measurement grew and became more or less consolidated, he prepared ex­tensive notes and preliminary drafts of some of the applications. Unfor­tunately, the increasing demands placed on him by his accession to the Chair of Physical Chemistry at the University of Melbourne made it impossible for him to complete his contribution. In the (northern) winter of 1978-79,1 had another study leave period, this time at the University of California at Berkeley in the laboratories of Professor Douglas Fuerstenau, and there had the opportunity of finishing off the manuscript, drawing on the notes which Tom Healy had prepared and made available to me. I am very pleased to be able to thank him for his generosity in that regard. A t the same time, I had the opportunity of discussing various ideas with Doug Fuerstenau and of drawing on his extensive knowledge of the application of zeta potentials in the study of mineral flotation and in his collection of reprints in the area. I am indebted to him for his personal assistance and for the facilities he placed at my disposal. In particular, I must express my deep appreciation to Mrs Gloria Pelatowski of the Berkeley laboratory for preparing most of the illustrations.

I am also indebted to Professors Ottewill and Rowell, who read the manu­script and have offered many valuable comments. I have tried to incorporate all of their suggestions but must, of course, take responsibility for any re­maining inconsistencies and misconceptions. Various parts of the manuscript were also read and corrected by Professor Bil l Russel of Princeton, Dr Di rk Stigter of the U.S.D.A. California, and Dr Neil Furlong of the University of Melbourne, and I thank them for their help.

Soon after I began this work, there appeared a book called Electrokinetic Phenomena by S. S. Duhkin and B. V. Deryaguin, in the series on Surface and Colloid Chemistry, edited by Professor Egon Matijevic. The appearance of a book in the same area by two such distinguished contributors caused us to think very carefully about whether we should proceed, but after examining the thrust of that work and noting the areas to which it had not addressed itself we decided to press ahead. Since then I have had the opportunity of showing some of my material to Professor Deryaguin and have been able to take advantage of his valuable comments.

Finally I thank the several typists who prepared various parts of the manuscript and the people at Academic Press for their continued interest in the work and their expedition in its publication.

April, 1981 Robert J. Hunter

Contents

Preface

Chapter 1 Introduction 1 1 • 1 Origin and classification of electrokinetic effects 1 1.2 The zeta potential and the surface of shear 4 1.3 Significance of zeta potential 6 1.4 Outline of the structure of this treatment 7 1.5 The basic equations 8

References 10

Chapter 2 Charge and Potential Distribution at Interfaces 11 2.1 The electrostatic potential of a phase 11

• 2.2 Mechanism of charge development at interfaces 17 2.3 The potential and charge distribution in the electrical

double layer (classical theory) 21 2.3.1 The flat plate model 22

2.3.1.1 The potential distribution 22 2.3.1.2 Surface density of charge 27 2.3.1.3 Ion distribution in the double layer 28 2.3.1.4 The electrokinetic charge density 29

2.3.2 The double layer around a sphere 30 2.3.2.1 The potential distribution 30 2.3.2.2 The particle charge 31

2.3.3 The double layer around a cylinder 32 2.4 Modifications to the Gouy-Chapman theory for

flat plates 33 2.4.1 The inner (compact) layer 33 2.4.2 The dielectric permittivity of the inner region 40 2.4.3 The discreteness of charge effect 42 2.4.4 The diffuse layer 44

2.5 The double layer around a sphere 46

vii

2.5.1 The exact solution of the Poisson-Boltzmann equation 46

2.5.2 Analytical approximations 47 2.5.3 Correction for finite ion size 50 2.5.4 The compact layer around a sphere 51

2.6 The double layer around a cylinder 52 2.7 The double diffuse double layer 52

References 55

Chapter 3 The Calculation of Zeta Potential 59

I . CLASSICAL THEORY

3.1 Electro-osmosis 59 3.1.1 Measurements on single capillaries 59 3.1.2 Behaviour in porous plugs 64 3.1.3 Electro-osmotic counter pressure 64

3.2 Streaming Potential 3.2.1 Measurements in single capillaries 64 3.2.2 Behaviour in porous plugs 67 3.2.3 Streaming current measurements 68

3.3 Electrophoresis 69 3.3.1 The Smoluchowski and Hiickel equations 69 3.3.2 Conduction effects 71

3.3.2.1 Effect of particle conductance and shape 71 3.3.2.2 Effect of surface conductance 73

3.3.3 Electrophoresis of emulsions 74 3.4 Sedimentation potential 75

I I . MORE RECENT DEVELOPMENTS

3.5 Electro-osmosis 77 3.5.1 Flow in narrow capillaries 77

3.5.1.1 Slit-shaped cross-section 77 3.5.1.2 Cylindrical cross-section 79

3.5.2 Flow in porous plugs 82 3.6 Streaming potential 82

3.6.1 Flow in narrow capillaries 82 3.6.1.1 Slit-shaped cross-section 82 3.6.1.2 Cylindrical capillaries 88

3.6.2 Electrokinetics in porous plugs 90 3.7 Electrophoresis 98

3.7.1 Introduction of the relaxation effect 98 3.7.2 Electrophoresis of a rigid insulating sphere 100 3.7.3 Electrophoresis of cylinders 112 3.7.4 Particles of arbitrary shape 114 3.7.5 Influence of particle concentration 115 3.7.6 Influence of Brownian motion 117

3.8 The sedimentation potential 117 3.9 Validity of the electrokinetic equations 119

References 121

Chapter 4 Measurement of Electrokinetic Parameters 125 4.1 Electro-osmosis 126

4.1.1 Electrical measurements 126 4.1.2 Measurement of liquid volume 127 4.1.3 Flow in a single closed capillary 131

4.1.3.1 Cylindrical capillaries 131 4.1.3.2 The rectangular cell 132

4.1.4 Electro-osmotic counter pressure 136 4.2 Streaming potential measurements 136

4.2.1 Cells for use with powders 137 4.2.2 The electrical measurements 142 4.2.3 Pressure measurement 144 4.2.4 The cell packing 145 4.2.5 Data treatment 145 4.2.6 Measurement of streaming current 145 4.2.7 Sinusoidal measurements 147

4.3 Electrophoresis measurements 150 4.3.1 Microelectrophoresis 152

4.3.1.1 Single cell designs 154 4.3.1.2 Optical problems 155 4.3.1.3 The electrical field 157 4.3.1.4 Two-tube cells 159 4.3.1.5 More recent developments 163

4.3.2 Moving boundary methods 165 4.3.3 Tracer electrophoresis 164 4.3.4 The mass transport method 165 4.3.5 Electrophoretic light scattering 167 4.3.6 Non-uniform field measurements 172 4.3.7 Other procedures 174

4.4 Sedimentation potential 174 References 175

Chapter 5 Electroviscous and Viscoelectric Effects 179 5.1 Porous plugs and capillaries 180

5.1.1 The primary electroviscous effect 180 5.1.2 The secondary electroviscous effect 183 5.1.3 The tertiary electroviscous effect 185 5.1.4 Experimental data on the electroviscous effect

in capillary systems 186 5.2 Suspensions of spherical particles 190

5.2.1 The primary electroviscous effect 192 5.2.2 The secondary electroviscous effect 194 5.2.3 Experimental evidence on the primary and

secondary electroviscous effects in suspensions 196 5.2.4 The tertiary electroviscous effect 200

5.3 The viscoelectric effect 204 5.3.1 Modified viscosity and permittivity in the

double layer 205 5.4 Position of the plane of shear 210

References 216

Chapter 6 Applications of the Zeta Potential 219

6.1 Ionic adsorption at interfaces 220 6.2 Simple inorganic ions as solutes 223

6.2.1 Potential-determining and indifferent ions 223 6.2.2 The point of zero charge 224 6.2.3 The isoelectric point 229 6.2.4 Specifically adsorbed ions 233

6.3 Charge and potential distribution for the Gouy-Chapman-Stern-Grahame (GCSG) model of the interface 236

6.4 Zeta potential and colloid stability 239 6.5 Sedimentation volume and settling time 246 6.6 Electrophoretic deposition 247 6.7 Flotation 248

6.7.1 Collector adsorption 248 6.7.2 Activation 250 6.7.3 The slime coating problem 251

6.8 Correlation of zeta with other properties 252 References 254

Chapter 7 Influence of Simple Inorganic Ions on Zeta Potential 258

7.1 Introduction 258 7.2 Surfaces obeying the Nernst equation 259

7.2.1 The silver halide-water interface 259 7.2.2 The calcium oxalate system 265

7.3 The polymer colloid-water interface 265 7.3.1 General electrokinetic properties of polymer

colloids 266 7.3.2 Site-dissociation models of the polymer

colloid-water interface 270 7.3.2.1 Single (acid) site dissociation 270 7.3.2.2 Two-site dissociation models 274

7.4 The oxide-water interface 278 7.4.1 General electrokinetic properties of oxide-

water interfaces 279 7.4.2 Ageing, solubility and leaching reactions 282 7.4.3 The porous gel model of the oxide-solution

interface 284 7.4.4 The site-dissociation model for oxides 286

- 7.4.5 Site-dissociation-site-binding models for oxides 290 7.4.6 The Stern model of the oxide-solution interface 295

7.5 Clay mineral systems 297 7.6 Application to other surfaces 299

References 302

Chapter 8 Influence of More Complex Adsorbates on Zeta Potential 305

8.1 Specific adsorption of simple metal ions 305 8.2 Surfactant adsorption 308

8.2.1 Ionic surfactant adsorption on A g l 309 8.2.2 Ionic surfactant adsorption on hydrophilic

surfaces (e.g. oxides) 316 8.2.3 Ionic surfactant adsorption on other surfaces 322 8.2.4 Non-ionic surfactant adsorption 324

8.3 Adsorption of hydrolysable metal ions 326 8.4 Adsorption of neutral polymers 334 8.5 Adsorption of polyelectrolytes 340 8.6 Adsorption of proteins 341 8.7 The interpretation of zeta potential 342

References 343

Appendix 1 Vector Calculus: The Equations of Poisson and of Navier and Stokes 345

A l . l Scalar and vector fields 345 A l .2 The gradient of a scalar field 345 A l .3 Divergence of a vector field 348 A1.4 Poisson's equation 349 A1.5 The Navier-Stokes equation 350 A1.6 Shear stress and viscosity 352 A l .7 The curl of a vector field 354 References , 356

Appendix 2 Electrical Units 357 Reference 359

Appendix 3 Viscous Flow of a Fluid 360

Appendix 4 The Stern Adsorption Isotherm 362

Appendix 5 Interaction between Colloidal Particles 363 A5.1 Interaction between approaching double layers 363

A5.1.1 The Debye-Hiickel approximation 366 A5.1.2 Small degrees of double-layer overlap 367 A5.1.3 For very large degrees of interaction

and high potentials 367 A5.2 The force between colloidal particles 368 References 369

Appendix 6 The Gibbs Adsorption Isotherm at Charged Interfaces 370 A6.1 The surface potential of the A g l crystal 371 Reference 372

Index 373

Chapter 1

Introduction

1.1 O r i g i n a n d c l a s s i f i c a t i o n o f e l e c t r o k i n e t i c e f f e c t s

When two phases are placed in contact there develops, in general, a difference in potential between them. I f one of the phases is a polar liquid, like water, its (dipolar) molecules wil l tend to be oriented in a particular direction at the interface and this wi l l generate a potential difference. I f there are ions or excess electrons in one or both phases, or ionogenic groups present, there wi l l be a tendency for the electric charges to distribute themselves in a non­uniform way at the interface. The reasons for this behaviour and the nature of the resulting distribution wil l be discussed below (Chapter 2). For the present we need only note that except under very special conditions, the region between two adjoining phases is always marked by a separation of electric charges so that near to or on the surface of phase I there is an excess of charge of one sign and the balancing charge is distributed in some way through the adjoining surface regions of phase I I (Fig. 1.1.).

I t would be difficult to overestimate the importance of this process, be­cause it is basic to an understanding of an enormous variety of natural phenomena, particularly in the fields of colloid chemistry and electro­chemistry. Phenomena such as electrode kinetics, electrocatalysis, corrosion, adsorption, crystal growth, colloid stability and flow behaviour (both of colloidal suspensions and of electrolytes through porous media) cannot be properly treated without a knowledge of the distribution of charges and dipoles in the interfacial region.

If the surface of phase I is positively charged, its electrostatic potential wi l l be positive with respect to the bulk of phase I I ; if phase I I is a liquid contain­ing dissolved ions, then as one moves into phase I I , the potential wil l decrease, more or less regularly, until i t becomes constant in the bulk liquid far from

1

Phase I I

Fig. 1.1. Possible distribution of charges at an interface between two phases. Only the excess charges in each phase are shown.

the surface of phase I . I t is customary to take this constant potential in the bulk of one of the phases (usually a liquid) as the reference or zero potential. For aqueous systems, with which we wil l be principally concerned, the words "far f rom" mean "at distances greater than about 5-200 nm (depending on the electrolyte concentration)" (Fig. 1.2). The region where the liquid has a positive electrostatic potential wil l accumulate an excess of negative ions and repel positive ions of the electrolyte. I t is this excess of negative ions which gradually lowers the electrostatic potential (and the electric field) to zero in the bulk electrolyte. The arrangement of (positive) charges on the surface of phase I and the charges in the liquid phase I I is referred to as the electrical double layer at the interface.

When one of these phases is caused to move tangentially past the second phase there are observed a number of phenomena which are grouped under the title of "electrokinetic effects". There are four distinct effects depending on the way in which motion is induced. They are: electrophoresis, electro-osmosis, streaming potential and sedimentation potential.

(a) Electrophoresis

I f one phase consists of a liquid or gas in which the second phase is suspended as particles of solid or liquid, then the particles can be induced to move by

O P x Fig. 1.2. A possible electrostatic potential distribution in the liquid (phase 11) near a solid surface. The point /"would be at a distance of about l - 5 0 n m in most colloid systems of interest.

Fig. 1.3. Electrophoresis. The motion of the particles can be followed by observing the move­ment of the boundary between the cloudy suspension and the clear supernatant.

applying an electric field across the system (Fig. 1.3). This is called electro­phoresis. Measurement of the velocity of the particles under a known external field gives information about their net electric charge, or their surface potential with respect to the bulk of the suspending phase.

(b ) E lec t ro -osmos is

When the solid remains stationary and the liquid moves in response to an applied electric field this is called electro-osmosis. I t occurs when the solid is in the form of a capillary or a porous plug which is filled with the liquid. The applied field acts upon the charges (usually ions) in the liquid, and as they move in response to the field they drag the liquid along with them. Measurement of the velocity of the liquid, or the volume of liquid transported per unit current flow, again gives information about the net surface charge or the electrical potential in the neighbourhood of the wall.

(c) S t reaming potent ia l

Instead of applying an electric field to cause liquid to move through a capillary or porous plug, one can force the liquid through under a pressure gradient. The excess charges near the wall are carried along by the liquid and their accumulation down-stream causes the build-up of an electric field which drives an electric current back (by ionic conduction through the liquid)

agamst the direction of the hquid flow. A steady state is quickly established and he measured potential difference across the capillary or plug is called

m the neighbourhood of the wall,

(d) Sed imen ta t i on potent ia l

When charge colloidal particles are allowed to settle (or rise) through a fluid under gravity or m a centrifugal field, a potential difference i generated this is the sedimentation potential. Since each particle is usually surrounded by a balancing atmosphere of opposite charge it might be expected that this movemen would not lead to a potential difference. As it moves, howeve

stabhshedbvaT ^ * ^ * ™ ° ™ 1 S continuously' established by a flow of charge into one side and out of the other (Fig 1 4)

moffo H T 6 5 ? U P 3 field W h i G h i s n e ^ a t i v e i n the direction of their motion and the steady state is established by a backflow of positive tons

v a S n T n f r ? 6 " " h a V S b e e n S t U d i e d f ° r a V e r y l 0 n g t u n e - T h e fcst obser-centurv n i t 7™™ * t h e b e g l n n i n S o f t h e ™ e t e e n t h ï o r hri g r l y S t e m a t l C S t U d i 6 S ° f e l e C t r i c i t y f i r s t b e c a m e Possible. (For a brief review of this early history see Dukhin and Deryaguin (1974)

Fig. 1.4. Schematic picture of the current flow which generates the sedimentation potential.

1.2 T h e z e t a p o t e n t i a l a n d t h e s u r f a c e o f s h e a r

In almost all electrokinetic phenomena a fluid moves with respect to a solid surface. (An exception is the electrophoresis of emulsions.) For the most nf T l ! be concerned with determining the relation between the velocity of the fluid (which wil l generally vary with distance from the solid) and the electric field in the interphase region. The electric field wil l be partly deter­

mined by the surface charges on the solid and in the liquid but may also include an externally imposed field, either generated deliberately by the experimenter (electro-osmosis and electrophoresis) or arising out of the motion of particles (sedimentation potential) or ions (streaming potential).

The relation between the potential (or the electric field) at any point and the number of charges is given by Poisson's equation (Appendix 1.4). The charges themselves wil l respond to three sorts of forces:

(i) the electrical potential;

(ii) the diffusion force, tending to smooth out concentration variations;

(iii) the bulk movement of charge carried along by the flow of the liquid (convective transport).

At the same time the liquid itself is subjected at each point to forces caused by pressure gradients in the system and the electrical charges it contains, as well as shear forces induced by neighbouring parcels of liquid moving with different velocities.

Even with the powerful tools of vector calculus and high-speed computer solution of the resulting differential equations it is still necessary to make a number of significant simplifications and to treat, at least in the first instance, highly idealized models of the real experimental systems. Nevertheless, a great deal of interesting and valuable information can now be obtained from electrokinetic measurements.

Theoretical treatments generally assume that the solid is either a sphere, a cylinder, or a large flat plate; more rarely it may be a disc or ellipsoid. The liquid is assumed to be Newtonian (i.e. its viscosity does not depend on shear rate (see Appendix 3) and moving sufficiently slowly so that turbulence and other non-linear effects are absent.

The most important concept which is introduced is that of the surface of shear. This is an imaginary surface which is considered to lie close to the solid surface and within which the fluid is stationary. In the case of a particle undergoing electrophoresis, the surface of shear forms a sheath which en­velopes the particle. A l l of the material inside that sheath forms the kinetic unit so that the particle moves along with a certain quantity of the surrounding liquid and its contained charge. Measurement of the electrophoretic mobility (i.e. the velocity per unit electric field) therefore gives a measure of the net charge on the solid particle.

The analysis of the forces on the solid or the liquid can be carried out in terms of either charge or electrostatic potential. In the latter case one calcu­lates the average potential in the surface of shear; this is called the electro­kinetic or zeta potential, and is universally given the Greek symbol, zeta (Q.

At the microscopic level it may be inferred from the finite dimensions of ions and solvent molecules that the real slipping surface is likely to be a

constantly varying one, with indentations fluctuating on a scale of the order of 10" 8 cm or more. Such fluctuations occur, however, on a time scale which is extremely short compared to the duration of a measurement and the mathematical abstraction of a "shear surface" then becomes a time-averaged quantity which wil l be physically very much smoother. On top of this, however, there may remain a more premanent "roughness" of the particle surface due to the existence of more or less permanent asperities on the particle surface.

Such particles, in extreme cases, may behave as though their real radius of curvature were very much smaller than their macroscopic dimensions would suggest. A regularly "crenellated" surface of this kind could occur, for example, on a "smooth" particle which was coated with a protein whose tertiary (partially denatured) structure was determining the local radius of curvature on the surface.

Another obvious limitation of the concept of the "surface of shear" is that it suggests that the fluid in the neighbourhood of the surface retains its bulk properties (especially permittivity and viscosity) right up to this surface and then suddenly, just inside the shear surface, the viscosity becomes in­finitely large. We shall find that this limitation can in many cases be replaced by a less restrictive one without too much trouble, but at the cost of some lack of definition of the zeta potential (Chapter 5). Nevertheless, i t seems prefer­able to go ahead with the simpler notion of the shear surface until we have established the basic theoretical structure of the electrokinetic effects and then to introduce this modification as an afterthought. This procedure is justified by the fact that the present consensus amongst colloid scientists is that the effect of the electric field on the viscosity is probably not as large as it was first thought to be.

1.3 S i g n i f i c a n c e o f t h e z e t a p o t e n t i a l

Many of the important properties of colloidal systems are determined directly or indirectly by the electrical charge (or potential) on the particles. Ad­sorption of ions and dipolar molecules is determined by, and also determines, this charge and potential distribution. The potential distribution itself determines the interaction energy between the particles, and this is in many cases responsible for the stability of particles towards coagulation and for many aspects of the flow behaviour of the colloidal suspension. I t is also possible to correlate the (-potential with the sedimentation behaviour of colloidal systems and with the flotation behaviour of mineral ores.

Since much of the theoretical basis of colloid chemistry demands a know­ledge of the potential distribution around the particles, it is essential that

we have access to accurate knowledge of that potential. Unfortunately, that knowledge is difficult, and in some respects impossible, to obtain, for reasons which we shall discuss below (2.1). The most important techniques used to acquire information about these potentials are (i) the measurement of volta potential differences in surface chemistry and (ii) the measurement of zeta potentials. Despite the very real limitations of both of these procedures they remain the most valuable ones we have at present. Some recent developments in the measurement of semiconductor properties (see e.g. Schenck, 1977) are very promising, and may ultimately provide us with an alternate accepted technique for measuring total double-layer potentials. In the meantime, there have been many improvements in the measurement and interpretation of the zeta potential.

I t is the purpose of this book to bring together the large amount of material, both experimental and theoretical, bearing on the question of interpreting zeta potentials and to place it in the context of general double-layer potential theory. The work up to the early 1950s was very ably reviewed by Overbeek (1952), and we shall take that work as essentially a base line although for the sake of completeness we have-included a good deal of material which Overbeek has already covered.

There are many situations in which the (-potential is used as a parameter in its own right—characterizing the outer, diffuse part of the double layer and hence valuable for discussing the interaction between particles or the flow of liquid through membrane pores or porous plugs. There are other situations in which one wishes to construct a picture of the charge and potential distribution across the entire interfacial region, and in this case (-potential values may be taken as part of the input information.

There are also many situations, especially in the field of biochemistry, in which one is more interested in separating a complex mixture of com­ponents with differing mobilities rather than in attempting to understand those mobilities in a fundamental way. Great strides have recently been made in this area and an excellent compilation of the modern techniques has recently appeared (Righetti et al, 1979), so we shall not attempt to cover this very important area.

1.4 O u t l i n e o f t h e s t r u c t u r e o f t h i s t r e a t m e n t

We begin with a review of double-layer theory in general, including a brief discussion of the nature of the electrical potential in the interfacial region (Chapter 2). Though this material has been reviewed many times it is necessary to. go over the ground again in order to have a consistent analysis to which we can refer when attempting to interpret the measured values.

In Chapter 3, theoretical relationships are derived to link the experi­mentally measurable quantities with the zeta potential, or charge. In this chapter the shear surface wil l be assumed to be a surface of constant potential, defined in terms of the boundary conditions for solving the differential equations which describe the electrical and hydrodynamic forces on the phases. The treatment is divided into two parts: the classical theory of each effect, which is sufficient for most purposes, followed by an examination of the various extensions and refinements which have appeared more recently.

Chapter 4 examines the experimental techniques required to determine as accurately as possible the parameters required in order to calculate (. In Chapter 5 the interaction between the electrical forces and the viscous prop­erties of the medium (the viscoelectric coefficient) are examined. This leads on naturally to an examination of the electroviscous effects and then to a consideration of the rheological (flow) properties of colloidal dispersions.

Chapter 6 is concerned with a number of applications of the (-potential including its use in the study of adsorption, stability behaviour, sedimentation and flotation. Finally, Chapters 7 and 8 take up, in more detail, the question of potential distribution and adsorption for simple ions and more complex species, respectively. In many cases this involves the postulation of models for the charge and potential distribution at the interface. The electrostatic potential at some distance ( > ~ 1 nm) from the surface wil l be assumed to be given by the classical Poisson-Boltzman equation, but at the surface itself the interplay between charge and potential (and the effect this has on the dielectric properties of the interface) is determined by the details of the mechanism of charge generation. The appropriate model description must be chosen in each case. Progress in this area has been very rapid in recent years although much remains to be clarified.

1.5 T h e b a s i c e q u a t i o n s

Any theoretical treatment of electrokinetics must start from the fundamental equations describing (i) the electrostatic potential, (ii) the fluid flow and (iii) the ionic current flows which are generated by the relative motion of the phases. The relevant equations are couched in the language of vector calculus (div, grad and curl (or rot, short for rotation)). Although a detailed explana­tion of these equations would be beyond the scope of this work an attempt is made in Appendix 1 to explain the meaning of these terms and the way in which the basic equations encapsulate our understanding of the behaviour of matter.

I t is not intended that the appendix should serve as an introduction to the solution of these equations, but rather it should indicate the nature of the

assumptions on which the basic equations rest. The most important of these is that each phase may be treated as a continuum and that quantum effects are negligible/

As noted above, the electrostatic potential is assumed to obey Poisson's equation (Appendix 1.4).

The ion distribution is assumed to be determined by the Boltzmann equation so the fundamental differential equation describing the potential distribution is the Poisson-Boltzmann equation eq (2.3.7). Among other things, this equation intrinsically assumes that (i) the ions are point charges, (ii) the solvent is continuous and is characterized by a constant permittivity which is not affected by the overall field strength or by the local field in the neighbourhood of an ion and (iii) the only work done in bringing an ion from the bulk up to a certain point in the field, where the electrostatic potential is

is the electrical work term z^eif/ where z{ is the valency of the ion. Other work terms like (i) the work done to displace some solvent to make room for the ion, (ii) the change in energy due to local reorientation of the solvent around the ion and (iii) the effect of the atmosphere of other ions on the electrostatic potential experienced by a given ion (the so-called self-atmos­phere potential of Debye-Hückel Theory) are all neglected in the initial treatment, though some discussion wil l be given of the significance of these approximations. The limitations of the Poisson-Boltzmann equation are most obvious in the immediate neighbourhood of the solid surface (within say 1 nm) and a considerable improvement is achieved by treating that layer (the so-called Stern layer) in a rather different fashion. The electrical potential in the outer (diffuse) part of the double layer is then well described by the Poisson-Boltzmann equation whilst the Stern layer can be approxi­mated as a molecular condenser, with explicit account taken of the dimen­sions and charge characteristics of the ions and (dipolar) molecules of which it is composed.

Likewise, when we come to consider the fluid flow we shall use the Navier-Stokes equation (Appendix 1.5). This assumes that the fluid is a simple Newtonian liquid (Appendix 3) with a viscosity independent of position (i.e. independent of shear rate and local electric field strength). Some relaxa­tion of the latter restriction can be tolerated (Section 5.1) but the analysis has so far been done only for the simplest geometry (a liquid near a flat plate). The particular form of the Navier-Stokes equation used here also assumes that the flow is slow and steady. This allows us to neglect the so-called inertial term (Appendix 1.5) which makes the equation non-linear (i.e. the full equation would predict that the velocity of the fluid, or particle, was a quadratic [or higher order] function of the electric field). I t is observed experimentally that the fluid velocity (or particle velocity) is always directly proportional to the applied field so that quadratic and higher terms can

safely be neglected. A striking proof of this assertion, for the case of particles in an insulating medium, is provided by the recent work of Vincett (1979), who established this proportionality for fields up to 3 x 10 7V m ~ 1 . Removing the non-linearity from the equation makes it a great deal easier to solve.

Although the adoption of these simplifying assumptions may not always be warranted we shall find that a great deal of the experimental evidence so far collected can be adequately rationalized on the basis of the theory so developed.

R e f e r e n c e s

Dukhin, S. S. and Deryaguin, B. V. (1974). Electro-kinetic phenomena. In "Surface and Colloid Science" Vol. 7 (E. Matijevic ed.) John Wiley, New York.

Overbeek, J. Th. G. (1952). In "Colloid Science", Vol. 1, pp. 194-244. (H. R. Kruyt ed.). Elsevier, Amsterdam and London.

Righetti, P. G., van Oss, C. J. and Vanderhoff, J. W. (1979). "Electrokinetic Separation Methods" Elsevier, New York.

Schenck, J. F. (1977). J. Colloid Interface Sci. 61, 569. Vincett, P. S. (1979). / . Colloid Interface Sci. 69, 354.

Chapter 2

Charge and Potential Distribution at loterffaces

2.1 T h e e l e c t r o s t a t i c p o t e n t i a l o f a p h a s e

Before we can describe the potential distribution at an interface in a satis­factory way, we must examine some basic concepts in the theory of electro­statics. Our description wil l be based on the distinctions introduced by Lange, as outlined by Overbeek (1952, p. 124), and subsequently discussed in some detail in a number of review articles (see, Grahame, 1947, and, particularly, Parsons, 1954) and texts (e.g. Davies and Rideal, 1963, Sparnaay, 1972a).

At the surface of any phase, even a pure metal in vacuo, there is a separation of positive and negative charge components so as to create a region of varying electrical potential which extends over a distance of the order of one or more molecular diameters. The potential differences generated across these layers are calculated to be of the order of a volt. Bardeen (1936), for example, calculated by wave mechanical methods that the electrons on a metal surface tend to protrude outwards from the surface, producing a negative charge layer which is compensated by a similar positive layer just inside the surface. (This conclusion had evidently been arrived at earlier by Frenkel (Frumkin and Pleskov, 1973)). By the same token, when two phases are in contact there is a tendency for any charged constituents, either electrons or ions, to be attracted, to different degrees, into the two phases and for surface dipolar molecules to be oriented selectively with respect to the two phases. The resulting electric field may also cause polarization effects in neighbouring molecules. Al l of these effects tend to produce a difference in the electrical potential between the interiors of the two phases. This difference is called the "inner" or Galvani potential difference, Ad>. Despite its theoretical sig­nificance and ubiquitous character, however, this difference in potential is impossible to measure unequivocally except when the two phases are chemi-

11

cally identical, in which case, by definition, most of the interesting interfacial effects disappear.

The impossibility of measuring Ad) in general is in no way related to the inadequacy of our measuring techniques but is quite fundamental. The potential difference referred to would measure the total work done when a test charge was moved from the interior of one phase to the interior of the second. As Guggenheim (1929) has so clearly pointed out, however, this is not a well-defined quantity, because the work done in such a case depends upon the nature of the electrical charge. From the experimental point of view the smallest and least disruptive charge which could be introduced to make such a measurement would be an electron, but an electron, when introduced into a material medium, has a significant effect on the electrical structure of its surroundings and hence on the work done. The theoretical test charge, on the other hand, as it is understood in electrostatic theory, is infinitesimally small and its introduction into the medium must not cause any changes in the distribution of charges or the orientation of dipoles. The distinctions and definitions introduced by Lange cannot eliminate the prob­lem but can clarify which kinds of potentials can be measured and which cannot.

The electrostatic potential near an isolated (macroscopic) charged object, in vacuo, is a well-defined quantity which measures the work done in bringing a unit charge from infinity up to the near neighbourhood of the object. Since it represents a potential difference between two points in the same medium (namely, a vacuum) it can easily be measured. The charge on the object may be assumed to be located on its surface and the potential falls off with distance from the surface in accordance with Coulomb's Law. If, for example, the object is a sphere of radius a, the potential at a distance r from its surface is given by (see Appendix 2):

¥ = (2.1.1) 47te0 {a + r)

A plot of ¥ as a function of r for a — 1 cm is given in Fig. 2.1 (after Parsons, 1954). I t is apparent that in the near neighbourhood of the surface (r < 10" 2

cm) the potential is essentially constant and equal to

¥ 0 = — ^ — (2.1.2) 47ie 0 . a

I f an attempt is made to measure the potential very close to the surface of the object (say, r < 1 0 _ 6 c m ) , then the measured value depends on the nature of the test charge because at such small distances the measured potential begins to be affected by the interaction between the test charge and the object. In the case of a conductor the force of longest range is the image

-8 - 6 - 4 -2 0 log 1 0(r/cm)

Fig. 2.1. Electrostatic potential in the neighbourhood of a charged sphere of radius 1 cm in vacuo (full line). The broken line indicates how the measured potential near a conductor would be affected by the image force if an electron were used as the test charge.

force (i.e. the attraction between the test charge and its image, of opposite sign, in the conductor) and its effect is shown by the broken line in Fig. 2.1. At still smaller distances (r < 10~ 7cm, say) the measured potential would depend on the details of the interaction of the test charge with, for example, any dipole layers in the surface of the sphere.

The plateau in Fig. 2.1 depends only on the charge on the sphere and not on the test charge, so it can be measured in a variety of ways. One simply has to ensure that the sensing probe, if one is used, is not brought too near the surface and also that the surface itself is perfectly clean. A contaminated surface (e.g. an oxide layer) could only be accommodated in eqs (1) and (2) by modifying the permittivity, and this would affect the potential. The plateau potential is called by Lange the outer potential of the phase.

If two dissimilar metals are placed in electrical contact there is a tendency for the electrons to redistribute themselves in accordance with the electron affinities of the two metals so that one becomes negatively and the other

0~

+ +

+ I

+ + \ E —

+ +

+ I

+ + \ E —

V V

Fig. 2.2. Measurement of contact potential or Volta potential difference between two metals.

positively charged. When a probe is brought into the neighbourhood of each surface (Fig. 2.2) a difference in the outer potentials A^o can be measured and this difference is called the contact potential or Volta potential difference between the two metals. In such a measurement it is again essential that the surfaces near the measuring probes be absolutely clean but it is not essential that the contact region ( I I I ) between the two metals be similarly cleaned (Grahame, 1947). Indeed so long as the two metals are connected by an electron path the result wil l be the same, since the measured potential depends only on the establishment of equilibrium for the electrons in the two metals (equality of what Grahame (1947) calls the electron potential—i.e. the electrochemical potential of the electron—in the two metals). The contact potential thus has a definite and characteristic value which is a measure of the difference in work function between the two metals.

In order to understand the nature of the inner potential, $ \ of a phase, a, we shall proceed as suggested by Parsons (1954). We note first that the work done in transferring a charged particle from infinity into the interior of phase a is equal to the electrochemical potential of the particle in a i.e. Jf . We would like to be able to break this total work down into a part which is "chemical" and a part which is "electrical", but we have already noted that that is not only impossible but meaningless since the "chemical" effects are themselves electrical in nature. Nevertheless, there is a useful differentiation to be made between the interactions which the charged particle makes with the phase as a whole and the other interactions due to charge and dipole layers at its surface.

Suppose that phase a is a sphere of material and that its charge and surface dipole layers can be assumed to be confined to a thin shell. We can then replace the original sphere by (i) a sphere of the same size but of homogeneous material with no charge and no surface dipoles and (ii) a thin, empty spherical shell of the same size with the charge and surface dipole arrangement of the original sphere (Fig. 2.3).

The total work done in transferring the particle i, of charge z;e, from a point at infinity to the point B can now be thought of as being made up of two

• o

(c ) (a ) ( b )

Fig. 2.3. (a) The original sphere with its surface layer; (b) A sphere of homogeneous composition; (c) Spherical shell with charge and dipolar layers.

contributions (IV = W' + W" = juf). The first (W) is the work involved in bringing the charge to the point B' in the interior of the homogeneous sphere (Fig. 2.3(b)). This measures the way in which the charge interacts with the bulk of the phase a and we wil l equate that with the chemical potential of the particle i in a, yU?. It should be reiterated, however, that this term will depend on the size and electric charge of the particle i. I f /' is a small sphere of radius r, this term will contain a contribution of the form:

An 2rt \e £ 0 j

from the Born effect, as Smith (1973) points out. The other term, W", measures an electrical potential in the interior of a

and this is the inner potential, (j)", so that W" = zfid)a. It is in this sense that we write:

R - t f + z f p (2.1.3) <pa is, of course, independent of the position of the point B' inside the sphere. It is convenient to break 4>" down further into the outer potential *¥* which is due to the overall charge on the phase and a chi- or jump-potential, %, due to oriented dipoles, polarizations, and charge separations at the surface:

<ƒ>« = * r + t (2.1.4)

Again, although *F* can, in principle, be measured, x" can only be estimated on the basis of a particular model.

When two phases are in contact the difference in their outer potentials, "A^T, can be measured quite easily, either by the method illustrated in Fig. 2.2 or, more commonly, by the Kelvin method in which one determines the e.m.f. that must be applied between the two phases to prevent a current flow when one phase is moved relative to the other (see e.g. Parsons, 1954).

The difference in the inner potentials, "Ap(f>, (the Galvani potential dif­ference) can be measured, however, only in the case where the two phases have identical composition. Then:

14 - fit = U4 + Z f i t f ) - (flf + z f i p ) = zfiW - 4>1 (2.1.5)

and the difference in the electrochemical potential of, say, an electron in the two phases can be measured by a potentiometer. In cases where one cannot set p.\ = jtif the Galvani potential difference remains inaccessible.

The relationship between the various potentials is illustrated in Fig. 2.4, (after Overbeek (1952, p. 124)). It is apparent that the potential difference across the interphase region is given by:

PAX4) = "Aax¥ + "A^x (2.1.6)

1 ^ 1 9

F/g. 2.4. Outer potentials, HP, inner potentials, and jump potentials, / , with two phases in contact.

It is tempting to regard the first term as due to the transfer of charge totally from one phase to the other and to lump together all of the microscopic changes at the interface itself into a pA"x term. Such a separation cannot be done in a rigorous way, but there is evidence that for many systems it is possible to treat changes in ^Aa</> in this way. Such changes can be induced by making small changes in the chemical composition of one phase or the other. Under these conditons d^A'p.^) can be estimated and since d(fiA*p.t) can be measured, d(pA"<f>) can be calculated from eq (5). Davies and Rideal (1963, Chapter 2) show, for example, that changes in distribution potentials between two immiscible phases, induced by changing the nature of the salt, are essentially measurements of changes in / }A c t ,F, as would be expected, since they arise from differences of solubility of the ions in the two bulk phases.

On the other hand, these same authors show that measurements of "sur­face potential", in the sense that that term is used in surface chemistry, are measurements of changes in the pA*x potential before and after the spreading of an adsorbed fi lm (see also Parsons, 1954, p. 116). In effect they correspond to measurements of the change in apparent Volta potential due to the presence of (controlled) surface contamination.

These fA*% potentials are interpreted in terms of all of the changes that are induced in the interfacial region by the presence of the f i lm : (i) the removal or modification of some of the original dipole orientations, (ii) the introduction of new dipoles due to the molecules of the spread f i lm and (iii) the redistribu­tion of charged species to produce an ionic double layer. In practice (i) and (ii) are treated together, and the net effect of the changed dipole orientation is assessed. The major contribution to the potential is usually (iii) and this is the calculation to which we will devote most attention.

In most situations of interest in colloid science there is no net charge on

the bulk phases, or if there is, i t is not of primary concern. In such cases one can regard the inner potential difference pA*<j) as a x potential difference due to charge and dipole arrangements confined to the interface (i.e. / )A a vP = 0). For the purposes of model calculations it is convenient to break PA"4> into components:

»A«* = »A'x = AiA + A Z d i p o I e (2.1.7)

where Ai// is due solely to the disposition of free charges (electrons, or more usually ions). More commonly we choose a reference point in the bulk of one phase from which to measure ij/ values and then write "

"A«(p = IAO + A Z d i p o I e (2.1.8)

where \j/0 is called the double-layer potential. In electrokinetic measurements, the potential which is measured is the

electrostatic potential difference between the interior of the liquid phase and a certain plane (or surface) in the interfacial region (the surface of shear). We have good reason to believe that the surface of shear lies in the diffuse part of the ionic double layer and a reasonable amount of evidence to support the idea that the potential developed in that diffuse layer is due almost entirely to a non-uniform distribution of ions in a solvent which differs little, i f at all, f rom the bulk solvent (Chapter 5). Since dipole orientation and polarization effects are small in this region we wil l write this potential as ip to indicate that it originates from a distribution of free charges. In deriving expressions for the electrokinetic potential, (, we wil l only need to consider a potential of this sort.

On the other hand, when we want to incorporate the results of (-potential measurement into more general theories of double layer structure we wil l have to pursue the calculation further. As we move closer to the adjoining phase, dipole orientation effects become more significant and a properly calculated potential must make allowance for them. To the extent that this is done successfully, the calculated potential comes closer to the true dif­ference pAa(j) between the phases, although it remains, of course, subject to the limitations imposed on it by the simplifications of the model used in its calculation.

2.2 M e c h a n i s m o f c h a r g e d e v e l o p m e n t a t i n t e r f a c e s

2.2.1. Ca lcu la t ion of t he d o u b l e layer potent ia l

The most important mechanisms which give rise to the spontaneous separa­tion of charge between two phases in contact are:

(1) differences in the affinity of the two phases for electrons,

(2) differences in the affinity of the two phases for ions of one charge or the other,

(3) ionization of surface groups, (4) physical entrapment of non-mobile charge in one phase.

As noted above, mechanism (1) is responsible for the development of the contact potential difference between dissimilar metals and it is no doubt of importance at the junction between a metal and a semiconductor. For the solid-liquid or l iquid-l iquid interfaces with which we are chiefly concerned, however, i t is unlikely to be of great importance. An exception is in the treat­ment of metal sols, including dispersions of mercury, but these are rather a special case in so far as their electrokinetic behaviour is concerned (see Chapter 3). The special case of a metal electrode in contact with an electrolyte has been extensively treated (Grahame, 1947); Parsons, 1954; Devanathan and Tilak, 1965; Payne, 1973; Sparnaay, 1972a), and we shall not discuss it in detail here although some of the important results f rom the work on the mercury-solution interface wil l be discussed below.

