in a similar way as i.e. analogous to eqn.(4.24). The combination of UP with other factors to form composite criteria will be discussed in section 4.4.

1 42

Figure 4.9: Variation of the product of peak-valley ratios, ( I IP) , the product of resolution values (IIR,) and the minimum resolution (R,,in) criteria as a function of the number of plates for the chromatograms shown in figure 4.1. P was calculated from eq~(4 .10 ) . Negative values for P were set equal to zero.

Although the value of R , for a given pair of peaks can be quickly transferred from one column to another by using the proportionality of R, and V N , this is not the case for the threshold criterion of eqn.(4.23). The problem is that if we know the boundaries of the area for which Rs.min > 1 using a column of 10,000 plates, we only know the boundaries of the area for which Rs,min > 0.5 for a column with 2,500 plates. We do not know what the boundaries for Rs.min > 1 are in the latter case, because we have no information on how the value of Rs.min changes with variations in the parameter settings. Only if the variation of the capacity factors as a function of the relevant parameters is known, can the boundaries of the area in which the resolution is adequate be calculated for different columns with different numbers of theoretical plates. Optimization methods in which this is the case (so-called “interpretive methods”) will be discussed in section 5.5.

The following conclusions can be drawn from the discussion in this section: 1 . The threshold criterion of eqa(4.23) can be used to establish acceptable areas for

adequate separation on a given column.

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2. The use of this criterion gives results that are similar to the use of IlP. 3. The areas defined by the threshold criterion are not transferable from one column to

another, unless this is done indirectly by means of known capacity factors.

4. Maximization of the minimum resolution value observed in the chromatogram (eqn.4.25) corresponds to aiming at the minimum number of plates required to eflectuate the separation.

5. This criterion can only be used ifthe overall capacity factors are roughly constant during the optimization process.

6. This criterion can readily be transferred from one column to another.

4.3.4 Other criteria

The first criterion to be suggested for the evaluation of the quality of an entire chromatogram was defined as the “total overlap” by Giddings in 1960 [426]. The definition reads:

cp = Z: exp -(2 R , ) . (4.27)

In figure 4.10 the function exp -(2 R,) for a single pair of peaks has been plotted as a function of R,. Also shown in this figure is the theoretical value for P (eqn.4.7).

Figure 4.10 shows that the two functions are roughly complementary, although the variation of P with R, is more abrupt. This is logical when we consider the theoretical

Figure 4.10: Variation of Giddings’ peak overlap criterion (p) for one pair of peaks and the peak-valley ratio (P) as a function of the resolution ( R J of the pair. P values were calculated from eqn.(dlO). Negative values for P were set equal to zero.

144

relationship between P and R, (eqn.4.10) in which the term 2 exp -(2 R:) appears. In comparison with eqn(4.27) the change of P with R, will be more abrupt than that of cp (for one pair of peaks) because of the ocurrance of the square of R, in eqn.(AlO).

Roughly speaking, for a complete chromatogram, the criterion Q behaves similarly to ZP. It functions as a threshold criterion with diffuse (and stepwise) boundaries, establishing areas for which adequate separation is obtained (cp=O). Because it is based on R, rather than on P, cp cannot easily be determined from the chromatogram. On the other hand, cp may more easily be calculated if the capacity factors and the plate number are known. Both cp and ZP should only be used for optimization processes run on the final analytical column. In the following discussions cp will not be considered as a separate criterion. Its merits correspond to those of the ZP criterion.

4.3.5 Summary

In the preceding four sections we have discussed various “sum”, “product” and “minimum” criteria. A schematic summary of these criteria and an indication of their applicability are given in table 4.7.

Table 4.7: Summary of sum, product and minimum criteria.

Glossary: C

tha,thf Threshold criteria, which locate boundaries at arbitrary (a) or fixed (f) degrees of

X

Continuous criterion, transferable to other columns

separation Not useful as an optimization criterion.

Criterion Sum Product Minimum

eqn. 4.25 eqn. 4.23

R s

S

P

(1) Normalized R, or S values should be used if the capacity factor of the last peak is expected to

(2) Only to be used if the capacity factor of the last peak is expected to be constant. (3) Use of l 7 P is to be preferred rather than ZP.

vary.

It is seen in the table that three groups of criteria are readily discernible:

comments. 1 . The useless criteria. This category comprises ZR, and ZS and needs no further

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2.

3.

The criteria which vary continuously between a low value of zero and a maximum value located at the optimum conditions. All these criteria allow a transfer of the resulting optimum from one column to another. The threshold criteria. which may be used to define the boundaries of areas in which an acceptable result can be achieved. The boundaries can be at some arbitrary value for R, or S (or at an arbitrary value for P< I ) , or at a fued value. The latter arises naturally from tht. use of ZP or (preferably) l 7P , or by setting Pmin = 1 . The use of l7P (or ZP) leads to di#ke boundaries.

4.4 COMPOSLTE CRITERIA

In many cases we have seen that the criteria in table 4.7 are adequate for judging the extent of chromatographic separation, but insufficient to account for the effects of chromatographic parameters on the separation. Two additional factors readily come to mind, and both have been used extensively in the literature. These factors are the number of peaks in the chromatogram and the analysis time.

Table 4.8 summarizes the requirements of the different criteria for these two additional factors.

Table 4.8: Requirements for additional parameters in the optimization criteria listed in table 4.7. The number of peaks present in the sample is asumed to be known.

Glossary: n Requires additional parameter to account for the number of peaks appearing in the

chromatogram. t Requires additional time parameter x Not useful as an optimization criterion

Criterion Sum Product Minimum

eqn. 4.25 eqn.4.23

R S

t - P t (n) t

(1) Time parameter is less necessary when normalized R,y or S values are used.

4.4.1 Number of peaks

One obvious conclusion from table 4.8 is that a provision for the number of peaks in the chromatogram is only required for the criteria which are not recommended for use in

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selectivity optimization. All product and minimum criteria become equal to zero once a peak disappears completely, while CP only approaches its limiting value of n-1 (n being the number of peaks) if all pairs of peaks are adequately resolved. If CP is used instead of nP, then a provision for the number of peaks in the chromatogram can easily be made by dividing ZP by the number of pairs of peaks in the chromatogram:

ZP n - I

c,=-. (4.28)

In eqn.(4.28) C, is the optimization criterion corrected for the number of peaks in the chromatogram*.

However, the above is only true if the number of solutes in the sample (i.e. the number of peaks that should appear in the chromatogram) is known, which is assumed to be the case in table 4.8. Obviously, if the number of peaks present in the sample is not known, complete overlap of two peaks may go unnoticed. This problem will affect all criteria, although not all to the same extent. Product criteria will often be affected more than will sum criteria. As soon as it is known that the number of peaks in a chromatogram equals at least the number n, then all the useful criteria in table 4.8 (groups 2 and 3) will automatically penalize chromatograms which show fewer than n peaks. Hence, if the number of peaks decreases during the optimization process, there is no need to correct any of the useful criteria in table 4.8 for the number of peaks present in the chromatogram.

A different situation may arise if the number of peaks increases during the optimization process, which will be more frequently the case if the process guides us in the right direction. In that case the situation may arise in which the calculated value for the criterion C is lower in a newly obtained chromatogram than it was in previous ones, while the number of peaks has actually increased. In that case we have clearly interpreted the previous chromatograms incorrectly. In many cases (for instance simultaneous methods, section 5.2 or interpretive methods, section 5.5) it is not necessary to introduce a separate correction for the criterion values if the observed number of peaks increases during the optimization process. This is because the calculation of the criterion (response) values is the final step in such a procedure. In each calculation step the number of peaks can be taken equal to the largest number of peaks observed in any of the chromatograms. If this number increases, then the results of the calculations are automatically adapted.

The situation is different if a sequential method of optimization (e.g. Simplex optimization, section 5.3) is used. In this case a criterion value is assigned to every chromatogram and the result is compared with previously obtained values. Hence, if the number of observed peaks increases, this may lead to incorrect comparisons. For example, if in one chromatogram three fully separated peaks were observed, the value of l7P for that chromatogram would equal one. However, if in the next chromatogram four peaks were observed which were not completely resolved (e.g. Pvalues between each pair of successive peaks of 0.9), then the resulting value for l7P would only be 0.73. However, the second chromatogram is clearly to be preferred to the first one.

* In this section we will generally use C for some function of the elemental criteria ( R , S or P), for instance one of the optimization criteria in table 4.8. C, refers to a criterion which has been corrected for the number of peaks in the chromatogram, while C, refers to a time-corrected criterion.

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To deal with this problem, it appears to be more correct to update the previously found criteria values than it is to increase the value of the new one. To do so, it is not only required to keep track of the criterion values of previous chromatograms, but also of the number of observed peaks. In the case of Simplex optimization this is especially easy, since only the criterion values of three chromatograms need to be remembered (see section 5.3). Hence, for the above example the previous result might be nP= 1 with n = 3. If the new result is nP= 0.73 with n = 4, then the previous result needs to be updated to nP= 0 with n=4.

It is extremely easy to update the old values for the criterion, because all product criteria become zero for chromatograms which contain less than the highest number of peaks, whereas all sum criteria remain unaltered. If a composite criterion is used, in which a time factor occurs, then the previous values for the optimization criterion (C,) may usually only be updated if the values for the individual contributions (the value of the criterion C and a time factor) are stored separately.

4.4.2 Analysis time

Analysis time has been incorporated into optimization criteria as a separate term

Typically, a separate term for the influence of the retention time appears as follows: [414,415,418] or as a separate factor [406,416].

(4.29)

where I, is the retention. time of the last peak (“analysis time”), t,,, is the maximum allowable analysis time and a and 6 are (positive) weighting factors*. The last three parameters can all be chosen arbitrarily by the user.

However, the actual influence of the user on the optimization process is limited to one parameter only, i.e. the ratio between the weighting factors a and 6. This can easily be understood from eqn.(4.29), once it is rewritten as

C, = a . { C + (Wa) t,,, - ( b / a ) t , }

= a . { C - ( b / a ) t , + C} (4.29a)

where c is a constant. Variations in c cause all values of C, to be increased by the same amount, and hence

the location of the optimum and the course of the optimization process are by no means affected. The same is true for variations in the weighting factor a, which cause all values of the criterion to be changed by a constant factor. Only variations in the 6 / a ratio that change the weighting of t, vs. C will affect the selectivity optimization process and the

* Note that for a meaningful summation in eqm(4.29) reciprocal time units (e.g. min- ’) are required for the parameter b. In this and subsequent equations we will tacitly assume that the correct dimensions have been assigned to the parameters.

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location of the optimum. Although users can easily be made to believe that the optimization process can be influenced by demands with regard to the maximum allowable analysis time, the fact is that a criterion that corresponds to eqm(4.29) completely ignores the maximum analysis time selected by the user.

Berridge [414] used a different term to incorporate the analysis time in the optmization criterion:

C, = ZR, - b It, - tm,,l - d (tmin - t,) (4.30)

where d is another constant, tmin the required “minimum retention time” and t, the retention time of the first peak. The last term is added in order to avoid overlap of the peaks of interest with solvent peaks and other signals around t = 1,. For the same reason as before, the value of tmin is completely irrelevant to the optimization process. However, t,,, has now become relevant. Eq~(4.30) can be divided into two equations. For t , < t,,,

C, = ZR, + b.t, + d.t, + c’ (4.30a)

where c’ is a constant equal to - (b.t,,, + d.tmin), and for t,> I,,,

C, = ZR, - b.t, + d.t, + C” (4.30b)

where the constant c” equals (b.t,,, - d.tmin). According to eqn.(4.30a) the value of the time-dependent criterion C, will increase with

increasing analysis time (td if the selected maximum analysis time (r,,,,,) has not yet been reached. According to eqn.(4.30b), an increase of t , above t,,, will result in a negative contribution to C,. Hence, t,,, serves as a desired value for the analysis time, rather than as a maximum value. The importance of the aim to realize the separation in a time that equals t,,, can be varied by varying the weighting factor 6.

The criterion C, is always increased if t , increases. In other words, a large value for t , will be one of the goals set by the selection of an optimization criterion according to eqm(4.30). Hence, the two time terms in eq~(4.30) join forces to bunch the peaks between a retention time of the first peak, which is as high as possible, and the desired maximum analysis time. It may be expected that the two terms will try to direct the chromatographic parameters included in the optimization process in opposite directions. The sum of the resolution factors contributes to this conflict of interest, as ZR, will tend to increase if the peaks are spread out over larger time intervals. The balance between the different factors is in principle decided upon by the user in choosing the values of the parameters b and din eqm(4.30). However, it seems impossible to establish an objective balance between the importance of resolution on the one hand and retention time on the other. This situation is disturbing, especially because the course and the result of a selectivity optimization process will depend on the arbitrarily selected weighting factors.

Smits et al. [416] and Wegscheider et al. [406] incorporated a time correction factor into their optimization criterion as follows:

c, = c / t, (4.31)

1 49

where t, again represents the retention time of the last peak*. Essentially, in this way the obtained separation (expressed in the criterion C ) per unit time becomes the optimization criterion. Again, a weighting factor may be added, i.e.

c, = c / t: (4.32)

where r is an arbitrary weighting factor. Nevertheless, the choice of r= 1 does seem to be a natural one.

Unlike the situation with the contribution of a time term (see the discussion above), C, can readily be expressed in inverse time units (e.g. min-’), so that a dimension problem will not arise.

Hence, both from the point of view of weighting factors and from that of dimensions a time factor as in eqn.(4.31) appears to be more appropriate than a time term as in eqn.(4.28).

A more fundamental advantage of the use of time factors may be that we no longer find ourselves in a position in which a compromise has to be established between two conflicting contributions to the optimization criterion. A proper balance between longer analysis times yielding higher resolution and shorter analysis times yielding lower resolution may be hard to find. The effect of a change in the chromatographic parameters will usually be such as to increase one term in eqn.(4.28), but to decrease the other. This may easily lead to oscillation effects in which the conditions are pushed back and forth, while the optimum is approached only very slowly. An example where such a problem seems to occur can be found in ref. [414].

An increase in the retention time accompanied by an increase in the resolution has the effect of increasing both the numerator and the denominator in eqn.(4.31), so that oscillations between high and low values are less likely to occur.

Nevertheless, the use of eqn.(4.32) may result in a slower optimization process than if a simple sum or product criterion were used. It is also unclear at the outset how the process would respond to chromatograms with the same value for C , but widely different values for C. In other words, the criterion cannot differentiate between a bad resolution in a short time and a good resolution in a long time.

This problem can most easily be circumvented by using a threshold criterion for C. If Cequalled one in acceptable regions of the parameter space and zero outside these regions, then the result of the use of eqn.(4.32) would correspond to the shortest possible chromatogram for which the resolution is acceptable (C= 1). A similar situation would occur if we were to use a threshold criterion with a diffuse change between zero and one,

* In fact, the time needed to elute 95% of the last peak was taken, in which case eqn.(4.31) would read for Gaussian peaks

C C c , = - = t , + 2 a, t ,( l+2/VN)‘

(4.3 1 a)

Clearly, the difference between eqm(4.31) and (4.31a) is small. If we assume that genuine chromatography columns have a minimum of 2500 theoretical plates, then the difference between the two equations is always less than 4’/0. Even for non-symmetrical (“tailing”) peaks eqn.(4.31) will almost always be an adequate approximation of eqn.(4.31a).

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such as DP. In fact, the criterion

c, = D P / t , (4.3 1 b)

is similar to the criterion of eqa(4.24). Because Pdecreases very rapidly once the resolution of a pair of peaks becomes less than 1, the use of eqn.(4.31b) will not usually result in chromatograms in which a poor resolution is compensated by a very short analysis time. The difference between eqns.(4.24) and (4.3 1 b) is that in the first case R, and k, are used, which are useful characteristics if the result of the optimization needs to be transferred to another column, while the use of P and t, and eqn.(4.31b) makes this criterion more appropriate for optimization on the final analytical column.

E q ~ ( 4 . 3 1 b) has the advantage over eqn(4.24) of being a continuous criterion rather than a combination of two separate ones used hierarchically. We have seen before that the use of P suffers from the insensitivity of this criterion to changes in the range of high P values (P = 1) and in the range of badly resolved peaks (P= 0). The use of eqn.(4.31b) will eliminate the first problem, but the latter problem will remain.

4.4.3 Column-independent time factors

The retention time is determined by three factors: 1. The capacity factor of the solute 2. The column dimensions (hold-up volume) 3. The flow rate.

Only the first factor is influenced by the physico-chemical separation process (the selectivity), while the other two factors are determined by the column and the operating conditions, respectively. If C is a continuous criterion (see table 4.7), then both C and C, can be transferred from one column to another. Both column dimensions and flow rate have trivial effects on the analysis time t,. However, if the final analysis is to be run on a different (optimized) column, then it is more logical to use the dimensionless, column-independent factor (1 + k,) in eqn.(4.31) instead of t,:

c,= C / ( l + k J . (4.33)

In the case, where the dimensions of the column are to be optimized after completion of the selectivity optimization process, another time factor may be even more appropriate. The first step to be taken after the completion of the optimization process is to establish the required number of plates (N,,). If the lowest resolution value encountered in the chromatogram is R,,,inr and if the required resolution is R,,,,, then

(4.34)

where N , is the number of plates available on the column used during the optimization process. If Smin is the lowest value for S in the chromatogram, then

(4.35)

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Since all continuous criteria (see table 4.7) require knowledge of the R, or S values, eqn.(4.34) or (4.35) can be used immediately to calculate the required number of plates. In many cases the required number of plates will only be used to estimate the length of the final analytical column, while all other parameters are being kept constant.

For example, in GC a capillary column with a given diameter may be used, operated at a given flow rate, with N being directly proprotional to the column length (L). In GC or in LC, packed columns with a given particle size may be used at given flow rates. In all these common cases, the following proportionality series applies:

t , cc to cc Vb cc L cc N,, (4.36)

and hence the final analysis time will be proportional to N,,. Of course, t, is also proportional to (1 + kd (eqn.l.6), and therefore an appropriate time corrected criterion is

(4.37)

where f and d denote constant flow rate and diameter (of open columns or of particles in packed columns), respectively.

The above simple proportionality between t , and N,, (eqn.4.36) is not always obeyed. For instance, at constant flow rate and particle size, the number of plates that can be achieved is limited by the maximum allowable column pressure. In that case, we are forced to vary the flow rate, the particle size, or both. If we do so, the analysis time ( tJ will no longer be proportional to the required number of plates ( N,,).

In chapter 7 (section 7.2.3) we will see that in the case where the pressure drop over a packed column is kept constant and both the flow rate and the particle size are allowed to vary in order to realize optimum operating conditions, the retention time will be proportional to the square of the required number of plates, i.e.

t , = p . N,: . (1 + kJ (4.38)

where pis a constant, the value of which depends on the viscosity of the mobile phase, the pressure drop and the quality of the packing.

According to eqn.(4.38) a time-corrected optimization criterion under constant pressure conditions (denoted by the subscript p) may be defined as

(4.39)

In the reality of LC practice, eqn.(4.38), which is based on a different optimum particle size for a different required number of plates, will not usually be realistic. For LC the truth may be somewhere in between the two extremes described by eqn.(4.36) (constant flow rate and particle size) and eqn.(4.38) (constant pressure drop). Relatively long columns with 10 pm particles may be used for difficult separations, requiring relatively large numbers of plates. For more convenient separations, somewhat shorter 5 pm particle columns may be used, and for relatively simple separations requiring modest numbers of plates 3 pm particles packed in very short columns may provide fast analysis.

152

If three or more different particle sizes are to be considered after the completion of the selectivity optimization process, then this may be an argument for the use of eqn.(4.39) instead of eqn.(4.37).

The required analysis time itself appears to be both a logical and an elegant choice for an optimization criterion, Either tne can be minimized, or, for reasons of consistency, 1 I t n e

can be maximized. The criterion of minimum required analysis time then corresponds to a constant value for C in eqn.(4.37):

4

(4.40)

and with eqn~(4.34) and (4.39, neglecting the constant factors which are irrelevant for optimization purposes,

(4.41)

An analogous expression can be found for the case of a constant pressure drop over a packed column with variable particle size from eqn.(4.39):

g i n a- ‘&in

CIlP = ( l + k J N ; ( I + k J ’ (4.42)

The equations in terms of Smin are especially attractive, since no estimate of the column plate count is required. If R,is used, an estimate of the peak width or plate count is required twice, but since R, a VN the plate count cancels in eqm(4.41) and (4.42). This becomes clear when the two equations are expressed in terms of S.

4.4.4 Time-corrected resolution products

In section 4.3.2 we have seen that the normalized resolution product criterion r aims at achieving a chromatogram in which all peaks appear at constant resolution intervals from the first one. If r* is used instead of r, then the regular intervals start at an imaginary peak at t = c,,. A chromatogram for which r* = 1 is one of a series for which the constant intervals can be found. Once the absolute value of S, the number of peaks and the plate number are known, the chromatogram is defined unambiguously.

The question we can now ask ourselves is whether all.members of the series are equally good chromatograms. In other words, is the criterion r* on its own sufficient for judging the quality of a chromatogram. To simplify the discussion, we will investigate the necessity of time correction factors for chromatograms for which r* = 1, and then extend the result to include the more realistic chromatograms for which r* < 1.

In the case of a chromatogram for which r* = 1, the value of k, can be calculated if the absolute value of S (S has the same value for all pairs of peaks in the chromatogram!) and the number of peaks is known. The capacity factor for the first peak can be found from

k, - ko S = -

and this leads for ko=O to

(k, + l)-(ko+ 1) - k 0 + k , + 2 ( k , + l ) + ( k , + l )

(4.43)

153

1+s 2s

I - s 1 - s k, = - (ko+l ) - 1 = -.

For the second peak we find in a similar way 1 + s 1 + s *

k“== ( k , + l ) - 1 = (GI - l

and in general terms

(4.44)

(4.45)

(4.46)

Eqm(4.46) allows us to calculate the k value for the nth peak in an “optimal” chromatogram (r* = I). Ifwesubstitute S = 2 R,/VNandassume S < 1, then theequation for the peak capacity (eqn.l.25) follows directly from eqm(4.46). Of course, in practice k values may also be obtained from the chromatogram instead of from eqr(4.46).

-2 -1 0 log s-

Figure 4.1 1: Calculated characteristics for optimum chromatograms (r* = 1) containing 10 equally resolved peaks as a function of the separation factor S. Plotted on a logarithmic scale are the capacity factor of last peak (1 + k, eqn.4.46), the required number of plates (Nne; eqn.4.47), the required analysis time under conditions of constant flow rate and particle diameter (tnelr.,,; eqn.4.48), and required analysis time under conditions of constant pressure drop t,&; eqn.4.49). For explanation see text,

154

In figure 4.11 the function log (1 + k,,) is plotted as a function of log S for chromatograms with r* = 1. k,, denotes the capacity factor of the 10th peak. Clearly, for very low values of S all k values are very low. At an S value of 0.03 (log S z - 1.5) all 10 peaks still elute before a k value of 1. Around an S value of 0.12 the 10th peak elutes at k=10, while at S=O.25 the capacity factor becomes equal to about 100. Roughly, optimum capacity factors are found around S = 0.1.

The number of plates required for realizing adequate resolution can be found from eqn.(4.35). If the required resolution (RS.,,) is unity, then

N,, = 4 / smin*. (4.47)

Again, choosing any other value for R,,, is totally irrelevant for the optimization of selectivity.

In figure 4.11 log N,, is also plotted against log S. In accordance with eqn.(4.47) a straight line results that has a slope of - 2. The number of required plates quickly decreases with increasing S.

Two other lines are drawn in figure 4.1 1. These correspond to the time correction factors in eqm(4.37) and (4.39). Under conditions of constant flow rate and constant diameter (of particles or open columns) the analysis time, neglecting constant factors, can be expressed by

(4.48)

while for packed columns under conditions of constant pressure drop and optimal particle size

(4.49)

Log t,,l,,, and log tnelp are illustrated in figure 4.11. It can be seen that under the conditions of eqa(4.48) a broad optimum exists around S=O.l. Over the range 0.03 < S < 0.2 the required analysis time varies by about a factor of 2. This range in Scovers a very large range in k values. For S = 0.03 the capacity factors for ten equally resolved peaks range from 0.06 to 0.82. For S = 0.2 the capacity factor ranges from 0.50 to 56.7. Hence, there is a broad optimum range around S = 0.1 in which the required analysis time does not vary considerably. In this range the criterion r* can be used to try and locate the best possible chromatogram.

Outside the optimum range this is no longer true. If ten peaks are equally resolved (r* = 1) with S values of 0.001, then according to eqn.(4.47), four million plates are required for adequate resolution. Moreover, we can see from figure 4.1 1 that the required analysis time is a factor of about 600 larger (under constant flow and diameter conditions) than it would be if S equalled 0.1. If S was 0.5, the analysis time would be a factor of about 200 larger than in the optimum. Hence, we may conclude that for optimization processes during which the capacity factors may be expected to vary dramatically, a time correction factor is required even when r* is used as the optimization criterion.

If we consider packed columns under constant pressure conditions, i.e. if we use eqn.(4.49) instead of eqn.(4.48), then the optimum that corresponds to the shortest analysis

155

time will be observed around S = 0.2. A variation in tne by a factor of two allows operation in the range 0.1 < S < 0.3. The corresponding ranges in k are 0.22 to 6.44 for ten peaks at S=O.1 and 0.86 to 487 for ten peaks at S=0.3. Again we see that the optimum range is quite broad. The range of k values usually considered as optimal, i.e. 1 < k < 10, is well encompassed in the optimum working ranges of both eqns.(4.48) and (4.49).

It should be noted that in the optimum ranges the number of plates required for “ideal” chromatograms that show constant resolution intervals throughout is always very small. The limiting values of S for the optimum ranges correspond to plate numbers of around 4500 (S=0.03) to 45 (S=0.3). When the number of peaks increases, the (1 + k d factor increases and the optimum shifts towards lower S values (to the left in figure 4.1 1). For instance, for 15 peaks the optimum S value shifts down from S = 0.1 (400 plates required) to S=O.O7 (800 plates) if eqn.(4.48) is used, and from S=O.2 (100 plates) to about 0.13 (250 plates) using eqn.(4.49).

Unfortunately, in practice “ideal”. chromatograms showing r* = 1 will be hard to find.

I (C)

0 5 10 k-

Figure 4.12: Three schematic chromatograms with equal values for the lowest separation factor (Smin. determined by the first pair of peaks) as well as for the capacity factor of the last peak (kd.

156

Therefore, for a chromatogram with ten peaks and an Smin value of 0.1, the capacity factor of the tenth peak is bound to be much higher then the value of 6.4 predicted by eqn.(4.46). In general therefore, the value for Smin in a real chromatogram will be shifted to the left (lower S values), the number of required plates will be higher and so will the analysis time. The time correction factors of eqn~(4.48) and (4.49) can readily be used with the lowest S value (Smi,,) and the largest k value ( k d observed in the chromatogram.

The use of the required analysis time as the optimization criterion (eqns.4.41-42 for criteria to be maximized, or eqns.4.48-49 for criteria to be minimized) yields a balanced comparison between the minimum resolution on the one hand and the retentjon on the other. However, in using the criterion the main disadvantage of the use of Smin as the function describing the resolution in the entire chromatogram remains. If Smin is used, no attention is paid to all but one pair of peaks in the chromatogram. By using the required analysis time as the optimization criterion care is taken that other pairs of peaks do not extend the length of the chromatogram with impunity. However, even when the Smin values and the k, values of different chromatograms are the same, these chromatograms can still be very different.

This is illustrated by the three chromatograms shown in figure 4.12. The Smin value, determined by the first two peaks is the same in all three chromatograms, and so is the k, value. The top chromatogram shows two peaks early in the chromatogram and a bunch of peaks between k = 5 and k = 10. The middle chromatogram shows four pairs of peaks and the bottom chromatogram shows good spacing of the peaks after the first pair.

Table 4.10 lists the required analysis times for the three chromatograms of figure 4.12, as well as the r* values. Constant flow rate and diameter are chosen as the conditions for table 4.10. Clearly, the required analysis times are the same for all three chromatograms. However, r* reveals large differences between the different chromatograms. These changes in r* are relatively large in comparison with possible changes in the time factor tne. We concluded from figure 4.1 1 that there were large ranges over which the required analysis time varied by less than a factor of two. The variation in relative peak positions in the chromatograms of figure 4.12 gives rise to changes in r* which amount to a factor of 50 between the top and the bottom chromatogram.

Table 4.10: Required analysis times and time-corrected resolution products for the three chromatograms of figure 4.12. Constant flow rate and diameter (of open columns or particles) (i.e. constant f,d conditions) have been assumed.

Mult. Criterion Eqn. Chromatogram factor

top middle bottom

103 tne1f.d 4.48 4.4 4.4 4.4 10-4 1 ’tne1f.d 4.41 2.21 2.27 2.27

r* 4.21 1.09 7.27 56.4 4.50 2.47 16.5 128

10-2 rr 10-6

10-4 ‘“1 4.52 1.30 1.64 2.12

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The ratio

r, = r / t,, (4.50) * *

could be used as an optimization criterion to try and accommodate both peak distribution and analysis time in one criterion. In that case an increase in rf by a factor of 50 would compensate for an increase in the required analysis time by the same factor. The bottom chromatogram in figure 4.12 may be more attractive than the top one, but it is quite obvious that a factor of 50 increase in analysis time is too high a price to pay for the improved peak distribution.

The reason for this is that changes in resolution are over-emphasized in the criterion r*, because n separate resolution factors for n pairs of peaks (n - 1 if r is used) occur in r*. In this way, resolution to the nth power is balanced vs. tne. Therefore, a more sensible criterion would be

rit = V7 / t,,

which for the conditions of eqn.(4.48) (constant f,d) becomes

and for the conditions of eqn.(4.49) (constant p)

(4.51)

(4.52)

(4.53)

The values of r&d and r:rlf,d for the three chromatograms of figure 4.12 are included in table 4.10. The values of r: show an increase of a factor of 50 as discussed before. The values of rif increase by about 60% in going from the top to the bottom chromatogram. Hence, the improved peak distribution observed in the bottom chromatogram may outweigh a 60% increase in analysis time.

4.5 RECOMMENDED CRITERIA FOR THE GENERAL CASE

The final recommendations for optimization criteria to be used in the general case (i.e. when all peaks are considered to be of equal importance) are summarized in table 4.1 1. The table shows recommended criteria for four different cases. Below the dashed line, an alternative criterion (second best choice) is given for each case.

4.6 SPECIFIC PROBLEMS

4.6.1 Limited number of peaks of interest

So far, we have only considered chromatograms in which all peaks were treated as being of equal importance. Now we will look at a chromatogram in which a number (n) of peaks appears, but only some peaks @; p < n) are of interest. An example is shown in figure 4.13. In this chromatogram, eight peaks occur, but only two of these (peaks 3 and 5 ) are of interest. Seven or eight (if a peak is assumed at t = to) S values can be obtained from the chromatogram. Four of these involve one of the peaks of interest.

158

Table 4.11: Recommended criteria for use in selectivity optimization processes in the general case (all peaks equally important).

Optimization on final analytical

ves U column? no

I I

I

sample overall

I I no 1 YFS

YFS no

W P , / t, l 7 P / t ,

I

* max max max r* max rn,

eqn.(4.31b) or max eqm(4.2 1) eqn.(4.52) LIP, / t" and (4.22) or (4.53) eqn.(4.31 b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

min t , n min t , n max Smin max l/ tne Rs.min > x Rs.min > x (1) (1) eqn.(4.24) eqn.(4.24) eqn.(4.25) eqn.(4.41)

or (4.42)

(1) Suggested value for x: 1.5.

The k and S values corresponding to the chromatogram of figure 4.1 3 are given in table 4.12. The relevant values are underlined in this table*.

The chromatogram in figure4.12 can be improved dramatically by using a multicolumn technique (see also section 6.1). The entire sample can be eluted from a short column to obtain a rough separation, and only the fraction that contains the relevant peaks (3 and 5) can be passed on to a longer column to realize the entire separation. The first column may be referred to as a clean-up column or pre-column and the second one as the analytical column.

However, with or without the use of multicolumn techniques, irrelevant peaks (e.g. peak no.4) will usually appear in real-life chromatograms and optimization criteria have to be developed to cope with them.

Most of the criteria used so far can readily be applied to cases where a limited number of peaks is of interest. The parameters Rs,min, Smin and Pmin retain their value, but now the lowest value should be selected from the relevant pairs of peaks (a pair of peaks is

* The number of relevant S values is never higher than 2p. I f two peaks of interest are adjacent in the chromatogram, then the number of relevant S values is decreased by one.

159

2

3 N

Figure 4.13: A chromatogram with eight peaks ( n = 8), of which only two peaks are of interest ( p = 2). The relevant peaks (3 and 5) are indicated by an asterisk.

Table 4.12: Retention and resolution data corresponding to the chromatogram of figure 4.13. The relevant peaks and the relevant values for k and S are underlined. !* was calculated from eqns.(4.55a) and (4.56a). f,$ was calculated from eqn.(4.57).

Peak no. k 5

0

1

0

0.5 0.20 Smin = 0.067

0.14

0.20

0.25

0.091

0.077

0.067

0.16

-

-

Smin = 0.077

160

relevant if either of the two peaks is relevant). We distinguish the appropriate criteria for the specific case again by underlining, hence13,,in,Smin andLmi, refer to the specific case. Even if 0.067 is the lowest separation factor observed in the chromatogram of fig 4.13 (Smin), then 0.077 is still the lowest relevant value (Smin). The analysis time is still determined by the capacity factor of the last peak (k,= lo), no matter whether or not this is a relevant peak. However, the lowest relevant value for Rs or S should now be used in eqn.(4.48) or eqn(4.49).

The criteria which may be used for the specific case in which only a limited number of peaks is of interest are listed in table 4.13.

Constant sample

composition?

Table 4.1 3: Recommended criteria for use inselectivity optimization processes for the specific case in which only a limited number of peaks is of interest (see also figure 4.13).

Constant overall

k value?

Yes no Yes

( 1 ) Suggested value for x: 1.5. (2) ine can be calculated from eqn.(4.48) or eqn.(4.49), using the lowest relevant value for S emin).

no

The square root should now be incorporated in all product criteria, i.e. also in the case in which lTP or I7P, is used instead of nP, This is because it is now now a sensible convention to incorporate two values for a peak-valley ratio (be it P, P , or P J into the criterion for every peak of interest. If we did not follow this convention, then a different situation would exist if two relevant peaks were adjacent (yielding one combined value for the peak-valley ratio) or separated by an irrelevant peak (which would lead to two different

161

P or P, values). Hence, two values may best be used for every peak of interest and the resulting product may be “normalized” by taking the square root. For example, we can write for the product of P values

P

i= 1 i7P = n vPi , i - , .P i . i+ l - (4.54)

where Pi, i - , refers to the separation factor between the ith relevant peak and the preceding one and Pi,i+ that between the ith relevant peak and the following one and where the number of factors in this product equals the number of relevant peaks @).

If peak-valley ratios are used as elemental criteria, then the separation between the first peak and the (imaginary) preceding one, as well as the separation between the last peak and the (imaginary) following one, may readily be characterized by a P value of one. The retention time of the last peak, which may be used in combination with a product of P values (see table 4.13) refers to the last appearing peak in the chromatogram, no matter whether or not this is a relevant peak.

The calculation of (normalized) resolution products if only a limited number of peaks is of interest requires some additional thought. In calculating normalized resolution products either R, or S may be used. We will use S in the following discussion. The ideal situation with regard to the normalized resolution product will be that all relevant S values are equal, while the irrelevant peaks contribute nothing to the overall length of the chromatogram (i.e. for the relevant values s=s and and for all other S=O). In that case the following product should equal unity:

whereSi.i- refers to the separation factor between the ith relevant peak and the preceding one and &+, that between the ith relevant peak and the following one and where

n - 1

i = 1

- S = l/@-I) c Si*i+l . (4.56)

When a hypothetical peak is assumed at t = to the calibrated normalized resolution product becomes:

with n - I

i = O

- s = l / p z si,i+l . (4.56a)

In these equations, p is again the number of peaks of interest. Hence, the product includes only the relevant S values, while the sum is taken over all pairs of peaks. If the sum is divided by the number of relevant peaks (p - 1 or p) , then a value of r= 1 or r* = 1 can only be reached if all irrelevant peaks appear “nowhere” in the chromatogram, i.e. coincide with the imaginary peak at t = r,,. If we divided the sum by the total number of peaks (n), then the resulting values for r would not be restricted to the range O < r< 1, and we would no longer be able to refer to r as a normalized resolution product.

162

The separation factor between the first peak and the (imaginary) preceding peak, as well as that between the last peak and the (imaginary) following one, do not take on a natural value, as was the case for peak-valley ratios (which in both cases could be said to equal one). In the case of the normalized resolution product the most logical choice is to take the optimum value, i.e. the average value 3, as the limiting separation factor of the chromatogram.

The time-corrected normalized resolution product now becomes:

:;t = / t,, (4.57)

where t,, can be found from eqm(4.48) or eqn.(4.49), using the lowest relevant value for S (Smin) in the calculations.

Weighting factors

The special case described above is in fact a particular example of the use of weighting factors in the optimization criteria. If some peaks are of interest whereas others are not, we could use weighting factors of one and zero, respectively, as we have in fact done above. However, we have seen that even the introduction of these simple weighting factors has caused some problems regarding the calculation of the various recommended criteria. These problems will be aggravated if weigthing factors other than one and zero are allowed.

If a product of peak-valley ratios is used as the optimization criterion, then two values would need to be used for every peak, one describing its resolution from the preceding peak in the chromatogram and the other one describing its resolution from the following peak. Because a product criterion is used, the weighting factors (8) will appear as powers in the product. If we assume the weighting factors to be positive, we may write

n

n P , = n (Pi.i-, . Pi.i+l)gi’* i = 1

or, equivalently,

nP, = exp { Z ( g / 2 ) In . P i , i + , ) } . n

i = 1

(4.58)

(4.59)

This product of peak-valley ratios can be normalized to the sum of weighting factors, so that the “true” value of the resolution product is less obscured by the arbitrarily selected values for g:

(4.60)

Eqn.(4.60) is equivalent to an equation suggested by Morgan and Deming [417], apart from the requirement to use two P values for each peak.

Eqns.(4.58) to (4.60) can be used with P , (if the sample composition is expected to be constant) or with P or P , if the sample composition is expected to vary.

163

The normalized resolution product (r or r*) can be obtained in an analogous way. In terms of S the product reads:

n-1 n

i = 1 i = 1

- s = x qi+, / x gi

or, with the inclusion of a (hypothetical peak at t = to)

with n - 1 n

i = O i = O

- s = x si,i+l / x gi

(4.61)

(4.62)

(4.61a)

(4.62a)

Table 4.14 Recommended criteria for use in selectivity optimization processes for the specific case in which weighting factors are used to indicate the importance of each individual peak.

Optimization on final analytical

column?

Y e no I i

Constant sample

composition ?

Constant overall

k value?

Yes I no Yes I no

max max max rg max r*,,.g

"p:.g 1 *" " P i / t ,

1 I

*

eqn.(4.60) or max eqn.(4.61a) eqm(4.63) W n . g 1 t , and (4.62a) eqn(4.60)

min t, n Rs.min x Rs.min 2 x (1) (1)

min t, n

eqn.(4.24) eqn.(4.24)

(1) Suggested value for x: 1.5.

164

Finally, the time-corrected normalized resolution product can be found from I n \

= exp In ri / z: g; t ; = o (4.63)

where t,, can again be found from eqn.(4.48) or eqn.(4.49), using the lowest value for S ( Smin) in the calculations. Table 4.14 lists the criteria that may be used in combination with weighting factors for each peak and refers to the appropriate equations.

The use of Smin or 1 /rne as alternative criteria when the optimization does not take place on the final analytical column (see table 4.1 1 and table 4.13) is not recommended in table (4.14), because these criteria are not compatible with the use of weighting fqctors.

4.6.2 Programmed analysis

The important aspect of programmed elution techniques with respect to optimization criteria is that the peak width does not increase with the retention time in a manner corresponding to eqm(l.16). In programmed analysis a constant peak width is wanted throughout the chromatogram (see section 6.1).

Because of their pragmatic definitions, the different P values are not at all affected by changes in the elution mode, i.e. they may be applied under programmed elution conditions in exactly the same manner as under constant conditions. Resolution (R,) factors are not fundamentally affected, i.e. the definition given in eqn.(l.l4) can still be applied. However, the relationship between R, and fundamental chromatographic parameters given by eqm(l.22) is no longer valid. The separation factor S loses its usual meaning, since its definition originates from the above eqn.(l.22). A simple solution to this problem is to use the difference in retention time (At) between two peaks as the sole criterion for resolution. This is justified by the fact that for ideal elution programmes the nominator in eqn.(l.l4) has a constant value. Hence, Atmin can be used instead of Rs,min or Smin. Also, a normalized resolution product may be defined as

r,, = n n-I

i = 1 - ti) / Z i }

n - l

i = 1 = n ( A r / z )

where

(4.64)

(4.65)

It should be noted that a constant peak width will only be achieved by approximation in most programmed elution chromatograms. Early eluting peaks (not subjected to gradient conditions during their migration through the column) may be considerably narrowed, while late eluting peaks (eluting after completion of the program) may be considerably broadened.

Harris and Habgood (ref. [427], p.123) have suggested a different definition for a

165

separation factor in programmed temperature gas chromatography. Their definition is based on the assumption that the width of a peak that is eluted from a gas chromatograph at a temperature T, during a programmed analysis is the same as it would have been if the same component had been eluted from the column under isothermal conditions at the temperature T,, Therefore, the isothermal retention volume at T = T, ( VT) may be used to characterize the peak width in the denominator of the separation factor, i.e.

VRj- VR,i - F ‘TPi+ ‘T,J ‘T ‘Tri+ ‘T,J

Trj- Tr,i - - S = (4.66)

where V, is the retention volume under programmed elution conditions, Fis the flow rate (for example expressed in mL/min) and rT the heating rate (e.g. OC/min). Harris and Habgood proceed to suggest that eqn.(4.66) can be used to explain the influence of the “programming rate” (r7/ F OC/mL) on the resolution in programmed temperature GC. However, this is a somewhat simplified picture because the retention temperature T, is affected by the programming rate. Nevertheless, their conclusion that the resolution in programmed temperature GC increases with a decrease in the programming rate is essentially correct.

Snyder [428,429] has suggested a way to relate resolution in solvent programmed (gradient elution) liquid chromatography to the fundamental parameters of the chromatographic process by defining a median capacity factor under gradient conditions (K), which is characteristic for the average speed of migration of the solute molecules through the column. In terms of % we can use an equation identical to eqn.(1.22) to describe the resolution R,:

(4.67)

It is important to realize two things in using eqn.(4.67). 1. The median capacity factor 5 is not directly related to the retention time under

gradient conditions. In fact, it can be shown that under someconditionsq has the same value for all the peaks in a chromatogram obtained under programmed conditions*.

2. In deriving eqn.(4.67), the relative retention a is assumed to be independent of the composition. In other words, plots of retention (In k) vs. composition (p) obtained under isocratic conditions are assumed to yield parallel lines. For components which are eluted under “ideal” gradient conditions (i.e. those

components that appear neither at the very beginning nor after the end of the actual gradient in linear solvent strength gradients**, it can be shown that the median capacity factor is inversely proportional to the gradient steepness parameter, defined as [428]

(4.68)

where S is the slope of the retention (In k) vs. composition (9) plots (see eqn.3.45), V, the

* The suggestion given before that in an “ideal” programmed analysis the p z k width should be the same for all solutes (see also figure 6.1~) corresponds to the assumption that k, is equal for all peaks. ** For a definition of linear solvent strength (LSS) gradients see section 5.4.

166

hold-up volume of the column and d@dt the programming rate. The latter can be related directly to the span (Ap) and the duration (1,) of a linear gradient.

It appears from eqn.(4.68) that if the flow rate and the span of the gradient are kept constant, the gradient steepness parameter (b) is inversely proportional to the duration time (t,) of the gradient, and, hence, that the median capacity factor (k,) is directly proportional to t,. Therefore, under these conditions, in gradient elution t, may take the place of the capacity factor in the resolution equation and eqn.(4.67) may be rewritten as

(4.67a)

where c is a constant. Cohen et al. [430] have demonstrated the validity of eqn.(4.67a) in practice for cases in

which a is constant. However, they have also shown that the equation is no longer valid if a varies with composition under isocratic conditions. Nevertheless, eqn.(4.67a) may serve as a good rule of thumb for the optimization of gradient duration times (see chapter 6).

Another aspect of programmed elution that will affect the quality of the chromatogram is the variation (“drift”) of the baseline during the program. Methods to reduce the baseline drift (or blank signal) and other aspects of programmed analysis will be discussed further in chapter 6.

4.6.3 Dealing with solvent peaks

In many chromatograms a “solvent peak” will appear. This is typically a large signal that appears early in the chromatogram. In GC, solvent peaks are usually highly non-symmetrical (tailing) peaks. In LC they may occur in many different ways, for instance as large negative and positive signals early in the chromatogram. In some forms of LC, especially the ionic separation methods (section 3.3) peaks induced by the mobile phase may occur as genuine (or negative) peaks much later in the chromatogram. This latter kind of solvent-induced signal can be dealt with as an additional (irrelevant) peak in the chromatogram, from which the (relevant) peaks need to be separated.

In this section we will discuss some aspects of the more common type of solvent peaks, i.e. large signals which appear early in the chromatogram. Figure 4.14 shows a typical chromatogram in which three solute peaks are preceded by a large solvent peak.

There are two fundamentally different ways in which we can deal with solvent peaks. I. Reduce the solvent signal. This can be done chromatographically, for instance by using

a column switching technique, that prevents the first part of the chromatogram from entering the analytical column (see also section 6.1). Reduction (or elimination) of the solvent signal may also be achieved mathematically by subtracting the signal of the pure solvent (“blank”) from the chromatogram.

2. By modifying the optimization criteria such that an optimum separation of the solutes from the solvent signal is achieved. This latter method, which is the relevant one in the context of this chapter, has not received much interest in the literature so far.

Solvent peaks are usually highly non-symmetrical, so that neither R , nor Scan be used.

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Moreover, neither P nor P, can be established from the chromatogram. The only criterion which maintains a realistic value is P, As is shown for the first peak in figure 4.14, this parameter may be estimated from the chromatogram in the usual way (cf. figure 4.2). Hence, criteria based on P, may be used for chromatograms that resemble figure 4.14. A great disadvantage of this is that the use of P , has been recommended only for optimization of samples of constant composition on the final analytical column (tables 4.4 and 4.1 1).

0 k- 5

Figure 4.14 A typical chromatogram containing a solvent peak followed by three small peaks. h, and v, can be used to estimate the peak-valley ratio of the first peak (see figure 4.2.c).

A possible way to deal with solvent peaks if the resolution (R, ) or the separation factor (S) is opted for as the elementary criterion (which is to be recommended if the optimization process does not take place on the final analytical column), is the introduction of (large) weighting factors for the solvent peak, using for example the criteria described by eqns.(4.61) to (4.63). For example, if a large weighting factor were assigned to the solvent peak in figure 4.14 (e.g. g= lo), while small factors were assigned to the (relevant) solute peaks (e.g. g= l), then the resulting criterion would have the effect of trying to enhance the separation between the solvent peak and the first peak in the chromatogram. From there on, a normalized resolution product criterion would again aim at a regular spacing of peaks in the chromatogram.