Mechanism (2) includes:

(a) the distribution of anions and cations between two immiscible phases, e.g. oil and water,

(b) the differential adsorption of ions from an electrolyte solution on to a solid surface,

(c) the differential solution of one type of ion over the other from a crystal lattice.

In each case equilibrium is established when the electrochemical potential is the same in both phases for any ion which can move freely between them. For the silver iodide/water system, for instance, the charge separation at the interface between the crystal and an aqueous electrolyte solution can be thought of as being due to either (b) or (c) above. The surface of the crystal may be treated as a separate phase and, at equilibrium, the electrochemical potentials of both A g + and I - ions must be the same in this phase as they are in the bulk aqueous solution:

^ s ( A g + ) + zed>s = / i a ( A g + ) + zefa

where subscript s refers to surface and a to the aqueous phase, i.e.

/z s°(Ag +) + kT\n[as(Ag+)] + zed>s = ^ a °(Ag + ) + kT In aa(Ag+) + zeqb, (2,2.1)

As the bulk activity of A g + ions is altered the relative proportions of A g +

and I ~ on the Agl crystal surface can be varied and so too can the potential differencè (0 S - <£a). At some particular value of a a ( A g + ) the numbers of A g + and I ~ ions wil l be exactly equal and there wi l l be no net charge on the

crystal surface. I t is customary to take this point of zero charge (p.z.c.) as also characterizing the point of zero potential difference {4>s = 4>a). Strictly speaking, however, i t is only the potential difference due to the free charges (Ai/0 which is zero at this point and there remains a difference A x d i p o l e

between the phases due to, for example, dipolar orientation at the interfaced Hence we can write at this point:

^ ° ( A g + ) + fcrin < ( A g + ) + z e A Z d i p o l e

= Ai a

0(Ag +) + / c r i n a ; ( A g + ) (2.2.2)

where ' refers to quantities at the point of zero charge. Subtracting equation (2) from (1) and assuming a s (Ag + ) = a's(Ag+) we obtain:

ze(cj>s - <pa) - z e A Z d i p o l e = kT In J ^ f ^ (2.2.3)

I f we can assume further that the A x ' d i p o I c potential does not change signifi­cantly as the bulk activity of A g + is altered we can use eq (2.1.8) to write

. . , kT « a (Ag+) Ai> = i o = — In

ze fla(Ag+) or

^ O (volts) = 0 - O 5 9 8 1 o g 1 0 j ^ (2.2.4)

at 25°C. An analogous Nernst-type equation can be derived for the potential generated by an ion distribution equilibrium between two immiscible phases (see Davies and Rideal, 1963, p. 59). The important assumption in deriving eq (4) is that a s ( A g + ) = a's(Ag+), i.e. that when the bulk activity of A g + is altered, the surface activity remains constant. The justification for this as­sumption is that the surface of the Agl crystal contains a large number of A g + and I " ions and the few extra ions which are adsorbed in order to establish the potential \j/0 are not likely to affect the activity of those surface ions. The special role of the crystal lattice ions is recognized by referring to them as the potential-determining ions for the system, to distinguish them from ions like K + and N O 3 which are not expected to enjoy a special interaction with the surface. These latter are called indifferent ions. Inter­mediate between these extremes are ions which appear to interact in some special (e.g. chemical) way with the surface and these are referred to as specifically adsorbed ions.

The potential \j/0 which we refer to as the double-layer potential is some­times called the "surface potential" by colloid scientists. This use of the term is a little confusing because, as noted above, surface chemists in their study of monolayers use the same term in a slightly different, though closely

related sense. Their surface potential is a measured difference in the Volta potential between a metal and an electrolyte solution before and after the spreading of a monolayer on the surface of the solution. The interpretation of the two potentials (double layer and surface) is, however, identical, if i t is carried through to completion (see e.g. Davies and Rideal, 1963, Chapter 2).

Mechanism (3) for charge development, i.e. the ionization of surface groups, is commonly observed with carboxylic acid, amine and oxide sur­faces. In these systems the degree of charge development (and its sign) depends on the pH of the solution, and the behaviour is very much like mechanism (2) with H + and O H - behaving as potential-determining ions. Oxide surfaces, for example, are considered to possess a large number of amphoteric hydroxyl groups which can undergo reaction with either H + or O H " depending on the p H :

M O H 2

+ ^ - + M — O H ° 4 ~ M ( O H ) J ( = M O " + H 2 0 ) (2.2.5)

Whether these reactions are regarded as adsorption reactions or as dissocia­tions is a matter of taste. There is, however, one significant difference from the silver iodide surface. The analogue of the Nernst eq (4):

kT <A0 = 2-303 — (pH. - pH) (2.2.6)

e

is no longer satisfactory for describing the surface potential because the assumption that a s (H + ) = a's(H + ) is clearly untenable. In these systems there are very few H + ions present on the surface at the p.z.c. and it is cer­tainly not true that the number of additional H + ions required to establish the charge is insignificant by comparison. The importance of this assumption was pointed out by Bérubé and de Bruyn (1968), and a modified Nernst equation was derived by Levine and Smith (1971). Essentially eq (6) must be replaced by

(2.2.7)

where the subscript z refers to the p.z.c. The additional term has the effect of lowering the expected double-layer potential, \j/0, which is in keeping with the experimental observations. Wright and Hunter (1973) showed that a simple ideal model of the adsorption process would lead to:

as(U + ) ^ H , ( H + ) Ns-nsJK + ) fls,z(H + ) • « 5 , Z (H + ) J V S - « S ( H + ) < • • ;

where N, is the number of adsorption sites available at the surface; a more satisfactory approach could be developed using the concept of a surface

kT\ iAo = 2-303— p H 2 - p H - log

a s ( H + ) fls,2(H + )

dissociation constant. We wil l return to these more elaborate procedures in Chapter 7.

Mechanism (4) for charge development, i.e. physical entrapment of charge in one phase, occurs commonly in solids where n and p type defects in the crystal may be so common that the charge distribution in the solid must be explicitly considered in a complete model (see e.g. Sparnaay (1972a, p. 145) and Section 2.7 below). A more extreme case occurs in many alumino-silicate clay minerals (e.g. montmorillonite and vermiculite) where a large negative charge is developed on the clay crystallite due to isomorphous substitutions in the lattice. The site of these substitutions varies, but many of them occur in the interior octahedral alumina layer and the balancing charges are provided by exchangeable counterions (usually alkali and alkaline earth ions and various aluminium-hydroxy species under natural conditions) (see e.g. van Olphen, 1963). In these systems the surface charge density cannot be reduced to zero although i t can possibly be modified by altering the p H to promote adsorption of O H " or H + ions onto the silicate surfaces.

Since the charges are about 0-5 nm from the physical surface of the crystal­lite and are about 1 nm apart, the field lines would tend to be as shown in Fig. 2.5 so that the surface of the crystal is approximately an equipotential surface and this is the way the system is usually treated. Childs (1954) has, however, presented a rather more consistent model for dealing with a pair of interacting crystals of this type.

2.3 T h e p o t e n t i a l a n d c h a r g e d i s t r i b u t i o n in t h e e l e c t r i c a l d o u b l e l a y e r

Classical theory. In the following analysis we begin, for the sake of com­pleteness, with the well-known Gouy-Chapman theory of the diffuse double layer as presented by, for example, Grahame (1947) and Overbeek (1952,

© ©

Fig. 2.5. Schematic arrangement of charges in a vermiculite or montmorillonite crystal. The negative charge results when a magnesium ion replaces an aluminium.

p. 128). We shall discuss the basic theory for various geometries (flat plate, sphere and cylinder) in this section and then go on in 2.4 to describe more recent developments.

2.3.1 The f lat p late mode l

2.3 .1 .1 . The potent ia l d i s t r i bu t ion The separation of charge that occurs at the interface between two phases is called an electrical double layer because it consists, ideally, of two regions of opposite charge. The simplest model of such a system is that usually attributed to Helmholtz in which both layers of charge are fixed in parallel planes to form a molecular condenser. As Overbeek (1952, p. 199) has pointed out, this model was actually first explicitly referred to by Perrin (1904). Although it can be used to derive some of the important electrokinetic relationships it has long been recognized as an inadequate representation of the situation. Whilst the charge on a metal surface may be assumed to be located in a plane, and the same is true for many solid insulating substances, it is unlikely to be so in a liquid phase where the electrical forces on the ions must compete with the thermal diffusive forces. The theory for such a diffuse double layer was developed independently by Gouy (1910) and by Chapman (1913).

According to this model, one layer of charge is assumed to be smeared out uniformly over a plane surface immersed in an electrolyte solution. This surface wil l have an electrostatic potential, \j/0. The compensating ions are regarded as point charges immersed in a continuous dielectric medium. The fundamental electrostatic equation for the system is Poisson's equation (see Appendix 1 for a discussion of the meaning of this equation):

div D = p

or div (— s grad = p (2.3.1)

where D is the dielectric displacement ( = sE where E is the electric field), e is the permittivity of the dielectric and p is the volume density of charge. Provided that the permittivity is independent of position, (1) can be written:

div grad i> = Ai/^ = - - (2.3.2) e

In Cartesian coordinates, this takes the more familiar form:

where D is the (dimensionless) dielectric constant or relative permittivity

(D = e/e0 where e0 is the permittivity of free space). The reason for expressing eq (2) in the manner shown is discussed in Appendix 2. In the SI system of units, which we shall use for the most part, the value of e0 is 8-854 x 1 0 " 1 2

farad m _ 1 (i.e. C V " ^ - 1 ) . A brief discussion of electrical units is given in Appendix 2.

The assumption that the wall is an equipotential surface is a good one for metallic conductors but is much less satisfactory for ionogenic surfaces, especially at low charge densities. The assumption of a permittivity which is independent of position is also questionable since the electrical fields gener­ated in the interfacial region are certainly high enough to modify solvent structure in many cases. Nevertheless, there remains an important situation in which eq (2) is still satisfactory and this wi l l become apparent when we have developed the model somewhat further.

At equilibrium the electrochemical potential of the ions must be constant everywhere (i.e. grad ju; = 0) or, put another way, the electrical and dif-fusional forces on the ion must balance out:

grad p.; = - zte grad i> (2.3.3)

where pt is the chemical potential and z ; is the valency of ion i. For a flat double layer the electrostatic potential and the chemical potential (or con­centration) are constant in planes parallel to the wall so that eq (3) can be written:

d}Xi= _z,e^t dx ' dx

and using the definition of the chemical potential (per ion) in the form

pi = pf + kT In « ;

where n; is the number of ions of type i per unit volume, we have

dlnnj^ 1 dn-, _ _z£aty dx n{ dx kT dx

Integrating this equation from a point in the bulk solution (where \j/ = 0 and B , = n°) leads to the Boltzmann equation:

« ; = « ? e x p ( - z ; e i A / / c r ) (2.3.5)

Equation (5) gives the local concentration of each type of ion in the double-layer region, provided that z ; has the correct sign attached to it. I f l// is positive (as i t is near a positively charged surface) then for positive ions nt < nf (i.e. positive ions are repelled from the double-layer region) whilst for negative ions > H? and negative ions are attracted to the double-layer region.

The volume charge density in the neighbourhood of the surface is given by:

P = £ w (2.3.6) i

where the summation is over all species of ion present. Substituting eqs (5) and (6) in (2) we obtain the complete Poisson-Boltzmann equation:

V V = S = - 4 ^ ^ Z » f ^ e x p ( - z ^ / f c r ) (2.3.7)

The Laplace operator (V 2 ) is defined in Appendix 1. In this case it is simplified to one dimension because \J/ varies only with x. Equation (7) is a non-linear differential equation, and although it can be solved analytically in this case we wil l first examine an approximate solution. I f y/ is small everywhere in the double layer (i.e. z;ei/f kT), we can expand the exponential (using the relation e~x = 1 — x for small x).

Then ^

dx* - £ = -2Zzfe2n?il,/kT]

The first term in the parentheses must be zero (to preserve electroneutrality in the bulk electrolyte) so the equation becomes (for a plane interface):

v 2 .> = ^ | = K v ax

where

The simplification of taking \jj as small in magnitude is referred to as the Debye-Hückel approximation because these two workers used the same approximation in their theory of strong electrolytes. The parameter K depends mainly on the electrolyte concentration (nf) and is very important in colloid science; it is referred to as the Debye-Hückel parameter. Its sig­nificance wil l be discussed in more detail below.

Equation (8) can be solved in a number of ways to obtain ip as a function of x. A method which can also be used on the complete eq (7) is to multiply both sides by 2{d\j//dx) and then integrate with respect to x, f rom some point in the bulk solution up to a point in the double layer:

f 2 # <fty dx dx2

2K2\l/-f- • dx dx

or

d_(<$\ .=aodx\dx J

) dx = 2K2 x/zdij/ l/> = 0

Then

(2.3.10)

since dxj/jdx = 0 and ip = 0 in the bulk solution (where x = co). The negative sign on the right hand side is chosen so that when \j/ is positive the potential wi l l decrease to zero in the bulk.

A second integration of eq (10) using the fact that ^ = uV0 at the plate, where x = 0 gives:

\j/ = \j/0e' (2.3.11)

This solution shows the approximate form of the potential distribution near the wall but the approximation on which it is based

* 0 < 2 Z ^ a t 2 5 " C

is not usually satisfied in colloid systems. The more general eq (7) can be integrated once using the same procedure

as above to yield:

dx J 8 exp( — zie\\ilkT)d\\i

or

dx J 2kT.

Znf[exp(-zieil//kr)-l] (2.3.12)

For reasons which wil l become clear later (see Section 2.3.1.3) it turns out to be possible to treat most electrolytes as though they were symmetric, with valency z, equal to the valency of the counterions (i.e. the ions which tend to be accumulated in the double layer). In that case, using the identity s inhp = (ep — e~p)/2, allows eq (7) to be written in the form:

d2ij/ 'abc2'' Am

where n° = n°+ •— n°_ and z = zH

then gives:

1 \%nn°ze . smh(zeil//kT) D

(2.3.13)

z_. The same integration procedure

# V An°kTr , I n r r , r I = [cosh zeip/kT - 1 ]

dx J or

dx

e

Sn°kT sinh(zei^/2fcr) (2.3.14)

(using the identity cosh p = 2 sinh 2 p/2 + 1). Using eq (9)

<MJ 2KKT -T= sinh zeiillkT (2.3.15) dx ze

where again the negative sign is chosen to allow xjr to fall to zero far from the wall. The integrated form of eq (15) can be expressed in a variety of ways, of which the most compact is probably:

tanh(zi///4) = tanh(zi// 0/4)exp(- KX) (2.3.16)

where i// = exjjjkT is a dimensionless potential parameter, which we wi l l refer to as the reduced potential. (At 25&C, $ = 1 when ij/ = 25-7 mV.) For small values of p we have sinh p « tanh p xp and so eq (13) reduces to (8) and eq (16) reduces to (11). A comparison between these latter equations is shown in Fig. 2.6. The approximation is more accurate than might at first be expected because eq (16) requires only that i// 4/z everywhere.

Note that the approximate expression always overestimates the potential.

Fig. 2.6. Electrical potential in the double layer according to Gouy and Chapman. The ful l curves are from eq (16) for various values of zij/0. The broken lines are f rom eq (11) for z\p0 = 2 and 4. (From Overbeek (1952, p. 131) with permission).

Far out in the double layer one can again assume that tanh p « p and so:

z l j } = 4ye-KX (2.3.17)

where y = tanh(zi//0/4)

= exp(zi//0/2) - 1 exp(zt//0/2) + 1

Since y approaches unity for high values of zi// 0 one can expect the potential far from the wall to resemble that for a wall of potential i / / 0 = 4/z irrespective of the wall potential provided it is sufficiently high, cf. eqs (11) and (17). Thus all measurements of highly charged systems at ordinary temperatures suggest il/0 « 100mV/z if they are based on methods which only sample the outer region of the double layer. By the same token, the likely behaviour of highly charged systems can be predicted assuming that t/>0 « 100mV/z if one is interested only in situations involving small degrees of double-layer overlap.

The electrolyte concentration enters through its effect on the parameter K, which can be written:

K = ( — — V / i n units of m _ 1 (2.3.18) \s0DRT J

= 3 -288V/(nm) _ 1

for water at 25°C. Here F i s the faraday of charge (i.e. the magnitude of the charge on a mole of electrons = 96485 coulombs), and R is the gas constant per mole. I is the ionic strength (mole l - 1 ) defined by the expression:

I = iZCizf (2.3.19)

where ci is the concentration of ion i in mole 1" 1 . Note that the dimensions of K are those of reciprocal length.

Increasing ionic strength causes an increase in K as a result of which the potential falls off more rapidly with distance. This is referred to as "com­pression of the double layer". The distance 1/K is referred to as the thickness of the double layer, although, as Fig. 2.6 shows, the region of varying potential extends to a distance of about 3/K before the potential has decayed to about 2% of its value at the surface. Values of 1/K range from 3-04 nm at / = 10" 2

to 96-17 nm at I = 10" 5 mole T 1 , in water at 25°C.

2.3.1.2. Sur face dens i ty of charge The charge per unit area on the surface, a0, must balance that in the adjacent solution:

<70 = - pdx (2.3.20)

and substituting for p from eq (2):

a0 = e 2dx o dx

= S ) (23.21) x=0 Kdx

For unsymmetrical electrolytes eq (12) for (d\j//dx) yields:

)4 = {2e fc rX"° [exp( - zMkT) - 1]} (2.3.22)

whilst for symmetrical electrolytes eq (15) gives:

4n°ze cr0 = sinh(zi/V0/2) (2.3.23)

K • •

= l l -74c 1 / 2 sinh(19-46zi>0) in uCcmT 2

for \j/0 in volts and c in m o l l - 1 . Note that because of the way z is defined (under eq 13), it is always positive so that cr0 has the same sign as \j/Q.

For very small potentials, eq (10) yields:

r j 0 = S0DKI]/0 (2.3.24)

and the integral capacity per unit area of the double layer

K = ^- = SK (2.3.25)

Under these conditions the system therefore behaves like a parallel plate condenser with a separation of 1/K between the plates (Appendix 2). The similarity to the Helmholtz model is then very close and the significance of the expression "thickness of the double layer" is most obvious.

2.3.1.3. Ion d is t r ibu t ion in the d o u b l e layer The ions in the diffuse double layer are, of course, not all of the same sign. The charge on the wall is balanced by an accumulation of charges of opposite sign (counterions) and a deficit of charges of the same sign (co-ions) compared to their concentrations in the bulk. At very low potentials when the ex­ponential in eq (5) can be expanded:

n+=n°{l-ze\p/kT); n_ = «°(1 + zexj/JkT)

and the two ion types are of equal significance. At higher potentials, when the full exponential must be used, the behaviour is quite different: almost all of the charge is balanced by the accumulation of counterions and relatively

_!_ 2

K X

( a )

2 KX

(b)

Fig. 2.7. Accumulation of positive ions and expulsion of negative ions from the neighbourhood of a negative surface (a) Debye -Hücke l (b) Gouy-Chapman model.

little by the expulsion of co-ions (Fig. 2.7). The counterions thus come to enjoy a greater significance than the co-ions (Verwey and Overbeek (1948, p. 30) and this is why it is possible to treat an unsymmetric electrolyte system as symmetric, with the valence of the counterion, without incurring too much error.

2.3.1.4. The e lec t rok inet ic charge dens i ty In the derivation of eqs (22) and (23) for the charge density, the integration process can be taken from the bulk solution up to any plane parallel to the surface. We then obtain the accumulated space charge per unit area up to that plane and this is balanced by the charge between the chosen plane and the particle surface. If, in particular, we choose to stop the integration at the shear plane, where \j> = £ we obtain a measure of the net charge per unit area over the shear plane which we wil l call the electrokinetic charge density, ae:

ae = | 2 e f c r X « ? [ e x p ( - z^jkT) - l ] j (2.3.26)

for unsymmetric electrolytes and

a = 4 — sinh(z(/2) (2.3.27) K

for symmetrical electrolytes. Here X — e£/kT is the reduced (dimensionless) zeta potential, c.f. eq (2.3.16). ae may be multiplied by the surface area to

obtain the electrokinetic charge, QE, on the particle as a whole. Such a pro­cedure is justified when the local radius of curvature of the particle, a, is large compared to the double layer thickness (i.e. KO. g> 1). I t should be noted that this can occur for quite modest particle sizes, well within the colloid range.

2.3.2. The doub le layer a round a sphere

2 .3 .2 .1 . The potent ia l d is t r ibu t ion Equation (2.3.7) remains true for the potential distribution around a sphere but the Laplace operator for a radially symmetric potential must be used so that:

W - ^ K - * * ) — ^ « P ( - i * • (2-3.28)

This equation cannot be solved analytically and recourse must be had to the Debye-Hückel approximation, valid for small values of the potential. Expanding the exponential to the linear term again gives, cf. eq (8):

ixf^W* (2-3-29) r2 dr\ dr ^

By substituting \p = ujr this equation can be transformed to:

cfu , = K u

dr

for which the general solution is:

u = AeKr + Be~Kr

so that

AeKr Be~Kr

$ = — + (2.3.30) r r

The constant A must be zero since \\i -» 0 as r -> oo and i f the particle has radius a and the potential on its surface is i/V0 then

Be~Ka

*o = ^ - (2.3.31)

Substituting for B in eq (30) gives: e - K ( r - a )

xj, = i]/0a (2.3.32)

2.3.2.2. The par t ic le charge The charge on the particle must balance that in the double layer so that

0 = - 4%r2pdr

and substituting for p f rom eqs (2) and (29):

Q = 4ne0DK2 r2\\idr (2.3.33)

Using the value of \J/ from eq (32) and integrating by parts we find

Q = 4ns0Da(l + Ka)\l/0 (2.3.34)

and substituting for \j/0 in eq (32) yields:

>A = -l - K ( r — a)

4ns0 D(l + Kd) r

The potential on the surface of the sphere, eq (34), can be written:

, 2 Q f 1 1' ïo=- — +

(2.3.35)

4ns0Da 4ne0D [a(I + KO) a

= _ e Q

4ns0 • Da 4ne0 • D{a + 1/K)

= <Aso + Vo

(2.3.36)

The first term in eq (36), ij/s

0, represents the potential on the surface due to the charge on the spherical particle itself. The second term, is the potential

Fig. 2.8. Apparent charge distribution around a spherical particle at low potential.

due to the atmosphere of Opposite sign. I t corresponds to the potential due to a spherical shell of charge — Q and radius (a + 1 / K ) so that once again the function 1/K can be interpreted as the thickness of the double layer (Fig. 2.8).

Just as in 2.3.1.4 we can calculate the electrokinetic charge on the particle from eq (34) by substituting \j/0 = (:

Qe = (4ns0)Da(\ + KO.% (2.3.37)

where a is now the radius of the particle out to the shear plane. Equation (37) can be used, of course, only when the potential is small.

2.3.3. The doub le layer a round a cy l inder

The Poisson-Boltzmann eq (7) in cylindrical coordinates, for a symmetrical electrolyte, reads:

. . \ d ( (ziA)\

where R = KT and cylindrical symmetry is assumed (i.e. end effects are neglect­ed). A general analytical solution of eq (38) is not available but a solution of the corresponding linear equation (valid for zt//< 1) was given by Dube (1943):

. aX0(Kr) <A= ^ , , (2.3.39)

ËKjf^Kü) '

where a is the radius of the cylinder and 0 and Xx are zero and first order modified Bessel functions of the second kind, respectively. I f the charge is assumed to be spread evenly over the cylinder (and its hemispherical ends)

TABLE 2.1 Bessel functions for computing potentials of long cylinders

(from Abramson et al., 1942, p. 129)

Ka ^ / ( / c a j f i ) Ka

006 2-950 0-60 0-9942 0-08 2-675 1-00 0-7176 010 2-463 1-40 0-5426 0-14 2-151 2-00 0-4071 • 0-20 1-835 3-00 0-2872 0-40 1-275 "4-00 0-2235

then o-0 = Q/(2nal + Ana2) where / is the length so that:

2Q^0{Ka) i> n = — ~ — (2.3.40) Y 0 4ne0DKa(2a + ^(KO)

Gorin (in Abramson et al., 1942, p. 126) stated this relation and has calculated a few values of the Bessel function ratio for use in calculating electrophoretic mobilities. These are reproduced in Table 2.1. Once again the electrophoretic charge Qe is obtained by putting i/ o = £ in eq (40). A more exact analysis of this problem is given in Section 2.6.

2.4. M o d i f i c a t i o n s t o t h e G o u y - C h a p m a n t h e o r y f o r f l a t p l a t e s

2 .4 .1 . The inner ( c o m p a c t ) layer

I t was soon recognized (Stern, 1924) that the assumptions (i) that the electrolyte ions could be regarded as point charges and (ii) that the solvent could be treated as a structureless dielectric of constant permittivity were quite unsatisfactory. A simple calculation shows that (i) certainly and (ii) very probably break down in the near neighbourhood of the wall, even at modest potentials and electrolyte concentrations. The finite size of the ions, whether hydrated or not, limits their maximum concentration at the wall, and their distance of closest approach to it, whilst the high electric fields near the wall ( ~ 106—10s V m " * ) must produce some ordering of solvent dipoles, especially for water.

Experimentally, this effect is most obvious when measured values of the electrical capacitance of an interface are compared with the theoretically calculated values. When the e.m.f, E, applied across a mercury-solution interface is changed, in the absence of a direct current flow, the charge per unit area on the metal, cM, changes in sympathy i.e. electrons move from the interior of the mercury to its surface (or vice versa) and a compensating adjustment occurs in the double layer. This arrangement of charges has a certain capacitance which could be written:

a™

(•^metal — ^solution)

K is called the integral capacitance of the double layer per unit area. We know, however, that absolute differences in galvani potentials, d>, cannot be measured between phases of different composition (2.1). I t is therefore preferable to define a differential capacitance (C = daM\dE) which can be measured experimentally by applying a small a.c. signal to the electrode system and measuring its impedance. I f we then assume that daM/dE is

approximately the same as da0/dij/0, we have from Gouy-Chapman theory (differentiating eq (2.3.23)):

do~n

C = -f-=EK cosh(zi>0/2) (2.4.1)

= 228-5zc1 / 2 cosh(19-4zi^ 0)uFcm" 2 for c in m o l l - 1

and \j/0 in volts. The very large values of C thus implied are realized only rarely (Grahame, 1947). We shall return to this point below in eq (2.4.8).

In current theories of the double layer the problem is alleviated, though not eliminated, by dividing the space charge in the electrolyte solution into two regions: (i) the compact or inner region very near the wall in which the charge and potential distribution are determined chiefly by the geometrical restrictions of ion and molecule size and the short range interactions be­tween ions, the wall and the adjoining dipoles and (ii) a diffuse layer, further out from the wall where the Poisson-Boltzmann equation may be expected to give a reasonable representation of the potential distribution. This separa­tion alone, assuming that anions and cations have different approach distances has been shown by Frumkin (1940) to remove many major anomalies from the Gouy-Chapman treatment of the mercury-solution interface, at least for simple ions.

A more elaborate theory, proposed by Stern (1924), allows the first layer of adsorbed ions to interact specifically with the wall, through a specific chemical adsorption potential,'6. In the modern version of Stern's theory (see Parsons, 1954, p. 153) the diffuse part of the double layer starts at a distance x2 from the wall and for x>x2 specific interaction forces are negligible. The distance of closest approach to the wall is xt so that the region 0 < x < xx is free of charge. The number of ions in the region xl < x < x2 is calculated using a Langmuir isotherm, modified by the incorporation of a Boltzmann factor (see Appendix 4 for an elementary derivation). The general result obtained for a binary z: z electrolyte incorporating the correction sug­gested by Esin and Markov (1939) for finite coverage is (Parsons, 1954):

_ zieNlxs{b+ - b . ) . l + (b++b_)x* ( 2 A 2 )

where b± = e x p ( - AG°±MJkT); Nx is the number of sites for adsorption, XS is the mole fraction of the electrolyte in the bulk solution and Stern wrote for the free energy of adsorption an expression corresponding to :

AG0

±Ms = z±eij/i + e± (2.4.3)

IHP I

OHP T I !

Fig. 2.9. Schematic representation of a possible potential distribution at an interface. Note that in some non-metallic systems (e.g. the oxides) the great majority of the countercharge (up to 90 %) appears to lie between the solid surface and the OHP. On the Agl surface on the other hand, there is usually little or no charge in the I H P in the presence of simple electrolytes like K N 0 3 . (After Bockris et at., 1963.)

Parsons (1954, p. 157) expresses some misgivings about this breakdown into chemical and electrical contributions but it is almost universal practice in colloid systems. Surface roughness and heterogeneities make it impossible to accurately define i>; in such systems anyway, so the additional degree of uncertainty in eq (2) (cf. the discussion about eq (2.1.3)) becomes incorpor­ated into estimates of 6± which is usually regarded to some extent as an empirical, adjustable parameter.

Where the specific adsorption effect is confined to one ion, either cation or anion, eq (2.4.2) simplifies to:

f j ^ Z i e N j ^ e x p f - {z±eij/i + 6±)/kT] (2.4.4)

since b±x* is usually much less than unity. Among the simple ions which have been studied at the mercury-solution interface, specific adsorption is more common among anions, presumably because they are less strongly hydrated. The potential distribution near a metal electrode (and possibly some colloidal particles) is therefore likely to look somewhat as shown in Fig. 2.9. The anions are shown unhydrated, at least on the side near the wall, whilst cations remain hydrated. Water dipoles are expected to be oriented to some extent depending on the charge on the wall and the perturbing effect of the specifically adsorbed anions. The plane of centres of the anions is called the Inner Helmholtz Plane (IHP) and that of the cations is called the Outer Helmholtz Plane (OHP). In Chapters 5-8 a considerable body of evidence wil l be adduced to suggest that the shear plane lies very close to the OHP.

In the transition region from the metal (with D = e/e0 -> oo) to the aqueous electrolyte (e = 80e0) the potential is determined by the fact that the di­electric displacement (D) must be continuous. Using the definition given in Appendix 1:

„ dip ax

in this case where gj refers to the region 0 < x < xv

For x = 0, eq (2.3.21) shows that this quantity is equal to the surface density of charge. For 0 < x < xu there is no other electric charge present, so from Poisson's equation (Appendix 1):

d i v ( - e ! grad \p) = Oor ^ ( _ e i ^ ) = 0

and s^ip/dx is constant in this region. Integration of eq (2.3.21) then gives

<Ao-<Ai = ^ ^ (2.4.5) £

Applying the same procedure to the region xt < x < x2 where the permit­tivity is s2 = D2e0 gives:

to-lh= - — ( X i - x t ) (2.4.6)

This procedure amounts to treating the layers as molecular capacitors, characterized by sharp changes of permittivity between successive pairs of plates.

More elaborate procedures, using a continuously variable dielectric permit­tivity £ (=£(,£>), have been examined by Buff and Goel (1969) and more

recently by Levine (1971) and Robinson and Levine (1973). Since the electric displacement, - s(dxp/dx), and £ are then both continuous, the ip function becomes smoother in the interfacial region (see Section 2.4.2).

The principal effect of introducing the compact layer is to lower the overall capacitance of the interfacial region, CT, since the compact and diffuse layers are in series. Hence:

_ L _ L _ L cy c i n n e r c d i f f u s e

where

Q i f f u s e =^r= 228 -5ZC 1 ' 2 cosh( 19-46*^) u F c m " 2

in water at 25°C, for c in m o l e l - 1 and ipd in volts. Except when ipd and/or c are very small, the diffuse layer capacitance is so

very large that it does not contribute effectively to the measured total. For very dilute solutions in the neighbourhood of the point of zero charge the measured capacitance approaches that given by eq (2.4.8) but at higher concentrations and away from the p.z.c. the much lower value attributable to C i n n e r is observed. Measurements at the dropping mercury electrode suggest a value for C i n n e r in aqueous solution from 16-30 uF c m " 2 depend­ing on the metal charge (Grahame, 1947). For a simple molecular condenser this means a value of x ; / D ; » 25-50 picometre in eqs (5) and (6). Using the model pictured in Fig. 2.9 (Bockris et al, 1963), this would correspond to values of, say, X j = 0-15nm, D x = 6, and x2 = 0-6nm, D2 = 20, which are not unreasonable considering the high electric fields in this region.

Equations (4), (5), (6) and (2.3.23), written in the form

trd = — sinh(z&,/2) (2.4.9) K

can be used, together with the electroneutrality condition:

ffo + ff. + <7<, = 0 (2.4.10)

and a suitable Nernst-type equation for \p0 (like eq (2.2.4)) or, if necessary, eq (2.2.7)) to solve for the six unknowns: \p0, ipb \pd, a0, a„ ad. The distances scu x2 and dielectric constants Dl and D2 have to be regarded as parameters for which reasonable values can be selected (xJDf m 10-100 picometres for aqueous systems) whilst eq (4) also requires the parameters JV1(1014"—1015

c m " 2 ) and 9 ( ± a few times W a t most). This approach was foreshadowed by Overbeek (1952) and has been used a number of times with limited success. Hunter and Wright (1971), for example, used it to demonstrate that eq (2.2.6) almost certainly overestimates ip0 in oxide systems and the refinement

(2.4.7)

(2.4.8)

incorporated in eq (2.2.7) is essential. They also showed that despite all of the adjustable parameters available it did not seem possible to reconcile electrokinetic and charge data on these systems with any reasonable values of the variables. The additional assumptions which may be used to effect such a reconciliation (Levine and Smith, 1971; Levine et al, 1972; Perram et al, 1974; Yates et al, 1974) wi l l be discussed in Chapter 7.

The approach outlined above is necessarily much cruder than that available to electrochemists concerned with adsorption at the dropping mercury electrode. In effect, i t assumes that the capacitances of the inner layer are constant and independent of the state of charge of the system, an assumption used by Devanathan (1954) for the mercury-solution interface but since abandoned in favour of the less objectionable procedures developed by Parsons and his co-workers (see e.g. Parry and Parsons, 1963). This is not to say that the inner layer capacitance always varies with charge. Indeed, Parsons (1959) has shown that for iodide ions adsorbed on mercury the experimental data indicate a linear dependence of the specific adsorption potential on the metal charge and this is consistent with a constant inner layer capacity (Parsons, 1963). The criticism of the Devanathan procedure (see e.g. Payne, 1973, p. 87) is that it is not necessary in general and can lead to conflict with experiment in certain cases. Unfortunately, in colloid systems we do not have access to some of the procedures applicable on a polarizable metallic electrode like mercury, so this criticism loses much of its weight in such situations. Nevertheless, it should be recognized that, on the silver iodide surface, where the accumulated evidence is much more trustworthy than on most colloidal systems, the inner layer capacity does vary with the surface charge. According to Lyklema and Overbeek (1961) it decreases almost linearly with increase in negative charge (Fig. 2.10), and Levine and Matijevic (1967) and Levine and Smith (1971) have built this sort of varia­tion into their models of the double layer on Agl and oxides respectively. The behaviour exhibited in Fig. 2.10 is, of course, what one would expect if the compact region were gradually changing from one in which anions were adsorbed at the I H P to one in which cations were adsorbed at the OHP as the surface charge moved from positive to negative.

The more elaborate isotherms which have been used on the mercury-solution interface have been reviewed by Parsons (1961), Frumkin and Damaskin (1964), Delahay (1965), Levine (1971) and Payne (1973), and we shall not discuss them in detail here. One important point, however, does warrant further discussion: the extent to which the specific adsorption potential, 6, is calculable. I f one regards the adsorbed ions in the I H P as charges occupying a finite (spherical or disc-shaped) region and considers the various electrostatic implications of such a model it seems that a good part of the magnitude of 9 can be accounted for. In particular, Levine et al.

Fig. 2.10. Dependence of differential capacity of the inner layer (<"„„„) on the surface charge of Agl for various electrolytes (From Lyklema and Overbeek (1961) with permission.)

(1962) were able to explain why the value of 9 for the chloride ion on mercury appears to depend on the electrode charge, rather than being a constant as suggested by eq (3). Andersen and Bockris (1964) performed a similar calculation for some other simple ions at the mercury-solution interface. One of the major contributors to 9 is the so-called discreteness of charge effect, which is a correction to the potential in eq (3) to make allowance for the fact that the ions in the compact layer are of finite size and the charge er,- is not uniformly spread over the IHP. When the true potential experienced by the ion (the micropotential i/cj) is put into eq (3) it reads

AG a° d s = zeM + 0; (2.4.11)

where Q\ is a function of pressure and temperature only (and is the true "chemical" specific adsorption potential) whilst = + qba where <f)a is called the self-atmosphere potential of the ion (by analogy with the use of that term in Debye-Hückel theory). We shall examine this approach in Section 2.4.3 below, but before doing so we must examine the behaviour of the solvent in the inner region.

2.4.2. The d ie lect r ic permi t t i v i t y of the inner reg ion

The potential drop across the inner region of the double layer is often 0-1-1 volt, and since this occurs across a distance of less than 1 nm the field strengths are ~ 10s—109 V m " 1 . Intuitively one would expect such high fields to cause considerable dipole orientation with a consequent decrease in permittivity. This expectation is borne out by more elaborate theoretical calculations (Booth, 1951b), by direct experiments at rather lower field strengths (Malsch, 1928, 1929) and by the measured inner layer capacitance at the dropping mercury electrode (Grahame, 1947) discussed above.

In the most elementary models of the compact layer, a single value is assigned to the permittivity e, corresponding to a dielectric constant of say 10-50. In slightly more elaborate models such as those of Grahame (1947) or Bockris et al. (1963), the compact layer is divided into two zones: one between the wall and the IHP and the other between I H P and OHP. The former is usually thinner and the field strength is higher so the chosen per­mittivity is lower. For water, the limiting value of D ^ 6, corresponding to zero orientation polarization, is often used for this zone. For the outer zone, a somewhat lower level of water organization, corresponding to D ^ 20-40, is commonly assumed, but since this zone is also assumed to be rather thicker the ratio of thickness to permittivity, which is required for the capacity calculation, remains much the same.

Values of the compact layer parameters, £>; and xh as used by various authors are tabulated by MacDonald and Barlow (1964), and a discussion

of the distance parameters is given by Levine et al. (1967). More recently Levine et al. (1969) calculated the permittivity of the inner layer on mercury using a two-state model for the water and Sparnaay (1972a, p. 93) has given an extensive discussion of the problem. A simple four-state model was developed by Parsons (1975) and Oldham and Parsons (1977), based on an earlier suggestion of Damaskin and Frumkin (1974). Salem (1976) and Damaskin (1977) also offer simple models of the same system; all these treatments attempt, with varying degrees of success, to account for both the inner layer capacitance as a function of applied e.m.f. and its dependence on temperature. A more general description would need to take account of the metal surface (Trasatti, 1971; Gardiner, 1975) and of the influence of the ions adsorbed in the I H P (Conway, 1977).

Attempts have also been made to overcome the limitations of these one-layer models and to avoid the discontinuities in the permittivity which they entail. Buff and Goel (1969) used a composite function—a cosh 2 and a cos2

function matched at the junction for both value and slope (Fig. 2.11a), and Robinson and Levine (1973) used a similar device in their calculations. The most significant impact of this refinement is to reduce the effect of image forces in the compact layer so that the calculations of the self-atmosphere potential which wil l be described in Section 2.4.3 will tend to be on the high side when the compact layer is assumed to consist of discrete layers of different permittivity.

Even in cases where the permittivity is permitted to vary over the compact layer, it is still possible to define integral capacities for the two zones (Levine, 1971). I f the dielectric displacement is constant in the inner region (i.e. there

IHP OHP • (a) (b)

Fig. 2.11. A possible profile for (a) the permittivity and (b) the potential in the inner region.

is no charge between wall and IHP), then integration of (2.3.21) gives:

r * ° ^ _ P" d x

<l>i , o £ i W

The integral capacity of the inner zone is then:

where &1 is given by:

'*> dx

o e i M j

dx

o ci

(2.4.12)

(2.4.13)

A similar expression can be written for the integral capacity of the outer zone of the compact layer K2, also involving the mean of the reciprocal of the permittivity in that region. The integral capacity of the whole compact layer, K, is then given by

I - _ L 1 K~ Kt

+ K (2.4.14)

We wil l need these expressions in Section 2-4.3 but wish to note here that models which employ equations like (2.4.5) and (2.4.6) do not require con­stancy of the dielectric permittivity for each capacitor but only that the charge between the wall and the OHP is confined exclusively to the IHP. This is, of course, an approximation but not such a drastic one as the equations might at first suggest.

The data on the mercury-solution interface permit quite elaborate models to be developed to describe the permittivity of the inner zone in terms of dipole orientation, as indicated above. This level of sophistication does not seem to be appropriate for colloid chemical systems as yet, chiefly because of the uncertainties caused by surface heterogeneity. I t may, however, be possible to apply these ideas to the l iquid-l iquid interface in emulsion systems. The more direct measurements of dielectric properties of interfaces, carried out by Dukhin (1970), show very large effects but the molecular dipoles referred to here probably play only a minor part in those situations.

2.4.3. The discreteness of charge ef fec t

The significance of this effect was first recognized by Frumkin (1933), and the first calculations of its magnitude were made by Esin and Shikov (1943) and Ershler (1946) in an attempt to explain the so-called Esin and Markov

effect (Esin and Markov, 1939). This refers to the fact that the point of zero charge of the mercury-solution interface, when measured against a reference electrode with liquid junction, depends on the electrolyte concentration i f the electrolyte is specifically adsorbed (see e.g. Delahay, 1965, p. 53), but not otherwise. I t is therefore used as a definitive indicator of the occurrence of specific adsorption in electrode systems, and Lyklema (1972) has shown that an analogous procedure can be used in colloidal systems. The importance of the discreteness of charge effect is that it appears to be able to explain a wide variety of effects which are not explicable using the simple version of the Gouy-Chapman-Stern theory (see e.g. Levine et al, 1967). In particular, i t is possible to explain the observation of Grahame and Soderberg (1954) that anion-specific adsorption can cause the (negative) potential in the OHP to pass through a maximum value as the potential \j/0 is made increasingly negative.