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REFERENCES

401. L.R.Snyder and J.J.Kirkland, An lntroduction to Modern Liquid Chromatography,

402. R.Kaiser, Gas-chromatographie, Geest und Portig, Leibzig, 1960, p.33. 403. 0.E.Schupp 111, Gas Chromatography, Wiley, New York, 1968, p.22. 404. A.B.Cristophe, Chromatographia 4 (1971) 455. 405. A.C.J.H.Drouen, Unpublished results, 1981. 406. W.Wegscheider, E.P.Lankmayr and K.W.Budna, Chromatographia 15 (2982) 498. 407. S.S.Ober in: V.J.Coates, H.J.Noebels and 1.S.Fagerson (eds.), Gas Chromatography,

408. P.Jones and C.A.Wellington, J.Chromatogr. 213 (1981) 357. 409. P.J.Schoenmakers, A.C.J.H.Drouen, H.A.H.Billiet and L.de Galan, Chromarogra-

410. A.C.J.H.Drouen, P.J.Schoenmakers, H.A.H.Billiet and L.de Galan, Chromatogra-

411. VSvoboda, J.Chromatogr. 201 (1980) 241. 412. J.E.Knoll and M.R.Midgett, J.Chromatogr.Sci. 20 (1982) 221. 413. H.J.G.Debets, B.L.Bajema and D.A.Doornbos, Anal.Chim. Acra 151 (1983) 131. 414. J.C.Berridge, J.Chromatogr. 244 (1982) 1. 41 5. J.L.Glajch, J.J.Kirkland, K.M.Squire and J.M.Minor, J.Chromatogr. 199 (1980) 57. 416. R.Smits, C.Vanroelen and D.L.Massart, Fresenius Zeitschr.Anal.Chem. 273 (1975) 1. 417. S.L.Morgan and S.N.Deming, J.Chromatogr. 112 (1975) 267. 418. M.W.Watson and P.W.Carr, AnaLChem. 51 (1979) 1835. 419. W.A.Spencer and L.B.Rogers, Anal.Chem. 52 (1980) 950. 420. R.J.Laub in: Th.Kuwana (ed.), Physical Methods in Modern Chemical Analysis,

Vo1.3, Academic Press, New York, 1983, pp.249-341. 421. S.N.Deming and M.L.H.Turoff, AnaLChem. 50 (1978) 546. 422. B.Sachok, J.J.Stranahan and S.N.Deming, Anal.Chern. 53 (1981) 70. 423. J.W.Weyland, H.Rolink and D.A.Doornbos, J.Chromatogr. 247 (1982) 221. 424. P.R.Haddad, A.C.J.H.Drouen, H.A.H.Billiet and L.de Galan, J.Chromatogr. 282

425. J.W.Weyland, C.H.P.Bruins and D.A.Doornbos, J.Chrornatogr. Sci. 22 (1984) 31. 426. J.C.Giddings, Anal.Chem. 32 (1960) 1707. 427. W.E.Harris and H. W.Habgood, Programmed Temperature Gas Chromatography,

428. L.R.Snyder, J.W.Dolan and J.R.Gant, J.Chrornatogr. 165 (1979) 3. 429. L.R.Snyder in: Cs.Horvath (ed.), HPLC, Advances and Perspectives, Vol.1, Academic

430. K.A.Cohen, J.W.Dolan and S.A.Grillo, J.Chromatogr. 316 (1984) 359.

Second edition, Wiley, New York, 1979.

Academic Press, New York, 1958, pp.41-50.

phia 15 (1982) 688.

phia 16 (1982) 48.

(1983) 71.

Wiley, New York, 1966.

Press, New York, 1980, p.207.

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CHAPTER 5

- OPTIMIZATION PROCEDURES The parameters that may be optimized have been discussed in chapter 3 and the criteria

for judging the quality of a chromatogram in chapter 4. In this chapter we will look at the actual optimization procedures.

A general introduction is given in section 5.1. The specific problems of selectivity optimization in chromatography are explained in this section and the terminology that will be used throughout the rest of the chapter is introduced.

Section 5.2 describes simultaneous methods of optimization. In these methods all experiments are performed according to a pre-planned schedule. After all the experiments have been performed, the optimum is located.

Section 5.3 describes sequential methods of optimization, in particular the Simplex method. In sequential methods the optimization procedure starts with some initial experiments, inspects the data and defines the location of a new data point which is expected to yield an improved chromatogram. The idea is to approach the optimum step by step in this way.

In section 5.4 methods are discussed that can be used to reduce the parameter space, i.e. to restrict the area which is searched for the optimum. Such methods may be used in conjunction with the optimization procedures described in sections 5.2 and 5.3, but they are more often used in combination with the optimization procedures described in section 5.5.

This section deals with interpretive optimization methods. In these.methods, the extent of chromatographic separation is predicted indirectly from the retention behaviour of the individual solutes. The data are interpreted to locate the optimum in terms of the complete chromatogram. The interpretive methods may involve a limited number of experiments according to a pre-planned experimental design (section 5.5.1) or may start with a minimum number of experiments in order to try and locate the optimum by an iterative process (section 5.5.2).

For all the interpretive methods described in section 5.5 it is essential to know the retention data of all the individual components in a sample. Section 5.6 deals with possibilities to obtain all this chromatographic information.

Finally, the different methods are summarized in section 5.7.

5.1 INTRODUCTION

In this chapter we will describe current procedures that aim at the actual optimization of selectivity. We will use the information contained in previous chapters. However, full knowledge of all the information provided in chapters 3 and 4 is by no means necessary. Clearly, the optimization of selectivity in GC does not require any knowledge of the parameters that are relevant for instance for ion-pair liquid chromatography. Moreover, there are some optimization procedures which do not rely on any knowledge of or information about the parameters to be optimized, nor on how they affect the selectivity. What the chromatographer needs, however, is sufficient knowledge to decide which

170

parameters he will be optimizing for and to establish sensible upper and lower limits for these parameters. In principle, this requirement can be circumvented by building knowledge bases (“expert systems”) into chromatographic instruments (see section 2.2.1). Although some activity in this area can be observed, it is unlikely that the requirement for some degree of chromatographic insight for those involved in method development will soon become obsolete.

The parameters together with their limits define the parameter space. The number of parameters considered is called the dimensionality of this space. Hence, two parameters form a two-dimensional parameter space. In this space a two-parameter optimization, or, alternatively, a two-dimensional optimization may be pursued.

In chapter4 the criteria which can be used for selectivity optimization have been defined and discussed extensively. In principle, any criterion can be used in combination with any optimization procedure. Some practical limitations will become obvious in this chapter. We will generally discuss optimization procedures independent of optimization criteria.

The selected criterion will vary as a function of the selected parameters. This function is called the response surface. Depending on the selected criterion, the optimization procedure will be aimed at locating either the maximum or the minimum value of the response surface. The optimum is defined by those values of the parameters that correspond to this maximum or minimum.

A very important characteristic of response surfaces in chromatography is their high degree of complexity. An example is shown in figure 5.1. In this figure a one-dimensional response surface is shown. The parameter considered is the (binary) composition of the

a i” min

1.10

1.05

0 0.1 0.2 0.3 0.L 0.5 0.6 0.7 0.8 0.9 1.0 ‘PA -

Figure 5.1 : One-dimensional response surface for the optimization of the stationary phase composition in gas chromatography. Horizontal axis: composition of binary stationary phase mixture. Vertical axis: lowest value for a (amin) observed in the chromatogram. For further explanation see section 5.5.1. Figure taken from ref. [501]. Reprinted with permission.

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stationary phase for a gas chromatographic separation. The criterion is the lowest value observed for the relative retention ( amin) in the chromatogram.

Typically for this kind of optimization, and, indeed, for all procedures used to optimize chromatographic selectivity is that at many different points in the parameter space complete overlap of two components occurs. In figure 5.1 this is the case at all points at which amin = 1. In two-dimensional optimization processes the conditions at which two particular components show complete overlap will form a line rather than a point, and so on.

Selectivity optimization is most useful if parameters can be selected which have a great effect on the selectivity, and hence complex response surfaces as in figure 5.1 are almost mandatory. The complexity of response surfaces will be further increased with increasing complexity of the sample, i.e. with an increasing number of sample components.

Response surfaces in more than one dimension (more than one parameter) are hard to visualize. Two representations are common for two-dimensional optimization problems, where the response surface as a function of the two parameters forms a three-dimensional picture. Figure 5.2 shows a pseudo-isometric three-dimensional plot of such a surface (figure 5.2a) as well as a contour plot (figure 5.2b).

THF

ACN MeOH

THF

ACN MeOH

Figure 5.2: (a) Pseudo-isometric three-dimensional response and (b) iso-response contour plot for a two-parameter optimization problem. Parameters (in triangular representation): quaternary mobile phase composition. Criterion: normalized resolution product (see section 4.3.2). 0, is the location of the optimum. For further details see section 5.5.2. Figure taken from ref. [502]. Reprinted with permission.

Figure 5.2a is difficult to interpret. Moreover, it may not reveal all the information that is present. Some information will literally be in the shadow of other parts of the surface and can only be made visible by turning the surface around (changing the position of observation). The contour plot on the right is more clear. The use of different tints of grey or different colours may further enhance the clarity of such a figure.

For three-dimensional optimization processes (leading to four-dimensional response surfaces) graphical presentation in a single two-dimensional figure is no longer possible. Any solution to this problem is bound to be difficult to construct and to interpret. A series

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of contour plots taken along constant intervals of one of the parameters is one of the least unattractive possibilities.

5.1.1 Univariate optimization

The complexity of the response surface is what makes the optimization of chromatographic selectivity stand out as a particular optimization problem rather than as an example to which known optimization strategies from other fields can be readily applied. This is illustrated by the application of univariate optimization. In univariate opti.mization (or univariate search) methods the parameters of interest are optimized sequentially. An optimum is located by varying a given parameter and keeping all other parameters constant. In this way, an “optimum” value is obtained for that particular parameter. From this moment on the optimum value is assigned to the parameter and another parameter is varied in order to establish its “optimum value”.

Univariate optimization is a common way of optimizing simple processes, which are affected by a series of mutually independent parameters. For two parameters such a simple problem is illustrated in figure 5.3a. In this figure a contour plot corresponding to the three-dimensional response surface is shown. The independence of the parameters leads to circular contour lines*. If the value of x is first optimized at some constant value of y (line 1) and if y is subsequently optimized at the optimum value observed for x, the true optimum is found in a straightforward way, regardless of the initial choice for the constant value of y. For this kind of optimization problem univariate optimization clearly is an attractive method.

However, this is no longer true in other cases. In figure 5.3b the response surface is still very simple, but because of a mutual interdependence (alternatively called “interaction”) between the two parameters the contour lines can no longer be transformed into circles. The simple procedure outlined above no longer results in the real optimum. The latter may still be reached by repeating the procedure several more times (possibly over smaller ranges in the parameter space, as indicated by lines 3 to 7 in figure 5.3b), but clearly the effectiveness and hence the attractiveness of the method are now lost. Moreover, in some unfortunate cases the variation of a single parameter at the time may not even give an indication of the location of the optimum [503].

Figure 5 . 3 ~ corresponds to a complicated three-dimensional response surface in which the two parameters x and y are mutually independent. Several optima appear to occur. Any optimum that occurs on such a response surface is a local optimum. The highest (lowest, if the absolute minimum is sought) of these optima is the global optimum. If we apply univariate optimization to this problem (lines 1 and 2 in figure 5.3c), we see that one of the local optima will be the result of the procedure. This is not necessarily the global optimum. Indeed, counting the contour lines in figure 5 . 3 ~ will reveal that the global optimum in figure 5 . 3 ~ is the one in the bottom left corner. The more the global optimum dominates the surface, the greater are the chances that it will indeed be found. However, this is merely a law of statistics, and hence no guarantee is implied.

* The contour lines are taken as circles for reasons of simplicity. Horizontal or vertical ellipses would yield the same result. These can be transformed into circles by a suitable transformation of the axes.

173

1

X-

Ib)

2

1

X-

Figure 5.3: Univariate optimization of individual parameters. (a) A simple response surface with two mutually independent parameters. (b) A simple surface with dependent parameters. (c) (opposite page) A complex surface with independent parameters. (d) (opposite page) A complex surface with dependent parameters. Drawn lines indicate the course of the optimization procedure. In figure b it is illustrated that for dependent parameters several reinitiations of the procedure are required to approach the optimum.

174

Finally, figure 5.3d illustrates the most difficult optimization problem, that of a complicated response surface in combination with mutually interdependent parameters. In this case not even a local optimum may result from the first two steps in the optimization process.

Unfortunately, for the optimization of chromatographic selectivity we will have to deal with response surfaces that correspond to figure 5 . 3 ~ or usually figure 5.3d [503], often with a considerably higher degree of complexity than shown there (compare figure 5.1).

5.1.2 Local vs. global optima

Given the likelihood that many different local optima exist on the response surface, an important question to ask is how necessary it is to locate the global optimum. It may be argued that a local optimum may provide an acceptable result in many cases. This will be true especially when the difference (in response value) between the global optimum and the local optimum considered is small. In some cases, the local optimum may be more attractive than the global one because of secondary considerations. For instance, a truly optimal resolution may be traded off against a lower temperature, a cheaper or less toxic solvent, or less critical conditions. Due to this last reason a broad local optimum may be preferred to a somewhat higher sharp global optimum.

There are three prevailing reasons to try and locate the true global optimum. In the first place, if an analysis needs to be performed a great number of times (e.g. in process monitoring, quality control), then it will be rewarding to spend time and effort on the optimization, in order that the analysis can be run quickly and cheaply on a routine basis. If the expected number of analyses to be run is small, then it is only necessary to reach sufficient resolution for all components in a reasonable time.

A second reason to aim for the global optimum is given by our inability to tell whether or not a particular chromatogram constitutes an acceptable result of the optimization process. Only if we have a sample containing a small number of known solutes, and if we are sure that no other components (e.g. impurities, degradation products, metabolytes) can possibly interfere with the chromatogram, only then will we be able to decide readily whether or not a given chromatogram forms an acceptable result. The danger is that we accept a chromatogram with say five well separated peaks, while in fact six (or more) major components were present in the sample. This danger is especially great if the optimization procedure does not create an impression of the entire response surface. Generally, we are not able to decide whether or not a local optimum is acceptable, unless we know the global optimum.

The third reason is that a procedure cannot be designed to find the global optimum only when it matters. In a number of cases we may give credit to an optimization procedure for yielding a satisfactory result, even if this result is a local optimum only. However, we cannot expect the same procedure to yield a satisfactory result in a case in which the global optimum is the only one at which an acceptable separation may be achieved.

We may conclude that local optima may be more acceptable as a result 1 . the closer the response at the local optimum approaches that at the global one, 2. the fewer the number of analyses that will have to be run on the optimized system, 3. the more information we have about the entire response su$ace, and

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4. the more we know about the sample. Finally, it is worth mentioning that we never establish the true global optimum, because

we can never study the effects of all possible parameters simultaneously over the entire possible range. Because of the parameters we choose and the limits we set for them, the best we can achieve is the global optimum within the constraints of the parameter space.

5.1.3 Characteristics of optimization procedures

Whatever the optimization procedure may be, it is safe to say that the length of the procedure, i.e. the number of experiments (chromatograms) required, will tend to increase with an increase in the number of parameters involved. Hence, increasing the number of parameters considered leads to 1. increased complexity of the response surface, i.e. more local optima, 2. reduced possibilities for graphical presentation, 3. lengthier optimization processes,

but 4. possibly a better overall result of the optimization.

This latter advantage needs to be balanced against the three considerable disadvantages. This makes the choice of which parameters to consider during the optimization process (chapter 3) even more important.

The process may be simplified by defining sensible limits for the parameters. For instance, GC columns have a specified maximum temperature, but for continuous operation a practical maximum well below that value is usually observed. Considering the parameter temperature at a value above the practical maximum is then a waste of time and effort, which may result in optimum conditions that will not be used in practice (e.g. an optimum temperature that equals the specified maximum).

After the relevant parameters have been selected and their limits have been defined, the actual optimization starts with some initial experiments. A sound a priori prediction of chromatographic retention behaviour is not usually possible for known samples (see chapter 3) and always impossible for unknown samples. Hence, the initial data for the optimization process will have to be provided by some sensibly selected experiments. The initial experiments to be performed will depend on the optimization procedure. They will be discussed in the following sections.

In some cases, the results of the initial experiments can be used to reduce the number of parameters considered or to narrow down the limits defined for the selected parameters. In both cases this can result in great simplifications for the optimization process. This will be discussed in section 5.4. After the reduction of the parameter space additional experiments may have to be performed to complete the initial data set.

For each chromatogram the value of the response can be calculated using the selected criterion (see chapter 4). These values may be used to search for the optimum conditions. Because of the complexity of the surface, many data points (chromatograms) are needed to form an impression of the response surface.

Alternatively, the response surface may be calculated indirectly by describing the capacity factor (or retention time) as a function of the considered parameters, using the

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data obtained from the initial experiments. We will use the word modelin referring to such a description of the retention sudace.

There is a distinct advantage in the use of models to describe retention surfaces rather than response surfaces. The important characteristic of the former is that they are much more simple than the response surface. Response surfaces form a combination of many (as many as there are components in the sample) different retention surfaces. In this way, fewer experiments (chromatograms) may be required to form an overall impression of the response surface.

The response data are then used to search for the optimum. This may result in the location of an optimum or in the location of one or more additional experiments. The decision as to whether or not the final optimum has been located can be incorporated in a stop-criterion, which decides whether or not additional experiments need to be performed.

We will refer to methods that try to characterize the response surface indirectly through the retention surfaces of the individual solutes as interpretive methods. They will be discussed extensively in section 5.5.

The above discussion can be summarized in a general outline for optimization procedures. This is shown in figure 5.4. The procedures to be discussed in subsequent sections can all be assigned a path in this figure.

[Define Liyits I

I I I 1 2000I 3000 I 1

Calculate Response

Characterize Response Surface1

Search for Optimum 5 - - - - - 7 2

REPORT Criterio

Figure 5.4 General outline for optimization procedures. The numbers shown in the figure can be used to characterize a particular path in the figure by adding up all numbers found along the way.

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The numbers shown in figure 5.4 can be used to characterize these paths. A particular path can be described by adding the numbers that are met along the way (see section 5.7).

At the end of this chapter we will discuss the merits of different optimization procedures. Some obvious criteria for this discussion have appeared in this introductory section. The following aspects appear to be relevant: 1. The length of the procedure, i.e. the number of chromatograms required. 2. The result of the procedure, whether it is a local or the global optimum. 3. The possibility to perform multi-parameter optimizations. 4. The complexity of the procedure and the required means of computation. 5 . The possibility of unattended automated optimization.

5.1.4 Definitions

We will end section 5.1 with a summary of the most relevant definitions encompassed in this introductory section.

Parameter Parameter limit Parameter space

Retention surface

Retention line Model Response

Response surface

Response line Initial experiments

Variable to be optimized Upper or lower limit of a variable All possible combinations of the variables within the parameter limits Retention (capacity factor) as a function of the variables in the parameter space Retention surface for one variable Description of the retention as a function of the variables Criterion value at a particular location in the parameter space Response as a function of the variables in the parameter space Response surface for one variable Chromatograms run at fixed locations in the parameter space

5.2 SIMULTANEOUS METHODS WITHOUT SOLUTE RECOGNITION

Simultaneous methods are those in which first a decision is made on which experiments should be performed, then in the second step all these chromatograms are recorded and in the third step the optimum is located. Of course, not all experiments can be done at the same time. The word simultaneous can be used because all measurements are done at the same stage in the optimization process without any calculations or interpretation taking place in between. In this section we will discuss methods in which the entire chromatogram is considered. In section 5.5.1 we will look at similar simultaneous methods, in which all the solutes in the sample are considered individually. In the present section it is not required to “recognize” the individual solutes in the different chromatograms, i.e. it is not relevant for the quality of the chromatogram (response value) in which order the peaks appear and the process is not disturbed by “cross-overs”, i.e. changes in the elution order.

The simplest form of a simultaneous method follows a path in figure 5.4 that can be characterized by 101 1. The term grid search forms an illustrative description of such a

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process. The parameter space is covered by a grid or raster of experimental conditions usually at regular intervals. The required number of data points (chromatograms) will be determined by the complexity of the response surface, the number of parameters considered and the limits set to these parameters.

The complexity of the response surface will tend to increase with the number of solutes and with a greater degree of variations in selectivity provided by the selected parameters. Ironically, we wish to optimize towards those parameters which most affect theselectivity, hence yield the most convoluted or "rugged" response surfaces and hence require the greatest numbers of experiments in a grid search approach. A reliable description of the response surface of figure 5.1 may require some 100 data points at regular intervals along the composition axis. Because of the selection of the criterion in figure 5.1 the local optima are most often sharp discontinuities in the response surface. Therefore, they could easily be missed in a grid search. Moreover, interpolation between the response values at different data points by means of some (digital) smoothing function around the optima is not feasible. The selection of a more continuous criterion might enhance the possibilities for sensible interpolation (see for example figure 5.5).

A greater number of experiments would provide a better description, but would create a greater experimental work load. A smaller number of points reduces the latter, but also the chances of locating the global optimum. It will also be clear that an initial coarse grid, followed by a finer grid located around the highest observed response value (path characterized by 1012 in figure 5.4) is no solution to this problem. The highest response found in the initial coarse grid may very well be close to a local optimum, while no data points happened to be taken in the vicinity of the global optimum.

Especially for the case of figure 5.1, where the composition of the stationary phase is the continuous parameter along the horizontal axis, a large number of experiments is extremely unattractive. Fortunately, there are much more effective procedures for this kind of optimization (see section 5.5.1).

For the important case ofthe Optimization of the mobile phase composition in reversed phase LC (RPLC), a typical two-dimensional response surface tends to be much less rugged, especially if the number of sample components is relatively small (n < 10). A typical example is shown in figure 5.5. The seiection of the normalized resolution product (c eqn.4.19) as the criterion has also contributed to the smoother appearance of figure 5.5 relative to figure 5.1. Note that the criterion r has been recommended in chapter 4 for optimization processes in which the dimensions of the column are to be optimized after completion of the procedure (table 4.1 1). Therefore, the grid search approach is more appropriate for this kind of optimization than for optimization processes on the final analytical column.

The one parameter in figure 5.5 is the mixing ratio of two iso-eluotropic binary mixtures (i.e. they are selected to provide roughly the same retention times: see section 3.2). The compositions are acetonitrile/water 46/54on the left hand side and THF/water 37/63 on the right hand side of figure 5.5. The resulting mixtures are ternary ones, but there is only onedegreee of freedom (one parameter). We will discuss the selection of the limiting binary mixtures extensively in section 5.4.

Some idea of the behaviour of the response surface in figure 5.5 might be obtained by recording chromatograms at regular intervals of lo%, i.e. 11 experiments along the entire axis. By interpolation between successive data points there is a reasonable chance that the

180

0 "THF - 0 37

Figure 5 . 5 Example of a response surface for the optimization of the mobile phase in RPLC. Horizontal axis: ternary mobile phase composition. Drawn line: response surface using the resolution product as the criterion. Dashed lines: retention surfaces for individual solutes (In k). For further details see section 5.5.2. Figure taken from ref. [504]. Reprinted with permission.

0' 4

1

I I

X I - x1-

Figure 5.6: Illustration of the data points required for a grid search in a two-parameter space at ten percent intervals. (a) no constraints; (b) sum of the two parameters not to exceed one.

global optimum will indeed be approached. However, the estimate of 11 experiments is rather optimistic and the number will have to increase rapidly if the number of components in the sample increases. For two parameters the number of experiments becomes 1 1 = 121. However, if the parameter space is constrained (for instance by the rule that the sum of the parameters should not exceed unity, which will be the case if x and y are volume fractions), this number may be reduced to 66. The 66 data points are spread out over an equilateral triangle rather than a square. This is illustrated in figure 5.6.

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Table 5.1 gives some figures on the required numbers of experiments for two- and three-parameter optimizations for cases in which the parameters are constrained or not, and for a 10% interval as well as for a 5% interval between data points.

Table 5.1: Number of experiments required for grid search optimization in various cases. The two numbers given for each situation refer to 10 and 5% intervals in the individual parameters.

Number of Constrained (1) Not constrained parameters

10% 5% 1 0% 5%

1 2 3

- 11 21 66 23 1 121 441

286 2925 1331 9261

-

(1) Sum of the parameters not to exceed one.

The recording of a series of chromatograms under varying conditions will require a finite amount of time. For the optimization of the major component composition of the mobile phase in RPLC the situation is relatively favourable due to the fast equilibration (see section 3.2.2.1). Nevertheless, for a conventional HPLC column with to= 2 min the actual chromatogram itself would take at least some 20 minutes (O< k< 10). Using ten minutes as a optimistic estimate for the equilibration time, well over half an hour will be required for each chromatogram. Of course, small columns with very small particles may be used for optimization purposes (so-called FAST-LC), and a total time of 20 minutes for each chromatogram has been claimed. Even if this figure of 20 minutes can be reduced further, the numbers in table 5.1 do suggest that a grid search approach has a very limited applicability , especially since many parameters will be much more difficult to control automatically than the solvent composition in LC.

We can formulate the following conclusions for the grid search approach: 1. The approach is feasible for response surfaces that are not too complicated. 2. Because of 1 ., it will be most useful for samples containing small numbers of components. 3. Also because of I . , the approach will benefit from the selection of criteria that give rise

to relatively smooth response surfaces. especially in the vicinity of the optima. 4. The criteria suggested by 3. imply that the grid search approach will be less useful for

optimization processes run on thefinal analytical column than it will for cases in which the column dimensions are optimized last.

5. The length of the procedure will increase rapidly with an increase in the number of parameters considered. Applications in which more than two parameters are considered do not appear to be feasible.

6. Once the chromatograms are recorded and the value of the criteria (response) calculated, very little effort or computation is required.

7. Complete automation of the method is easily possible. 8. Zf the result of a grid search approach is a local optimum rather than the global one, then

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the result is probably still acceptable, because the procedure provides a good impression of the entire response sur$ace.

5.3 THE SIMPLEX METHOD

In contrast to the simultaneous optimization procedures described in the previous section, the Simplex method is a sequential one. A minimum number of initial experiments is performed, and based on the outcome of these a decision is made on the location of a subsequent data point. This simplest form of a sequential optimization scheme can be characterized by the path 1012 in figure 5.4.

The number of initial data points is one more than the number of parameters considered in the optimization process. These initial experiments define a geometrical figure in the parameter space which is called a Simplex. A two-dimensional Simplex is a triangle (often equilateral). A three-dimensional Simplex is a tetrahedron. The description of Simplexes in more dimensions is somewhat more difficult to envisage, but is mathematically straightforward.

The initial experiments yield a set of chromatograms, each of which can be assigned a (criterion) value. Any of the criteria of chapter 4 that yields a single number for each chromatogram can be used. It will be assumed in the following that a criterion has been selected for which a maximum value needs to be obtained from the optimization procedure.

The next step in the Simplex algorithm is to reject the lowest point, i.e. the chromatogram that yields the lowest response value, and the location of the next data point is found by reflecting the Simplex in the opposite direction. This process can then be repeated.

Variable 1

Figure 5.7: Illustration of a two-dimensional Simplex optimization. Dotted lines are contour lines: figures represent the response. Figure adapted from ref. [505]. Reprinted with permission.

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For a two-dimensional optimization this is illustrated in figure 5.7. The initial Simplex is the triangle ABC at the bottom of the figure. Because point B yields the lowest response, it is rejected and the triangle is reflected towards point D. Out of the three remaining data points, point A is now the lowest and this point is rejected. The Simplex is seen to “walk up” in the direction of increasing response and will eventually approach the optimum (0).

Around theoptimum, the situation will arise in which the reflection of the triangle results in a position at which a measurement has already been performed. This will be the case in the triangle MLN, in which N yields the lowest value. Instead of rejecting N and returning to measure K, the point with the second lowest response (L) is now rejected and the triangle is reflected towards point P. This procedure can be repeated until a measurement has been obtained at point R. Thereafter, no new measurements wilt be suggested from rejecting either the lowest or the second lowest response value and the optimization process comes to a halt.

The highest response has been obtained at point M, and hence this is the predicted optimum. Note that point M has been retained in the Simplex ever since it was first measured, so that no tables of high values need to be kept and the optimum is one of the three last remaining points.

It is seen in figure 5.7 that point M is not exactly located at the optimum (point 0). A decrease in the size of the initial Simplex will imply that the optimum will be approached more closely, but also that the number of experiments will increase further. Since the simple optimization described in figure 5.7 has already required 17 experiments, the latter prospect is not very attractive.

A more rewarding solution to this problem is the use of modified Simplex procedures, such as first described by Nelder and Mead [507]. Such modified algorithms allow other operations besides reflecting the triangle, such as contraction or expansion. The manner in which such a modified Simplex algorithm proceeds is illustrated in figure 5.8 for a

A 20 60 60 80 100

%Water

Figure 5.8: Illustration of a two-dimensional optimization using a modified Simplex algorithm. A ternary mobile phase for RPLC is being optimized. The third component is acetonitrile. Figure taken from ref. [505]. Reprinted with permission.

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two-dimensional chromatographic optimization problem. This example is taken from the work of Berridge 15051. The two parameters are two of the three volume fractions in a ternary mobile phase mixture for RPLC. Because the sum of compositions is required to equal one, only two parameters can be chosen to define the composition of the mixture. A boundary condition is used to avoid that the sum of the two parameters considered exceeds one.

Because the initial parameters are taken very close to the edges of the parameter space, the Simplex necessarily contracts immediately. It typically approaches the optimum area quickly, but spends much time locating its final optimum around a composition of 50% water, 20°/0 methanol and (hence) 30% acetonitrile. The advantage of this is that most measurements are obtained in the location of the optimum. This could also turn into a disadvantage, because it implies that little information is obtained about the rest of the response surface, so that no idea can be formed about the merits of the located local optimum with respect to other optima.

If the response surface is simple, the true optimum can be approached without the need for a compromise between the required accuracy of the resulting optimum and the number of experiments required, as was the case for the use of a constant step size in figure 5.7. On the other hand, a stop criterion for the Simplex needs to be defined carefully, because it will be clear from figure 5.8 that many experiments can easily be wasted in the close vicinity of the optimum if the requirements are too tight. For example, to locate an optimum with an accuracy of 0.1% in composition will require much more time (many more experiments) than if the procedure is stopped when the changes in the composition in successive steps start to fall well below 1%.

From the point of view of the operation of the Simplex it is advantageous to span the entire parameter space as much as possible with the initial experiments. From a chromatographic point of view this is often less attractive. The initial experiments of figure 5.8 had to be performed at mobile phases containing 100% acetonitrile (point A) and 80% methanol plus 10% acetonitrile (point C). Usually, both these compositions will give rise to chromatograms with very small k values and hence very little resolution (eqn.1.22). These chromatograms will be rapid, but hard to characterize by any criterion. At point B, on the other hand, a mobile phase containing 90% water is likely to yield ample resolution but with impractically (and maybe even immeasurably) high capacity factors. The choice for the initial points will have to be made on the basis of considerations regarding the demands of the Simplex on the one hand and the practical aspects of the chromatography on the other.

Figure 5.9 shows the result of the optimization procedure illustrated in figure 5.8. Given that the sample contains four solutes (and not more), the result is reasonable, regardless of whether it represents a global or a local optimum. A different and perhaps better chromatogram might have been obtained if a different optimization criterion had been selected. However, figure 5.9 clearly shows that Simplex optimization may be used with some success in chromatography.

An important advantage of the Simplex method is that it does not rely on any chromatographic model and does not require any chromatographic insight. This implies that a Simplex optimization program can be applied to LSC as well as to RPLC without any modifications [506]. This is not true for many other methods as will be discussed in section 5.5.1.

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1

I , I I I

0 2 L 6 8 t lmin -

Figure 5.9: Resulting chromato ram from the optimization procedure of figure 5.8 obtained at a

[505]. Reprinted with permission. mobile phase composition of 52 d /o water, 21°h methanol and 27% acetonitrile. Figure taken from ref.

The Simplex procedure has some important advantages over other methods: I. It requires no (chromatographic) information or insight beyond a sensible selection of the

parameters. 2. I t only requires the recording of a chromatogram and the calculation of the response at

each data point. No information regarding the behaviour of the individual solutes is required.

3. I t can be performed with any number of parameters of any kind. Of course, the number of required experiments will rapidly increase when an increasing number of parameters is considered.

However, the Simplex procedure also has some considerable disadvantages, one of which we have already touched upon. A large number of experiments is usually required to locate the optimum. Typically, about 40 experiments appear to be required [508,509]. If the parameter space is reduced before the Simplex procedure is started, this number might be brought down to about 25 (see ref. [510] and section 5.4).

A second, and more serious problem involves the complexity of the response surface. A simple response surface with one broad optimum, such as the one in figure 5.1, is neither very common, nor very desirable for the optimization of chromatographic selectivity (see section 5.1). In the more common case in which the global optimum is the highest of a series of local optima, the result of the Simplex may very well be one of the latter. Again, it will be logical that the chances of finding the global optimum.are greatest for simple response surfaces in which the global optimum is dominant. The fact that Simplex optimization is most useful for simple response surfaces makes it most useful for simple samples, containing a limited number of solutes. Also, the inclusion of non-selective parameters,

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such as the flowrate [508] or to some extent the water content of the mobile phase in RPLC (figure 5.8) renders the response surface more compatible to Simplex optimization.

The result of the Simplex procedure is a local or possibly the global optimum. To have any indication of the relevance of the optimum found, the procedure should ideally be restarted several times from different sets of initial data points (chromatograms) [51 11. This is the more true since the Simplex procedure provides very little insight into the overall character of the response surface. However, having to restart the algortithm several times is in conflict with the large number of experiments required by the Simplex process. This creates a circle which forms the main objection against the application of Simplex procedures for the optimization of chromatographic selectivity. This circle is depicted in figure 5.1 0.

The disadvantages of the Simplex method can be summarized as follows: I. A large number of experiments is required. 2. A local optimum may be the result. 3. Little insight into the response surface is obtained.

Restarting the Simplex from different initial experiments will decimate problems 2. and 3. above, but will aggravate point 1. Because of the likelihood that the Simplex optimization will lead to a local optimum, the use of an initial coarse Simplex in order to find a suitable area in which an experimental design can be located [512] cannot be recommended.

Statistical optimization methods other than the Simplex algorithm have only occasionally been used in chromatography. Rafel [513] compared the Simplex method with an extended Hooke-Jeeves direct search method [514] and the Box-Wilson steepest ascent path [515] after an initial 23 full factorial design for the parameters methanol-water composition, temperature and flowrate in RPLC. Although they concluded that the Hooke-Jeeves method was superior for this particular case, the comparison is neither representative, nor conclusive.

Drawback of simplex optimization for LC

Many experiments to locate

optimum

procedure

A 2-l

Figure 5.10: Figure illustrating the main problem of the use of Simplex procedures for the optimization of chromatographic selectivity.

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5.4 REDUCTEON OF THE PARAMETER SPACE

5.4.1 Fall factorial desigm

In section 5.1 we have stressed the importance of selecting the most relevant parameters for the optimization process and of defining sensible limits for these parameters. The maximum temperature in GC optimization was discussed as an example of an upper limit that could be selected sensibly before the optimization is s t a d . Without knowing anything about the sample, we can guess which parameters will most likely have the greatest effect on the selectivity. This was done in general terms in chapter 3. The selected chromatographic system will impose its own limits on the parameter space, e.g. the maximum temperature of GC columns and the pH limitations of silica-based LC columns. These constraints are also independent of the sample.

Once we know more about the sample we may be able to narrow down the parameter space further. Certain parameters may not show any selectivity effects for a certain sample and may therefore be neglected on the basis of factual knowledge (e.g. pH and ion-pairing reagents for non-ionizable solutes in LC) or after some initial experiments.

A systematic procedure for the latter has been described by Lindberg et al. [516]. They suggest the use of a full factorial design to cover the parameter space. If np parameters are considered and if each parameter is considered to take on lvalues (levels), then the number of experiments (ne) for the full factorial design is [517]

This exponential relationship prompts us to be careful in the application of full factorial designs to problems involving many parameters and to minimize the number of levels for each of them [517). The lowest possible value of 1 is two. If each parameter takes on only one value (level), then the full factorial design condenses into a single point and loses its significance. If the number of levels is two, then the study of 3 parameters involves 8 experiments. For 4 parameters this becomes 16 and for 5 parameters 32. Hence, full factorial designs should only be used if chapter 3 does not provide sufficient insight into which parameters are the most relevant, and if the optimization problem is important enough to warrant a very thorough approach.

- ++8 7+++

“1 -.-1 2+--

Figure 5.11: Full factorial design for three parameters at two levels. At each comer the level of the parameters is indicated. The centre of the cube forms the origin of the design (OOO for parameter values). The arrows on the left illustrate the three different parameters. Figure taken from ref. [516]. Reprinted with permission.

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Figure 5.1 1 illustrates a full factorial design for three parameters at two levels. The eight experiments prescribed by eqa(5.1) are positioned on the comers of a cube. The three parameters each take a high ( + ) and a low (- ) value. Each parameter can be assigned a direction in the cube.

Figure 5.1 1 shows that parameter 1 ( v , ) will have a low value on the left face of the cube and a high value on the right face. Parameter 2 (v2) will be low in the front and high in the back, and parameter 3 (v,) low at the bottom and high at the top. The levels of the three parameters are indicated at each comer of the cube in figure 5.1 1.

The mean effect of one particular variable can now be estimated by subtracting all experimental results from the points at which that parameter is low from those results at which the parameter is high, with all other values equal. Hence, using the design of figure 5.1 1, the function values u> on the left face of the cube may be subtracted from those found on the right face to yield the mean effect of parameter 1 (e,):

el = (5.2)

where4 is the function value obtained at the jth corner of the cube. Besides an estimate for the mean effect of a parameter, estimates can also be made for the mutual interaction effects between different parameters, as described in refs. [518] and [519].

Lindberg et al. [516] studied the effects of four parameters for the optimization of a separation in ion-pairing RPLC. The parameters considered, together with their high and low values, are given in table 5.2a. Four parameters at two levels lead to 16 data points (eqn.5.1) and the mean effect for each parameter can be estimated from eight differences between two data points.

For the optimization of chromatographic separations it is not useful to compare response (criterion) values at different points in the factorial design. Due to the convoluted character of the response surface it is unlikely that a good estimate of the influence of one individual parameter on the response may be obtained from a few data points. If we look at figure 5.5, the replacement of some methanol by the corresponding amount of THF in a ternary mixture (i.e. moving from the left to the right in figure 5.5) will sometimes lead to an increase in the response (r), and sometimes to a decrease, depending on where the high and the low levels of the parameters are located along the axis in figure 5.5.

For chromatographic separations it is more sensible to compare k values, because retention surfaces are easier to characterize than response surfaces. Hence, f , in eqm(5.2) is the capacity factor of the solute at thejth data point. Each solute will have its own values for kq), and hence a different mean effect can be defined for each sample component and each parameter. The results for the four solutes and four parameters studied are given in table 5.2b. To allow a rapid comparison, the mean relative effects are given, i.e. the difference in

percentage points between two k values at positionsj(high) and h (low) for solute i is found from

cu2-A) + u-f,> + u,-f,> + U,-f,,} ,

200(k,,- ki.J

k i j + ki.h Ae, = (5.3)

and the mean effect is found as the average of eight values for Ae,. It is clear from table 5.2b that the methanol-water ratio and the CSA concentration have

the largest effect on the capacity factor. On this basis Lindberg et al. [516] selected these

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Table 5.2: a. Parameters considered for the optimization of a four component mixture in ion-pair RPLC [516].

Parameter High Low value value

1. Methanol-water composition (v/v) 38-62 32-68 2. pH 4 2 3. Phosphate buffer concentration (mM) 90 10 4. CSA (1) counter ion concentration (mM) 10 0

(1) Camphor sulfonic acid

b. Calculated mean relative effects of the four parameters of the capacity factors of four individual solutes. Calculated from data in ref. [516].

Parameter Mean effect on k ('10)

Morphine Codeine Noscapine Papaverine

1. Methanol 2. pH 3. Buffer 4. CSA

- 33 - 43 - 80 - 84 0 - 1 6 2

- 9 - 1 1 - 15 - 10 81 74 75 66

c. Calculated mean relative effects of the pH and the buffer concentration at different levels of the CSA concentration. Calculated from data in ref. [516].

Parameter CSA Mean effect on k (%) conc.

(mM) Morphine Codeine Noscapine Papaverine

2. pH 0 20 17 27 17 2. pH 10 - 21 - 18 - 14 - 13 3. Buffer 0 12 5 - 4 3 3. Buffer 10 - 31 - 28 - 25 - 23

two parameters as the most relevant ones. As expected (see section 3.3), the effect of the methanol concentration increases with increasing k values. The solutes in table 5.2b are listed in order of increasing retention, with kMORpH < kcoD 4 k,,,, < kPAp Because the effect of the CSA concentration is about the same for each solute, an increase in the

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methanol concentration in combination with an increase in the CSA concentration can be used to narrow the big gap between morphine and noscapine in the chromatogram, while moving the first two solutes away from the solvent front or reducing the analysis time.

The parameters in table 5.2b do not show considerable specific effects towards the individual solutes. Ideal parameters for optimization procedures would reveal more differences along a horizontal line in table 5.2b. The small effect of the pH seems logical from the selection of a pH range over which the solutes are fully ionized to enhance the ion-pairing mechanism. However, there is a strong interaction between the pH and the CSA concentration, which is not revealed in table 5.2b. When the CSA concentration is 0, an increase in the pH has the effect of increasing the retention. If the CSA concentration is 10 mM, then the pH has the opposite effect. The effect of the buffer concentration is dependent on the CSA concentration in a similar way. The effects of the pH and the buffer concentration are shown under the two different conditions with respect to the CSA concentration in table 5 . 2 ~ . Because of the averaging, the true effects of the pH and the buffer concentration on the retention are concealed in table 5.2b. It should be noted that the differences observed in table 5 . 2 ~ would have been much less dramatic if the lower CSA level had been higher than zero (e.g. lmM), and that pH effects and buffer concentrations in the absence of a counter ion are not very relevant for ion-pairing RPLC. Nevertheless, table 5 . 2 ~ carries a warning for applying factorial designs in the selection of the most relevant parameters.

The following conclusions can be formulated: 1 . Full factorial designs can be used to select the most relevant parameters. 2. Quite a few experiments are necessary. Therefore, this strategy should only be applied if

the information provided in chapter 3 is insuficient and ifthe analysis to be optimized warrants a great effort.

3. Individual capacity factors for all solutes need to be measured. This requires individual injection of each sample component (if known and available) under all conditions, or advanced detection techniques (see section 5.6).

4 . Care must be taken to consider the mutual interaction of individualparameters in order to avoid an erroneous interpretation of averaged data.

5.4.2 Scouting techniques

In chapter 1 (section 1.5) we have seen that optimum elution conditions require the solute capacity factor to be in a limited range. Hence, even if we have selected the most relevant parameters for optimization on the basis of chromatographic knowledge (chapter 3) or a series of carefully selected experiments (section 5.4.1), a large part of the parameter space may still be irrelevant for optimization purposes, because the capacity factors in these regions are either too high or too low. Especially those parameters which have a large effect on retention in chromatography (such as temperature in GC or mobile phase composition in LC) will show narrow margins. We identified such parameters as “primary parameters” in chapter 3 (table 3.10). It will be highly beneficial for the efficiency of the optimization procedure to establish realistic limits for the primary parameters at an early stage.

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This approach is based on eqn.(l.22). It is assumed that the three factors in the resolution equation can be optimized independently. In this philosophy the retention (k) is first optimized so that roughly optimal k values are obtained for all solutes. Hence, we are looking for a few initial experiments, which allow us to narrow down the search to a limited part of the parameter space. The complicating factor is that we are especially interested in the primary parameters, i.e. those parameters that have a large effect on the magnitude of the capacity factors. Therefore, it is not possible to define a fixed set of conditions which will allow us to elute all samples conveniently. It is highly probable that such a set of fixed conditions will result in excessively large or small capacity factors.

One solution is therefore to allow a series of conditions to be used, until some idea of the optimum working range is obtained. For instance, a series of isothermal gas chromatograms may be recorded, starting at a high temperature and then descending at regular intervals of (for example) 25 OC, until capacity factors are obtained that are roughly in the optimum range. A similar method may be used with many other parameters, such as the mobile phase composition, pH, or concentration of ion-pairing reagent in LC.

It is essential to start a series of such scouting experiments under conditions at which very low retention may be anticipated for all solutes. In this way, no late eluting peaks will be overlooked. It is much more practical to increase short capacity factors than it is to decrease large ones.

The main disadvantage of such a series of sequential scans is the large number of experiments required to establish the area of optimum capacity factors (see for example refs. [520] and [521]).

Another possibility for performing a series of isocratic scouting experiments is the application of thin layer chromatography [522,523]. The data obtained on thin layer plates may readily be related to capacity factors in column LC (see e.g. ref. [524], p.383). Thin layer chromatography is especially useful for investigating the possibilities of a series of stationary phases for the separation of a particular sample, keeping the mobile phase constant. Different experiments can be run simultaneously on a series of different stationary phases in TLC. TLC is less attractive when a series of mobile phases has to be tested on a given stationary phase. An additional advantage of the use of TLC as a scanning technique is that there is no problem with highly retained compounds. These would be recognized in TLC as spots around the point of injection. In column chromatography, they might stay on the column and lead to erroneous interpretation of the data and eventually to column pollution and degradation.

Programmed analysis

The requirement of a large number of initial experiments can be avoided by using programmed analysis as a scouting or scanning technique. In GC this is conveniently done with a temperature program. This is typically realized by a gradual increase of the oven temperature after the injection of the sample. Linear temperature programs are almost exclusively used. The great advantage of such a program is that a large number of solutes can be made to elute from the column under optimum conditions in one experiment. Compounds that would yield optimum capacity factors at low temperatures occur early in the chromatogram, while the components that require a higher temperature elute later

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(see also section 6.1). Hence, no knowledge of the sample is required to select the initial conditions.

A convenient rule of thumb is the following [525]: In order to achieve the same degree of separation in an isothermal analysis as in a

programmed temperature run, a temperature 45 OC below the retention temperature of a pair of peaks should be selected.

Expressed as a formula, if T,is the retention temperature, i.e. the temperature of the oven at the time of elution, and To the recommended isothermal temperature, then

To = Tr-45. (5.4)

According to eqn.(5.4), if the result of a programmed temperature scanning experiment in GC is a bunch of peaks eluted around a column temperature of 195 OC, then a chromatogram in which all the peaks appear with roughly optimal capacity factors may be expected to result from an isothermal experiment at 150 OC.

In different forms of LC there may be different primary parameters (see table 3.10). The term “gradient elution” is generally used for a chromatographic experiment in which the composition of the mobile phase is varied during the analysis. Salt gradients as well as pH gradients have been used, especially in IEC [526]. However, the most popular application of gradient elution involves the composition of the mobile phase. This typically involves the addition of increasing amounts of a strong solvent (B) to a weak solvent (A). Common examples involve gradients of water (solvent A) with methanol, acetonitrile or THF (solvent B) in Reversed Phase LC (RPLC). In Normal Phase LC (NPLC), increasing amounts of di-isopropyl ether, methylene chloride or chloroform (B) can be added to n-hexane (A) [527].

A simple way to estimate the appropriate isocratic conditions from the result of a gradient elution chromatogram is provided by the theory of linear solvent strength (LSS) gradients of Snyder (for a review, see ref. [528] or [527]). By definition, an LSS gradient obeys the following relationship:

In this equation kin is the capacity factor, which the solute would show under isocratic conditions (i.e. an elution at a constant mobile phase composition) corresponding to the composition at the inlef of the column at the time t that has elapsed since the start of the gradient. k, is the capacity factor at the start of the gradient ( t = 0), b the gradient steepness parameter, and to, as usual, the hold-up time of the column.

Clearly, eqn.(5.5) arises as the combination of two effects: 1. The composition of the mobile phase varies as a function of time at the column inlet.’

We refer to this as the gradient program. 2. The capacity factor of the solute varies with the composition of the mobile phase. This

aspect is related to the mechanism of retention (chapter 3). The shape of the gradient program should be adapted to the mechanism of retention

(i.e. to the particular form of LC) in order to achieve an LSS gradient (eqn.5.5). For

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example, the variation of retention with mobile phase composition in RPLC can be approximated by

I n k = I n k , - S p . (3.45)

Hence, to make the logarithm of the capacity factor linearly dependent on time, the composition can simply be varied linearly with time, according to

p = A + B t . (5.6)

The combination of eqm(3.45) and (5.6) yields

log kin = log k, - S ( A + B t)/2.303

S A SBt

2.303 2.303 = logk, - - + - * (5.7)

A comparison of this equation with eqn.(5.5) shows that the gradient steepness parameter b is a function of the solute (through S), the gradient program (through B ) and of the column (through to):

b = S B to / 2.303 . (5.8)

Eq~(5.6) defines a so-called linear gradient. Indeed, linear gradients are most popular in RPLC [527]. In LSC, retention varies much more strongly with mobile phase composition than in RPLC, especially when small amounts of organic modifier are added to the mobile phase (see section 3.2.3). Therefore, concave gradients are to be preferred [527].

The following is a very convenient rule of thumb for estimating optimum isocratic conditions from the result of a gradient run [527]: The mobile phase composition at the column inlet, at a time twice the value of the hold-up time before the elution of a sample component from the column, may be expected to yield a capacity factor of three for that component under isocratic conditions.

As an example, we assume a gradient from 100% water to 100% methanol in 20 minutes, on a column with a to value of 1.5 min. Now a solute that elutes with a retention time t , = 15 min ( t , is the retention time under gradient conditions) is expected to yield k = 3 at the composition that was reached at the column inlet at t = 15 - 2 x 1.5 = 12 min, which is 60% methanol, 40% water. Assuming that there is no delay time due to instrumental considerations, this is the composition at the start of the column, but not at the end. One and a half minutes (to) later, this composition will have reached the end of the column.

If the instrument incorporates a delay, for instance because of the presence of mixing chambers, this can easily be accounted for. For example, if the delay time is 2 min, then the composition at the column inlet at t = 12 min. does not equal 60% methanol, but rather 50°/o methanol.

The rule given above is indeed a rule of thumb and not an accurate estimate for the isocratic behaviour of the solute. It cannot be, since we have seen before that the gradient

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0 0.5 1 b-

Figure 5.12: Expected capacity factor (k) under isocratic conditions that correspond to the composition at the column inlet at t = t - 2t,, as a function of the gradient steepness parameter b. Figure calculated according to ref. [528!.

steepness parameter b will vary with the nature of the solute, because different solutes show different values of S in eqn.(3.45). However, the rule is a rather robust one, as is illustrated in figure 5.12. In this figure, the expected capacity factor under isocratic conditions at a mobile phase composition corresponding to the value at t = t , - 2t, is plotted as a function of b. It is seen that the resulting capacity factor is indeed around three over a wide range of b values, incorporating the range of 0.2 < b< 0.4, which Snyder demonstrated to be optimal [528]. In the range of low b values the simplified model used to construct figure 5.12 is no longer valid, but is is obvious that in the range of very small b values (very slow gradients) the rule of thumb loses its significance.

If a more accurate prediction of the isocratic elution behaviour is required, then the use of two [529] or more [531,532] different gradients may provide a possibility. The disadvantage of this method, besides the need to perform additional experiments, is that instrumental factors can give rise to quite considerable errors [529,530], so that extreme precautions may be required.

For the important case of optimizing the solvent eluotropic strength in RPLC, a more elegant alternative is available. We have seen in chapter 3 (section 3.2.2) that eqn.(3.45) is a good approximation for the retention behaviour of solutes in RPLC in the range of optimum capacity factors (1 < k< 10). In chapter 3 we also discussed the validity of the empirical equation

S = p + q l n k,, (3.46)

which seems to be closely followed, especially for the methanol-water system. If we

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consider the combination of eqns.(3.45) and (3.46), we observe that knowledge of only one parameter (i.e. either k, or S) or the capacity factor (1 < k < 10) at one composition is sufficient to describe retention as a function of composition over the optimum range. A combination of the two relevant equations yields:

I n k = I n k , - @ + q I n k , ) p (5.9)

If the value of k has been determined at a certain mobile phase composition, we can obtain k, from

In ko = (In k + p @/(l - qp) (5.10)

and subsequently S from eqn43.46). Indeed, eqn. (5.10) has been applied with some success even to extrapolate existing

retention data to find an estimate fdr the retention in pure water, assuming eqn.(3.45) to be roughly valid even for k + 10 [533]. However, such extrapolations cannot generalIy be expected to yield reliable results [534].

In RPLC, values for S are usuaIIy between 5 and 10 for small solutes and increase with the size of the solute molechles [535]. Hence, as a rule of thumb, retention may be expected to vary by 50 to 100% for a change of 0.1 in p (10% change in composition). This rule applies to sm 11 sol in the range 1 < k < 10. Reckoning with the possibility that nothing is known about e pJe, there is only a small range (typically 25-40°/o for small solutes, but even much Q sm e for larger molecules) over which optimum k values are observed for each individual solute. If we consider all solutes in the sample, the working range is further restricted. Therefore, the one data point that is required to use the combination of eqns.(5.10), (3.45) and (3.46) to estimate the retention behaviour of a solute may not be obtained in a straightforward manner. For the same reason as above, therefore, we will have to rely on a gradient elution run to provide us with the information needed.

The retention behaviour under gradient conditions, assuming both eqm(3.45) and (3.46) to be valid, can be calculated mathematically 15361. On the basis of such calculations, we can Fonstruct a diagram that allows us to predict the retention behaviour under isocraticlconditions from the result of a single gradient run. An example of such a plot is given in (igure 5.13, for a linear gradient from 1 W0/o water to 100°/o methanol in 15 minutes. The to value of the column is relevant in the calculations. For the present case, it was equal to 125 s. To apply figure 5.13 to other columns, it suffices to adapt the flow rate such that a similar value for to is obtained. If another gradient or another to value is used, a new plot should be constructed by the method described in the literature [536]. . Figure 5.13 gives a plot of the volume fraction of the strong solvent that is required to obtain a given capacity factor for a solute that elutes at a net retention time tk under gradient conditions. Lines have been drawn that correspond to a series of isocratic capacity factors (k = 0.5,1,2,5,10 and 20). The optimum range (1 < k < 10) appears grey in the figure, If we obtain the retention time of a solute under gradient conditions, we can find from figure 5.13 a series of compositions for different isocratic k values. Hence, figure 5.f3 allows us to estimate the isocratic retention behaviour of a solute from a single chromatographic experiment.

An example of the application of figure 5.13 is shown in figures 5.14 and 5.15. The

196

k=0.5 1 2 5

s

05

0 0 5 10 15 20 25

tA/min - Figure 5.13: Curves relating the isocratic composition (pJ to the net retention time under gradient conditions for various values of the isocratic capacity factor. Curves calculated on the basis of eqm(3.45) and (3.46). Linear gradient 0 - 100°/o methano1 in water. to= 125 s. Figure taken from ref. [536]. Reprinted with permission.

1 Orcinol 100%MeOH 2 Phenol 3 p-Cresol L 3.L -Xy lenol 5 3.5-Xylenol 6 2.L-Xylenol

3

0 5 10 15 tklmin -

Figure 5.14 Gradient elution chromatogram of a mixture of phenolic solutes. The six numbered peaks refer to the sample. The remaining signals to the blank. Linear gradient 0 - 100% methanol in water. ro=125 s. Figure taken from ref. [536]. Reprinted with permission.

former figure shows the result of a gradient elution chromatogram of six phenolic solutes. Figure 5.15 shows a simplified version of figure 5.13, in which only the curves for k= 1 and k = 10 have been drawn. Both figures correspond to the same linear gradient as figure 5.13. We identify six peaks in the chromatogram under gradient conditions. The other peaks in the chromatogram correspond to the blank signal*. The gradient program is

* A blank signal is usually inevitable in a gradient run, due to contaminations in the weaker solvent. Therefore, a blank experiment needs to be performed and the results of the actual experiment need to be compared with the blank (see also figure 6.6).

197

\

tktmin - Figure 5.15: Application of figure 5.13, relating the composition required for k = 1 and k = 10 to the net retention time under gradient conditions, for the gradient elution chromatogram of figure 5.14. The first (a) and last (0) peaks that occur in this chromatogram are indicated in the figure. The required isocratic compositions fall in the range between (pA and cpz Linear gradient 0 - 100 '10 methanol in water. to= 125s. Figure adapted from ref. [536]. Reprinted with permission.

illustrated in the figure, by means of the composition at the column outlet. The first peak elutes at a net retention time of 9 minutes, the last (sixth) after 13.5 minutes. Our task is now to identify proper isocratic conditions for the bunch of six peaks.

In figure 5.15 we have drawn two vertical lines corresponding to the first peak (a) at a net retention time of 9 minutes and the last peak (0) after 13.5 minutes. The intersections of these lines with the lines for k = 1 and k = 10 give us an indication on the optimum isocratic composition. The first peak is expected to be eluted with k = 1 at a composition (qA) of 63% methanol in water. If more methanol is used, then the retention of the first peak will be lower than the optimum range. The last peak is shown to be eluted with k = 10 at a composition (qd that contains 59% methanol in water. If less methanol is present in the mobile phase, then the last peak in the sample will show a capacity factor that is too high. Hence, in the range of compositions between qa and qz, coloured grey in figure 5.1 5, the six peaks are expected to be eluted with capacity factors in the optimum range.

We have learned the following from the application of figure 5.15: 1. The sample of figure 5.14 can be eluted under isocratic conditions, with all capacity

factors in the optimum range. 2. The required compositions contain between 59 and 63% methanol in water.

We may now proceed by investigating either a single composition in the range between qA and qa or the entire practical range. The former is more commonly done. In that case the result of a gradient scan is a binary mixtures, which defines the optimal eluotropic strength of the mobile phase for theelution of the sample. Iso-eluotropic mixtures of other compositions (using other organic modifiers) may subsequently be exploited for the optimization of selectivity (see section 5.5).

If only mixtures of a given eluotropic strength are considered as the result of a gradient scan, then a further optimization of the primary parameter (solvent eluotropic strength) is not contemplated and the number of parameters involved in the optimization process is effectively reduced by one. In the optimization of a ternary mobile phase composition one of the three volume fractions is defined by the two others, as their sum must equal one.

198

If we limit ourselves to iso-eluotropic mixtures, a one-parameter optimization problem remains. As was described in section 3.2.2, a binary mixture of 60% methanol and 40% water corresponds approximately to 48% acetonitrile in water or 37% T H F in water. We may proceed with the optimization procedure by considering these binary mixtures as pure solvents (e.g. solvent A equals 60140 methanol/water) and refer to them as pseudosolvents [537] or pseudocomponents [538].

Limitations of the gradient scanning approach

The main disadvantage of the gradient scanning techniques for LC described above, is the requirement to use selective detectors. Universal detectors, i.e. detectors which register any solute, will necessarily show a gigantic signal for the change in mobile phase composition. This background renders the detection of solute molecules impossible. Hence, selective detectors are required, which do not react to changes in solvent composition. The UV detector is the most common detector in HPLC. It is compatible with gradient elution if eluent components are selected which are transparent in the UV. Unfortunately, a number of solutes will not be detected. For example, aliphatic hydrocarbons are completely UV-inactive. UV detection can be almost universal, however, if short wavelengths are selected (e.g. 210 nm). In this case we may talk about “near universal detection” [543].

The possibility of using short wavelengths will depend on the nature and the purity of the solvents. From this point of view, acetonitrile may be preferred to methanol for RPLC. However, as discussed above, methanol-water gradients offer the possibility to estimate the isocratic retention behaviour fairly accurately from a single gradient run, because of the validity of eqm(3.46). In mixtures of THF and water eqn.(3.46) is only approximately observed, whereas it is completely invalid in mixtures of acetonitrile and water (see table 3.1).

In many cases, selective detection is an advantage rather than a disadvantage. This is generally the case as long as a detection method is selected which is sensitive to all the relevant components in the sample.

5.5 INTERPRETIVE METHODS

Interpretive methods of optimization can be described as follows: 1. The chromatographic data is interpreted in terms of the retention surfaces of the

individual components. 2. These surfaces are described by some kind of model. This model may be graphical or

algebraic and based on mathematical or statistical theories, but preferably on chromatographic insight.

3. Use the model for the retention surfaces of the individual solutes to calculate the response surface for the complete chromatogram.

4. Search the response surface for the optimum. Interpretive methods owe their existence to the relative simplicity of the retention

surfaces in comparison to the response surface. Indeed, attempts to describe the latter by a mathematical model [539,540,541,542] have never been successful. The general idea behind interpretive methods is is that whereas many experiments are necessary to describe

199

the response surface (see section 5.2), the retention surfaces may be described by an accurate model on the basis of a small number of experiments.

In the following two sections we will describe two kinds of interpretive methods. In section 5.5.1 we will discuss simultaneous methods, which involve a fixed experimental design. In the iterative procedures of section 5.5.2, an initial design that consists of a minimum number of experiments is used and the location of the next data point is calculated during the optimization process.

5.5.1 Simultaneous interpretive methods

In this section we will describe several optimization procedures which are simultaneous in the sense that all experiments are performed according to a pre-planned experimental design. However, unlike the methods described in section 5.2, the experimentai data are now interpreted in terms of the individual retention surfaces for all solutes. The “window diagram” is the best known example of this kind of procedure.

Window diagrams

Window diagrams were developed by h u b and Purnell for the optimization of the composition of mixed stationary phases for GC (for a review see ref. [Sol] or ref. ISM]). An example of a window diagram is given in figure 5.16. This figure will be explained below.

Figure 5.16a is a plot of the retention against the composition. These retention lines (surfaces) are required for the construction of the actual window diagram (figure 5.16b). In figure 5.16a the distribution coefficient (0 is shown on the vertical axis. If the total volume of the stationary phase is kept constant, then the phase ratio is constant and K is directly proportional to the capacity factor k (eqn.l.10). On the horizontal axis is the mixing ratio of the two components of the stationary phase (9). On the two extremes are the pure stationary phases S (left) and A (right).

Figure 5.16a can be constructed once the retention data of all solutes have been measured on the two pure phases. It is assumed that retention (K) varies linearly with composition (q see section 3.1). Figure 5.16a shows four straight lines, which represent the (expected) variation of retention with stationary phasecomposition for the four solutes W,X, Y and Z.

From this figure alone it is possible to get some indication of the optimum conditions for separation. Any vertical line drawn in figure 5.16a corresponds to a chromatogram that might be obtained with a particular binary stationary phase mixture. The separation between individual solutes can be estimated from the intersections of such a vertical line with the retention lines of the solutes. The two vertical axes represent the chromatograms on the purestationary phases. On pure S (cp= 0), solute Zis expected to elute first, followed by Wand Y. Thelatter two solutes will appear as asingle peak. Component X will be eluted last. If the stationary phase is pure A, then W will elute first, X and Y will co-elute completely and Z will elute last. At every point where the retention lines for two solutes intersect in figure 5.16a, the corresponding composition of the stationary phase will give rise to a complete overlap of two peaks. In figure 5.16a this can be seen to occur around cp NN 0.2,0.5 and 0.7. At every other composition some separation is predicted, the extent

200

fa1

2500’

a2 0.L 06 0.8 1.0 0

9 A

0 0.2 0.4 0.6 0.8 1.0 ‘PA -

Figure 5.16: Example of a window diagram for optimizing the stationary phase composition in GLC. (a) (top): variation of the retention (distribution coefficient K) with composition for the individual solutes W,X, Y and Z. (b) (bottom) window diagram showing grey areas (“windows”) at compositions where all components may be separated. Figure taken from ref. [545]. Reprinted with permission.

of which may be estimated by moving a ruler through figure 5.16a, parallel to the vertical axis.

This “ruler method” of optimization will suggest that a good separation may be obtained at compositions around rp z 0.15 or around Q z 0.25. The ruler method is the simplest and by far the cheapest optimization procedure. It encounters a great deal of scepticism, because it does not involve the use of a microprocessor. Of course, the ruler method suffers from severe limitations: 1 . Only a qualitative idea about the optimum chromatogram is formed. 2. I t can only be used for one-parameter optimization problems. 3. The retention lines for the individual solutes need to be known.

Window diagrams, such as figure 5.16b, overcome at least the first of these three problems. Moreover, they can in principle be expanded to cover two-dimensional optimization problems (see below). In figure 5.16b lines have been constructed that

201

represent the relative retention (a) of two solutes as a function of composition. For four solutes a maximum of six values can be defined. If n peaks occur in the chromatogram, then the number of a values (nJ is generally given by [501,542]

n - (n- 1) - n ! na = - 2(n-2)! 2

By definition

aji = kJk, = K / K i .

For the distribution coefficient we can write

K i = Ki.s + rp, A K i ,

(5.11)

(5.12)

(5.13)

where

A K , = Ki ,a - Ki,s . (5.14)

Hence, a combination of eqm(5.12) and (5.13) yields the following equation for the example of the solutes Z and Y:

K z = K Z , s + P A A K ,

K Y K Y . s + P A A K Y

azy = -

if K z > K and

- K Y = K Y , s + q A A K Y

K Z K Z . . s + q A A K Z ayz - -

(5.15)

(5.1 5a)

if K , > K , Eqns.(5.1 5) and (5.1 5a) describe a hyperbolic function with a discontinuity at the point

where K y = K , i.e. where the retention lines for the two solutes Y and Zintersect in figure 5.16a. This is seen to occur at qA=0.5. Hence, we see the hyperbola for ayz descend towards a value of 1 at pa = 0.5 in figure 5.16b, and then rise again slowly towards the right (as azy). Similar hyperbolic lines can be constructed for all five other possible pairs of solutes and this is done in figure 5.1 6b. Values of asmaller than 1 do not exist by definition.

The final step in the construction of the window diagram is to identify the lowest value of a which occurs at any composition. In other words, the a,,,,, criterion (see section 4.3.3) is used to characterize the separation. In figure 5.16b the areas between the axis for a= 1 and the amin value arecoloured grey. Wheregrey areas occur there is a chance of separating all solutes, provided that the number of plates is sufficiently large. The grey areas, therefore, are the so-called windows.

It can be seen in figure 5.16b that the highest value of amin is predicted to occur for q,=O.12 (amin= 1.23). Local optima occur at qA values of 0.25, 0.62 and 0.85.

One problem associated with the simple window diagram in figure 5.16b is the use of amin as the criterion. This was first pointed out by Jones and Wellington [546], who suggested the use of S,,,,, instead of a,,,. In table 5.3 it is shown that the highest possible

202

value for amin found from the window diagram does not necessarily correspond to the highest possible value for Smin, and hence is not necessarily the global optimum in terms of the required number of plates (eqn.4.48 or 4.49).

Table 5.3: Characterization of the optima predicted by the window diagram of figure 5.16b in terms of several criteria.

- 'PA amin k ( 1 ) Smin Nne (2) k m tnel,,, (3)

0.12 1.23 0.5 0.034 30,000 1.1 1,800 0.25 1.21 0.7 0.039 24,000 1.25 1,500 0.62 1.05 1.5 0.01 5 168,000 1.6 12,000 0.85 1.04 1.8 0.01 3 227,000 1.85 1 7,000

(1) For critical pair assuming phase ratio to be 0.001 (2) Eqm(4.35) with R,= 1.5 (3) Eqn.(4.48)

It should be noted that the predicted optima may shift slightly if Smin is used instead of amin. It is also shown in the last column of table 5.3 that under conditions of constant flow rate and constant diameter (of open columns or of particles in a packed column) the optimum at cpA = 0.25 requires a shorter analysis time than the one at cpA = 0.12.

The above discussion illustrates that the window diagram method can be applied with a variety of criteria. Indeed, the use of amin was not recommended in chapter 4. For the case of optimization of the stationary phase in GC, a new column will necessarily have to be prepared either by physically mixing the two stationary phases in the correct proportion and then coating the column with the mixture, or by combining calculated lengths of individual columns. In either case, the length of the column should be adapted to the result of the optimization process. Also, the overall capacity factor may be expected to vary considerably, as is shown in figure 5.16a. Hence, the recommended criteria are rz, or l / t n e (see table 4.1 1). If the window diagram method is applied to other problems, for instance to the optimization of the temperature in GC, then the column may be a fixed entry and other criteria may be considered.

Laub [544] has suggested the use of l / N n e as a criterion. Noyes [547] suggested that for optimization on a given column log t , / t , (where retention times and not net retention times are used) might relate more easily to R, than does amin. However, Smin [546] may be obtained just as easily from a chromatogram and this quantity is exactly proportional to R,, as long as the plate count (N) is constant.

The window diagram method also lends itself to the optimization of different parameters. However, in order to construct the window diagram it is necessary to know the retention lines or surfaces of the individual solutes. For the optimization of the stationary phase composition in GC a linear relationship may be assumed between retention (K or k) and composition (volume fraction cp; see section 3.1). Also, the window diagram method may be very useful for optimizing the stationary phase composition of

203

a mixed phase for LC [548] or for optimizing the temperature (plotting retention vs. 1/ T 15471). Constanzo I5491 applied the method for the optimization of the mobile phase composition in ion-pairing LC using a mixture of two pairing-ions (pentane sulfonate and

so.

tRlmin t a 30

20

10

6or----l

2.00-

1.75

I a 1.50

1.25

1.00L

-

-

-

O' i f 3 3'5 Qo Q5 i.0 45 6.0 PH -

31) 3.5 LO L.5 5.0 ZS 60 - PH- Figure 5.17: Application of the window diagram method for optimizing the pH in RPLC. Solutes: S = scopoletin, U = umbelliferone, TF = trans-ferulic acid, TC = trans-p-coumaric acid, CF = cis-ferutic acid and CC = cis-p-coumaric acid. (a) retention surfaces, (b) window diagram. Figure taken from ref. 15521. Reprinted with permission.

204

heptane sulfonate). The retention times of the solutes turned out to vary linearly with the ratio of these two ions.

However, a simple linear relationship does not usually exist. A clear example is the optimization of the pH in RPLC. The window diagram approach was applied to this problem by Deming et al. [550,551,552]. They measured the retention of each solute at a series of pH values (9 in ref. [550], 4 in refs. [551,552]) and fitted the experiments to eqn.(3.70). This is a three-parameter equation and hence a minimum of three experiments is required for it to be applied as a description of the retention surface. If more data points are available, the equation can be fitted to the data by regression analysis.

An example is shown in figure 5.17. In figure 5.17a the retention time is shown as a function of the pH for six solutes. It can be seen from this figure that four of these solutes have a p K , value in the pH range studied. Two other solutes do not show a great variation of retention with pH. Generally speaking, pH optimization is most useful when the different components in the sample mixture show considerable differences in behaviour, in other words when the p K , values are different.

From the retention lines in figure 5.17a, the response line (response surface in one dimension) can be calculated. This is shown in figure 5.17b. Again, a,,,,, has been used as the criterion. In order to perform the regression analysis on the retention data and to subsequently calculate the response surface, a computer should be used. This also facilitates a more efficient calculation of the response. At each value of the pH, the retention of each solute should be calculated from eqn.(3.70), using the coefficients obtained from the regression analysis. If the six capacity factors are subsequently arranged in increasing order, then only five avalues need to be considered to select the value of a,,,,,. According to eqn.(5.11), the total number of a values for six solutes is 15.

An example of the application of the window diagram method to a similar problem using Smin as the criterion instead of a,,, may be found in ref. [546].

Hsu et al. [553] applied the window diagram method to the optimization of the composition of a binary mobile phase in RPLC. However, a straight line was not obtained by plotting l / k vs. composition and therefore more than two experimental locations were required.

The latter was also the case for the optimization of the composition of a ternary mobile phase in RPLC by Issaq et al. [554]. The ternary mixture was formed by mixing two limiting (non iso-eluotropic) binary mixtures and a fourth order polynomial equation was fitted through five equally spaced data points.

The characteristics of the window diagram method can be summarized as follows: 1 . I t is a graphical method, to locate areas (windows) in which allsolutes may be separated. 2, I t can be used in conjunction with a series of optimization criteria. Use of amin is not

recommended. 3. The original window diagram method can be applied to a single parameter only*. 4. The retention surfaces (lines) for all solutes need to be known. 5. Linear relationships are preferred, but not mandatory. For non-linear retention lines

more than two initial (sets of) experiments are required.

* See, however, the discussion below on two-dimensional window diagrams.

205

6. Ifthe number of experimental data points exceeds the minimum requirements to describe the retention line by a selectedfunction, then the coeficients may be calculated using regression analysis. The method may then be referred to as a regressive method and the initial experiments form a regression design.

7. The response surface needs to be calculatedfrom the retention surfaces.

Critical band method

Colin et al. [555] have described a different method to construct a diagram that allows the prediction of optimum conditions. Their approach is based on the calculation of so-called critical bands. If the retention surface of a solute j is known, then a forbidden zone may be defined below the capacity factor ki. If the preceding solute i has a capacity factor k, which falls in this critical band, then the resolution between iandjis insufficient. Eqn.(l.20) relates the resolution to the capacity factors of the individual solutes:

kj-ki VN .- Rs = ki+kj+2 2

We can rewrite this equation and find*

ki = kj(VN-2RJ -4R,

v N + 2 R ,

(1.20)

(5.16)

On a column with a given (average) number of plates, the critical bands can be calculated for any value of the desired resolution R,. If the retention line (in the case of a one parameter optimization problem) for solute j is straight, then eqn.(5.16) describes another straight line. Two applications of this approach are shown in the figures 5.1 8 and 5.19.

In figure 5.1 8a the retention lines are shown for five aromatic solutes, together with their shaded critical bands. The vertical axis represents the logarithm of the capacity factor in RPLC, while the horizontal axis shows the mixing ratio of two (iso-eluotropic) binary mobile phases, which together constitute a ternary mixture. Separation with all R, values above 1.6 (the value selected for the construction of figure 5.18a) can be achieved at compositions at which none of the critical bands overlap. The optimum composition can be located using the “ruler method (see above), and it is indicated in the figure at a mixing ratio of 0.83. This corresponds to a mixture containing 33% (0.83 x 40) acetonitrile, 8.5% (0.17 x 50) methanol and the remaining 58.5% water. The chromatogram obtained with this mixture is shown in figure 5.18b. All components in the mixture are shown to be well separated. Clearly, this critical band method is a graphical procedure for the optimization of a single parameter using a threshold minimum resolution value as the criterion for separation.

Alternative to the ruler method, other secondary criteria may be applied once separation windows have been located. For instance, the composition may be selected at which all

* Eqn.(5.16) differs slightly from the original equation [5551, because an alternative equation for the resolution was used in that work.

206

LO AC N 50 H20 60 H20

1.0 I

I

1 . I

I

1.0

0.5

0.0

I I I I I 1

B

L

11

L I

1111 0 1 2 3 L 5 6 7 8

t / m i n - Figure 5.18: (a) Figure showing the retention surfaces for some aromatic solutes in RPLC. Critical bands have been constructed according to eqn.(5.16) below each solute. The dashed line indicates the optimum ternary mobile phase composition. (b) Chromatogram obtained at the predicted optimum composition. Figures taken from ref. (5551. Reprinted with permission.

resolution values exceed the threshold (1.6 in the example above), while the capacity factor of the last eluting peak (kd is the lowest. Another secondary criterion may be a minimum solvent viscosity (ref. [555], see also ref. [556]).

207

60 H20 1 .o

QO

-J I

- I - 0.0 I I , I ,

I log k

I '7 I 1 1

I I

- 0 1 2 3 L 5 6 7 8 9

t lmin - Figure 5.19: (a) Figure showing the retention surfaces for some aromatic solutes in RPLC. Only solute nr.11 is assumed to be of interest. Critical bands have been constructed according to eqw(5.16) and (5.17) below and abovetheretention line for this solute. Thedashed line indicates theoptimum ternary mobile phase composition. (b) Chromatogram obtained at the predicted optimum composition. Figures taken from ref. [SSSL Reprinted with permission.

Figure 5.19 shows another example, in which the method is applied t o a specific case in which only the separation of one solute (3,4-dimethylphenol, no.11) is assumed to be of interest. In such a case a second line can be drawn above the one for the solute of interest, which can be characterized by

208

kj ( V N + 2 RJ + 4 R ,

V N - 2 R S k, = (5.17)

The solute of interest can be separated from all the other solutes at compositions at which no other retention lines fall within the critical band. This is illustrated in figure 5.19b.

Eqn(5.17) can also be used to allow for the occurance of a solvent peak in the chromatogram. For example, a large resolution may be demanded between an imaginary peak at k=O and the first peak in the chromatogram. In that case all solutes may be assigned a critical band above (eqn.l.17) rather than below (eq.l.16) their retention lines.

Additionally, the method also lends itself to a convenient use of weighting factors in terms of different resolution requirements for different solutes.

The two applications shown here concern the optimization of the mobile phase composition in RPLC. However, the method may easily be adapted to other problems. It is most practical if straight retention lines can be obtained. It should be noted that this is not usually the case for retention as a function of mobile phase composition in RPLC. In fact, Colin et al. [555] adapted the value of the hold-up time ( to) such as to obtain straight lines. The fact that they succeeded in doing so for all of 11 solutes considered at the same time is remarkable, but it may not always be possible. In any case, adapting to in order to linearize the retention lines will be an awkward practice.

Toon and Rowland [557] used a similar method for the optimization of the composition of a binary mobile phase in RPLC. However, they did not use eqns.(5.16) and (5.17), but plotted lines for the observed front and back of the peak. Since these quantities are affected not only by the capacity factor and the peak width of the solute, but also by the sensitivity of the detection, the method of Colin et al. is to be preferred.

We may summarize the critical band method of Colin et al. as follows: 1 . I t is a simple, graphical method to locate areas where separation may be achieved. 2. Minimum resolution is used as a threshold criterion. 3. The method can be applied to a single parameter only. 4 . The retention surfaces (lines) for all solutes need to be known and preferably linear. 5. Minimal computational means are required. 6. The method lends itself readily to be adapted from the general case to specific

optimization problems.

Extension to multidimensional optimization problems

Window diagrams and related methods may in principle be applied to optimization problems in more than one dimension. The main difference compared with one-parameter problems is that graphical procedures become much more difficult and that the role of the computer becomes more and more important. Deming et al. [558,559] have applied the window diagram method to the simultaneous optimization of two parameters in RPLC. The volume fraction of methanol and the concentration of ion-pairing reagent (1 -octane sulfonic acid) were considered for the optimization of a mixture of 2,6-disubstituted anilines [558]. A five-parameter model equation was used to describe the retention surface for each solute. Data were recorded according to a three-level, two-factor experimental

209

design. This implies that the two variables were each assigned three different values (40, 50 and 60% methanol; 0 ,3 and 6 mM ion-pairing reagent).

Experiments were performed at the nine possible combinations of the values of the two parameters (see eqn.5.1) The model was fitted to the data by regression analysis. From the retention surfaces, the response surface was calculated.

The same method was applied for optimizing the separation of nine acidic solutes, using pH (between 3.6 and 6.0) and the concentration of n-octylamine (between 0 and 6 mM) as the variables [559]. The response surface for the latter application is shown in figure 5.20. The minimum value for the relative retention (amin) was used as the criterion for this figure. Although the uSe of this criterion has serious disadvantages (see chapter 4), it does not form an objection for the application of the optimization procedure itself, since once the retention surfaces are known, the response surface can readily be calculated for a variety of optimization criteria. The response surface of figure 5.20 suggests a broad global optimum at a pH close to 5.8 and a reagent concentration around 3.2 mM. The chromatogram obtained under these conditions is shown in figure 5.21.

I 1.3

1.2

1.1

1.0

Figure 5.20 : Response surface (“two-dimensional window diagram”) for the separation of a mixture of nine acidic solutes by RPLC. Variables are pH and the concentration of an “ion interaction reagent” (NOA = n-octylamine). The vertical axis represents the lowest value of a observed for any combination of two solutes in the sample (amin). Figure taken from ref. [559]. Reprinted with permission.

Weyland ef al. [560,561] used this method to optimize ternary mobile phase compositions for the separation of sulfonamides by RPLC. They fitted the retention surfaces to a quadratic model similar to eqn.(3.39), and also used a combination of a threshold resolution and minimum analysis time (min t , n Rs,min > 1.25; eqn.4.24) [560]. This criterion may yield a good optimum if the optimization is performed on the final analytical column (see table 4.1 1).

Otto and Wegscheider [562, 5631 applied the window diagram method for the simultaneous optimization of the (binary methanol-water) mobile phase composition, the ionic strength and the pH for the separation of ionic solutes in RPLC. They fitted the experimental data to a semi-empirical 13-parameter equation based on eqn.(3.45) for the composition effect, eqn.(3.71) for the effect of the ionic strength and eqn.(3.70) for the

210

0 25 50 tlmin-

Figure 5.21 : Resulting chromatogram at the optimum conditions predicted by figure 5.20. pH = 5.8; concentration of n-octylamine = 3.2 mM. ODS column; mobile phase methanol-water (20180) with 0.010 M acetate buffer. Solutes: E = phenylethylamine, P = phenylalanine, V = vanillic acid, C = trans caffeic acid, M = trans p-coumaric acid, F = trans ferulic acid, A = phenylacetic acid, H = hydrocinnamic acid and N = trans cinnamic acid. Figure taken from ref. [559]. Reprinted with permission.

effect of the pH*. The solutes included an amino acid, a weak diprotic acid, two dipeptides (zwitter ions) and three isomeric amino-benzoic acids.

In ref. [562] the data were collected according to a 6 x 3 x 2 factorial design (36 experimental locations at six values for the pH between 2 and 7, three values for the methanol content between 10 and 30% and ionic strengths of 0.1 and 0.2M). Additionally, data were taken along a vector in the parameter space for which only the pH varied.

A limited experimental design was used in ref. [563], in which a citrate buffer was used instead of the phosphate buffer in ref. [562]. The two different buffers yielded markedly different optimum conditions.

It was found that in order to locate the global optimum the entire parameter space had to be searched with a computer. A grid with 0.1 unit steps in pH, 2% steps in methanol concentration and 0.01 M steps in ionic strength [562] meant that over 6000 points had to be calculated. This indicates that whereas the window diagram method can be applied in more than one dimension, a considerable price has to be paid in terms of both the number of experiments (depending on the model) and the computation time required. The validity of the calculated optimum will mainly depend on the accuracy of the model that is used to describe the data.

Clearly, the response surface for the three-parameter optimization problems discussed

* For equations for diprotic acids and basis see ref. [316], p.239 et seq..

21 1

above are four-dimensional (hyper-) surfaces, which cannot be visualized. Hence, the term window diagram refers to a strictly mathematical, rather than a graphical procedure.

Sentinel metbad

A similar method has been developed for the optimization of the mobile phase composition in RPLC or LSC by Glajch et al. [542,564]. In their scheme, an optimum quaternary mobile phase composition is the goal of the optimization process. After reduction of the parameter space to solvents of equal eluotropic strength with k values in the optimum range, a fixed experimental design, referred to as a Simplex design,* is applied. This design consists of seven experiments, which are illustrated in figure 5.22 and listed in table 5.4a. Because the sum of the volume fractions of the three binary solvents always equals one, we are dealing with a two parameter optimization problem.

Figure 5.22: Experimental design as used in the Sentinel method of Glajch et al. [542]. Figure taken from ref. [565]. Reprinted with permission.

Praton Acceptor

Proton donor Xn -- Dipole interaction

Figure 5.23: Illustration of preferred modifiers for NPLC (dashed triangle) and for RPLC (dashed and dotted triangle) in the Snyder selectivity triangle (see section 2.3.3). Figure taken from ref. [542]. Reprinted with permission.

* The term Simplex design (or Simplex lattice design) is unfortunate. It creates confusion between the Simplex method for optimization (section 5.3) and the Sentinel method of Glajch et al., which is a very different method by all accounts. To avoid further confusion, we will not use the word Simplex in connection with the Sentinel method.

212

The points on the sides of the triangle (A, B, and C) represent three iso-eluotropic binary mixtures (solvents A, B and C). The composition of one of the three binary mixtures (i.e. the appropriate eluotropic strength) should be determined by either a scanning gradient or a stepwise series of isocratic scans (see section 5.4). Once one of the compositions is known, the compositions of the two iso-eluotropic binary mixtures can be calculated using the conversion factors given in table 5.4b.

The three mixtures A, B and Care mixed in the ratios listed in table 5.4a to yield three ternary mixtures (in the middle of each of the sides) and one quaternary one (in the centre of the triangle).

Table 5.4 Summary of mobile phase compositions to be used in the experimental design for the Sentinel method.

a. Solvent composition at each experimental location.

Experiment no. Volume fractions of binary solvents

'PA 'Pi3 'PC

0 0 0 112 1 /2 0 1 /3

0 1 0 1 /2 0 112 113

0 0 1 0 1 /2 1 /2 1 /3

b. Preferred solvents and required volume fractions in iso-eluotropic binary mixtures (1).

Column Base solvent Modifiers O/O(V/V)

RPLC Water

LSC n-Hexane

A Methanol B Acetonitrile C THF

50 (2) 41 36

A Diethyl ether (3)

C Methylene chloride 45

50 (2) B Chloroform 36

(1) Percentages given correspond to section 3.2.2 and not to the original publication ref. [542]. (2) Arbitrary value (3) Methyl t-butyl ether is to be preferred for practical reasons (ref. [5241, p.366).

213

Glajch ef al. characterize the retention surface by a quadratic model (similar to eqn.3.39). In such an equation only six coefficients appear, so that the seventh experiment allows the coefficients to be estimated by regression analysis. An advantage of that may be that random experimental errors in one or more of the data points become less significant. There is, however, also a potential disadvantage. If the quadratic equation is not adequate for the description of the response surface, then the regression analysis creates errors in the description of the retention surface throughout the parameter space. Therefore, the model induces an error in the description of the data even at the the experimental locations.

In general, the following procedure should be recommended. If the main source of error is in the inaccuracy of the experimentation, then it may be advantageous to fit a model equation to the data by means of regression analysis. If, however, the limiting factor is the inaccuracy of the model, then regression analysis should not be applied. An alternative to regression analysis is a division of the parameter space in segments (see section 5.5.2). Glajch et al. [542] have used three additional experiments to improve the accuracy of the optimum predicted by the quadratic model. We will return to the problem of model inaccpracies in section 5.5.2.

Once the retention surfaces are known, any criterion may in principle be used to calculate the response surface and to locate the optimum composition. One of the criteria used by Glajch et al. is the threshold minimum resolution criterion (section 4.3.3). This is done by means of a graphical procedure, referred to as overlapping resolution mapping or ORM. This procedure involves the location of areas in the triangle where the resolution R, exceeds a certain threshold value. This is repeated for all pairs of solutes and the results are combined to form a single figure.

An example of the procedure and the resulting overlapping resolution map is shown in figures 5.24 and 5.25. Because of the simple retention surfaces, each pair of peaks yields

2- 3 6-7

3-L 7- 8

L- 5 8-9

Figure 5.24 : Overlapping resolution maps for eight relevant solute pairs for the separation of nine substituted naphthalenes. Figure taken from Glajch et al. [542]. Reprinted with permission.

214

pair 2-3 \

ACN

Figure 5.25 : Overlapping resolution map (ORM) for all nine solutes of the sample in figure 5.24. Figure 5.25 is the result of superimposing all eight triangles of figure 5.24. Figure taken from Glajch ef al. [542]. Reprinted with permission.

0.00 2.25 L.50 6.75 9.00 t lmin -

Figure 5.26 : Resulting chromatogram corresponding to the ORM procedure illustrated in figures 5.24 and 5.25. Chromatogram recorded at the optimum composition indicated in figure 5.25 (32'/0 ACN, 15% THF, 53% water and 0% MeOH). Figure taken from Glajch et al. [542]. Reprinted with permission.

a simple figure in which two areas may be identified. The area where the resolution exceeds the threshold value is left white, the remaining area in the triangle is grey. Eight different plots are shown in figure 5.24. Potentially (eqn.5.1 I), there are 36 of such triangles for 9 solutes, but the remaining pairs of peaks are irrelevant (white triangles) and not shown in the figure. All the different plots may then be combined to form the final ORM (figure 5.25).

There is a clear analogy between this type of figure and a window diagram. In the white area, which may be called a window, the resolution will be at least 1.5 (for the example in figure 5.25) for all pairs of peaks. This is illustrated in figure 5.26, which shows the

215

chromatogram that is obtained at a composition within the window. This composition is indicated with a circled x in figure 5.25.

A disadvantage of the ORM method relative to the window diagram method is that no idea can be formed of the magnitude of R , within the window area. Hence, the exact optimum cannot be located. This follows as a logical consequence of the use of R , as a threshold criterion (see discussion in section 4.3.3). ORM may be used to select areas for operation in the parameter space, using the same column for the optimization procedure as for the analysis to be performed later. Because the retention surfaces of the individual solutes have been characterized during the procedure, a new optimum can be located on a different column (or different flow rate) by reinitiation of the calculation procedure using another value for the threshold. The calculation step needs to be repeated completely, but no new experiments are required. Because the number of solute pairs increases very quickly with the number of sample components (see the discussion on window diagrams above), the Sentinel method is most useful for relatively simple samples.

Issaq et al. [566,567] used the ORM method with a ten-point design, in which additional experiments were performed at compositions containing the three pseudosolvents in 4 1 : 1 ratios.

Application of the Sentinel method to LSC

Snyder, Glajch and Kirkland [568,569,570,571] have paid much attention to the possibilities of using a similar experimental design for optimizing the mobile phase selectivity in LSC. Unlike the situation in RPLC, it cannot be assumed that any mixture of two iso-eluotropic mixtures will yield a new mixture which is in turn iso-eluotropic. Snyder and Glajch [568] conducted a theoretical study on the possibility of calculating the eluotropic strength for binary solvent mixtures in LSC with a sufficient accuracy. This approach was expanded by Glajch and Snyder [569] to include ternary and quaternary mixtures.

Snyder, Glajch and Kirkland [570] introduced two new parameters to describe the selectivity effects in the optimization triangle for LSC. If methylene chloride (MC), acetonitrile (ACN) and methyl t-butyl ether (MtBE) are used as the preferred modifiers in n-hexane, then an empirical solvent selectivity parameter ( m ) can be defined which is low for methylene chloride and can be made equal for the other two binary solvents. The latter is achieved by adding the appropriate amount of methylene chloride to the hexane-ACN binary. Addition of MC is required at any rate, because hexane and ACN are not miscible in all proportions. By definition we can assume m to equal zero for the hexane-MC binary mixture and m to equal one for the two other binaries.

A second parameter can be defined as the ratio of the concentrations of the two localizing solvents:

R = [MtBE]/([MtBE] + [ACN]).

The selectivity parameters for the various solvents in the experimental design of figure 5.22 are listed in table 5.4.5. The combination of pseudosolvents for each experiment is also listed in the table.

21 6

Table 5.5: Summary of mobile phase compositions to be used in the experimental design for the Sentinel-method as applied to LSC. Selectivity parameters for LSC solvents according to the experimental design of figure 5.22.

solvent A : Dichloromethane (Methylene chloride; MC) solvent B : Methyl t-butyl ether (MtBE) solvent C : Acetonitrile (ACN) base solvent : n-Hexane

Experiment no. Selectivity parameters Solvent combination

m R

0 1 1 1 /2 1 /2 1 213

- 0 1 0 1 1 /2 1 /2

A B c (1) A / B A / C B / C A / B/ C

(1) Small amount of MC added to promote miscibility.

A serious disadvantage of using these new selectivity parameters is that they are not related to volume fractions (or mole fractions) in a straightforward way. Procedures have been described which can be used to calculate the eluotropic strength of binary 15681 and more complicated mixtures [569] and the selectivity parameters 15691. However, these already complicated (iterative) procedures are only applicable to solvents of known composition and calculating the composition once the required values of the solvent strength (8) and of the selectivity parameters (m and R) are known is highly complicated. Therefore, it seems that simplifications are required to create a useful system for the rapid estimation of iso-eluotropic binary, ternary and quaternary solvents using the preferred modifiers for LSC.

So far, an empirical approach that neglects the specific problems of LSC has appeared more feasible. Antle [572] demonstrated the applicability of the Sentinel method to LSC, using mixed mobile phases corresponding to table 5.4a, i.e. mixing the individual binary mixtures according to their volume fractions. This yielded some success, although admittedly not all solvents were iso-eluotropic.

Use of different stationary phases /

Glajch et al. [573] have expanded the Sentinel method to include the “simultaneous” optimization of the stationary phase. They applied the complete seven-point experimental

21 7

design in figure 5.22 to three different columns (chemically bonded alkyl, cyano and phenyl phases). Three separate ORM maps were constructed for the different columns and the highest optimum was selected. In this way the scope of the method could be expanded without a dramatic increase in the number of experiments required. However, no attempt was made to correlate the data obtained with equal mobile phases on different columns. Only if there is a considerable interaction between the mobile and the stationary phase (e.g. because of specific absorption of solvent components) will all 21 data points be significant. If the reverse is true and the stationary phase effects are independent of the mobile phase, then only one experiment is required on each additional stationary phase. Hence, nine experimental locations (7 + 1 + 1) would be sufficient to investigate the behaviour of three different stationary phases. An optimum number of data points for a complete optimization using three modifiers and three stationary phases may be somewhere in between the minimum number of 9 and the maximum number of 21.

It was claimed [573] that for complicated samples, such as the separation of 20 phenylthiohydantoin (PTH) derivatives of amino acids, the optimization of many parameters simultaneously is required to achieve sufficient selectivity. However, in ref. [573] the pH was optimized separately, before starting the complete three-parameter optimization with two continuous parameters and one discrete one.

A more or less opposite goal was pursued by de Smet et al. [574], who attempted to reduce the number of stationary phases to a single one, by choosing a cyanopropyl bonded phase of intermediate polarity, which can be used in both the normal phase and the reversed phase mode (see figure 3.8). Furthermore, because of a clever choice of modifiers, the total number of solvents required was restricted to six: n-hexane, dichloromethane, acetonitrile and THF for NPLC and the latter two plus methanol and water for RPLC. A variety of drug samples could be separated with a selected number of binary and ternary mobile phase mixtures.

The advantage of the simplified procedure described by de Smet et al. is the use of only one column and six solvents, which enhances the possibilities for fully automated optimization on relatively simple commercial instruments. The disadvantages are that the column of intermediate polarity could lead to a reduced general selectivity in both modes (see figure 3.8) and the long equilibration procedure (about 2 hours, involving several gradients), which is required to switch from the reversed phase to the normal phase mode and vice versa.