The calculation of the discreteness effect has been fully reviewed by MacDonald and Barlow (1964), by Levine et al. (1967), and more briefly by Levine (1971). A simplified analysis of the effect is given by Smith (1973) who shows how it can explain a maximum in i/>d as i/>0 changes monotonically. We are more concerned with the problem of calculating a "true" chemical adsorption potential, Q[, which as indicated in eq (11) above should be independent of electrical effects at the interface. In this section we apply the correction to the Stern layer ions, cr,-. I t should be noted, however, that on insulators it can be applied to the surface charge, cr0, as well, and it is also of importance in the diffuse layer, though the method of calculating the potential 4>a must, of course, be related to the detailed circumstances of the ion con­cerned.

Although a number of approaches have been made to the problem they have a considerable amount in common. According to Levine et al. (1967, p. 290) they can all be regarded as modifications or refinements of the "cut­off disc" method in which the potential is calculated by dividing the I H P into circular discs, one for each ion, and calculating the potential at the centre of each (charge-free) disc due to the atmosphere of the surrounding ions in the IHP. The different treatments essentially differ only in the estimate of the radius of the disc associated with each ion, and this is determined to a large extent by the dielectric properties assumed for the wall and the diffuse layer. The calculation is rather insensitive to the size of the disc at low values of <7j, and an infinite disc can be used with little loss of accuracy (Levine, 1971). For larger values of cr,-, however, the size of the disc does become important and so therefore do the assumed dielectric properties, because they determine the extent to which the ion is screened from the effects of its self-atmosphere. Levine (1971) has explicitly taken the properties of the diffuse layer into account and has obtained for an infinite disc and a

conducting wall :

- op (1 + K2/Cd) . * - - J C A ( l + K / Q ) ( 2 A 1 5 )

where K is the integral capacity of the inner layer, and K2 refer to the inner and outer zones (see Section 2.4.2) and Cd is the differential capacity of the diffuse layer as given by eq (2.4.8). The use of an infinite disc over­estimates the size of 4>a, but the error, according to Levine (1971), is not too serious. I t should be offset in the case of many colloids by the fact that the wall is a non-conductor which tends to increase the size of 4>a because of poor screening, but there is evidence (Wright and Hunter, 1973) that even in those cases eq (15) may overestimate the magnitude of <j>a.

At high concentrations or away from the p.z.c. the value of Cd wil l dominate over K and K2 so that eq (15) becomes

* - - ï £ ( 2 A 1 6 )

Equations (2.4.3), (11), (14) and (16) imply that

6-'ff--& ( 2 A 1 7 )

and since z ; and cr; are of the same sign this means that the apparent chemical adsorption potential, 0;, is more negative than the true value, 9\. The dis­creteness of charge effect therefore leads to higher adsorption densities in the IHP than would be expected in its absence. This semi-quantitative conclusion is, however, not as important as the fact that application of the discreteness correction to 6 yields a chemical adsorption potential, 8', which is more or less independent of the state of charge of the surface (Levine, 1971).

2.4.4. The d i f fuse layer

The major approximations and assumptions that are involved in the use of the Poisson-Boltzmann equation (2.3.7) for the diffuse part of the double layer have been ably reviewed by Haydon (1964). Their importance generally decreases with decrease in the potential and concentration, and hence it is not surprising that the criticism which can be levelled at the original Gouy-Chapman theory carries much less weight when applied to the diffuse layer in the Gouy-Stern-Grahame theory.

The improvements examined by Haydon include (i) the introduction of the finite volume of the ions (ii) the modification of the permittivity of the medium to include the effects of the double-layer field and of the ions present,

(iii) the effect of ionic polarization in the double-layer field, and (iv) the self-atmosphere effect. The first two corrections act in the opposite direction to the second two which explains why it is essential to attempt to take into account all of the significant effects at the same time. Where only the relative sizes of the different effects is being considered (e.g. Sparnaay, 1972) no great problems arise, but when one effect is singled out for examination (e.g. Ravina and Zaslavsky, 1972; Babchin, 1974) quite erroneous conclusions can be drawn (Bell and Levine, 1976; Deryaguin, 1976). When all the major effects are treated at the same time (Levine and Bell, 1966) the overall correction to the Poisson-Boltzmann equation is quite small. For 1:1 electrolytes at modest concentrations ( < 1 0 - 2 M ) the corrections to the simple theory are less than 2% for \\J/d\ < 100mV. Since this is the region in which most electrokinetic data is collected, the uncertainties from this source would seem to be negligible.

Nevertheless, there remains some disquiet at the more fundamental level: the statistical mechanical basis of the Poisson-Boltzmann equation and of the Gouy-Chapman-Stern-Grahame model of the double layer has fre­quently been questioned over the years. The fact that the complete equation (2.3.7) violates the superposition principle (i.e. it does not satisfy the funda­mental physical requirement that the electric fields from the individual charges should be linearly superposable) throws an initial doubt on its validity. Casimir argued that it was better to ignore that problem than to overcome it by using the linear form (eq 2.3.8) because the error involved in that approximation (i.e. assuming zt//0 < 1 everywhere) was certain to be large in many colloidal systems (see Verwey and Overbeek, 1948, p. 23).

The problem can be traced back to the partitioning of the electrochemical potential p.h into a chemical and an electrical part in eq (2.3.3). The potential which occurs in the Boltzmann equation (2.3.5) is then identified with the mean (i.e. time averaged) electrostatic potential at each point; this is the quantity which appears in the Poisson equation (2.3.2). A more exact statistical mechanical treatment requires that \p should be the potential of the mean force experienced by the ion as it is brought from the bulk electrolyte up to the point in question (see Fowler and Guggenheim (1952) or the treat­ment of the corresponding problem in the theory of strong electrolytes by Robinson and Stokes (1959)). One way to overcome this problem is to try to make a better estimate of the potential of the mean force by introducing various corrections to the local electrostatic potential. Onsager (1964), among others, has done this for electrolyte solutions, and the work of Levine and Bell (1966), already referred to, offers a similar approach for colloid (or double-layer) systems. A more satisfactory approach would be to tackle the problem using a self-consistent statistical mechanical procedure.

The early attempts in this direction (Buff and Stillinger, 1963; Krylov and

Levich, 1963; Martynov, 1966) showed that the behaviour in concentrated electrolyte solutions would be significantly different f rom that predicted by the Poisson-Boltzmann equation: this is hardly surprising but of little consequence to the theory of electrokinetics since most electrokinetic phenomena disappear in concentrated electrolyte solutions. More serious was the sort of criticism levelled by Blum (1977) and Cooper and Harrison (1977) , who objected to the breakdown of the system into a compact and a diffuse layer and demanded a more consistent treatment of ion size effects. An effective rebuttal of this criticism is given by Levine and Outhwaite (1978) for the case where there is no specific adsorption. Their conclusion has recently been strikingly confirmed by Torrie and Valleau (1979) whose Monte Carlo calculations are in essentially quantitative agreement with the Stern-layer-modified Poisson-Boltzmann equation with g, = 0 at an electrolyte concentration of 0-1 M . I t may be assumed, therefore, that for the dilute electrolyte solutions with which we wil l be concerned, the more exact statistical mechanical treatments now made possible by improvements in liquid state theories (see e.g. Henderson and Blum, 1978) do not really challenge the validity of the Poisson-Boltzmann equation which can also be justified using the methods of continuum mechanics (Cade, 1978). We shall return to the question of dielectric saturation (i.e. reduction of the permittivity of the dielectric as a consequence of the electric field) in the diffuse layer in Chapter 5 when we examine the concept of the (-potential in a little more detail.

2 .5 . T h e d o u b l e l a y e r a r o u n d a s p h e r e

2 .5 .1 . The exact so lu t i on of the P o i s s o n - B o l t z m a n n equa t i on

As noted in Section 2.3.2, there is no exact analytical solution of eq (2.3.28). A few numerical solutions were worked out by Muller (1928), and a more extensive tabulation was compiled by Hoskin (1953) using electronic com­puting techniques. More recently Loeb et al. (1961) compiled a much more comprehensive set of tables covering a wide variety of electrolyte types, concentrations and surface potentials. To facilitate interpolation from those tables Stigter (1972) developed a double power series representation which describes the numerical results of Loeb et al. (1961) very accurately.

For our purposes, the principal result of the numerical calculation is a tabulation of the integrals I+ and /_ where ƒ+ + ƒ_ = I(q), since the surface density of charge cr(q) is given by:

kT K o-(q) = e0D—-I(q)

ze A (2.5.1)

where q = Kr/X and X2 = (z+ + |z_|)/2|z_|. For a symmetrical electrolyte, X = l and q = Kr in which case the accumulated space charge outside a sphere of radius r is given by

kT Q(Kr) = 4nr2cr(q) = ( 4 T I £ 0 ) D — Kr2I(icr)

The electrophoretic charge on the sphere is then:

kT Qe(Ka)={4m0)D—Ka2I(Ka) (2.5.2)

ze

where a is the radius out to the plane of shear. The function I(KO) depends on both Ka and £ and according to Stigter (1972) is given very accurately by the following expression:

I(Ka,0 = z l { l + K a ) \ \ + [(2/zD sinh(zr/2) - 1] x Ka I

k=0 1=0

= z f + ™ ) m (2.5.3) Ka

where g = log KO, p = (z£/4 - 1) and the coefficients Bkl of the double power series are given in Table 2.2 (from Stigter, 1972). Stigter claims that eq (3) reproduces the data in the Loeb et al (1961) tables to an accuracy of 1 in 104, save for the region 0-1 s= Ka < 0-4 and 5 < 7 where the error may be a little higher ( ~ 1 in 103). Comparison of eq (2) and (3) with the earlier expression for the electrophoretic charge (2.3.37) shows that the corrected value of the charge is given by:

Qe = 4ns0Da(l + KaKF(Q (2.5.4)

where F(() is the function appearing in braces in eq (3). As z\ becomes small (sinh zlll^zlll) the correction obviously approaches unity. Except for very small values of Ka, however, it is doubtful whether there is any need to go to these lengths to obtain reasonably accurate estimates of Qe. For Ka > 1 the simpler empirical relations to be described in the next section give values of Qe accurate to better than 1 % in most cases (Table 2.3).

2.5.2. Ana ly t i ca l app rox ima t ions

The first attempt to improve on the Debye-Hückel approach with an analytical approximation was made by Gronwall et al. (1928), whose series

r~- 0 \ O oo CN —i -O CT\ O CN CN f- — o\ m (N - ' l o m v O s o O o ^ -*mm(Ncs ó ó ó ó ö ó ö ó

' I I I I

"~ï ^ — ' ^ - C ^ t ~ - ^ O T — < \ o ^ o o c ~ - o - » - o o \ o oo—<m\omr^rO(N Or0i /1O(N(N00i / ->

Ó Ó Ó Ó Ö Ö Ó Ö I I I I

' } m - i o M o c o o \ o o O ) OO C \ - ' 0 0 0 * 0 1 0

p p O O — < - H O O O Ó Ó Ó Ö Ó Ó Ó I I I I

mooo>ooo^i-o>o p o p ^ — < — Ó Ó Ó Ó Ó Ó Ó Ó I I I I I

p o p o o o o o Ó Ö Ó Ó Ó Ö Ö Ó

I I I I

^ o o m o o \ o o » r ^

p p ^ o o o o o Ó Ö Ó Ó Ö Ö Ó Ó

1 I

CT\0'-<OT)--o-^-m

Ö Ó Ö Ó Ó Ó Ö Ó I

I TABLE 2.3

Ratio, of (Qc) eq ( 3 ) / Q C (eq 5) as a function of £ and Ka, for 1:1 electrolytes (from Loeb et al, 1961 , p. 37 )

r 0 1 0-2 0-5 1 2 5 1 0

0 1 0000 1 0 0 0 0 1-0000 1 0000 1 0000 1-0000 1-0000 1 1-0158 1-0126 1-0073 1-0039 1-0016 1-0004 1-0001 2 1-0600 1-0464 1-0249 1-0127 1-0051 1-0009 1-0003

3 1-1232 1-0903 1-0448 1-0203 1-0075 1-0016 1-0004 4 1-1902 1-1285 1-0550 1-0223 1-0076 1-0015 1-0004

6 1-2599 1 1 3 6 0 1 0 4 0 3 1-0130 1-0038 1-0007 1-0002

expansion method was also used by Booth (1951a). Levine (1939) showed how the convergence of this series could be improved, but it remains a cumbersome procedure. More recently Parlange (1972), Abraham-Shrauner (1973) and Brenner and Roberts (1973) have provided analytical approxi­mations of use under some conditions.

There is some controversy concerning the general validity of the first two of these latter procedures (Semenikhin and Sigal, 1975; Abraham-Shrauner, 1975; Parlange, 1975). A procedure which appears to circumvent the main problems and avoids the arbitrariness of the Brenner and Roberts variational method has recently been proposed by White (1977).

For our purposes the conditions are not very stringent because electro­kinetic potentials are seldom very high f j f | < 150mV usually), and we are concerned in the present context only with the problem of relating a measured (-potential to the charge on the particle. For that purpose the empirical relationship proposed by Loeb et al. (1961) for the surface charge wil l often be adequate:

kT Qe = 4ns0D—Ka2{2 sinh(z£/2) + — tanh(z(/4)

ze Ka (2.5.5)

This equation can readily be shown to reduce to the approximate expression (2.3.37) for low values of Z (when sinh p ad tanh p at p). The accuracy of eq (5) can be judged from the accompanying Table (2.3), which is abridged from Loeb et al. (1961). At higher values of % ( > 6), it improves still further but such values are not of much interest in electrokinetic phenomena. The reason for the improvement at high ( i s that eq (5) bears a strong relationship to the expression for flat plates (indeed it tends to eq (2.3.23) for large Ka), and Loeb et al. (1961, p. 36) show that high values of the surface potential on a sphere tend to make the double layer contract towards the surface so it behaves more like a flat plate. Theoretical justifications of eq (5) have been

developed by Dukhin et al. (1970) and by Stokes (1976). More recently, Semenikhin and Rulev (1975) have extended the approach

of Dukhin et al. (1970) to obtain an approximate analytical expression for the potential around an ellipsoidal particle.

2.5.3. Cor rec t ion for f in i te ion size

Gorin (1939) claims that for very small particles (e.g. proteins) one could find that the radius of the counterion, ah was comparable with that of the particle a. If, then, the shear plane is regarded as impenetrable to the counter­ions, one would have a charge-free region surrounding the particle (Fig. 2.12)

This would alter the boundary conditions for the evaluation of the constant B m eq (2.3.30). One would then write for the balancing charge:

r tydr (2.5.6) a + ai

QE = (4KS0)DK2

= (4US0)DK2B I re~Krdr (2.5.7)

and integration by parts leads to

QE = (4ne0)DB. e - K ^ + a < \ l + K a + K a . ) (2.5.8)

The potential in the plane of closest approach of the counterions is

Fig. 2.12. The large sphere represents the shear surface and the small spheres represent the counterions. (After Gorin in Abramson et al. (1942)).

obtained by using (8) to substitute for B in eq (2.3.30) with r = (a + af) and A = 0:

^ < r = a + a i ) (47ie 0)D(a + a ; ) ( l + Ka + Ka,) ( 2 ' 5 ' 9 )

As Gorin points out, this is not the zeta potential because, by definition, that remains the potential at the plane of shear (where r = a). To find ( we must subtract from \p the part due to the central particle. This leaves the part, rp', due to the atmosphere which must be the same everywhere inside the sphere of radius (a + a,). Adding this to the potential due to the particle at the surface r = a gives the (-potential:

4ns0D(a + at) 4ns0Da

(2.(1 + *ad 4ns0Da(\ + Ka + Ka,)

(2.5.10)

The function (1 +- Ka-)/(l + KO + Ka-) replaces the simpler \/(1+KÜ) in eq (2.3.37), and Gorin has shown (see Abramson et al., 1942, p. 123) that in unfavourable cases it could involve a correction of as much as 25%. This is, however, for very small particles ( ~ 2-5 nm), so the effect is not usually very obvious. Hunter (1966) has, however, used a similar argument to explain some results of Haydon (1960) at the oil-water interface, and we shall discuss this work in more detail in Chapter 5.

2.5.4. The c o m p a c t layer a round a sphere

It is obvious that all of the remarks concerning the effect of high field strengths and finite ion size on the Poisson-Boltzmann equation in the near neigh­bourhood of a flat surface apply with almost equal force to the surface of a sphere. The fact that the field lines diverge from a sphere means that the field itself falls off a little more rapidly with distance than for a flat plate, but this effect is more noticeable when the diffuse layer is reached. The capacitance of a simple spherical molecular condenser per unit area of the inner sphere is given by:

— =s0D — (2.5.11)

where a0 is the radius of the inner sphere and ö is the distance between the (concentric) spherical surfaces. In a very simple model of a colloidal particle one might take a0 as the radius of the particle, <5 as the thickness of the com­pact layer, and use an equation like eq (11) in place of eq (2.4.5 and 2.4.6).

2 .6. T h e d o u b l e l a y e r a r o u n d a c y l i n d e r

Philip and Wooding (1970) provided an improved description of the double layer around a cylinder by breaking it into two parts. For sufficiently large distances from the axis of the cylinder, the potential must fall to values such that zi> < 1 and in this region the linear solution (eq (2.3.39)) applies. In the inner region, only one of the exponential terms in the sinh function is re­tained (eq (2.3.38)). The solutions to the two regions are matched at the boundary so that \]/ and aty/dr are continuous. The resulting analytical approximation has been shown by van der Dr i f t et al. (1979) to match the exact numerical solution of eq (2.3.38) with an accuracy which is always better than 3-5% and usually much better than 1 % for z = 1. They applied the solution to the treatment of electrophoresis of certain polyelectrolytes and proteins (see Section 3.7.3).

This same problem was examined in a somewhat different fashion by Stigter (1975), who used a numerical solution of eq (2.3.38) to obtain the potential function. His results for the case z = 1 are presented in the form of a correction factor lj$ multiplying the right-hand side of eq (2.3.40). Values of for various values of \j/0 and Ka are given in Table 2.4. Note that ft is always greater than unity, which means that the linearized equation always overestimates the potential corresponding to a given charge on the cylinder.

2.7 . T h e d o u b l e d i f f u s e l a y e r

In some situations it is thought that the space charge may be distributed in a diffuse layer in both phases. This would certainly be expected to be the case. at. the oil-water interface, but i t is also thought to be so,at some, solid-liquid interfaces, either because of the presence of a space charge of defect crystal sites in the solid or because of a disordered gel layer on the surface.

The oil-water interface was treated by Verwey and Niessen (1939), and that treatment is discussed by Verwey and Overbeek (1948) and, more recently, by Sparnaay (1972a). Essentially one regards the total double-layer potential, \j/0, to be the difference in potential f rom the interior of one phase to the interior of the other (Fig. 2.13) and applies the Poisson equation to both phases. I f \j/s

0 is the total potential drop in one phase and \p% that in the other, then i> 0 = ifo + i> 0. The first integral of eq (2.3.1) then gives, at the boundary between the two phases:

£ s ( ^ ) * = o ~ e a ( ^ ) * = o ( 2 ' 7 , 1 )

and introducing eq (2.3.14) for (d\jjjdx), which of course, also introduces the

TABLE 2.4 Highly charged colloidal cylinders

Correction factor /? for the charge potential relation, eq (2.3.40) as a function of Ka and \j/0 (from Stigter, 1975)

1

1/128 1/64 1/32 1/16 1/8 1/4 1/2 1 2 4 8 16

100140 1-00212 1-00331 1-00529 1 00845 1-01309 1-01908 1-02560 1-03147 1-03586 1-03870 1-04034 1-04219

1-00575 1 00876 1-01378 1-02214 1-03550 1-05511 1 08028 1-10746 1-13165 1-14962 1-16113 1-16779 1-17520

1-01354 1-02082 1-03313 1-05373 1-08670 1 13489 1-19616 1-26139 1-31864 1-36062 1-38726 1-40256 1-41952

1-02569 1-04011 1-06476 1-10632 1-17279 1-26914 1-38992 1-51632 1-62540 1-70429 1-75390 1-78221 1-81343

1-04385 1-06979 1-11473 1-19086 1-31210 1-48557 1-69919 • 1- 91863 2- 10497 2-23811 2-32114 2-36829 2-42008

1-07085 111538 1-19339 1-32561 1-53413 1- 82755 2- 18208 2-53999 2- 83979 3- 05195 3-18342 3-25780 3-33929

111150 1-18620 1-31788 1-53986 1- 88486 2- 36147 2- 92725 3- 49027 3- 95705 4- 28504 4-48738 4-60156 4-72646

1-17400 1-29780 1-51589 1- 87934 2- 43465 3- 18840 4- 07026 4- 93860 5- 65306 6- 15266 6-45989 6-63327 6-82248

Boltzmann equation, we may write

V<X°£S) smh(ze\j/s

0/2kT) = J{n°aza) sinh(zei>0/2/c7) (2.7.2)

The potential distribution in the two phases is evidently determined by the ratio

1/2 (2.7.3)

and Fig. 2.14 shows the relation between a and the potentials ij/s

0 and \jfQ for two particular values of ^/0(z\j/0 = 4 and z $ 0 = 8).

Since low values of e tend to be associated with low electrolyte concentra­tion (n°), the value of a can diverge significantly from unity so that the potential drops in the two phases may differ markedly. At the oil-water interface where n°e for oil is certainly much smaller than for water, almost the entire potential drop wil l occur in the oil phase. On the other hand, the thickness of the double layer in the two phases is determined by the ratio

5L K,

1/2 (2.7.4)

so variations in e tend to be offset to some extent by corresponding variations in n°. Nevertheless, the double layer can be very much more extended in the

250

v//0=205 mV

Fig. 2.14. Distribution of the total potential drop \ji0 between two phases as a function of log a. (From Verwey and Overbeek (1948) p. 36 with permission.)

oil phase (extending to well over 1 mm) in the case of a non-polar (pure hydrocarbon) oil.

This sort of approach has more recently been applied to the Agl-solution interface where the total potential drop \ji0 may be equated with the Nernst potential of eq (2.2.4). This amounts to treating the solid Agl crystal as a semiconductor with a space charge made up of defects obeying the Boltzmann equation. Combining eq (2) with the condition <Ao = <Ao + ^o, 1 1 is n o t

difficult to show that:

exp(i>0) = ; (2.7.5) oc + y

where

y = exp( - # 0 /2) = ( j ^ f J ' (2-7-6)

Now, if we identify the superscript s with the solid phase and a with the aqueous phase we might expect to be approximately equal to the £-potential, since the compact layer may be expected to resemble the solid phase in many ways. Equation (5) then suggests that if a is large, as seems likely from eq (3), then wil l not depend very much on y; in other words the (-potential wi l l not be strongly determined by the silver ion concentration. This type of argument was used by Davies and Halliday (1952) and by Ottewill and Woodbridge (1964), and we shall return to it in Chapter 7. I t might be noted in passing, however, that the argument is not universally accepted (see e.g. Smith, 1973).

When measurements are made of the (-potential and charge on oxide surfaces it is generally found that very high values of surface charge are associated with quite modest values of (-potential. I t is possible that this behaviour is due to the presence of a highly disordered (gel-like) layer on the oxide surface (Lyklema, 1968), and mathematical models of this type of boundary, not unlike the simple model treated above, have recently been developed (Levine et al., 1972; Perram et al, 191 A); these too wil l be treated in Chapter 7.

R e f e r e n c e s

Abraham-Shrauner, B. (1973). J. Colloid Interface Sci. 44, 79. Abraham-Shrauner, B. (1975). J. Colloid Interface Sci. 53, 499. Abramson, H . A. , Moyer, L . S. and Gorin, M . H . (1942). "Electrophoresis of Pro­

teins." Reinhold, New York. Andersen, T. N . and Bockris, J. O ' M . (1964). Electrochim Acta 9, 347. Babchin, A. J. (1974). J. Colloid Interface Sci. 41, 390. Bardeen, J. (1936). Phys. Rev. 71, 717.

Barlow, C. A. Jr. and MacDonald, J. R. (1967). In "Advances in Electrochem. and Electrochem. Eng." 6, 1 (P. Delahay ed.) Interscience, New York.

Bell, G. M . and Levine, S. (1976). J. Colloid Interface Sci. 56, 218. Bérubé, Y . G. and de Bruyn, P. L . (1968). J. Colloid Interface Sci. 28, 92. Blum, L . (1977). J. Phys. Chem. 81, 136. Bockris, J. O 'M. , Devanathan, M . A. V. and Miiller, K . (1963). Proc. Roy. Soc.

(London) A274, 55. Booth, F. (1951a). / . Chem. Phys. 19, 821. Booth, F. (1951b). J. Chem. Phys. 19, 391, 1327, 1615. Brenner, S. L . and Roberts, R. E. (1973). J. Phys. Chem. 77, 2367. Buff, F. P. and Goel, N . S. (1969). J. Chem. Phys. 51, 4983, 5363. Buff, Frank P. and Stillinger, Frank H . , Jr (1963). / . Chem. Phys. 39, 1911. Cade, R. (1978). J. Colloid Interface Sci. 66, 358. Chapman, D . L . (1913). Phil. Mag. 25, 475. Childs, E. C. (1954). Trans. Faraday Soc. 50, 1356. Conway, B. E. (1977). Soviet Electrochem. 13, 695. Cooper, I . L . and Harrison, J. A . (1977). Electrochim. Acta. 22, 519.. Damaskin, B. B. (1977). J. Electroanal. Chem. 75, 359. Damaskin, B. B. and Frumkin, A. N . (1974). Electrochim. Acta 19, 173. Davies, J. T. and Rideal, E. K. (1963). "Interfacial Pheonomena" Chapter 2. Academic

Press, London and New York. Davies, K . N . and Halliday, A. K . (1952). Trans. Faraday Soc. 48, 1061, 1066. Delahay, P. (1965). "The Double Layer and Electrode Kinetics" Chapter 5. Interscience,

New York and London. Deryaguin, B. V. (1976). / . Colloid Interface Sci. 56, 492. Devanathan, M . A . V. and Tilak, B. V. K . S. R. A. (1965). Chem. Revs. 65, 635. Dube, G. P. (1943). Indian J. Phys. 17, 189. Dukhin, S. S. (1970). Spec. Disc. Faraday Soc. No. 1, p. 158. Dukhin, S. S., Semenikhin, N . M . and Shapinskai, L . M . (1970). Dokl. Akad. Nauk

SSSR, 193, 385. Ershler, B. V. (1946). Zhur. Fiz. Khim. 20, 679. Esin, O. A. and Markov, B, F. (1939). Zhur. Fiz. Khim. 13, 318. Esin, O. A. and Shikov, V. M . (1943). Zhur. Fiz. Khim. 17, 236. Fowler, R. H . and Guggenheim, E. A. (1952). "Statistical Thermodynamics." Cam­

bridge University Press, Cambridge. Frumkin, A . N . (1933). Phys. Zhur. Soviet Union 4, 256. Frumkin, A. N . (1940). Trans. Faraday Soc. 36, 117. Frumkin, A. N . and Damaskin, B. B. (1964). In "Modern Aspects of Electrochemistry"

Vol. 3, pp. 149-223, (J. O 'M. Bockris and B. E. Conway eds.). Butterworths, London. Frumkin, A. N . and Pleskov, Yu . V. (1973). Soviet Electrochem. 9, 1642. Gardiner, C. L . (1975). J. Electroanal. Chem. 61, 113. Gorin, M . H . (1939). / . Chem. Phys. 7, 405. Gouy, G. (1910). J. Phys. Chem. 9, 457. Grahame, D . C. (1947). Chem. Rev. 41, 441. Grahame, D . C. and Soderberg, B. A . (1954). J. Chem. Phys. 22, 449. Gronwall, T. H . , La Mer, V. K . and Sandved, K . (1928). Physik. Z. 29, 358. Guggenheim, E. A. (1929). / . Phys. Chem. 33, 842. Haydon, D . A. (1960). Proc. Roy. Soc. (London) 258A, 319. Haydon, D . A. (1964). In "Recent Progress in Surface Science" Vol. 1, pp. 94-158

(J. F. Danielli, K . G. A . Pankhurst and A. C. Riddiford eds.), Academic Press,

New York and London. Henderson, D . and Blum, L . (1978). J. Electroanal. Chem. 93, 151. Hoskin, N . E. (1953). Trans. Faraday Soc. 49, 1471. Hunter, R. J. (1966). J. Colloid Interface Sci. 22, 231. Hunter, R. J. and Wright, H . J. L . (1971). J. Colloid Interface Sci. 37, 564. Krylov, V. S. and Levich, V. G. (1963). Russian J. Phys. Chem. 37, [10], 1224. Levine, S. (1939). J. Chem. Phys. 7, 831. Levine, S. (1971). J. Colloid Interface Sci. 37, 619. Levine, S. and Bell, G. M . (1966). Disc. Faraday Soc, 42, 69. Levine, S. and Matijevic, E. (1967). J. Colloid Interface Sci. 23, 188. Levine, S. and Outhwaite, C. W. (1978). J. Chem. Soc. Faraday II, 74,1670. Levine, S. and Smith, A. L . (1971). Disc. Faraday Soc. 52, 290. Levine, S., Bell, G. M . and Calvert, D . (1962). Canadian J. Chem. 40, 518. Levine, S., Mingins, J. and Bell, G. M . (1967). J. Electroanal. Chem. 13, 280. Levine, S., Bell, G. M . and Smith, A . L . (1969). J. Phys. Chem. 73, 3534. Levine, S., Smith, A . L . and Brett, A. C. (1972). Seventh Intern. Cong. Surface Activity

(Zurich) Proceedings, p. 603. Loeb, A. L . , Wiersema, P. H . and Overbeek, J. Th. G. (1961). "The Electrical Double

Layer Around a Spherical Colloid Particle", M I T Press, Cambridge, Mass. Lyklema, J. (1968). J. Electroanal. Chem. 18, 341. Lyklema, J. (1972). / . Electroanal. Chem. 37, 53. Lyklema, J. and Overbeek, J. Th. G. (1961). J. Colloid Sci. 16, 595. MacDonald, J. R. and Barlow, C. A. , Jr (1962). J. Chem. Phys. 36, 3062. MacDonald, J. R. and Barlow, C. A. Jr (1964). In "Electro-Chemistry", pp. 199-247.

Proc. 1st Australian Conf. in Electrochem (1963). Pergamon, London and New York. Malsch, J. (1928). Physik. Zeits. 29, 770. Malsch, J. (1929). Physik. Zeits. 30, 837. Martynov, G. A . (1966). "Research in Surface Forces" Vol . 2 pp. 75-102 (B. V. Dery­

aguin ed.). Consultants Bureau, New York. Miiller, H . (1928). Kolloidchem. Beihefte 26, 257. Oldham, K . B. and Parsons, R. (1977). Soviet Electrochem. 13, 732. Onsager, L . (1964). / . Amer. Chem. Soc. 86, 3421. Ottewill, R. H . and Woodbridge, R. F. (1964). J. Colloid Sci. 19, 606. Overbeek, J. Th. G. (1952). In "Colloid Science" Vol . 1, (H. R. Kruyt, ed.) Elsevier,

Amsterdam and London. Parlange, J.-Y. (1972). J. Chem. Phys. 57, 376. Parlange, J.-Y. (1975). J. Chem. Phys. 63, 1699. Parry, J. M . and Parsons, R. (1963). Trans. Faraday Soc. 59, 241. Parsons, R. (1954). In "Modern Aspects of Electrochemistry" Vol. 1, pp. 103-179.

(J. O. M . Bockris and B. E. Conway eds.) Butterworths, London. Parsons, R. (1959). Trans. Faraday Soc. 55, 999. Parsons, R. (1961). In "Advances in Electrochem. and Electrochem. Engineering",

Vol. 1, p. 1-64 (P. Delahay, ed.). Interscience, New York. Parsons, R. (1975). J. Electroanal. Chem. 59, 229. Payne, R. (1973). In "Progress in Surface and Membrane Science". Vol. 6, pp. 51-123

J. F. Danielli, M . D. Rosenberg and D . A. Cadenhead eds.), Academic Press, New York and London.

Perram, J. W., Hunter, R. J. and Wright, H . J. L . (1974). Aust. J. Chem. 27, 461. Perrin, J. (1904). J. Chim. Phys. 2, 601. Philip, J. R. and Wooding, R. A . (1970). J. Chem. Phys. 52, 593.

58 ZETA POTENTIAL I N COLLOID SCIENCE

Ravina, I . and Zaslavsky, D . (1972). IsraelJ. Chem. 10, 707. Robinson, K . and Levine, S. (1973). J. Electroanal. Chem. 47, 395. Robinson, R. A . and Stokes, R. S. (1959). "Elect rolyte Solut ions ." But terworths ,

L o n d o n . Salem, R. R. (1976). Russ. J. Phys. Chem. 50, 656. Semenikhin, N . M . and Sigal, V . L . (1975). / . Colloid Interface Sci. 5 1 , 215. Semenikhin, N . M . and R u l ë v , N . N . (1975). Colloid J. (U.S.S.R.) 37, 815. Smith , A . L . (1973). In "Dispersions o f Powders in L i q u i d s " Chapter 3. ( G . D . Par f i t t

ed.). A p p l i e d Sciences Pub., Bark ing . Sparnaay, M . J. (1972a). The electrical double layer In " In te rna t Encyclopaedia o f

Phys. Chem. and Chem. Phys." Top ic 14, V o l . 4 ( D . H . Everett, ed.) Pergamon, O x f o r d and N e w Y o r k .

Sparnaay, M . J. (1972b). J. Electroanal. Chem. 37, 65. Stern, O. (1924). Z. Elektrochem. 30, 508. Stigter, D . (1972). / . Electroanal. Chem. 37, 61 . Stigter, D . (1975). J. Colloid Interface Sci. 53, 296. Stokes, A . N . (1976). / . Chem. Phys. 65, 261 . Tor r i e , G . M . and Valleau, J. P. (1979). Chem. Phys. Letters 65, 343. Trasat t i , S. (1971). J. Electroanal. Chem. 33, 351. van der D r i f t , W . P. J. T . , de Keizer, A . and Overbeek, J. T h . G . (1979). J. Colloid

Interface Sci. 7 1 , 67. van Olphen, H . (1963). " I n t r o d u c t i o n to Clay C o l l o i d Chemis t ry" p . 67. Interscience,

N e w Y o r k and L o n d o n . Verwey, E. J. W . and Niessen, K . F . (1939). Philos. Mag. 28, 435. Verwey, E. j . W . and Overbeek, J. T h . G . (1948). "Theory o f Stabi l i ty o f Lyophob ic

Co l lo id s " p . 34. Elsevier, Amste rdam and N e w Y o r k . W h i t e , L . R. (1977). J. Chem. Soc. Faraday II, 73, 577. W r i g h t , H . J. L . and Hunter , R . J. (1973). Aust. J. Chem. 26, 1183, 1191. W r o b l o v a , H . and M i i l l e r , K . (1969). J. Phys. Chem. 73, 3528. Yates, D . E. , Levine, S. and Healy, T . W . (1974). J. Chem. Soc. Faraday, I , 70, 1807.

Chapter 3

The Calculation of Zeta Potential

Since the earliest theoretical treatments of electrokinetic phenomena a variety of modifications and refinements of greater or less validity and generality have appeared. We shall therefore present the classical (and simple) description of each electrokinetic process first (Part I) and go on to the re­finements later (Part I I ) . The classical treatment follows closely that given by Overbeek (1952) with some additional suggestions made by Dukhin and Deryaguin (1974).

I. CLASSICAL THEORY

3.1. Electro-osmosis

3.1.1. Measurements on s ing le capi l lar ies

The theory of the process was first given in its present form by von Smolu-chowski (1921), who considered the movement of the liquid adjacent to a flat, charged surface under the influence of an electric field applied parallel to the interface. I f the surface is negatively charged, there wil l be a net excess of positive ions in the adjacent liquid and as they move under the influence of the applied field they wil l draw the liquid along with them. The surface of shear in this case may be taken as a plane parallel to the surface and distant 5 from it.

The velocity of the liquid in the direction parallel to the wall, v., rises from a value of zero in the plane of shear to a maximum value, veo, at some distance from the wall (Fig. 3.1), after which it remains constant; veo is called the electro-osmotic velocity of the liquid.

59

The forces on an element of volume of the liquid are shown in Fig. 3.2 (see also Appendices 1 and 3) and evidently:

E._.Q = EzpAdx = rjA ) - r\A ( ^ f \ \ d x j x

1 \ d x j x . dx or

dx E:. pdx = — rj-j-^dx (3.1.1)

Substituting for p using Poisson's equation (2.3.7) gives:

D cfip (tPv.\ E.(4ne0)-——=-dx = ri\—-=-)dx (3.1.2)

4n dx \ebr J

This equation can be integrated twice from a point far from the solid where i/r = 0 and vz = veo up to the shear plane where v. = 0 and i// = C, using the fact that for the first integration both dxjjjdx and dv.jdx are zero far from the surface. The result is:

^ = " * = - 4 ™ o ^ (3-1.3)

= - — (3.1.3a) n

if it is assumed that, throughout the double-layer region, both y\ and D retain their normal bulk values. The quantity uE may be called the electro-osmotic mobility, and the negative sign indicates that when £ is negative the space charge is positive and so the liquid flow is towards the negative electrode (i.e. from bottom to top, which is the positive direction in Fig. 3.2).

Note that in deriving eq (3) no special assumptions are made about the potential distribution in the electrical double layer, save that it obeys Poisson's equation. Also notice that the integration is carried out from the bulk solution up to the shear plane. The result is therefore unaffected by the details of the potential distribution between the shear plane and the wall, provided it can be assumed that in that layer the liquid is uninfluenced by the applied electric field.

Direct measurement of the electro-osmotic mobility is possible using the ultramicroscope technique (Section 4.3), but it is more common to measure the total volume of liquid transported in an electro-osmosis experiment (Section 4.1). For experiments conducted in capillaries of the usual size ( 1 0 ~ 3 < r < 1 0 _ 1 cm) the electrical double layer is very thin compared to the capillary radius (xr > 1), and it is permissible to regard the capillary surface as flat. Furthermore, the bulk of the liquid in the capillary moves with velocity ve0 since the region of varying velocity extends only through the double layer

-*•

\ / \ / \ /

\ /

\ / \ / \

/ -\ / -

\ /

\ /

\ /

/ \ /

Fig 3 3. Velocity profile in a capillary during (a) electro-osmosis and (6) electro-osmotic counter pressure measurement or closed tube electro-osmosis. The thickness of the S r of

S L " H H T j f hT b e e " g r e a t l y I t is not visible with an ordinary microscope and all of the liquid near the wall appears to move with the velocity v

(Fig 3.3a). (The first integration of eq (2) shows that the velocity gradient and the potential gradient are always proportional to one another, as shown in Mg 3.1). Thus for a capillary of constant cross-section, the volume dis­placed per unit time is :

V=nr\0 = nr\Am0)^E._ ( 3.1.4)

In order to eliminate the area of cross-section, we introduce the electric current, i, transported by the l iquid:

E~_==nrX° (3.1.5)

where A0 is the electrical conductivity. Substituting in eq (4) then yields:

V DC l = i 4 n £ o ) 4 ^ T 0 (3.1.6)

£C ( 3- ]-6a)

The alternative expressions (3a) and (6a) are the rationalized equations (see

Appendix 2), which despite their evident simplicity have only recently come into common use. In SI units (Appendix 2), e0 = 8-854 x 1 0 " 1 2 C V - 1 m - 1 , and if ( i s in volts, r\ in Pa s (N m " 2 s) and 1 0 in ohm" tm~1,Vfi is obtained in m 3 C - 1 or m 3 s - 1 per ampere of current passed. In the usual mixed units

F ( c m 3 s - 1 ) _ 8-854 x 1 0 - 1 3 £ > x q m V ) i'(mA) y\(poise) x A 0 ( o h m - 1 c m - 1 )

Equations (5) and (6) are valid only if all or almost all of the current is transported by the bulk liquid. I t frequently happens, however, that a sig­nificant proportion is transported by the layers near the wall (or through the solid). The accumulated charge in the double layer may then lead to an unusually high conductivity, especially at low salt concentrations. Equation (5) must then be written

i n

= nr2A0 + 2nrAs

where As is the specific surface conductivity. (Note that A0 refers to the con­ductance of a cylindrical block of liquid of unit cross-sectional area and unit length and is measured in o h m - 1 m _ 1 (or o h m - 1 c m - 1 ) ; As refers to the conductance of a square sheet of material of unit area and constant, though negligible, thickness, measured along the length of the square and is measured in o h m - 1 . )

Equation (6) now becomes:

V DC T=(47t6 0 ) » (3.1.7) i 4nr](A0 + zAJr)

The term in parentheses is sometimes rendered more generally as (A0 + j l s ) where ƒ is a "form factor" for the capillary, equal to the ratio of its circum­ference to its cross-section. Specific surface conductivity values are of order 1 0 - 9 - 1 0 - 8 o h m " 1 for water in glass capillaries (Overbeek, 1952, p.237), so that significant effects on ( potential can be expected in 1 mm capillaries at concentrations below about 10 - 3 5 M.