Since it is easily possible to change columns automatically with the aid of selection valves, it appears that an approach involving a minimum of two colums (one for NPLC and one for RPLC) is generally to be preferred.

Expanding the parameter space to non-iso-eluotropic solvents

The parameter space in the original Sentinel method is restricted to a series of iso-eluotropic solvents, which means that only a very small fraction of all possible quaternary mixtures is considered. This is illustrated in figure 5.27a.

Poyle [575] and d'Agostino et al. [537] have shown that a higher optimum might be located outside the iso-eluotropic plane in figure 5.27a. The exclusive use of iso-eluotropic solvents may be justified on the following two grounds:

21 8

(a 1

B

Figure 5.27 : (a) Illustration of the location of the Sentinel experimental design in the tetrahedron representing all possible quaternary solvents. (b) Illustration of a 12 point experimental design in which a range of solvent strengths is considered. Figures taken from ref. [537]. Reprinted with permission.

1. The possibilities of varying the eluotropic strength of the eluent, while keeping all k

2. If an extra parameter is considered, the optimization procedure becomes much more

d’Agostino et al. [537] modified the method of Glajch et al. [542] so that a total of twelve experiments is performed in a “truncated pyramid”, i.e. a “slice” of the tetrahedron located around the iso-eluotropic plane of figure 5.27a. This design is illustrated in figure 5.27b. On a microcomputer it took d’Agostino et al. one hour of computation time to locate the optimum with a grid search of the entire response surface at 4% intervals. To find the optimum with 1% steps (corresponding to steps in the solvent concentrations between 0.1 and 0.7%) took no less than 14 hours of calculation time [537].

values in the optimum range, are usually very limited.

time-consuming.

Summary

The advantages of the use of fixed regression designs, including multi-dimensional window diagrams and the Sentinel method, can be summarized as follows: 1. The experimental procedure is straightforward and may easily be automated. 2. A good impression is formed of the entire response surjiace. 3. Any desired criterion may be used for the caldlation of the response surface. 4. Because the retention surjiaces are known, the calculation step of the procedure can be

5. The method is relatively fast and simple, because only a limited parameter space is repeated using diflerent conditions, such as another criterion or another column.

considered.

The disadvantages are: 1. Experiments are spread out over the entire parameter space. The accuracy of the

219

description may therefore be expected to be roughly equal at each location in the parameter space. Extra experiments may be required i f a more accurate description of the response surface around the optimum is required. Zfthe retention surfaceis characterized by a modelequation, then the accuracy with which this equation describes the true surface becomes a limiting factor. To characterize the retention surfaces, individual capacity factors need to be obtained for all solutes. If the parameter space is reduced to a two-dimensional triangle (Sentinel method), then a better optimum outside this plane may be neglected.

5.5.2 Iterative designs

The first two of the above disadvantages of fixed design methods can be overcome by the use of iterative designs. These are methods in which an initial design that contains a minimum number of data points is used, then the results are investigated and the results of that investigation are used to conclude whether or not one or more new experiments are required, as well as where these additional experiments should be located in the parameter space.

The meaning of this complex definition is illustrated in figure 5.28. The procedure starts with a (small) set of initial experiments. The next step is the application of a model to the data. This model can be a graphical or a mathematical one, but may also be a simple linear interpolation between the individual data points. Typically, the model is applied to the retention surfaces of the individual solutes, and not to the response surface. Alternatively [537], it may describe relative retentions with respect to a reference component in the

Initial exps.

Figure 5.28 : Illustration of the operation of iterative designs for the optimization of chromatographic selectivity.

220

sample. The simplicity of the retention surfaces allows a reasonable approximation to be made from only a very limited number of experiments.

The model is then used in a calculation step to predict the location of the optimum. This step involves the calculation of the response surface from the retention surfaces using a suitable criterion, the location of the (predicted) optimum on this surface, and a decision about new experiments to be performed. Only this last aspect distinguishes iterative designs from the previously described fixed experimental designs.

New experiments may not be required if, for instance, the optimum is located at a position in the parameter space where an experiment has already been performed. If this is not the case, then the location of one or more additional experiments will be the result of the calculation step. Subsequently, a new set of experiments is run and added to the existing database. The model can then be refined using all the available data, and a new optimum can be predicted.

The procedure may be stopped not only if experiments have already been run at the suggested locations, but also if the predicted optimum is the same as it was before, or if it can be established in the calculation step that no further improvement may be expected from an additional iteration cycle.

The philosophy of iterative designs is to locate the true (global) optimum using a minimum number of experiments and making maximum use of available insight and experimental data. Such a philosophy can be justified if 1. the required number of experiments is indeed less than it is using other optimization

2. the time and effort needed to analyse the data (calculation step) is small compared to

3. The global optimum is found.

procedures,

the task of performing a new set of experiments, and

Phase selection diagrams

The method of phase selection diagrams was developed by Schoenmakers et al. [504] for the optimization of the composition of ternary mobile phases in RPLC. The starting point of an iterative design may be the same as for a window diagram. We will consider the optimization of the composition of a ternary mobile phase in RPLC. A very simple example involving six aromatic solutes is shown in figures 5.29,5.30 and 5.3 1. The ternary mixture is prepared by mixing two iso-etuotropic binary mixtures (see the discussion on the Sentinel method in the previous section).

In the present example these mixtures contain 50°/o methanol and 32% THF in water, respectively. The two chromatograms obtained with the binary mobile phases are shown in figure 5.29. From the capacity factors observed in these chromatograms, the phase selection diagram of figure 5.30 can be constructed. On the horizontal axis in figure 5.30 is the mixing ratio between the two limiting binary mixtures. The logarithm of the capacity factor is plotted on the vertical axis in this figure, and the (dashed) straight lines connect the two capacity factors observed for each solute. Using this linear interpolation for the retention lines, the response surface may then be calculated. In figure 5.30 the response line is drawn using the product resolution criterion.

It can be seen from figure 5.30 that co-elution of three solutes, and hence a product . resolution of zero, is predicted at compositions of around 35'/0 methanol and 10°/o THF

22 1

1.2

I 50% MeOH 50%H20 2 Phenol 3 3-Phenyipropanol L 2.4-Dimethylphenol 5 Benzene

!

I L 5

0 10 20 - tlmin

1

32% THF 68% H20

5 L

L I I I I I I I I 1

0 10 20 - tlmin

Figure 5.29 : Initial chromatograms for the construction of the phase selection diagram shown in figure 5.30. Figure taken from ref. [504]. Reprinted with permission.

(55% water). Optimum selectivity for all solutes is predicted to occur at a composition of 10% methanol and 25% THF (65% water). The two chromatograms that can be obtained at these compositions are shown in figure 5.31, and it can be seen that the phase selection diagram method is very useful in this case.

Figure 5.30 is a so-called phase selection diagram. It is essentially the same as a window diagram. However, figure 5.30 is the simplest phase selection diagram, which is the starting point for an iterative procedure, while a window diagram is the final stage of an optimization procedure using a fixed experimental design. The above example is a very favourable one, because a phase selection diagram as described above does not usually give a correct prediction of the optimum composition.

The reason for this is that the linear relationship between In k and composition (mixing ratio of two iso-eluotropic binary mixtures) is not rigorously valid. A careful examination shows that the observed lines for In kvs. composition are slightly and systematically curved [576,577].

222

0 '4'-THF - 0.32

Figure 5.30 : Phase selection diagram constructed from the chromatograms shown in figure 5.29. Dashed lines are retention surfaces, the drawn line is the response surface. Figure taken from ref. [504]. Reprinted with permission.

One way to solve this problem is to choose conditions such that accurate linear interpolation is possible for all solutes. This approach has been followed by Colin et al. (ref. (5551; see section 5.5.1), who suggested adapting the value for the hold-up time (to) as a function of composition. It is questionable whether such an approach can be applied successfully to a large number of solutes. Moreover, the procedure used to estimate the appropriate to values may be quite time-consuming.

Alternatively, an iterative method may be applied. The phase selection diagram may be used to predict the optimum composition. The chromatogram obtained at this composition may then be compared with the predicted values for the capacity factors. If the experimental optimum corresponds to the predicted chromatogram (in terms of response and capacity factors), then apparently the linear interpolation was justified and the result is a reliable global optimum. If the resulting chromatogram differs from the predicted one, then the newly obtained set of capacity factors can be used to refine the phase selection diagram and to predict a new optimum composition. This iterative procedure can be repeated until the predicted capacity factors no longer differ from the experimental ones.

Such an iterative procedure has been worked out in detail by Drouen et al. [576]. Refinements of the method using the phase selection diagram discussed above include the use of normalized resolution products (see section 4.3.2), shifted compositions and confidence ranges.

223

I , - tlmin 0 10 20

1

lQ%MeOH 25%THF 65%H20

0 10 20 - t/min

Figure 5.31 : Chromatograms run at compositions predicted by the phase selection diagram af figure 5.30 (a) to yield oo-elution of three peaks and (b) to yield optimum separation conditions. Figure taken from ref. [504]. Reprinted with permission.

The use of shifted compositions encourages a good distribution of the experimental data over the parameter space. The optimization procedure directs the search to a certain area in the parameter space (around the predicted optimum), but the use of shifted compositions ensures that the maximum amount of new information is obtained from each next data point. The shift in composition (for a one-parameter optimization problem) can be described by

X’ = x + 2 A { 0.5 - ( x - x ~ ) / ( x ~ - x ~ ) } (5.18)

where x‘ is the shifted composition, x the predicted optimum composition, A is a constant, and x1 and x2 are the locations of the closest data point previously measured, below and above x, respectively. A typical value for A is 0.2 [576], so that

X’ = x + 0.4 (0.5 - (x-x1)/(x2-x1)) . (5.18a)

224

The effects of the shift in composition prescribed by eqn. (5.18a) can be illustrated by considering the initial phase selection diagram, in which two experimentat chromatograms are incorporated, i.e. at x, = O and at x2= I. Hence, eqn.(5.18a) becomes

X’ = x + 0.4 (0.5 - X) = 0 . 6 ~ + 0.2 . (5.18b)

Eqn.(S.l8b) shows that if the predicted optimum is located at one of the two limiting binary mixtures, the experimental verification will be performed at a composition 0.2 x-units away, i.e. at a mixing ratio of 4 1 ( A : B ) if x = 0 (x’ = 0.2), or at a ratio of 1:4 ( A : @ if x= 1 (x’ =0.8). The further the predicted optimum composition is removed from the existing data points, the less the shift in composition will amount to. If x= 0.5, no shift in composition will occur.

Confidence ranges may be defined around each experimental data point by the following equation:

d = Ax/2 - 1/2 v(A2-46/IAI) (5.19)

where d is the confidence range, Ax the distance between two available data points, 6 the allowed uncertainty in In k and IAI the (absolute) curvature of a quadratic equation describing In kas a function of x in the area around the data point. Eqn.(5.19) can be used if linear interpolation between successive data points is used as a model for the variation of retention with cornposition. It describes the difference between a linear interpolation and a quadratic one. A value of 0.025 has been suggested for 6 [576] and in the example of optimization of a ternary mobile phase composition in RPLC, A is usually smaller than 1. If we take IAl= 1 and consider the initial situation, then the size of the confidence intervals that extend above x= 0 and below x = 1 is

d = 1/2 - 112 V0.9 w 0.03.

Hence, in the initial situation the confidence ranges stretch from 0 to about 0.03 and from about 0.97 to 1.

However, when more data points become available, the size of the confidence ranges quickly increases. From eqn.(5.19) we see that the confidence interval will equal Ax/2 when the square root equals zero, i.e. when

Ax2 = 46/IAI. (5.20)

Using the same estimates for Sand IAI as before, we find that Axz0.32. Hence, when IAI equals 1, a total of four data points (x = 0,0.33,0.67 and 1) is suficient to describe the capacity factor within 2.5% (an error in In k of 6= 0.025 corresponds to an error of about 2.5% in k). When more than two data points are available, a better estimate for A may of course be obtained from the data. For instance, when the verification of the first predicted optimum yields exactly the same capacity factors as were predicted, then apparently all A values are equal to zero and the confidence intervals extend over the entire parameter space.

225

A considerable advantage of the iterative procedure is that the accuracy of the predicted retention times is increased at each stage of the procedure. This is not true when a fixed experimental design is used. For example, d'Agostino et al. [579] obtained a precision of 6% between the predicted and experimental retention times using a fixed experimental design (corresponding to figure 5.27b). The figure of 6% gives an indication of the reliability of the result in terms of the predicted optimum conditions. It may be compared with the present iterative procedure if the value of 6 in eqn.(5.19) is set equal to 0.06.

Confidence intervals may be used to define a stop criterion, i.e. they can be used to judge whether the optimization process should be continued or halted. If the predicted optimum falls within one of the existing confidence intervals (calculated for 6= 0.025), then the experimental capacity factors will be within 2.5% of the predicted values. It should be noted that an error of 2.5% in k can make a big difference if the relative retention (a) of a pair of peaks is close to one. It may therefore be required to use a lower value for 6 in eqn.(5.19).

The full procedure is illustrated in figures 5.32, 5.33 and 5.34 for the separation of a mixture of five diphenylamines by RPLC. Figure 5.32 contains the three phase selection diagrams that can be constructed if the initial experiments involve three iso-eluotropic binary mixtures*. The three initial chromatograms needed to construct this figure are shown in figure 5.33 (chromatograms a, b and c). The methanol-water (65135) binary mixture appears at the far left and at the far right of the picture. The two other binary mixtures (THF-water, 40160 and acetonitrile-water, 50150) occur once, on the two vertical axes in the centre of the figure. The top half of the figure represents the (interpolated) linear retention lines for the five solutes. The bottom half represents the response surface, using the normalized resolution product (eqn.4.19) as the criterion.

0 THF- LO - T H F 0 MeOH-

Figure 5.32 : Initial phase selection diagrams for three possible ternary mobile phase systems applied to the separation of five diphenyl amines. Top: (Initial) retention lines. Bottom: (initial) response line. Criterion: normalized resolution product (c eqn.4.19; drawn line) Also shown is the response surface using the product resolution criterion (IIR; eqn.4.18; dashed line). The required chromatograms are shown in figure 5.33 (a, b and c). Figure taken from ref. [576]. Reprinted with permission.

* Methods usedtotimate the correct eluotropic strength (methanol-water ratio) have been described in section 5.4. Methods used to calculate corresponding compositions of other (iso-eluotropic) binary mixtures were discussed in section 3.2.

226

LO THF

O N 50 ACN

19.5 MeOH I

nRs 15.2 @ , ::2:

1 I

0 200 100 6 o O t l s 80rJ

1 26 L THF

17 ACN r R s 190

@ r 072

31 ACN nRs 149

2L1 MeOH 25 2 THF nRs 212 ”; r 025 fi

I I I I I 0 200 LOO 6 0 0 _ t l s 800

Figure 5.33 : Chromatograms obtained during the optimization of the composition of a ternary mobile phase for RPLC for the separation of five substituted diphenyl amines (DPAs). Solutes: (1) N-nitroso-DPA, (2) 4-nitro-DPA, (3) Z,rl‘-dinitro-DPA, (4) DPA and (5) 2-nitro-DPA. Stationary phase: Hypersil ODs. Figure taken from ref. [576]. Reprinted with permission.

In figure 5.32 the optimum composition predicted from a combination of all three phase selection diagrams is a mixture that contains 10.4% methanol and 33.6% THF (x= 0.84). Eqn.(5.18b) then prescribes a shifted composition of 19.5% methanol and 28% THF (x’ = 0.7). Obviously, this composition does not fall within one of the (small) confidence regions, and therefore an experimental chromatogram is recorded at the shifted composition. This chromatogram is shown in figure 5.33 (chromatogram 6).

Although theseparation is better than in any of the three binary mixtures, it is not nearly as good as we expected from figure 5.32. The obvious reason for this is curvature of the retention lines. Therefore, the new data are entered in the phase selection diagram and the iterative procedure is started. The optimum composition is now predicted to be in a completely different part of figure 5.32, namely at 30.6% THF and 11.7% acetonitrile (x= 0.25). According to eqn.(5.18b), this composition is then shifted to 26.4% THF and 17% acetonitrile (x’ = 0.34). The resulting chromatogram is shown as chromatogram e in figure 5.33. Clearly, the separation is now much improved. In fact, it is better than expected from figure 5.32, again due to curvature of the retention lines.

227

From here, the iterative procedure takes two more steps, shown as chromatogramsfand g in figure 5.33. The composition of chromatogram g falls within the confidence range around the composition used to record chromatogram e. Hence, the procedure has advised us to stop before recording chromatogram g, but this final chromatogram has been run to verify the optimum. Chromatogram g is the final result of the procedure, and it does indeed yield a satisfactory distribution of the peaks over the chromatogram.

The actual retention and response lines, constructed using the data obtained from the chromatograms in figure 5.33 and from some additional experiments, are shown in figure 5.34.

-30

t d

-0 c-. !2

0 ACN- 50 -ACN o 0 THF- LO -THF 0 MeOH-

Figure 5.34 : Final phase selection diagrams for the ternary optimization problem illustrated in figures 5.32 and 5.33. Top: retention lines approximated by linear interpolation. Bottom: response lines; dashed line: resolution product ( lTR, eqn.4.18), drawn line: normalized resolution product ( r ; eqn.4.19). Figure taken from ref. [576]. Reprinted with permission.

The non-linearity of the retention lines is apparent from this figure. The response lines have been drawn for two different criteria: the normalized resolution product r (drawn line; eqn.4.19) and the product resolution function IIR, (dashed line; eqn.4.18). It is seen that the product resolution criterion would in fact have guided us to a completely different optimum at a composition of 24.1% methanol and 25.2% THF. The chromatogram that we would have obtained at this composition is shown in figure 5.33h. Clearly, this chromatogram is less attractive than the one of figure 5.338. Obviously, the normalized resolution product is to be preferred to the resolution product itself (see the discussion in section 4.3.2).

Figure 5.34 also carries a warning. As we saw with chromatogram d (the response of which was lower in practice than was expected from figure 5.32) and with chromatogram e (yielding a higher response), the initial predictions of a phase selection diagram should be approached with some care. The same conclusion can be drawn if we compare the phase selection diagram of figure 5.32 with the final diagram (figure 5.34). It is seen that the two figures are markedly different. This is not only true in the two ternary systems which were considered during the optimization procedure (methanol-THF-water and THF-acetoni-

228

trile-water), but also for the third system (acetonitrile-methanol-water). It is seen from figure 5.34 that the actual response line in this figure is even worse than was predicted in figure 5.32. Hence, in retrospect it was quite correct to neglect this system entirely during the optimization process. However, as in the THF-acetonitrile-water system, the reverse might also have been the case. If the actual response line in figure 5.34 had been much higher, instead of much lower than the one predicted from figure 5.32, then it is in principle possible that a mixture in this phase system would have yielded a higher response than the optimum THF-acetonitrile-water composition. In other words, this system might have encompassed the global optimum.

The warning contained in this example is that highly non-linear retention lines may give rise to global optima that remain unrevealed during the course of an iterative optimization process. Hence, unlike the situation in which the curvature coefficient A is equal to zero, a window diagram or a phase selection diagram offers no guarantee that the global optimum can indeed be located. Usually, however, it it quite simple to verify the result of an interative optimization procedure by performing one additional experiment.

Billiet et al. [578] have illustrated the importance of the curvature for the optimization procedure of Drouen et al.. They optimized the pairing-ion concentration for the separation of a synthetic sample containing both anions and cations. By plotting the logarithm of the capacity factor against the logarithm of the concentration of the ion-pairing reagent fairly smooth curves were obtained and the optimization could be completed within a few chromatograms. However, if the reagent concentration is not logarithmically transformed, the curves are extremely non-linear (especially in the low concentration region) and the procedure fails.

The chances that the global optimum will not be found increase when 1. the curvature of the retention lines increases, 2. large areas of the parameter space remain unsearched, and 3. smaller differences exist between the responses at the different (predicted) optima. Therefore, it should be recommended that one or more additional chromatograms are recorded after the completion of the optimization process if 1. a local optimum, only slightly inferior to the global one, is predicted to occur, and 2. large areas of the parameter space remain unsearched, in which severe curvature of the

retention lines cannot be excluded.

There appears to be more reason to record extra chromatograms if the result of the optimization process is not satisfactory. For instance, if the chromatogram of figure 5.338 has been obtained and if it can safely be assumed that there are no more than five solutes present in the sample, then there is no reason to record an additional chromatogram in the middle (x= 0.5) of the large unsearched area corresponding to the acetonitrile-methanol-water system.

Linear interpolation vs. model equations

In figure 5.34 the retention lines have been approximated by a series of linear line segments, rather than by a smooth curve. The alternative is to fit a mathematical equation to the data, for example a quadratic function for In k vs. the mixing ratio x. If more than

229

three data points are available for one ternary system, then the coefficients for the equation can be found from regression analysis. The same argument holds here as was used in the discussion of the Sentinel method. If the largest source of deviation from a mathematical model equation is experimental error, then the use of regression analysis may be beneficial. If it is lack of fit between the model and the experiments, then it may be detrimental. In the absence of experimental error, the linear line segments will give rise to interpolation errors in between data points, but will be correct at those points where experimental data are available. If a mathematical model equation is used, which does provide an exact description of the retention behaviour, then the experimental errors are spread out over the entire parameter space.

In summary, linear interpolation between successive data points should be preferred if the experimental error in the data points is expected to be small relative to the error involved in the description of the data with a mathematical equation.

A mathematical model equation is preferred if an equation is available that yields a quantitatively accurate description of the data within experimental error.

It is clear from the above, that model equations for the description of retention surfaces have to meet high demands. Preferably, equations should be used that relate to reliable chromatographic theory, such as the one used to describe the retention behaviour as a function of pH in RPLC in the window diagram approach described in section 5.5.1. The use of such a chromatographic equation was clearly better in that case than a statistical approach using (for example) polynomial equations.

An optimization procedure that involves the use of mathematical equations to model the retention surfaces within an iterative design has been described by Lankmayr et al. [580,581]. Using statistical or, preferably, chromatographic model equations, it is straightforward to extend an iterative design method to cover more than one parameter. In order to describe the retention behaviour in RPLC using ternary and quaternary mixtures, a two parameter quadratic equation may be used. In this case, there is hardly any difference between a model based on chromatographic theory and a purely mathematical approach. This becomes more obvious if other parameters are considered, such as the combined optimization of mobile phase composition and temperature in RPLC, where eqn.(3.58) or eqn.(3.59) may be used as a (chromatographic) model equation, or the simultaneous optimization of methanol content, pH and ionic strength [562,563] described above.

Naturally, the number of initial experiments required to start the optimization procedure will increase if either the number of parameters considered or the complexity of the model equations increases. As far as the number of parameters is concerned, we have seen this to be true with any optimization procedure, and hence the number of parameters should be carefully selected. In order to avoid a large number of initial experiments, the complexity of the model equations may be increased once more data become available during the course of the procedure. For example, retention in RPLC may be assumed to vary linearly with the mixing ratio of two iso-eluotropic binary mixtures at first. When more experimental data points become available, the model may be expanded to include quadratic terms. However, complex mathematical equations, which bear no relation to chromatographic theory (e.g. higher order polynomials [537,579]) are dangerous, because they may describe a retention surface that is much more complicated than it actually is in practice. In other words, the complexity of the model may be dictated by experimental

230

error rather than by the underlying retention mechanism. In that case all experimental data points will be accurately described by the model, but the interpolation between them may not be correct.

Extension to multi-dimensional optimization problems

Lankmayr and Wegscheider [580,581] have developed a flexible iterative design method which allows the use of a variable number of parameters. The retention surfaces are approximated by means of a modified moving least squares algorithm. The procedure is started with the parameters and their step sizes (e.g. one percent steps in composition) as selected by the user. During the optimization process the number of parameters may be decreased by deleting those that appear to be irrelevant, or increased by adding new parameters. Also, the step sizes can be changed during the optimization process. Unfortunately, this method has so far not been published in the literature.

The linear segmentation method described by Drouen et al. may also be expanded to include two-parameter optimization problems. They described the application of an iterative design method to the optimization of the composition of quaternary mobile phase mixtures in RPLC [502]. However, the division of a two-dimensional parameter space (in this case a triangle, similar to the one shown in section 5.5.1) into segments, and the approximation of the retention surfaces with a series of triangles, is not as straightforward as the use of a series of linear line segments in a one-parameter optimization problem. In order to avoid the occurrence of a series of awkward (i.e. long and narrow) triangles, Drouen et al. established a series of complicated “shift rules” that shift the experimental location from the predicted optimum towards the edge, along the edge or towards the centre of a triangular segment in the parameter space [502].

The problem encountered by Drouen et al. is that the procedure has become so complicated that it requires a large computational effort. Half an hour of computer time is reported to be needed for each computation step [502]. Another half an hour may be needed to plot the available information in the form of diagrams. However, this is not strictly required at intermediate stages of the optimization procedure. If the computer time becomes excessive, the underlying philosophy of the iterative design, as discussed at the beginning of this section, comes under pressure. It may then no longer be argued that the time required for computation is much less than the time required to record another chromatogram. In that case, several other approaches should be considered: 1. a fixed experimental design might be used, such as the one used in the Sentinel method,

2. two-parameter optimization procedures should not be used until the possibilities with

3. the computational procedure should be reconsidered in order to meet the philosophical

to make the first predicted optimum more reliable;

single parameter optimization have been fully exploited;

requirements of the method.

The first approach may always be followed. A drawback is that it might imply that a larger number of experiments will be required, some of which may turn out to be of little value in the end. The required calculations may sill be lengthy, but they only need to be performed once during the entire optimization procedure. A major advantage is that the chances of overlooking the global optimum are greatly reduced.

23 1

The second approach is the one followed by Drouen et al. [502]. It is based on the experience that only in very few cases does the optimization of a quaternary mobile phase composition in RPLC yield an optimum that is truly quaternary, i.e. contains all four solvents. Hence, the procedure discussed before for ternary solvents usually leads to the global optimum. This argument, correct though it may be, only applies to the particular problem of mobile phase optimization in RPLC, and prohibits the application of the same method to other two-parameter optimization problems [582].

The third solution to the problem may be found in the use of more efficient computers, algorithms and computational methods. For instance, if segmentation of the parameter space (linear interpolation) is used, large parts of the retention surfaces and hence of the response surface may remain unaltered when a new data point is added to the existing set. The use of simple model equations instead of linear segmentation may also be more efficient from a computational point of view. However, such simple equations may only be used for the description of the retention behaviour in a limited number of cases and if the model equations become more complex the advantage quickly disappears. For example, d’Agostino et al. used up to sixth order polynomial equations [537] and their procedure also led to excessive calculation times.

Another possibility to reduce the computation time for the location of the optimum is the use of the Simplex algorithm. In section 5.3 we discussed the severe limitations of the Simplex method as a stepwise approach towards the chromatographic optimum. The main problem associated with the method turned out to be that many experiments were required to locate an optimum and that this might be a local one, so that the procedure has to be started repeatedly from different starting points. Hence, the procedure might have to be run for say ten times with an average number of say 40 data points, thus requiring 400 chromatograms to be recorded.

However, if the Simplex algorithm is only used in the computation step to locate the optimum in the response surface with all the retention surfaces being known, then 400 chromatograms may be calculated rather than measured. Although the average number of steps required for each application of the algorithm turned out to be closer to 100 than to 40 in practice, Svoboda I5171 demonstrated that the Simplex apprach may still compare favourably with a grid search approach during the calculation step, especially for multi-parameter optimization problems.

In the long run, it seems likely that improvement in instrumentation will speed up the computation at a faster rate than the chromatography, so that the present discussion will lose more and more of its relevance. Indeed, it has already turned out that the use of more efficient computational procedures may succeed in a very considerable reduction of the calculation time required in the procedure of Drouen et al. [593].

We may summarize the characteristics of iterative design methods as follows: 1 . A minimum number of experiments is required. 2. A good idea of the response surface is obtained, especially in the area around the

optimum. 3. A disadvantage of this last aspect is that large areas may remain unsearched. In some

cases, this could imply that the global optimum is overlooked. 4. Linear interpolation between individual data points should be preferred ifthe experimen-

232

tal errors in the data points are small and no reliable equation to describe the data mathematically is available.

5. The use of model equations should be preferred if the data can be described within experimental error over the entire parameter space.

6. When several parameters need to be optimized simultaneously, the use of simple model equations (ifpossible) seems to have advantages over linear interpolation methods. If the required equations become more complicated, however, this advantage is rapidly losr.

7. Model equations based on chromatographic theory should be preferred to strictly mathematical ones.

8. The individual capacity factors of all solutes are required to calculate the retention surfaces.

5.5.3 Summary

Table 5.6 gives a summary of the interpretive methods described in sections 5.5.1 and 5.5.2. The general characteristics of all interpretive methods are the following: I . The capacity factors of all the individual solutes need to be obtained at each experimental

location. 2. The retention surfaces must be approximated by some kind of model. 3. A generally small number of experiments is required. 4. A good overall impression of the response surface can be obtained. 5. For single-parameter optimization both graphical and mathematical methods may be

used. 6. In principle, all methods can be adapted to include multi-parameter optimization, but

graphical methods are then no longer possible. 7. The number of required experiments and the computation time will increase when the

number of parameters increases. In section 5.7 these characteristics will be compared with those of the other optimization procedures described in this chapter.

5.6 PEAK ASSIGNMENT AND RECOGNITION

By definition, all interpretive methods of optimization require knowledge of the capacity factors of all individual solutes. This is the fundamental difference between the simultaneous and sequential methods of optimization (sections 5.2 and 5.3, respectively) and the interpretive methods of section 5.5. Moreover, in the specific cases in which only a limited number of components is of interest o r in which weighting factors are assigned to the individual solutes (see section 4.6.1)* it is also necessary to recognize the individual peaks (at least the relevant ones) in each chromatogram. In section 5.5 we have tacitly assumed that it would be possible to obtain the retention data (capacity factors) of all the individual solutes at each experimental location.

The problem of recognizing the individual solutes in the chromatograms during the

* An exception to this is when weighting factors are only used to deal with a solvent peak in the chromatogram (see section 4.6.3).

233

optimization process is complicated because of the large amount of overlap that may be expected to occur between the various peaks in the chromatograms. If this overlap would not occur, then there is not much need for optimization procedures. In other words,

Table 5.6 Selection of interpretive methods applied for selectivity optimization in chromatography.

Ref. Author(s) No. No. Design Method (1) par. exp. (1)

501 549 550 552 546 547 554 553

555 557

558 559 560 583

542 572 512 566

562 563 537 517

504 576 578 582 502

580

Laub/Purnell Constanzo Deming/Turoff Price/Deming Jones/ Wellington Noyes Issaq et al. Hsu et al.

Colin et al. Toon/Rowland

Sachok et al. Sachok et al. Weyland et al. Gant et al.

Glajch et al. Antle Weyland et al. Issaq et al.

Otto/ Wegscheider Otto/ Wegscheider d’Agostino et al. Svoboda

Schoenmakers et al. Drouen et al. Billiet et al. Drouen et al. Haddad et al.

Lankmayr et al.

1 1 1 1 1 1 I 1

1 1

2 2 2 2

2 2 2 2

3 3 3 var

1 1 1 2 2

var

2 2 9 4 6 3/4 5 4

2 2

9 9 9 4

7 7 7 10

36+6 9+6 12 var

3-1 0 3-8 3-8 4-10 4-10

var

f f f f f f f f

f f

f f f f

f f f f

f f f f

1

1

1

1

1

1

window diagram window diagram window diagram window diagram window diagram window diagram window diagram window diagram

critical band critical band

full factorial full factorial full factorial full factorial

Simplex lattice Simplex lattice Simplex lattice extended lattice

full factorial limited factorial modified lattice quadratic design

iterative design iterative design iterative design iterative design iterative design

sequential global opt.

(1) f = fixed design (simultaneous interpretive method); i = iterative design

234

optimization procedures are most useful when the task of recognizing the individual peaks is most complicated.

In this section different ways to measure the required data will be discussed.

Table 5.6 (Continued)

Ref. Model Application (compositions refer to mobile phase unless stated otherwise)

501 549 550 552 546 547 554 553

555 557

558 559 560 583

542 572 512 566

562 563 537 517

504 576 578 502 582

580

empirical (linear) empirical (linear) chrom.mode1 (eqn.3.70) chrom.model (eqn.3.70) linear interpol. linear / curved empirical (4th order) chrom.mode1

linear ( to adapted) empirical (linear)

semi-empirical semi-empirical semi-empir.(2nd order) chrom.model (eqn.3.58)

empirical (2nd order) visual comparison semi-empir.(2nd order) empirical

semi empirical semi empirical empirical (to 6th order) empirical (2nd order)

linear interpolation linear interpolation linear interpolation linear interpolation linear interpolation

moving least squares

stationary phase composition GLC mixed pairing ions LC pH opt. RPLC pH opt. RPLC pH opt. RPLC temperature / bin.comp. RPLC ternary comp. RPLC binary comp. RPLC

ternary comp. RPLC binary comp. RPLC

bin.comp. and pairing ion RPLC binary comp. and pH RPLC quaternary comp. RPLC temp. and binary comp. RPLC

quaternary comp. RPLC and LSC quaternary comp. LSC ternary comp. RPLC quaternary comp. RPLC

3 param. RPLC ionic solutes 3 param. RPLC ionic solutes quaternary comp. RPLC/LSC pH/comp./pairing reagent RPLC

ternary comp. RPLC ternary comp. RPLC pairing-ion concentration quaternary comp. RPLC binary comp. and pH RPLC

various

23 5

peak area (counts)

0 , 1 , , 2 , 3 , 4 , 5 K 6 , 1 , 8 , 9 , ~ ,

k-

Figure 5.35: Three simulated chromatograms in which peaks have been assigned the numbers 1 to 5 (corresponding to the elution order in chromatogram a) on the basis of the peak areas. The areas are shown in the chromatograms. For chromatogram (c) see opposite page.

5.6.1 Single channel detection

The most obvious solution to the problem described above is the injection of all solute components separately. Clearly, this allows an accurate determination of the capacity factors, provided that tbe chromatographic conditions (e.g. flow rate, temperature, mobile phase composition) are adequately controlled. However, there are two clear disadvantages of this method: 1. injection of each solute separately greatly increases the number of experiments to be

performed, and

236

I I I I 1 8 1 I I , l a I , I I I , I , I .

0 1 2 3 L 5 6 7 8 9 10 k-

2. in many cases, not all the components in the sample are known, or they are not available in pure form (or solution).

The first disadvantage implies that lengthier optimization procedures are required. Hence, one of the most attractive characteristics of interpretive methods, the small number of experiments required, is sacrificed. However, in fully automated systems little effort is required from the operator.

In contrast, the second objection is fundamental. In most cases it will be impossible to inject all sample components separately and hence an alternative method will have to be found if interpretive methods are to be used.

The problem is illustrated in figure 5.35. In this figure three simulated chromatograms are shown, which we will presume have been recorded under different conditions. A chromatogram contains information in two directions. So far, we have almost exclusively made use of the information in one direction: the retention times (capacity factors) of the solutes. The information in the other direction, peak height or peak area, may be used to assist in the assignment of peaks. In figure 5.35a a chromatogram is shown that contains five peaks. The peaks have been numbered 1 to 5 (circled numbers) and the area of each peak is indicated in the figure. Figure 5.35b shows another chromatogram, also containing five peaks. Apparently, however, the elution order has changed in going from chromatogram a to chromatogram b. It is seen in the figure that the peaks may be assigned the numbers 1 to 5 in a different order on the basis of the peak areas.

In figure 5 . 3 5 ~ the problem has become more difficult, because now only three peaks occur in the chromatogram instead of five. On the basis of the areas we may conclude that the first peak consists of solutes 2 and 3, the small second peak of solute 1 and the third peak of solutes 4 and 5.

For several reasons figure 5.35 represents a favourable example. In the first place, all solutes show markedly different areas. The difference in area between each two solutes is at least 20°/0. In the second place, we have assumed that there are no more than five solutes present in chromatogram a, an assumption which has not been proved wrong (but neither

237

has it been proved right) in the following two chromatograms. It would clearly have been wrong’if the chromatograms had been obtained in the reverse order and if we had concluded from figure 5.35~ that only three solutes were present in the sample. The problem would have been more complicated if none of the chromatograms had shown as many peaks as there are solutes present in the sample.

There are other major problems with peak assignment on the basis of the areas. These problems relate to the reproducibility of peak area measurements under widely varying conditions. Ideally, the area of a peak remains constant even if its capacity factor varies. However, varying the conditions may affect the peak areas. If the column temperature is changed in GC, then the flow rate may be affected. Peak areas will change (by a constant factor) if concentration-sensitive detectors such as the hot wire detector (HWD; katharo- meter) are used, but not with mass flow sensitive detectors (such as the flame ionization detector, FID).

When the mobile phase composition is changed in LC, the sensitivity of the detector to different solutes may be altered. For example, the UV spectra of solutes may shift upon a change in the composition of the mobile phase [584]. Especially if the detection wavelength is on the flank of an absorption band, this may easily lead to variations in the peak area that exceed the 20% difference which we used to discriminate between the different solutes in figure 5.35.

An additional problem is the measurement of peak areas itself. The integration will give rise to errors, especially if the peaks are not completely resolved and if the baseline varies during the analysis.

Hence, we conclude that peak area measurements may be of some help in the assignment of peaks, but there are a series of major limitations, so that only in some very favourable (simple) cases will a satisfactory result be obtained. Of course, in such simple cases there may not be much reason to optimize the separation.

Issaq and McNitt [585] published a computer program for peak recognition on the basis of peak areas. They investigated the reproducibility of the area of some well-separated peaks for three solutes (anthraquinone, methyl anthraquinone and ethyl anthraquinone) in the 10 solvents used for their optimization procedure. The solvents inctuded binary, ternary and quaternary mixtures of water with methanol, acetonitrile and THF. The areas were found to be reproducible within about 2 percent. The wavelength used for the UV detector in this study was not reported.

One final comment may be made regarding figure 5.35. This figure forms a good illustration of the problem of peak assignment as it occurs during optimization procedures. We tried to assign numbers to peaks, once arbitrary numbers had been assigned in one chromatogram (figure 5.35a). During this process, none of the peaks was identified. For the general problem of selectivity optimization, this is quite sufficient. Only in specific cases, where a limited number of sample components is of interest, are we required to recognize (but not to identify) the relevant peaks in the chromatogram. To do this, standard solutions containing one (or more) of the components of interest are required in the case of single channel detection. If multichannel detectors are used (section 5.6.3), the components of interest may also be recognized on the basis of information (e.g. spectra) obtained in independent experiments.

238

5.6.2 Dualchannel detection

The assignment or recognition of peaks in different chromatograms may be aided if extra qualitative information about the solutes is obtained from the detector. The simplest way to obtain more information is to combine two detectors in series. Many combinations are possible, but some limitations arise (see for example ref. [586]):

The time delay between the two detectors should be minimal. The first detector should be non-destructive and should not contribute significantly to the band-broadening. The two detectors should have similar types of sensitivities (e.g. two concentration sensitive detectors or two mass flow sensitive detectors). Ideally, the degrees of sensitivity of the two detectors should also be comparable.

Because of this last reason, it is not attractive to combine a hot wire detector with a flame ionization detector in GC (a combination that also conflicts with the third limitation above) or a differential refractometer with a UV spectrometer in LC.

For all the above reasons, it is to be preferred to measure two solute properties in one detector, especially if both measurements can be performed simultaneously. An example of this is the application of dual-wavelength absorption detection in LC. The application of this technique for the purpose of selectivity optimization has been investigated by Drouen et af. [584]. For the purpose of peak assignment or recognition, ratio recording may be used. The principle of this technique is based on Beer’s law and may be explained from the following equation for the absorption ratio R,:

(5.21)

In this equation, A , is the absorbance at one wavelength (A,) and A , the absorbance at the second wavelength (A,). a, and a2 are the respective molar absorptivities (extinction coefficients) at the two wavelengths, b is the optical path length and cis the concentration of the solute in the detector cell. It can be seen from eq~(5.21) that the absorbance ratio R , is independent of the concentration and hence that R , is a constant throughout the elution of a single peak. The value of R, is determined by the ratio of the two absorptivities at the different wavelengths and therefore it is characteristic of the solute. In practice, it is necessary to introduce a threshold for both absorbances (because R, is not defined on the baseline), whereas it is attractive to use slightly different definitions for the absorbance ratios, for example [584]:

R A T = A , / A , for A , > A , > A (5.22)

RAT = 2- A , / A , for A, > A , > A (5.23)

R A T = 0 for A , < A n A , < A (5.24)

R A T = -0.1 for A , < A fl A , > A (5.25)

R A T = -0.1 for A , > A n A , < A (5.26)

239

In these equations RAT is the modified absorbance ratio and A is the threshold absorbance. In fact, it would be beneficial to differentiate between the last two cases. This may easily be realized by defining RAT= - 0.2 instead of - 0.1 in eqn.(5.26), i.e.

RAT = -0.2 for A , > A n A,<A (5.26a)

An example where the above definitions are used is shown in figure 5.36. In this figure, three chromatograms are shown, each of them recorded at two different wavelengths (signals A, and A*). The corresponding “ratiograms” are shown above the chromatograms. Figure 5.36a shows the typical block shaped signals that are obtained for pure peaks. Each of the six peaks yields a specific vaIue for RATand, fortunately, the values are all different. This allows a rapid assignment of all peaks in figure 5.36b, where again six peaks are observed in the chromatogram. The peak sequence is indicated by the letters a to f in the chromatograms.

The situation is much more complicated in figure 5.36c, in which only four peaks are observed. On the basis of the RAT values and peak areas, peaks a and d can readily be assigned in this chromatogram. However, the second peak in figure 5.36~ consists of two partially overlapping peaks (indicated by the deformed shape of the block) and the fourth peak contains two completely overlapping solutes (indicated by the perfect block shape, in combination with a not previously observed RAT value). In this particular nasty (but not necessarily uncommon) example, the problem is almost impossible to solve using RAT

C

Figure 5.36: Three chromatograms recorded at two different wavelengths, accompanied by the ratiograms (top of the figure). For explanation see text. Figure taken from ref. [584]. Reprinted with permission.

240

values alone. However, the ratiogram reveals the presence of two partially overlapping peaks in figure 5.36c, which is not revealed in the chromatogram itself. On an only slightly better column, the two solutes in the second peak in figure 5.36~ might have been sufficiently resolved to allow for a straightforward assignment of all the peaks.

Figure 5.36 illustrates the potential of ratio-recording (chromatogram b) as well as its shortcomings (chromatogram c) as an aid in the assignment of chromatographic peaks for the purpose of selectivity optimization. Ratio recording may help to assign peaks that are well resolved, or at least sufficiently resolved to allow an estimate to be made of the separate RAT values.

According to Drouen e t d [584], this implies that two Gaussian peaks should be more than one standard deviation apart ( R , = 0.25). For such a pair of peaks only one peak is observed in the chromatogram, since a valley is only observed if the R , value exceeds 0.59 (eqn.4.11). In order to make a reliable estimate of the peak area, a much larger resolution (R, > 1, or preferably R , > 1.5) is required.

Hence, the use of dual-channel detection increases the possibilities for peak assignment in optimization processes. However, complications are caused by variations in the baseline [584] and the effect of solvent composition on the UV spectrum may now become even more serious than in the case of single-channel detection. This may occur for instance if one wavelength is selected on a rising flank in the UV spectrum, and the second one on a descending flank. In any case, the selection of the two most suitable wavelengths is one of the most critical factors. Unfortunately, the wavelength selection is usually quite arbitrary.

5.6.3 Multichannel detection

The logical next step to consider is the application of multichannel detection. The combination of more than two detectors in series is unattractive, because of cost considerations and because of increasing band broadening effects. One possibility is the combination of several different detection principles into one detector, which has recently been demonstrated by Cant and Perrone [587], who described a three-channel LC detector that allows simultaneous monitoring of UV absorbance, fluorescence and conductivity. However, both of these last two detection principles are fairly specific (i.e. they will be aplicable to only a limited number of solutes) and hence the true three-channel capability may only be available in some rare cases. True multichannel detection is obtained by the combination of the chromatographic separation with a spectroscopic identification technique. The most successful application of such a “hyphenated method is the use of a mass spectrometer in the GC-MS combination. The mass spectrometer yields (almost) universal detection, very high sensitivity and a large amount of qualitative (spectral) information.

Unfortunately, the LC-MS combination is less successful. In part, this may be due to technological interfacing problems, but even if these are solved, LC-MS is unlikely to provide the same degree of universality (large molecules will remain a problem), spectral information and reproducibility as the GC-MS combination. For the moment, the combination of LC with a multichannel UV absorption detector is a more realistic proposition.

Both in GC-MS and in LC-UV true multichannel detection may be obtained, and

241

200 LOO Wavelengthlnm

(b)

0 100 200 300 timels

Figure 5.37: An example of a spectro-chromatogram recorded with a multichannel UV absorbance detector in LC. The sample contains a series of dipeptides. (a) (top): pseudo-isomeric three-dimensional plot; dimensions are time, wavelength and absorption. (b) (bottom): contour plot with constant absorption lines. Figure taken from ref. 15881. Reprinted with permission.

three-dimensional so-called “spectro-chromatograms” may be recorded. An example of such a three-dimensional figure is shown in figure 5.37, both as a pseudo-isomeric and as a contour plot. As was the case for the representation of response surfaces (figure 5.2), the latter are to be preferred if an objective interpretation of the data from the figure is required.

The problem of peak recognition may be seen as a simplified version of the problem of peak identification. The spectra found during the elution of a chromatogram can be subjected to automated retrieval systems [589]. In this case the peak may be tentatively identified by comparing its spectrum to a (potentially large) number of reference spectra,

242

contained in a “library”. The spectra which resemble the spectrum of the peak may be listed in order of decreasing probability. Although many of such library search techniques work well for GC-MS in particular, there are some limitations. In the first place the peaks are identified on the basis of probabilities, which should by no means be mistaken for certainties. Also, only those components which are represented in the library by a reference spectrum are considered, which limits the search to a finite number of possibilities. Moreover, such retrieval systems may be too large a tool for the purpose of peak recognition only.

One possibility is to build a miniature library from the spectra observed during the chromatograms in an optimization process, and then to assign peak numbers on the basis of previously stored spectra. Such a library search method requires that representative spectra of all solutes are obtained at some stage of the optimization process (i.e. ideally each component should appear as a single, well-resolved peak in at least one of the chromatograms [590]). Moreover, the method needs to be very flexible in the sense that sum spectra may occur once two components start to overlap, but also new spectra may be found when components start to be resolved. MS spectra often yield sufficient information to allow peaks to be recognized on the basis of their spectra alone, given that only a very small library is created during the optimization process. However, there are cases (e.g. geometrical isomers) in which different solutes may show very similar MS spectra.

In general retrieval systems, additional information such as retention data may be required for a conclusive peak recognition using UV spectra [590]. For obvious reasons, however, retention data cannot be used as an aid for peak recognition for optimization purposes. Related solutes, such as isomers and homologues, may show similar spectra, which are hard to differentiate. In case of doubt, one may have to refer to peak areas for additional information. Again (see section 4.6.1), this requires very good resolution of the individual peaks.