Many of the early measurements of C show a maximum in absolute value at about this concentration because the absolute magnitude of C is under­estimated at lower concentrations if surface conductance is not taken into account. Rutgers (1940) has shown that the true (-potential can be obtained by measurements on capillaries of the same material but of different radii since As can then be eliminated, assuming that it is constant at constant electrolyte concentration. I t should be noted, however, that the values of As

obtained by Rutgers on Jena glass were shown by later work (Rutgers and de Smet, 1947; Wijga, 1946) to be rather too high.

3.1.2. Behav iour in po rous p lugs

Porous plugs may be made of (i) particles of quite irregular geometry, (ii) nearly spherical particles, (iii) flat plates, more or less well oriented, (iv) fibres, more or less well oriented, or (v) bundles of capillaries. Provided that the local radius of curvature of the particles is always (or almost always) large compared to \/K and that the size of the pores is also large compared to 1/K and the effects of surface conduction are negligible then Smoluchowski has shown that eq (6) can be used to describe the electro-osmotic flow through the plug (Overbeek (1952) p. 202). The flow must be linear and laminar, a condition which is probably always met in electro-osmosis experiments.

The methods that have been developed for dealing with systems exhibiting surface conduction wil l be discussed below.

3.1.3. È lec t ro -osmot i c coun te r pressure

Instead of determining the volume of liquid transported under zero pressure difference across the capillary or plug, i t is possible to measure the electro-osmotic flow by applying a counter pressure until the flow is exactly com­pensated (Fig. 3.3b). The back flow is given by Poiseuille's equation, so

V=^-=.— %r\l nl0

where p is the back pressure and / is the capillary length. Hence

„ P 8eC

This equation can readily be extended for single capillaries exhibiting surface conduction and can be applied to porous plugs in the absence of surface conduction (Abramson, 1934, p. 59).

One advantage of the counter-pressure measurement is that it avoids the errors which can result from the movement of the meniscus in the observation capillary (Section 4.1). The availability of accurate pressure transducers should make it possible to design a direct recording apparatus which would plot output pressure against applied voltage.

3.2. Streaming potential

3 . 2 . 1 . Measuremen ts in s ing le capi l lar ies

When liquid is forced through a capillary, the charges in the mobile part of the double layer near the wall are carried towards one end. This constitutes

a streaming current, I0 and the accumulation of charge sets up an electric field. The field causes a current flow in the opposite direction through the bulk of the liquid and when this latter conduction current, Ic, is equal to the streaming, current, a steady state is achieved. The resulting electrostatic potential difference between the ends of the capillary is the streaming poten­tial. I t must be measured, as a function of applied pressure, with a high im­pedance electrometer so that the current flows, / s and / „ are not disturbed.

The relation between the streaming potential, Es, and the zeta potential may be derived as follows. The linear velocity of the liquid at a distance y from the axis of the capillary is given by Poiseuille's equation:

vziy) =p j r 2 - y 2 )

4nl (3.2.1)

and the streaming current is, by definition (Fig. 3.4):

v:(y)p(y)dV

2%yv,(y)p(y)dy

Since the double layer is assumed to be confined to a thin region near the wall of the capillary, only values of y near to y = r are of importance in deter-

/ / / / / / / / / / / /

v z ( y )

T

11 11

i I i !

! I I 1

r I

\ \ \

Fig. 3.4. Calculation of the streaming current, I r

mining the current (i.e. the bulk of the moving liquid carries no net charge). We substitute y = (r - x) and, hence,

So

. pr prx v. == — (r — v) = -—

L - - 2n(r-x).~p{x)dx Ir] l

nr p xp(x)dx (3.2.2)

Substituting for p(x) from Poisson's equation (2.3.2) and integrating by parts we have:

nr2p f ° < ty , — r - XE—^rdx

1' Jr a*

_Tir2zp{( d\p dx

°#

7tf"2£p nl

dx

dtp = 7T7"2D 0 >7

(3.2.3).

The first term in the brackets disappears because dip/dx is zero when x = r (and, indeed, long before x = r). I t might be objected that the approximations leading to eq (3) wi l l not hold over the whole integration range. I t should be noted, however, that the contributions to the integral are confined entirely to the thin layer near the wall (x -4 r).

The streaming potential, Es> generated by this current causes a conduction current in the reverse direction given by:

I __ nr2Es. A0 + 2nrEs. Xs

I (3.2.4)

if we take account of both bulk and surface conduction. (Note that E f l - E is the electric field strength in this case.) When a steady state has been estab­lished I, + L = 0 and so

E±_E1_ s0DC P P r,U0 + 2XJr)

f](X0 + 2XJr)

(3.2.5)

(3.2.5a)

where P is the pressure gradient (pressure drop per unit length) through the capillary. In the usual mixed units system:

£ s (mV) _ 8-854 x l O ' ^ C V - 1 m - 1 ) x P x C(mV) 0-1 x /?(dynecm - 2 ) ~ 0-1 x ^(poise) x (A 0 + 2 A » ( o h m ~ 1 c m " l ) x 10

so that, at 20°C in water:

C= 1-055 x 10 s— (k0 + 2XJr) P

where C and Es are measured in the same units, p is in cm of Hg, and the conductivity is in o h m " 1 c m " 1 . Comparing eqs (3.1.7) and (3.2.5) we note that

p J i = o W P = O

This is a fundamental relationship which has been shown by Mazur and Overbeek (1951) to be a direct consequence of Onsager's principle of reciprocity, as applied in the thermodynamics of irreversible processes.

Rutger's (1940) procedure for dealing with surface conduction (Section 3.1.1) is, of course, equally applicable to the treatment of streaming potential in single capillaries. A simpler procedure suggested by Briggs (1928) (actually for the case of porous plugs) is to measure the actual resistance of the liquid in the capillary ( R e x p ) and to compare this with the value expected from measurements at high concentration ( i? e a l e ) where surface conduction can be expected to be negligible. Equation (5) then becomes:

Es eC R e x p (3.2.6) P ^ 0 -Rcalc

Very accurate estimates of ( potential of vitreous silica capillaries have been obtained by Jones and Wood (1945), Wood (1946) and Wood and Robinson (1946) using this relation.

3.2.2. Behav iour in porous p lugs

Overbeek (1952, p. 204) shows that eq (3.2.5) can be applied to porous plugs provided that Xs = 0. The proof again requires that the flow must be linear and laminar, and some care is advisable on this score because in streaming potential measurements an external pressure must be applied. Although turbulence is not a problem under normal conditions, non-linearities occur very readily in porous media. Philip (1960) points out that the non-linear terms in the Navier-Stokes equation (Appendix 1.5) can be significant in

porous media at Reynolds' numbers of the order of unity. (The Reynolds number (Re) is a dimensionless number which measures the ratio of inertial to viscous forces in the fluid. Re = dvrjr] where d= density, v = velocity, r = some characteristic length, and n = viscosity. Turbulence occurs in straight pipes at Re ~ 2000.)

The effects of surface conduction wil l be discussed in Section 3.6.

3.2.3. S t reaming cur rent measurement

Instead of allowing the streaming current to establish a potential difference across the plug or capillary it is possible to measure the current directly by drawing it off through a low impedance path which short-circuits the return path through the conducting liquid. The current involved is very small ( ~ 1 0 " 1 0 - 1 0 - 1 1 amp) at low electrolyte concentration (Hurd and Hacker-man, 1955) so that electrode polarization effects are negligible, and since no appreciable current flows back through the plug or capillary, surface con­duction effects are unimportant. For a single capillary of length /, the £-potential can then be calculated from eq (3.2.3) in the form:

' = Z 3 = — - - P (3.2.7) nr r\ '

where P is the pressure gradient ( = p/l). Streaming current measurements thus afford a means of avoiding the surface conduction problem at low con­centrations. The measurement must, however, be made under conditions of essentially zero potential difference between the measuring electrodes (see Section 4.2).

We conclude this section by noting that the theory of the thermodynamics of irreversible processes can be used to show (Mazur and Overbeek (1951)) that for porous plugs as well as for single capillaries:

= [ y J (3.2.8)

and also m ( v

and

(F)E_=0

=~(&

(3.2.9)

(3.2.10)

where V' is the volume displaced per unit time per unit cross-sectional area. These equations apply even in systems involving surface conduction.

3.3. Electrophoresis

The early work in this field has been extensively reviewed by Overbeek (1950), and by Booth (1948b, 1953). More recently, Overbeek and Wiersema (1967) and Overbeek and Bijsterbosch (1979) reviewed the modern theory, including the first computer solution of the problem. An extensive discussion is given by Dukhin and Deryaguin (1974) in their monograph, but since then a quite new approach to the problem has appeared (O'Brien and White, 1978).

3 . 3 . 1 . The S m o l u c h o w s k i and H i i cke l equa t ions

The analysis of electro-osmosis given in Section 3.1.1 can equally well be applied to the electrophoretic motion of a large particle with a thin double layer simply by regarding the liquid as fixed, so that the particle moves in the opposite direction. From eq (3.1.3):

U E = ( A m Q ) - p - = ^ (3-3.1)

where uE is now the particle mobility. This is known as Smoluchowski's equation for the electrophoretic mobility. Soon after the publication by Debye and Hiickel of the theory of the behaviour of strong electrolytes, Hiickel (1924) re-examined the electrophoresis problem and obtained a sig­nificantly different result:

, DC 2sC n * j . uE = (47te0) • - — = — • (3.3.2)

An equation similar to eq (2) can be obtained by balancing the electrical force on the particle ( f e = Q. Ex) against the Stokes frictional resistance (ƒ„ = 6nvEar]) tó obtain:

( 3 3 3 )

where a is the particle radius. Substituting eq (2.3.37) for the particle charge, Q, gives

u =4ne0p-(l+Ka) (3.3.4) 671??

which is obviously identical with eq (2) in the limit of very small values of KO (i.e. small particles with relatively thick double layers). A more rigorous derivation of eq (2) is given by Overbeek and Bijsterbosch (1979). The action of the electric field on the double layer ions, causing the liquid to move in

accordance with eq (3.1.3), is called electrophoretic retardation because it causes a reduction in the velocity of the migrating particle. Smolchowski's treatment assumes that this is the dominant force and that the particle's motion is equal and opposite to the liquid motion. Hiickel, on the other hand, also made proper allowance for electrophoretic retardation in his analysis (see Abramson et al. (1942), p. 115). I t turns out however, that his equation-our eq (2)—is valid only for small values of tea when electrophoretic retarda­tion of the particle is relatively unimportant and the main retarding force is the frictional resistance of the medium. This is why the development of eq (4) generates eq (2) under those conditions. A detailed analysis of the situation is given by Dukhin and Deryaguin (1974, p. 59), who show that the ratio of the retardation force ( f r ) to the viscous resistance (ƒ„) is of order ica. Thus for small particles, although the retardation force acts across the whole double layer very little of it is transmitted to the particle.

That electrophoretic retardation remains important in the description of electrolyte conduction (where tea is very small indeed) must then be attributed to the fact that in that case one is interested in (i) the movement of ions of both positive and negative sign and (ii) the calculation of interaction effects for large numbers of ions. In electrophoresis we consider only the particle and regard it as isolated in an infinite medium (see Section 3.7.5, however). For large particles with thin double layers, essentially all of the electrophoretic retardation is communicated directly to the particle (Dukhin and Deryaguin, 1974, p. 59).

The reason for the discrepancy between eqs (1) and (2) was traced by Henry (1931) to the different ways in which account was taken of the electric field in the neighbourhood of the particle. Hiickel had disregarded the deformation of the applied field by the presence of the particle, and Smoluchowski had assumed the field to be uniform and everywhere parallel to the particle surface. As Fig. 3.5 shows, these assumptions are justifiable in the extreme situations of tea -4 1 and tea > 1 respectively.

(b)

Fig. 33. Effect of a non-conducting particle on the applied field, (a) KCI 4 1; (b) Kafr 1. The broken line is at a distance 1/K f rom the particle surface.

Henry showed that when the external field was superimposed on the local field around the particle the mobility could be written

2eC uE = (4TT£0) • ^— • / j (xa) = -fi Oca)

onn in. (3.3.5)

The function f t Oca) depended on the particle shape and, for a sphere, was given by:

for tea < 1

. ( K a ? 5{Ka)3 {Kaf (tea fx0ca)= 1 + — — +

16 48

'Qcaf (Kaf

~1T 96

96 96

—dt t

and for m > 1

3 9 75 330 fx (KO.) = — — - 1- „ 7 7 r ^ f

J 1 K ' 2 2KÜ 2K2 a1 K3a3

(3.3.6)

Values of the exponential integral are given in tables of mathematical func­tions (e.g. Jahncke and Emde, 1945) and values of the function fx Oca) are given in Table 3.1. Note that / i (KO) approaches 1 for small KO and § for large tea.

The smooth transition from the Hiickel to the Smoluchowski equation as Ka increases is shown in Fig. 3.6, curve a. In the transition region one must take account of not only the electrical force on the particle but also both the frictional force and the electrophoretic retardation, since all are of comparable magnitude for Ka ~ 1 (Dukhin and Deryaguin, 1974, p. 64).

TABLE 3.1 Values of the Henry (1931) correct ion factor/^Kfl) to be used in eq (3.3.5) as a func t ion

of tea ( f r o m Abramson et al, 1942, p. 121)

fx(m) KO Mica)

0 1-000 1 1-027 2 1-066 3 1-101 4 1-133

5 1-160 10 1-239 25 1-370

100 1-460 oo 1-500

3.3.2. C o n d u c t i o n ef fects 3 .3 .2 . 1 . Effect o f par t ic le c o n d u c t a n c e and shape

The effect of the conductivity of the particle can be discussed conveniently at this point because it, too, determines how the presence of the particle

X

0-75

100 1000

Fig 36 Value of Henry s (1931) function for particles of various shape- (a) Non-conducting sphere (K = 0); (b) conducting sphere (K' = oo); (c) and (d) Non-conducting cylinder <K' = 0) with axis perpendicular and parallel to field respectively (e) particle of any shape with K! = 1 K is the relative conductivity of the particle compared to that of the medium IK' = X (From Overbeek (1952) with permission.) "' 0 > '

distorts the applied electric field. A fuller discussion of the problem is given by Dukhin and Deryaguin (1974, p. 65). Henry (1931) considered the case of spheres of different conductivity and established the relation:

where

F(Ka, K') = l+ U[f1(Ka)-l]

X = (1 - K')/{2 + K') and K' = Xp/X0

(33.7)

(3.3.8)

(3.3.9) where Xp is the particle conductivity and X0 is that of the medium. For non­conducting spheres K' = 0 and F(KÜ, K') =f1(Ka). This is shown as curve a

in Fig. 3.6. For a highly conducting sphere the behaviour is unaffected for Ka < 1 since the effect of deformation of the field is only important if electro­phoretic retardation is important, and it is not for KO. < 1. As Ka increases and electrophoretic retardation becomes more significant, the mobility decreases until ultimately, at very large Ka i t falls to zero, since X = — 1 and f x (tea) approaches § (curve b).

For K' = 1, although the field is distorted by the double layer it is not at all distorted by the particle, and since this corresponds to Hückel's model it is not surprising that the Hiickel equation is then valid for all Ka (curve e). For non-conducting cylinders aligned parallel to the field, the field lines run parallel to the surface irrespective of the value of Ka, so Smoluchowski's equation is valid in that case for all Ka (curve d). Finally for cylinders oriented perpendicular to the field, Gorin (1939) has extended Henry's preliminary calculations and his conclusions are shown as curve c. Although the effects of high particle conductivity have been demonstrated (Henry and Brittain, 1931), it is doubtful whether in ordinary circumstances there is any need to take it into account. Even metallic dispersions appear to exhibit normal electrophoretic behaviour, which Overbeek (1952, p. 209) attributes to the effects of polarization. Unless there is evidence to the contrary, then, all particles may be treated as insulators. The one exception is the liquid metallic dispersion (mercury), and this wi l l be discussed in Sections 3.3.3 and 3.7.

3.3.2.2. Effect of sur face c o n d u c t a n c e As in the case of streaming potential and electro-osmosis, the effect of specific surface conductivity Xs can be introduced by modifying the conductivity parameter X to read

• \-K'-2k'Ja 2 + K' + 2X'Ja

where X's = XJX0 (3.3.10)

This expression was established by Henry (1948) and a similar one proposed by Booth (1948a) but is seldom used because the modern theory of electro­phoresis takes the surface conductance into account by dealing explicitly with the mobility of the double-layer ions. Provided the surface conductivity is "normal" (i.e. equal to what would be expected from the known concen­trations and mobilities of ions in the double layer), its effects are taken into account in the treatment of the relaxation effect. We shall postpone examina­tion of this more elaborate treatment until Section 3.7. I t must be noted, however, that the use of the simple equations of Hiickel and of Smoluchowski wil l lead to significant errors ( > 5 %) except for values of Ka < 0-1 and Ka > 500 respectively, especially for large values of C

3.3.3. Electrophoresis of emu ls ions

I f the electrophoretic particle is a fluid of finite viscosity (e.g. an emulsion droplet or an air bubble), it is necessary to consider how its motion is affected by the motion of the fluid within the particle. Such motion would affect the velocity field in the surrounding fluid (for example, because the tangential component of the velocity at the surface of shear need no longer be zero). Not only would the Stokes friction be reduced but the electrophoretic re­tardation and relaxation effects would also be affected (Overbeek and Wiersema, 1967).

A detailed theoretical analysis was given by Booth (1951), assuming various possible charge distributions in the particle but neglecting the relax­ation effect. His results are reviewed briefly by Overbeek and Wiersema (1967), and we shall not discuss all of them again here because there is con­siderable doubt about their applicability. I t seems that in most cases the fluid particle behaves more like a solid than a liquid, especially if there is surfactant present in the solution (as is usually the case). Thus Linton and Sutherland (1957), from their microscopic observations of drops of a few millimetres diameter, concluded that even a trace of surface active impurity was sufficient to prevent internal circulation. Likewise, Anderson (1959) did not observe a change from solid to liquid behaviour when the electrophoretic mobility of «-octadecane was measured in surfactant solution at temperatures below and above the melting point of octadecane. Finally, although Dery­aguin and Dukhin (1971) adduced theoretical evidence that a liquid drop should behave very differently from a solid, even in the presence of sur­factants, they subsequently concluded (Dukhin and Deryaguin (1974) pp. 310-314) that the experimental evidence did not in general support such a conclusion.

The above remarks apply to an insulating particle. For a conducting f luid particle the behaviour can be very different indeed. Booth (1951) derives the following equation:

2eC «* = ! -

'3n'(l + A) + 2i7(l -22.) 3rj' + 2n (3.3.11)

for small potentials and KÜ |> 1. Here ri is the viscosity of the fluid drop and X is defined in eq (3.3.9) or (3.3.10). To investigate the behaviour of charged mercury drops we set X = - 1 in eq (11) (corresponding to K' - X P / X 0 -* oo in eq (9)). Then:

4e[ M B O O U , = „ , , ~ > (3.3.12) {3n +2n)

This same problem has been examined from an entirely different standpoint by Frumkin (1946) and Frumkin and Levich (1946) (see e.g. Levich, 1962) who arrive at a very different result:

(3*7' + 2rj + oi/X)

where ae is the charge density in the shear plane. The mobilities predicted by these two equations are in the ratio (w^Levich:

("E)Booth a s Ka:4, and since Ka can be in excess of 1000 the discrepancy is enormous. The much higher mobility predicted by Frumkin and Levich and observed in Frumkin's experiments is attributed by them to an increased mobility of the liquid mercury, brought about by electrical forces which act in different directions in different parts of the drop. Levine and O'Brien (1973) have made a detailed comparison of the two approaches and conclude that the failure of Booth's treatment in this case can be attributed to his not allowing for the possibility of a redistribution of charge on the surface of the mercury drop. This redistribution results in a significant departure from superposition of the equilibrium electric field (due to the particle) and the applied field. The result is a large additional electric force on the particle which accounts for the much higher mobility observed.

The Frumkin-Levich approach has been extended by Dukhin, Semenikhin and Deryaguin in a series of papers which wil l be referred to again in Sections 3.7 and 3.8.

3.4. Sedimentation potential

When a particle is sedimenting through a liquid there occurs a continuous flow of ions from the bulk solution into the lower half of the particle double layer. These ions flow around the particle surface and return to the bulk from its upper surface. The net effect is that the sedimenting particles constitute a current in one direction whilst the ion flow is a current in the opposite direction, and in the steady state the two are equal.

An approximate relation can be derived by equating the two currents:

nv:Q = E . X Q (3.4.1)

where n is the number of particles per unit volume, of charge Q settling with velocity v„ and X 0 is the bulk conductivity. Substituting for vz from Stokes's Law:

v: = 2a2{d-d1)g/9t1 (3.4.2)

where d and d1 are the densities of particles and medium respectively, and for Q from eq (2.3.37) gives:

_(4ns0)2DCa3(d-dl)gn(l + KO)

- 9^0 ( 3 A 3 )

Smoluchowski's (1903) more rigorous analysis gives:

(4ne0)DCa3(d-dl)gn h - = 5 1 3.4.4)

which coincides with eq (3) for Ka = 0-5. Equation (4 ) is valid for a solid insu­lating particle with Ka > 1 and in the absence of surface conduction. A more complete analysis, for arbitrary values of Ka is given by Booth (1954).

The electric field E. is measured by inserting reversible electrode probes at two different heights in the column of settling particles and then £ . = A 0 / L where A0 is the potential difference and L i s the separation between the electrodes. Equation (4) suggests that Ez is proportional to the amount of material between the electrodes and this has been confirmed by Quist and Washburn (1940). Elton and Peace (1957) used an equivalent version of eq (4) to determine the particle size distribution in a sample of glass spheres and two samples of carborundum. Their method was based on the idea that as the larger particles settle below the bottom electrode they no longer contributed to the sedimentation potential. The resulting size distributions were rather narrower than those determined more directly, possibly because too high a volume fraction of particles was used.

When the falling particle is a conducting liquid, the same considerations as were discussed in Section 3.3.3 apply. The velocity of the falling drop is not given by eq (3.4.2) but by (Frumkin and Levich, 1946):

= 2a2(d-d1)g 9>7

> + 3g + <T2/A0

2n + 3r]' + <T2/A0_ (3.4.5)

The corresponding value for E, is then, in our notation (Frumkin and Bagotskaya, 1948):

_(4ns0)DCa3(d-d1)gn : 3A 0(2^ + 3?7' + ( 7 2 M 0 ) ( i A 6 )

which has been applied to the fall of mercury drops through glycerol. More recent developments of this analysis wi l l be discussed in Section 3.8.

II. MORE RECENT DEVELOPMENTS

3.5. Electro-osmosis

3 . 5 . 1 . Flow in narrow capillaries

3 . 5 . 1 . 1 . Slit-shaped cross-section The most important assumptions in the derivation of eq (3.1.3) for the electro-osmotic mobility are: (i) that the capillary surface is flat and (ii) that the double layer is able to develop fully so that the potential in the middle of the capillary is zero. For a slit-shaped capillary of depth 2h assumption (i) is true and (ii) wi l l also hold provided xh is large enough. At first sight one might expect a value of Kh > 5 to be ample since the potential in the mid plane is then small compared to the zeta potential, but this is not so. The calculations of Burgreen and Nakache (1964) show that the correction to the Smolu­chowski eq (3.1.3) is significant even for KH values of about 10.

Burgreen and Nakache used the complete Poisson-Boltzmann equation for the potential and placed no restriction on the value of 2KL The potential profile for small h when the double layers from both sides of the capillary overlap is the same as that between two approaching flat colloidal particles undergoing coagulation. That problem is treated in detail in the Deryaguin-Landau-Verwey-Overbeek (DLVO) Theory of Colloid Stability (Verwey and Overbeek, 1948), a brief outline of which is given in Appendix 5. The potential profile is symmetric about the median plane and, for high (-potentials, is expressed in terms of elliptic integrals of the first kind. For small potentials or in the case of small degrees of double layer overlap there are adequate approximate expressions available (see Appendix 5).

In the more complete treatment, equation (3.1.2) is replaced by (see Appendix 1.5):

d2t\i d2v. dp / o c i v

where p is the hydrostatic pressure, which may include an externally applied pressure or merely be the consequence of the osmotic flow, if the flow is impeded. The field strength, E., is also the result of externally applied and locally generated fields. The mean velocity, v, is found to be:

- v = - ^ . ^ . + i.Etai-G(zr,Kh)] (3.5.2) 3rj dz r]

where Z= ei\kT. The first term is the normal hydraulic conductivity term

and the next is the classical value of vE given by eq (3.1.3). The correction Junction, G, is defined by Burgreen and Nakache as:

1 fKhé G(C,Kh) = — ^d[K(h-x)]

(3.5.3)

where \J,{x) is the potential profile in the capillary and ^ is thus the integrated mean potential over the capillary cross-section. As noted above the value of HW can only be accurately expressed in terms of elliptic integrals but i f the potential is very small everywhere the approximation (Overbeek 1952 p. 249) (see Appendix 5):

cosh K(h - x) cosh Kh (3.5.4)

U j 1 1 4 8 12

ZKh H f l d ^ h n o ™ 1 ™ , / c o r r e c t i o n f u n c t i o n G(zl Kh) with electrokinetic radius. (From Hildreth (1970) with permission, © Amer. Chem. Soc.)

may be used. Substituting eq (4) in eq (3) and integrating gives

G = ^ L (3.5.5) Kh

under these conditions. Thus G ranges from unity, when h = 0, to zero for very large Kh. At modest values of Kh ( > 3), G is given to better than. 1 % by G = 1 /K/Z for small potentials. Figure 3.7 shows the exactly calculated value of G as a function of Kh for various value of I, the reduced (-potential. As would be expected, G only falls below about 0T for values of Kh greater than 10, for small values of f.

The approximate equation (5) corresponds almost exactly with the curve for Z= 1 but obviously grossly overestimates the correction for larger (-potentials, especially at small values of the electrokinetic radius.

The value of JC, in distilled water, is of order 3 x 104 cm" \ so that even in that case significant corrections to the classical theory are not required until the slit width decreases below about 5 um. For higher salt concentrations the slit width can be still further reduced before recourse must be had to the more elaborate calculation. The proper description of flow through oriented plates of clay minerals like vermiculite, however, would require this more exact treatment.

Another approximate expression for G could be obtained for the case when the double layers overlap only slightly. Then the potential profile is approximately given by a linear superposition of the potentials due to the two separate surfaces. An extension of this approach was used by Sigal and Alekseyenko (1971), who used the result obtained by linear superposition as a first approximation, f 0 , to the solution of the Poisson-Boltzmann equation. Then, expressing the true solution, ƒ in the form

ƒ = f o +fi

they set about obtaining an accurate approximation for f x by substitution in the original equation with appropriate boundary conditions. Their result is described by Dukhin and Deryaguin (1974, p. 111). The potential profile can be expressed in terms of elementary functions, though the expression is far from simple, and its integration to yield an analytical expression for G, even if possible, would be algebraically very complicated. (Note that both r and z in Dukhin and Deryaguin's formulae should be replaced by the number 2.)

3.5.1.2. Cy l indr ica l c ross-sec t ion For capillaries of cylindrical cross-section the potential profile is more diff i -

cult to evaluate. It must satisfy the cylindrical form of the Poisson-Boltzmann equation (c.f. eq (2.3.38)):

1 d ( d\j/\ Inez y-y dy\ dy

'-smh(ze\j//kT) (3.5.6)

for which, unfortunately, there is no exact solution in closed form. Approxi­mate analytical solutions can be obtained using the method of "joining of solutions", in which separate solutions are obtained for the region near the wall and for the outer part of the diffuse layer and then a matching performed on the boundary surface between the two solution regions. Both the method and its limitations are discussed briefly by Dukhin and Deryaguin (1974), p. 132), and we shall not consider it further since most of the electrokinetic work has been confined to the case where ( is not too large. Equation (6) can then be linearized (Rice and Whitehead, 1965) by setting sinh zei///kT = z&j/jkTso that:

1 d( é / f \ (3.5.7)

y dy\ dy

The solution of this equation is:

i// = BI0(Ky) (3.5.8)

where Tiis a constant and I0 is the zero-order modified Bessel function of the first kind.

At y — r we require that \jf = £ and so:

-fo(KV)

I0(Kr)

where y is measured from the capillary axis and r is the capillary radius.

1 0

(3.5.9)

l og Kr Fig. 3.8. Values of the correction function F(Kr) (eq (3.5.10)) as a function of log Kr. (From Rice and Whitehead (1965) with permission, © Amer. Chem. Soc.)

The velocity profile is obtained by solving the equation for fluid motion in the capillary, and when this is integrated across the tube Rice and White­head (1965) find that equation (3.1.4) for the electro-osmotic volume trans­ported must be modified by multiplying the right-hand side by the correction factor:

F{KV) = 1 2 / 1 ( K T )

KrI0(Kr) (3.5.10)

where Ix is the first order modified Bessel Function of the first kind. Values of this expression, as a function of Kr are shown in Fig. 3.8.

Rice and Whitehead also attempted to calculate the corrected value of the ratio Vji (cf. eq (3.1.6)) and arrived at the relation:

V i

where

and

F,(KrJ)

Fircr)

F{Kr) • n(rcr) lliKr)

(3.5.11)

P = »X0

Unfortunately, as Dukhin and Deryaguin (1974, p. 118) point out, they used an expression for the current which is not strictly correct. The total current 1= It + Ic where I, is the current transported by electro-osmotic flow and Ic is the conduction current. Rice and Whitehead set

I, = 2n

and

v.(r)p(r)rdr

X0E,A

(3.5.12)

(3.5.13)

where A is the cross-sectional area of the capillary. Equation (12) is un­objectionable but equation (13) ascribes a constant conductance to the liquid across the capillary and makes no allowance for the modified ionic concentrations in the double layer near the walls. Such a procedure is really only satisfactory when the pores are so large that the conduction current is essentially carried by liquid which is, overall, electrically uncharged and this is certainly not the case in fine capillaries.

I f one wishes to use the known mobilities of the separate ions, one must take explicit account of the separate ion concentrations across the capillary,

as given by the Boltzmann equation. The axial current is then given by (Morrison and Osterle, 1965; Churayev and Deryaguin, 1966a):

ƒ « / , + / , - [vs(r)p(r) + X 0 E . cosh $(r)]dA (3.5.14) A

where X 0 ( = 2bzen0) is the bulk conductivity of the liquid and b is the mobility of both cation and anion (assumed equal). Unfortunately, the analysis in this case has not yet been carried out but the corresponding development for a slit-shaped capillary wil l be treated below.

An alternative method has been proposed by Oldham et al. (1963) and wil l be discussed when we come to examine the streaming potential, since that is the case they treated. A separate treatment of electro-osmosis is unnecessary because the equality of the ratios EJp and Vji (see Section 3.2) is quite general (Mazur and Overbeek, 1951). Irmer (1973) has also analysed the electro-osmotic flow through cylindrical capillaries of various sizes.

3.5.2. Flow in porous plugs Narrow capillaries of uniform bore are very difficult to construct, and most of the interest in the analysis of flow in narrow capillaries stems from attempts to understand the behaviour of porous plugs or membranes with small pore size. Where the plug or membrane can be considered to be composed of a collection of parallel tubes of uniform cross-section it is obvious that the analysis for a single capillary can readily be extended. Since both volume output and electrical current are in parallel, eq (3.5.11) wi l l be unaffected.

The more usual type of plug, consisting of granular material, is a little more difficult to deal with and wil l be considered separately when discussing streaming potential (Section 3.6.2).

3.6. Streaming potential

3 . 6 . 1 . F low in na r row capi l lar ies

3 .6 .1 .1 . S l i t - shaped c ross-sec t ion Burgreen and Nakache (1964) continued their analysis of electrokinetic processes in thin slit-shaped pores (see Section 3.5) by calculating the stream­ing potential. They showed that the classical expression must be modified by multiplying the right-hand side by a correction factor so that

where

and

„ 1 - G(zt, Kh) F2 = -

1 + IJIc

4j8 (3.6.2) 7 r ^ r h © - c o t 0 o + £ ( 0 o ' ^ 4 ^

I, and Ic are again the transport (convection) current and the conduction current; /? was defined in connection with eq (3.5.11); s= 1/cosh (z# e/2) where \j)c is the reduced potential at the centre of the slit and sin 0O = cosh (zt£ c/2)/cosh(zf/2); Eis the elliptic integral of the second kind.

Burgreen and Nakache (1964) plotted the function F2 against the electro­kinetic radius 2Kh for various values of z f a n d p\ The values chosen for ft were, however, rather unsatisfactory (/? = 1 and 10 for z f values of 1, 4 and 10). The parameter /? is related to I by definition:

Y]XQ

nX0

Taking typical values for aqueous solutions at room temperature, p « 0-3z2^ and so we wil l use the value adopted by Hildreth (1970)—0-338z2p —for purposes of illustration.

In deriving their equation (our eq (1)), Burgreen and Nakache (1964) also used eq (3.5.13) for the conduction current Ic, so that their analysis is unsatisfactory when the conductivity varies significantly across the capillary bore. I t has recently been extended by Hildreth (1970), who also introduces the possibility of different mobilities for cations and anions. Hildreth's expression for F2 is:

_A0 A 0 y cJ

where A r/A 0 respresents the mean conductance of the fluid in the capillary when the ions have equal mobility and X B / X 0 is the contribution due to dif­ferences in ion mobility.

To obtain an idea of the importance of these conductance effects we first set the ionic mobilities equal so that X B / X 0 = 0 and then, for moderate to large values of Kh the ratio X R / X 0 is given by Hildreth as:

^ = i + ^ / c o s h z r _ \ ( 3 6 5 )

X 0 Kh\ 2 J

When this is substituted into eq (4) it is apparent that the Burgreen and Nakache relation for F2 is altered by the addition of the term 2(cosh — 1)/ Kh in the denominator. That this has a substantial effect on the result is clearly demonstrated in Fig. 3.9. The ful l lines were calculated by Hildreth from eq (4) with Xb = 0 and the broken line is calculated from eq (2) for zl = 4 and /? = 0-338z2£"2 = 5-41. The use of the complete eq (3.5.14) for the current is obviously essential to obtain an accurate assessment of the effect, especially at low values of Kh.

The importance of the individual ionic mobilities is demonstrated in Fig. 3.10 for the same value of ^and /? and various values of y' = (b_ — b + )/ (b_ + b+) where b± is the ionic mobility. As Hildreth points out, the effect can be greatly magnified or diminished by suitable choice of the ionic

1-0

0-5

0-2

0-1

0-05

0 0 2

0-01

0 -005

0 -002

0 -001

0 - 0 0 0 5

0 0 0 0 2

0 - 0 0 0 1

0 2 4 6 8 10 12

Fig. 3.9. The correction factor for the streaming potential. The full lines are for the more exact theory of Hildreth (1970) with equal ionic mobilities. The broken line is f rom eq (3.6.2) and the dotted line is f rom eq (3.6.8), each for z£ = 4. (Adapted f rom Hildreth (1970), © Amer. Chem. Soc.)

0-7

0 - 6 k -

0 - 5 r —

0-4

0 - 3 r —

0-2 r—

0-1 t—

Z K / I

Fig. 3.10. The effect of anion and cation mobility on measured streaming potential. ( / (ft_ - ft+)/(ft_ + ft+)). (From Hildreth (1970) with permission, © Amer. Chem. Soc.)

mobility ratio in the salt. Values of ƒ equal to + 1 or - 1 are obviously impossible but even for modest variations in cation and anion mobility one could observe variations in streaming potential over a factor of about 3 for fine capillaries.

An alternative treatment of the behaviour of a slit-shaped capillary, taking account of eq (3.5.14) for the current, has been given by Churayev and Deryaguin (1966b) but only for the case of low potentials. They used eq (3.5.4) to describe the potential profile and obtained, for the streaming potential, an expression like eq (1) in which the correction factor was given by:

Kh — tanh Kh p72[tanh Kh - (K /Z /COSI I 2 Kh)] + Kh{\ + 4)

(3.6.6)

where 4 = ba0jl0h; b is the ionic mobility; <r0 is the surface charge density. 4 measures the ratio of true surface conductance effects (i.e. those due to Stern layer ions) to bulk conduction mechanisms, and Churayev and Deryaguin concluded that it is usually much less than unity for the simple Stern layer model they chose. Bearing in mind eq (3.5.5) we can write eq (6) in the form

1 - G ^3 = g (3.6.7)

1+4 + yG- l/(cosh 2K/z]

In a more elaborate treatment of the conductivity effect Dukhin and Dery­aguin (1974, p. 116) have established, again for the case of low potentials, a relation which would require that the correction function be written:

F3 = « — — (3.6.8) 1 + —{G + 1/cosh2 Kh) + £[G - 1/cosh2 KK\

(The actual function given by Dukhin and Deryaguin as equation (104) is not consistent with their equation (99). Assuming that the latter is correct, their function 3>3 should read (for <5 = 0)

^ \ i_ r j i 7i/sinh 2K/Z + 2KK\ d>3 = cosh xjjdx = Kh + l 2 \ — — u 2 3.6.9)

Jo V Scosh Kh )

and this is the relation we have used to write eq (3.6.8).) Equation 3.6.8. proves to be a good deal closer to the exact solution than

that of Burgreen and Nakache as shown by Fig. 3.9. Although eq (3.5.4) is used to derive eq (8), its limitations for f = 4 are apparently much less important than the failure to take account of the conductivity in the double layer near the capillary walls. For small potentials and moderate to large values of Kh, eq (5) (Hildreth, 1970) would yield for the conductivity term (1 + l2/4Kh), and it is apparent that at large values of Kh, the conductivity term

r z2 i 1 + j(G + 1/cosh2 Kh)

tends to this same limit. Equation (1) in its complete form, with eq (4) substituted, is not very

easy to use, although interpolation from Fig. 3.9 should be accurate enough for most purposes for aqueous solutions at room temperature. For the complete expressions the reader is referred to Hildreth (1970). To use Fig. 3.9 to calculate Z from the measured value of EJp one would have to

assume a value for F2 and calculate a first approximation to Z, then proceed by successive references to Fig. 3.9 until the value converged. Unfortunately in most cases the result wil l not be unique for narrow capillaries, as is clearly shown by Fig. 3.11, where we have plotted the function F2Z, as interpolated approximately from Fig. 3.9. According to eq (1) this should be proportional to the streaming potential so i t is apparent that a particular value of the ratio EJp can be associated, at any particular value of Kh, with two possible (and very different) values of the zeta potential.

0-05

0-02

0-01

Fig. 3.11. F2%as a function of I, for various values of Kh, interpolated approximately from Fig. 3.9. Unique values of l cannot be associated with particular values of EJp (oc F2l).

A method of overcoming this problem has been suggested by Oldham et al. (1963). Although they treated only the case of cylindrical capillaries, the method can be extended to capillaries of any shape. A general expression for the current per unit area through a slit-shaped capillary is given by Hildreth (1970) as:

where E. is the electric field due to the streaming potential and P is the

pressure gradient. When 1=0, as is required for a streaming potential measurement, we have

£ z _ - E s _ _ £ £ _ ( l - G )

p~p~rt*W^K^S (3'6'n)

\A0 A0 IJ

which is the expression obtained when eq (4) is substituted in eq (1). If, however, we measure independently the conductivity of the liquid in the capillary AC!Lp using, say, an a.c. measuring technique under zero pressure gradient, we obtain a value for

We can, therefore, write eq (11) in the form:

Es <£ p nK

( l - G ) (3.6.13)

where the function ( l - G ) takes account of the change in the formula for the flow current in a narrow capillary, and A c a p takes account of the change in the conductance (cf. Dukhin and Deryaguin, 1974, p. 118). Since G(z£ KK) is a monotonic decreasing function of £ the product Z(\ - G) wi l l always increase with f and the ambiguity which one encounters when using eq (11) alone is removed.

Levine et al. (1975) were evidently unaware of Hildreth's work and have extended Burgreen and Nakache's analysis in essentially the same way, and with very similar results. Their work wil l be discussed again in connection with the electroviscous effect (Chapter 5).

3.6.1.2. Cy l indr ica l capi l lar ies Rice and Whitehead (1965) in their analysis of the streaming potential developed in cylindrical capillaries gave, for the correction factor in eq (1), the expression

F2 = F1(Kr, fi)

where Ft(Kr, /?) is given in eq (3.5.11). This result is, however, inadequate on two counts: firstly, it would hold only for low values of the potential; secondly, it does not take proper account of the current flow in the capillary. Again, one can overcome the second problem by making a direct measure­ment of the conductance of the liquid in the capillary and using that in

eq (3.6.1). The streaming potential is then given by:

F(Kr) (3.6.14) P Vkap

where F(KT) is given by eq (3.5.10) and Fig. 3.8. In order to extend this equation to the region of high potentials it is

necessary to adopt this same procedure of measuring both the streaming potential and the actual conductance in the capillary, at the same time using a better description of the potential profile in the capillary. This task was undertaken by Oldham et al. (1963), who used a power series representation for the potential satisfying eq (3.5.6). Their result can yield an alternative expression for F(KY) in eq (14):

1 » a„(Kr)2n+~2

(Kr)2zZni n + 1 (3.6.15)

where the coefficients a„ are tabulated by Oldham et al. (1963) in terms of sinh and cosh functions of the leading term a0. A comparison of the alterna­tive expressions for F{r) from the theory of Rice and Whitehead (1965) and that of Oldham et al. (1963) is given in Fig. 3.12, from Dukhin and Deryaguin (1974). As would be expected the two results are in fairly good agreement for values of I less than about 2, i.e.