Clearly, what is required for a reliable recognition of all the peaks during the optimization procedure is information on the pure component spectra and the pure component peaks (elution profiles). A method to obtain both the spectral and the chromatographic data involves the application of a mathematical technique called “principal component analysis” (PCA) [592]. This method is based on the additivity of spectra according to Beer’s law. The absorption (A) at a time t and wavelength A is given by

n

j = 1 ,qt,n) = I: a p ) b q t ) (5.27)

where u,(A) is the molar absorptivity of componentj at the wavelength A, b the optical path length and c,(t) the concentration of j at time t in the detector cell. n is the total number of solutes present in the sample.

Eqn.(5.27) shows that the total absorption is the result of the contributions of a series of factors which depend either exclusively on the wavelength (aj factors = spectra) or on the time (cj factors = elution profiles). The individual factors may be obtained with PCA, but an unambiguous solution for the mathematical problem may only be obtained in a small part of the chromatogram (“peak cluster”), in which three or fewer components contribute to the absorption [592].

243

However, the spectra as well as the elution profiles may be very similar (i.e. closely related, ill-resolved solutes). Figure 5.38 shows an example of a result of PCA applied to a cluster of three peaks using LC-UV detection. The problem involves the separation of three proteins with very similar UV spectra. The chromatogram obtained is shown in figure

D,

c 0 f 5: 0

3 300 A/nm LUJ 250 300 200 250 0

tlmin

Figure 5.38: An example of the application of principal component analysis to obtain the individual spectra and elution profiles of ill-resolved proteins. (a): Illustration of the chromatogram obtained at a low wavelength (e.g. 200 nm); (b), (c) and (d): Spectra of the three pure components identified in the peak cluster by PCA (drawn lines) and true pure component spectra (dashed lines); (e): pure component elution profiles from PCA (drawn lines) and estimated pure component profiles (dashed lines). Figure adapted from ref. [592]. Reprinted with permission.

244

5.38a. The three spectra of the individual solutes as obtained by PCA are shown in figures 5.38b to d. The pure component spectra are indicated in these figures by dashed lines. Figure 5.38e shows the individual elution profiles obtained from PCA, as well as estimated pure component elution profiles (obtained from separate injections; dashed lines). Clearly, the potential of the PCA method to obtain both retention and spectral information for ill-resolved peaks is impressive.

The use of multichannel analysis for the recognition and identification of individual peaks in a chromatogram is a very rapidly developing area and it may be anticipated that complex mathematical techniques (such as PCA) will soon become available as a standard tool for the chromatographer.

A remaining problem for all spectrometric peak recognition methods is the reproducibility of the spectra recorded under different chromatographic conditions. For example, if the differences between the UV spectra for a given solute induced by variations in the mobile phase in RPLC are larger than the differences between the UV spectra of different solutes recorded under identical conditions, then clearly the application of multichannel UV detection, with or without the use of PCA techniques, will be of limited use.

5.7 SUMMARY

In the last section of this chapter we will summarize the different optimization

Table 5.7a: Summary of optimization procedures

Method: Univariate optimization

Includes: -

Section of this book Path in Figure 5.4

Optimum found Accuracy of optimum Number of experiments Criteria to be used Impression of response surface Knowledge required

Multi-dimensional optimization Computational requirements Complete automation Solute recognition

: 5.1.1 : 2111

: Local (1) : Low(1) : Fairly high (1,2) : Any : Very limited : Selection of parameters

: Straightforward : Low : Possible (2) : Not required

(1) Repeated initiation required. (2) Dependent on procedure used in one dimension.

245

procedures that have been described in the preceding sections. An overview of the methods is given in table 5.7.

All different optimization methods discussed in this chapter have been classified in five different parts of this table. The characteristics of the different classes of methods can be compared using the tables.

Univariate optimization (table 5.7a) is not a good method for the optimization of chromatographic selectivity. This will be clear from the table, since despite a fairly high number of experiments only a local optimum will be located on the kinds of response surfaces typically encountered in chromatography (see figure 5.3). Moreover, the local optimum may be of little value, because no overall impression of the response surface is obtained during the process. Once this has been established, the other (favourable) characteristics of this method are no longer relevant.

Table 5.7b: Summary of optimization procedures

Method: Simultaneous methods without solute recognition

Includes: Grid search




: 5.2 : 1011,1012 (1)

: Global (2) : Moderate (High with (1)) : Very high : Any (3) : Very good : Selection of parameters

: Impractical above 2 parameters : Low : Straightforward : Not required

(1) Initial coarse grid followed by finer one around the predicted optimum. (2) For complex samples the global optimum may not be located. (3) Smooth response surfaces are to be preferred.

Simultaneous optimization methods (table 5.7b) do provide a means of establishing the global optimum. However, a large number of experiments is required, especially if we wish to locate the optimum with good accuracy, using a fine grid or an initial coarse grid followed by a finer one in the area of the optimum. If this latter strategy is employed, or if the response surface is very complicated (complex samples; inappropriate criteria), the

246

global optimum may not be found. However, because of the very good impression that is obtained of the overall response surface, it is unlikely that a local optimum, if this results from the procedure, will be much worse than the global one.

Therefore, the large number of experiments required, which becomes excessive for the simultaneous optimization of more than two parameters, is the main disadvantage of this method.

Table 5 .7~: Summary of optimization procedures

Method: Simplex

Includes: Simplex, Modified Simplex, Other statistical search methods


Optimum found Accuracy of optimum Number of experiments Criteria to be used Impression of response surface Knowledge required Multi-dimensional optimization Computational requirements Complete automation Solute recognition

: 5.3 : 1012,2012

: Local (1) : High : High : Single-value criteria : Poor : Selection of parameters : Straightforward : Fairly low : Possible : Not required

(1) Global optimum may be found after repeated initiation, requiring large numbers of experiments.

The Simplex method (and related sequential search techniques) suffers mainly from the fact that a local optimum will be found. This will especially be the case if complex samples are considered. Simplex methods require a large number of experiments (say 25). If the global optimum needs to be found, then the procedure needs to be repeated a number of times, and the total number of experiments increases proportionally. A local optimum resulting from a Simplex optimization procedure may be entirely unacceptable, because only a poor impression of the response surface is obtained.

On the other hand, the practical characteristics of the Simplex method show that its application is usually staightforward (even for multi-parameter optimizations) and requires little knowledge or computational effort. This explains the popularity of the Simplex methods for the optimization of chromatographic selectivity, despite its obvious fundamental shortcomings.

247

Table 5.7d: Summary of optimization procedures

Method: Simultaneous interpretive methods

Includes: Window diagrams (one or more dimensions) Critical band method Sentinel method




5.5.1 4021,2121,3221

Global Low (1) Low Any (2) Moderate to good (1) Selection of parameters and model

Possible (3) Variable (4) Complicated (5) Required

(1) Dependent on accuracy of the model. (2) Criteria based on R, or S are to be preferred over criteria based on P (see section 4.2.4). (3) Possible for computational procedures, not possible for graphical ones. (4) Low for graphical procedures to high for multi-dimensional computational procedures. (5) Complete automation is complicated because of requirement for solute recognition.

Simultaneous interpretive methods (table 5.7d) provide a way to locate the global optimum from a relatively low number of experiments. The price that should be paid for this very important advantage of these methods is an increased effort from the chromatographer to provide knowledge (to model the retention surfaces), increased computational requirements and the necessity to recognize all the individual solutes in each chromatogram. The reliability of the final result will depend on the accuracy of the model.

Table 5.7d suggests that simultaneous interpretive methods are highly promising for selectivity optimization, but that there is still much room for improvement if research effort is directed at 1. the formulation of models based on sound chromatographic theory, 2. improvement of computer techniques to locate the global optimum in a multi-dimen-

3. automatic procedures to recognize the individual solutes in each chromatogram. sional parameter space,

248

Table 5.7e: Summary of optimization procedures

Method: Iterative designs

Includes: Phase selection diagrams




5.5.2 4022,2122,3222

: Global (1) : High : Low : Any : Moderate : Selection of parameters and model (3)

: Possible (4) : Variable (5) : Complicated (6) : Required

(1) Global optimum may be overlooked if large areas remain unsearched. (2) Criteria based on R,7 or S are to be preferred over criteria based on P (see section 4.2.4). (3) No model required if linear interpolation is used. (4) Possible for computational procedures, not possible for graphical ones. (5) Low for graphical procedures to high for multi-dimensional computational procedures. Reduc-

(6) Complete automation is complicated because of requirement for solute recognition. tion of the computation time appears to be possible (see section 5.5.2).

Iterative designs (table 5.7e) have two main advantages over the simultaneous interpretive methods described above. 1. The optimum can be located with a high degree of accuracy (defined by the user). 2. The accuracy of the model used to describe the retention surfaces is not the limiting

However, there is a risk that the global optimum will not be found if large areas remain unsearched. Therefore, a combination of the two different interpretive methods, a fixed experimental design followed by an iterative procedure to refine the location of the optimum, may be the best possible approach, even though a slightly larger number of experiments may be required than for either of the two methods separately.

factor.

Conclusions

Very briefly, the conclusions of this chapter can be summarized as follows: 1. Simultaneous methods (without solute recognition) may be used for selectivity optimiza-

tion in chromatography, but require large numbers of experiments.

249

2. Simplex optimization also requires many experiments, especially if the global optimum is sought. This is a relatively straightforward method to apply in practice.

3. Simultaneous interpretive methods are a good way to locate the global optimum in a small number of experiments, but the requirements are shifted towards good models, computers, and peak recognition methods.

4. Iterative interpretive methods allow a more accurate location of the optimum and do not rely on the accuracy of models. However, the global optimum may not always be found.

5. An interpretive method, which combines an initial fuced experimental design with an iterative refinement of the optimum, appears to be the most promising approach.

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252

CHAPTER 6

PROGRAMMED ANALYSIS Programmed analysis can be defined as a chromatographic elution during which the

operation conditions are varied. The parameters that may be varied during the analysis include temperature, mobile phase composition and flow rate.

In many respects programmed analysis does not differ from chromatography under constant conditions. Retention is still determined by the distribution of solute molecules over the two chromatographic phases and the selectivity of the system is still determined by differences between the distribution coefficients of the solutes. However, if the operation conditions are changed during the elution, then the distribution coefficients may change with time, thus affecting both retention and selectivity.

In this chapter we will take a look at some aspects of programmed analysis, particularly those which bear relation to the chromatographic selectivity. The parameters involved in the optimization of programmed analysis will be divided into primary or program parameters and secondary or selectivity parameters. These parameters will be identified for different chromatographic techniques and procedures will be discussed for the optimization of both kinds of parameters.

6.1 THE APPLICATION OF PROGRAMMED ANALYSIS

The general elution problem

In chapter 1 (section 1.6) we have seen that only a limited number of sample components can be eluted with optimum capacity factors in a chromatogram (see eqn.l.25). Real-life samples often confront us with the problem that some of the components are bunched together (and ill-resolved) early in the chromatogram, while some other components are eluted in the optimum range of k values (see figure 6.la). If we change the conditions so as to increase the capacity factors of the early eluting components, then the later eluting ones will tend to give rise to impractically high k values. This so-called “general elution problem” (see ref. [601], pp.54-55) is illustrated in figure 6.1 (chromatograms a and 6).

The idea of programmed analysis is to vary the operating conditions during the analysis, so that all components of the sample may be eluted under optimum conditions. Such an ideal situation is illustrated in chromatogram c of figure 6.1. Although such an ideal situation may not always be realized, figure 6.lc provides a good illustration of the aim of programmed analysis.

The analysis program may be defined as the function that describes the variation of the operating conditions (or elution parameters) with time. Most often, only one parameter is varied during the analysis. Many different programs may be used. The simplest program is a single step (figure 6.2a) in which the parameter x changes instantaneously at a certain time t. Other possible elution programs are illustrated in figure 6.2.

253

1

, 0 50 100

k-

Figure 6.1: Illustration of the general elution problem in chromatography. Chromatograms a and b: constant elution conditions. Chromatogram c (opposite page): programmed analysis.

When to apply programmed analysis?

In general, programmed analysis may be applied to samples that give rise to the general elution problem, for example samples with a wide volatility (boiling point) range in GC or samples with a wide polarity range in LC. The different ways in which programmed analysis can be applied are summarized in figure 6.3.

The first field of application involves the use of programmed analysis as a scanning or scouting technique for unknown samples. In this case the (volatility or polarity) range of

254

1 2 3 d 5 6 7 8 (C)

I

10 tlmin 0

the sample is not necessarily large, but the sample components may fall anywhere in a large range. This application of programmed analysis has been discussed extensively in section 5.4.

I t -

t -

t-

t -

t-

Figure 6.2: Different shapes of elution programs in chromatography. Description of programs: (a) step; (b) linear; (c) convex; (d) concave; (e) multisegment.

255

The second field of application involves the occasional analysis of wide range samples. In this category we find samples which only occur in the laboratory occasionally and in small numbers, so that only a small number of chromatographic analyses have to be performed. For samples of this kind it is usually sufficient to realize a separation and it is not rewarding to try and optimize the selectivity, not even if the analysis time is rather long and the required number of plates high..

The third field of application in figure 6.3 concerns a routine situation, in which a large number of similar samples needs to be analyzed. It is in this field that it is usually worthwhile to optimize the program.

The use of programmed analysis in a routine situation is not attractive. The application of programmed analysis 1. requires more complicated and therefore more vulnerable equipment, 2. leads to reduced analytical reproducibility, 3. leads to increased detection limits because of variations in the baseline*), and 4. will add to the analysis time because of the time required to return to the starting

conditions.

Figure 6.3: Schematic illustration of the fields of application of programmed analysis in chromatography.

* Detection limits under programmed conditions compare unfavourably to those obtained with isocratic elution, provided that optimum k values can be obtained in the latter case.

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Hence, ironically, the best possible result of the optimization of a programmed analysis is a non-programmed one, i.e. a set of conditions where an optimum separation (or at least optimum elution of all components) can be achieved without the need to change parameters during the analysis.

Multicolumn analysis

One way to avoid the need for programmed analysis in a routine situation is to use of “multicolumn” or ”column-switching” methods. In these techniques more than one column is used to realize optimum capacity factors (and optimum separation) for all sample components. For example, if we look at the chromatogram of figure 6.1 a, we may use a short column to separate the later eluting components, but a column with a higher phase ratio ( VJ V,J is required to separate the early eluting components. Also, columns with different stationary phases may be used, as long as the columns are all compatible with the mobile phase. If different stationary phases are used, then the selectivity may be optimized using the fixed experimental designs described in section 5.5.1.

Multicolumn analysis requires careful optimization. However, the effects of column length, phase ratio and particle size are all predictable, so that the separation that will be achieved on a multicolumn system can be predicted almost exactly. A different set of columns is usually required for every different analytical problem.

The effort needed to develop and optimize a multicolumn method will become the more justified the larger the number of analyses that needs to be performed.

More information on theoretical [602] and practical [603] aspects of multicolumn techniques in GC can be found in the literature. Ref. [604] contains a review of column-switching methods in LC.

6.2 PARAMETERS AFFECTING SELECTIVITY IN PROGRAMMED ANALYSIS

The effects of changes in a parameter during a programmed elution will generally be the product of two independent factors: 1. the relationship between the parameter that is being programmed and the retention

under non-programmed conditions, and 2. the variation of the parameter as a function of time.

The relationships referred to in the first factor have been discussed extensively in chapter 3. Two important examples are the variation of retention with temperature in GC and with mobile phase composition in LC. If we use programmed analysis to separate wide range samples, then the parameters which are varied during the elution should have a large effect on retention. Hence, the most relevant parameters to be considered for programmed analysis are the primary parameters, which have been listed in table 3.10 for the various chromatographic techniques. Table 6.1 summarizes programmed analysis techniques for various forms of chromatography.

An important characteristic of primary optimization parameters is that whereas they have a large effect on the capacity factors of the solutes, they have a relatively minor effect on the selectivity (a). This implies that the factors involved in optimizing an analysis program (the initial and final conditions, programming rate and shape of the program) do not affect the chromatographic selectivity to a large extent. This will be even more true

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for parameters which have no effect whatsoever on the selectivity under non-programmed conditions. Hence, techniques which involve the programming of such parameters (e.g.

. * flow rate programming) will not be discussed in this book.

Table 6.1: Programmed analysis methods for various forms of chromatography.

Method Primary parameters (1)

Programmed analysis

GC Temperature Temperature programming

RPLC Mobile phase polarity Solvent programming (gradient elution)

PH pH gradients

LSC

I EC

I PC

SFC

Eluotropic strength Solvent programming

Ionic strength PH

Salt gradients pH gradients

Various (2)

Mobile phase density

Mobile phase composition Solvent programming

Density programming; pressure programming

~ _ _ ~ ~~ ~

(1) See table 3.10. (2) Not compatible with programmed analysis owing to slow equilibration.

The second factor that determines the effects of programmed analysis, the variation of the elution parameter(s) with time, is usually referred to as the program, for example a temperature program in GC. In LC, the program is often referred to as a gradient. However, we will see below that a programmed analysis in LC involves more than just a gradient and therefore it is better to speak of a program or a gradient program.

6.2.1 Temperature programming in GC

We have seen in section 3.1 that the primary parameter in both GLC and GSC is the temperature. We have also seen that retention in GC varies very strongly with the temperature. The following equation was found to describe the relationship in quantitative terms:

I n k = I n T + A / T + B , (3.10)

where k is the capacity factor under isothermal conditions at an absolute temperature T and A and B are constants.

258

Figure 6.4 shows a schematic example of the variation of retention with temperature in GC for a number of solutes, which could, for example, form part of a homologous series.

The vertical lines a and b correspond to temperatures at which chromatograms would be obtained which are similar to the chromatograms a and b in figure 6.1. Hence, we are confronted with the “general elution problem”.

This is further illustrated by the two (almost) horizontal lines, which enclose the optimum elution range (1 < k< 10). Apparently, there is no single temperature at which all components can be eluted from the column under optimal conditions.

Figure 6.5a shows a typical temperature program for GC. The relevant parameters of the program are also explained in this figure. Temperature programs in GC are almost exclusively linear programs, i.e. during the actual heating step in the program the temperature varies linearly with time. Occasionally a program may be comprised of several linear segments.

Figure 6.5b shows the typical variation of the baseline with time during a programmed temperature run according to the program of figure 6.5a. The two main sources of baseline drift in programmed temperature GC are increased bleeding of the stationary phase at elevated temperatures and variations in the gas flow rate. The use of two identical columns and two detectors in a parallel configuration (baseline subtraction), of accurate flow controllers and, especially, the use of stable stationary phases are factors which may be used to reduce the blank signal.

Harris and Habgood, in their standard work on programmed temperature GC [605] have shown that the retention time of a component under programmed temperature conditions is a function of the retention behaviour of the solute under isothermal conditions and the programming rate. The latter they defined as the heating rate ( r T :

3 1 0 ~ 1 ~ -

a b

Figure 6.4: Schematic example of the variation of retention with temperature in gas chromatography. Retention lines are drawn for a group of 8 solutes (e.g. homologues). Vertical dashed lines (a and b) correspond to chromatograms (a and b) in figure 6.1. “Horizontal” dashed lines indicate the range of optimum capacity factors.

259

0 ti t- 4

start ( inject I

,I,, I ‘b’ - t-

Figure 6.5: (a) Schematic illustration of a temperature program for gas chromatography. The relevant parameters of the program and the units in which they are typically expressed are as follows: Ti = initial temperature (“C); 7’’ = final temperature (“C); rT = heating rate (OC/min); ti = initial time (min); 9 = final time (min). (b) Typical variation of the baseline as a function of time in programmed temperature GC.

“C/min) divided by the flow rate ( F ; ml/min). Because of this, it may be hard to reproduce retention data in temperature programmed G C exactly, because whereas it may be possible to accurately control the heating rate, it may be more difficult to reproduce the flow rate F within 0.5%.

Resolution in programmed temperature GC is enhanced if the programming rate ( r T / F ) is decreased and if the initial temperature (Ti) is decreased. Giddings [606] suggested that the first peak in a programmed analysis should not appear within about five times the hold-up volume of the column. Since the temperature has little effect on the selectivity in GC (see section 3.1.1), the optimization of temperature programs is a process that may be seen as resolution optimization rather than as selectivity optimization.

6-22 Gradient elution in LC

We have seen in chapter 3 (table 3.10 b-d) that the composition of the mobile phase is a primary parameter in various forms of LC (LLC, RPLG, LSC). Gradient elution is only relevant for the latter two techniques, because the LLC system is not compatible with mobile phase gradients. Figure 6.6a shows a typical gradient program for LC. The complete program can be divided into a number of segments.

The program starts and ends at the purge segment (P). The reason for this is related to the typical baseline observed in a gradient elution LC experiment (figure 6.6b). Unlike the situation in GC, the main cause of the blank signal in programmed solvent LC* is formed

* By analogy with the term “programmed temperature GC” [605] we will use the term “programmed solvent LC”, although “solvent programmed LC” is also commonly used.

260

t Ip

# t t - f/

I

I

t-

Figure 6.6: (a) Schematic illustration of a solvent program (or gradient program) for LC. (p = mobile phase composition: P = purge: R = reverse: E = equilibrate; 1 = inject; G = gradient. (b) Typical variation of the baseline as a function of time in programmed solvent LC.

by impurities in the mobile phase, especially in the weaker solvent. Because of the high capacity factors in this solvent, impurities tend to be concentrated at the top of the column when the weak solvent is run through the column in the equilibrate segment (E). If a gradient is subsequently applied, then the impurities will be washed from the column and appear as peaks in the chromatogram. In order to minimize the background signal, the equilibrate segment (E) should be kept as short as possible.

A second factor that contributes to the baseline variation is the difference in the background signal (absorption; fluorescence) between the two solvents. This effect causes the difference in the baseline level between the left and the centre in figure 6.6b. A more extensive discussion on baseline variations in programmed solvent LC can be found in ref. [607].

The actual gradient is denoted by G in figure 6.6a. Because large instantaneous variations in the composition may reduce both the reproducibility of the analysis and the lifetime of the column, a reverse segment (R) is also necessary in a gradient program. A reproducible blank signal can only be obtained if the duration of the reverse, equilibrate and gradient segments, as well as the time of injection (I) and the flow rate, are accurately controlled. The duration of the purge segment is not relevant in this respect. Therefore, it is to be recommended that a solvent program in LC is built up from a minimum of four segments, starting and ending at the purge level.

In RPLC retention varies exponentially with the composition of the mobile phase, i.e. approximately straight lines are obtained in a plot of In k vs. cp (see section 3.2.2). If we look at the retention behaviour of each individual solute, then the optimum conditions (LSS gradient, see section 5.4) correspond to a linear gradient (figure 6.2b). Linear gradients will indeed be optimal when acetonitrile-water mixtures are used as the mobile

26 1

t I(c' t

-3 -2 0.8 0.85 log 9 - cp-.

Figure 6.7: Schematic illustration of the variation of retention with mobile phase composition in LC. (a) RPLC with acetonitile-water mixtures; (b) RPLC of small molecules with methanol-water mixtures; (c) LSC; (d) RPLC of large molecules with methanol-water mixtures.

phase. The typical variation of retention with mobile phase composition for some low molecular weight solutes in this system is illustrated in figure 6.7a. Linear, roughly parallel lines are obtained in a plot of In k vs. 'p (see also section 3.2.2).

However, if methanol-water mixtures are used as the mobile phase, the retention lines for individual solutes tend to diverge towards 'p= 0, as is schematically illustrated in figure 6.7b (see also figure 3.14). In this system, a linear composition gradient would result in a series of peaks with decreasing intervals. This can easily be understood by following the horizontal dashed line for which In k= 2.3 (k= 10) from left to right in figure 6.7b. As a consequence, slightly convex gradients are optimal for RPLC with methanol-water (and THF-water) mixtures [608]. Nevertheless, for most practical purposes linear gradients are acceptable for RPLC.

In LSC, an approximately linear behaviour is observed if In k is plotted against In 9. This is schematically illustrated in figure 6.7~. Hence, in order to obtain the same effect of the gradient program as in RPLC (figure 6.7a for the simplest case of a linear gradient), we should aim at a linear variation of In 'p with time, i.e.*

* Eqn.(6.1) arises from the definition equation of LSS gradients (eqn.54, if a linear relationship is assumed to exist between In k and In cp.

262

l n ( c p + d ) = a r + b

or

cp = c exp (a t ) - d (6.la)

In eqm(6.1) and (6.la) a, 6, c and d are constants. Because cp should increase with time, a is a positive constant in both equations and hence a concave gradient (figure 6.2d) is optimal for LSC.

Figure 6.7d shows the variation of retention with composition for some large molecular weight solutes in RPLC. In this case, the mobile phase composition has a very strong effect on the retention of the solutes (note the tenfold expansion of the horizontal axis). For low molecular weight solutes, the slope in the retention vs. composition plots is typically around 7 (see section 3.2.2) and therefore a typical solute can be eluted with a capacity factor in the optimum range (1 < k < 10) at mobile phase compositions which span a range of about 30% (2.3 x 100/7). Hence, for a limited number of low molecular weight solutes, there is a good chance that an isocratic composition can be established that is within the optimum range of each individual sample component. An example is given by the vertical line in figure 6.7a.

The situation is quite different for high molecular weight solutes, as is illustrated in figure 6.7d. For large, polar molecules that may be eluted with RPLC (e.g. proteins), retention may be expected to be an exceptionally strong function of mobile phase composition [609]. In this case, every individual sample component will have a very narrow composition range over which optimum capacity factors will occur. If a number of different large molecular weight components are present in the sample it may be almost impossible to find a constant (isocratic) composition that will give rise to optimum capacity factors for all sample components, and hence the use of gradient elution may be hard to avoid.

It turns out [609] that the slope in the In k vs. cp plots is mainly determined by the molecular weight of the solute. Solutes with very large molecular weight show very steep lines. The lines denoted by 1 and 2 in figure 6.7d form two examples. Retention, however, is mainly determined by the polarity of the solute. Therefore, a component with a much lower molecular weight but also a lower polarity than solutes 1 and 2 in figure 6.7d may have a similar retention time, but show a much shallower retention vs. composition line (solute 3 in figure 6.7). Consequently, the regular picture of figure 6.7b is disturbed and a much less structured pattern remains.

It will be clear from figure 6.7 that the nature of the mobile phase (compare figures 6.7a and 6.7b) and the stationary phase (compare figure 6 . 7 ~ with figures 6.7a and 6.7b) have a great effect on the character of the retention vs. composition plots and hence on the shape of the required (optimum) gradient. It will also be clear that, unlike the situation in GC, the selectivity may be greatly influenced by variations in the mobile phase.

The situation becomes more complex if we realize that figure 6.7 only provides schematic illustrations of the typical retention behaviour in different forms of LC. Examples of anomalous behaviour will not be hard to find. For example, figure 6.8 shows the variation of retention with composition for 23 phenylthiohydantoin (PTH) derivatives

263

k

0 0.10 0.20 0.30 0.U 0.50 0.60 9-

Figure 6.8: Experimental variation of the retention of 23 phenylthiohydantoin (PTH) derivatives of amino acids with mobile phase composition in RPLC. Mobile phase: mixtures of acetonitrile and 0.05M aqueous sodium nitrate buffer (pH = 5.81). All mobile phases contain 3’/0 THF. Stationary phase: ODS silica. Solutes: D = aspartic acid; C-OH = cysteic acid; E = glutamic acid; N = asparagine; S = serine; T = threonine; G = glycine; H = histidine; Q = glutamine; R = arginine; A = alanine; METS = methionine sulphone; ABA = a-aminobutyric acid; Y = tyrosine; P = proline; V = valine; M = methionine; NV = norvaline; I = isoleucine; F = phenylalanine; L = leucine; W = tryptophan; K = lysine. Figure taken from ref. [610]. Reprinted with permission.

of amino acids in RPLC using acetonitrile-water mixtures that contain a small amount (3’10) of THF as the mobile phase. The retention behaviour of these solutes under isocratic and gradient conditions was studied by Cohen et al. [610]. In figure 6.8 we recognize a rough pattern of parallel In k vs. cp lines, but we also see that many lines intersect (“cross-over”) due to variations in the slopes for individual solutes.

Another complication may arise if we choose to vary the selectivity of a gradient program in LC by varying more than one parameter at the same time. For example, the concentration of two organic modifiers may be varied independently in RPLC (so-called ternary gradients) or both the mobile phase composition and the temperature may be programmed.

In figure 6.9 we take a closer look at some ternary gradients in which the composition (cp) of two solvent components (B and C) is varied with time*. For simplicity, figure 6.9 has been limited to linear gradients.

In figure 6.9a the concentration of both organic modifiers is seen to increase with time.

* The third and weakest solvent A is assumed to make up the solvent to loO%, i.e. p,, + p ~ + pc= 1.

264

t- t -

t cp

t - t-

Figure 6.9: Examples of linear ternary gradients in which the concentration of two modifiers (Band C) is varied simultaneously. The concentration of the base solvent ( A ) is not indicated in the figure.

In figure 6.9b we see that one organic modifier is gradually being replaced by another. In this particular kind of gradient, the two limiting compositions (in figure 6.9b 60% B in A and 40% C in A ) may be of equal eluotropic strength, so that only the selectivity and not the eluotropic strength of the eluent is varied during the elution. Glajch and Kirkland [611] refer to this kind of gradient as “isocratic multi-solvent programming”, because the elution pattern of the solutes and the resulting chromatogram will represent isocratic elution much more closely than typical gradient elution. Although it may be possible to vary the selectivity in different parts of the chromatogram by “isocratic multi-solvent programming”, it should be noted that this technique features all the disadvantages of programmed analysis described in section 6.1. Hence, if “simple isocratic mixtures” (mixtures of constant composition [61 I]) can be used, the use of “isocratic multi-solvent programming” should be avoided.

In figure 6 . 9 ~ a ternary gradient is shown in which a small concentration of the second modifier Cis present throughout the elution. Figure 6.9d shows a gradient that runs from 100% A to 100% B and subsequently from 100% B to 100% C. This may be a sensible program if C is a considerably stronger solvent than B.

Although all the gradients in figure 6.9 are ternary gradients in that two parameters (the concentrations of two modifiers) are varied at the same time, we may interpret three of the four gradients as special forms of binary gradients (solvent A‘ -solvent W ) , in which A’ and B’ are mixtures of the pure components A, B and C. We may refer to A‘ and B‘ as pseudo-solvents and to the ternary gradients as pseudo-binary gradients. The last gradient in figure 6.7 (figure d) can be seen as a combination of two binary gradients

265

(solvent A' - solvent B' -, solvent C'). The compositions of the different pseudo-solvents for the gradients shown in figure 6.7 are listed in table 6.2.

Table 6.2: Compositions of the solvents in pseudo-binary gradients (A'-B' or A'+B'+C' )which are equivalent to the ternary gradients of figure 6.7.

Gradient Solvent A' Solvent B' Solvent C' Figure 6.7

'/o B 90 c '/o B 'lo c '/o B '/O c

0 0 60 40 N/A 60 0 0 40 N/A 0 10 90 10 N/A 0 0 100 0 0 100

From the above we may conclude that many of the ternary gradients which may be used in LC can be seen as special forms of binary gradients. Of course, this conclusion is no longer correct if we do not restrict the discussion to linear gradients and allow the shape of the gradient for one solvent to be different from that for another. However, it may be difficult to find applications for which such complicated ternary gradients can be proved to yield better results than the simpler (pseudo-) binary ones.

Summary

In this section we have come to the following conclusions. I . A solvent program (or gradient program) in LC should be comprised of at least four

segments (purge, reverse, equilibrate and gradient; seefigure 6.6). The program should begin and end at the purge stage.

2. The pattern of the variation of retention with composition in LC is aflected by the choice of both the stationary and the mobile phase. The optimum shape of the gradient for unknown wide range samples is dictated by the phase system. Linear or slightly convex gradients are optimal for RPLC. Concave gradients are optimal for LSC.

3. For specijk samples the optimum shape may deviate from this general rule. The retention and selectivity under gradient conditions may not follow the expected pattern because of anomalies in the isocratic retention vs. composition relationships.

4. The selectivity in programmed solvent LC may be varied by varying the solvents used or by the application of ternary or even more complicated gradients. However, most ternary gradients can in fact be reduced to binary ones using mixed (pseudo-) solvents.

6.3 OPTIMIZATION OF PROGRAMMED ANALYSIS

There are two aspects involved in the optimization of programmed analysis. The first one is the optimization of the parameters of the program. These parameters include the initial and final conditions, the shape of the program (see figure 6.2) and the duration of

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the program segments, for instance the heating rate (in programmed temperature GC), or the slope of the gradient in programmed solvent LC. Programmed analysis almost always involves the variation of primary parameters during the analysis. These parameters (and others such as the flow rate and the length of the column) will affect the separation, but the selectivity (a) is only slightly (or not at all) affected. Nevertheless, the program parameters form a relevant part of the optimization of programmed analysis. For instance, the choice of the initial conditions will affect the resolution for peaks that appear early in the chromatogram and the shape of the gradient will determine the overall distribution of the peaks over the chromatogram and may affect the selectivity for some pairs of peaks. Multisegment programs (see figure 6.2e) may allow the optimization of the resolution throughout the entire chromatogram. Examples of this will be given below.

The second aspect of optimization in programmed analysis involves adapting the selectivity by variation of secondary parameters. The various secondary parameters listed in table 3.10 may be used to vary the selectivity of a chromatographic system without affecting retention to a great extent (see the discussion in section 3.6.1).

The situation in programmed analysis is similar to the one described above for chromatographic elution under constant conditions, in that retention and selectivity may be optimized more or less independently. However, under constant elution conditions the optimization of the retention only involves adapting the primary parameters such that all capacity factors fall into the optimum range (1 < k< 10). In programmed analysis the optimization of the retention involves optimizing the characteristics of the program (initial and final composition, slope and shape) in conjunction with the physical parameters (e.g. flow rate and column dimensions, see section 3.6).

If the program is optimized so that all sample components are eluted under optimal conditions, then other (secondary) parameters may be used for the optimization of the selectivity. However, changes in the secondary parameters may imply that the parameters of the program need to be re-optimized. For example, if the selectivity in a temperature programmed GC analysis is insufficient, then another stationary phase may be used to enhance the separation. However, the optimum program parameters obtained with one stationary phase cannot be transferred to another column that contains another stationary phase. The re-optimization of the temperature program for the other column will require at least one additional experiment to be performed.

The primary and secondary parameters that may be used for the optimization of the program and the selectivity in programmed analysis, respectively, are listed in table 6.3 for the various chromatographic techniques.

It can be seen in table 6.3 that the optimization of selectivity in programmed temperature GC involves variation of the (nature or composition) of the stationary phase. To vary this parameter, a different column and re-optimization of the (primary) program parameters will be required. This is clearly not a very attractive proposition and therefore the optimization of programmed temperature GC is usually restricted to optimizing the program.

In programmed solvent LC the nature of the modifier(s) in the mobile phase is the most common secondary parameter that may be used for the optimization of the selectivity. This is an attractive parameter, because different modifiers may be selected and programmed automatically on various commercial instruments. Therefore, the possibilities for selectivi-

267

Table 6.3: Parameters for the optimization of programmed analyses in various chromatographic techniques. Primary parameters may be used to optimize the program parameters (initial and final conditions, slope and shape). Secondary parameters may be used to optimize the selectivity.

Method Primary parameter(s) (program)

Secondary parameter(s) (selectivity)

GC Temperature Stationary phase

RPLC Mobile phase polarity; pH Nature of modifier(s); stationary phase

LSC Eluotropic strength

I EC Concentration of counterion; pH

Nature of modifier(s); stationary phase

Nature of modifier(s), counterion or buffer

SFC Mobile phase density Nature of mobile phase; stationary phase Nature of modifier(s) Mobile phase composition

ty optimization in programmed analysis are much greater in programmed solvent LC than they are in programmed temperature GC.

Selectivity optimization vs. multisegment programs

Two ways are open that may lead to the optimization of the resolution of all pairs of peaks in the chromatogram. The first is to use the primary (program) parameters in designing a multisegment gradient, the second relies on the optimization of secondary (selectivity) parameters. In the first case, the resulting programs will be generally complex and consist of many segments. In the second case, relatively simple, continuous programs will be obtained. The latter is generally to be preferred, for the following reasons: 1. With simple, continuous elution programs the elution conditions for the individual

peaks (in terms of peak width and detector sensitivity) will either be constant throughout the chromatogram, or will vary in a continuous way.

2. Simpler instrumentation may be used and the effect of the quality of instrumentation on the resulting chromatogram is reduced.

3. The reproducibility of the analysis will be enhanced. 4. Column lifetime will be increased. 5. Selectivity optimization of simple, continuous gradients will be easier than the

optimization of complex multisegment programs, because there are bound to be serious

268

discrepancies between theory and practice, which will prohibit the exact calculation of programs comprised of many “subtle” segments.

6. Optimization of the primary parameters of the gradient program may only lead to sufficient separation if the selectivity is sufficiently large. If the a values between one or more pairs of solutes are low, resolution may be enhanced by a reduction of the programming rate and by increasing the number of plates. However, this will be at the expense of increased analysis times and the resolution will never be better than under constant elution conditions.

Therefore, selectivity optimization, is in principle to be preferred over multisegment gradients. In GC, where selectivity optimization is not attractive because of the requirement to use different columns, one may wish to resort to multisegment gradients for practical reasons. In LC, where selectivity optimization is readily possible by using different modifiers in the mobile phase, multisegment gradients are of little practical interest.

We have seen in section 6.1 that a programmed analysis in chromatography generally requires more time than an experiment under constant elution conditions. Therefore, optimization procedures that require large numbers of experiments are the least attractive for the optimization of programmed analysis. The procedures that were found to require the largest numbers of experiments under constant elution conditions in chapter 5 were the simultaneous (“grid search”) optimization methods (see section 5.2). For this reason, such procedures have not been contemplated for the optimization of programmed analysis and they will not be discussed in this section. Attention in this section will be focussed on the choice between sequential methods as described in section 5.3 and interpretive methods as described in section 5.5.

6.3.1 Optimization of programmed temperature GC

6.3.1.1 Sequential methods

Simplex optimization

Walters and Deming [612] have used a Simplex procedure for the optimization of the program parameters in programmed temperature GC. We have seen in chapter 5 (section 5.3) that one of the main advantages of Simplex optimization procedures is that no knowledge is required about the relationships between the parameters considered on the one hand and the retention and selectivity on the other. Hence, a Simplex program that can be used for the optimization of chromatographic separations under constant elution conditions may be used equally well for the optimization of programmed analysis. All that is necessary to adapt the Simplex program for this purpose is to select an appropriate optimization criterion for programmed analysis. This subject has been discussed in section 4.6.2.

The two parameters considered by Walters and Deming [612] were the initial temperature and the heating rate. They used a composite optimization criterion (see section 4.4.2) and imposed a time constraint of 5 minutes on the system by assigning a very unfavourable

269

(infinite) value to the criterion when the analysis time was longer*. The procedure required 13 experiments, two of which could not be performed because negative heating rates were suggested by the optimization program.

Because this optimization only concerned program parameters and not selectivity parameters, the response surface will have been relatively simple. Therefore, the probability that the Simplex procedure would arrive at the global optimum rather than at a local one was greater than it was in section 5.3, where wedescribed the use of the Simplex method for selectivity optimization.

Systematic sequential optimization

Stan and Steinbach [613] have described a sequential optimization procedure for programmed temperature GC that searches for an optimum multisegment program in a systematic way. This procedure can be divided into three different stages: 1. Separation of a maximum number of peaks by adapting the programming rate of each

segment, as well as the length of the preceding isothermal period**. 2. Increasing the resolution (R,; eqn.l.14) values to exceed 1.5 for each pair of peaks by

reducing segment slopes and inserting isothermal periods. 3. Reducing the resolution values to be less than 1.5 for each pair of peaks by increasing

segment slopes and shortening isothermal periods. The first stage is the actual sequential optimization procedure. It involves the

optimization of the heating rate of each segment followed by the optimization of the duration of the preceding isothermal period. As an example, a program was described in ref. [613] that started with (splitless) injection of the sample at 100 O C . The injection period was followed by a rapid (30 OC/min) heating to the initial program temperature (150 "C). The total span of the program from 150 to 250 OC was divided in five non-isothermal segments, each spanning 20 OC. Isothermal segments could be inserted before each of these, so that a total of ten segments was considered during the first stage of the process.

The procedure starts by recording a first chromatogram in which the maximum heating rate (30 OC/min) is applied throughout the program. After the injection period, the temperature is raised from 100 OC (injection temperature) to 250 "C. The resulting chromatogram is shown in figure 6.10a. In this example, 28 peaks can be registered from the chromatogram.

The sequential optimization procedure now starts by optimizing the last segment of the program (230-250 "C) aiming to increase the number of peaks observed in the chromatogram. To this end, the slope of this segment is successively reduced from 30 OC/min to 8, 4 and 2.67 OC/min. If reducing the slope does not result in an increase in the observed number of peaks, then the value is rejected and the previous one is retained. A similar procedure is followed for the next segment (isothermal at 230 "C). The duration of this

* This time constraint is required because, as we have seen in section 4.4, the incorporation of a (weighted) time term into the optimization criterion is not an effective way to constrain the analysis time (see eqn.4.29 and subsequent discussion). ** In this optimization procedure a segment usually refers to a part of the temperature program at which heating takes place. Such segments may be separated by isothermal periods, during which the temperature is kept constant. In the present discussion we will refer to the two kinds of segments as non-isothermal and isothermal, respectively.

270

period is increased in steps from 0 to 1 ,2 or 3 minutes, until there is no further increase in the observed number of peaks.

For the optimization of each segment a minimum of one and a maximum of three experiments needs to be performed. Every experiment involves a complete temperature program from 100 O C (injection) to 250 O C . For the optimization of the entire ten-segment program, 11 to 31 experiments (including the initial run) are required. Figure 6.10b shows the resulting chromatogram after 16 experiments were performed following the procedure described above. The temperature program is shown underneath the chromatogram. During the procedure, the number of registered peaks has been increased from 28 to 36,

During the second stage of the procedure, each pair of peaks is checked for sufficient resolution. If R,< 1.5 (eqn.l.l4), then depending on the elution temperature observed for the peak pair, either an isothermal segment may be inserted in the program, or the slope of a non-isothermal segment may be reduced. This procedure may be followed simultaneously for every ill-resolved pair of peaks. Therefore, few additional experiments are required*.

Figure 6.1 Oc shows the resulting chromatogram and temperature program after two more injections. It is seen that the program is now much more complicated and that the analysis time has been increased from about 25 to about 43 minutes.

During the second stage of the optimization process the number of registered peaks was increased from 36 to 38. Whereas it was claimed in ref. [613] that 38 is the actual number of peaks present in the sample, it seems that at this stage of the procedure additional peaks may only be found accidentally. This will be the case if, in striving for sufficient resolution of one particular pair of peaks somewhere in the chromatogram, a hidden peak is suddenly revealed. In a second optimization cycle these peaks may subsequently be resolved with R,> 1.5. However, if peaks are hidden in parts of the chromatogram in which the resolution appears to be sufficient for all registered peaks, they will not be found during stage 2 of the optimization process. The fact that the presence of two more peaks was revealed during the this stage suggests that additional peaks may be “hidden” in the chromatogram. Therefore, it may illustrate one of the shortcomings rather than one of the advantages of the method.

Finally, in the third stage of the process, a procedure similar to that of the second stage may be followed to reduce the resolution of abundantly resolved pairs of peaks (R,>2). During this stage, slopes may be increased and isothermal periods shortened, leading to a reduction of the required analysis time. Figure 6.10d shows the result obtained after an additional two chromatograms. It is seen that the analysis time has been reduced from about 43 to about 37 minutes.

The entire procedure illustrated in figure 6.10 involved 21 (1 + 16 + 2 + 2) chromatograms and took about 10 hours. Because of the sequential nature and because of the selection of the criteria, automation is relatively easy. A serious disadvantage of the method, besides the large number of required experiments and the complexity of the resulting program, is the dependence of the result on the column used. Possibly, a different

* I t may not be possible to achieve sufficient resolution for all the pairs of peaks in the chromatogram on the particular column. Therefore, a stop criterion is needed in the optimization procedure, for instance a maximum of two attempts to separate a particular pair of peaks. If it is difficult to recognize (pairs of) peaks, then a maximum of two or three optimization cycles each for stage 2 and stage 3 of the optimization procedure may be considered.

27 1

t lmin -c tlmin-

I

250 -

0 5 10 15 20 25 30 35 LO L5 50 55 60 t lmin -

Figure 6.10: Application of the systematic sequential optimization procedure of Stan and Steinbach [613] for the optimization of a temperature program in capillary GC. Column: 25 m x 0.2 mm I.D. coated with dimethylsilicone bonded phase BP-1 (S.G.E.); Carrier gas: helium; Detector: electron capture; Sample: halogenated pesticides (residue analysis). (a) Initial chromatogram; (b) Resulting chromatogram after stage 1 (maximum number of peaks); (c) Resulting chromatogram after stage 2 (increased resolution); (d) (opposite page) Final chromatogram after stage 3 (reduced resolution). For explanation see text. Figure taken from ref. [613]. Reprinted with permission.

272

100, 0 5 10 15 20 25 30 35 LO 15 50

tlmin - result may even be obtained on the same column under different operating conditions (e.g. flow rate). This is due to the use of the column-dependent R, criterion during the second and third stages of the optimization process (see discussion in section 4.3.3). Finally, a reasonable estimate for the initial and final program temperatures should either be made on beforehand, or established from the first chromatogram.

6.3.1.2 Interpretive methods

The obvious alternative to the sequential optimization methods is the use of an interpretive optimization method. In such a method a limited number of experiments is performed and the results are used to estimate (predict) the retention behaviour of all individual solutes as a function of the parameters considered during the optimization (retention surfaces). Knowledge of the retention surfaces is then used to calculate the response surface, which in turn is searched for the global optimum (see the description of interpretive methods in section 5.5). For programmed temperature GC the framework of such an interpretive method has been described by Grant and Hollis [614] and by Bartd [615].

All interpretive optimization methods are by definition required to obtain the retention data of all sample components at each experimental location. If the sample components are known and available they may be injected separately (at the cost of a large increase in the required number of experiments). For unknown samples, for samples of which the individual components are not available, and in those situations in which we are not prepared to perform a very large number of experiments (as will usually be the case in the optimization of programmed analysis) we need to rely on the recognition of all the individual sample components in each chromatogram (see section 5.6).

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If many peaks occur in a chromatogram this appears to be a very difficult proposition. However, the requirement of solute recognition may not give rise to insurmountable problems in the optimization of programmed temperature GC for the following two reasons: 1. The optimization procedure may be carried out on the basis of a limited number of

2. Changes in elution order (”component cross-overs”) are unlikely to occur. For the interpretive optimization of the primary (program) parameters in the programmed analysis of complex sample mixtures it may well be sufficient to optimize for the major sample components. This may be done if it is assumed that the primary parameters do not have a considerable effect on the selectivity, so that if the major sample components are well spread out over the chromatogram, the minor components in between these peaks will follow suit automatically, and if it is assumed that the minor peaks are randomly distributed over the chromatogram. The major chromatographic peaks can be separated to any desired degree if optimization criteria are selected which allow a transfer of the result to another column.

Changes in elution order are unlikely to occur, because temperature is not a truly selective parameter (see section 3.1). To a first approximation, the elution order of the peaks, and certainly the elution order of the major components, may therefore be assumed constant.

(major) components in the sample [614].