50 £ <—mV

z

k

1 2 I I I I I I I I I I I I I I

/cr = oo

1 0

0-8 * r = 1 0

0 - 6 - Kf-5 —

0 - 4 — /cr = 2 — -

0 - 2 — -^r = 1 _ — « " * "

0 I I L - ^ l l I I I I 10 40 0-1 0 -4 1 0 4

I = zt,/kT Fig. 3.12. Comparison of the correction function F(KT) from the work of Oldham et al. ( 1 9 6 3 ) (heavy lines) with that from the theory of Rice and Whitehead ( 1 9 6 5 ) (fine lines). (From Dukhin and Deryaguin ( 1 9 7 4 ) with permission.)

100

Oldham et al. (1963) applied their treatment to existing literature data for which A c a p was not available. They therefore approximated its value by using equation (3.2.5) in the form:

is, sC FUcr, O - = ~ tl v (3- 6 - 1 6 ) p r\ ( A 0 + 2AJr)

and estimated values of the specific surface conductivity, As, from other literature data (McBain and Foster, 1935; Klinkenberg and van der Minne, 1958). In this way they were able to show that the streaming potential data of White, et al. (1936) and that of Rutgers, et al. (1959) which appeared to show a C value which depended on capillary radius, could be explained using eq (16) with a ( value independent of the radius.

Equation (16) implies that the surface conduction itself is a constant, independent of the radius of the capillary. In a later paper Morrison and Osterle (1965) showed that this, too, is an approximation and that when due account is taken of the potential distribution in a capillary, the specific surface conductivity is given by:

r , D2

r'lcosh $ - \)dr' + (4ne0)2— o W

- (4ns0)2-^-rjr o

(3.6.17) Using the values of i^(r') obtained by numerical solution of the cylindrical Poisson-Boltzmann equation, Morrison and Osterle (1965) calculated values of As as a function of capillary radius for water in glass. Their results are shown in Fig. 3.13, which shows that As becomes fairly constant for capil­laries of radius about 0-1-1 um.

3.6.2. E lect rok inet ics in po rous p lugs

As pointed out above the treatment of electrokinetic phenomena in porous plugs presents some problems in situations where surface conduction be­comes important (i.e. at low concentrations). The situation is discussed at considerable length by Dukhin and Deryaguin (1974, p. 143 et seq.), and an excellent summary of the literature is given there.

The first attempt to tackle the problem was made by Briggs (1928), who defined the surface conductivity, Ks, as the difference between the conductivity of the liquid when it was in the plug or diaphragm, Ap, compared to its value outside, A0:

Ks — Ap — A0 (3.6.18)

Radius / m

Fig. 3.13. Specific surface conductivity of water in glass as a function of capillary size. (From Morrison and Osterle (1965) with permission.)

To determine Ap one must establish the "cell constant" of the diaphragm, and this is done by measuring its resistance, R°, when filled with a liquid of high salt concentration ( > 0-1 M ) with conductivity A0 (when the surface conduction can be assumed to be negligible). The value of Ap at other con­centrations is then given by:

A°R° A P = — (3-6.19)

vexp

where i ? e x p is the measured resistance. Briggs was able to show that if one used the measured value of -Ap to express the conductivity in equation (3.2.5), one obtained a (-potential which, at constant electrolyte concentration, was independent of the density of packing of the plug. On the other hand, if one simply used the value of A0 and ignored As, the apparent ( value decreased with decrease in the pore size of the diaphragm. Similar effects are observed in the data of Bull and Gortner (1932) and Samartsev and Ostroumov (1950).

The most common manifestation of surface conduction effects occurs when a plot is made of the dependence of apparent ( potential, (i.e. the value obtained directly from equation (3.2.5)) on concentration. Whether the data is obtained by measurements in diaphragms or in capillaries, the surface conduction effects produce a maximum in the absolute value of ( ' at a concentration of about 1 0 _ 3 M , since below this concentration the surface conduction becomes increasingly important. I t adds greatly to the

J _ 5 0

I" Concentration L mol r'

Fig. 3.14. Dependence off-potential of collodion membranes of different pore size on the electro­lyte concentration f rom the work of Zhukov and Fridrikhsberg (1950). Contiguous lines show values calculated by the Smoluchowski formula; broken lines are corrected for surface conduc-tion using the Briggs (1928) procedure. Curves 1-7: Radius (in nm): 28 4 9 14 25 98 257 (From Dukhin and Deryaguin (1974) with permission.)

back flow of current and so reduces the magnitude of the streaming potential which can be built up. When eq (3.2.6) or eq (19) is used for the conductivity, the calculated (-potential usually shows a monotonic decrease in absolute value with increase in electrolyte concentration as is shown by the work of Zhukov and Fridrikhsberg (1950), reproduced in Fig. 3.14. The fact that the curves for different porosities all merge into one curve suggests that this procedure eliminates the apparent dependence of ( on pore size and ade­quately accounts for surface conduction effects.

There does remain, however, an element of doubt about this procedure. Whilst the extension of eq (3.2.6) to bundles of parallel and equal capillaries is quite straightforward (Overbeek and van Est, 1953), it has been shown that its application to a set of capillaries of different radii connected in series/ parallel leads to low values of ( (Overbeek and Wijga, 1946); Rutgers and Janssen, 1955). This latter model appears at first sight to be a better way of describing a porous medium. Fridrikhsberg has, however, been able to establish, using methods similar to those of Overbeek and Wijga (1946), that if the capillaries are conical in shape, so that there are no rapid changes in radius, the error in using eq (19) is not serious (Dukhin and Deryaguin, 1974, p. 150). Although this result has been challenged by Tikhomolova and Boikova (1974) it is not clear that the model on which their calculations are based (Tikhomolova, 1974) is sufficiently general to justify their conclusion (see below).

A more elaborate method of correcting for surface conduction has been proposed by Ghosh and his co-workers (Ghosh, 1954; Ghosh et al, 1954). They suggest that the apparent (-potential, (', is related to the true value by an equation of the form:

£ - l + - - 2 » - i (3.6.20)

This corresponds to writing the surface conductance in eq (18) in the form:

r

instead of the usual expression for a cylindrical capillary which would require a = 1. Ghosh (1955) interprets the constant a as a correction for pore shape: it may be thought of as the ratio of the surface area of a particle to the surface area of a pore, per particle. Ghosh et al. (1953) have shown that a relation of the form of eq (20) can be derived by applying certain restrictions to the equations derived by Overbeek and Wijga (1946) for series/parallel capillary combinations. (They assume that large pores wil l be more likely to collapse and so wil l be shorter.) An alternative, but closely related, theoretical inter-

pretation of the parameter ot is given by Tikhomolova (1974). By using measurements on porous plugs of different (but uniform) particle size and plotting 1/C' against l/rt where rp is the particle radius (assumed proportional to the pore radius), Ghosh et al. (1953) showed that a linear relation could be obtained (Fig. 3.15) using the data of Bull and Gortner (1932). From the intercept they obtained the true value of £ for the particular electrolyte concentration involved (2 x 10~ 4 MNaCl). I f one assumes rp/rx5, then the slope of their curve corresponds to a Xs value of about 1-2 x 10" 9 o h m " 1

for the quartz-solution interface, and this is in excellent agreement with the calculations cited by Overbeek (1952, p. 237).

T

Fig. 3.15. Determination of true ([-potential. £' is the apparent value and rp is the radius of particles in the plug. (From Ghosh, Rakshit and Chattoraj (1953) with permission.)

More recently Ghosh and Ghosh (1958) and Ghosh et al. (1965) measured (' on a series of diaphragms with varying particle size and at various electro­lyte concentrations. Their plots of l / ( ' against \/rp (or (Xp — X0)/X0) were linear and led to values of ( in agreement with those found by Wijga (1946) for capillaries of the same material. Tikhomolova and Duda (1974) per­formed similar experiments on crushed quartz diaphragms with aqueous KC1 solution and concluded that the Ghosh procedure could produce re­liable values of (-potential with a values of the order of 1-25-1-35. The Fridrikhsberg procedure—corresponding to the use of eq (19)—gave results which were satisfactory in all but the most extreme cases, with errors in ( o f less than 10%.

3. T H E C A L C U L A T I O N O F Z E T A P O T E N T I A L 95

Though further evidence on the efficacy of this procedure would be useful (Dukhin and Deryaguin, 1974, p. 152) it seems that, at least for values of Kr > 5, measurements on plugs of granular material of regular shape and fairly uniform particle size can give reliable results. Of course, there may be cases where the crushing process itself leads to different ( potential values (Dukhin and Deryaguin, 1974, p. 148) and in that case the Ghosh procedure cannot be applied. One is then forced to use the simpler (Briggs) procedure, employing eq (19). This wi l l probably give satisfactory results if the material forms a plug with fairly uniform (preferably cylindrical) pores.

One can always overcome the surface conduction problem itself by meas­uring the streaming current, since surface conduction is not involved in that case (Hurd and Hackerman, 1955). There are no problems involved in measurements on single capillaries, and the success of the Briggs method and the work of Zhukov and Fridrikhsberg and Ghosh et al. suggest that i t could probably solve most of the problems in measurements on porous plugs, provided they are composed of particles of granular material of fairly uniform size and shape.

For measurements on compressible plugs, especially those composed of fibres, a more elaborate procedure has been suggested. Neale (1946) has shown that eq (3.2.3) should be replaced by:

/s= AP(1 - <p)Xr (3.6.21)

where (j> is the volume of solids per unit volume of plug, A is the cross-sectional area of the plug, and T c is an orientation factor which depends on the average value of cos20 where Q is the angle between the pore direction and the axis of the plug. T C = T ~ 2 where T is the tortuosity ( = effective path length/length of plug) as that term is used in the theory of flow through porous media (see e.g. Scheidegger (1957) p. 93 et seq.). Neale took for T C

the value 0-79, as suggested by Kraemer (1942) for a plug of cylindrical fibres oriented at random. Goring and Mason (1950) have since used an essentially identical relation with T ? = 0-5, the figure suggested by Sullivan and Hertel (1942) for randomly packed cylindrical particles at void fractions below 0-8.

The factor (1 — 0) in eq (21) is the porosity or effective void fraction, y, and Goring and Mason (1950) compared the value obtained from eq (21) with that obtained from hydrodynamic measurements using plugs which could be compressed to different degrees. Their results were not concordant, and in a subsequent paper Biefer and Mason (1959) showed that the dis­crepancy could be removed if one recognized a dependence of T { on the porosity. They established the empirical relationship:

x; = by3'2 (3.6.22)

for 0-5 < y ^ 0-8. Combining eqs (21) and (22):

• = - Cby5'2 (3.6.23) sPA

Plots of the function [Isn/ePA]215 against porosity (in the range 0-5 ^ y ^ 0-8) could be used to evaluate the product b£, but Biefer and Mason (1959) claimed that absolute values of £ could be obtained only by making some rather arbitrary assumptions about the influence of packing on

The parameter b which proves such a problem in this case is a measure of the extent to which the geometry of the pores is modified when a plug of fibres is compressed. The problem is much less severe for plugs composed of incompressible, near-spherical particles where one would expect the pore orientations to remain fairly constant as the porosity is altered, since this can be done only by altering the breadth of the size distribution of particles.

I t might be thought that the orientation factor could be determined by electrical measurements, but unfortunately this is not so in the general case, although there is evidence (see below) that it is a reasonable procedure for plugs made of granules of fairly uniform particle size and possibly also for fibres. The problem is that the tortuosity of the path for electrical conduction is not the same as that for fluid flow and the difference depends upon the detailed geometry of the system (Philip, 1957). Biefer and Mason (1959) show that the corresponding orientation factor derived from conductance measurements for their fibrous plugs is proportional to the porosity, y (c.f.eq(22)):

t c = ky (3.6.24)

where k = 0-93 for the systems they studied. Conductance measurements at high concentrations, where surface conduction is negligible can be used with eq (24) to calculate y and, hence, using eq (23) the value of K, can be obtained.

A much more promising, though still empirical, procedure has since been suggested by Chang and Robertson (1966), who work in terms of the con­centration of solid material in the plug c, which is related to the porosity by:

d - y ) C = ~ J -

where jS is the specific volume of the solid. In place of eq (23) they suggest:

^ = -A&xp{-Bc) (3.6.25)

and they show that plots of In (I/i/eP) against c are quite linear for a variety of fibrous plugs (beaten and unbeaten cellulose fibres, Dacron, nylon, Orion and glass wool). From the slopes of these plots i t is possible to evaluate £

directly and Jaycock (1979, priv. comm.) has since shown that the values obtained by this procedure are in good agreement with those obtained by particle electrophoresis on small fibres of the same material.

An alternative approach is provided by the work of Ghosh and Pal (1960), who claim that true values of the (-potential can be extracted from measure­ments on fibre plugs, even when the ion distribution across the capillary should be taken into account. In other words they have developed what appears to be a satisfactory description of the way the particles influence the conductivity of the pad, despite the theoretical limitations. This is not really surprising when one considers the structure of a pad composed of fibres. Alignment of the individual particles should lead to flow paths for both liquid and electric charges which are not unduly tortuous.

Ghosh and Pal (1960) consider two cases: (i) No surface conductance. I f the surface conduction effects are negligible,

then the solid merely has a geometric effect on the flow and the conductivity. They write for the effective area of flow: A' = A/(I + m^), which is very similar to Neale's expression (c.f. eq (3.6.21)):

A'= A{\ - <t>)zr

They show that with this modification the data of Glixelli and Stolzmann (1931) on gelatine gels can give a single value of ( which is independent of solids concentration.

(ii) With surface conduction. The streaming current is in this case affected by the gross geometrical effect and also to some extent by the ion distribution in the pores. The other major effect, however, is the influence of surface conduction on the electrical measurement of the plug porosity. Ghosh and Pal give an approximate analysis which leads to a relation which replaces eq(21):

I5= - - • APU[\+m4>l{\-<pWx

n

where m and x are constants. A plot of the function sP/riIs against (b/(\ - <b) for a given fibre sample and electrolyte solution should then give a straight line from which ( can be calculated by measuring the slope and intercept. They showed that when applied to Biefer and Mason's data, straight lines were indeed obtained and consistent values of ( could be calculated.

Finally, again with reference to Section 3.2.3, we note that because of the general relation between streaming potential and streaming current (eq (3.2.10)) the Ghosh and Pal (1960) procedure can also be applied to streaming potential data, even in the presence of surface conduction, provided one has information on the behaviour of plugs of varying porosity.

The work of Dresner and Kraus (1963) suggests that a plug formed by a bundle of fibres could be expected to give results corresponding to cylindrical capillaries of uniform cross-section, and one would expect the same to be true of plugs of uniform spherical particles.

In situations where Kr < 5 one would have to use the procedure of Oldham, et al. (1963) with eq (13) or (14), but making a direct measurement of rather than an estimate from As data. The value of r would depend on the packing geometry assumed for the porous medium and precise results could not be expected for packed plugs.

3.7. Electrophoresis

A great deal of work has been done on the theory of electrophoresis to improve upon the models discussed in Section 3.3. above. The mobility of an isolated, solid, spherical non-conducting particle of arbitrary (-potential has been calculated for arbitrary Ka by Wiersema, et al. (1966) and by O'Brien and White (1978). The most important aspect of the problem is the incor­poration of the relaxation effect, and that wil l be examined first. ^ The same approach has recently been applied to cylindrical particles by Stigter (1978) and that work wil l be discussed after we have considered the sphere. As usual we shall concentrate on presenting the equations and pro­cedures which are necessary to calculate \ from the measured mobility and not attempt to present more than a very brief outline of the underlying theory. We shall then go on to examine the effect of particle shape. The limitations of the models used in the calculation of (—from electrophoresis, electro-osmosis and streaming potential—will be examined in Chapter 5.

3.7.1. Introduction of the relaxation effect

Henry's (1931) calculations were based on the assumption that the external field could be superimposed on the field due to the particle and that the latter could be described by the linearized version of the Poisson-Boltzmann equation. The treatment can therefore only be valid for particles of low potential (( < 25 mV). I t also fails to take account of the distortion of the field induced by the movement of the particle—the relaxation effect. As the particle moves, the surrounding atmosphere must re-form by suitable motions of the double layer ions. The extent to which the symmetry of the atmosphere is destroyed by the movement of the particle depends on the mobility and charge of the counter ions. A satisfactory analysis must also include a simultaneous treatment of the retardation effect.

The early efforts to analyse this situation are described by Overbeek (1950),

who made the first successful attack on the problem (Overbeek, 1943). The general features of Overbeek's results were confirmed by Booth's (1950a) calculations. Both authors used a series expression to represent the potential in the neighbourhood of the spherical particle (see Section 2.5) so that their results would be more reliable for higher values of (. Their calculations emphasize the importance of the relaxation effect in the region of W~ 1, especially for electrolytes of valency greater than unity. A detailed descrip­tion of Booth's analysis is given in Rice and Nagasawa (1961), and an excel­lent review of the modern theory of electrophoresis has already been given by Overbeek and Wiersema (1967).

To obtain an expression for the mobility which is valid for all Ka i t is necessary to take account of the geometrical effect considered by Henry (1931) and, at the same time, both the relaxation and retardation effects. This involves solving several differential equations simultaneously. Firstly the potential around the particle must satisfy the Poisson equation:

div grad \jf= (2.3.2)

where p = e(n+z+ -ri-zJ) as usual. (Note that in this formulation the valency z_ is to be used as a positive number), is in this case the to tal electro­static potential, including that due to the applied field. Inside the particle the potential, \ j f b satisfies Laplace's equation ( V 2 ^ ; = 0) since there is no space charge there.

The concentrations n+ and «_ are determined by the balance between the electrical forces, the diffusional forces and the bulk fluid transport. They must satisfy a generalized form of the Poisson-Boltzmann equation, of the Nernst-Planck type:

d i v [ + (R+z+e) grad \j/ - fcTgrad n± + n.J±v] = 0 (3.7.1)

where ƒ+ are the frictional coefficients and v is the velocity of the liquid with respect to the particle. The terms in the brackets correspond to the ionic flows due to (i) the electric field, (ii) the diffusion force and (iii) convective transport, respectively. The divergence of the ion flow is zero, for a co­ordinate system based on the particle, because in the steady state the ion distribution around the moving particle remains constant. (Note that for v = 0 this equation reduces to eq (2.3.3) for the equilibrium double layer.) A slightly different form of eq (3.7.1) was proposed by Pickard (1961) in his analysis of the problem, but the error involved in that work is discussed by Dukhin and Deryaguin (1974, pp. 293-295).

The fluid motion is described by the time independent form of the Navier-Stokes equation for an incompressible fluid (see Appendix 1.5):

- rj curl curl v + grad p + p grad i// = 0 (3.7.2)

which is the vector form of eq (3.5.1). These three equations are quite general and apply to a particle of any shape. Their solution is only possible when the boundary conditions appropriate to a particular shape of particle are in­troduced.

3.7.2. Electrophoresis of a rigid insulating sphere

Solutions to the eqs (2.3.2), (3.7.1) and (3.7.2) for the case of a spherical particle were obtained by Overbeek (1943) and by Booth (1950a) expressing the mobility as an infinite series in powers of the zeta-potential. Owing to the complexity of the problem, only the first few coefficients were evaluated More recently a computer solution to the problem, for a wide range of (-potentials, was provided by Wiersema, et al. (1966). A comparison of the results of these three more exact treatments is given in Fig. 3.16, which shows a plot of the dimensionless quantity E, defined by:

as a function of Ka. In terms of this variable, the Hiickel equation takes the form

E=Z (3.7.4)

o - o i 1000

toi3itrIhr%^nenTTi0nleSSumO,bi,1it

n

y f U n C t i ° n ' E' f r o m e q ( 3 ' 7 - 3 > ' a s a f u n c t i o n ° f ™ according S i ^ S T S ST w ( V'V B ° ° t h U 9 5 0 ) a n d I V : T h e C O m P u t e d s o l u t i o n °f wiersema et al. (1966). Calculations are for f = 5 and m ± = 0-184.

and the Smoluchowski equation reads

3l

E = q (3.7.5) I t is apparent from the figure (taken from Overbeek and Wiersema, 1967) that the approximate analytical solutions are in good agreement but that they somewhat overestimate the magnitude of the relaxation correction for intermediate values of Ka, at least for this value of Z ( = 5). For smaller values of Z, the relaxation correction decreases significantly and curves I I , I I I and IV all approach the Henry curve, I, which is valid for all Ka i f X < 1 (i-e. ( < 25 mV at 25°C). The magnitude of the relaxation correction depends on the parameter m± which represents the dimensionless mobility of the co-ions ( + ) and counter ions (—) respectively:

2 e R T z± 171^.=——•—5- (3.7.6)

3rj Al

where A° ± is the limiting molar conductance of the ions at infinite dilution. Highly mobile ions (i.e. low m values) give smaller relaxation corrections. The value of m± at 25°C in water is given by

m + = 12-86 x 1 0 - 4

when A+ is in o h m - 1 m 2 m o l - 1 or 12-86(z ± /A°) when A°± is in o h m - 1

c m 2 equ iv . - 1 . The calculation of (-potential from electrophoretic mobility using the

Wiersema (1964) procedure is not an easy task. I t has been described in detail already by Wiersema et al. (1966) and that description will not be repeated here. Suffice it to say that one normally has to prepare a number of graphs in order to interpolate the values of the functions needed, especially if one departs from the simplest electrolyte systems. There are also limitations on the maximum values of (which can be dealt with, especially for z± > 1. The general features of the solution can, however, be seen in Fig. 3.17, where E is plotted against Ka for various values of Z taking the value of m± as 0-184 (corresponding to a 1:1 electrolyte with limiting ionic conductivities of 7 0 o h m - 1 c m 2 m o l - 1 , e.g. KC1). Note that (i) E approaches (Tor low values of Ka (the Hiickel equation (3.7.4)); (ii) E approaches 3(/2 (the Smolu­chowski equation (3.7.5)) for high values of Ka; (iii) the relaxation correction becomes quite large for values of Ka in the region 1-10 and a unique assign­ment of ( to a particular mobility may not be possible.

A numerical tabulation of the relation between ( (mV) and mobility (in

102 Z E T A P O T E N T I A L IN C O L L O I D S C I E N C E

E

Ka Fig. 3.17. E(KO) for various values of the reduced zeta potential according to Wiersema et al. (1966). Note that E approaches ('for small Ka (the Hiickel solution) and that unambiguous assignments of (-potential become impossible for high (-potentials in the neighbourhood of na = 10.

urn cm s " 1 V " x ) based on the Wiersema et al. (1966) computer solution has been given by Ottewill and Shaw (1972) for the case where:

(i) m+ = m_ = 0T84 (i.e. a 1:1 electrolyte with limiting conductivity for each ion equal to 70 o h m " 1 cm 2 m o l " 1 ;

(ii) m + = 50-15; m_ =76-35 o h m - 1 cm 2 m o l " 1 (i.e. NaCl); (iii) 2:1 electrolyte with limiting conductance of 140 and 70ohm" 1 cm 2

m o l " 1 for cation and anion respectively.

A much more effective numerical solution of the electrophoresis problem for a sphere has now been provided by O'Brien and White (1978). Starting with the same set of equations, they show that the problem can be broken down into two simpler problems: the calculation of (i) the force fx required to move the particle at a velocity v with no applied field and (ii) the force f2

required to hold the particle fixed in the presence of the applied field E. Provided the fields and velocities are small enough to ensure linearity—

and this is always so in practice (see Booth, 1950b)—these forces can be written:

/ ! = a « and f2 = (3E (3.7.7)

3. T H E C A L C U L A T I O N O F Z E T A P O T E N T I A L 103

where the parameters a and /? are independent of v and E. The sum of these two forces corresponds to the net force on the particle when it is undergoing electrophoresis and since this must be zero we have

av = — fiE

so that the electrophoretic mobility uE= — plot. O'Brien and White (1978) have plotted E against Z ( f < 10) for a large

number of Ka values. Their results are shown in Figs 3.18 and 3.19. Again the limiting behaviour at low Ka and high Ka is clear as is also the strange behaviour for Ka in the region of 3-10. The plots are for m± =0-184 and they agree with Wiersema's results in the region where the latter could obtain solutions.

The new feature which becomes clear from this more extended work is

2h

Fig. 3.18. The reduced mobility, E, versus reduced (-potential for various (small) values of Ka. (After O'Brien and White (1978) with permission.)

7

6

2

O'

• / / Ka'-oo/250/ 1 i ff / /.150

l t È

>\ \ i j 100

KCl lij/ 70

11/ / 5 0 ^ ^ f f / / 50 \

l jij I / /

\ \ - /f 40 \ \

/ 1 / I/V ^ N N

lm l i f f f ^ \ \

•

laf fff / \ N

/ f f / / / / 30 \ * \ * i v \ \ ^ \

V \ x

20 //////// y*"" N

W/s. I 9 ^ \

\ \

\ N \ V \ N

V \ \ V

s \ \ \ v- \ \ \ >N \ \ \ x» \ \ \ \ \ \ > \ \ - \ \ \ V

i

i j Ui

l i i _ — i 1 1 1 1_

Fig. 3.19. As for Fig. 3.18 but for large Ka. Note the hiatus for 3 < Ka < 10 in the region J ss 5 and the pronounced maximum in E for large Ka values. The broken curves show eq (3.7.13) for Ka = 150, 50 and 30 respectively.

that for all values of Ka ^ 3 the mobility function has a maximum value which becomes particularly pronounced at high Ka. The maximum occurs at Z~ 5-7 (i.e. ( ~ 125-175mV); it is, therefore, possible that some measured mobilities correspond to (-potentials on the high side of the maximum. This could possibly explain some results in which anomalously low (-potentials are reported in systems of very high charge density (e.g. clay minerals).

The O'Brien and White computer solution is so much more rapid than the earlier one that it is possible to apply it to individual sets of experi­mental data. A program is available from the authors (Department of Applied Mathematics Research School of Physical Sciences, A.N.U., Canberra ACT, Australia) which permits the experimenter to input any arbitrary collection of ion valencies, concentrations and limiting ionic conductivities (the latter

measured at any temperature) and to calculate ( for any given particle radius and experimental temperature given the mobility in aqueous solution. Extensions to other media require a fairly simple modification of some sub­routines. In practice, the program outputs a set of mobilities with the equiva­lent (-values (up to ( = 250 mV) for the particular system being studied.

Although this new computer solution is said to be rapid and simple to use there may be some experimenters who do not have access to computing facilities or who prefer to obtain an estimate of the magnitude of corrections using the approximate analytical expressions for the relation between ( and uE or E. I t is, therefore, worth while to examine the solutions of Overbeek and of Booth.

Overbeek (1943) gave, for symmetrical electrolytes:

E = U,{Ka) - Zz[z%{Ka) + i (m+ + m_)/ 4 0«z)] (3.7.8)

and for unsymmetrical electrolytes:

"z+m+ + z_m. E=Zf1{Ka) - Z\z_ - z + ) f 2 { K a ) - ?

z+ + z. MKa) (3.7.9)

In eq (8), z = z+ = z_; eq (9) holds for a positively charged colloid but a consistent series of sign changes on both the colloid and the small ions yields the correct result for a negative particle. The function fi(Ka) is Henry's function (from eq (3.3.6) and Fig. 3.6 curve a), and both equations (8) and (9) approach Henry's expression, E=Zfl(Ka), for Z<1. Note that the quadratic term does not appear for symmetric electrolytes. The functions f2{Ka) and f4{Ka) are plotted in Overbeek's paper.

Booth (1950a) treated only the symmetric case and took the analysis to the next term. His equation, in the present terminology is:

E=Zf1(Ka) + P[z 2 / 3 *(Kfl) + 3(m + + m_)Zf (KO)] + (" 4[3z(m+ - m_)ZX(Ka)] (3.7.10)

and again, the functions/ 3* ( = X* + F3*); Z% and Z% are plotted in Booth's paper and reproduced below.

Figure 3.20, adapted from Overbeek and Wiersema (1967), shows the regions in which the various analytical approximations can be used, if one is content to accept an error of 1 mV (full lines) or 2-5 mV (broken lines) in the absolute value of (. The figure implies that if, for example, the combination of £ a n d Ka correspond to the bottom left hand area then one may use the Hiickel equation if an error of 1 mV is tolerable. For conditions which fall below the lower broken line, nothing more complicated than the Henry equation need be used if an error of up to 2-5 mV is acceptable. For somewhat more difficult situations, the Overbeek and Booth solutions are required and, unfortunately, they are not in a very suitable form for computation. One normally wishes

0-01 0-1 1 10 100 1000 KO

Fig. 3.20. Range of validity of the various analytical approximations for calculating the electro­phoretic mobility for a 1:1 electrolyte. Solid lines correspond to 1 mV error in (-potential, broken lines to an error of 2-5mV. The accuracy of eqs (3.7.13 and 14) is best assessed by refer­ence to Tables 3.2 and 3.3 and Fig. 3.19. Equation (14) is far simpler to use than eq (12) and should be considered for any KO > 50. (Adapted from Overbeek and Wiersema, 1967.)

to calculate the (-potential f rom the measured mobility, which would in this case involve solving a cubic or quartic equation. Stigter and Mysels (1955) suggested an inverted form of eq (10), but a more accurate version (Stigter, 1967) has since been given by Hunter (1962). His equation may be obtained by applying the Newton-Raphson procedure to Booth's equation, with Henry's equation used as a trial root. The result, for symmetrical electrolytes, is:

h + 3 C 3 ( i i / / 1 ) 2 + 4 C 4 ( £ / / i ) 3 J ( X J A l )

where f x is the Henry function of Ka referred to previously. The coefficients C„ are the coefficients of Z" in eq (10). For the case of a single symmetrical electrolyte, to which eq (10) refers, at 25°C in water:

C 3 = z2f3*(Ka) + 38-6z ^ + -^\z%(rca)

and

C 4 = 3 8 - 6 z 2 ( ^ - ^ W K a )

. i 1-2

Fig. 3.21. The Henry (1931) function, ft(Ka), measuring the influence of an insulating sphere on the imposed electric field.

The functions f ^ f f , Z% and Z% are reproduced in Figs 3.21 and 3.22 for easy reference. At 25°C in water £ = 7520w£ (cm 2 v o i r 1 s _ 1 ) and £(mV) = 25-7(".

Comparing eqs (8) and (10) it is apparent that, for the case where m + = m_, the two would be identical i f / 4 (Overbeek) = - 6Zf (Booth). The value of - / 4 / 6 is shown in Fig. 3.22, and although it has the same form as Z%, Wiersema (1964) points out that differences of 20% occur between the two functions.

For unsymmetrical electrolytes, eq (9) yields an expression analogous to e q ( H ) :

C'^E/f,)2 + C'^E/f,)3

where

and

1 fx _ / 1 + 2C2(yY// 1) + 3 C 3 ( £ / / 1 ) 2

(z+ -z_)f2(Ka)

(3.7.12)

z + m+ -I - z_m. z+ + z_

I M K O )

X \J\J

O

- 0 0 1

- 0 0 2

I I 1 I I 1 1 1 1 1 1 1 1 1 1

7 * 1

i i i i i j

j SS

*^ / * /

i i n j

-

-

1 1 1 1 1 1 1 1 1 1 i i i i i i i i i i -1 0 0 1-0 2-0 3-0

l o g 1 0 ( « : o )

Fig. 3.22. The Booth (1950a) relaxation correction functions X%, Y% 2 f , and Z J . The broken line is the value of — fJ6 where / 4 is Overbeek's (1943) relaxation function. This is to be com­pared with Booth's Z * .

The funct ion/ 4 can be obtained approximately from Fig. 3.22 and considera­tion of Fig. 4 of Overbeek (1950) suggests that little error is involved in taking f2{Ka) =/ 4 (Ka). This expression might be expected to be useful in much the same region of ( Z — KO) as eq (11). Unfortunately, this is not so. Wiersema et al. (1966) point out that eq (9) is significantly less accurate than eq (8) because the coefficient of Z3 is incomplete in that case (Overbeek, 1943). In such cases it is much better to use the complete computer solution of O'Brien and White (1978). The alternative approach of interpolating from Wiersema's (1964) calculations is outlined by Wiersema, et al. (1966).

In a series of works, Dukhin and his collaborators (Dukhin and Shilov, 1969; Dukhin, 1970, 1971; Dukhin and Deryaguin, 1974) have sought to develop an alternative analytical approximation for spherical particles undergoing electrophoresis. Their method, which they refer to as the "polar­ized double layer treatment", amounts to the incorporation of a relaxation correction into the Smoluchowski equation. This is done using the assump­

tion, which is carefully justified by Dukhin and Deryaguin (1974, Chapter 3), that for large values of Ka, even when the relaxation effect is occurring, the ions in the double layer maintain a local equilibrium with the adjoining (neutral) fluid. This is necessarily so because in a large thin double layer it is impossible for tangential ion flows to balance any radial influx of ions. Coupling this with the so-called boundary layer method (Dukhin and Deryaguin, 1974, p. 297) it is possible to obtain the following expression4: for the mobility:

3Z , (IM sinh 2 Z + [2z~ 1M sinh 2Z - 3mZ] In cosh Z E = — — 6

2 V Ka + 8M sinh 2 Z - (24m/z2) In cosh Z (3.7.13)

where M—1 + 3m/z2 and Z = zZ/4. To simplify the expression it has been assumed that the ions are of equal mobility and valency. Dukhin and Dery­aguin (1974) were not able to compare their equation with the Wiersema computations because of the limitations in the range of the latter. Even the maximum in the mobility predicted by eq (13) was not readily apparent from Wiersema et al. (1966) point out that eq (9) is significantly less accurate and Shaw (1972) clearly shows that there is a maximum. That maximum is, of course, amply demonstrated by the more extended calculations of O'Brien and White (1978); indeed, eq (13) turns out to be very close to the computer solution for large Ka (see Table 3.2). Even at KO -> 30 the error is unacceptably large only for values of Z which are unlikely to be encountered in practice (and which would, in any case, not be recognised from mobility measure­ments because they occur beyond the maximum). Values of E as a function of Z f rom eq (13) are superimposed on O'Brien and White's (1978) data for KO = 150, 50, and 30 in Fig. 3.19.

O'Brien (1979, priv. comm.) points out that eq (13) can be simplified by eliminating terms of order ( K G ) - 1 (O'Brien and Hunter, 1981) to yield:

| - ^ ( l - e x p ( - z ö )

2 „ K a 2 + M 6 X P

(3.7.14)

This very simple expression reproduces the results of eq (13) with an accuracy of the order of (1/ïca), and Table 3.3 shows clearly that for z = 2 it reproduces the computer solution more faithfully than the more elaborate eq (13) except at the very smallest values of Z when eq (13) reduces to the Smolu-

:j;I am indebted to Dr Richard O'Brien for drawing my attention to an error in the original . statement of eq (13) in the text by Dukhin and Deryaguin (1974): the term z~l in eq (13) was

there rendered as z.

_ T A B L E 3.2

1 9 0 7 8 W n ° P

n n ^ ^ . ^ P " ^ . solution.of mobiHty equations (O'Brien and White, 1978) with eq (3.7 13) based on the "polarized double layer" model of Dukhin et al and showing values of E (=7520H„ tan*v-i.-itt v f \ = 0 a84) for v ï £ s

£ v - " * T u » V J I ^ W l l

values of C( = C(mV)/25-7 at 298 K)

JCC= 150

9-9997 6-8705 5-6589 4-8761 4- 2880 3-8109 3-4054 3- 0497 2-7303 2- 4385

Ka= 100

10-000 5- 6640 4- 2927 3- 4096 2-7340 2-1712 1-6776 1-2267 0-8036 0-3980

KO = 50

9-9995 6-8894 5-6790 4-8960 4-3070 3-8289 3-4224 3-0655 2-7451 2-4521

Ka = 30

9-9993 6-9078 5-6987 4-9153 4-3256 3-8466 3-4390 3-0811 2-7595 2-4653

J(comp) "(eq 13) % error -(eq 14)

5- 2194 7-1103 6- 8490 6-3229 5-7764 5-2577 4-7747 4-3253 3-9054 3-5105

4- 3574 6-3045 5- 5166 4-6335 3-8200 3-0849 2-4077 1-7722 1-1657 0-5785

3- 3835 4- 9304 5- 2567 5-1815 4-9455 4-6417 4-3105 3-9709 3-6320 3-2979

2- 9788 4-1057 4-4866 4-5426 4-4334 4-2356 3- 9892 3-7164 3-4301 3-1375

5- 1845 7-1394 6- 9070 6-3932 5-8514 5-3333 4-8485 4-3960 3-9720 3-5725

4- 2862 6-3557 5- 6104 4-7352 3-9184 3-1699 2-4778 1-8254 1-2013 0-5965

3- 2074 4- 7948 5- 2056 5-1994 5-0124 4-7411 4-4301 4-1015 3-7669 3-4321

2- 7124 3- 8356 4- 2997 4-4490 4-4212 4-2876 4-0889 3-8494 3-5842 3-3029

-0-7 + 0-4 + 0-8 + 1-1 + 1-3 + 1-4 + 1-5 + 1-6 + 1-7 + 1-8

- 1-6 + 0-8 + 1-7 + 2-2 + 2-6 4-2-8 + 2-9 + 3-0 + 3-1 + 3-1

-5 -2 -2 -7 - 1-0 + 0-3 + 1-4 + 2-1 + 2-8 + 3-3 + 3-7 + 4-1

-8-9 -6 -6 -4 -2 - 2 - 1 -0-3 + 1-3 + 2-5 + 3-6 + 4-5 + 5-3

5- 2532 7-0875 6- 8232 6-3083 5-7744 5-2666 4-7921 4-3494 3-9341 3-5426

4- 3837 6-2788 5- 5184 4-6618 3-8668 3-1377 2-4611 1-8200 1-2029 0-6001

3- 3446 4- 8807 5- 1984 5-1373 4-9241 4-6442 4-3351 4-0141 3-6904 3-3677

2- 8695 4-0165 4-3954 4-4682 4-3872 4-2216 4-0068 3- 7627 3-5007 3-2277

, error

+ 0-6 -0-3 -0 -4 -0 -2 - 0 0 3 + 0-2 + 0-4 + 0-6 + 0-7 + 0-9

+ 0-6 - 0 - 4 -0-03 + 0-6 + 1-2 + 1-7 + 2-2 + 2-7 + 3-2 + 3-7

- 11 - 1 - 0 - 11 -0 -9 - 0 - 4 + 0-05 + 0-6 + 1-1 + 1-6 + 2-1

-3 -7 -2 -2 -2 -0 - 1-6 - 1-0 -0-3 + 0-4 + 1-2 + 2-1 + 2-9

T A B L E 3.3

Comparison of exact computer solution of mobility equations (O'Brien and White (1978) with eqs (3.7.13) and (3.7.14) for z_

(m ± = 0-4287) 2 and A° = 60 ohm 1 cm 2 equiv. 1

i p -^(comp) ^(eq 13) % error E{tq 14) % error

Ka= 150

0-1335 0-1965 0-2002 + 1-9 0-2011 2-3 0-4054 0-5956 0-6072 + 1-9 0-6073 2-0 0-6929 1-0143 1-0346 + 2-0 1-0305 1-6 1-0107 1-4698 1-5001 + 2-1 1-4877 1-2 1-3841 1-9875 2-0306 + 2-2 2-0038 0-8 2-1879 2-9742 3-0450 + 2-4 2-9756 -05 3-3266 3-7150 3-8053 + 2-4 3-6893 -0-7 6-4102 1-7909 1-8026 + 0-6 1-7718 - 11

Ka = 100

0-1335 0-1947 0-2002 + 2-9 0-2016 3-5 0-4054 0-5897 0-6067 + 2-9 0-6069 2-9 0-6927 1-0026 1-0320 + 2-9 1-0261 2-3 1-0103 1-4484 1-4918 + 3-0 1-4740 1-8 1-3830 1-9471 2-0075 + 3-1 1-9696 1-2 1-8635 2-5250 2-6060 + 3-2 2-5360 0-4 2-1834 2-8475 2-9397 + 3-2 2-8478 0-01 3-3073 3-3504 3-4437 + 2-8 3-3165 - 1-0 6-0177 1-7457 1-7552 + 0-5 1-7165 -1-7

KO = 50

0-1335 0-1899 0-2002 + 5-4 0-2028 6-8 0-4053 0-5740 0-6052 5-4 0-6057 5-5 0-6923 0-9715 1-0244 5-5 1-0138 4-4 1-0091 1-3919 1-4677 5-5 1-4361 3-2 1-3798 1-8438 1-9432 54 1-8795 1-9 1-8555 2-3244 2-4443 5-2 2-3363 0-5 2-1702 2-5572 2-6821 4-9 2-5498 -0-3 3-2529 2-7168 2-8085 3-4 2-6608 - 2 - 1 5-3391 1-6784 1-6751 -0 -2 1-6198 -3-5

chowski result. For z = 1, Table 3.2 shows that eq (14) is, rather fortuitously, superior to eq (13) over the entire range. The analytical approach of Dukhin has recently been extended by Semenikhin (1976) to even lower values of Ka, but the resulting expressions are far from simple (see also Semenikhin and Dukhin, 1975a, b).