The retention behaviour of solute molecules under programmed temperature conditions is completely characterized by 1. the parameters of the program, and 2. the variation of the retention with the temperature under isothermal conditions. If we leave out of account the delay that both the column and the packing material may cause in the temperature program inside the column relative to the program followed in the column oven [615], then the program parameters are naturally known.

In principle, the description of the retention vs. temperature relationships requires two experiments, because a straight line can be obtained by plotting In (k/T) vs. 1 /T (eqn. 3.10).

Grant and Hollis [614] assume a linear relationship between In k and 1/T:

I n k = A + B / T (6.2)

where A and Bare solute-dependent coefficients. They assume that the intercept A remains “sensibly constant”, and that the slope B is proportional to the (absolute) boiling point for solutes within a given class. Therefore, the isothermal retention data for some “typical” solutes from a class at a minimum of two different temperatures is thought to be sufficient to describe the retention behaviour of all solutes within that class under programmed temperature conditions. Unlike eqn.(3.10), eqn.(6.2) is not a fundamentally linear relationship. Since both equations require two experimental data points two establish the coefficients, the use of the former is to be preferred.

BartQ [615] uses a different relationship to describe the retention vs. temperature relationship. His equation is also not fundamentally linear and requires a minimum of three parameters:

274

In(t,-C') = A + B / T . (6.3)

In this equation t , is the retention time under isothermal conditions at the temperature T and A, B and Care constants. An analogous expression is used to describe the variation of the peak width at the location of half the peak height ( w , , ~ ) with temperature.

The two experiments required to describe the isothermal retention vs. temperature relationships through eqn.O.10) or eqn.(6.2), or the three required by eqn.(6.3), may either be performed isothermally or under programmed conditions. However, in the latter case the calculations to obtain the coefficients A, B, and, if eqn.(6.3) is used, C, will be more complicated and more than two or three experiments may be required to estimate the coefficients with sufficient accuracy. The latter aspect suggests the use of an iterative interpretive method, in which the values of the coefficients are updated after each new experiment until the accuracy of the predicted optimum turns out to be sufficient.

Neither the procedure described by Grant and Hollis [614], nor that of Bartd [615] is a complete optimization procedure. They do not provide a generally useful strategy for unknown or ill-known samples. The application of either approach in practice has not been described.

6.3.1.3 Discussion

Simplex optimization of the primary (program) parameters in programmed temperature GC analysis has been demonstrated [612]. A systematic sequential search [613] may be used as an alternative. The Simplex method may be used to optimize a limited number of program parameters, whereas the latter approach was developed for the optimization of multisegment gradients. The use of interpretive methods has so far only been suggested [614, 6151.

As was the case in its application to the optimization of chromatographic selectivity under constant conditions, the Simplex algorithm appears to require a rather large number of experiments. This is also true for a systematic sequential procedure.

If interpretive methods are used, the calculations involved may be complicated and it is necessary to recognize the individual solutes in each chromatogram. Because the optimization procedure may be carried out for a limited number of major sample components and because the elution order is not likely to vary, this will not usually be a serious problem.

It certainly would not have been a problem in the example for which the Simplex program was demonstrated in ref. [612]. In this sample only four components were present. The selection of this particular example to demonstrate the applicability of Simplex optimization for programmed temperature GC was unfortunate in any case, because a straightforward isothermal separation of the sample at 70 OC also appeared to be possible.

The example shown in figure 6.10 (ref. [613]) for the optimization of a multisegment temperature program was more impressive. Unfortunately, the required number of experiments was large (21). The selection of simple criteria (e.g. maximum number of peaks) may greatly enhance the possibilities for fully automatic optimization.

If an interpretive method is used, then the number of experiments required to allow an accurate prediction of the global optimum may be somewhat larger than the theoretical minimum of two experiments. However, this still compares favourably with the 21

275

experiments performed by Stan and Steinbach [613b who used a systematic sequential procedure, and to the 11 experiments performed by Walters and Deming [612] to locate the optimum with a Simplex method. Moreaver, Wdters and Deming performed 8 additional experiments in the vicinity of the qtimurn to enhance the accuracy oithe result.

6.3.1.4 Selectivity optimization

In order to optimize the selectivity in programmed temperature GC, the parameter to be varied is the nature or composition of the stationary phase. If this kind of optimization is to be pursued, then the Simplex procedure will be especially unattractive, because it will require large numbers of experiments using different stationary phases and, consequently, different columns. Therefore, interpretive methods are to be preferred for optimizing the selectivity in programmed temperature GC. Because of the experimentally observed linear relationship between retention and composition in isothermal GC using mixed stationary phases (eqn.3.14), fixed experimental designs may be used, similar to those employed for optimizing the stationary phase composition in isothermal GC (window diagrams, see section 5.5.1).

6.3.1.5 Summary

1. Simplex optimization of the primary parameters in programmed temperature GC analysis is possible.

2. As with other applications of the Simplex algorithm in chromatography (see section 5.3), a large number of experiments is required.

3. The response surface for the optimization of the primary (program) parameters in programmed temperature GC is less convoluted than a typical response surface obtained in selectivity optimization procedures (see section 5.1). This will increase the possibility of a Simplex procedure locating the global optimum.

4. A systematic sequential optimization procedure may be used to establish an optimum multisegment temperature program.

5. Such a procedure requires a large number of experiments, but may readily be automated. 6. For simple samples, in which the individual components can be recognized, interpretive

methods should ailow the prediction of the (global) optimum from a small number of experiments.

7. For complex samples the separation of the major components may be optimized by an interpretive method. The resulting optimum program presumably corresponds to the optimum for the entire sample.

8. Optimization of the selectivity in programmed temperature GC requires the application of diflerent stationary phases or stationary phase mixtures.

9. In that case, interpretive methods based on fued experimental designs (window diagrams) may be used.

6.3.2 Optimization of programmed solvent LC

The (primary) program parameters may be used to optimize the separation in programmed solvent LC in a non-selective way. Since this involves optimization of the

276

retention rather than the selectivity, this kind of optimization will only be adressed briefly in this section. The optimization of the program parameters has been discussed extensively by Snyder [616,6171 and more recently in an excellent book by Jandera and ChurhEek [618].

The most useful secondary parameter for the optimization of the selectivity in programmed solvent LC is the nature of the modifier(s) in the mobile phase. The selectivity can be varied by selecting various solvents (pure solvents for binary or ternary gradients; mixed solvents for pseudo-binary gradients). Analogous to the situation in isocratic LC, it is possible to use different modifiers (and hence to obtain different selectivity),. while optimum retention conditions are maintained for all solutes. This possibility to optimize the selectivity in programmed solvent LC will be discussed below.

6.3.2.1 Simplex optimization

As with programmed temperature GC, the application of the Simplex optimization procedure to programmed solvent LC is relatively straightforward. The same procedure can be used both for isocratic and for gradient optimization, as long as an appropriate criterion is selected for each case*.

After earlier applications of the Simplex algorithm for the optimization of programmed solvent LC by Watson and Carr [619] and by Fast et al. [6201, the possibility of applying (slightly) different versions of a single Simplex program for the optimization of isocratic and programmed solvent analysis in LC was demonstrated by Berridge [621]. He used the Simplex procedure to optimize three program parameters: the initial and the final composition and the duration of a linear gradient. The convergence of the Simplex algorithm to the final optimum was said to be rapid, but still 15 experiments were required to arrive at the optimum. A reason for such a “rapid convergence was suggested to be the location of the resulting optimum on the edge of the parameter space (final composition: 100 %B). Another reason may be the relative simplicity of the response surface in comparison to isocratic optimization in which the selectivity (secondary parameter: nature and concentration of modifiers) is varied.

An indication of this latter effect can be found in figure 6.1 1, which shows the result of the Simplex optimization procedure applied to the programmed solvent LC separation of three antioxidants [621].

The sum of peak-valley ratios was used as the resolution term in a composite optimization criterion, which otherwise corresponds to eq~(4.30). Berridge also added a term to describe the contribution of the number of peaks (n). With this, the complete optimization criterion became

The desired analysis time (t,,,) was set equal to 4 min., whereas the value of the minimum time ( tmin, which is irrelevant for the optimization process; see section 4.4.2) was taken to be 1.5 min.

* For criteria based on the peak-valley ratio ( P ) no modification of the criterion used for isocratic optimization may be necessary (see section 4.6.2).

277

0 1 2 3 tlmin-

Figure 6.1 1: Resulting chromatogram from a Simplex optimization procedure applied to the separation of three antioxidants. Solvents: 5’h acetonitrile in water (A) and 5Oh water in acetonitrile (B). Linear gradient 44 to 100% B in A in 1.5 min. Column: 10 cm x 5 mm I.D. 5 pm Lichrosorb C-18. Flow rate: 2.0 mL/min. Solutes: 1 = propyl gallate; 2 = 2-t-butyl-p-methoxyphenol (BHA); 3 = unknown; 4 = 2,6-di-t-bytul-p-cresol (BHT). Figure taken from ref. [621]. Reprinted with permission.

It can be seen in the chromatogram of figure 6.1 1 that four peaks (the three antioxidants plus an unknown impurity) are amply resolved to the baseline. This implies that all values for the peak-valley ratio Pare equal to 1 and that the criterion has become very insensitive to (minor) variations in the resolution between the different peak pairs. In the area of the parameter space in which four well-resolved peaks are observed, the only remaining aim of the optimization procedure is to approach the desired analysis time of 4 minutes. The irrelevance of the “minimum time” tmin is illustrated by the occurrence of the first peak in figure 4.9 well within the value of 1.5 min chosen for this parameter.

The application of the Simplex procedure for the optimization of the selectivity in programmed solvent LC (e.g. for the application of ternary gradients) has not yet been reported. However, there is no apparent obstacle to the applicability of the Simplex procedure for this purpose.

Of course, the simultaneous optimization of different (primary) program parameters (initial and final composition, slope and shape of the gradient) and secondary parameters (nature and relative concentration of modifiers) may involve too many parameters, so that an excessive number of experiments will be required to locate the optimum. This problem may be solved by a separate optimization of the program (primary parameters) and the selectivity (secondary parameters) based on the concept of iso-eluotropic mixtures (see section 3.2.2). This will be demonstrated below (section 6.3.2.2). However, the transfer of

278

the program parameters optimized with one modifier to an analysis program using another modifier (or a combination of two modifiers in a ternary gradient) requires more knowledge and understanding of the relationships between chromatographic retention and the parameters considered in the optimization procedure than is usual for Simplex optimization.

6.3.2.2 Systematic optimization of program parameters

Optimization without solute recognition

The concept of linear solvent strength (LSS) gradients developed by Snyder (see also sections 5.4.2 and 6.2.2) incorporates optimization of both the shape and the slope of gradient programs. The shape of an LSS gradient is determined by 1. the definition equation of LSS gradients, i.e.

log kin = log k , - b ( t / t o ) ,

where kin is the capacity factor of the solute under the isocratic conditions at the column inlet at time r, k , the capacity factor under isocratic conditions corresponding to the initial composition of the gradient program, b the gradient steepness parameter, t the time elapsed since the start of the gradient (or, more precisely, the time elapsed since the arrival of the gradient at the inlet of the column) and to the hold-up time of the column.

2. The relationship between retention and composition under isocratic conditions, i.e. the function

k = f ( q ) . (6-5)

The combination of these two factors determines the required shape of an LSS gradient. Linear gradients were shown to result for RPLC in section 5.4, whereas a concave gradient was found to be optimal for LSC in section 6.2.2.

The optimal slope of the gradient also follows from the LSS concept, since it was shown by Snyder et al. I6161 that optimum values for the gradient steepness parameter b are in the range 0.2 < b < 0.4. If the function f(q) is known, then the optimum slope of the gradient can be calculated. For example, in RPLC the relationship between retention and composition over the range 1 < k< 10 can be described by

Ink = Ink, - S q . (3.45)

In RPLC an LSS gradient is a linear gradient that can be described by

q = A + B t . (5.6)

Combination of eq~(3.45) with eqa(5.5) (see also section 5.4) yields

b = S B to / 2.303 . (5.8)

279

Typical S values for small solutes using methanol-water mixtures as the mobile phase are in the range 5 < Sc 10 [608]. The value of to is determined by the column and the flow rate. For example, if a column of 15 cm length with an internal diameter of 4.6 mm is used, the hold-up volume ( Vo) is around 1.5 ml, so at a flow rate of 1.5 ml/min the hold-up time ( f a ) is about 1 min. An optimal gradient with a b value of 0.3 then leads to a range of B values in eqm(5.6) given by

0.069 < B -= 0.138,

where B is expressed in min-I. The optimum programming rate is seen to be between about 7 and 14 %/min. For a 0-100% gradient this corresponds to gradient durations (t,) in the range

where f , is expressed in minutes. Snyder et al. [616,622] recommend a simple trial-and-error approach for the optimiza-

tion of the remaining two parameters of the program, i.e. the initial and the final composition. These parameters should be adapted such that solute bands are eluted neither too early, nor too late in the chromatogram.

If larger solute molecules (e.g. proteins) are to be separated by programmed solvent LC, then much higher S values may be expected and consequently (eqn.5.8) a lower B value (shallower gradient) will be required [609].

The Snyder procedure would have led to a quick solution of the separation problem shown in figure 6.1 1. However, the answer would have been different from that obtained with the Simplex optimization program. If we assume an S value of about 7 for the solutes involved and estimate the hold-up volume of the column to be around 1.18 mL (60% of the volume of the empty column), then we can estimate the b value for the gradient used in figure 6.1 1

6 = S B to / 2.303 = S 19 V, / (2.303 F) = 7 x 0.373 x 1.18 / (2.303 x 2) = 0.67.

This shows that the very fast gradient (t,= 1.5 min.) used in figure 6.1 1 was indeed two or three times steeper than the optimum conditions suggested by Snyder.

Following Snyder’s approach, the first experiment could have been a gradient of 0 - 100 %B in A in 6 minutes ( b = 0.30). As a result of this gradient, the initial concentration could then have been increased to yield (after one or two experiments) an optimum program with a gradient from about 50 to 100% B in 3 minutes. The overall analysis time (retention time of the last peak) would not have been much longer than the 3 minutes observed in figure 6.11, whereas all peaks would have been eluted under optimum conditions with roughly equal peak widths. The last peak in figure 6.1 1 is considerably broader than the other ones, because it is eluted after the completion of the gradient program.

However, the most important difference between the Simplex procedure and a systematic approach such as the one suggested by Snyder is not in the quality of the

280

resulting chromatogram but is the number of experiments required. For the optimization of the primary (program) parameters the former required 15 experiments, whereas the latter would not have required more than 2 or 3.

Optimization with limited solute recognition

In the Snyder approach to gradient optimization the characteristics of the individual solutes are largely neglected. The optimum shape of the gradient is determined by the phase system and the optimum slope is usually estimated from simple rules for the retention behaviour of the solutes (e.g. assuming S = 7 for small solute molecules as we did above). Only the initial and the final conditions are adapted to the requirements of the sample.

A strategy for the optimization of gradient programs based on the actual retention behaviour of some sample components has been described by Jandera and Churaeek [623, 6241. This approach relies on the possibility to calculate retention and resolution under gradient conditions from known retention vs. composition relationships and plate numbers. Both typical RPLC (eqn.3.45) and LSC (eqn.3.74) relationships can be accommodated in the calculations and linear, convex and concave gradients are all possible because of the use of a flexible equation to describe the gradient function. This equation reads

Q = + B V)

where A is the initial concentration, B the slope of the gradient and V is the volume of eluent delivered since the start of the gradient. Vis related to the elapsed time t and the flow rate F by V = Ft. K characterizes the shape of the gradient. If K= 1 the gradient is linear. K< 1 corresponds to a convex gradient and K > 1 to a concave one.

1. a preset (required) value for the resolution (R,) between two arbitrary solutes, and 2. a minimum retention volume for another arbitrary solute. We can summarize this optimization goal in a way that is consistent with the criteria described in chapter 4 (section 4.3.3) as follows:

Optimum gradients were defined by Jandera and ChuraEek [624] to yield

In eqn.(6.7) the pair of solutes for which a minimum resolution of x is required is denoted by i and i + 1. j denotes the sample component for which the retention volume under gradient conditions ( Vg) is to be minimized.

If the retention vs. composition relationships for the solutes i, i + 1 andjare known, then the gradient parameters A, B and K can readily be calculated for the optimum gradient according to equation 6.6. Not unexpectedly, the value of the shape parameter K turns out to be of little significance for an optimization procedure in which only three solutes affect the result [624]. Therefore, it may be sufficient to optimize the parameters A and B for a linear gradient (K= 1).

Figure 6.12a shows the resulting optimal chromatogram for the separation of a mixture of seven barbiturates by programmed solvent RPLC. This figure was obtained with the following optimization criterion:

281

5 0 V/mllO - 5 0 - Vlml 10

10 Vlml -

Figure 6.12 Resulting chromatograms from the Jandera and ChuraEek gradient optimization method. (a) requiring a minimum resolution ( R 3 between solutes 1 and 2 of 1.7 and minimizing the retention volume ( Vg) of solute 7 (eqn.6.7a); (b) requiring a minimum resolution between solutes 6 and 7 of 1.75 and minimizing the retention volume of solute 1 (eqn.6.7b); (c) linear gradients used to obtain the chromatograms a and b (gradient a: 9 = 0.368 + 0.061 V; gradient b (p = 0.523 + 0.0082 V). Mobile phase components: water (A) and methanol (B). Stationary phase: Lichrosorb ODs. Solutes: 1. barbital; 2. heptobarbital; 3. allobarbital; 4. aprobarbital; 5. butobarbital; 6. hexobarbital; 7. amobarbital. Figure taken from ref. (6241. Reprinted with permission.

Figure 6.1 2b shows the resulting chromatogram obtained under the conditions

RJ7,6 > 1.75 n min Vg,, .

(6.7a)

(6.7b)

The two different linear gradients are shown in figure 6.12~. It can be seen in figure 6.12 that the two different criteria described by eqns.(6.7a) and

(6.7b) result in different gradient profiles and different chromatograms. In figure 6.12a the resolution between the last two peaks is clearly insufficient. In figure 6.12b the resolution of these last two peaks has increased, but a t the expense of a decreased resolution of the first two peaks. In the first chromatogram the gradient is too steep to obtain sufficient resolution. In the latter chromatogram the initial concentration may be slightly too high.

Clearly, neither in chromatogram a nor in chromatogram b is the resolution optimized

282

throughout the chromatogram. This is a disadvantage of the procedure. Another disadvantage is that a choice needs to be made as to which two components will be the most difficult to separate (“critical pair”) and for which solute the retention volume should be minimized. The two above chromatograms illustrate that a different choice for the solutes involved in the optimization criterion will lead to a different result. Apparently, in order to improve the method other optimization criteria need to be considered. For example, the resolution could be optimized for both the first two and the last two peaks in the chromatogram.

Advantages of the procedure are that the calculations are relatively simple and that only the retention vs. composition relationships of the three solutes involved in the optimization criterion need to be known.

Complete mathematical optimization

If the retention vs. composition plots of all solutes are known, then it is in principle possible to calculate the optimum program parameters for a simple, continuous gradient (figure 6.2a-d). In such a procedure an appropriate optimization criterion can be selected such that the distribution of all the peaks over the chromatogram, as well as the required analysis time, can be taken into account (see chapter 4).

However, the calculations required for such an optimization are quite involved. This is caused by the requirement to calculate the retention times of each solute (and the resolutions of each pair of adjacent peaks) from the isocratic retention vs. composition relationships. In order to characterize the response surface, these calculations need to be performed a number of times. Finally, the optimum needs to be found on the response surface. If all four program parameters (initial and final concentration, slope and shape) are considered, the number of calculations would be large, even though the response surface may be simple compared with those encountered in selectivity optimization (see the discussion in section 6.3.2.1).

Multisegment gradients

A procedure that avoids the lengthy calculation procedure mentioned above is the one described by Noyes [625]. She designed a multisegment gradient program on the basis of visual interpretation of the isocratic retention vs. composition relationships for a number of phenylthiohydantoin (PTH) amino acids. It was claimed that the mixture could not be separated by a continuous linear gradient, but no further details on the design of the multisegment gradient were given.

Issaq et al. [626] have reported on a method for the optimization of a multisegment gradient program for the optimum resolution of all pairs of peaks in a programmed solvent LC chromatogram. In their procedure a number of programmed solvent experiments are performed, either a series of linear gradients between two solvents A and B of variable duration ( tG), or a series of linear gradients with constant tG, but a variable final concentration of B in A. For each pair of adjacent solutes the gradient which yields the best resolution is then selected and the different linear segments are combined into a multisegment program.

The exact procedure in which the multisegment gradient is built up from the optimum

283

gradients for the individual pairs of peaks is not clarified, however, and it remains to be seen whether the calculated program wifl indeed result in optimum separation for all pairs of peaks in the chrumatogram. It appears that this goal can only be achieved if the elution pattern of a pair of peaks through the column is only affected by the particular segment designed for the optimum resolution of this pair. Unless the different solute pairs are very far apart in the chromatogram (in which case the overall distribution of the peaks over the chromatogram would be far from optimal!), the resolution of a pair of peaks is likely to be much affected by the preceding segments of the program. No examples to demonstrate the applicability of the method are given in ref. €6261.

6.3.2.3 Znterpretive methods for selectivity optimization

Glajch and Kirkland [627] have extended the Sentinel optimization method (see section 5.5.1) to include the optimization of the selectivity in programmed solvent LC. This optimization procedure allows the use of linear gradients in RPLC using one or more organic modifiers in water. The relative concentration of the modifiers does not change during the analysis (so-called iso-selective muiti-solvent gradients (61 11, see figure 6.7a). This allows a straightforward extension of the Sentinel method.

For the optimization of programmed solvent LC the Sentinel method starts by establishing a suitable binary methanol-water gradient. The appraa& of Snyder described above may be used for this purpose. For example 16271, a gradient from 20 to IO@h methanol (in water) in 20 minutes may be the result.

THF

ACN

Figure 6.13: Figure illustrating the 7 linear gradients used in the Sentinel optimization method for programmed solvent U=. Initial and final compositions of the gradients are listed in table 6.4.

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Next, the concept of iso-eluotropic mobile phases is used to determine the binary acetonitrile-water and THF-water mixtures that correspond to the initial and the final composition. For example, 20% methanol corresponds [627] to 17% acetonitrile and to 12% THF, whereas 100% methanol corresponds to 84% acetonitrile and to 59% THF.

By analogy with the isocratic Sentinel optimization procedure a series of 7 gradients (all of the same duration time) can now be defined. These gradients are shown in figure 6.13 and the initial and final compositions are listed in table 6.4.

The individual retention times of all solutes in a 14-component sample mixture were measured and used to calculate resolution values (eqn.1.14, because eqn.l.22 is invalid) between each pair of peaks in the chromatogram. The largest value for the limiting resolution (max Rs.min; eqn.4.25) was used as the optimization criterion.

Table 6.4 Initial and final compositions of the 7 linear gradients shown in figure 6.13. All 7 gradients have the same duration.

Gradient number

Mobile phase composition (% v/v)

Water MeOH ACN THF

1 Initial Final

80 0

20 100

0 0

0 0

2 Initial Final

83 16

0 0

17 84

0 0

3 Initial Final

88 41

0 0

0 0

12 59

4 Initial 81 10 9 0

Final 8 50 42 0

5 Initial Final

85 28

0 0

9 42

6 30

6 Initial Final

84 20

10 50

0 0

6 30

7 Initial Final

83 19

7 33

6 28

4 20

8 (1) Initial 83 2 14 1 Final 16 10 69 5

(1 ) Predicted optimum gradient.

285

tlmin -

al In C 0

a, :: K

I

(b) I

:e 0.0 L O 8.0 12.0 160

tlmin - Figure 6.14 Result of a Sentinel optimization of programmed solvent LC. Experimental design according to figure 6.13 and table 6.4. (a) Predicted optimum linear gradient and (b) chromatogram obtained with the optimum linear gradient. Stationary phase: Zorbax alkylsilica. Flow rate: 3.0 ml/min. Solutes: A = resorcinol; B = theophylline; C = phenol; D = benzyl alcohol; E = caffeine; F = methyl paraben; G = benzonitrile; H = nitrobenzene; I = cortisone; J = propyl paraben; K = ramrod L = butyl paraben; M = chloro-isopropyl-N-(3-chlorophenyl) carbamate (CIPC); N = progesterone. Figure taken from ref. [627]. Reprinted with permission.

Figure 6.14 shows the resulting quaternary gradient and the resulting chromatogram for the 14-component mixture to which the 7 gradients described in figure 6.13 and table 6.4 have been applied.

It will be clear that the interpretive procedure described here allows the recalculation of the resolution surfaces (and the response surface) after the retention times of the individual solutes have been obtained from the chromatogram at the predicted optimum (figure 6.14), so that an iterative optimization procedure, in which the accuracy of the resulting optimum is improved, is also possible.

The Sentinel gradient optimization method, by analogy with the isocratic Sentinel method, requires a minimum of 7 chromatograms to be recorded before the optimum conditions can be predicted and it requires the retention data of all solute components to be established at each experimental location.

286

la1 Step 3

100 I I

tlrnin-

t t b i:I 8.0

tlmin - Figure 6.1 5: (a) Step-selectivity gradient program designed after “visual interpretation” of the 7 chromatograms obtained during a Sentinel gradient optimization procedure (figure 6.13). (b) Chromatogram obtained with the step-selectivity gradient of figure 6.15a. Sample and conditions as in figure 6.14. Figure taken from ref. 16271. Reprinted with permission.

Advantages are that the selectivity is optimized (secondary parameters) so that optimum resolution can be obtained and that all components of the sample are considered in the optimization procedure. Unlike the result of the gradient optimization procedure suggested by Jandera and ChuraEek, (section 6.3.2.2) the lowest value for the resolution in the chromatogram is maximized and not the resolution of an arbitrary pair of solutes.

However, because of the selection of the max Rs,min criterion the distribution of the peaks over the rest of the chromatogram (other than the critical pair of peaks) is not optimized (see discussion in section 4.3.3). This was realized by Glajch and Kirkland [627] who therefore tried to optimize a “selective multi-solvent’’ gradient, in which a series of segments is allowed in order to try and optimize the resolution in various parts of the chromatogram. They did not describe a formal procedure for the optimization of such step-selectivity gradients, but used “visual interpretation” of the seven chromatograms obtained during the optimization procedure described above to design the gradient shown in figure 6.15a. The chromatogram obtained with this gradient is shown in figure 6.15b.

287

The chromatogram in figure 6.1 5 is only marginally (if at all) better than the one shown in figure 6.14. However, Glajch and Kirkland correctly state that very few of the possibilities of exploiting various selective gradients have yet been explored. If the relative concentrations of the organic modifiers are allowed to vary and if the variation of composition with time is not restricted to linear relationships, then the distribution of the peaks over the chromatogram may still be greatly improved. However, the use of simple continuous gradients is to be preferred to the use of complex multisegment gradients for a number of reasons outlined in the introduction of section 6.3.

predictive optimization method

The Sentinel method of GIajch and Kirkland described above involves the measurement of retention data under gradient conditions and the direct optimization of the selectivity, i.e. the differences between these retention times for different solutes. Jandera et al. [628] have described a predictive optimization method in which 1. retention vs. composition relationships are obtained under isocratic conditions using

2. the retention data using ternary gradients are predicted from the isocratic data, and 3. an adequate ternary gradient is selected based on the predicted retention times.

According to Jandera et al. [628], the isocratic retention behaviour of solutes in ternary solvents in RPLC may be predicted from data obtained with binary mixtures. However, such predictions are only accurate within about 5%. This accuracy is insufficient for the purpose of selectivity optimization, where small differences in retention times between adjacent peaks are of critical importance. Therefore, ideally, binary as well as ternary mixtures should be used in the isocratic experiments. The selection of an adequate ternary gradient takes place largely on a trial-and-error basis. However, instead of trial experiments, trial calculations are performed until a satisfactory result is predicted. Only then will a trial experiment be performed.

Figure 6.16 illustrates the application of the method of Jandera et al. for the selection of a satisfactory linear gradient for the separation of a mixture of 9 phenolic solutes. It is seen in figure 6.16a and figure 6.16b that the mixture is not completely separated using either a binary methanol-water (chromatogram a) or a binary acetonitrile-water gradient (chromatogram 6). Also, an ”iso-selective” linear gradient, in which the ratio between the concentrations of methanol and acetonitrile is kept constant, provides insufficient resolution. Figure 6.16d shows the chromatogram obtained with a linear ternary gradient which was predicted to provide a satisfactory separation. Indeed, the resolution is better than in any of the previous chromatograms (a, b and c) and is sufficient with the column and conditions used in figure 6.16.

Figures 6.16e and 6.16f show two chromatograms using gradients which were predicted to yield insufficient separation. Using the optimization procedure of Jandera et al., a number of gradient programs can be tested by calculating the resulting chromatograms, so that the number of experiments required can be greatly reduced.

It is interesting to note that the gradient predicted by Jandera et al. could not have been arrived at using the Sentinel method described in figure 6.13.

The predictive optimization method of Jandera et al. is designed to yield an “adequate” result. In other words, a threshold optimization criterion is used (eqn.4.23). Once a certain

several modifiers,

288

1 I I

60 V/ml 30 20 10 -0

I

LOV/ml 30 20 10 - 0 8.9

I I

LOVlml 30 20 10- 0

8 6 3

I I I 1

V/ml 30 20 10 - 0

I I I

Vlml 30 20 10 - 0 (fl

8.9

7 II 3

I

Vlml30 20 10 -0

Figure 6.16: Illustration of the predictive optimization method for ternary gradients in RPLC of Jandera et al. [628]. All figures were recorded with linear gradients from 100°h solvent A to l0O0/o solvent B in 60 min. Stationary phase: Lichrosorb C18. Flow rate: 1.0 mllmin. Solutes: 1 = 4cyanophenol; 2 = 2-methoxyphenol; 3 = 4-fluorophenol; 4 = 3-fluorophenol; 5 = m-cresol; 6 = 4-chlorophenol; 7 = 4-iodophenol; 8 = 2-phenylphenol; 9 = 3-t-butylphenol. Mobile phase components: (a) solvent A: 20% methanol (in water), solvent B: 100°h MeOH; (b) A: 100% water, B: 100°/o acetonitrile (ACN); (c) A: 100°h water, B 60°h MeOH + 40°/o ACN; (d) A: 20°/0 ACN, B: 100°h MeOH; (e) A: lOoh ACN, B lOOoh MeOH; ( f ) A: 30°h ACN, B lOOoh MeOH. Figure adapted from ref. [628]. Reprinted with permission.

289

minimum resolution is predicted for all the pairs of peaks in the chromatogram this is said to be an adequate or sufficient result, provided that it can be verified experimentally.

A disadvantage of the method of Jandera et al. is the requirement to know the isocratic retention vs. composition relationships. If these data are not already known, which will most likely be the case in the optimization of real-life samples, the experimental effort needed to obtain sufficient data of sufficient accuracy will be very large.

6.3.2.4 Discussion

We have seen that the primary (program) parameters can be optimized in one of several ways. If the actual gradient consists of a single segment, four parameters may be considered, of which two (the slope and the initial composition of the gradient) are most relevant for the result in terms of resolution. The final composition may affect the required analysis time (the program should not extend beyond the chromatogram), whereas the shape of the gradient will have an effect on the overall distribution of the peaks over the chromatogram.

The Simplex optimization procedure allows different optimization criteria to be used, so that a good distribution of all the peaks over the chromatogram may be aimed at. However, the Simplex method does require a large number of experiments, and therefore seems to be very inefficient for optimization of the primary parameters alone.

Without knowing much about the sample, the Snyder approach may also be used to optimize the program parameters. This is an empirical approach in which the sample properties are largely disregarded, but it does lead to the formulation of reasonable working conditions after only one or two chromatograms have been obtained.

The approach of Jandera and ChuraEek allows the optimization of the resolution of one given (arbitrary) pair of sample components and the minimization of the retention volume of another (arbitrary) solute. It requires knowledge of the isocratic retention vs. composition relationships of these three solutes. The information needed may be acquired from gradient elution experiments performed as part of the optimization procedure, or from separate isocratic experiments. The selection of the three arbitrary solutes considered during the optimization process appears to have a large effect on the result and the resolution cannot be optimized throughout the chromatogram. In principle, the retention behaviour of all sample components under gradient

conditions can be calculated once two experimental retention times have been obtained (either under isocratic or under gradient conditions) [616,618,623,629]. Therefore, in principle, it ought to be possible to calculate optimum gradient parameters from two solvent programmed experiments [630]. However, to account for inaccuracies in the gradient elution data [630,631,632], a few more experiments may be required. Procedures to obtain isocratic retention vs. composition relationships from a series of gradient experiments have also been described by Jandera and ChuraEek [633,634]. Determination of the optimum program parameters based on the retention vs. composition relationships for all (or all major) sample components will require quite complicated and extensive calculations. It is the charm of the methods described in section 6.3.2.3 that the required computational effort is either minimal (i.e. a few computations, which can easily be performed on a pocket calculator for the Snyder method) or small (i.e. a limited number

290

of computations involving more complex but analytical expressions for the method of Jandera and ChuraEek).

A second reason not to become involved in extensive calculations for the complete mathematical optimization of the (primary) program parameters is that a more powerful way to optimize the separation of all sample components in the mixture may be to optimize the selectivity of the gradient by varying the nature of the mobile phase components (secondary parameters).

Three methods appear to be available for optimizing the selectivity in programmed solvent LC: 1. the Simplex procedure, 2. interpretive methods, and 3. the predictive optimization method. The Sentinel method is the outstanding exponent of the group of interpretive methods, as it has already been applied successfully for selectivity optimization in programmed solvent LC. However, other interpretive methods, based either on fixed experimental designs or on iterative procedures, can be applied along the same lines. It was seen in section 6.3.2.3 that the extension of the Sentinel method to incorporate gradient optimization was fairly straightforward.

For the Simplex optimization procedure the common disadvantage of the large number of required experiments weighs more heavily for programmed analysis, because more time is required for each experiment (see section 6.1). Also, the response surfaces encountered in the optimization of selectivity in programmed solvent LC appear to be no less convoluted than the ones encountered in isocratic selectivity optimization [627], so that there is again a large chance that the Simplex algorithm will arrive at a local rather than the global optimum.

The advantages and disadvantages of interpretive methods are also fully analogous to those listed in chapter 5 (section 5.5). Fewer experiments are needed, but the recognition of the different sample components is required in each experiment. Contrary to the complete optimization of the (primary) program parameters, interpretive methods for the optimization of the selectivity under programmed conditions do not require more complicated calculations than do their isocratic analogs. This was amply demonstrated by Glajch and Kirkland [627], who used the same computer program for the two optimization processes.

The predictive method of Jandera et al. [628] requires knowledge of the isocratic retention data of all solute components in binary and (preferably) ternary mobile phase mixtures. Once these data are available, the method may be very helpful in obtaining an “adequate” (but not an optimum) separation with a ternary gradient. Unfortunately, the data required for the application of this predictive method are almost never available, and hence a large number of experiments need to be performed before any predictions can take place. When this is the case the method is of very little practical use.

The final question we need to address in this discussion is the general need for gradient optimization procedures, both for optimizing the program parameters and for optimizing the selectivity. In section 6.1 several disadvantages of programmed analysis were described and it was concluded that its application should be avoided if possible. Especially for large

29 I

series of samples, the use of alternative (multicolumn) techniques should be considered. In isocratic analysis, the general motivation is that the larger the supply of a particular

kind of sample, the more optimization effort is warranted. In programmed analysis this is not true. In that case, the larger the supply of samples, the larger the urge to look for alternativemethods. Therefore, gradient optimization procedures are only relevant if they represent a limited effort. It yet remains to be established just how far the word "limited" will reach.

6.3.2.5 Summary

The characteristics of the different methods for gradient optimization are summarized in table 6.5. In table 6Sa, the different methods for the optimization of the program parameters are compared. Bearing in mind that a large effort is generally not warranted for the optimization of programmed analysis (see section 6.3.2.41, we shouldconclude that the Simplex method is not suitable because of the large experimental effort required, and

Table 6.5: Summary of the characteristics of gradient optimization methods. a. Optimization of primary (program) parameters

Simplex Snyder Jandera Complete method method method mathematical

optimization

No-experiments Large

Computational effort Moderate

Resolution YeS optimization (1)

Time Yes optimization (3)

Recognition None requirements

1 o r 2 few few

Minimal Small Large

No One All solutes (2) pair

YeS One YeS (4) solute (3)

None Three All (major) solutes solutes

Complete automation Ea4y Easy Difficult Difficult

(1) Any optimization criterion can be selected that assigns a single criterion value to each

(2) Optimum slope is selected to provide optimum elution conditions for "average" solutes. (3) Time factor may be incorporated in optimization criterion. (4) Initial and final conditions may be adapted to first and last peaks to minimize analysis time.

chromatogram.

292

that a complete mathematical optimization is unattractive because of the large computational effort involved. The method proposed by Jandera and ChuraEek requires somewhat more effort than that of Snyder. It requires some calculations, the recognition of three solutes, and knowledge of the isocratic retention vs. composition relationships for these solutes, obtained either during the optimization procedure or from independent (isocratic) experiments.

Table 6.5: Summary of the characteristics of gradient optimization methods. b. Selectivity optimization.

Simplex Interpretive methods

method Fixed Iterative Predictive design design optimization (Sentinel) method

No.experiments

Computational effort

Optimum found

Accuracy of optimum

Impression of response surface

Optimization criterion

Recognit ion required

Complete automation

Large

Small

Local

High

Poor

Single value

No

Easy

7

Moderate

Global

Low

Good

Any

Yes

Partly easy (5)

5-10

Moderate

Global(2)

High

Moderate

Any

Yes

Difficult

Large (1)

Moderate

“Adequate” (3)

-

Poor

Rs.min ’ x (eqn.4.23)

Yes (4)

-

(1) Large number of isocratic experiments required. (2) Global optimum may be overlooked if large areas remain unsearched. (3) This method aims at achieving an adequate rather than an optimum result. (4) Recognition of the peaks is required during the isocratic experiments to establish the retention

( 5 ) Experimental part may easily be automated. vs. composition relationships.

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On the other hand, this method does take into account the resolution of the most critical pair of solutes. If this pair can easily and unambiguously be identified, then the method of Jandera and ChuraiSek may be worth the extra effort.

In table 6.5b the methods for selectivity optimization are compared. Again, the Simplex method turns out to be unattractive, because of the large number of experiments required. Also, the resulting optimum may well be a local one.

Interpretive methods will generally arrive at the global optimum after a limited number of experiments. However, (by definition) the recognition of the individual solutes is required in each experimental chromatogram. Also, the computational requirements are relatively high, especially if the simultaneous optimization of several parameters is considered. For example, (linear) ternary gradients (one parameter) will be much easier to optimize than quaternary gradients (two parameters).

Interpretive methods may possibly be used for the complete optimization of selectivity in solvent programmed LC. If any gradient program (multisegment gradients, see figure 6.2e) is allowed, then it may be possible to optimize the resolution of each pair of peaks in the chromatogram. This possibility has been largely unexploited. However, it also appears to be of limited practical interest, because of the disadvantages of multisegment gradients compared with simple, continuous gradients (see introduction section 6.3).

REFERENCES

601. L.R.Snyder and J.J.Kirkland, Introduction to Modern Liquid Chromatography, 2nd

602. J.F.K.Huber, E.Kenndler, W.Nyiri and M.Oreans, J.Chromatogr. 247 (1982) 21 1. 603. W.Blass, K.Riegner and H.Hulpke, J.Chromatogr. 172 (1979) 67. 604. C.J.Little, D.J.Tompkins, O.Stahel, R.W.Frei and C.E.Goewie, J.Chromatogr. 264

605. W.E.Harris and H.W.Habgood, Programmed Temperature Gas Chromatography,

606. JCGiddings in: N.Brenner, J.E.Callen and M.D.Weiss (eds.), Gas chromatography,

607. V.V.Berry, J.Chromatogr. 236 (1982) 279. 608. P.J.Schoenmakers, H.A.H.Billiet and L.de Galan J.Chromatogr. 185 (1979) 179. 609. L.R.Snyder, MStadalius and M.A.Quarry, AnaLChem. 55 (1983) 1421A. 610. K.A.Cohen, J.W.Dolan and S.A.Grillo, J.Chromatogr. 316 (1984) 359. 61 1. J.L.Glajch and J.J.Kirkland, AnaLChern. 54 (1982) 2593. 612. F.H.Walters and S.N.Deming, AnaLLett. 17 (1984) 2197. 613. H.-J.Stan and B.Steinbach, J.Chromatogr. 290 (1984) 31 1. 614. D.W.Grant and M.G.Hollis, J.Chromatogr. 158 (1978) 319. 615. V.BartO, J.Chromatogr. 260 (1983) 255. 616. L.R.Snyder, J.W.Dolan and J.R.Gant, J.Chromatogr. 165 (1979) 3. 61 7. L.R.Snyder in: Cs.Horvath (ed.), HPLC, Advances and Perspectives, Vol.1, Academic

61 8. P.Jandera and J.ChurhEek, Gradient Elution in Column Liquid Chromatography.

edition, Wiley, New York, 1979.

(1983) 183.

Wiley, New York, 1966.

Academic Press, New York, 1962, pp.57-77.


Theory and practice, Elsevier, Amsterdam, 1985.

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619. M.W.Watson and P.W.Carr, Anal.Chem. 51 (1979) 1835. 620. D.M.Fast, P.H.Culbreth and E.J.Sampson, Clin.Chem. 27 (1981) 1055. 621. J.C.Berridge, J.Chrornatogr. 244 (1982) 1. 622. J.W.Dolan, J.R.Gant and L.R.Snyder, J.Chromatogr. 165 (1979) 31. 623. P.Jandera and J.ChuraEek, J.Chromatogr. 192 (1980) 1. 624. P.Jandera and J.ChuraEek, J.Chromatogr. 192 (1980) 19. 625. C.M.Noyes, J.Chrornatogr. 266 (1983) 451. 626. H.J.Issaq, K.L.McNitt and N.Goldgaber, J.Liq.Chromatogr. 7 (1984) 2535. 627. J.L.Glajch and J.J.Kirkland, J.Chromatogr. 255 (1983) 27. 628. P.Jandera, J.ChuraEek and H.Colin, J.Chrornatogr. 214 (1981) 35. 629. P.J.Schoenmakers, H.A.H.Billiet, R.Tijssen and L.de Galan, J.Chrornatogr. 149

630. M.A.Quarry, L.R.Grob and L.R.Snyder, J.Chrornatogr. 285 (1984) 1. 631. M.A.Quarry, L.R.Grob and L.R.Snyder, J.Chromatogr. 285 (1984) 19. 632. P.Jandera and J.ChuraEek, J.Chromatogr. 192 (1980) 37. 633. P.Jandera and J.ChuraEek, J.Chrornatogr. 91 (1974) 223. 634. P.Jandera and J.ChuraEek, J.Chrornatogr. 93 (1974) 17.

(1978) 519.

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CHAPTER 7

SYSTEM OPTIMIZATION In the preceding chapters we have dealt with the various stages of the process of

developing methods for chromatographic analysis. We discussed the selection of the appropriate chromatographic method in chapter 2. Chapters 3,4 and 5 described the parameters, the criteria and the procedures, respectively, that may be used to optimize the retention and the selectivity. In chapter 6 this approach was extended to include the optimization of programmed analysis methods.

At the end of the selectivity optimization procedure, we have established the optimum combination of a mobile and a stationary phase (the optimum phase system). In some cases, the procedure has been conducted on the column and instrument on which the analysis will eventually take place (“final analytical column”). For example, if we have optimized the mobile phase composition for a particular separation of inorganic anions on a dedicated ion chromatography system, we may not be able to vary the dimensions of the column or to select different pieces of instrumentation.

Preferably, however, we may still optimize the dimensions of the column after we have established an optimum phase system. The available instrumentation puts constraints on the column that may be used and hence, ideally, we should also have the possibility to select the most appropriate instrumentation for a given application.

In this chapter we will briefly discuss the selection of optimum columns and instruments, in other words the final optimization of the complete system.

7.1 INTRODUCTION

This chapter describes the final configuration of the chromatographic system (column and instrument) after the optimization of the phase system (the combination of the stationary and the mobile phase) has been completed. The entire optimization process is illustrated in figure 7.1. This figure shows the different stages in the process from the moment at which it has been decided (either on the basis of literature information or on the basis of figure2.1) which chromatographic method should be used. For example, it may have been decided that RPLC is the method of choice. It should also be decided what kind of detector will be used. For instance, we may choose to use a UV absorption detector.

The “build instrument” stage in figure 7.1 implies that a system should be assembled that contains the appropriate column and detector. For the optimization of separations with (capillary) GC we may also have to decide upon the type of injector to be used. However, at this stage only a workable system (one in which all relevant components can be injected and detected) and not an optimized system has to be assembled or “built”.

The method development process will be aided if we are able to use sophisticated instrumentation (see also section 1.7.2). Automated injection and data handling will allow a number of experiments to be performed without the requirement of an analyst being present. Moreover, we have seen in chapter 5 (section 5.6) that the use of sophisticated detection techniques (dual-channel or multi-channel detectors) may be of help in the optimization process.

296

The inclusion of programming options (temperature programming in GC, solvent programming in LC) in the instrument may also be helpful, not only if a programmed analysis may be the result of the Optimization procedure (chapter 6), but also to provide a scanning (or “scouting”) facility for unknown samples (section 5.4).

In many cases, the “build instrument” stage only involves the insertion of the column of choice in an existing instrument configuration for method development.

Bui Id Instrument

time - Figure 7.1: Different stages in the process of developing methods for chromatographic analysis. The individual stages of the process are located on a curve that indicates the sequence of events (horizontal axis) and the relative importance of the various steps (vertical axis). See text for further explanation.

The next two stages in figure 7.1 (“optimize k” and “optimize a”) represent the selectivity optimization process, which we have discussed extensively in previous chapters. It is assumed in figure 7.1 that most time (and effort) is spent at this stage.

We then come to the final two stages, the optimization of the column and the instrument. These will be the subject of this chapter. Both of the final stages can be realized in a relatively short time.

In figure 7.1 a relative importance is assigned to each stage of the process. Because it is (as yet) impossible to predict the chromatographic behaviour of solutes from structural information alone and, moreover, because the structure of all sample components is usually not known, we have to rely on chromatographic experiments for the optimization of the selectivity. Consequently, the first step is that an instrument should be “built”.

The next most important factor is to bring the capacity factors into the optimum range. At the same time or immediately thereafter, we should try to optimize the selectivity (a). Both are very important stages in the method development process, because no separation will be obtained if either k = 0 or a= 1 (see section 1 S), no matter how efficient the column and how good the instrument. Very large k values should also be eliminated at this stage, because of both time and sensitivity considerations (see e.g. figure 6.lb).

The column and instrument optimization stages are not of the same degree of importance as the preceding stages. However, this by no means implies that they are irrelevant. Analyses may be performed in a fully adequate way on “overdesigned” columns with large numbers of theoretical plates, but this will usually involve long analysis times

297

and moderate sensitivities. By optimizing the column dimensions and adapting the characteristics of the instrument to those of the column, the speed and the sensitivity of the analysis may be greatly enhanced.