The notion of the polarized double layer is obviously a very fruitful one, and we shall return to it in Section 3.8. The term "polarization" implies that the double layer around the particle is regarded as being distorted from

T A B L E 3.4 Values of the function g a v ( K f l ) to be used in eq (3.7.16) for calculating the average mobility of long arcular cylinders of random orientation (from Stigter, 1978a). Values for7= 4 and 5 are from Sügter (1979). The friction factor for the small ions' m , is defined in

eq (3.7.6)

Ka

0-1 0-2 0-5 1 2 5 10 00

I m+ = = 0-184 m_ = = 0-184 0 1-001 1-006 1024 1-057 1113 1-223 1-312 1-5 1 0-995 0-997 1-011 1-039 1-091 1-198 1-290 1-5 2 0-973 0-970 0-974 0-991 1-031 1-129 1-227 1-5 3 0-940 0-931 0-924 0-928 0-952 1-032 1-130 1-5 4 0-902 0-888 0-871 0-866 0-877 0-933 1-018 1-5 5 0-862 0-844 0-823 0-812 0-813 0-846 0-914 1-5

m+ = 0-368 w_ = 0-184 3 0-931 0-922 0-914 0-913 0-942 1-022 1-121 1-5 4 0-890 0-876 0-859 0-855 0-866 0-923 1-009 1-5 5 0-848 0-831 0-811 0-801 0-803 0-838 0-906 1-5

m + = 0-184 m_ = 0-368 3 0-926 0-916 0-907 0-909 0-930 1-004 1-100 1-5 4 c

0-880 0-866 0-849 0-843 0-850 0-899 0-978 1-5 5 0-835 0-819 0-799 0-789 0-788 0-815 0-873 1-5

its equilibrium shape by the motion of the particle. The effects of this distor­tion are significant in electrophoresis (because they give rise to the relaxation eliect) but they are much more significant in the theory of sedimentation potential (Dukhin and Deryaguin, 1974).

3.7.3. Electrophoresis of cylinders

The mobility of a charged cylinder in an electric field depends upon its orientation with respect to the field. Experimental values therefore represent some kind of average over the different orientations. In Section 3 32 we saw that Henry's calculations could be represented by:

2eC UE = ^-g{KU) {5.1.15)

or

E = Z-g{Ka) (3.7.16)

where g(m) was equal to 1-5 for all Ka i f the cylinders were aligned parallel

to the field and g(Ka) varied from 0-75 to 1-5 as Ka varied from low to high values if the cylinders were aligned perpendicular to the field. Henry's calculations are valid only for small values of the potential where the relaxa­tion effect can be neglected.

Recently, Stigter (1978a) has recalculated the mobility taking account of the relaxation correction, following the same approach as Wiersema (1964). He first calculated E for cylinders aligned perpendicular to the field and then assumed the Smoluchowski expression would still hold for cylinders parallel to the field.- This amounts to assuming that the relaxation effect is unim­portant in that case. Stigter showed that, even when the relaxation effect is considered, the average mobility is still given by the expression suggested earlier by de Keizer et al. (1975):

£av=(£n+2£J/3 (3.7.17)

Values of the function g{Ka) to be used in calculating Em from eq (16) are given in Table 3.4 for various values of Ka and for Z= 0, 1, 2, 3, 4 and 5.

Overbeek and Wiersema (1967) using this same averaging procedure show that for a cylinder for which / = 20a, the average mobility is close to that of a sphere of the same volume, if relaxation effects are neglected. Harris's (1970) calculations add some weight to this procedure.

In a subsequent paper, Stigter (1978b) analyses the behaviour of cylinders of ellipsoidal cross-section and concludes that for randomly oriented short elliptic cylinders, the electrophoretic mobility is the average of the mobilities of the particle along each of its three main axes. This average applies in the presence of the relaxation effect, and to all particles with the same or higher symmetry than the elliptic cylinder as long as the system obeys Ohm's law and there is no appreciable double-layer overlap. I f the semi-axes of the ellipse are of length a and b then

011 + g{Ka) + g{Kb) (3.7.18)

where the appropriate values of g(Ka) and g(Kb) are obtained from the values of gav in Table 3.3 by multiplying by 1-5 and subtracting 0-75 (c.f. eq (3.7.17)).

Van der Drif t , et al. (1979) also examined the electrophoretic mobility of a cylinder using the Philip and Wooding (1970) analytical approximation for the potential (see 2.6). Though they did not explicitly include the relaxa­tion correction, they pointed out that the semi-empirical approach of Möller, et al. (1961) could be used to make an approximate correction. They also pointed out that Stigter's (1978a) calculation of the relaxation effect is probably an underestimate because of the neglect of double-layer polarization for cylinders parallel to the applied field. Certainly the experimental data of van der Dr i f t and Overbeek (1979) on the transport properties of poly-

electrolytes suggest that the effect is large, as does the theoretical analysis of Dukhin leading to eq (3.7.13).

3.7.4. Part icles of arbi t rary shape

Most regular particles can be likened to either elliptical cylinders or spheroids. The case of elliptical cylinders was discussed above (Stigter, 1978b), and the spheroid (obtained by rotating an ellipse around one of its axes) can be treated in the same way by using the average mobility calculated using (3.7.18) but now with the mobility of a sphere of radius equal to the major and minor axes of the ellipse. For an oblate spheroid, which would be the best representation of a disc-shaped particle, the relevant average would be:

_ \g{Kd) + 2g(icb)~ 9 a w ~ l 3 _

where a and b are the minor and major axes of the spheroid. I t should be noted again that Smoluchowski's equation holds for a non­

conducting particle of any shape provided that the radius of curvature at all points on the surface is large compared to the double-layer thickness (1/K). Furthermore, Overbeek (1946) has shown that the relaxation correction can be neglected for any particle shape if the double layer is thin compared to any radius of curvature of the particle. Equation (3.7.13) or (3.7.14) can be used to check the point at which Ka becomes large enough for relaxation to be neglected.

At the other end of the scale, when Ka -4 1, we noted in Section 3.3 that the dominant force opposing the movement due to the field is the frictional resistance of the medium (Dukhin and Deryaguin, 1974, p. 59). In such cases the mobility is given by:

v. Q

where Q is the particle charge, which may be related to the £ potential by an appropriate equation, depending on the geometry of the system. The factor ƒ is a friction factor which must be obtained from the hydrodynamics of the situation, either theoretically or from diffusion or sedimentation studies. A number of situations have been treated theoretically with varying degrees of sophistication. For the spherical case, eq (2.3.37) together with Stokes's equation for ƒ wil l yield eq (3.3.4). The relevant relations for cylinders and discs are given by Gorin (see Abramson, et al., 1942, p. 130). For cylinders, the charge is related to the (-potential by putting £ = \j/0 in eq (2.3.40), provided £ is sufficiently small to permit use of the linearized Poisson-

Boltzmann equation, and Gorin gives a table of the ratio of the frictional coefficient, ƒ to the corresponding value for a sphere for various values of the asymmetry parameter Sr = l/2a. The radius a is, of course, measured out to the plane of shear.

A similar type of procedure has been used by Noda et al. (1964) for poly-electrolyte systems. They defined an elementary segment of the polymer as that part of the chain which contains one elementary electronic charge. By treating this segment as a prolate ellipsoid with a diameter independent of its length they were able to explain the mobility of various sodium poly-acrylates in O T M NaCl using an equation like eq (19). The method has been taken up by Prokopova and Bohdanecky (1973), who used various hydro-dynamic relations to calculate the frictional coefficient,/.

There is, of course, much interest in the calculation of the electrophoretic mobility of randomly coiled polyelectrolytes, for which the concept of a clearly defined shear plane and a £-potential are inappropriate. Overbeek and Wiersema (1967) have reviewed this subject, and a more detailed dis­cussion is provided by Rice and Nagasawa (1961, Chapter 11). More recently Overbeek and Bijsterbosch (1979) reviewed the electrophoretic behaviour of polyelectrolytes and pointed out that many polyelectrolytes can be approxi­mated as randomly kinked cylinders surrounded by a thin (coaxial) double layer. For such systems the treatment developed by Stigter (1978a, b) or by van der Drif t , et al. (1979) would be most appropriate.

3.7.5. In f luence of part ic le concen t ra t i on

In the regime where Smoluchowski's equation holds (large Ka), electro­phoresis and electro-osmosis are simply opposite expressions of the same relative motion of particle and liquid. I t would therefore be expected that for large values of Ka (i.e. thin double layers) a collection of particles would behave just like a single particle (Reed and Morrison, 1976). After all, the electro-osmotic flow through a porous bed is not affected appreciably until the double layers on opposite sides of a pore begin to overlap (Section 3.5).

A more detailed investigation of this phenomenon, using a cell-model for the particles, was undertaken by Levine and Neale (1974a). A single particle was considered to be surrounded by a sphere of such a size that the fraction of liquid within the sphere was the same as the porosity of the swarm of particles. The detailed motion of the fluid inside this sphere was then cal­culated using appropriate boundary conditions. Two possible cell-models were examined: the "free-surface" model of Happel (1958) and the "zero-vorticity" model of Kuwabara (1959), both of which had been shown to give good results for the relative movement of liquid and uncharged particles (e.g. in gravitational settling). Although the results from both models were

in reasonable agreement for large values of Ka, only the Kuwabara model gave a result which tended to the Smoluchowski (or Henry) result as Ka -»• oo. The Happel model only approached this limit for very large values of the sphere radius, b (i.e. in the limit of a single particle). For this reason Levine and Neale (1974a) considered the Kuwabara model to be superior. They attributed this to the fact that the electrical contribution to the vorticity vector on the surface of the outer sphere can be shown to be zero, and since the model also assumes that the hydrodynamical contribution to the vorticity is zero on the sphere, the hydrodynamical and electrical boundary conditions are more consistent in this model than in Happel's. The vorticity vector is a measure of the circulation of the liquid velocity field (see Appendix 1.5).

Extension of the analysis to smaller values of Ka is therefore best carried out using the Kuwabara model. The analysis is essentially an extension of Henry's (1931) approach and is valid only in the limit of low potentials (lx 1) where the relaxation effect can be neglected. The result may be expressed in the form of a correction factor to the Smoluchowski equation:

Fig. 3.23. The dependence of the function ƒ (Ka,y) on Ka for various values of the porosity (y) (From Levine and Neale, 1974a.) This function corrects the Smoluchowski equation for the effects of particle concentration.

where y is the porosity (y = 1 - <j> where <j> is the volume fraction of solids). A plot of/fjca, y) is shown in Fig. 3.23. In the limit as y -> 1 (i.e. a single isolated sphere) f(Ka, y) = | A ( K a ) of Henry (defined in eq (3.3.6)). The influence of double-layer overlap is apparent for smaller values of y, especially as Ka becomes smaller; a small y value then corresponds to many small spheres with large double layers overlapping strongly. (Note that the minimum value of y for uniformly sized spheres is 0-26.) According to Levine and Neale (1974b), these results are in substantial agreement with the earlier assess­ments of the concentration effect by Möller et al. (1961) and Long and Ross (1965); the conclusions are also confirmed by the calculations of Reed and Morrison (1976). I t should be noted here, however, that Hall and Sculley (1977) have suggested that as particles are pushed closer together the change in electrochemical potential of the ions in the system causes a higher pro­portion of them to be adsorbed behind the shear plane so that the interpre­tation of C in these cases may need some modification.

The problem of calculating the fluid velocity field in a system containing many particles wi l l be discussed further when we consider the second electroviscous effect (Chapter 5).

3.7.6. In f luence of B r o w n i a n m o t i o n

The electrophoretic motion of a particle wil l , of course, be superimposed on its Brownian motion. In the normal measurement procedures the effect of this motion is eliminated, either by averaging over a number of particles (microelectrophoresis) or by following the combined movement of a large number of particles (moving boundary or mass transport methods). How­ever, as Overbeek and Wiersema (1967) point out, the colloid particle, by its Brownian motion, takes part in the relaxation of its own atmosphere and thereby reduces the relaxation effect. An approximate calculation of the upper limit of the Brownian motion correction has been made by Wiersema (1964), and the results are discussed by Wiersema et al. (1966) and Overbeek and Wiersema (1967). In general, i t can be concluded that Brownian motion can be neglected in most practical cases, partly because the correction is small but also because the next refinement in the theory (introduction of the self-atmosphere potential of the ions in the double layer) acts in the opposite direction.

3.8. The sedimentation potential

The generation of a potential difference as the result of the fall (or rise) of a charged particle in a fluid is referred to as the D o m effect. We discussed

briefly in Section 3.4 the behaviour predicted by Smoluchowski for rigid particles and the modifications which were suggested by Frumkin and Bagotskaya (1948) for a liquid drop. A much more detailed investigation of this effect has since been given by Dukhin (1966), using an extension of Levich's procedures for examining the hydrodynamics of "polarized" double layers. Dukhin sets up a more general description of the ion flows than that given by equations like (3.7.1). He argues that the "polarization" of the double layer around a moving particle, which is responsible for the relaxation effect in electrophoresis, is not merely a correction term in the theory of the D o m effect but is, indeed, the dominant process.

The mechanism by which the potential is developed depends upon the magnitude of the Péclet number (a dimensionless quantity which measures the relative importance of convective and molecular diffusion: Pe = au/@ = au/cokT where u is the particle velocity, 3) is the ion diffusion Coefficient, and co is the velocity of the ion per unit force).

When Pe 1 and the particle is solid and the diffusion coefficients of the double layer ions are equal. Smoluchowski's result, eq (3.4.4), is obtained. I f the particle is fluid, added to the effect of the (-potential is another contri­bution to the sedimentation potential which depends on the extent of ad­sorption occurring at the interface. I t is this term which, under certain simplifying assumptions, generates the Frumkin-Levich (1946) result.

When Pe > 1 (i.e. large particles or very slow counterions—particularly relevant to the behaviour of air bubbles) there is again a contribution from the (-potential given by Smoluchowski's equation, together with contri­butions which depend on the difference between the bulk diffusion co­efficients and the surface diffusion coefficients of the double-layer ions and the extent of adsorption at the interface. A detailed description of this work is beyond the scope of the present analysis but it should be made clear that any measurements on fluid drops should not be interpreted simply on the basis of the Smoluchowski equation (3.4.4). I t is this misconception which has led Usui and Sasaki (1978) to record very large (-potentials for air bubbles in surfactant solution (see Section 4.4).

The effect of particle concentration on the sedimentation potential of solid spheres is discussed by Levine, et al. (1976) with the aid of a cell model (c.f. Section 3.7.5). The theory is derived for KO > 10 and for low potentials and in general it agrees with the Smoluchowski result for large Ka and low particle concentrations. In the limit of a single sphere it generates a result intermediate between those calculated by the Booth (1954) and by the Smolu­chowski equations (3.4.4). In a subsequent paper Levine, et al. (1978) elaborate their cell model by using a rather more complicated function to describe the potential due to the distortion of the electrical double layer around the particle.

3.9. Val idity of the e lectrokinet ic equations

Absolute tests of the equations developed in the foregoing pages are not available because there is no independent measurement of the electrokinetic potential. One can demonstrate that different methods applied to the same surface give the same result (provided they are truly comparable) but un­fortunately that provides no proof of the validity of the equations for calcu­lating (. As we noted in Section 3.2, i t can be shown, using the theory of the thermodynamics of irreversible processes, that, for example:

( V - ) J * ) (3.9.1) V V P = O \Pji=o

so that the measured quantities in electro-osmosis and streaming potential wil l produce the same result no matter what the relation between these ratios and the ( potential may be. Furthermore, equation (i) and its analogues hold even when surface conduction and double layer overlap are important so that one cannot test the validity of the more elaborate equations of Part I I of this chapter by making comparisons between the different measurement methods.

There is, however, one kind of test which can be applied. We can test whether the elaborate expressions of Sections 3.5 and 3.6, when applied to very fine capillaries, are able to rationalize the apparent changes in the ratio EJp or Vji for different values of r even when the surface properties of the capillaries remain unchanged. This is referred to as the electroviscous effect and we shall examine it in considerable detail in Chapter 5.

A less stringent test of the surface conduction equations for electrophoresis (Section 3.3.2) is provided by the work of Ghosh (1954) and his co-workers who claim that "true" (-potentials can be calculated from measured (-values by taking account of the surface conduction in an empirical way (see e.g. Gangopadhyay, 1961).

Another test, which can be applied to the equations for electrophoretic mobility, is to examine whether they provide a satisfactory description of the dependence of mobility on Ka for a system of constant surface properties. One cannot simply alter K because this would change the (-potential by altering the adsorption equilibrium at the interface and/or compressing the double layer. Rather one must alter the particle radius without altering its surface properties. For an unequivocal test one must also be able to prepare particles which are, indeed, spherical and of known, constant particle size. That ruled out most crystalline materials (at least until the recent develop­ment of techniques for growing spherical monodisperse metal oxide particles (see e.g. Matijevic, 1977). Indeed, the only evidence which has appeared on this question comes from the work of Shaw and Ottewill (1965), who pre-

0-1 1 10 100 «•£7

Fig. 3.24. Experimental test of the equations for electrophoretic mobility. Theoretical curves obtained from Wiersema's computations for (-potentials of negatively charged particles in NaCl solution: (1) 20 (2) 34 (3) 40 (4) 50 (5) 55 and (6) 90mV. Experimental results for various Iatices: (A) A; (n)_B; (•) C ; (O) D ; (•) E . Concentrations of NaCl: bare symbols 5 x 10- 3 M-horizontal bar: 10" 3 M; vertical bar: 10" 2 M; sloping bar: 5 x 10" 2 M. (From Ottewill and Shaw (1972) with permission.)

pared five different polystyrene Iatices ranging in diameter from 60 to 420 nm and containing — C O O H groups on their surfaces. Three of them had very similar numbers of carboxyl groups per unit area whilst the other two had a larger number (Ottewill and Shaw, 1972). A comparison between the ex­perimental mobility in NaCl solution and the theoretical values at various values of tea is shown in Fig. 3.24. I t is apparent that at the same electrolyte concentration, the low-charge Iatices (A, B and C) can all be accommodated with a common value of the (-potential, whilst the more highly charged Iatices (D and E) clearly have a higher (-potential under the same conditions. The fact that Iatices D and E at 1 0 ~ 2 M fall on curves 4 and 5 merely indicates that they exhibit a (-potential of 50 and 55 mV respectively under these conditions. The lower-charged Iatices A, B and C only acquire such a high (-potential at concentrations of 5 x 10" 5 M and 10" 3 M respectively. Clearly it would be advisable to extend this work with systems having a wider range of values of a and exactly the same surface properties with K constant and Ka in the range 1-50 where the effect is most marked. One disturbing feature of the Ottewill and Shaw (1972) results was that the experimentally deter­mined mobility in barium nitrate solutions was higher than the theoretical

maximum value for the given value of Ka. Ottewill and Shaw did not use the exact limiting ionic mobilities for B a 2 + and N 0 3 " and thought that this might have been responsible for the discrepancy but the exact calculations of O'Brien and White (1978), at least for Ka = 5 (their Fig. 6), indicate that if anything Ottewill and Shaw's mobility estimates are slightly on the high side. Discrepancies between theory and experiment are, of course, much more likely in solutions of multivalent salts since the Poisson-Boltzmann equation is likely to break down at lower potentials. Extension of the theory to di-and trivalent systems should, therefore, be done cautiously and should be limited to the low potential region.

References Abramson, H . A. (1934). "Electrokinetic Phenomena and their Application to Biology

and Medicine." Chemical Catalogue Co, New York. Abramson, H . A. , Moyer, L . S. and Gorin, M . H . (1942). "Electrophoresis of Proteins."

Reinhold, New York. Anderson, P. J. (1959). Trans. Faraday Soc. 55, 1421. Biefer, G. J. and Mason, S. G. (1959). Trans. Faraday Soc. 55, 1239. Bikerman, J. J. (1935). Kolloid-Z. 72, 100. Booth, F. (1948a). Trans. Faraday Soc. 44, 955. Booth, F. (1948b). Nature (London) 161, 83. Booth, F. (1950a). Proc. Roy. Soc. London. A 2 0 3 , 514. Booth, F. (1950b). J. Chem. Phys. 18,1361. Booth, F. (1951). J. Chem. Phys. 19,1331. Booth, F. (1953). Progr. Biophys. & Biophys. Chem. 3 , 175. Booth, F. (1954). J. Chem. Phys. 22 . 1956. Briggs, D . K . (1928). J. Phys. Chem. 32 , 641. Bull, H . B. (1943). "Physical Biochemistry" p. 162. John Wiley, New York. Bull, H . B. and Gortner, K . (1932). J. Phys. Chem. 36 ,111. Burgreen, D . and Nakache, F. R. (1964). J. Phys. Chem. 68, 1084. Chang, M . Y. and Robertson, A. A. (1966). Pulp and Paper Res. Inst. (Canada) Tech.

Report No. 485. Churaev, N . V . and Deryaguin, B. V. (1966a). Dokl. Akad. Nauk. S.S.S.R. 169, 369. Churaev, N . V. and Deryaguin, B. V. (1966b). Kolloidnyi Zhurn. 28, 751 (Colloid J.

U.S.S.R. 28, 608). Deryaguin, B. V. and Dukhin, S. S. (1971). "Research in Surface Forces" (B. V. Dery­

aguin, ed) Vol . 3, p. 269. Consultants Bureau, New York. Dresner, L . and Kraus, K . A . (1963). J. Phys. Chem. 67, 990. Dukhin, S. S. (1966). In "Research in Surface Forces" Vol . 2, p. 54. (B. V. Deryaguin ed).

Consultants Bureau, New York. Dukhin, S. S. (1970). Special Discussions Faraday Soc. [1], 158. Dukhin, S. S. (1971). "Research in Surface Forces" (B. V. Deryaguin ed) Vol. 3, p. 313.

Consultants Bureau, New York. Dukhin, S. S. and Deryaguin, B. V. (1974). In "Surface and Colloid Science" (E.

Matijevic ed) Vol . 7, pp. 52-72. John Wiley, New York. Dukhin, S. S. and Semenikhin, N . M . (1970). Colloid J. (U.S.S.R.) 32 , 298. Dukhin, S. S. and Shilov, V . N . (1969). Colloid J. (U.S.S.R.) 31 , 564.

Elton, G. A . H . and Peace, J. B. (1957). Proc. 2nd Inter. Cong. Surface Activity Vol. IV, p. 177. Academic Press, New York.

Frumkin, A . N . (1946). / . Colloid Sci. 1, 277. Frumkin, A . N . and Bagotskaya, J. (1948). J. Phys. Colloid Chem. 52, 1. Frumkin, A. N . and Levich, V. I . (1946). Acta Physiocochim URSS 21, 193. Gangopadhyay, A. K . (1961). Kolloid-Zeit. 178, 108. Ghosh, B. N . (1954). J. Indian Chem. Soc. 31 , 273, 393. Ghosh, B. N . (1955). Naturwiss. 42 , 121. Ghosh, B. N . , Choudhury, B. K . and De, P. K . (1954). Trans. Faraday Soc. 50, 955. Ghosh, B. N . and Ghosh, A . K . (1958). KolloidZ., 161, 110. Ghosh, B. N . , Moulic, S. P. and Sengupta, M . (1965). J. Electroanal. Chem. 9, 372. Ghosh, B. N . and Pal, P. K . (1960). Trans. Faraday Soc. 56, 116. Ghosh, B. N . , Rakshit, S. C. and Chattoraj, D . K. (1953). / . Indian Chem. Soc. 30, 601. Glixelli, S. and Stolzmann, R. (1931). Roczniki Chem. 11, 690. Goring, D . A . I . and Mason, S. G. (1950). Canad. J. Res. B 2 8 , 307, 323. Gorin, M . H . (1939). In Abramson, H . A. , Gorin, M . H . and Moyer, L. S. Chem. Revs.

24, 364. Hall , D . G. and Sculley, M . J. (1977). J. Chem. Soc. Faraday 11, 73 , 869. Happel, J. (1958). Amer. Inst. Chem. Eng. J. 4, 197. Harris, L . B. (1970). / . Colloid Interface Sci. 34, 322. Henry, D . C. (1931). Proc. Roy. Soc. (London) Ser. A. 133, 106. Henry, D . C. (1948). Trans. Faraday Soc. 44, 1021. Henry, D . C. and Brittain, J. (1931). Proc. Roy. Soc. (London) A 1 4 3 , 130. Hildreth, D . (1970). J. Phys. Chem. 74, 2006. Hiickel, E. (1924). Physik. Z. 25, 204. Hunter, R. J. (1962). J. Phys. Chem. 66, 1367. Hurd, R. M . and Hackerman, N . (1955). Electrochem. Soc. J. 102, 594. Irmer, G. (1973). Z . Phys. Chem. (Leipzig) 254, 137. Jahncke, E. and Emde, F. (1945). "Tables of Functions with Formulae and Curves".

4th ed. Dover, New York. Jones, G. and Wood, L . A. (1945). / . Chem. Phys. 13, 106. Keizer, A. de., van der Dr i f t , W. P. J. T. and Overbeek, J. Th. G. (1975). Biophys. Chem.

3, 107. Klinkenberg, A. and van der Minne, J. L . (1958). "Electrostatics in the Petroleum

Industry." Elsevier, Amsterdam. Kuwabara, S. (1959). J. Phys. Soc. Japan 14, 527. Levich, V. G. (1962). "Physicochemical Hydrodynamics." Prentice-Hall, Englewood

Cliffs, N . J. Levine, S., Marriott, J. R. and Robinson, K . (1975). J. Chem. Soc. Faraday II, 71 ,1 . Levine, S. and Neale, G. H . (1974a). J. Colloid Interface Sci. 47, 520. Levine, S. and Neale, G. (1974b). J. Colloid Interface Sci. 49, 332. Levine, S., Neale, G. and Epstein, N . (1976). J. Colloid Interface Sci. 57,424. Levine, S., Neale, G. and Epstein, N . (1978). In "Emulsions, Latices and Dispersions"

(Paul Becher and M . N . Yudenfreund eds), p. 289. Marcel Dekker, New York. Levine, S. and O'Brien, Robert N . (1973). / . Colloid Interface Sci. 43, 616. Linton, M . and Sutherland, K . (1957). 2nd Internat. Congress on Surface Activity Vol. I ,

p. 494. Academic Press, New York and London. Long, R. P. and Ross, S. (1965). J. Colloid Sci. 20, 438. McBain, J. W. and Foster, J. F. (1935). / . Phys. Chem. 39, 331. Matijevic, E. (1977). J. Colloid Interface Sci. 58, 374.

Mazur, P. and Overbeek, J. Th. G. (1951). Rec. Trav. Chim. 70, 83. Móller, W. J. H . M . , van Os, G. A. J. and Overbeek, J. Th. G. (1961). Trans. Faraday

Soc. 57, 325. Morrison, F. A. , Jr and Osterle, J. F. (1965). J. Chem. Phys. 43 , 2111. Neale, S. M . (1946). Trans. Faraday Soc. 42 , 473. Neale, S. M . and Peters, R. H . (1946). Trans. Faraday Soc. 42 , 478. Noda, I . , Nagasawa, M . and Ota, M . (1964). J. Amer. Chem. Soc. 86, 5075. O'Brien, Richard W. and Hunter, R. J. (1981). Canad. J. Chem. (in press). O'Brien, Richard W. and White Lee, R. (1978). / . Chem. Soc. Faraday II74, 1607. O'Connor, D . J., Street, N and Buchanan, A . S. (1954). Austr. J. Chem. 7, 245. Oldham, I . B., Young, F. J., and Osterle, J. F. (1963). J. Colloid Sci. 18, 328. Ottewill, R. H . and Shaw, J. N . (1972). J. Electroanal. Chem. 37 , 133. Overbeek, J. Th. G. (1943). Kolloid Beih. 54, 287. Overbeek, J. Th. G. (1946). Philips Res. Reports 1, 315. Overbeek, J. Th. G. (1950). Adv. Colloid Sci. 3 , 97. Overbeek, J. Th. G. (1952). In "Colloid Science" (H. R. Kruyt ed.). Vol . 1, pp. 197-209.

Elsevier, Amsterdam and London. Overbeek, J. Th. G. and Bijsterbosch, B. H . (1979). In "Electrokinetic Separation

Methods". (P. G. Righetti, C. J. van Oss and J. W. Vanderhoff eds.) Elsevier/Nth Holland Biomedical Press.

Overbeek, J. Th. G. and Van Est, W. T. (1953). Rec. Trav. Chim. Pays-bas 72, 97. Overbeek, J. Th. G. and Wiersema, P. H . (1967). In "Electrophoresis" ( M . Bier ed.).

Vol. I I , Chap. 1. Academic Press, New York and London. Overbeek, J. Th. G. and Wijga, P. W. O. (1946). Rec. Trav. Chim. 65, 556. Philip, J. R. (1957). 2nd Aust. Conf. in Soil Sci. (Melbourne). Paper No. 63. Philip, J. R. (I960). J. Hyd. Div. Proc. Amer. Soc. Civil Eng. 86, Hy5, 179. Philip, J. R. and Wooding, R. A. (1970). J. Chem. Phys. 52, 593. Pickard, W. F. (1961). Kolloid Z.Z. Polym. 179,117. Prokopova, E. and Bohdanecky, M . (1973). Coll. Czech. Chem. Comm. 38, 1502. Quist, J. D . and Washburn, E. R. (1940). J. Amer. Chem. Soc. 62, 3169. Reed, L . D . and Morrison, F. A . Jnr. (1976). / . Colloid Interface Sci. 54,117. Rice, C. L . and Whitehead, R. (1965). / . Phys. Chem. 69, 4017. Rice, S. A. and Nagasawa, M . (1961). "Polyelectrolyte Solutions" Chap. 4. Academic

Press, New York and London. Rutgers, A . J. (1940). Trans. Faraday Soc. 36, 69. Rutgers, A . J. and de Smet, M . (1947). Trans. Faraday Soc. 43 , 102. Rutgers, A. J., de Smet, M . and Rigole, W. (1959). / . Colloid Sci. 14, 330. Rutgers, A . J. and Janssen, R. (1955). Trans. Faraday Soc. 51, 830. Samartsev, H . G. and Ostroumov, V . V. (1950). Kolloidnyi Zhur. 12, 136. Scheidegger, A. E. (1957). "Physics of Flow through Porous Media", Univ. of Toronto

Press, Toronto. Semenikhin, N . M . (1976). Colloid J. (U.S.S.R.) 38, 56. Semenikhin, N . M . and Dukhin, S. S. (1975a). Colloid J. (U.S.S.R.) 37, 1013. Semenikhin, N . M . and Dukhin, S. S. (1975b). Colloid J. (U.S.S.R.) 37, 1017. Shaw, J. N . and Ottewill, R. H . (1965). Nature 208, 681. Sigal, V. L . and Alekseyenko, A . M . (1971). Kolloid. Zhur. 33, 737. Slawinski, A. (1926). J. Chim. Phys. 23, 710. Smoluchowski, M . von (1903). Bull. Akad. Sci. Cracovie, Classe Sci. Math. Natur. 1,182. Smoluchowski, M . von (1921). In "Handbuch der Electrizitat und des Magnetismus"

(Graetz) Vol . I I , p. 366. Barth, Leipzig.

Stigter, D. (1967). J. Colloid Interface Sci. 23, 379 Stigter, D. (1978a). / . Phys. Chem. 82, 1417 Stigter, D. (1978b). J. Phys. Chem. 82, 1424 Stigter, D. (1979). J. Phys. Chem. 83, 1670. Stigter, D. and Mysels, K. J. (1955). J. Phys. Chem. 59, 45 Sullivan and Hertel (1942). Adv. in Colloid Sci. 1, 37. Interscience, New York Tikhomolova, K. P. (1974). Colloid J. (U.S.S R ) 36 715 Tikhomolova, K. P. and Boikova, L. A. (1974). Colloid J. (U.S.S.R) 36, 659 Tikhomolova K. P and Duda, L. V. (1974). Colloid J. (U.S.S.R.) 36, 999 ' Urban F., White, H. L. and Stassner, E. A. (1935). J. Phys Chem 39 311 Usui S. and Sasaki, H. (1978). J. Colloid Interface Sci. 65, 36

*£^Vè'.T'de Keizer'A- "* °verbeek'J-Th-G (1979>-j- c m

Van der Drift, W.P.J. T. and Overbeek, J. Th. G. (1979). J. Colloid Interface Sci. 71,

V , SS£- ' l£ T ° V e ? e e k ' J - T h - G ' ( 1 9 4 8 ) ' " T h e o r y of Stability of Lyophobic Colloids. Elsevier, London and Amsterdam. p

White, H. L., Monaghan, B. and Urban, F. (1936). J. Phys. Chem 40 207 Wiersema, P. H. (1964). Ph.D. Thesis, Univers.ty of Utrecht pTs Wgsema, P. H., Loeb, A. L. and Overbeek, J. Th. G. (1966). J. Colloid Interface Sci.

W S t A S 946^ f ' D - thesis, State U n i v e r s i t y U t r e c h t , N e t h e r ] a n d , wooa, JL. A. (1946). J. Amer. Chem. Soc. 68, 437 Wood, U A and Robinson, L. B. (1946). Chem. Phys. 14, 251 Zhukov, P. N . and Fridrikhsberg, D. A. (1950). Kolloidnyi Zhur. 12, 25.

Chapter 4

Measurement of Electrokinetic Parameters

In this chapter we shall examine the various experimental techniques which have been developed for studying electrokinetic effects. In the study of capillary surfaces, granules and fibres, the early interest in electro-osmotic measurements has largely given way to the measurement of streaming potentials. This is no doubt mainly due to the ready availability of electrical measuring devices of very high input impedance ( ~ 1 0 1 4 o h m ) which eliminates one of the principal problems of the streaming potential method. No comparable advance has occurred in the observation of the transport of small volumes of liquids.

By far the greatest interest, however, remains in electrophoresis because i t can be used for materials throughout the whole colloid size range. Even when quantitative estimates of zeta potential are impossible it retains a great practical utility as a means of separating materials like proteins.

Since Overbeek's (1952) discussion of electrokinetics there have been few attempts to review the entire field either in articles or books. The review by Sennett and Olivier (1965) is pitched at an introductory level, as is Shaw's (1969) book. The much more detailed work of Dukhin and Deryaguin (1974) does not pay a great deal of attention to experimental procedures, and only in the field of electrophoresis has there appeared an advanced treatise covering both theoretical and experimental aspects. This is the text edited by Bier (1959, 1967), which concentrates almost exclusively on electro­phoresis in biological systems, as does the review by Strickland (1970). More recently Ball and Fuerstenau (1973) have reviewed the literature on stream­ing potentials with particular reference to asymmetry potentials, and Mac-Kenzie (1971) has collected together a great deal of information on the zeta potential of mineral particles and its relevance in the flotation process. Recently, a review of the microelectrophoresis procedure has been presented

125

by James (1979) but it has not been possible to incorporate that material in the present work. We shall therefore examine experimental procedures in some detail, since the acquisition of reliable data in this area is not at all an easy task.

4 . 1 . Electro-osmosis

The equations developed in Chapter 3 indicate that for measurements of electro-osmotic velocity we require a means of following the movement of the liquid itself and also a measure of the electric field applied along the measuring capillary. The most effective method of following the liquid movement is to suspend in it some colloidal particles and to follow their motion by direct (ultra) microscopic observation. We shall discuss the motion of the fluid in a simple cylindrical or rectangular capillary in Section 4.1.3 and show how the electro-osmotic velocity can be determined. The details of the observation technique and the method of determining the electric field or voltage gradient along the capillary wil l not be discussed until we examine the microelectro­phoresis procedure, since the remarks made there apply equally well to the osmotic measurement.

For measurements of electro-osmotic volume flow, the equations developed in Chapter 3 indicate that we require knowledge of (i) the current flow, (ii) the conductivity of the liquid in the capillary or plug, and (iii) the volume of fluid transported. In addition, the apparatus must provide a means of delivering a steady electric field and, in the case of a plug, an arrangement which permits packing of the plug and maintenance of a fixed geometry for each series of measurements. For work on aligned fibres it may also be desir­able to vary the packing density by applying some pressure around the plug. We shall discuss the various designs for porous plugs in Section 4.2.1 because much the same design considerations apply to electro-osmosis and to stream­ing potential.

4 . 1 . 1 . Electr ical measurements

The current measurement presents no problem, but the electric field should be applied through a reversible electrode system because it is necessary to pass current in one direction for quite a lengthy period, in order to observe sufficient volume transport. The electrode system must be capable of doing so without the gradual development of polarization effects which would alter the effective field across the sample. I t is also important that the electrode reactions and the products of these reactions should not interfere with the surface properties of the plug, so the electrodes are usually situated some

distance from the plug. Ancillary electrodes are then placed near to the plug fo the conductivity measurement. These should be of platinum gauze or expanded platinum so that they do not impede the liquid flow. As noted in Section 3 6 2 it is essential to measure the conductivity of the material m the plug and to compare this with the expected value in order to account properly for surface conduction effects.

4.1 .2 . Measu remen t of l i qu id v o l u m e

Finally it is necessary to measure the volume transported, and this is usually done by observing the movement of a meniscus m an adjouung eapdtoy (Fig 4 1) I t is, of course, essential that the apparatus be absolute y liquid-tight and of fixed volume, because the volumes transported are not large and would be seriously underestimated by leakage or volume expansion m the

" S e f n i m s t r a t e s a typical solution to the problem (Ham and Douglas 1942) The modern use of ground joint apparatus makes construction of uch celis a great deal easier than it was, but it is still difficult to develop an

Z angemenf w h i c h satisfactorily links the parts together but std retams the abihty to change both the plug and the electrolyte (see Ham and Hodgson, 1942; Douglas and Walker, 1950).

using the capillary G.

I

A more recent version developed by Stigter (1964) (Fig. 4.2) for use with fibres uses two procedures for determining the electro-osmotic volume flow: (i) for most situations, stopcock (8) was closed and the amount of liquid flowing through stopcocks (7) and the plug was measured by direct weighing, with the d.c. field applied successively in each direction; (ii) for very small' C-potentials, stopcocks (7) were closed and (8) was opened so that the volume flow could be followed by monitoring the movement of the liquid meniscus in the capillary (6). Adequate temperature control must be provided in the latter case, since temperature fluctuations wi l l directly affect the position of the liquid meniscus. There may also be problems created by the fact that in this apparatus the meniscus must advance along a dry tube. Apart from the problem of sticking caused by minute traces of impurities there is a difference in the contact angle of the liquid when it is advancing compared to when it is receding. The problem is reviewed by Dukhin and Deryaguin (.1974, p. 80) who concluded that great care must be exercised in cleaning the observation capillary and that when examining situations where the volume transport is small a closed system is to be preferred. The error is reduced in the usual

Fig. 4.2. Electro-osmosis cell for fibrous material (Stigter (1964)). In this case Ag/AgCl elec­trodes (5) are used to apply the field to the plug (2).

-ptzzzzz zzzzzzzzzzzzzzzzzzzzzzzzzzzz Fig. 4.3. The bubble in the observation capillary of an electro-osmosis apparatus.

apparatus (Fig. 4.1) where the transport is measured by the movement of a bubble (though later in this section we shall examine an alternative method).

The bubble method has its own problems, which were also discussed by Dukhin and Deryaguin (1974, p. 160), who concluded that the errors in its use are significant and may well be insurmountable, although Biefer and Mason (1954) were able to correct their results and obtain excellent agree­ment with streaming current measurements. The bubble is surrounded by a thin f i lm of liquid of thickness, t, (Fig. 4.3) which depends upon the velocity with which it is moving. Fairbrother and Stubb (1935) suggested the empirical relation:

<-!(?)"* whilst Deryaguin (1943) gave a rather different value:

; = l - 3 2 * ( ^ ) 2 / 3 (4-1.2)

where U is the bubble velocity and y the surface tension. When osmotic flow begins to occur a hydrostatic pressure builds up on the outflow side of the plug causing a backflow of liquid through the diaphragm or plug and also down the observation tube. One aims, of course, to have all, or almost all of the backflow occur down the observation tube, and the fraction which does so depends upon the relative hydraulic conductivity of the plug and the capillary. Mason and co-workers (Goring and Mason, 1950; Biefer and Mason, 1954) write for the correction to the measured velocity of electro-osmotic f low:

V = V l n J l + ^ ] (4-1.3) (obs)\ * '

where 4>„ is the permeabihty of the plug and $ c is the permeability of the

observation tube, complete with bubble. <S>C is calculated by adding the pres­sure drop across the capillary as given by Poisseuille's equation:

Wrilc

to the pressure drop across the bubble due to the flow through the annulus of liquid around i t :

2Vnlb

Pb:

KR*t Then

Vr\

b+Pc/

, y \ 1 / 2

+24 (4.1.4)

\ + 9 / 1

where we have substituted for t from eq (1). The question of whether eq (1) or (2) should be used for the thickness, t,

however, remains an important one because the dimensionless quantity (y/Uri) has a value of 10 3 -10 4 for water when U = 1 mm s~ r . The thickness estimates using the two equations, therefore, vary by 2-300 % at this bubble velocity. Biefer and Mason (1954) present some experimental results in support of their estimates of t but the best way to minimise the problem is to make the hydrodynamic resistance of the plug much higher than that of the capillary so that OP/<S>C 1 (Dukhin and Deryaguin (1974, p. 162)).

I t has also been suggested, by Fairbrother and Stubb (1935), that the volume flow velocity should be calculated by multiplying the bubble velocity by the bubble cross-section rather than the capillary cross-section." This procedure assumes that the annulus of fluid around the bubble remains stationary and the relative error involved is (Dukhin and Deryaguin, 1974, p. 162):

(2R-t)t It fUriV'2

for / 6 p R. The error is larger for smaller bubble lengths because of the end effects, and Biefer and Mason (1954) suggest using 4. ~ 3/? as a compromise to avoid reducing <5C too much (cf. eq (4)) and so increasing the importance of 4>p in eq (3). Dukhin and Deryaguin (1974, p. 163) point out, however, that eq (5) assumes that the liquid is stationary in the annulus whereas in fact it will be undergoing very rapid electro-osmotic motion on its own account.

In an actual measurement one is not so much interested in the bubble

velocity as in the total bubble movement over a known period of time during which a steady current flow occurs. At the very least one should conduct a series of measurements under various imposed fields so that the bubble velocity varies significantly. The degrees of constancy of the ratio of velocity to current then affords a check on the precision of the result. The consistency of Biefer and Mason's (1954) results indicates that the method can be made reliable.

4.1.3. F low in a s ing le c losed capi l lary

A more satisfactory method of following the electro-osmotic flow in a closed capillary is available if i t is permissible to introduce some colloidal particles into the capillary to act as markers for the flow. Direct observation of the particle velocities when a field is applied then gives information on both the electrophoretic mobility of the particles and the electro-osmotic behaviour of the liquid near the capillary wall. I t is not necessary for the particle surface to be identical with that of the wall, although if that is the case it simplifies the analysis and provides a rapid check on the measurements. The method is particularly suited to the study of adsorbed polymers, like proteins, which can be made to cover the capillary walls and also some suitable "inert" colloidal particles.

4 . 1 . 3 . 1 . Cy l indr ica l capi l lar ies The observed motion of the particle when the field is applied, vp(y) is the algebraic sum of its true velocity, vE, and the local liquid velocity, v,(y). This latter wi l l vary with the distance, y, from the axis in accordance with Poiseuille's equation (3.2.1), so the observed particle velocity is parabolic and symmetric about the tube axis. We can therefore write (cf. Mattson, 1933; Alexander and Johnson, 1948):

vp(y) = vt(y) + vE = ay2 + c (4.1.6)

where the linear term is dropped because of the symmetry of the system. The velocity profile is illustrated in Fig. 3.3(b), from which it is clear that the liquid velocity at the wall :

v,(r) = ar2 + (c - vE) = veo (4.1.7)

where ve0 is the electro-osmotic velocity. Since the tube is closed, the overall volume flow is zero:

f r

2nyvi(y)dy = 0 (4.1.8) Jo

and substituting for v{(y) f rom eq (6) and integrating gives:

a -£+(c-vE)— = 0

so that - 2(c - vE)

Q = ?

Using eqs (7) and (9) the liquid velocity becomes:

and this is obviously zero when y = r/J2 = 0-707r. This point in the liquid is called the stationary level, and measurements of

the particle velocity, vp, at that level give the electrophoretic velocity, vE, directly.

The observed velocity at the wall is, from eqs (6) and (10), for y = r:

vp(r) = vE + veo (4.1.11)

and the observed velocity on the tube axis:

vp(0) = v E - v e o (4.1.12)

One can obtain veo by eliminating vE between eqs (11) and (12), but it is better to use data from a complete set of measurements across the whole capillary tube. I f x is the relative distance measured into the tube as a fraction of the diameter, then x= (1 —y/r)/2 and substituting this in eq (6) and rearranging gives, using eq (10):

; vp(x) = (vE+veo)-8veo(x-x2) (4.1.13)

A plot of vp(x) against the function (x — x2) should give a straight line from the slope of which the electro-osmotic velocity is obtained. In practice, this method cannot be easily applied to the cylindrical cell design of Mattson (1933) because optical distortions, introduced by the wall, make it difficult to view particles deep inside the tube. The method can, however, be readily applied to the modern thin-walled cells of the van Gils type (see Section 4.3.1).

(4.1.9)

(4.1.10)

4.1.3 .2 . The rectangular cell The detailed hydrodynamic behaviour of a rectangular cell is quite compli­cated and is discussed in Section 4.3.1.4. For the case where the breadth to depth ratio is large ( > ~ 20), the velocity profile in a closed cell is again parabolic and symmetrical about the axis so that eq (6) is still valid (Fig. 4.4).

• J 1 — Stationary levels —I

Fig. 4.4. Velocity profiles in a rectangular capillary. Full line: l iquid velocity. Broken line: apparent particle velocity. (The two cross-hatched areas are equal.)

Then eq (8) becomes:

vi(y)dy = 2 {ay2 + c — vE)dy = 0 (4.1.14)

from which a = 3{vE - c)/h2. Again at the wall v,(h) = veo and so

Substituting these expressions in eq (6):

3y2

- 1 (4.1.16)

which replaces eq (10).

The stationary levels occur a ty = ± or, in terms of the relative depth x (= ( 1 ~ y/h)/2) they occur at:

*o=.0-5 ± ( & ) 1 ' 2

( 4 J 1 7 )

= 0-211 and 0-789

Near the wall the observed particle velocity must again be:

vp(+h) = vE + veo (4.1.18) but on the axis of the cell:

»»(0) = % — (4.1.19)

In the special case when the walls and the particles are covered with the same material and behaving in the same way, we have from eqs 3.1.3 and

vE = - v.

so that the apparent velocity at the walls of the tube is zero. This provides a z&szzszzad5orba,e (such as a protd">* Substituting for x in eq (16) and using eq (6) we f ind:

VpW = (vE + veo) - 6veo(x - x2) (4.1.20)

v n f v l T " ° f e \ ( 1 3 ) - a d V a D t a g e h e r e i s t h a t t h e r e 1 S " ° « a l difficulty in­volved in making measurements through the entire cell and plotting the

w E T h e H t l t y a g a m S t ^ d C P t h f U n C t i ° n ( — 2 ) ' from slope o f which the electro-osmotic velocity is obtained. In the particular case where veo - - v E the apparent particle velocity becomes (Abramson et al., 1942):

vp(x) = 6vE(x - x2)

= 4vll2(x-x2) (4.1.21)

where v1/2 is the apparent velocity at the midpoint of the cell. This means that:

3vE

vv2~~2 (4.1.22)

Fig. 4.5. A typical curve illustrating the apparent particle velocity in a cell where all surfaces are covered with the same material (in this case a protein). The broken lines would have been ob­tained if the ratio vElv„ were not 1-0 but 0-67 or 0-5. (From Abramson et al. (1942) with per­mission.)

This is a useful relation because it enables the electrophoretic velocity to be measured at the axis of the tube where the velocity gradient is minimal and it wil l be further discussed in Section 4.3. There are other checks which can be used to determine whether the velocity profile is, indeed, following eq (6) exactly and these too wil l be discussed in Section 4.3. At the very least one should always check that the particle mobilities at the two stationary levels are the same, within permissible experimental error. Such a check wi l l reveal anomalies in the hydrodynamic flow pattern due to differences in the electro-osmotic character of the cell walls or to convection currents caused by temperature gradients in the fluid. I t wi l l not, however, enable one to determine whether the flow pattern obeys eqs (14) to (22). Departures from these equations can be anticipated if the width to depth ratio is less than about 20. For smaller ratios the more elaborate hydrodynamic calculations of Komagata (1933) must be used. These wil l be discussed in more detail in Section 4.3.1.4, but at this point we wil l note that eq (17) must be replaced by:

11/2

x0 = 0-5 ± 1 f±\ .1 l2 + \n k

(4.1.23)

where k is the ratio of breadth to depth of the cell. I t is preferable to achieve a large k by increasing the breadth rather than by reducing the depth, because a very shallow cell produces a very sharp velocity profile with a consequent increase in error due to the finite depth of field of the microscope objective used for the mobility measurement. I f cells of modest k value must be used,

T A B L E 4.1 of x0 calculated from eq (23)

x0 0-823 0-806 0-801 0-798 0-792 0-789 0-177 0-194 0-199 0-202 0-208 0-211

the stationary level is determined from eq (23) and measurements at that point give values of vE. The velocity of the liquid at the wall is still equal to veo so its value can be obtained from the extrapolated value of the apparent particle velocity at the wall, after subtracting vE. Some values of x0, calculated from eq (23) are given in Table 4.1, from which it is apparent that even at k = 20, the correction is significant.

4.1.4 . E lec t ro -osmot ic coun te r pressure

Very little work has been reported on this procedure. Biefer and Mason (1954) made some preliminary measurements in order to overcome the un­certainties introduced by the bubble in electro-osmotic studies using a capillary flowmeter. They used a capillary differential manometer of small bore ( ~ 1 mm) in order to reduce the volume of liquid transported. Tem­perature fluctuations and some lack of reproducibility in the system rendered the results inaccurate, and obviously much more needs to be done on this technique.

The application of an alternating potential and the resulting sinusoidally varying pressure was examined theoretically by Cooke (1955), who showed that meaningful results can be obtained even when the electro-osmotic flow is not zero. His experiments were concerned, however, mainly with a.c. measurements of the streaming potential so we shall delay consideration of them until Section 4.2.7.

4 .2 . S t r e a m i n g p o t e n t i a l m e a s u r e m e n t s

Streaming potential measurements require a knowledge of: (i) the applied pressure, (ii) the conductivity of the liquid in the capillary or plug, and (ii) the streaming potential developed. Again the apparatus must supply a constant stream of the (uncontaminated) electrolyte to the capillary or plug and should permit easy replacement of both the streaming liquid and the plug while retaining a fixed geometry for the plug packing when an experiment is in progress. Measurements on capillaries should not present any real difficulties (Bull, 1934), and an upgraded version of the device used by Jones and Wood (1945) should suffice for most purposes.

Ground glass joint

Diaphragm

Pt wire

Tungsten seal

Fig. 4.6. Solid-l iquid streaming potential cell after Gortner et al. (From Alexander and Johnson,

1948, p. 341.)

4 . 2 . 1 . Cells for use w i t h p o w d e r s

A typical cell for use with powders is that designed by Gortner etal. (Martin and Gortner 1930; Lauffer and Gortner, 1938) and reproduced schematically in Fig 4 6 More recent designs are provided by Buchanan and Heyman (1948) Hazel (1953), Biefer and Mason (1954), Gaudin and Fuerstenau (1955)! Fuerstenau (1956), Pravdic and Mirn ik (1958), Watanabe et al. (1961), Robinson et al. (1964), Somasundaran and Kulkarm (1973), Horn and Onoda (1977) and Srivastava and Yadav (1979). None offers a complete solution to all the problems involved, not the least of which is that of provid­ing electrode systems to measure accurately both the conductivity and the potential. I t is common practice to use a pair of platinum gauzes to contain the plug and also to function as the conductivity measuring electrodes, as is done in Fig 4 6. Wi th dense materials like mineral oxides and sulphides a vertically oriented plug is easily made up and, provided streaming is only conducted in the downward direction, the geometry of the plug remains constant. I f streaming in both directions is required (see below), the packing

Fig. 4.7. Horizontal streaming cell. The L .H . electrode is fixed and the R.H. one removable to allow the plug to be flushed.

must be carefully done so that the upper electrode maintains a sufficient positive pressure on the plug material. In cells of all glass construction it is easier to establish, and to change a plug, using a horizontal design like that shown in Fig. 4.7. The plug material is introduced through the upper socket whilst the assembly is ful l of fluid. Entrapped air can be removed by replacing the stopper A with a vertical tube and backwashing through either or both electrodes. Applying a vacuum through the cone A has a similar effect and avoids the formation of a bed which is graded from top to bottom in particle size.

I t is possible to use the platinum gauze electrodes for both conductivity and potential measurement, but separate assemblies are preferred when platinum is used. Even with a separate pair of platinized platinum electrodes to measure the potential, Hunter and Alexander (1962) found an asymmetry potential, EA, which under working conditions was 20-40 mV. They showed that it could be allowed for by streaming in alternate directions or, more easily, by backing off the asymmetry potential when the liquid was stationary. The early workers (Martin and Gortner, 1930; Bull, 1935) avoided this problem by working at higher driving pressures when EA is negligible, at least at low electrolyte concentrations. More recently Lewis et al. (1964) sought to eliminate the problem by using calomel electrodes with salt bridges (cf. Hazel, 1953) to pick up the potential. This made it necessary for them to provide a mechanism for preventing contamination from the salt bridge on the upstream side. Their design, however, does not permit a measurement to be made of the conductivity of the plug, although it does make for easy removal of the plug material.

In an extensive review of the problem of asymmetry potentials, Ball and Fuerstenau (1973) conclude that one should always measure Es as a function oïp over a sufficiently wide range to obtain an accurate estimate of the slope, (EJp). I t is this slope which is used in eq (3.2.5) to calculate the {-potential. Ball and Fuerstenau show that in many cases the literature data reveal the existence of an asymmetry potential since the streaming potential shows a

finite intercept at p — 0. The ratio EJp at various values of p wi l l then not be strictly constant. Their criticism of the "back-off method of eliminating the asymmetry (Hunter and Alexander, 1962) is, however, quite unjustified. The remaining intercept revealed in the data of Hunter and Alexander (1962) is less than 0T mV which is much less than could be detected in most experi­mental runs. The high degree of scatter in their data is also easily understood when it is realized that in those experiments the asymmetry potential was about 30-40 times as large as the streaming potential. Under normal cir­cumstances (that is, for pressures of the order of 5-20 cm Hg) the asymmetry potential would be much smaller than Es, and at worst of comparable magnitude. The asymmetry potential can then be backed-off quite accurately enough for most purposes. Rather than backing off the EA value, Horn and Onoda (1977) store it on a very large capacitor and then subtract it from the measured signal when flow first begins; if i t is small, i t can sometimes be nullified simply by keeping the electrodes short circuited between measure­ments (Ottewill, priv. comm.).

The very small currents involved in these measurements usually call for careful shielding of the electrode system and the capillary or plug assembly, though some workers have not found this essential.

Some of the more obvious cell fabrication problems can be overcome by the use of plastic materials rather than glass. The cell designed by Parriera and Ottewill (see Parriera and Schulman, 1961) using Perspex and Teflon has been improved upon in a number of respects by Joy et al. (1965), whose design is reproduced in Fig. 4.8. They use rather thick perforated discs of silver coated with silver chloride to retain the plug and to measure both the potential and the conductivity. They overcome most, if not all of the asym­metry and other stray potential problems (which can be present even when Ag/AgCl electrodes are used (Pravdic and Mirnik, 1958) by machining the complete electrode and its lead-in rod from a solid piece of silver rod and electrodepositing a matt silver coating over its entire surface.

Instead of screwing the electrodes down on to the plug (as Parriera and Ottewill did), Joy et al. slid the electrodes through a leak proof "0" - r ing assembly which avoids scratching the electrode surface against the plug material. The chief objection to this type of apparatus is that surface con­taminants may be dissolved from the Perspex body, but Joy et al. claim that a preliminary soaking in sodium carbonate (5 % for 24 hours) followed by a detergent wash and warm water rinse eliminates this problem. I t must be remembered that the glass apparatus usually favoured is not without its own contamination problems: some workers have observed effects attributable to adsorbed silica on oxides stored in glass containers at low particle con­centrations (i.e. small specific surface area conditions) (Wright, 1972, p. 84). More serious in some situations is the introduction of adsorbed Cr 3 + ions

if chromic acid is used as a cleaning agent. Rinsing with concentrated mineral acid followed by N a O H is absolutely essential if this reagent is used, and some would doubt that even this procedure is adequate. There is no reason why surfactants cannot be used for some cleaning purposes, provided storage bottles and apparatus can be rinsed with a colloidal suspension of the sample material of high area to remove any residual adsorbate.

Joy et al. (1965) also use plastic tubing (polythene and silicone rubber) and silicone rubber bungs in the ancillary apparatus and again use the same cleaning technique. The justification here is less obvious since the dangers of releasing plasticizers and other contaminating surfactants must be greatly increased when flexible tubing is used; a really successful procedure for removing the plasticizers often renders the tubing so inflexible that one might as well resort to glass.

A rather different form of streaming potential apparatus was constructed

by Street (1964) for determining the {-potential of flat sheets of material, in the form of discs. Figure 4.9 shows schematically an improved version of that apparatus (Lyons, 1979) which permits flow of electrolyte in both directions between the discs (1 and 2 in the figure). Entry or exit of the fluid occurs through a hole in the centre of the lower disc, and the streaming potential is measured between two Ag/AgCl electrodes (6).

The analytical expressions for the convection current {Is) and the conduc­tion current {Ic) in this apparatus are analogous to those developed in eqs

Manometer

Tap _ ^

O f \ Compressed N 2

y

Gate

D2

J i

5 7

Faraday cage

Electrometer Recorder pH meter -

Fig. 4.9. The radial streaming potential apparatus for measurements on solid sheets (Lyons, 1979). (1) Lower disc with central hole, (2) upper disc attached to micrometer, (3) electrolyte reservoir ( P M M A ) , (4) stainless steel barrel, (5) electrolyte reservoir (boro-silicate glass), (6) Ag/AgCl electrodes, (7) combined pH electrode, (8) thermometer. The solenoid-operated gate allows either side of the apparatus to be opened to the atmosphere and pressure to be applied to the other side.

142 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

(3.2.3) and (3.24) but with the capillary cross-sectional area (nr2) replaced by the area of a cylinder (2nrh0, where h0 is the distance between the discs). The resulting expression for the streaming potential is therefore identical to the Helmholtz-Smoluchowski equation (3.2.5).

A quite different form of apparatus for measuring the streaming potential of flat surfaces is described in the Soviet literature (see e.g. Zhukov, 1974). This is the rotating disc method in which the liquid wells up through a hole in the (horizontal) disc which rotates at a constant angular velocity co. The height of the liquid diminishes with distance from the centre, and the rota­tional motion creates an effective driving pressure given by P = co2r2d/2 where d is the density of the fluid. The streaming potential, E„ is measured between two concentric ring electrodes at distances rx and r2 from the central hole and is given by:

e0DC [(Q2(r2-r2)d] s ril0 2

Zhukov (1974) discusses the corrections necessary in this case if surface con­duction is significant.

I t should also be noted that for streaming potential measurements the very low ratio of solid surface area to solution volume makes it essential to use highly purified solutions and to pay particular attention to the water. Purification by exchange resins (especially the cationics) has been found to leave traces of low molecular weight surfactants in the water.

4.2.2. The electr ical measurements

We noted above that one of the principal design problems was to eliminate stray potentials in the measuring system. Fabrication of the electrode from a single piece of silver rod, as Joy et al. (1965) suggest, may solve the problem, but many wil l no doubt continue to use more conventional methods. We shall therefore discuss some of the problems which can be expected to arise in such cases. The asymmetry potential, EA, already mentioned is made up of contact potential differences between the metals in the electrode circuit, thermo-electric e.ml.s, and also real differences in the response of the supposedly identical electrode surfaces when placed in a common solution. This latter is presumably caused by differences in the adsorption charac­teristics of the electrode, surface due to small differences in preparation pro­cedure or stress effects induced during fabrication of the electrode shape. Ideally the contribution from this source should be less than 1 mV, but it can be several tens of millivolts and quite persistent. Hunter (1961) found that some pairs of electrodes, after platinizing, could generate as much as 200 mV when immersed in a common solution, and although this could be

4. M E A S U R E M E N T O F E L E C T R O K 1 N E T I C P A R A M E T E R S 143

eliminated by an appropriate prolonged electrolysis regime it rapidly re­turned to values of around 20-40 mV when the electrodes were left to stand in alkaline or acid solutions.

Wright (1972) found evidence of an additional electrochemical effect generated in platinum electrodes made up as shown in Fig. 4.10a. I t is well known that the large difference in thermal expansion between platinum and borosilicate glass makes it impossible to obtain a good Pt-glass seal. I t is usual to form the seal with tungsten wire as in Fig. 4.10a, but this procedure may allow liquid to penetrate back along the platinum wire to generate an e.m.f. at the P t - W junction. A better arrangement is shown in Fig. 4.10b. The joint between the platinum and copper wire is separated from the inner walls of the glass tube so that it remains dry. The remote end of the glass tube is left open so that significant amounts of liquid cannot accumulate in the tube due to capillary leakage along the platinum-glass "seal".

Contact potentials and thermal effects can be reduced by maintaining symmetry of the arrangements on each side of the measuring system. The potentials due to fabrication stresses can be reduced by using a separate platinum wire, coated with silver and then with silver chloride (Robinson and Stokes, 1959) to measure the potential. The platinum gauze electrodes which form the cell are then used only for the conductivity measurement. Any remaining e.ml. should be constant and, as noted above, can be backed off before the streaming potential measurement is made or eliminated by flowing in both directions (Hunter and Alexander, 1962).

Many of these stray potential effects have been aggravated by the use of high input impedance millivolt amplifiers, like the Keithley 600 series. The combined capacitance of the leads and the input circuitry and the very high input resistance ( ~ 10 1 4Q) of these devices makes them capable of respond­ing to very small charge flows which in less sensitive instruments would go undetected. For this reason it is important when using them to provide a short unimpeded path from one electrode to the high input side of the meas­uring instrument. The backing-off arrangement for the elimination of the asymmetry potential should then be on the low terminal side which is well earthed and screened.

The use of a recorder on the output of the electrometer makes it easier to examine electrode symmetry, rate of attainment of equilibrium, spurious potentials due to pressure fluctuations and constancy (or absence) of the asymmetry potential (Joy et al., 1965). Parriera (1965) gives details of an automatic instrument which uses a pressure transducer and feeds the output to an X - Y recorder.

The resistance of the plug is usually measured with an a.c. bridge; the modern transformer bridges (such as the Wayne-Kerr) are both accurate and convenient to use but it must be noted that they apply a substantial a.c.

144 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

Fig. 4.10b. Pt/Cu electrode connection. Leakage of electrolyte around the platinum is un­important as long as the junction remains dry.

voltage to the system during a measurement. The platinum gauze electrodes, if used, must therefore be well platinized or converted to Ag/AgCl electrodes to eliminate polarization effects. Joy et al. (1965) recommend that the measurement be made whilst the liquid is actually streaming in which case the conductance bridge must be isolated from the d.c. streaming potential by a 4uF paper capacitor in each lead. Most workers, however, have been content to make the resistance reading whilst the liquid is stationary.

4.2.3 . Pressure measurement

The usual practice is to use a simple manometer (mercury or water) for the pressure measurement and to supplement the hydrostatic pressure of the streaming liquid with pressure from a source of clean, C0 2 -free gas (usually nitrogen). As noted above, however, (Parriera, 1965) a pressure transducer can be incorporated.

4. M E A S U R E M E N T O F E L E C T R O K I N E T I C P A R A M E T E R S 145

4.2.4. The cel l pack ing

For granular material the pore size and packing density are determined by the particle size distribution. I t should be obvious that the plug must be filled in such a way as to eliminate air, but it should also be remembered that the particles may act as nucleation sites for the formation of bubbles due to gases coming out of solution when the temperature is raised. Gas evolution from the electrodes can also present problems, so it is advisable to be able to flush the cell freely before measurements. Alternatively, as noted above, the system may need to be evacuated temporarily.

When working with heterogeneous material the procedure recommended in Fig. 4.7 for backwashing the plug, in order to remove gases and to form a reproducible plug, may cause a particle size segregation to occur. This wi l l mean that the porosity of the upper part of the plug wil l be smaller than that of the lower part, with consequent sampling errors if the material is unevenly charged.

Several designs have appeared for cells which permit a pressure to be applied to the plug material (Goring and Mason, 1950; Biefer and Mason, 1954; Stigter, 1964). This permits an examination of the effect of plug porosity on electrokinetic behaviour, and it is relevant for fibrous or platelike material (like clays) but is not normally applied to granular beds of incompressible materials.

4.2.5. Data t rea tment

The very first test of the reliability of streaming potential data comes from the plot of Es against applied pressure. The linearity of that relation is a necessary first step though i t does not ensure accuracy of the resulting £-potential unless adequate steps are taken to overcome surface conduction problems. In the case where the plug is composed of easily compressible material i t is possible for the applied pressure to influence the plug geometry. The simplest procedure in that case (Joy et al, 1965) is to plot EJR^V against pressure which amounts to applying the Briggs (1928) method via eq (3.2.6) to a porous pad. The limitations of that approach have been discussed at length by Biefer and Mason (1959) (see Section 3.6.2), but the appearance of a linear plot of EJRap against pressure would go some way towards justifying the Briggs procedure.

4.2.6. Measuremen t of s t reaming current

The principles behind this measurement have been discussed in some detail by Hurd and Hackerman (1955), who used it to study the electrokinetic

146 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

ZZZZZ7

zzz

W / / / / / / / / / / / T Z Z ZZ2

TZZ1

i o 3 - i o " a.

VWVNAAAAAA

Amplif ier

Recorder

Fig. 4.11. Measurement of streaming current. The electrodes should be of platinized platinum gauze to avoid polarization and impedance of the liquid flow.

behaviour of metals in contact with aqueous solutions. The streaming cur­rent at low electrolyte concentrations is ~ 1 ( T 1 0 - 1 0 " 1 1 amp and the re­sistance of a typical capillary is of order 107£2 or more. To provide a low resistance by-pass (see Section 3.2.3) the cell can be shorted with a resistor of the order of l O M O 4 ^ across which a potential of order 1 uV wil l develop (Fig. 4.11). One must therefore use a stable microvolt amplifier with an input impedance of about 1 M£2 to achieve a precision of better than 1 %. Hurd and Hackerman (1955) show that the measured current in their systems was independent of the size of the shunting resistance in the range 10 4 -10 6 Q (Fig. 4.12). One can readily test whether the electrical measurement pro­cedure is adequate by calculating the anticipated streaming potential. Pro­vided the voltage measured across the shunting resistor is much smaller than this (say < 0T %) there should be negligible error f rom this source. I t is noteworthy that in the range 1 0 ~ 6 M KC1 upwards, the {-potential of all three metals studied (Au, Ag and Pt), showed a monotonic decrease in (absolute) value (Hurd and Hackerman, 1956) with no indication of the maximum which occurs around 10" 3 M when surface conduction effects are possible. Pravdic (1963) describes an improved form of the Hurd and Hackerman apparatus for use with powders.

4. M E A S U R E M E N T O F E L E C T R O K I N E T I C P A R A M E T E R S 147 4.2 .7 . S inuso ida l measurements

The first reported study of electrokinetic effects using a sinusoidal pressure was apparently made by Williams (1948), who described a suitable trans­ducer but did not attempt a mathematical analysis. Some more extensive studies were undertaken by Ueda et al. (1950, 1951a, b), who demonstrated that the peak-to-peak value of the a.c. potential depended on both the fre­quency and amplitude of the pressure variations. The dependence on ampli­tude is linear and can be accommodated merely by referring all measure­ments to a single (arbitrary) amplitude value. The dependence on frequency is more complicated, although Ueda et al. were able to show that the quali­tative effects were similar at different frequencies. Figures 4.13a and b indi­cate the significance of the amplitude and frequency effects. I t is apparent that a suitable extrapolation procedure is required to obtain the absolute value of Es at zero frequency. The Japanese workers have preferred to work at an arbitrary frequency, however, and to obtain what may be called a relative zeta potential. Their approach has been reviewed in some detail by Watanabe (1963). A more recent application of this empirical procedure is given by Beck et al. (1978), in the study of paper-making components.

Cooke (1955) has presented the necessary theoretical relationships for deriving the {-potential from measurements of the a.c. streaming potential

8 h

v — t , ^ e a —

1 0 4 1 0 5 1 0 6 1 0 7 1 0 8 • 1 0 9 1 0 1 0

Shunt resistance (ohms) Fig. 4.12. Influence of shunting resistance on measured streaming current. (From Hurd and Hackerman (1955) with permission.)

developed when a sinusoidally varying pressure is applied to the system. Strictly speaking, Cooke measured a "pseudo streaming potential" across a shunting resistor which was not very large, but given its value, the true streaming potential can be calculated. The total current passing through the cell may be written:

/ = L n £ + L 1 2 p (4-2.2)

where L n is the ordinary electrical conductance under zero flow conditions and L 1 2 measures the electrokinetic effect. For a true streaming potential measurement, 1=0 and then

n

from eq (3.2.5). The sign change involved here is explained by Cooke, who also shows that if the current is not zero then L 1 2 is given by

(4.2.3)

4. M E A S U R E M E N T O F E L E C T R O K I N E T I C P A R A M E T E R S 149

where Rt is the total resistance between the two electrodes and E's is the pseudo streaming potential. The pressure is applied using a diaphragm which passes a sound wave through the sample and the sinusoidal potential is dis­played on a cathode ray tube. Details of the apparatus and the limitations of the method are contained in Cooke's (1955) paper. Values of Es independent of frequency are obtained on porous plugs only up to frequencies of about 20 Hz. Above that the r.m.s. value of Es falls off regularly with increase in frequency.

A theoretical analysis of the frequency effect has been given by Packard (1953) who obtained good agreement between his theoretical calculations and data on capillaries, for which the effect is quite pronounced. The cor­rection to eq (3.2.5) is given as a ratio of Bessel functions, and the magnitude of the correction depends on the parameter Ya, where

Here a is the capillary radius, p and rj are the fluid density and viscosity and co is the frequency. The correction is 0-50 (i.e. the r.m.s. voltage is decreased to half its value at co = 0) for Ya = 3-50 and 01 for Ya = 20. Thus for water in capillaries of radius 1 mm one would expect the measured a.c. streaming potential to be only half of its static value when co is about 2 Hz. Extrapolation of data to zero frequency is clearly essential in this case. I t is not clear why the behaviour is quantitatively different in plugs (though it is qualitatively similar (Cooke, 1955)), but in that case the extrapolation to zero frequency is simple since, as noted above, the a.c. streaming potential is constant below about 20 Hz.

Rather than extrapolating to zero frequency Sears and Groves (1978) calculated the correction factor as a function of Ya, using the theoretical analysis of Uchida (1956). They showed that consistent values of {-potential can be calculated for capillaries (radius 0-5-1-0 mm) over a frequency range from 0-4-21 Hz when the correction factor varies from near unity to about 0-3. This gives support to Uchida's analysis and suggests that, at least for simple capillaries, i t might be possible to design much more compact, closed systems for streaming potential measurement. Note that this procedure yields absolute values for the {-potential and does not require calibration against the direct flow method; unfortunately it does not seem to be possible to distinguish the sign of the {-potential in the Groves and Sears (1975) apparatus; this would require a knowledge of the phase relationship between the applied sinusoidal pressure wave and the output voltage which should not be impossible to determine.

The principal advantages of the a.c. method are that it eliminates electrode polarization effects, permits easier amplification of the output signal, and

150 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

presents a smaller input impedance problem. Thus Ueda et al. (1956) were able to show that the relative zeta potential in non-aqueous systems increases with the dipole moment of the liquid phase molecules and is zero for benzene and carbon tetrachloride. The very high impedance of such systems would make d.c. measurements well nigh impossible.

Another method of obtaining a.c. estimates of interfacial parameters is to set the capillary itself into, mechanical vibration and to measure the output voltage. Watanabe (1963) describes a number of very interesting experi­ments using this technique which can be used to determine the integral double-layer capacity at the mercury-solution interface. Indeed, by using a capillary containing many ( ~ 40) beads of mercury separated by drops of electrolyte solution, the mechano-electrical effects can be amplified so markedly that the device can function as a suitable transducer for a gramo­phone pick-up or supersonic hydrophone. The very flat frequency response of some of these devices indicates that their mechanism of operation is not the same as those referred to earlier where the a.c. effects are induced by varying the driving pressure at constant interfacial area. In these U-effect-II devices, as the Japanese workers call them, the interfacial area changes sinusoidally and at constant capacity per unit area. This involves an a.c. current flow into and out of the interfacial region, determined by its capacity. The (-potential is not involved, however, in this type of measurement, since the double layer is not undergoing shear.

Alternating-current measurements have also been applied to the measure­ment of adsorption onto a colloid surface using a so-called streaming current detector (Cardwell, 1966). The apparatus was originally designed by Gerdes (1966) and operates by driving a dilute colloidal suspension into and out of a narrow annulus between two cylindrical plastic surfaces. The output current is determined by the charge on these plastic surfaces which is in turn deter­mined by competitive adsorption of an adsorbate between the surfaces and the colloidal dispersion. Though the device is unsuitable for quantitative studies of (-potential i t does permit the establishment of an adsorption iso­therm in cases where the adsorbate has a significant effect on (, and it can evidently follow fairly rapid changes. Since the frequency of alternation is about 4 Hz, the lower limit of kinetic studies would presumably be of the order of one minute or so.

4 .3 . Electrophoresis measurements

Electrophoresis is by far the most common procedure for determining (-potential, and a large number of different techniques have been developed for application to different types of material. The most important, from a

4. M E A S U R E M E N T O F E L E C T R O K I N E T I C P A R A M E T E R S 151

colloid chemical viewpoint, is the microelectrophoretic procedure (Section 4.3.1) in which the movement of individual particles under the influence of a known field is followed directly in an ultramicroscope assembly (or dark-field microscope). Even for small particles like proteins and colloidal electrolytes (surfactants), which are invisible in such systems, much useful information can be obtained by adsorbing them on to "neutral" substrates of reasonable size—commonly either pure paraffin oil drops (Nujol), solid paraffin wax, or a solid substrate of known properties like silica. We shall therefore concentrate initially on this procedure. The alternative moving boundary method (Section 4.3.2) has been used more extensively in bio­chemistry (see e.g. Henley and Schuettler, 1955); Strickland, 1970) for the separation of proteins; attention then is focused on mobility rather than on (-potential, and the theory and practice of that procedure have been very fully reviewed in the monographs edited by Bier (1959, 1967). The micro­electrophoresis method discussed here has also been extensively reviewed recently by Seaman (1975) and James (1979).

Smith (1973) points out that the microelectrophoretic method has the following advantages over the moving boundary method:

(i) the particles are observed in their normal sol environment; (ii) very dilute sols can be studied so that flocculation rates, even at high electrolyte concentration or near the p.z.c. are negligible; (Hi) the high magnification of the ultra-microscope system leads to very short ob­servation times and high sensitivity; and (iv) in a polydisperse sol the particles in a chosen (though necessarily wide) size range can be observed while others are ignored.

With reference to (ii) it should be noted that only very dilute sols can be studied by the microelectrophoretic technique. The particle concentration should be so low as to induce only the slightest haziness in the solution, otherwise interparticle interference and multiple light scattering may vitiate the results.

The rather crude adaptations of ordinary light microscopes for micro­electrophoresis measurements have given way in recent years to instruments designed specifically for the purpose. A very rugged device, suitable for use by unskilled technicians, was developed by T. M . Riddick and is marketed by Zeta Meter Corp. (New York). A highly competitive device, designed by G. V. F. Seaman (see Blume et al, 1978) and an improved version (The Mark II) designed by A. L. Smith are marketed by Rank Bros. (Bottisham, Cam­bridgeshire, England). The notes compiled by Dr Smith for this latter in­strument have been used in some of the following discussion.

A much more elaborate series of instruments is now being developed by the Pen-Kem Company of New York. These are aimed at reducing or elimi­nating the tedium associated with the velocity measurement of individual

152 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

particles. Sophisticated electronic procedures make it possible to collect frequency distribution curves for mobilities of particles in a fraction of the time required for manual data collection (see Section 4.3.1.5).

Electrophoresis of particles can also be followed by tracer methods and by measuring mass transport (Section 4.3.4). For all sizes of particles, in­cluding proteins and surfactant micelles the more recent technique of laser light scattering in the presence of an electric field provides a very promising method of determining mobility (Section 4.3.5). Finally, there remains the possibility of determining the mobility in an a.c. field (Section 4.3.6), although this has found little favour to date.

Equations (3.3.1), 3.3.3), and (3.3.5) indicate that £ can be calculated directly from the particle mobility, i.e. the velocity per unit field. A suitable apparatus must therefore be able to impose a known electric field on the particles and provide a means of measuring the actual particle velocity due to that field.

4 . 3 . 1 . M ic roe lec t rophores is

The electrophoretic velocity of colloidal particles can be determined by direct observation of a colloidal sol in a closed capillary (Section 4.1). Both the particles and the liquid are affected by the imposed field, and a proper estimate of the particle velocity, vE, can only be obtained when account is taken of the fluid motion. vE is obtained by direct observation at the so-called stationary level where the liquid velocity is zero, or by a consideration of the entire velocity profile. The positions of the stationary levels in cylindrical and rectangular capillaries have already been calculated (Section 4.1.3) and in most cases only measurements at those levels are necessary. A check on the accuracy of the measurement and on whether the cell is behaving in accordance with the equations derived in Section 4.1.3 can be made by making observations at both stationary levels (i.e. on the near and far side of the capillary).

From time to time, however, and certainly when setting up for the first time, it is advisable to establish that the entire velocity profile is parabolic and that the value of vE obtained at the stationary level is the same as that obtained by extrapolation using eqs (4.1.13) or (4.1.20), or other special relations like (4.1.22).

An alternative procedure, suggested by Black and Smith (1966), is to calculate the mobility from the integrated area under the velocity profile divided by the apparent cell depth, da. The mean velocity of the particle with respect to the liquid may be defined as:

1 f d " ,

4. M E A S U R E M E N T O F E L E C T R O K I N E T I C P A R A M E T E R S 153

Apparent depth ( f i m )

Fig. 4.14. Comparison of position of stationary level with that calculated using the Komagata correction for a rectangular cell (from Black and Smith (1966)). The stationary level is marked by a broken vertical line and is determined as the point at which the observed mobility equals that calculated f rom eq (4.3.1). This value is shown on the figure in c m 2 V _ 1 s _ 1 . Reprinted from Journal American Water Works Association, Vol . 58, (1) (January 1966), by per­mission. Copyright (1966), American Water Works Association.

where z is the apparent depth coordinate across the tube. For a rectangular cell:

l r*" i r«° t > £ = j - | (vE + v!)dz = j \ Vp(x)dz (4.3.1)

since the integral of u, over the whole tube is zero if the tube is closed. Figure 4.14 shows a comparison of the theoretical and observed stationary level positions using this latter procedure on a rectangular cell for which the Komagata correction (eq (4.1.23)) was necessary.

An alternative procedure, much favoured by those working with biological material, is to eliminate (or at least reduce).the electro-osmotic flow in the cell by coating its surface with a suitable (polymeric) material. Van Oss et al. (1974) show that by coating the inside walls and the ends of the capillary with agarose (a polysaccharide), the electro-osmosis can be entirely eliminated and the particle motion is then independent of position in the cell. Nordt et al. (1976) achieved similar results with a number of specially prepared polymers, the best of which was a diethylaminoethyl derivative of methyl-cellulose. Although highly successful in eliminating the velocity gradients which cause significant errors in mobility determination, these procedures are not recommended for general use. I t appears that the polymers used do not adsorb on biological surfaces, but the fact that they do adsorb on glass makes them useless for work on inorganic systems. Even "irreversibly" adsorbed material could not be guaranteed not to transfer to receptive colloidal particles in the cell.

154 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

4 . 3 . 1 . 1 . S ing le cel l des igns The construction of a cell which is easily cleaned, rugged and reliable is not a simple task, though many of the problems have been minimized in the commercial instruments which have been on the market for some years and to which reference was made in Section 4.3.1.

The salient features of the various microelectrophoretic cells have been described by Shaw (1969), who presents line drawings of the designs by Mattson (1933), van Gils and Kruyt (1936), Shenkel and Kitchener (1958) and Douglas (1957); the reader is referred to that work for further details. Gittens and James (1960) suggest a number of special features for a rec­tangular cell, and Black and Smith (1965) suggest some improvements in the usual procedures. More recently, Shergold et al. (1966) described a de­mountable rectangular cell for work with mineral particles. The cell is based on a commercially available spectrophotometer cell (cf. Marshall, 1966) with no ends, and this facilitates cleaning considerably.

The original cylindrical cell of Mattson (1933) was made by grinding two flat surfaces on a thick-walled capillary of uniform bore (Fig. 4.15a). The illuminating light beam (focused on the appropriate level) was sent in hori­zontally and the particles were viewed from above. Alexander and Saggers (1948) reversed this arrangement so that settling particles remained in the field of view a little longer. Their design, however, does not prevent settling particles from moving into regions where the liquid velocity changes, and for rapidly settling particles it is preferable to use a rectangular cell mounted laterally (Fig. 4.16b) so that the particles always maintain the same hydro-dynamic environment. This arrangement also has the advantage that the settled particles coat a surface which is not important in determining the electro-osmotic flow pattern. In both the cylindrical cell and the horizontally mounted flat cell, the settled particles can cause a significant upset in the liquid flow pattern, especially if they are very different in charge from the glass surface.

s

Fig. 4.15. Cross-section of (a) the Mattson cell and (b) the van Gils cell. H' is the apparent posi­tion nf the noint H due to refraction effects.

4. M E A S U R E M E N T O F E L E C T R O K I N E T I C P A R A M E T E R S 155

(a) (b) (C)

Fig. 4.16. Rectangular cell mounted in the (a) horizontal (b) lateral and (c) vertical mode. For (a) the microscope axis would be in the plane of the paper and for (b) and (c) it would be at right angles. Dark field illumination is usually provided by a cardioid condenser (see Hartman et al (1952) for details).

The flat cell does have some disadvantages, however: it is not suitable for use with laser illumination and its larger cross-sectional area requires larger current flows, which places greater stress on the electrode system. Heat transfer is also poorer, and the illuminating beam and the electric current can therefore cause convection currents which disrupt the hydrodynamic flow pattern. For these reasons the very thin-walled van Gils cell (Fig. 4.15b) is preferred in all cases where settling is not an important consideration.