The result of one of the optimization procedures described in chapter 5 is a chromatogram with the best achievable separation of peaks. Depending on the optimization criterion (chapter 4), this optimum chromatogram may have been defined in one of several ways. For example, the separation may be optimized so as to require the shortest possible analysis time on a given column, or to require the lowest possible number of plates on a tailor-made column.

1. The lowest value for the resolution ( Rsemin) or for the separation factor (Smin) observed in the chromatogram.

2. The capacity factor of the last peak ( k J . Once a column and flow rate have been selected for the analysis, this parameter determines the required analysis time (10).

The lowest value of R, or S observed in the chromatogram determines the number of plates that is required for the adequate resolution (characterized by R,,,,) of all the peaks in the chromatogram. This became clear in chapter 4 (section 4.4.3), where we found

The resulting optimum chromatogram may be characterized by two parameters:

or

(4.35)

(4.34)

where N , is the number of plates available on the column used during the optimization procedure.

The required number of plates (Nnd is the most relevant factor for the selection of the type of column and the column dimensions. However, there are various other factors which we need to consider in the selection of the most suitable column for a given analysis: 1. The instrumental constraints, such as the maximum acceptable pressure drop over the

column. 2. The typical size of the sample (injection volume) and the concentrations of the

components to be analyzed. 3. The combination of column and instrument, for instance the selection of injectors and

detectors that allow the optimized separation to be performed. In packed columns, there are two parameters which may be varied independently in

order to optimize the column characteristics, i.e. the diameter of the column and the diameter of the particles. In open columns, only the column diameter may be varied. Additionally, the phase ratio may be varied by changing one of the “capacity parameters” (see section 3.5). For packed columns these parameters include the surface area of the packing material, the column porosity and the stationary phase film thickness. For open columns only the latter parameter is relevant.

In subsequent sections we will discuss the implications of the optimization of the efficiency and the sensitivity for optimum dimensions of the column. In section 7.4 we will address the consequences of these optimum dimensions in terms of instrumentation requirements.

298

7.2 EFFICIENCY OPTIMIZATION

The optimization of the efficiency of the chromatographic system involves the selection of a column with a sufficient but not excessive number of plates. If a column is used with twice the required number of plates, then the observed resolution would exceed the required value by 40%, but both the analysis time and the pressure drop over the column would be double the required value. Moreover, the sensitivity of the detection would be decreased by 40% (see section 7.3). It is clear that we should aim to use a column with the optimum (i.e. the required) number of plates.

7.2.1 Open columns vs. packed columns

The main fundamental difference between open columns and packed columns is in the required pressure drop, for which Darcy's law provides a general equation:

A p = q L u / B ,

where Ap is the pressure drop over the column, q the viscosity of the mobile phase, u the (average) linear velocity, and B, a constant representing the specific permeability coefficient of the column. For an open column*

B , = d: / 32,

whereas for a packed column

B, z d p ' / 1000. (7.2a)

Consequently, if we compare a packed column with an open column for which dp= d , , we find a 30 times lower pressure drop over a capillary column of the same length. Conversely, if we keep the pressure drop constant, much longer capillary columns may be used, yielding a much higher number of plates.

For the comparison of different kinds of columns it has become increasingly common to use the reduced plate height (h), defined as

h = H / d

and the reduced linear velocity (v), defined as

v = u d / D ,

(7.3)

(7.4)

where H is the conventional plate height (in units of length) and D, the diffusioq coefficient of the solute in the mobile phase. In eqm(7.3) and (7.4) the diameter d represents the diameter of the particles in a packed column (d,) or the diameter of an open

* The constants in eqns.(7.2) and (7.2a) are dimensionless if p is expressed in Pa (N/m2 ), q in Ns/m2, L, d, and d,, in m and u in m/s.

299

column (dJ An equation for the analysis time (td can be found if we combine eqns.( 1.3), (1.6) and (1.18):

N H t, = r,fl + kJ = L/u(1 + kJ =-(I + kd.

U

With eqna(7.3) and (7.4) we find

(7.5)

For packed columns, typical values [701] are h = 3 and v= 3, so that h/ v= 1. For open columns typically h = 1.5 and v= 5, so that h/v=0.3. Consequently, capillary columns will lead to analysis times that are about three times shorter (for dp= dJ for thesame separation ( N and k constant). Therefore, in principle, capillary columns are superior to packed columns. Unfortunately, capillary columns cannot always be used. This arises from the occurrence of the diffusion coefficient (D,J in eqn.(7.6). Typically, D, is I~,OOO times larger in gases than it is in liquids. This necessitates the use of very small particles (typically 5-10 pm) in HPLC columns. If we compare packed and capillary columns with d,,= d, , which is a reasonable assumption for GC [7021, then capilIary columns with very small internal diameters need to be considered for LC [703]. Such very narrow columns impose extreme demands on the instrumentation, and at present open tubufar columns cannot be used for practical LC separations.

7.2.2 Gas chromatography (open columns)

In GC we have a real choice between packed columns (d,= 100-200 p; 150-65 mesh) and open columns (d,= 50-500 pm). Capillary columns have the advantage of enhanced speed of analysis (eqn.7.6). In order to exploit this advantage, “narrow-bore” capillaries (d,< 100 p) should ideally be used. However, such columns require relatively high inlet pressures (especially for high plate counts)* and considerable experimental modifications and have a very low sample capacity [702].

Because of all these reasons, so-called “wide-bore” capillaries (d,w 500 pm) have recently gained considerable popularity. These columns, which are usually provided with a thick (about 1 pm) film of stationary phase, behave in a fairly similar way to packed columns. They show low pressure drops (allowing them to provide a much higher efficiency than packed columns), may easily be installed in most instruments and have a high sample capacity. However, they also behave similar to packed columns in terms of separation speed. Therefore, the current capillary cotumns with diameters between 100 and 300 pm form a reasonable compromise between instrumental limitations and theoretical promises.

Despite the current popularity of “wide-bore” capiIIary columns, it is to be expected that advances in instrumentation and column technology, combined with the increased

* For high pressure drops, because of the compressibility of the mobile phase, the favourable effect of reducing the column diameter is less than that suggested by eqm(7.6). In that case t ,a d, is a better approximation than r,a d,‘ [702].

300

acceptance of capillary columns amongst practical chromatographers, will lead to a further reduction of the diameter of the column in the future.

Packed columns may still be used in GC as a robust tool to effectuate simple, routine separations. They have two fundamental disadvantages relative to capillary columns: 1. longer analysis times, and 2. limited efficiency due to a high pressure drop per unit length.

Because of these two reasons the use of packed columns should be limited to simple separations. In that case there are some practical advantages: 1. 2.

3.

4. 5.

Large sample capacity. Because the entire sample can be brought onto the column, the accuracy of quantitative analysis may be enhanced. High contamination capacity, i.e. “dirty” samples can be injected onto the column without causing rapid degradation. Low instrumental requirements and easy operation. Detectors which intrinsically require a large volume (e.g. (FT)IR spectrometers) may be used more readily.

Therefore, despite the theoretical superiority of capillary columns, a place remains for the use of packed columns in GC.

It is important to notice at this stage that the result of a selectivity optimization procedure is often a separation that can be realized with a limited number of theoretical plates. For example, we have seen in chapter 4 that the complete resolution of 10 equally distributed peaks requires only 400 plates in the optimum situation at which the lowest analysis time can be achieved (see figure 4.1 1 and related discussion). Large numbers of theoretical plates are more appropriate for very complex samples, which contain large numbers of peaks, making selectivity optimization an unrealistic proposition.

If the required number of plates is moderate (say severaI thousands), then short capillary columns may be used to provide fast analysis of the sample. The required column length and retention time can easily be calculated from eqm(7.3) and (7.6). For example, if we operate a capillary column with a diameter of 200 pm at a reduced velocity of 5 with a reduced plate height of 1.5, then 2000 theoretical plates require a column with

L = N h d, = 2,000 x 1.5 x 0.02 = 60 cm

and, using eqm(7.6) with a diffusion coefficient of 0.1 cm2/s and a capacity factor of 3 for the last solute

1.5 (0.02)2 t, = 3 N ( l + k J = x 2,000 x 4 = 10 s.

V D m 5 x 0.1

This illustrates the kind of separations that may be realized on conventional capillary columns (d,= 0.2 mm) if the selectivity has been optimized. The instrumental implications of such rapid analysis will be addressed briefly in section 7.4.

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7.2.3 Liquid chromatography (packed columns)

In LC, packed columns are used for practical separations. Eqn.(7.6) shows the speed of analysis to be proportional to the required number of plates and to the square of the (particle) diameter. We used this equation in chapter 4 to derive an expression for the required retention time under conditions of constant flow rate and particle size ( t,,elf,d; eqn.4.48). However, eqn.(7.6) suggests that the speed of analysis may always be increased by decreasing the particle size. In other words, it suggests that the smallest available particles should always be used in packed columns. This interpretation is too simple.

In LC (and to a lesser extent also in GC) the limiting factor is the maximum allowable pressure drop over the column. As an example, we will look at the dimensions of packed columns for LC when the operating pressure (Ap) is fixed at the maximum value allowed by the instrumentation [704].

The pressure drop is given by Darcy’s law (eqn.7.1). Optimum flow rates on columns with different particle sizes can be related by using the same reduced velocity (v. eqn.7.4) for each column. Since the diffusion coefficient D, is a constant, we find for the ratio of the linear velocities on two columns (1 and 2)

U , / U , = dp.2 Idp, , . (7.7)

If the reduced velocities on the two columns are equal, then the reduced plate height (h eqn.7.3) may also be expected to be equal, and, hence, the column length varies according to

L = H N,,, = h N,,, d p . (7.8)

Substitution of eqns.(7.2a), (7.4) and (7.8) in eqn.(7.1) yields

All parameters on the right-hand side of eqn.(7.9) are constants, and therefore the optimum particle size for a given separation is proportional to the square root of the number of plates required.

Substitution of eqn.(7.9) in eqn.(7.6) now yields

1000 qh2 t, = . N:, (1 + kd= /3 N i e (1 + kJ

AP (7.10)

where /3 is a constant. An important conclusion from eqn.O.10) is that the analysis time is proportional to the

square of the required number of plates. Eqn.(7.10) also reveals that the analysis time may be decreased by three factors: 1. A decrease in h. This is achieved by using good column packing techniques. 2. A decrease in q. This can be realized by using low viscosity solvents [705] or elevated

3. An increase in the pressure drop over the column. temperatures.

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From eqn.(7.9), we may easily obtain a good estimate for the optimum particle size for HPLC separations. If the maximum operating pressure of the instrument is 400 bar (40 MPa), the viscosity of the eluent 1 CP ( low3 Pas; e.g. water at 20 "C) and the diffusion coefficient m2/s, then we find around the optimum conditions (v=3,h= 2) from eqm(7.9)

with d, in m. If we express d, in pm we find

d, = v(Nn,/6500) M K / 8 0

This shows that theoretically a particle size of about 1.2 pm should be used for a separation that requires 10,000 plates. The associated column length is 2.5 cm (eqn.7.8) and the analysis time for k= 1 about 20 s.

In practice, we operate LC systems still well above the optimum flow rate. We work well below the maximum operating pressure of the pump (say 200 bar), at a higher reduced velocity (e.g. v= 10) and with a correspondingly higher reduced plate height (h = 4). Under these practical conditions we find:

20.106 = 0.5.10'5 Nne - - -

dg lo3 * - 4 . 1 0 . lop9

with d,, in m, and again with dp in pm

dp = l / ( N n e / 5 0 0 ) M K / 2 0

Therefore, under practical conditions a particle size of about 5 pm may be used to realize 10,000 theoretical plates. The corresponding column length is 20 cm and the retention time for k = 1 is 200 s.

Another practical consideration is that only a limited number of particle sizes and column lengths is available. Figure 7.2 illustrates the practical situation using the same estimates as for the latter calculation above. In a logarithmic plot, the required column length as a function of the required number of plates forms a straight line with a slope of 1. Such straight lines are shown for four different particles sizes (3, 5, 10 and 20 pm) assuming a reduced plate height of 4. The maximum allowable pressure limits the column length. This is illustrated in figure 7.2 by the thin lines with slope 3/2. The two lines represent pressure limits of 200 and 400 bar and are calculated for v= 10, D, = m2/s and q= 1 mPa.s. Eqns.(7.8) and (7.9) readily allow lines to be constructed for other conditions.

Three columns are indicated with heavy dots in figure 7.2. With these columns separations may be performed that require up to 10,000 plates (under the conditions of figure 7.2). The dimensions and characteristics of these columns are listed in table 7.1 (columns 1-111).

A possible fourth column in illustrated in figure 7.2 by an open circle. This 80 cm long column packed with 10 pm particles is not an attractive column in practice, because of both its length and the required analysis time. However, it should be noted that shorter columns

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(L < 40 cm) packed with 10 pm particles are of even less practical interest, because the number of plates is less than that obtained on the 20 cm long column packed with 5 pm particles (column 111). The only advantage of columns packed with larger particles is the lower pressure drop (see table 7.1).

t Id Llcm

lo2.

10

1

lo2 lo3 loL lo5 N-

Figure 7.2 Example of the relationship between the required number of plates and the required column length (h = 4) for four different particle sizes in HPLC. Thin lines indicate the maximum column lengths for pressure drops of 200 and 400 bar, with v= 10, D,= m2/s and q= 1 mPa.s. Heavy dots indicate possible columns for separations requiring up to 10,000 plates. Open circle represents a column with 20,000 plates.

Table 7.1: Example of a set of columns (1-111) that may be used to realize separations in HPLC requiring up to 10,000 theoretical plates. Column IVmay be used to extend the set to 20,000 plates. Conditions are given in figure 7.2.

I 3 3 2,500 111 9

I1 5 3 4,200 185 15

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The set of columns in table 7.1 is a reasonable one for practical purposes. Reasonable assumptions have been made both for h and for v. However, the pressure drop over the column may vary quite considerably, depending on the viscosity of the eluent. A viscosity corresponding to that of water was used for the calculations of figure 7.2 and table 7.1, but in the practice of RPLC a viscosity that is double that value may be observed for some mobile phases (e.g. methanol-water 60-40 [705]). For this reason, columns have been selected for which the pressure drop is below 200 bar under the conditions of table 7.2.

7.2.4 Summary

1.

2.

3.

4.

5.

6.

7.

The following conclusions may be formulated after the discussion in this section. In GC, open (capillary) columns provide faster analyses and allow larger numbers of plates than do packed columns. However, in some cases packed columns may still have prevailing advantages. In LC, open columns cannot yet be used forpractical separations. GCseparations for which the selectivity has been optimized may often be performed very rapidly on short capillary columns with a conventional diameter (0.2 mm). “Narrow-bore” capillary GC columns will lead to shorter analysis times, but impose stringent demands on the instrumentation. “Wide-bore” capillary GC columns allow easy operation and have a large sample capacity. However, they give rise to relatively long analysis times. The optimum particle size for packed columns in HPLC varies with the (square root of the) number of plates required for the separation. The time of analysis in HPLC may be reduced by using well-packed columns, low viscosity eluents and high column pressure drops. A small numbpr (e.g. 3) of HPLC columns allows the realization of separations that require up to 10,000plates under conditions that form a reasonable compromise between theory and practice. Relatively large particles (dp> 10 pm) are of limited practical importance.

7.3 SENSITIVITY OPTIMIZATION

An important aspect of the chromatographic process is the sensitivity of the detection. For most common detectors, the recorded signal is directly proportional to the concentration of the solute in the effluent from the column*. We have seen in chapter 1 (eqn.l.15) that the observed peak height (h,) can be related to the peak area (A) by

h o = A / c T ~ (7.1 1)

where CT is the standard deviation of a Gaussian peak. We may express the peak height in concentration units:

ho = cmax (7.12)

* For some detectors (e.g. flame ionization, FID and mass spectroscopy, MS) the detected signal is proportional to the mass flow of the solute entering the detector, a quantity which equals the product of the concentration and the volumetric flow rate. Therefore, the detector sensitivity is still directly related to the solute concentration.

305

where the subscript max indicates the concentration at the peak maximum, and the standard deviation in volume units (CT= 0,). In that case the area (A) should be expressed in units of mass (concentration x volume). The law of mass conservation prescribes that the mass of solute leaving the column should equal the mass of solute entering the column. The latter can easily be expressed in terms of the concentration of the solute in the sample solution (ci) and the injected volume ( Vinj):

A = ci Fnj

and with eqn~(7.11) and (7.12)

(7.13)

(7.14)

Using the common equations for the plate count and the retention volume we now find

c,,, - -.- - ci vinj fi G V R

- ci vinj -\rN I& v, ( l + k J

--.-.- (7.15)

Eqn.(7.15) is the key equation for the optimization of chromatographic sensitivity. Naturally, the peak height is proportional to the concentration of the solute in the sample and to the volume of the injected sample. However, this proportionality holds over a limited range and we cannot increase these two quantities indefinitely without having to sacrifice another vital characteristic of the system, the linearity of detection. The proportionality between c,,, and the product ciVinj ends when N may no longer be considered as a constant. Consequently, the aforementioned product may be increased until the plate count starts to be affected.

Injection of large volumes

A series of “tricks” has been devised for the injection of large volumes of samples, all of which aim at increasing Vinj without affecting N.

In GC the injection may take place at a temperature that is lower than that of the column oven. The solute bands will be concentrated in a small volume and may be brought into the column by a subsequent heating of the cold zone. If this zone is part of the column itself we talk about “cold (on-column) injection”, if it is part of a separate injector unit we talk about “cold trap” injection. A similar “band compression” effect may be achieved in a different way by leaving the first part of the capillary column “uncoated” (i.e. no stationary phase present). The solute band will then be compressed at the point where the stationary phase starts to be present in the column. This band compression technique is usually referred to by the unfortunate term “retention gap” [706].

In LC the solute bands may be concentrated on a pre-column, which is eluted with a weak eluent (low eluotropic strength). A simple way in which the effective injection volume may be reduced considerably is by dissolving or diluting the sample in a solvent that is much weaker than the eluent [707]. This has the effect that the initial capacity factor during

306

the injection of the sample is high, so that the solute bands are effectively compressed at the top of the column (compare “cold” injection in GC).

Effect of column efficiency

Eqn.(7.15) suggests that the sensitivity (cmaX) increases with increasing plate count (N). However, this is only true if all other factors, in particular V,, are kept constant. A better way to look at the effect of the column efficiency on the sensitivity is to consider the ratio

(7.16)

where L is the column length, H the plate height, d, the column diameter and E the column porosity ( ~ = l for open columns). The implications of eqn.(7.16) become clear if we introduce the reduced plate height (h).

First for a packed column, where

h = H / d , (7.3a)

and therefore - 4 1 1 1 - ---.-.-.- V, nsdh d,‘ d L ddp

(7.17)

Eqn.(7.17) shows that an increase in the column efficiency by way of a decrease in the reduced plate height (h) has a positive effect on the sensitivity. Therefore, well-packed columns should be used at (or just above) the optimum flow rate*. If that is the case, then h may be considered as a constant (2 < h < 3), i.e. considered to be independent of the column diameter and the particle size.

Eqn.(7.17) shows that an increase in the column efficiency simply by increasing the column length ( L increases with h constant) has an adverse effect on the sensitivity.

For open columns the reduced plate height may be defined as

h = H / d , , (7.3b)

where, at the optimum flow rate h z 0.7. With E= 1 we find G 4 1 1 - -_-.-.- V, ndh d;l2 dL‘

(7.18)

Again, the sensitivity is enhanced by working at the optimum flow rate (lowest value for h) and decreased by increasing the column length.

Effect of column diameter

For packed columns eqn.(7.17) suggests a quadratic increase of the sensitivity with a decreasing column diameter (dJ This is a cause of much confusion, as it suggests that

* The optimum flow rate is the flow rate at which the plate height (H and h) is minimal.

307

columns with a small diameter (“narrow-bore’’ or “micro-bore’’ columns) show superior sensitivity. Generally speaking, this statement is incorrect. This becomes evident if we return to eqn.(7.15). A combination of this equation and eqn.(7.17) yields:

cmaX a ci Vinj / d: . (7.19)

Therefore, narrow-bore columns yield a much higher sensitivity if, but only if, the amount of solute injected is kept constant. The amount of stationary phase in a packed column is proportional to the square of the column diameter (keeping other factors constant).

If we define the reduced mass of the solute injected in the column ( Q,) as

then we find

and

(7.20)

(7.21)

(7.22)

Consequently, if Q, is kept constant, then the sensitivity is independent of the diameter of a packed column.

The following may serve as an example to illustrate the point. We may inject 5 y1 of a sample solution into a column with an inner diameter of 5 mm. If we inject the same amount of sample into a column with a diameter of 1 mm, then (eqn.7.19) the sensitivity would be increased by a factor of 25. However, the sample loading ( Q,) would be increased by the same factor and the result may be increased peak broadening, necessarily combined with loss of detector linearity.

If, on the other hand, we keep the value of Qs constant, then only 0.2 pl of the sample should be injected on the smaller column. This would lead to the same sensitivity as obtained on the 5 mm column. In fact, provided that the linear velocity of the mobile phase and the sensitivity of the detector (defined as the observed signal divided by the concentration of the solute) remain constant, identical chromatograms may be obtained.

In the case in which no more than 0.2 yl of the sample solution is available for analysis, the narrow-bore column will give rise to a larger sensitivity. in other words, reduction of the diameter of a packed column leads to an increased “mass sensitivity” in situations in which the amount of sample available is the limiting factor.

We may summarize the two real advantages of narrow-bore columns over conventional columns as follows: 1. Increased “mass sensitivity” in situations in which a limited amount of sample is

available. 2. Decreased consumption of mobile and stationary phase.

For open columns we find from a combination of eqm(7.15) and (7.18) that

cmaX a ci ynj / d:‘*. (7.23)

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In open columns the volume of the stationary phase ( Vs) will be proportional to the column diameter and to the thickness of the stationary phase film (ds). Thus, we find with eqn.(7.20)

C, ‘inj 0~ Q , , dc ds (7.24)

and hence

According to eqn(7.25) the sensitivity in capillary chromatography increases with decreasing column diameter.

For several reasons, the drastic effect predicted by eqm(7.25) is not usually observed. In the first place, the maximum allowable injection volume may not only be determined by the amount of stationary phase, but also by the amount of mobile phase present in the column (or, to be precise, the volume of mobile phase that occupies one plate in the column)*.

In the second place, the film thickness (d,) tends to be increased in the practice of GC if the column diameter is increased (“thick-film wide-bore” capillary columns), which leads at constant Qs to an increase in sensitivity (eqn.7.25).

Most importantly, however, detectors which show a signal that is only proportional to the Concentration and independent of the flow rate (so-called “concentration-sensitive detectors”) are hardly used in contemporary GC.

Most of today’s popular detectors are so-called “mass-flow sensitive detectors”, i.e. the recorded signal (h,) is proportional to the mass of solute passing through the detector per unit time, in other words proportional to the product of the solute concentration (cmax) and the volumetric flow rate (0. Therefore, we find for “mass-flow sensitive detectors” with eqm(7.25):

(7.26)

where h, is the observed peak height. Consequentiy, if “mass-flow sensitive detectors” are used, a reduction of the diameter

of capillary columns will give rise to a decrease in sensitivity. This effect is enhanced by the tendency to use thicker films of stationary phase (larger d 2 in columns with larger diameters. The limited sensitivity of the detection is a disadvantage of the use of “narrow-bore” capillary columns in GC.

Effect of particle size

If we reduce the size of the particles in a packed column, then the column length may be reduced by the same factor in order to keep the number of plates constant (i.e. Lcc dp;

* Especially in open tubular liquid chromatography (OTLC) the solubility of the sample components in the mobile phase rather than in the stationary phase may be the limiting factor.

see eqn.7.8). Thus, according to eqm(7.15) and (7.17) ciy”j Ciyinj

Cmax cc - oc - d L V d , dp

(7.27)

Eqn.(7.27) shows that the sensitivity may be increased by reducing the particle diameter, if the sample size is kept constant. A constant sample size will correspond to the same reduced sample size (QJ if the specific surface area is constant. Again, the advantage may disappear if the solubility in the mobile phase is the limiting factor, because the volume of mobile phase that occupies one plate decreases proportionaly with the particle size.

In LC practice, we do observe an increase in the sensitivity for columns packed with small (3 pm) particles (“FAST LC” or “High speed LC columns) in comparison to conventional columns (5 pm or 10 pm particles).

Summary

1.

2.

3.

4.

5.

The sensitivity (observed peak height) in a chromatographic system can be increased by the injection of large volumes of samples, as long as this can be done without a decrease in the observed number of peaks. The sensitivity is increased by using good columns (small values for h) operated at the optimumflow rate. The sensitivity decreases upon increasing the column length. Reducing the diameter of packed columns leads to enhanced sensitivity ifthe amount of sample available is the limiting factor. Reducing the diameter of capillary columns leads to an enhanced sensitivity if “concentration-sensitive’’ detectors are used, but to a reduced sensitivity in combination with “mass-flow sensitive” detectors. If the limiting factor for the maximum allowable sample size is the capacity of the stationary phase rather than that of the mobile phase, then a reduction of the particle diameter in a packed column will result in an increase in sensitivity.

7.4 INSTRUMENT OPTIMIZATION

In previous sections we have discussed the optimum dimensions of a column with respect to efficiency, time of analysis and sensitivity. However, the resulting optimum dimensions may not be practical. For example, we rejected the theoretically predicted optimum particle size of 1.2 pm for an HPLC separation requiring 10,000 plates. After changing the conditions, we arrived at a practical “optimum” particle size of 5 pm. Although 1.2 pm is the true optimum value, particles of this size cannot (yet) be used.

In this section we will identify other constraints imposed by the instrumentation on the dimensions of the column. Two factors need to be considered in this respect: 1. the extra-column contributions to the peak broadening, and 2. the time constant of the detection.

Extra-column dispersion

In chapter 1 (section 1.4) we have expressed the width of a (Gaussian) chromatographic peak in terms of the standard deviation 0. Using the additivity of variances, we may

310

interpret the observed standard deviation (a& as the result of two independent contributions:

.’,= 4 4- 4,- (7.28)

where 0, is the contribution from dispersive processes in the column and a,, the sum of all other contributions to the peak width (“extra-column dispersion”).

In general, every piece of instrumentation from the injector upto and including the detector will contribute to oe.. Thus, contributions are included from 1. the injector, 2. the connection between the injector and the column, 3. the connection between the column and the detector, and 4. the detector.

The injector may contribute to the extra-column dispersion in several ways. Ideally, the sample is injected “instantaneously” onto the column as is illustrated in figure 7.3a. This ideal situation may only be approached in practice if one of the “band compression”

t-

i c )

t-

t- t-

Figure 7.3: Illustration of possible injection profiles. (a) ideal profile (theoretical); (b) ideal situation (practical): (c) practical profile; (d) practical profile with tail “cut off‘.

31 1

techniques described in section 7.3 is used. Otherwise, it is more realistic to consider the “block-shaped injection profile shown in figure 7.3b as the ideal situation. In this case the sample is injected during a time tinj and theconcentration of any solute (cinj) is constant during this time. tinj is related to the injection volume by

tinj = qnj / F , (7.29)

where F is the volumetric flow rate. The less ideal situation illustrated in figure 7 . 3 ~ is a typical injection profile in a practical

situation, although the extent of “tailing” and the characteristic delay time (rinj; see figure 7.3~) will vary for different injection methods and for different forms of chromatography. An injection profile corresponding to figure 7 . 3 ~ can only be tolerated if the contribution of rinj to the width and to the “non-symmetry” or “tailing” of the chromatographic peaks can be neglected (see also the discussion on the detection time constant below). If this is not the case, then the “tail” of the injection profile needs to be cut off (e.g. by switching the injection valve). The resulting profile is illustrated in figure 7.3d. A disadvantage of such a profile is that the quantitative accuracy of the chromatographic method is reduced if the fraction of the total sample that is cut off in the tail is not completely reproducible.

The connections of the column to the injector and to the detector are critical for the performance of the chromatographic system. They become the more so if the dispersion of the column itself is decreased (eqn.7.28). All volumes outside the column need to be kept to a minimum, and those connections in particular that involve diameter changes (e.g. to and from the column in HPLC) may give significant contributions to sex.

Many detection principles require a finite volume of eluent. For example, a UV absorption detector yields a signal that is directly proportional to the optical pathlength (Beer’s law, see eqn.5.21). The volume of the detector flow cell is usually well-defined and its contribution to a,, and hence its effects on the observed dispersion a,, can be discussed in quantitative terms (see section 7.4.2).

We may assume as a general rule that the extra-column dispersion should not lead to an increase in the observed peak width of more than 10%. In other words

a, = 1.1 a,. (7.30)

With eqn(7.28) this yields

so that

a,, = v0.21 a, = 0.46 a,

(7.31)

(7.32)

Therefore, as a rule of thumb, we may say [707] that a,, should be less than half the value of a,:

a,, < a y 2 . (7.33)

Eqr~(7.33) can be used as a guideline for the design of chromatographic instruments.

312

Detection time constant

The time constant* ofthe detection is the combined effect of the detector (“detector time constant”) and the data handling or recorder system. The time constant of the detector may be partly due to the fundamental kinetics of the detection (e.g. in polarographic detection), but is usually determined by the amplifier and other electronic components.

The time constant (T) affects both the height and the width of the observed peaks. It can be shown [708,709] that (for .r<O.2 0,) the observed standard deviation in time units (or) increases with 7 according to

where the subscript t ,c indicates that the column dispersion should also be expressed in time units. If we again accept a 10°/o increase in the observed peak width (eqn.7.30), then

7 < or,/lO. (7.35)

Eqn.(7.35) should be a guideline not only for the design of chromatographic systems, but also for the selection of peripherals such as recorders. It can easily be seen that eqn(7.35) imposes fairly stringent demands. For example, if we generate 10,000 plates in 100 seconds on an HPLC column of conventional dimensions (column I11 in table 7.1), then or.,= I s and should be 0.1 s or less. Regular recorders have time constants of 0.5 s or sometimes even 1 s. It is not easy to find recorders which are compatible even with the conventional LC columns.

The situation is certainly not more favourable in GC, where 80,000 plates may be generated in 100 s with a capillary column of conventional diameter (0.2 mm). This corresponds to or,,% 0.4 s and therefore the time constant of the detection system should be less than 40 ms.

This necessitates the abolition of recorders as the routine practical tool for the representation of chromatograms. Since analog devices of adequate speed are expensive, digitization of the data is the logical solution to the problem. In that case, we may assume as a rule of thumb that at least 20 data points are needed to describe a chromatographic peak. Using 4or., as a measure of the peak width in time units (seconds), this implies a sampling frequency cf; expressed in Hz) of

f = 5 / a r . (7.36)

According to the above examples this implies that for LC columns of conventional diameters a sampling frequency of 5 Hz is required, whereas for contemporary capillary G C the required frequencies are between 10 and 15 Hz.

One final comment regarding the time constant in chromatographic detection limits concerns the noise level observed on the baseline. Large time constants serve as efficient filters for (high frequency) noise. Therefore, if a great reduction of the time constant is not

* The time constant of an exponential decay corresponds to the time needed for the signal to decrease to 37’/0 of its initial value.

313

to result in much increased detection limits, other methods should be used to keep the noise level down. Because of the trend towards digitization, digital filters seem to be needed to meet these requirements.

7.4.1 Gas chromatography (open columns)

For the dispersion in volume units in open tubular chromatography we can write

Expressed in the number of plates we find

cr, = ( d 4 ) h (I + k ) d f f i

(7.37)

(7.37a)

Clearly, the column dispersion in capillary chromatography is a very strong function of the diameter of the column. If the column diameter is decreased, then the column dispersion will decrease strongly and therefore, according to eqn.(7.33), the demands imposed on the maximum allowable extra-column dispersion will become increasingly severe.

Schutjes (ref. [702], p.45) has suggested “compensating” for a decrease in d , by an increase in L. In this way, “narrow-bore’’ columns might be used to effectuate separations that require very large numbers of plates. However, eqn.(7.37) reveals that in order to keep the column dispersion constant, a reduction in the column diameter by a factor of 2 would require an increase in the column length by a factor of V32 w 5.7. The efficiency (N) would be increased by a factor of 11. Since at constant v the linear velocity is inversely proportional to the column diameter, the analysis time would only be increased by a factor of 2.8. However, the pressure drop over the column (eqns. 7.1 and 7.2) would increase in proportion to L, w and d;*, i.e. by a factor of 5.7 because of the increased column length, a factor of 2 because of the increased velocity, and a factor of 4 because of the reduced column diameter, resulting in a total increase by a factor of 45!

Although the above calculation is somewhat oversimplified because the effects of the compressibility of the gas have been neglected, it serves to illustrate that a reduction of the column diameter cannot be fully compensated by an increase in the column length to keep the column dispersion constant. Therefore, when “narrow-bore’’ capillary columns are to be used in GC, the extra-column contribution to band broadening will need to be reduced.

In contrast, the use of “wide-bore’’ capillary columns allows a considerably larger extra-column dispersion. Consequently, these columns may readily be used in instruments that are compatible with conventional capillary columns. Moreover, they may be used instead of packed columns to yield greatly increased plate numbers without the requirement of major modifications to the instrument. One of the major reasons for the popularity of “wide-bore’’ capillary columns may therefore be their role as intermediates in the gradual replacement of packed columns by capillary columns in GC.

Table 7.2 lists the characteristic parameters of various columns for GC. The table shows the typical characteristics of capillary columns of conventional diameters for two

314

situations: efficient columns for complex samples and short columns for separations for which the selectivity has been optimized. These columns are compared in the table with “wide-bore” and “narrow-bore” capillary columns. Typical figures for a packed column have been added for comparison. The data for the “narrow-bore” columns have been taken from experiments described in the literature [702]. All other data were calculated using the equations given in this chapter and the conditions listed above the table.

It is clear from table 7.2 that in terms of extra-column dispersion a “wide-bore’’ capillary column requires instrumentation similar to that used for the packed column. However, the capillary column provides eight times as many plates (in a fifteen-fold analysis time). Conventional capillary columns require a reduction in the dispersion by about an order of magnitude, whereas “narrow-bore’’ columns require a further reduction by a factor of about 100. This, combined with the high pressures required, puts “narrow-bore’’ columns out of reach for current G C instruments.

Table 7.2: Possible column dimensions for GC. Data calculated from eqns.(7.1), (7.2), (7.6), (7.33), (7.359, (7.36) and (7.37) for all columns, except “narrow-bore” (literature data; [702]). Conditions: h = 1.5, v = 5 (open columns); h=3, v = 3 (packed columns); q=1 mPa.s (hydrogen); D, = 10 - m2/s.

Parameter/ Packed Open units

Conventional Wide Narrow

Long Short Long Short (1) (2)

d/pm L/m N / 103

Ap/bar

D ” 4 1 Dex/P1

t , /s a,/ms(4) dms(4) f/Hz N,,JI 03 (5)

200 3 5

1

1 800(3) 900 (3)

20 280 28 15 1.5

250 30 80

3

170 80

150 500 50 10 7.5

250 3 8

0.3

50 25

15 170 17 30 2.25

500 30 40

0.4

1000 500

300 1500 150 3 4.5

65 95 1,000

19

0.2 0.1

1800 1800 180 3 54

55 22 35

10

0.3 0.2

2 10 1 500 10

(1 ) Experimental data from ref. [702], appendix 4, carrier gas: hydrogen. (2) Experimental data from ref. [702], pp.59-60, carrier gas: helium. (3) for d,=5 mm and ~ = 0 . 6 . (4) for k = 0. ( 5 ) Number of datapoints recorded for a digitized isothermal chromatogram for 0 < k < 4 .

315

For each of the columns in table 7.2 the hold-up time ( to) can readily be calculated from eqn.(7.6) using k = 0. The standard deviation in time units follows from to after division by I6 (eqn.l.16). Eqns.(7.35) and (7.36) then provide the maximum allowable detection time constant (T), the required sample frequency for digital data handling u> and the required number of datapoints ( Ndat) for recording a chromatogram (0 < k< 4). All these characteristics are shown for the different columns in the bottom part of table 7.2.

Clearly, both packed column GC and conventional capillary GC already put serious constraints on the maximum allowable time constants. Ironically, the easiest separations, requiring the lowest numbers of theoretical plates and hence requiring short columns, are the most difficult ones to perform in an optimum way. The generation of lo6 plates (in about an hour) allows the highest value for z (180 ms) in the entire table!

Table 7.2 also lists the required data storage for a typical isothermal chromatogram (0 < k < 4) recorded on each of the columns. It is seen that conventional capillary columns already require a five times larger facility than do conventional packed columns. The use of “long” narrow-bore columns would lead to a further increase in the number of datapoints by about an order of magnitude.

An increase in the number of datapoints recorded per chromatogram also leads to increased demands with respect to data handling facilities and computing time. For instance, 54,000 datapoints require about 100k computer memory (double precision).

The increased computation demands form another obstruction for the application of narrow-bore capillary columns in routine GC.

7.4.2 Liquid chromatography (packed columns)

Because of the low diffusion coefficients in liquids, the particle size for packed columns in LC needs to be very small (see section 7.2.1). For the same reason, all external volumes and diameters need to be minimized. This may easily be understood if we express the standard deviation in volume units (a,) in the parameters that represent the dimensions of the column:

v, VdI+ k) fi d(L/H)

a = - =

(7.38)

For conventional HPLC columns with E= 0.6 and h = 4 we find for k = 0

a, = ( d 4 ) x 0.6 x 4 fi da d , w 1.8 I6 d: d, (7.39)

where a, is expressed in p1 if d , and d, are both expressed in mm. As an example, a conventional HPLC column of 20 cm length packed with 5 pm (0.005

mm) particles yields 10,000 plates (column I11 in table 7.1). With the conventional column inner diameter of 5 mm we find

a, = 1.8 x 100 x 25 x 0.005 w 23 pI .

Eqn.(7.33) shows that the total extra-column dispersion for a conventional HPLC system should be less than about 12 pl.

316

The detector flow-cell, the contribution of which to CJ” is approximately equal to its volume [707], represents a considerable and recognizable contribution to the extra-column band broadening. Typical conventional flow-cells have a volume of 8 pl, which is quite substantial compared with the maximum allowable extra-column dispersion.

Table 7.3a lists the maximum allowable extra-column-dispersion for the first three columns listed in table 7.1, using three different internal diameters. It is seen that the contribution from the detector flow-cell (as well as other contributions) will have to be reduced considerably if short columns packed with 3 pm particles are to be used. An even larger reduction in the extra-column dispersion is required for the use of columns with a reduced inner diameter.

Presently [707], we may be able to reduce the extra-column band broadening by modifying conventional HPLC equipment to a total of 1 or 2 3. This implies that with these modifications short “bulky” 3 pm columns or narrow (2 mm) 5 pm columns may be used.

Table 7.3: a. Maximum allowable extra-column dispersion for the first three columns of table 7.1. Calculated from eqn~(7.33) and (7.39) for three different column diameters.

Column N dP a, (max) / pl (mm)

do= 5 d,=3 d,=l mm

I 2,500 0.003 3.4 0.5 0.14 I1 4,200 0.003 4.4 0.7 0.17 I11 10,000 0.005 11.3 1.8 0.5

b. Maximum allowable detection time constants for the first three columns of table 7.1. Calculated from eqm(l.16) and (7.35) for three different k values using the data from table 7.1. Minimum required sampling frequencies for the same three columns were calculated from eqn.(7.36) for k = 0. Number of recorded datapoints calculated for the range 0 < k < 4.

t0 T (max) / ms f Ndat (HZ) (x 103)

dP tPm) (9

Column N

k=O k = l k = 4

I 2,500 3 9 18 36 90 30 1.4 I1 4,200 3 15 23 46 116 20 1.5 111 10,000 5 100 loo 200 500 5 2.5

In practice, it may be useful to calculate the minimum required inner diameter for a given column. This may easily be done by combining eqn.(7.33) with eqn.(7.39):

(7.40)

317

For instance, if the available instrumentation has an extra-column dispersion of 5 p1 and we have opted for a 5 pm column of 10 cm to realize 5,000 plates, then we find from eqn.(7.40)

v5 I d

x 3.8 mm. dc ', v(5.10-3)

Therefore, in this situation a column with an internal diameter of 4 mm or more should be selected.

Table 7.3b shows the calculated maximum allowable detection time constants for the first three columns of table 7.1 for three different values of the capacity factor, using the values for N and to given in this table.

It appears from table 7.3b that modern HPLC columns impose very stringent demands on the detection (and recording) system. Typical time constants of current LC detectors are in the range of 0.3 to 0.5 s [710], which is not even sufficient to allow the use of a 20 cm, 5 pm column (column 111 in table 7.3b).

Therefore, a reduction in the time constant of current detection systems, without the accompanying effect of a great increase in the noise level, should have at least the same priority as the reduction of the extra-column dispersion in the design of future HPLC systems.

7.4.3 Summary

In this section we have derived rules of thumb for the maximum allowable extra-column dispersion and detection time constant and for the minimum required sample frequency for digital data handling. In GC the use of "wide-bore" capillary columns allows the use of instruments designed to accommodate packed columns (in terms of extra-column dispersion). For capillary columns of conventional diameter a reduction of the extra-column dispersion by a factor of 10, and for narrow bore columns a reduction by a factor of 1000, are required. The extra-column dispersion in HPLC should be further reduced to allow the use of columns packed with very small particles and/or columns with small internal diameters. Both in GC and in LC the detection time constant needs to be reduced, even for the application of conventional columns. A great reduction is required to follow modern developments in column technology and the use of digital data handling appears to be unavoidable.

REFERENCES

701. G.Guiochon, AnaLChem. 50 (1978) 1812. 702. C.P.M.Schutjes, Ph.D. Thesis, Eindhoven Technical University, 1983. 703. J.H.Knox and M.T.Gilbert, J.Chromatogr. 186 (1979) 405. 704. J.H.Knox and M.Saleem, J.Chrornatogr.Sci. 7 (1969) 614. 705. %.van der Wal, Chromatographia u) (1985) 274. 706. K.Grob, Jr., J.Chromatogr. 237 (1982) 15.

318

707. P.J.Naish, D.P.Goulder and C.V.Perkins, Chromatographia 20 (1985) 335. 708. L.J.Schmauch, AnaLChem. 31 (1959) 225. 709. G.McWilliam and H.C.Bolton, AnaLChem. 41 (1969) 1755. 710. G.Guiochon in Cs.Horvath (ed.), High Performance Liquid Chromatography; Ad-

vances and Perspectives, V01.2, Academic Press, New York, 1980, pp.1-56.

319

SYMBOLS AND ABBREVIATIONS

Use of symbols is limited to certain section of the book when indicated.

Symbol Description

a a a b b b b b' c c c c C

C

c' d d d d d c

dP d., d e e f f fm f g g g* h h h

ho

k,

1

k

activity solute parameter constant gradient steepness parameter optical pathlength solute parameter constant constant concentration cohesive energy density solute parameter constant continuous criterion continuous parameters constant confidence range diameter solute parameter constant column diameter particle diameter film thickness discrete parameters effect of variable solute parameter experimental value measure for peak separation measure for peak separation f i xed design measure for peak separation weighting factor measure for peak separation partial molar enthalpy peak height reduced plate height peak height iterative design capacity factor capacity factor under gradient conditions

Sections

3.2 2.3.2

5.6 2.3.2

2.3.1 2.3.2

chapter 4 3.5

5.5.2 chapter 7

Introduced

eqn.(3.24) eqn.(2.5) various eqn.(4.68) eqn.(5.21) eqiL(2.5) various various - eqn42.1) eqm(2.5) various table 4.7

eqn.{4.30a) eqn.(5.19) eqn.Q.3)

2.3.2

3.5 5.4.1 2.3.2 5.4,1

4.6.1

eqn.(2.5) various table 4.2

table 4.2

eqn.( 5.2) eqn.(2.5) eqn45.2) eqn.(4.3) eqn.(4.4) table 5.6 eqn.(4.3) eqn.(4.58) eqn.(4.4)

-

chapter 3 eqq.(3.7) ,1.4 eqn.(l.l5) chapter 7 eqn.(7.3) chapter. 7 eqn.(7.11)

table 5.6 eqn.( 1.5) eqtk(4.67)

321

Symbol

k i n

kobs

- ko k 1 n n n

"e

"e

nP

nP n nf "a

P P P P

nb

P' Po 4 9 r

r A

r B

TT r* S

S

S

f

5 t ??I

tG

t R

t0

t i

t

tha thf U

U

322

Description

isocratic capacity factor observed capacity factor extrapoiated capacity factor average capacity factor number of levels noise level number of moles number molecular size of solvent chain length number of experiments number of parameters peak capacity correction for number of peaks required critical chainlength number of peakpairs number of relevant peaks pressure regression parameter vapour pressure critical pressure pure component vapour pressure quantity regression parameter normalized resolution product dissociation ratio buffer dissociation ratio heating rate calibrated normalized resolution product partial molar entropy stationary phase parameter stepwise parameters time isothermal retention time

gradient duration time retention time hold-up time time correction required net retention time threshold criterion, arbitrary boundaries threshold criterion, fixed boundaries mobile phase linear velocity stationary phase parameter

hoid-up W

Sections

5.4.1 4.2.1 chapter 3

3.2.3

5.4.1 5.4.1 1.6

4.6.1

3.2.2.1 3.1.1 3.4

3.2.2.1

3.2.2.1 3.2.2.1

Introduced

eqn.( 5.5) eqn.(3.70b) eqn.(3.45) eqn.(l.22) eqn.(5.1) eqn.(4.12) eqn(3.6) various eqn.(3.73) figure3.13 eqn.(5.1) eqn.( 5.1 ) eqn.(l.25) table 4.8 figure3.13 eqn.(5.11) eqn.(4.54)

eqn.( 3.46) eqn.(3.3)

eqn43.4)

eqn .( 3.46) eqn(4.19) eqn.(3.62) eqa(3.68) eqn.(4.66) eqn.(4.21)

-

-

-

chapter 3 eqn43.7) 2.3.2 eqn.(2.5) 3.5

6.3.1.2 eqm(6.3) eqn.( 1.3) eqn.(4.68) eqn.( 1.2) eqn.(l.3) table 4.8 eqn.( 1.7) table 4.7 table 4.7 eqn.( 1 .I) eqn.( 2.5)

-

2.3.2

Symbol Description Sections Introduced

V

V

V

V

W

ws

w1/2 X

X

X

X

' d

xf? X" x X'

Y Y Z

Z

A A A A A

A' ACN B B

B' C C C C' CBP D D E F FO GC GLC

A ,

Bo

measure for peak separation migration speed molar volume vector in parameter space peakwidth weight of stationary phase peakwidth at half height mole fraction optimization variable stationary phase parameter threshold value proton donor parameter proton acceptor parameter strong dipole parameter useless criterion shifted composition optimization variable stationary phase parameter charge of solute ion stationary phase parameter absorption adsorption area curvature coefficient peak area constant adsorption surface area constant acetonitrile gradient slope constant specific permeability coefficient constant optimization criterion constant capacity parameters constant chemically bonded phase constant distribution coefficient cohesive energy flowrate fractional overlap criterion gas chromatography gas-liquid chromatography

1.2.1

5.1.1 2.3.2

5.5.2 5.1.1 2.3.2 3.2.2.1 2.3.2 5.6 3.2.3

1.4

3.2.1

chapter 4

3.5

3.3 2.3.1

eqn.(4.5) eqn.(l.l)

figure5. 1 1

eqm(3.11) eqn.( 1.1 6a)

figure 5.3 eqn(2.5) eq~(4.23) eqn(2.16) eq~(2.15) eqn.(2.17) table 4.7 eq~(5.18) figure 5.3 eqn(2.5) eqn.(3.71) eqn.(2.5) eqn.(5.21) eq~(3.72) eqm(3.38) eqn.(l .15) various eqm(3.17) various

-

-

-

eqn.(5.6) various eqn.(7.1) various

various -

various

various eqm(3.76) eqn.(2.1) eqn.(4.66) eqa(4.13)

323

Symbol

GPC GSC H I I IEC IPC K K" K a ' a

K b

K* KO

K ,

K t h

L LBPC LC LLC LSC

M MeOH MtBE MC N

MS

Nb

Description

gel permeation chromatograph gas-solid chromatography Henry's adsorption coefficient ionic strength retention index ion-exchange chromatography ion-pairiag chromatography distribution coefficient n-alkane distribution coeffient acid dissociation constant adsorption coefficient base dissociation constant distribution coefficient (GC) n-octane distribution coefficient thermodynamic distribution coefficient corrected distribution coefficient corrected distribution coefficient column length liquid-bonded phase chromatography liquid chromatography liquid-liquid chromatography liquid-solid chromatography stationary phase molecular weight mobile phase parameters methanol methyl t-butyl ether methylene chloride number of plates mole fraction of strong solvent number of data points Ndat

NP-IPC normal phase IPC NPLC ODS P

prn P" P P' PCA PTH

R Qs

' a

RS

324

normal phase LC octadecyl silica peak valley ratio median peak valley ratio valley-to-top ratio physical parameters polarity parameter principai component analysis phenylthiohydantoin reduced sample size gas constant absorbance ratio resolution

Sections Introduced

3.1.2 eqn(l.17) 3.2.2.1 eqm(3.71)

eq~(2.3)

eqn.( 1.9) 2.3.3 eqa(2.13) 3.2.2.1 eqn.(3.61) 3.1.2,3.2.3) eqr~(3.16) 3.2.2.1 eq~(3.65) 2.3.3 eqa(2.11) 2.3.3 eqa(2.13)

eqm(3.24) 2.3.3 eqa(2.11) 2.3.3 eqm(2.12)

-

eqm(3.2) 3.5

eqn.(l.16) 3.2.3 eqn(3.73)

-

eqm(4.3) eqm(4.4) eqn.(4.5)

3.5 2.3.3 eqn.(2.14)

eqn.(7.20) eqn.(3.3)

5.6 eqa(5.21) eqn.( 1 .f4)

Symbol Description

RAT absorbance ratio RP-IPC reversed phase IPC RPLC S' S S

3 S SEC SFC T

Tc Tc T o Tr T THF TLC V va

v g

vg V R

vo V R Z

a a a B B 7 Y 6 6

8 tl

L c1

&

K

V

reversed phase LC separation factor corrected number of plates separation factor slope (RPLC) surface area adsorption energy Stationary phase parameters size exclusion chromatography supercritical fluid chromatography temperature compensation temperature critical temperature recommended isothermal temperature isothermal temperature thermodynamic parameters tetrahydrofuran thin layer chromatography volume volume of adsorbed stationary phase retention volume gradient conditions specific retention volume retention volume hold-up volume net retention volume compressibility coefficient

constant relative retention (selectivity) adsorbent activity constant constant activity coefficient constant solubility parameter allowed uncertainty column porosity eluotropic strength (LSC) viscosity gradient shape parameter wavelength thermodynamic potential reduced linear velocity

Sections

5.6

3.2.3 3.5.

3.2.2.1 3.4

3.5

3.2.3 chapter 6 chapter 3

eqn.(3.59)

3.2.3 eq~(3.59) 4.4.3

eqn.(3.59)

5.5.2

3.2

Introduced

eqn(5.21)

eqn.(4.15a) eqm(4.15) eqn.(3.45) table 4.2 ' eqn.(3.72)

- eqn.(3.58) eqn.(3.58) eqn.(5.4) eqn.(4.66)

- eqm(3.72) eqn.( 6.7) eqn.(3.11) - - - qn43.3)

eqn.( 3.59) eqn.(l.ll) eqn.(3.72) eqn.(3.59) eqn.(4.38) eqn .( 3.4) eqn.(3.59) eqm(2.1) eqm(5.19) table 4.2 eqm(3.72) eqn.(7.1) eqn46.6) eqn.(5.27)

eqm(7.4) -

325

Symbol Description Sections Introduced

density standard deviation observed standard deviation time constant weighting factor total overlap criterion volume fraction isocratic volume fraction threshold absorption pressure drop eluotropic strength (RPLC) constant

D, diffusion coefficient V selectivity of phase system

SUBSCRIPTS

Symbol Description

a b C

C

C

d d e ex

f f,d g g I

1

i+ 1 i - 1 ind inj j j j i m max

326

acid base column concentration number of carbon atoms in chain dispersion proton donor (dioxane) proton acceptor (ethanol) extra-column final conditions of constant flow and diameter gradient weighted solute initial peak following i peak preceding i induction injection solute modifier pair of solutes mobile phase maximum

chapter 7 4.4.2 4.3.4

5.6

5.5.2

Sections

2.3.1 2.3.1

chapter 3 3.2.2 2.3.1 2.3.3 2.3.3

4.6.1

6.2.1

2.3.1

eqn.(3.34)

eqn.(3.2) eqn.(l .I 6) eqn.(7.28) eqn.(7.34) eqn.(4.32) eqn.(4.27)

figure5.13 eqm(5.22) eqm(7.1) eqn.(3.52) eqn.( 5.1 8)

-

eqn.(7.4) eqn.(3.32)

Symbol Description Sections

min n

n nP n + l n+l n-I ne nt 0

n

0% P P P ref t

V

V

C E IE Me R T T W

a w

minimum preceding n-alkane strong dipole (nitromethane) corrected for number of peaks non-polar following n-alkane peak following n peak preceding n required corrected for number of peaks and analysis time orientation 2.3.1 organic phase constant pressure conditions polar 2.3.2 programmed elution 4.6.2 reference corrected for analysis time evaporation 3.1.1 volume units

number of carbon atoms in chain excess ion-exchange methanol retention total tetrahydrofuran (THF) water

first peak last peak

2.3.2 2.3.3 chapter 4 2.3.2 2.3.2

2.3.1

Underlining of symbols indicates relevant peaks (section 4.6).

SUPERSCRIPTS

Symbol Description Sections

aq aqueous org organic 03 infinite dilution G gas phase 0 standard state

Lines above symbols indicate average or median values.

3.1.1

327

AUTHOR INDEX Numbers in this index refer to references in the text.

Abbott, S. 204 d'Agostino, G. 537, 579 Allen, A.C. 567 Antle, P.E. 572 Asche, W. 388 Bajema, B.L. 413, 561 BaYke, S.T. 334 Barker, J.A. 307 Barth, H.G. 318 Bartha, A. 381, 386 Barton, A.F.M. 320 Bartb, V. 615 Berendsen, G.E. 317, 319 Berridge, J.C. 414, 505, 506, 508, 509,

Berry, V.V. 520, 543, 607 Bidlingmeyer, B.A. 202 Billiet, H.A.H. 207, 209, 303,311, 314,

510,621

321, 322, 323,324, 335, 336, 341, 373, 374,383,386,409,410,424,502,504, 534, 535, 536, 576, 578, 582, 584, 590, 608,629

Blass, W. 603 Board, R.D. 387 Bolton, H.C. 709 Bosman, Th. 592 Bounine, J.P. 555 Bower, J.G. 503 Bower, K.D. 503 Box, G.E.P. 515,518 Bradley, M.P.T. 521 Brenner, N. (ed.), 606 Brinkman, U.A.Th 332,533 Bruins, C.H.P. 327,425,560, 561 Budna, K.W. 406 Calfen, J.E. (ed.), 606 Campbell, D.E. 339 Carr, P.W. 418, 619 Castagnetta, L. 537, 579 Charikofsky, J.G. 573 Chen, B.-K. 338 Chien, C.-F. 304 Chu, C.H. 343

ChuriEek, J. 354,357,377, 526,531, 532,

Clark, B.J. 588 Clerc, J.T. 591 Coates, V.J. (ed.) 407 Cohen, K.A. 430,610 Cohen, M.J. 348 Colin H. 325, 351, 555, 577, 628 Conlon, R.D. 575 Constanzo, S.J. 549 Cowie, C.E. 368 Cristophe, A.B. 404 Crombeen, J.P. 382 Crommen, J. 379 Culbreth, P.H. 620 Davis, J.M. 105 Davis, O.L. (ed.) 519 Debets, H.J.G. 413, 561 Deelder, R.S. 345 Deming, S.N. 340, 417,421, 422, 503,

511,540,550,551,552,558,559,612 Dolan, J.W. 331, 337,428, 430, 528, 583,

610,616,622 Doornbos, D.A. 327,413,423,425,512,

560, 561 Driscoll, J.N. 586 Drouen, A.C.J.H. 336,383,405,409,410,

424,502,504,576,578,582,584,590 Edens, R. 551 Eksteen, R. 371 Elyashberg, M. 589 Eon, C. 208,312,351 Essers, R. 592 Everett, D.H. 307 Fagerson, I.S. (ed.) 407 Fast, D.M. 620 Feibush, B. 348 Fell, A.F. 588 Fransson, B. 379 Frei, R.W. 332,533,604 Galan, L.de 209,311,314,319,321, 322,

323, 324, 335, 336,341, 373, 374, 383, 386,409,410,424,502,504,534,535,

618,623,624,628,632,633,634

329

536, 576, 578, 582, 584,590, 593,608, 629

616,622 Gant, J.R. 331,337,428,528,583,587,

Giddings, J.C. 104,105,426,606 Gilbert, M.T. 366,703 Gillen, D. 521 Glajch, J.L. 359, 360,361,362, 363,415,

542,564,565 568,569,570,571,573, 611,627

Gluckman, J.C. 573 Goewie, C.E. 332,533,604 Goldgaber, N. 626 Goldsmith, P.L. (ed.) 519 Golkiewicz, W. 353, 522, 523 Goulder, D.P. 707 Grant, D.W. 614 Gribov, L.A. 589 Grillo, S.A. 430, 610 Grob, K., Jr., 706 Grob, L.R. 529,530,630,631 Grob, R.L. (ed.) 525 Guiochon, G. 102, 325,351,555,577,

Habgood, H.W. 427,605 Haddad, P.R. 368,376,425,582 Hafkenscheid, T.L. 333 Haky, J.E. 566 Hamilton, P.B., 369 Harbison, M.W.P. 305 Harris, W.E. 427 Harris, W.E. 605 Hearn, M.T.W. 344 Heckenberg, A.L. 376 Hendrikx, L.H.M. 345 Hendrix, D.L. 551 Hildebrand, J.H. 206,313 Hollis, M.G. 614 Hoogewijs, G. 574 Hooke, R. 514 Horvath, Cs. (ed.) 316,429,527,617, 710 Horvath, Cs. 316,326,329,339,342

Huber, J.F.K. 315,370,372,602 Hulpke, H. 603 Hunter, J.S. 518

701,710

HSU, A.-J. 553

Hunter, W.G. 518 Issaq, H.J. 538, 554, 566, 567, 585, 626 Jacques, C.H. 541 Jandera, P. 325, 354, 357, 377, 526, 531,

532, 577,618,623,624,628,632,633, 634

Janderovh, M. 357 Janini, G.M. 554 Jeeves, T.A. 514 Jefferies, T.M. 385 Johansson, E. 516 Johansson, K. 516 Jones, P. 408, 546 Kaiser, R. 402 Kalashnikova, E.V. 309 Kaplar, L. 302 Karger, B.L. 208, 312, 348 Karnicky, J. 204 Kastelan-Macan, M. 539 Kateman, G. 203,592 Kaur, B. 366 Kenndler, E. 602 Kirkland, J.J. 103,201, 347, 349, 359,

362,401,415, 524, 542, 564, 565, 570, 601,611,627

Kiselev, A.V. 309 Klaessens, J. W.A. 203 Klein, J. 310 Klose, J.R. 566 Knoll, J.E. 412 Knox, J.H. 365,366,380,703,704 Kong, R.C. 559 Kopecni, M.M. 304 Kraak, J.C. 371,382 Krstulovic, A. 555 Krull, I.S. 586 Kuwana, Th. (ed.) 306,420,501 Laird, G.R. 380 Lankmayr, E.P. 406,580,58 Laub, R.J. 304,305,306,420,501,544,

Lauer, H.H. 387 Laurent, C.J.C.M. 341,364,373,374 Lindberg, W. 516 Linsen, P. 371 Little, C.J. 604

545,553

330

Littlewood, A.B. 301 Lochmueller, C.H. 318 Locke, D.C. 308 Lu Peichang, 330 Lu Xiaoming, 330 Lurie, I.S. 567 Madden, S.J. 553 Majors, R.E. 318 Marcus, Y. (ed.) 378 Marinsky, J.A. (ed.) 378 Markl, P. 3 15 Martire, D.E. 305 Massart, D.L. 416, 574 May, W.E. 548 McCann, M. 328 McManigilI, D. 387 McNitt, K.L. 566, 585, 626 McReynolds, W.O. 213 McWiIliam, G. 709 Mead, R. 507 Melander, W.R. 316, 326, 329, 338, 342 Midgett, M.R. 412 Minor, J.M. 415, 542, 573 Minor, J.M. 573 Mitchell, F. 537, 579 Modin, R. 384 Molnar, I. 342 Morgan, S.L. 417, 540, 541 Morrisey, E.C. 510 Mueller, H. 365 Mulik J.D. (ed.) 375 Muschik, G.M. 554, 566 Naegli, P.R. 592 Naish, P.J. 707 Nelder, J.A. 507 Noebels, H.J. (ed.) 407 Noyes, C.M. 547, 625 Nyiri, W. 602 O’Hare, M.J. 537, 579 Ober, S.S. 407 Olacsi, I. 302 Oreans, M. 602 Otto, M. 346, 562, 563 Pawlowska, M. 315 Perkins, C.V. 707 Perrone, P.R. 587

Phillips, G.S.G. 301 Pietnyk, D.J. 343 Poile, A.F. 575 Poppe, H. 356,382 Poshkus, D.P. 309 Prausnitz, J.M. 206,313,389 Price, D.T. 301 Price, W.P. 551, 552 Purnell, J.H. 305,328, 545 Puttemans, M. 574 Quarry, M.A. 529,530,609,630,631 Rabel, F.M. 367 Rafel, J. 5 13 Rajcsanyi, P. 302 Randall, L.G. 392 Reid, R.C. 389 Reijnen, J. 592 Riegner, K. 603 Riley, C.M. 385 Rogers, L.B. 419 Rohrschneider, L. 205, 210, 216 Rolink, H. 423, 512 Rowland, M. 557 Sachok, B. 422,558,559 Saleem, M. 704 Sampson, E.J. 620 Sander, L.C. 548 Saunders, D.L. 358 Sawicki, E. (ed.) 375 Sawicki, E. 375 Schill, G. 378, 379, 384 Schill, R. 525 Schlabach, T. 204 Schmauch, L.J. 708 Schneider, G.M. 391 Schoenmakers, P.J. 207, 209, 303, 311,

314,321,322,323,324,335, 390, 409, 410, 504, 534, 535, 536, 576,608,629

Schupp, O.E. 403 Schutjes, C.P.M. 702 Schwartz, M. 586 Scott, H.P. 588 Scott, R.L. 206,313 Shcherbakova, K.D. 309 Sherwood, T.K. 389 Smet, M.de 574

33 1

Smits, R. 416 Snyder, L.R. 101, 103,201,208,214, 215,

312, 331,337,349, 350, 356, 359, 360, 361, 362,363,401,428,429, 524, 527, 528,529,530,564,568,569,570,571, 573,583,601,609,616,617,622,630, 63 1

Sonewinski, E. 352,353,355 Spencer, W.A. 419 Squire, K.M. 415, 542 Stadalius, M. 609 Stahel, 0. 604 Stan, H.-J. 613 Steinbach, B. 613 Stranahan, J.J. 422, 558 Supina, W.R. 212 Svoboda, V. 41 1,517 Swaid, I. 391 Takacs, J. 302 Tijssen, R. 207, 303,321,629 Tomlinson, E. 333,385 Tompkins, DJ. 604 Toon, S. 557 Trbojevic, M. 539

Turina, S. 539 Turoff, M.L. 340,421, 550 Unger, K.K. 365 Vandeginste, B.G.M. 203,592 Vanroelen, C. 416 Venne, J.L.M. van de 345 Vigh, Gy. 381, 386 Wal, Sj.van der 204,370, 372, 556, 705 Walters, F.H. 511,612 Warren, F.V. 202 Wasen, U.van 391 Watson, M.W. 418,619 Wegscheider, W. 346,406,562,563,580,

Weiss, M.D. (ed.), 606 Wellington, C.A. 328,408, 546 Weyland, J.W. 327, 423,425, 512, 560,

Widdecke, H. 310 Williams, P.S. 305 Wilson, K.B. 515 Wise, S.A. 548 Wittgenstein, E. (ed.) 375

581

56 1

332

SUBJECT INDEX

Absorbance ratio - see: ratio recording Accuracy of predicted optimum, in itera-

tive designs 226 Acid-base interactions 25 - 26 Activity coefficient -, definition 38 -, effect on retention -, -, in GLC 38 -, -, in LC 48 Adsorbent activity, in LSC 76 Adsorption area, of solute in LSC 76 Adsorption coefficient -, definition 43 -, in LSC 76 Adsorption energy, of solute in LSC 76 Adsorption isotherm, definition 4 Alkaloid drugs -, retention in IEC 92-93 -, retention in I F C 190-191 Alkyl chain length -, effect on selectivity in RPLC 58 - 59 -, see also: chain length Alumina 70,77,81 -, use in GSC 45 -, use in IEC 92-93 - , see also: polar adsorbents Analysis program -, definition 253 - , possible shapes 253 - 255 Analysis time, effect on optimization cri-

Antioxidants, separation by programrned

Applications of SFC 103 Background signal, in programmed sol-

vent LC 261 Baseline drift, in programmed temperatu-

re GC 259-260 Binary mixtures 60 Blank signal, in programmed solvent LC

Boiling point separation 41 Bonded phase chromatography 56 - 75

teria 136-137,146, 148-151

solvent LC 278

261

-, see also: chemically bonded phases Buffer(s) -, dissociation ratio 71 -, effect on retention in RPLC 70-71 -, effect on retention in IPC 100 Calibrated normalized resolution product

-, correction for analysis time 157 - 158 -, correction for number of peaks 158 -, definition 139 -, for limited number of relevant peaks

-, need for time correction factors 155 -, use of solute weighting factors 164 Capacity factor -, definition 3 - , effect on resolution 10 - 14 -, measurement 3 -, optimum range 11 - 12,16- l7,62,

- , optimum value 1 1 - 12 -, use as retention parameter 37 Capacity parameters, definition 105 Capillary columns - see: open coiumns - see: column(s), capillary Carbon dioxide, as solvent for SFC 103 Carbon stationary phase(s) 52, 70, 77,

Chain length, of pairing ion in IPC 99 Chemically bonded phases (CBPs) 20,53,

-, perfluorinated 52, 74 -, polar 51,74-75 -, -, retention mechanism 75 -, - , stability 75 -, polymeric 57 -, see also: end-capping -, see also: silica, reaction with silanes Chromatograph, schematic 1 Chromatographic methods -, classification 20-21 - , nomenclature 20 - 21

153

162

192,253

81 -82

56 - 59

333

-, selection 16, 21 -23 Chromatography, definition 1 Cold injection in GC 306 Column diameter -, effect on phase ratio 6 - , effect on sensitivity 307 Column independent time factors

Column porosity, definition 6 Column(s) - , capillary, for GC 300 - 301 -, -, detection sensitivity 309 - , - , evaluation 3 15 -, -, fast analysis 301 - , -, narrow-bore 300 -, -, - , extra-column dispersion

- , -, wide-bore 300 -, -, - , extra-column dispersion

-, effect on optimization criteria

- , effect on overlapping resolution map-

- , effect on sum criteria 132 - 133 - , effect on threshold criteria 143 - , factors affecting selection 298 -, importance of optimization 297 - 298 -, packed, for GC 301 - , packed, for LC 302 - 305 -, -, evaluation 317-318 -, pzicked vs. open 299 - 300 -, practical dimensions for LC 303 - 304 - , pressure drop 299 -, temperature limit 21 Column-switching - see: multi-column

Compensation temperature in RPLC 68 Competition model, in LSC 76 - 77 Complete mathematical optimization of

progammed solvent LC 283 Composite criteria 146 - 158, 277 - 278 Computation time 219,231 -232,290 Concentration sensitive detectors 305,

Concentration of solute in effluent

151 - 153

314-315

314-315

145 - 146

ping 216

techniques

309

305 - 306 Confidence ranges, in iterative designs

Contamination capacity 55 Continuous parameters, definition 109 Contour plot, of response surface 172 Counterion, definition for IEC 82 Counterion concentration -, effect on retention in IPC 100 -, effect on retention in IEC 84-87 Counterion type, effect on retention in

Critical band method 206 - 209 -, application to specific cases 208-209 - , characteristics 209, 248 Critical band, calculation of 206, 209 Critical chain length 58 - 59 Critical point, definition 101 Critical properties, of solvents for SFC

Darcy’s law 299 Density of mobile phase, effect on reten-

Dependent variables - , effect of temperature and compostion

in RPLC 68 - 69 - , general 173 -, in IPC 191 -, mobile and stationary phase in LC

-, optimization in IPC 209 - 21 1 Depressants, separation by programmed

Desired analysis time in optimization cri-

Detection time constant 313 -314 - , effect on noise level 3 13 - 3 14 -, in GC 315-316 -, in LC 317-318 Detection volume, effect on extra-column

Detection - , dual detectors in series 239 - , dual-wavelength UV 239 - 240 -, in SFC 103 - , linearity 306

225,221 - 228

IEC 87

102

tion in SFC 103 - 104

218

solvent LC 282

teria 149

dispersion 3 12

334

Detector flowcell, -, effect on extra-column dispersion in

Diachoric model -, for GLC 41 -43

Differential migration 1 Digitization, need for - in chromato-

Dilute solutions 37 - 38 Diphenyl amines, retention in RPLC 226 Dipole induction interaction 25 Dipole orientation interaction 25 Discrete parameters, definition 110 Dispersion interaction 25 Dissociation constant, definition 69 Dissociation ratio, definition 69 Distribution coefficient -, definition 4 -, in IEC 84 -, in LSC 76 -, use for solvent classification 32 Distribution constant - see: distribution

Distribution isotherm, definition 4 Dual-channel detection, for peak recogni-

tion 239-241 Dynamic LLC 53 - 55 Efficiency optimization 299 - 305 Elemental criteria - , characteristics 130 - , comparison 127 - 131 -, definition 119 -, recommendations 131 Eluotropic series, for IEC 87 Eluotropic strength parameter for LSC

-, nomogram 80 -, values of 77 -, for binary mixtures 78, 80 Eluotropic strength, in LSC 217 Elution program - see: analysis program End-capping 57 - 58 Equilibration time, in LLC 55 Equilibrium constant - see: distribution

LC 317

-, for RPLC 61

graphy 3 13

coefficient

76

Excess quantities 38 Exclusion 22 Experimental design(s) 21 1 -, for Sentinel method 212-213 - , see also: fixed experimental designs -, see also: full factorial designs Expert systems 23 - 24,171 Extra-column dispersion 3 10 - 3 12 -, in GC 314-315 -, in LC 316-317 Factorial designs - see: full factorial de-

Film thickness 6 Fixed experimental design(s) 200 - 220 - , characteristics 219 - 220 -, see also: critical band method - , see also: full factorial designs - , see also: window diagrams - , see also: Sentinel method Flexible equation, for gradient shape 281 Fractional overlap - see fractional peak

Fractional peak overlap 123 - 125 -, definition 124 - , evaluation 127 - 129 - , measurement 124- 125 Full factorial design(s) 188 - 191,

209-210 -, evaluation 191 FAST LC columns 310 Gas-liquid chromatography (GLC)

- , relevant parameters 106 Gas-solid chromatography (GSC) 43 -45 - , relevant parameters 106 Gaussian peak -, characteristics 8 -, mathematical description 8 Gel permeation chromatography 22 - 23 General case, definition 119 General elution problem 253 - 255 Global optimum, definition 173 Gradient duration times, calculation of

optimum range 280 Gradient elution 193 - 199

signs

overlap

37 - 43

coefficient -, blank signal 197

335

-, in IEC 91 -, of proteins 263,280 - , see also: programmed solvent LC Gradient program 260 - 261 - , optimum shape in LSC 262 -, optimum shape in RPLC 261 - 262 Gradient scanning 193 - 199,290 -, for RPLC 195 - 199 -, -, graphical procedure 197 - 198 -, limitations 199 -, rule of thumb 194 Gradient shape parameter 281 Grid search -, as computation method 211, 219 -, as optimization procedure 179- 181 - , - , evaluation 182 - 183 -, required number of points 181 - 182 GC-MS combination 241 Height equivalent of theoretical plate

- , see: plate height Henry’s adsorption law 43 Henry’s law 38 Hierarchic criteria 141 - 142, 206 - 207,

-, for programmed solvent LC 281 Hold-up time -, definition 2-3 -, effect on retention surfaces in RPLC

Hyperbolic equation, for retention in

Hyphenated methods 241 Ideal chromatograms, calculation of ca-

Immiscibility of phases for LLC 52 - 53 Initial experiments 177 - , for iterative designs 220,230 Injection delay time 31 1 - 312 Injection profiles 311 -312 Injection time 311 -312 Inorganic anions, retention in IEC 87 -, see also: ion chromatography Instrument optimization 310- 31 8 - , importance of 297 - 298 Instrumentation

(HETP)

210

209,223

RPLC 61

pacity factors t 53 - 154

-, build instrument stage 2% - 297 - , for method development 18 - 19,

-, requirements in GC 314-316 -- requirements in LC 316-318 Interaction Chromatography, definition 1 Intermediate polarity, phases for LC 52,

Interpretive methods 199 - 235 - , characteristics 233,248 - 249 -, definition 178 -, description 199 -, evaluation 234 - 235 - , for programmed soIvent LC 284 - 290 - , for programmed temperature GC

- , optimization criteria 130 Ion chromatography 87,91 Ion exchange chromatography (IEC)

-, stationary phases 82 - 84 -, of proteins 87 - , relevant parameters 1 f 0 - , retention mechanism 86 - 87 - , stationary phase -, - , microparticulate 84 - , -, pelliculars 83 - 84 Ion exchange equilibrium constant 85 Ion-pair chromatography (IPC) 53,

- , full factorial design 189 - 191 - , normal phase 95 - 96 - , relevant parameters 11 1 -, retention equation 94 - , retention mechanism 94- 95 -, reversed phase 95 - 96 - , simple mechanism 93 - 94 Ion-pair extraction - see: ion pair chro-

Ionic separation methods 23 -, see also: ion-pair chromatography -, see also: ion exchange chromatograp-

Ionic strength, effect on retention in

Iso-eluotropic mixtures 198- 199, 206,

296 - 297

218

275 - 276,273 - 275

82 - 93

93 - 101

matography

hy

RPLC 73

336

/

212-213,218-219,221,226,278, 284 - 285

-, in LSC 80-81, 216, 217 - , in RPLC 63 - 67 - , -, experimental composition 65 - , - , multicomponent 66- 67 - , -, prediction of composition 65 - 67 Isocratic composition -, optimum range 198 -, prediction - see: gradient scanning Isocratic multi-solvent programming 265 Isoelectric point 73 Isothermal conditions, prediction from

temperature program 193 Iterative design(s) 220-233 -, characteristics 232 - 233, 249 -, definition 220 -, effect of local optima 228 -229 - , multi-dimensional 231 - 232 Iterative optimization methods - see: ite-

Library search techniques 242 - 243 Linear equation, for retention in RPLC

Linear interpolation, of retention surfaces

Linear retention relationships 203 - 205,

Linear segmentation, multi-dimensional

Linear solvent strength (LSS) gradients

-, definition 279 -, optimum slope 280 -, shape of 279 Liquid-liquid chromatography (LLC) 48,

-, characteristics 55 - 56 -, relevant parameters 107 Liquid-solid chromatography 76 - 82 -, relevant parameters 109 Literature search 16 Local optimum, definition 173 Local vs. global optima 176- 177 LC-MS combination 241 Mass flow sensitive detectors 305, 309

rative designs

62

229 - 23 1

209

231 -232

166,193-195,261-262,279-280

52 - 56

McReynolds constants 31 Mean effect of variable, estimation 189 Mean relative effect of variable, estima-

Median peak-valley ratios, definition 121 Method development -, general approach 15 - 18 -, in laboratory 18- 19 -, instrumentation for 18- 19, 296, 297 Micro-bore columns - see: narrow-bore

Migration speed, definition 2 Minimum a criterion 140- 141,

Minimum column diameter in LC, calculation 317-318

Minimum criteria 140 - 144 - , evaluation 143 - 144 Minimum resolution criterion 141 - 142,

-, as threshold criterion 207,214 Minimum separation factor criterion 141,

Mixed stationary phases -, for GC 41 -43,200 -, for LC 75 Mobile phase effects -, in LSC 77-78 - , in RPLC 59 - 67 -, in SFC 103-104 Mobile phase parameters, definition 105 Mobile phase time - see: hold-up time Mobile phase(s) -, definition 2

Model(s) -, equations vs. linear interpolation

- , for response surface 199 - 200 -, for retention surface 178, 199-200,

-, from chromatographic theory 230 -, moving least squares 231 - , polynomial 230 - 23 1 -, regression analysis 230 Modified Simplex optimization procedure

tion 189

columns

202-203,210

270 - 273,285,287

142,202 - 203,205

-, for SFC 102

229-231

220,214

337

-, see Simplex optimization Modifier - see: organic modifier Modulators in LSC 79-80 Molecular interactions 25 - 26 Monofunctional reagents 56 Multichannel detection, for peak recogni-

Multi-column techniques 159, 167, 257 Multi-dimensional window diagrams - , see: window diagrams, multi-dimen-

Multisegment gradients 287 - 288 Multisegment programs 268 - 269 - , disadvantages 268 - 269 - , for programmed temperature GC

-, in LC 283 - 284 - , -, systematic optimization 283 - 284 Multivalent ions, retention in RPLC

Mutual independence of parameters 173 -, see also: dependent variables Narrow-bore capillary columns for GC

- , extra-column dispersion 3 14 - 3 15 - , see also: column(s), capillary Narrow-bore columns for LC 308 -, advantages 308 Near-universal UV detection 199 Net retention time, definition 4 Normal phase liquid chromatography

-, preferred modifiers 21 2 - 21 3 Normalized resolution product 153,

-, definition 138 -, for limited number of relevant peaks

- , in programmed analysis 165 -, use of solute weighting factors 164 Nucleobases, retention behaviour in IEC

Nucleosides, retention behaviour in IEC

Nucleotides, retention behaviour in IEC

tion 241 -245

sional

270 - 273

72 - 73

300

(NPLC) 23,49,51- 52

223

162

90

90

86

Number of (theoretical) plates - see:

Number of datapoints - , for recording gas chromatograms

- , for recording liquid chromatograms

Number of experiments -, for full factorial designs 188 -, for optimizing LSS gradients

- , for predictive method in programmed solvent LC 288

- , for sequential optimization of programmed temperature GC 271

- , for Simplex optimization 186 -, -, in programmed solvent LC 277-

-, initial, for iterative designs 230 Number of parameters -, in iterative designs 230 - , in optimization procedures 177 Number of peaks, effect on optimization

Number of plates -, effect on minimum criteria 142- 143 -, effect on product criteria 135 -, effect on sensitivity 307 Observed capacity factor, of ionized spe-

cies in RPLC 71 -72 Octadecyl silica 58 Octyl silica 58 Open columns 6 - , see also: column(s), capillary Optimal particle size in LC 303 Optimization criteria - , comparison 137 -, effect on optimum gradient 282 - 283 - , evaluation 145 - 146 -, for limited number of relevant peaks

- , - , recommendations 161 - 162 - , for programmed analysis 165 - 167 - , recommendations for the general case

- , recommendations using solute weigh-

plate number

315-316

317-318

280-281

278

criteria 1 4 6 - 148

158-163

158-159

338

ting factors 164 Optimization procedures - , characteristics 177 - 179, 245 - 249 - , conclusions 249 - 250 -, general outline 178 - , for programmed temperature GC -, -, evaluation 275 - 276 Optimization process, overview 296 - 298 Optimization, of programmed analysis

Optimum temperature, for GC stationary

Optimum, definition 171 Organic modifier(s) 59 - , effect in programmed solvent LC 277 -, effect on retention in IEC 90-93 -, effect on retention in IPC 99 - , effect on retention in SFC 103 - 104 Organic polymers -, use in GSC 45 -, use in RPLC 70 Overlapping resolution mapping (ORM)

Packed columns 6 Packed columns - see also: column(s),

packed Pairing ion -, chain length 99 - , concentration -, -, effect on retention in IPC 94,

-, definition 93 -, distribution isotherm 97 -, examples of 98 -, nature of, effect on retention in IPC

Parameter limits 177 Parameter space -, definition 171 - , reduction of 188 - 199 Partial polarities 25 - 27 Particle size, effect on sensitivity

Peak area -, measurement 238 -, reproducibility 238

266 - 294

phases 41

141,214-215

96 - 98

97 - 99

309 - 3 1 0

Peak assignment 233 - 245 - see also: peak recognition Peak capacity -, statistical 15 -, theoretical 14- 15 Peak height (relative) -, effect on elemental criteria 127- 129 -, effect on peak-valley ratio(s)

- , effect on resolution 11 7 Peak identification 238 - , using spectroscopic techniques

Peak recognition 233 - 245 -, based on peak areas 236 - 238 -, -, computer program 238 - , in programmed temperature GC

- , using principal component analysis

- , using separate injections 236 - 237 - , with dual-channel detection 239 - 241 -, with multichannel detection 241 - 245 -, with single channel detection

Peak shape, effect on elemental criteria

Peak width, definition 7 Peak-valley ratio(s) 1 19 - 123 - , correction for baseline noise 123 -, definitions 119-121 -, measurement 122 -, product criteria 135-138, 142 - , -, for limited number of relevant pe-

-, -, time correction factors 151 -, -, use of solute weighting factors 163 -, relation to resolution 122 - 123 -, sum criteria 132-133, 137-138 -, -, correction for number of peaks

-, theoretical 122 - , threshold value 123 Permanent gases, analysis of 22,44 PH -, effect on retention in IEC 87 -90

122-123

241 - 243

273 - 275

243 - 245

236 - 238

129

aks 162

147

339

-, effect on retention in IPC 100 -, effect on retention in RPLC 69-73 -, -, equation for 71 -, working range, for RPLC 70 Phase ratio -, definition 4 -, parameters affecting 5 Phase selection diagram(s) 180- 181,

-, construction 221 -, two-dimensional 231 -232 Phenylthiohydantoin - see: PTH Physical parameters, definition 105 Pilot techniques - see: scouting techni-

Plate count - see: plate number Plate height, definition 9 Plate number -, definition 9 -, effect on resolution 10- 14 -, measurement 9 Polar adsorbents -, use for RPLC 51 -, see also: silica -, see also: alumina Polarity 24-27, 32-33 -, see also: solubility parameter(s) Polarity difference 53 - , see also selectivity of phase systems in

Polarity range of samples 254 Polyelectrolytes 73 -, see also: proteins Pre-column, for sample concentration in

Predictive optimization method -, for programmed solvent LC 288 - 290 Pressure limited conditions 155 - 156,

-, see also: required analysis time, pres-

Pressure, effect on retention in SFC

Primary parameters 17,191 - , definition 108 - , in programmed analysis 257 - 258,

221 -231

ques

LC

LC 306

302 - 303

sure limited conditions

103 - 104

266 - 268 Principal component analysis, for peak

Probe solutes - , McReynolds 31 - , Rohrschneider 29 -, Snyder 32 Product criteria 134- 140 -, literature 134 -, see also: peak-valley ratio(s), product

Product resolution criteria 134- 135, 226,

- , evaluation 137 - 138 -, see also: time corrected resolution

- , see also: calibrated normalized resolu-

- , see also: normalized resolution pro-

Program parameters -, definition 266-267 - , optimization 269 - 270,270,273 Programmed analysis 17,253 - 295 - , advantages of simple programs

-, applications 253 - 257 -, as scouting technique 192- 199 -, definition 253 -, disadvantages 256 - , factors affecting retention 257 - 258 - , factors affecting selectivity 257 - 266 -, in routine situations 256 -, optimization criteria 165 - 167 -, primary parameters 257 - 258 Programmed elution - see: programmed

Programmed solvent LC 260 - 266 -, optimization 276 - 294 - , optimization of primary parameters

-, optimization procedures - , - , characteristics 292 - 294 -, -, evaluation 290-294 - , primary parameters 276 - 277 -, secondary parameters 277

recognition 243 - 245

criteria

228

products

tion product

duct

268 - 269

analysis

292 - 293

340

- , selectivity optimization 293 - 294 -, ternary gradients 264 - 265 Programmed temperature GC 258 - 260 -, optimization 269 - 276 - , resolution 260 -, retention 259 - 260 Proteins 73,263 -, analysis by IEC 87 Proton acceptor parameter 32 - 33 Proton donor parameter 32 - 33 Pseudo-binary gradients 265 - 266 Pseudo-components - see: pseudo-sol-

Pseudo-isomeric plot, of response surface

Pseudo-solvents 265 -, definition 199 PTH amino acids -, retention behaviour in RPLC 264 -, separation by programmed solvent

Quadratic equation(s), for retention in

Quaternary mixtures 60 Ratio recording 239 - 241 -, limitations 241 Ratiograms - see: ratio recording Reduced linear velocity, definition 299 Reduced plate height, definition 299 Reduced sample size, definition 308 Relative retention -, definition 5 - , see also: selectivity Reproducibility -, in programmed solvent LC 261 -, of spectral information 245 Required analysis time 152,203 - , as optimization criterion 153,

-, calculation 155, 300 -, factors affecting, in LC 302 -, pressure limited conditions 152, 302 Required number of plates 126, 203,298 -, calculation 151, 155 - , for ideal chromatograms 156 Residual silanols 58

vents

172

LC 283

RPLC 60,214

156-157

Resolution 116 - 117 -, definition 7 -, factors affecting 10-14 -, fundamental equation 10 -, in programmed analysis 165-166 -, in programmed temperature GC 260 -, relation to peak-valley ratio(s)

-, with solvent peaks 168 Resolution criterion -, characteristics 117 -, measurement 117 -, see also: resolution Resolution mapping - see: overlapping

~ resolution mapping Response surface -, definition 171 -, representations 172- 173 Retention gap, in capillary GC 306 Retention index -, definition 27 - 28 -, polar contribution 28 - , use for GC optimization 45 - 47 -, variation with stationary phase com-

-, variation with temperature 46 Retention line - see: retention surface Retention mechanism -, in IEC 86-87 -, in IPC 93-94,94-95 -, in LSC 76-77 -, in RPLC 56 Retention surface -, definition 177-178 -, using iso-eluotropic mixtures in

Retention time, definition 2 Retention -, fundamental equation 3 -, in programmed temperature GC

Retrieval systems 243 Reversed phase liquid chromatography

-, characteristics 74 -, fexibility 49, 56

122-123

~

position 46 - 47

RPLC 222 - 223

259-260274-215

(RPLC) 20,23,49,51-52,5644

341

-, for large solute molecules 262 - 263 -, relevant parameters 108 - , preferred modifiers 21 2 - 21 3 -, selectivity 56 Rohrschneider classification scheme

Routine analysis -, instrumentation for 18- 19 -, with programmed elution 246 Ruler method 200-201, 206 Sample capacity -, in GSC 44 -, in LLC 55 Sample composition, effect on elemental

Sample molecular weight 22 - 23 Sample solvent 23, 306 - 307 Sample volatility 21 -22 Samples, information about 15 - 16 Sampling frequency 3 13 - , required for GC columns 3 15 - 3 16 - , required for LC columns 3 17 - 3 18 Scanning techniques - see: scouting

Scouting techniques 191 - 199 - , graphical procedures for RPLC

Secondary parameters 18 -, definition' 108 - 109 - , in programmed analysis 268 - 269 Selectivity classes, of solvents 35 Selectivity optimization . - , definition 17 - , in programmed solvent LC 284 - 290 - , - , evaluation 290 - 294 Selectivity -, definition 5 -, effect on resolution 10- 14 - , in programmed solvent LC 263 - 264 - , in programmed temperature GC 269,

-, in LC 50-52 - , in LLC 54-55 -, of phase systems in LC 52 Sensitivity optimization 305 -310 Sentinel method 212 - 220

27-31

criteria 129

techniques

197-198

276

- , application to programmed solvent

- , - , experimental design 284- 285 -, application to LSC 216-217 -, characteristics 219 - 220, 248 - , expansion to non-iso-eluotropic sol-

-, optimization criteria 214 Separation factor 125 - 127,153 - 154 - , characteristics 127 - , correction for plate count 127 - , definition 126 -, in programmed analysis 166 -, optimum range 154- 157 -, with solvent peaks 168 Sequential methods - , for optimizing programmed tempera-

ture GC 269 - 273 Sequential scanning 192 Shape of gradient programs 194 -, for RPLC 194 Shift rules - see: shifted compositions Shifted compositions - , in iterative designs 224- 225,

- , multi-dimensional 231 Silica 77 -, characteristics 81 - , reaction with silanes 56 - 57 -, use in GSC 45 -, see also: polar adsorbents Simplex design 212 Simplex lattice design - see: Simplex de-

Simplex method, as computation proce-

Simplex optimization 183 - 187 - , advantages 186 - , basic method 183 - 184 - , definition 183 -, disadvantages 187 -, initial experiments 185 -, modified method 184- 185 - , number of experiments 186 -, characteristics 247 -, optimization criteria 147 - 148

LC 284 - 288

vents 218-219

227 - 228

sign

dure 232

342

-, for programmed solvent LC 277 - 279 - , for programmed temperature GC

Simultaneous interpretive methods

-, see also: fixed experimental designs Simultaneous optimization procedures -, definition 179 - , without solute recognition 179 - 183 - , - , characteristics 246 - , - , see also: grid search -, with solute recognition - , - , see: interpretive methods Single channel detection, for peak recog-

Size exclusion chromatography 22 - 23 Slope, of retention lines in RPLC 62 - 63 - , variation with solute 62 - 64 Snyder classification scheme 31 -35, 212 Snyder theory for LSC - see: competi-

Soap chromatography - see: IPC, rever-

Soczewinski equation for retention in - LSC79 Solubility parameter(s) 24 - 27 -, definition 24 - , effect on retention -, -, in GLC 40-41 -, -, in LC 48-50 - , effect on selectivity in LC 50 - 52 -, limitations 52 -, of mixtures 60 -, relation to eluotropic strength in LSC

-, units 24 - , use for selection of LC phase systems

Solute, definition 2 Solvent generated LLC - see: dynamic

Solvent localization in LSC 81 Solvent peaks, effect on optimization cri-

Solvent program 260 - 261 - , see also: gradient program

275 - 276,269 - 270

200 - 220

nition 236 - 238

tion model

sed phase

77 - 78

48 - 52

LLC

teria 167 - 168

Solvent selectivity parameters, for LSC

Solvent strength in LSC - see: eluotro-

Solvent(s) -, classification 33 - 34, 77 -, polarity 32-33 - , selectivity 32 - 35 Specific cases, definition 1 19 Specific permeability coefficient 299 Specific retention volume, definition 39 Specific surface area 43 - 44 -, definition 6 Spectro-chromatograms 242 Stability of LLC systems 53 Staight phase liquid chromatography -, see: normal phase liquid chromato-

Standard deviation, of Gaussian peak 9 Standard state 48 Stationary phase characterization (GC)

Stationary phase parameters, definition

Stationary phase(s) -, definition 2 - , effect on selectivity in programmed

-, for GSC 45 -, for SFC 105 -, optimization in LC 217-218 -, polarity (GC) 27-31 Stepwise parameters, definition 110 Stepwise scanning - see: sequential

scanning Stop criteria - , definition 178 -, for iterative designs 225 - , for Simplex optimization procedure

Strong dipole parameter 32 - 33 Sum criteria 131 - 133 -, evaluation 133 -, see also: peak-valley ratio(s), sum cri-

-, see also: sum resolution criterion

216-217

pic strength

graphy (NPLC)

27-31

105

temperature GC 276

185

teria

343

Sum resolution criterion 117 - 119,

Supercritical fluid chromatography

-, applications 103 -, detection 103 - relevant parameters 1 12 Supercritical fluid, definition 101 Surface area, effect on phase ratio 6 Systematic optimization in programmed

- , primary paramems 279- 284 -, with limited solute recognition

-, without solute recognition 279 - 281 Systematic sequential optimization - , for programmed temperature GC

Temperature control in LLC 53,55 Temperature program 259 - 260 - , see also: programmed temperature

Temperature - , effect on retention in GLC 38 -40 -, effect on retention in GSC 44-45 -, effect on retention in IEC 89-90 -, effect on retention in IPC 101 -, effect on retention in LSC 82 -, effect on retention in RPLC 67 - 69 Ternary gradients 264- 265,278 Ternary mixtures 60,67 - , prediction of retention from binary

Thermodynamic parameters, definition

Thermodynamic phase diagram - , pure component 101 -, ternary mixture 54 Thin layer chromatography (TLC), as

Threshoid criteria 136,141 - 143,145,

Threshold resolution 129 - , see also: threshold separation -, see also: peak-valley ratio@), thres-

131 - 133,137- 138,149

(SFC) 20,101 - 105

solvent LC

281 - 283

270 - 273,275 - 276

GC

data 288

105

scouting technique 192

150-151,207,214,288-290

hold value

Threshold separation 121 - , see also: pea&-valley ratids), thres-

Time corrected resolution products

- , definitions 158 - , for limited number of peaks 163 Time correction factors in optimization

- , see also: analysis time, effect on opti-

- , see also: column independent time

Time correction terms in optimization

Time required for optimization experi-

Total overlap criterion 144- 145 Trial calculations 288 Trifunctional reagents 57 Two-dimensionai window diagrams 210 Type of cdumn, comparison 299 - 300 Univariate optimization 173 - 176 - , characteristics 245 -, in IEC 92-93 Unretained time - see: hold-up time Valley to top ratio - , definition 12 1 - , use in optimization criteria 140 -, with solvent jxaks 168 Vapour pressure, effect on retention in

Viscosity, of LC eluents 305 Volatility range, of samples 254 Water, polarity of 25 Weak acids and bases, retention behavio-

ur in IEC 88-89 Weak acids, effect of pH on retention in

RPLC 204 Weighting factors for optimization crite-

ria 148-149, 150 -, for solutes 163 - 165 Wide wage samples 254 - 255 Wide-bore capillary columns for GC 300 -, extra-column dispersion 314-315 Window diagram@) 17 1 - 172,200 - 206

hold value

153-158

criteria 149 - 15 1

mization criteria

factors

criteria 148 - 149

ments 182

GLC 38

344

- , application areas 203 - 205 - , characteristics 205 - 206, 219 - 220,

-, for pH optimization in RPLC

- , multi-dimensional 209 - 21 2 -, -, applications 210 - 21 1 -, optimization criteria for 203

248

204 - 205

345

ion of chromatographic selectivity_1986

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