4.3 .1 .2 . Opt ica l p rob lems The chief disadvantage of the Mattson cell is the optical distortion that i t induces in the particle images. The cell acts like a planocylindrical lens which has the effect of shifting the apparent position of the stationary level. A correction was first proposed by Henry (1938) and Hunter and Alexander (1962) showed that for a glass cell (refractive index = 1-51) viewed with the microscope objective in air, Henry's correction meant that the apparent depth of the stationary level below the apparent position of the inner wall was 0-316a' where a' was the apparent cell radius. Without Henry's correction it would be expected at (1 - V2)a' = 0-293a'. For a cell of radius 0-1 cm this amounts to a correction of 23 urn, which is somewhat larger than the depth of field of the usual microscope objective. More recently, Hall (1964) has shown that this analysis is inadequate because cells of this type suffer from a complex form of astigmatism. Each point within the cell gives rise to two virtual images, both of which are in the form of caustic surfaces with cuspidal edges. The position of one caustic surface depends on the radius of the cell; the other does not. When the microscope is focused just inside the cell cavity' the two images are sufficiently close together to allow a particle to appear as a point source. Deeper in the cell these two surfaces become so widely spaced that the particles appear greatly elongated and poorly defined; along the axis of a cell of radius 0-1 cm the image has an apparent length of 11 um but at the far wall its apparent length is 430 um. Hall claims that this invalidates

156 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

the use of the Henry correction altogether, and certainly that is true if one attempts to use data collected across the entire cell. The more usual practice, however, is to collect data at a number of points up to, and just beyond, the axis of the cell. I f this data is optically corrected before substitution in eq (4.1.13), a higher degree of consistency should be attained. The correction can be obtained from Henry's equation in the form (Hunter and Alexander, 1962):

E'H' = n i { q ~ k ) d (4.3.2) "3U —q + k) + n2(q — k)

where E and H' are the apparent positions of the points E and H (Fig. 4.15(a)) and qa = PH; ka = PE and a' is the apparent radius ( = EA'). I t should be said, however, that this refinement is rarely, if ever, attempted.

For the rectangular cell the only optical effect is the alteration of all distances in the cell in the ratio of the refractive index of the filling solution to that of air, i.e.:

4-eal — apparent X (4.3.3) "1

• where n3 is the refractive index of, say, water. Equation (3) applies, irrespec­tive of whether the cell is itself immersed in water, provided that the micro­scope objective is in air.

For the van Gils cell the image of a point is moved as a consequence of refraction effects at the water-glass interfaces, but the effect can be kept negligibly small if the wall is sufficiently thin ( < 100 urn). Smith (1973) has shown that if the aperture of the microscope is small then the distances EH=u=(q- k)a and EH' = v=(p — k)a (Fig. 4.15(b)) are related by the e q U a Ü O n S : « 3 _ ^ = ^ ( 4 3 A )

(4.3.5) v' + ö v + 5 a+ 5

where v' is the depth of the image of the point H due to refraction at the first water-glass boundary. These equations can be used also for the Mattson cell, in which case the right hand side of eq (5) is zero, since the radius of curvature of the outer surface is then infinitely large. Eliminating if between eqs (4) and (5) then leads to the Henry relation:

4. M E A S U R E M E N T O F E L E C T R O K I N E T I C P A R A M E T E R S 157

from which eq (4.3.2) was derived. Smith has shown that the overestimation of the diameter of the van Gils cell due to this optical effect is at most only

apparent 2 ? 2 K n n a r c n , where

t = S apparent = <5 _«3 n 2 _

As a multiplicative factor, the correction would be t2~

«3 «2/

and since S/d~ 0T and (1 - n3/n2) ~ 0-1 the error is quite negligible.

4.3 .1 .3 . The electr ical f ie ld The electric field must be applied by a power supply stabilized against mains fluctuations. The voltage rather than the current should be held constant, especially if one is using the effective length method for determining the field strength (method (a) below). Maintaining constant current could overcome the effects of minor degrees of electrode polarization if one used method (b) to calculate the field, but it is a questionable procedure because the steadily increasing applied voltage required to maintain the current only aggravates the polarization effects.

Choice of electrodes Platinized platinum electrodes are satisfactory and convenient for work at low electrolyte concentrations. The platinum black layer is deposited on the electrodes by electrolysis in chloro-platinic acid solution containing a little lead acetate (see Robinson and Stokes, 1959). For intermediate concentra­tions the effects of polarization can be minimized by using palladium elec­trodes, precharged with hydrogen (Niehof, 1969).

At high concentrations it is necessary to use reversible electrodes of the Ag/AgCl/KCl (0-1-1 M ) or Cu/CuS0 4 (0-1-1 M ) type. These must be iso­lated from the observation cell by porous plugs; they also introduce some problems into the determination of the voltage gradient (see below). It is, in any case, standard practice to measure velocities of about ten particles, whilst reversing the current flow after each measurement in order to minimize polarization effects.

Measuring the voltage gradient (a) The effective length method. The cell, complete with electrodes is filled

with a solution of known electrical conductivity, A 0, and its resistance, R, determined, preferably with an a.c. resistance bridge. The effective length, /,

158 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

is then given by:

l=RX0A (4.3.7)

where A is the cross-sectional area of the cell. A may be determined by direct microscopic observation of the cell dimensions, or by the mercury thread method. The latter gives an average value over the whole cell; the former method can be used to give a value at the plane of viewing. I f there are variations in the cell cross-section, the value at the plane of viewing is more appropriate because it is the field strength in that region which is required. For those measurements which require a traverse through the cell, the meas­ured dimensions must be corrected for the refractive index of the liquid in the cell, irrespective of whether or not it is surrounded by liquid.

The voltage gradient is then obtained simply by dividing the applied voltage, V, by the effective length:

Field strength = E = V/l (4.3.8)

The accuracy of this relation is not affected by the quite dramatic changes which occur in the cross-sectional area outside of the capillary region. It is wise, however, to ensure that the electrodes are always inserted in the same orientation. This method is, of course, inapplicable if the end electrode com­partments are isolated by porous plugs.

{b) Current measurement. I f the current flow can be accurately measured, the voltage gradient at any point in the cell is given by:

E=i/(X0A) (4.3.9)

In this case one must determine the electrical conductivity of each solution under study, using an auxiliary conductivity cell. Such a cell can be incor­porated into a flow-through apparatus without too much difficulty. The method is not as convenient as method (a) but does eliminate errors due to slight polarization effects. I t is essential, of course, that there be no leakage path for currents through the thermostat bath. This method can also be applied when the end compartments are separated by porous plugs. I t is, however, preferable in that case to use a special set of electrodes inside the observation cell to determine the voltage gradient (Smith, 1973, p. 122).

(c) The four-electrode cell. A further two electrodes may be incorporated near to the ends of the capillary in the van Gils or Mattson cell to determine more precisely the voltage gradient in the observation region. The potential difference between them can be measured with a high-impedance ( > 10MQ) voltmeter, and their effective separation can be calculated using eq (7) if the cell is provided with a known d.c. current source so that R can be estimated. Once completed, this calibration need not be repeated unless the cell has to

4. M E A S U R E M E N T O F E L E C T R O K I N E T I C P A R A M E T E R S 159

be reconstructed after breakage. The effective length should also be very' close to the physical length of the capillary.

I t is also recommended (by the manufacturers of a four-electrode van Gils cell (Rank Bros.)) that with these electrodes there is some advantage in using a constant current, rather than constant voltage, source because heating effects in the observation capillary can be readily detected by the sudden drop in voltage gradient to which they give rise.

4.3 .1 .4 . T w o - t u b e cel ls One major limitation of the cells discussed above is that the particle velocity yaries rapidly with depth in the region of the stationary level so that accurate estimates of mobility are very difficult to make, especially if the microscope has a significant depth of focus. The method of overcoming this problem was first suggested by Smith and Lisse (1936), who showed that it was pos­sible to set up two tubes in parallel in such a way that the stationary level occurred along the axis of the observation tube where the velocity gradient is zero (Fig. 4.17). This is achieved by arranging the radii and lengths of the two tubes in such a way that the pressure produced by the electro-osmotic flow is just sufficient to cause the necessary volume flow of liquid down the axis of the observation tube to counterbalance the forward electro-osmotic flow. This condition is satisfied when:

^ 1 - 2 (4.3.10)

where l2, lx are the lengths and a2, ax, are the radii of the by-pass and obser- -vation tubes respectively.

An alternative method of achieving the same liquid flow behaviour in the centre of the observation capillary was described by Rutgers et al. (1950) but does not seem to have proved very popular, despite the fact that it uses a single capillary.

Hunter and Alexander (1962) described two procedures for accurately determining the voltage gradient in the Smith and Lisse type cell; the cell's

Electrode Electrode

Fig. 4.17. The two-tube electrophoresis cell of Smith and Lisse (as modified by Bradbury and Jordan (1942)).

160 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

only drawback is that optical distortion becomes significant even at the axis of the observation tube. For this reason Hamilton and Stevens (1967) developed a similar two-tube version of the flat electrophoresis cell in which optical distortion is unimportant. Their cell has the additional advantage that it can be made shallower than a single flat cell so that the Komagata correction becomes less important (eq (4.1.23)). As noted in Section 4.1.3.2 it is unwise to make single flat capillaries too shallow because this increases the velocity gradient at the stationary level.

Hamilton and Stevens (1967) used Komagata's (1933) complete treatment of the hydrodynamics of the rectangular capillary. The velocity, u, along any (y, z) streamline is given by:

u = uh + — ( h 2 - y 2 ) + 2n

cosh(2n+ l)7tz cos (2«+ l)ny 00

E c - i : n = 0

n+l \6Ph2 Muh-ub)

J77i3(2n + l ) 3 >77t(2s+l)J

2h 2h cosh(2« + \)nb

2h (4.3.11)

where b and h are the half-width and half-depth of the tube; uh and ub are the electro-osmotic velocities at the xz and xy boundaries respectively (Fig. 4.18), P is the hydrostatic pressure gradient and n is the viscosity. Applying the flow condition for a closed cell and assuming that k = b/h is large, the velocity equation for z = 0 reduces to:

uhJn k_ *z=0 = 1 2 1

192 %5k

1 h2 (4.3.12)

I f ub = uh = ve0 (as would be so for an all-glass capillary) we have from eq(12):

' = 1 -31 1 - —

h2

2 -384 T?k

(4.3.13)

and so the depth y0 at which the velocity is zero in the median plane of the capillary is given by

yl = h2 U + 4 (4.3.14)

(A i fnllnws from the definition of x0.

4. M E A S U R E M E N T O F E L E C T R O K I N E T I C P A R A M E T E R S 161

0

Fig. 4.18. Coordinates for the rectangular cell (Hamilton and Stevens, 1967) .

Equation (4.1.22) requires a similar modification when the Komagata correction is necessary (Abramson et al., 1942, p. 51):

128 "r?k '1/2 (4.1.22a)

Hamilton and Stevens applied the comparable flow condition for a pair of closed cells in parallel:

uldzdy + -b,

f>2 iPdzdy = 0 i2 J - b 2

and introduced the relations P2 = Pxlx\l2 and aJJ, = v[0. l j l 2 where l u l2 were the lengths of the two capillaries. Hamilton and Stevens concluded that if the capillaries were of equal width, the liquid flow velocity would be zero along the axis of tube I i f the following condition held:

H3

6L • +

i _ 6 4 / ^ _1_ 3 Tv'Xk^L k2

= 0 (4.3.15)

where H= hjh2; L = l j l 2 and kt = bj^. For /c; = 20 and L = 1, eq (4.3.15) is satisfied by H = 0-730. Figure 4.19 shows a comparison of the velocity profile of a single flat cell and a double flat cell containing the same system.

Considering all the advantages of the van Gils cell, at least for non-settling particles, it is obvious that the most generally satisfactory electrophoresis cell would be a two-tube cylindrical cell after the design of Smith and Lisse (1936) but using as an observation tube the thin-walled van Gils cell. The bypass tube could then be used as one of the strengtheners for the very fragile observation cell.

162 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

x10"

>

0-2 1 0 0-4 0 -6 Relative depth

Fig. 4.19. Particle velocity profiles through gelatine-coated single tube ( - 0 - O - ) and double tube ( • • ) flat electrophoresis cells obtained with a suspension of gelatine-coated glass particles in 0-01 M acetic acid. uE = electrophoretic mobility (from Hamilton and Stevens (1967) with permission).

Tison (1975) suggests a number of techniques which should help to estab­lish more reproducible and accurate assessments of field strength in these systems.

4.3.1.5. M o r e recent deve lopmen ts The most time-consuming aspect of the microelectrophoretic procedure stems from the fact that many particles must be individually monitored in both directions so that a suitable average is obtained. A novel method of achieving this averaging has recently been introduced by the Pen Kem Co. (P.O. Box 364, Croton-on-Hudson, N.Y. 10520). Their apparatus (the Laser Zee Meter Model 400) incorporates a prism which rotates with a speed controlled manually by the observer, so that the motion of the particles is just compensated by the prism rotation and the particles appear to be stationary. The control knob can be directly calibrated or the zeta potential (using the Smoluchowski equation) displayed on a digital voltmeter. An averaging computer can also be incorporated to provide an average value of the setting of the zeta potential knob over a 30-second interval. The illumina­tion is by a laser which can be focused with an accuracy of 1 um which eliminates any errors due to the depth of field of the microscope objective.

4. M E A S U R E M E N T O F E L E C T R O K I N E T I C P A R A M E T E R S 163

The thickness of the laser beam is also equal to the depth of field of the micro­scope so that only "in-focus" particles are illuminated.

A more elaborate apparatus (Laser Zee"" System 3000, available from the same source) is an automatic device which uses a multistage photo-multiplier tube and a frequency tracker circuit to determine the average particle velocity. Incorporation of a real-time spectrum analyser (an optional extra) makes it possible to determine the entire frequency distribution of mobility. The manufacturers claim some advantages for this system over .the laser doppler technique (Section 4.3.5). I t can, for instance, be used with any light source (not necessarily a monochromatic one), does not require such critical alignment, and does not need to be worked at small scattering angles. It also permits direct viewing of particles and therefore can be used for mixed systems. A typical mobility histogram for human red blood cells in 0-03 M NaCl (44 % sorbitol) is shown in Fig. 4.20 (from Goetz and Penniman, priv. comm.).

4.3.2. M o v i n g bounda ry me thods

The theory and application of these procedures were reviewed by Longs-worth (1959) and wil l not be dealt with again here. The relative mobilities of the various components can be compared with the mobility of the fastest moving one, and i f its electrophoretic properties are well known, the electro­phoretic properties of the remainder can be inferred using a procedure

164 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

developed by Dole (1945). A more recent formulation of the mobility equations has been given by Tokiwa (1972) using the same principles, and the reader is referred to that review article for further details. An introductory treatment is also provided by Shaw (1969).

Apart from the "free" electrophoresis method, Longsworth also refers to the related procedures of zone electrophoresis (see also Shaw, 1969) and electrophoresis in porous media (paper, ground glass and starch or poly-acrylamide gels) where adsorption and electro-osmotic effects are of im­portance. An adequate treatment of these more qualitative procedures would take us too far afield for the present purpose. Although usually applied to proteins, apparatus for use with colloidal dispersions has been described by Kumar et al. (1966) and by Ottewill and Shaw (1967).

4.3.3. Tracer e lect rophores is

The measurement of the mobility of very small particles or micelles always presents problems that are not completely solved by the moving boundary method because it commonly suffers f rom boundary anomalies and the presence of foreign leading ions. Analytical procedures, based on the same principle as the measurement of transference numbers, wi l l be discussed in the next section, and the more recently developed scattering techniques are described in Section 4.3.5. The tracer method was first successfully applied by Brady (1948) and more recently by Mysels and his collaborators (Hoyer et al, 1954; Stigter and Mysels, 1955). I t can in principle be used with radio­active tracers, if they can be irreversibly incorporated into the colloid, but Mysels et al. used water-insoluble dyes which could be transported only within the micelles whose mobility they wished to determine. Their apparatus is shown in Fig. 4.21. Initially the central horizontal tube is filled with the micellar solution containing the tracer. After equilibrium is established, an electric current is passed for a known time interval (2-4 h), and the average concentration of the material in the central tube is re-determined, either by flushing it from the tube with tracer-free solution or by direct optical density measurements after homogenizing it.

The electrophoretic mobility, vE, is given by (Hoyer et al, 1954):

Xoxim0-m)v ( 4 3 1 6 )

it m0

where X0 is the solution conductivity, i is the current flowing for time t, V is the volume of the liquid between the stopcocks and m 0 and m are the initial and final concentrations of the tracer.

The method depends for its success on the provision of a stable current over a long period and hence requires a suitable pair of reversible electrodes.

4. M E A S U R E M E N T O F E L E C T R O K I N E T I C P A R A M E T E R S 165

0 1 0 0 1 I I I I I

Fig. 4.21. The free liquid tracer electrophoresis cell. The central horizontal tube is initially the only region containing tracer (Hoyer et al., 1954).

Some care must also be exercised in avoiding excessive heating and in the choice and pretreatment of the stopcock grease, but Stigter and Mysels (1955) report data of high precision for sodium lauryl sulphate micelles in water using orange OT and oil red N 1700 as tracer dyes.

An alternative apparatus, more closely resembling that of Brady, has been described by Mysels and Hoyer (1955), who also explain why electro-osmotic flow in the porous frits used to isolate the central compartment does not interfere with the measurement.

4.3.4 . The mass t ranspor t m e t h o d

This procedure is based on the same principles as the Hit torf method of determining transport numbers. The large mass of colloidal ions makes the method at first sight very attractive because it should be possible to deter­mine the mass transported by direct weighing of a suitable electrode com­partment. Overbeek (1952) briefly reviewed the early work of Paine (1928), Pauli and Engel (1928) and Tattje (1941), who calculated the mobility, vE, from the quantity of colloidal material transported, g, using the relation:

vE = —t (4.3.17) m0tt

More recently Olivier and Sennett (1966) revived the method with a specially designed cell which they claim allows results of quite high reproducibility to be obtained. The collection chamber (Fig. 4.22) is constructed of clear plastic and may be slowly rotated during a measurement to reduce gravita­tional settling. A known and constant current must be passed through the cell for periods up to about 5 minutes, and a pair of reversible electrodes is clearly essential. Olivier and Sennett claim that Joule heating is unimportant at low concentrations and that even for more concentrated systems (A0 ~ 10" 3 o h m - 1 c m - 1 ) i t causes a temperature change of only a few degrees.

166 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

RESERVOIR COLLECTION CHAMBER

Shutter actuating arm

O 2 5 5 0 7-5 cm

Fig. 4.22. The collection chamber for the mass transport mobility apparatus. (From Olivier and Sennett (1966) with permission.)

They suggest compensating for this by calculating the product £>( which is essentially independent of temperature, a procedure suggested by Guggen­heim (1940) for rather different reasons. The operation of their apparatus is described by Long and Ross (1965).

When the current has been passed for a certain time the collection com­partment is removed, carefully dried and weighed. The mobility, vE, is calculated by taking account of both the particle velocity, vE • E and the velocity vw of the liquid which is displaced from the collection chamber (Stigter, 1954). The result is:

V e = ^ (4.3.18)

where tSV is the net change in mass, <f> is the volume fraction of particles of density ps, and pw is the density of the suspension medium.

This whole procedure has been criticized on a number of counts by James (1978). The main problems stem from the necessity to pass a significant current through the system in one direction for such a long time, though Visca et al. (1978) claim to have solved this problem by using a salt bridge to isolate the collection compartment from the electrode so that a concentrated re­versible electrode solution can be used. Their apparatus is an improvement on that described by Tison (1975a).

Olivier and Sennett (1966) claim that the special advantage of their apparatus is that it enables measurements to be made at rather large particle concentrations, more easily correlated with the concentrations at which the colloid is commonly used. For moderate particle concentrations ( < 25 %

4. M E A S U R E M E N T O F E L E C T R O K I N E T I C P A R A M E T E R S 167

v/v) Homola and Robertson (1975) have shown that the Stigter correction to the apparent mobility is satisfactory and can in fact be used to determine the effective size of the particles. The mobility obtained by them in a Hit torf type cell was the same as that obtained by microelectrophoretic measure­ments.

At higher particle concentrations the particle double layers begin to over­lap and there are then some objections to regarding the resulting mobility as a fundamental and easily interpretable quantity. I t would seem to be preferable to make measurements under conditions in which the mobility can be related to the true (-potential (i.e. at low particle concentration) and then attempt to look at particle-particle interactions as a separate issue (see e.g. Napper and Hunter, 1974). The question of mobility of particles at high particle concentrations was discussed in Section 3.7.5.

4.3.5. E lec t rophoret ic l igh t scat ter ing

Electrophoretic light scattering is a technique which has recently been de­veloped for the study of the electrophoretic mobility of macromolecules, micelles and colloidal particles. Laser light scattering can be used to measure diffusion coefficients of small colloidal particles by measuring the Doppler broadening of the frequency of the scattered light due to the velocity of the scattering centres. I f an electric field is placed at right angles to the incident light and in the plane defined by the incident and observation beam (Fig. 4.23), the line broadening is unaffected but the centre frequency of the scattered light is shifted to an extent determined by the electrophoretic mobility. The shift is very small compared with the incident frequency ( ~ 100 Hz for an incident frequency ~ 6 x 10 1 4 Hz) but with a laser light source it can be detected by heterodyning (i.e. mixing) the scattered light

Light trap

— = )

PMT power] supply

— Analog out - scope, plot — Digital out - computer analysis

Fig. 4.23. A schematic diagram of an electrophoretic light scattering apparatus. (After Ware, 1974.)

168 ZETA POTENTIAL IN COLLOID SCIENCE

with the incident beam and detecting the output of the difference frequency. The method was first suggested by Flygare and the theory and some pre­

liminary results were presented by Ware and Flygare (1971), who subse­quently (1972) presented some improved measurements on bovine serum albumin (BSA) in which it proved possible to distinguish the mobilities of three components (BSA, BSA dimers and fibrinogen). Experimental work has also been reported by Uzgiris (1972, 1974) and by Uzgiris and Kaplan (1974, 1976); some extensions have been made to the theory by Phillies and by Berne. The early theoretical and experimental developments have been very ably reviewed by Ware (1974), and only a brief summary wil l be pre­sented here. References to the more recent work of Uzgiris and Kaplan are given in the works referred to above: in particular Uzgiris (1974) gives a detailed description of the apparatus.

The theory is developed by first calculating the effect of random diffusion of the scattering centres on the aurocorrelation function of the scattered light ( C ( K , T ) ) :

C(K, T) = A e x p ( - ico0x) e x p ( - <2)K.2x) (4.3.19)

where x is the time, co0 the initial frequency, 3) is the diffusion coefficient of the scattering centres and K is the wave vector whose magnitude is:

K = {Annjl) sin(0/2) (4.3.20)

where X is the wavelength of light in the scattering medium, n is an integer, and 6 is the scattering angle. The spectrum of scattered light is obtained by a Fourier transformation of eq (19):

2n C(K, x) « p ( t o r ) * = A . — ( 4 - 3 . 2 1 )

which shows that the scattered light has a Lorentzian frequency distribution with a half-width at half-height given by

Aco 1 / 2 = ®K2 (4.3.22)

The shape of these functions is shown in Fig. 4.24. The broadening is usually of order 103 Hz and again can be detected by heterodyning the scattered signal with the incident light or by homodyning (i.e. mixing the signal with itself) which also produces a Lorentzian spectrum centred at zero frequency

. but with Aco 1 / 2 = 2 0 K 2 . • In the presence of an electric field the electrophoretic velocity vE = uE.E,

is superimposed on the random thermal diffusive motion. The autocorrela­tion function for the scattered light then becomes:

C(K, x) = A e x p ( - ico0x) e x p ( ± iKvEx) e x p ( - @K.2x) (4.3.23)

4. MEASUREMENT OF ELECTROKINETIC PARAMETERS 169

/ ( " )

Fig. 4.24. The autocorrelation function and power spectrum of the light scattered f rom a solution oi macromolecules. The diagrams are the anticipated forms for a heterodyne experiment. -

which corresponds to a superimposed oscillation which is damped by the function given in eq (19) (see Fig. 4.25). The new spectrum is obtained by Fourier transformation of C(K, T ) :

2>K2

m = A T ^ - c o 0 ± K n

V E )

2

+ ^ K 2 ) 2 <4-3"24)

which is identical with eq (21) but shifted by an amount

K . vE = KvE cos(0/2) = KuEE cos(6/2) (4.3.25)

The choice of sign in eq (24) depends on the direction of the applied field and the sign of the charge on the scattering centre.

The success of the method depends on the relative magnitudes of the Doppler shift (K. vE and the Doppler broadening (Aco 1 / 2) and the resolu­tion, R, is defined as the ratio of these two quantities. Substituting from eqs

170 ZETA POTENTIAL IN COLLOID SCIENCE

O w

Fig. 4.25. As for Fig. 24 but with an electric field applied perpendicular to K.

(25), (22) and (20) we have:

provided 9 is small enough (tan 9/2 ss 0/2). For macromolecules, micelles or very small colloidal particles, a resolution of order 20 can be obtained for very small scattering angles (9 ~ 3°) provided that electric fields of about 150 V c m - 1 can be used. This is about ten times higher than the fields used in ordinary electrokinetic measurements and for measurement in aqueous solutions at moderate salt concentrations pulsed fields must be used. Ware and Flygare (1972) find that pulsing the field alternately in opposite direc­tions with an off to on ratio of 10 to 1 eliminates the effects of excessive Joule heating, polarization and the development of concentration gradients in the solution. The "on" time used was about 0T2s, which is about 20% longer than the time required to collect the frequency spectrum and allows time for the system to achieve a steady state before the analysis begins. A typical

4. MEASUREMENT OF ELECTROKINETIC PARAMETERS 171

correlation function and frequency spectrum are shown in Fig 426 The resolution is much larger for larger colloidal particles and much larger scattering angles and lower fields can then be employed (see e.g. Uzgiris,

The particle velocity is measured with respect to a coordinate system which is tixed on the apparatus so it is important that it is unaffected by electro-

E = 175 V / c m

0 2 4

0 -00 1 2 0 0 0 2 4 0 ^ 0 0 §60~ÖÖ 4 8 0 0 0 ' 6 0 0 0 0 ' 7 2 0 0 0 w ( H z )

Fig. 4.26 Correlation function and power spectrum from Bovine Serum Albumin at 175 V cm " 1

(From Ware and Flygare (1972) with permission.) '

172 ZETA POTENTIAL IN COLLOID SCIENCE

osmotic motion of the liquid. This problem is obviated in the very small cell used by Uzgiris (1974) because the electrodes and the particles are far re­moved from any surfaces.

The method is as yet in its infancy but has already attracted considerable interest. The development described above assumes that the system consists of identical spherical, non-interacting particles. I t can readily be extended to mixtures of spherical, non-interacting particles and some theoretical work has also been done on its application to reacting systems (Berne and Gininger, 1973). For these and other developments, both theoretical and experimental, the reader is referred to the reviews by Ware (1974) and Flygare and Ware (1974).

The main advantages of the method over other electrophoretic techniques is its very high speed ( ~ thousands of times faster) and high resolution which, as usual, are purchased at higher cost.

4.3.6. N o n - u n i f o r m f ie ld measurements

Colloid particles are able to respond fairly rapidly to changing electric fields (Sher and Schwan, 1965), and i f they are charged, their motion is dominated by the interaction between the charge and the field (Morrison, 1969). The more elaborate analysis of Vorob'eva et al. (1970) reveals that microelectro­phoretic measurements using a low-frequency a.c. field ( ~ 10 Hz) provide considerable advantages over the d.c. method and that, at such frequencies, the electro-osmotic velocity of the liquid in a capillary has an almost con­stant amplitude over the central quarter of the cross-section of the cell (i.e. y < all). Their theoretical predictions were confirmed by experimental studies on an aqueous dispersion at 30 Hz (see Ottewill, 1973). An a.c. micro­electrophoresis apparatus has also been described by Delatour and Hanss (1976). They have modified the Rank apparatus so that the upper vernier micrometer drum can be mechanically driven; this raises or lowers the flat cell which is in the lateral orientation (Fig. 4.16b). If, at the same time, a burst of a.c. is applied to the electrodes at a frequency of 0-02 to 10 Hz, the particles trace our sinusoidal paths which can be photographed and sub­sequently analysed (Fig. 4.27). An alternating rectangular wave gives rise to a triangular motion of amplitude {vE/co) x n/2 whilst an alternating t r i ­angular wave gives rise to a parabolic motion of amplitude (vEjco) x n/4. For the sinusoidal applied field the amplitude is vE/co where co is the applied frequency. Delatour and Hanss (1976) applied voltages of 600 V peak-to-peak which induced currents of order 40 mA (r.m.s.) in their systems, but Joule heating was minimized by keeping the duration of the voltage burst to 1-5-2 cycles. Though the preliminary results on erythrocytes look promising the measurements were not directly compared with reliable d.c. measurements.

4. MEASUREMENT OF ELECTROKINETIC PARAMETERS 173

Fig. 4.27. Principle used in the determination of particle mobility by the a.c. method. (After Delatour and Hanss, 1976.)

A related procedure was used by Kuz'mina et al. (1976), who applied a 5-10V rectangular pulse to a flat cell (10 x 10 x 1 mm) at a frequency of 0-1 Hz and measured the total particle displacement during the pulse with an 8 urn grid. The field strength is f rom 2 to 12 V c m - 1 and the mobility is constant over that range.

Even uncharged particles can be affected by the application of non­uniform fields. Elul (1966) and Chen et al. (1971) have studied the motion of both charged and neutral particles of AgBr in non-uniform a.c. fields. The. motion is frequency dependent and is a maximum for frequencies near 1 M H z which correlates with the frequency dependence of dielectric dis­persion in AgBr. Such measurements should provide some information about the structure of the diffuse double layer within the solid crystal. Wilkins and Heller (1963), on the other hand, claim that at some frequencies the applied field is able to discharge polystyrene particles, temporarily. Since the charge only returns after standing for some hours, it appears that, for some surfaces at least, the application of a.c. fields may produce very significant surface changes. More detailed descriptions of these phenomena, which are collectively described as dielectrophoresis, are given by Ottewill (1973) and by Pohl and Pickard (1969).

174 ZETA POTENTIAL IN COLLOID SCIENCE

4.3.7 . Other procedures

The electrophoretic motion of suspended fibres has attracted occasional attention since the early work of Sumner and Henry (1931). The usual, rather elaborate procedure (Mossman and Rideal, 1956) has been simplified considerably by Landwehr and Stigter (1974), who obtain at least a quali­tative measurement of the sign of charge and a reasonable estimate of the i.e.p. by following the motion of the tip of a freely suspended wool fibre in an electric field. Quantitative results are evidently extremely difficult to obtain by such methods.

A novel method of measurement of (-potential using the viscous damping of an oscillating pendulum has been proposed by Cometta and van Rutten (1967).

A quite different phenomenon has recently been observed by Stotz (1978), who studied the deflection of sedimenting particles by a very high electric field, placed at right angles to the gravitational field. These experiments, which are conducted in non-aqueous media, give mobilities much higher than the usual ones because the particle is stripped of its double layer by the high field. An analogous effect (the Wien effect) is observed in the conductivity of strong electrolyte solutions.

4.4 . S e d i m e n t a t i o n p o t e n t i a l

According to eq (3.4.4) the estimation of (-potential by this procedure re­quires only a measurement of the potential developed between two points at different heights in a column of sedimenting monodisperse particles of known particle concentration. As noted in Section 3.4, the restriction to mono­disperse systems was removed by Elton and Peace (1957), who used the time dependence of the sedimentation potential to estimate the size distribution of a sample of glass spheres and two samples of carborundum.

Since the early work of Quist and Washburn (1940), the method has been used only rarely, because most workers who have examined it have found it to be too unreliable for general use. Nevertheless, Benton and Elton (1953) and Elton and Mitchell (1953) obtained some significant results for carborundum in a variety of electrolyte solutions. The results were compared with those of Fairbrother and Mastin (1924, 1925), using electro-osmosis, and found to be much higher in absolute magnitude, suggesting that the electro-osmotic data suffered from some errors due to surface conduction effects. A better correlation was achieved by Peace and Elton (1956, 1960) in a comparison with more recent streaming potential data.

The magnitude of the electric field generated by the sedimentation process

4. MEASUREMENT OF ELECTROKINETIC PARAMETERS 175

is typically about 0-1-0-2Van" 1 , and its measurement with modern high impedance millivolt meters should present no problem. A schematic diagram of an apparatus for measuring the field generated by the rise of a stream of air bubbles is given by Usui and Sasaki (1978); the difficulty of interpreting their data m terms of (-potential was alluded to in Section 3.8.

R e f e r e n c e s

Abramson R A . , Mover, L. S. and Gorin, M . H. (1942). "Electrophoresis of Proteins." Reinhold, New York.

A1presnsdOxford' ^ J ° h n S ° n ' ? ' " C ° U O i d S d e n C e " ' V ° L 1 ° x f ° r d U n i v e r s i t y Alexander, A . E. and Saggers, L. (1948). J. Scient. Inst. 25, 374 Ball, B. and Fuerstenau, D. W. (1973). Miner. Sci. Engng 5, 267 Beck, U. , Zana, R. and Rohloff, E. (1978). T.APPI 61 63 Benton, D. P. and Elton, G. A . H. (1953). J. Chem. Soc. p 2096 Berne, B. J. and Gininger, R. (1973). Biopolymers 12, 1161 Biefer, G. J. and Mason, S. G. (1954). J. Colloid Sci. 9, 20 Biefer, G. J. and Mason, S. G. (1959). Trans. Faraday Sci. 55, 1239

I f 1 ' S (fdA ffi- Electrophoresis" Vol. 1. Academic Press, New York and London.

R I A ( t J ( ! f I 1 H ^ P t o r e s i s " Vol. 2. Academic Press, New York and London B adc, A . P. and Smith, Annie L. (1965). J. Amer. Water Works Assoc. 57 (4) 485 Back, A^P and Smith, Annie L. (1966). J. Amer. Water Works Assoc. 58 (41|W

SnOO A " K n ° X ' R ' J ' S C a m a n ' °- V- F- ( 1 9 7 8 ) - C M c a l Chemistry Bradbury and Jordan (1942). Biochem J. 36, 287 Brady, A . P. (1948). / . Amer. Chem. Soc. 70, 911 Briggs, D. K. (1928). J. Phys. Chem. 32, 641. Buchanan, A . S. and Heymann, E. (1948). Proc. Royal Soc. (London) A195, 150 BU11,H.B .(1934).A: 0 / / 0 WZ.66 ,20. Bull, H. B. (1935). / . Amer. Chem. Soc. 57, 259. BullI, H. B. (1971). "An Introduction to Physical Biochemistry", 2nd ed. F A Davis

Philadelphia. Cardwell, P. H. (1966). J. Colloid Interface Sci. 22, 430 Chi-Shih Chen Pohl, H. A . , Huebner, J. S. and Bruner, L. J. (1971). J. Colloid Inter­

face Sci. 37, 354. Cometta, C. and van Rutten, F. (1967). J. Colloid Interface Sci. 25, 322 Cooke, Claude E., Jnr (1955). J. Chem. Phys. 23, 2299 Delatour, E. and Hanss, M . (1976). Rev. Sci. Instruments 47, 1531 Deryaguin, B. V. (1943). Doklady Akad. Nauk. SSSR. 39, 11 Dole, V. P. (1945). / . Amer. Chem. Soc. 67, 1119 Douglas, H. W. (1957). J. Appl. Bad. 20, 390. Douglas, H. W. and Walker (1950). Trans. Faraday Soc. 46, 559 DukhinL S. S and Deryaguin, B. V. (1974). "Surface and Colloid Science" (E. Matijevtc

erf.), Vol. 7. Electrokinetic Phenomena". John Wiley Interscience, New York and London.

Elton, G. A . H. and Mitchell, J. W. (1953). J. Chem. Soc. preprint No. 741. Elton, G. A . H. and Peace, J. B. (1957). Proc. 2nd Inter. Conf. Surface Activity, Vol

IV, p. 177. Academic Press, New York and London.

176 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

Elul, R. (1966). Trans. Faraday Soc. 62, 3484. Fairbrother and Mastin (1924). / . Chem. Soc. 125, 2319. Fairbrother and Mastin (1925). / . Chem. Soc. 127, 332. Fairbrother, F. and Stubb, A. E. (1935). J. Chem. Soc. p. 527. Flygare, W. H. and Ware, B. R. (1974). Electrophoretic light scattering. In "Molecular

Electro-Optics" (C. T. O'Konski ed.). Dekker, New York. Fuerstenau, D. W. (1956). Mining Eng. (August). Gaudin, A. M. and Fuerstenau, D. W. (1955). Mining Engr. 7, 1. Gerdes, W. F. (1966). 12th Nat. ISA Analysis Instrumentation Symposium, Houston,

Texas (May 1966). Gittens, G. J. and James, A. M . (1960). Analytical Biochem. 1, 478. Goring, D. A. and Mason, S. G. (1950). Canad. J. Res. 28B, 323. Groves, J. N . and Sears, A. R. (1975). J. Colloid Interface Sci. 53, 83. Guggenheim, E. A. (1940). Trans. Faraday Soc. 36, 139. Hall, E. S. (1964). Nature, 203, 1371. Ham, A. J. and Douglas, H. (1942). Trans. Faraday Soc. 38, 404. Ham, A. J. and Hodgson, W. (1942). Trans. Faraday Soc. 38, 217. Hamilton, J. D. and Stevens, T. J. (1967). J. Colloid Interface Sci. 25, 518. Hartman, R. S., Bateman, J. B. and Lauffer, M. A. (1952). Arch. Biochem. and Biophys.

39,56. Hazel, J. F. (1953). J. Electrochem. Soc. 100, 65. Henley, A. and Schuettler, C. L. (1955). "Electrophoresis Bibliography", 2nd Ed.

Amer. Instrument Co., Silver Spring, Ma.) Henry, D. C. (1938). J. Chem. Soc. p. 997. Homola, A. and Robertson, A. A. (1975). J. Colloid Interface Sci. 51, 202. Horn, J. M., Jr and Onoda, G. Y., Jr (1977). J. Colloid Interface Sci. 61, 272. Hoyer, H. W., Mysels, K. J. and Stigter, D. (1954). J. Phys. Chem. 58, 385. Hunter, R. J. (1961). Ph.D. Thesis, University of Sydney. Hunter, R. J. and Alexander, A. E. (1962). / . Colloid Sci. 17, 781. Hurd, R. M . and Hackerman, N. (1955). J. Electrochem. Soc. 102, 594. Hurd, R. M . and Hackerman, N . (1956). J. Electrochem. Soc. 103, 316. James, A. M . (1979). "Electrophoresis of Particles in Suspension" in Surface and

Colloid Science (E. Matijevic ed) 11, pp. 121-185. Plenum Press, New York and London.

James, S. D. (1978). J. Colloid Interface Sci. 63, 577. Jones, G. and Wood, L. A. (1945). J. Chem. Phys. 13, 106. Joy, A. S., Watson, D. and Botten, R. (1965). Research Techniques and Instrumentation

1(1), 6. Komagata, S. (1933). Researches Electrotech. Lab. Tokyo. Comm. No, 348. Kumar, A., Bhatnagar, P. and Bhattacharya, A. (1966). Colloid J. (USSR) 28, 338

(KolloidnyiZhur. 28(3), 416). Kuz'mina, G. L., Girfanova, T. F., Marynov, G. A. and Muller, V. M . (1976). Colloid

J. (U.S.S.R.) 38, 717. Landwehr, R. C. and Stigter, D- (1974). Textile Res. J. 44, 45. Lauffer, M . A. and Gortner, R. A. (1938). J. Phys. Chem. 42, 641. Lewis, A. F., Nammari, M . Z. and Myers, R. R. (1964). Lab Practice 13, 50. Long, R. P. and Ross, S. (1965). J. Colloid Sci. 22, 438. Longsworth, L. G. (1959). "Electrophoresis" (Milan Bier ed.) Chap. 3 and 4. Academic

Press, New York and London. Lyons, J. (1979). M.Sc. Thesis, University of Melbourne.

4. M E A S U R E M E N T O F E L E C T R O K I N E T I C P A R A M E T E R S 177

MacKenzie, J. M . W. (1971). Minerals Sci. and Eng. (July), 25 Marshall, J. K. (1966). J. Sci. Inst. 43, 769. Martin, W. McK. and Gortner, R. A. (1930). / . Phys. Chem. 34, 1509 Mattson, S. (1933). J. Phys. Chem. 37, 223. Morrison, F. A. (1969). J. Colloid Interface Sci. 29, 687. Mossman, C. E. and Rideal, E. (1956). Canad. J. Chem. 34, 88. Mysels, K. J. and Hoyer, H. W. (1955). J. Phys. Chem. 59, 1119. Napper, D. H. and Hunter, R. J. (1974). "Hydrosols" In MTP Inter. Rev. of Sci. Series

2, Phys. Chem. Vol. 7, Chap. 3. pp. 161-213. Butterworths, London. Neihof, R. (1969). / . Colloid Interface Sci. 30, 128. Nordt, F. J., Knox, R. J. and Seaman, G. V. F. (1976). In "Hydrogels for Medical and

Related Applications", Amer. Chem. Soc. Symposium Series No. 31, p. 225 (J. D. Andrade ed.). A. C. S., Washington, D.C.

Olivier, J. P. and Sennett, P. (1966). Paper presented at the Third Annual Meeting Clay Minerals Society, October 1966, Pittsburgh, Pennsylvania.

Ottewill, R. H. (1973). Chem. Soc. (London), Specialist Perioidical Report on Colloid Science 1, 211.

Ottewill, R. H. and Shaw, J. N . (1967). KolloidZh. 218, 34. Overbeek, J. Th. G. (1952). "Colloid Science" Vol. 1, Chap. V. (H. R. Kruyt ed.).

Elsevier, Amsterdam and London. Packard, Robert G. (1953). J. Chem. Phys. 21, 303. Paine, H. H. (1928). Trans. Faraday Soc. 24, 412. Parriera, H. C. (1965). J. Colloid Sci. 20, 1. Parriera, H. C. and Schulman, J. H. (1961). Advan. Chem. Ser. 33, 160. Pauli, W. and Engel, L. (1928). Z.physik. Chem. 126, 247. Peace, J. B. and Elton, G. A. H. (1956). J. Chem. Soc. p. 22. Peace, J. B. and Elton, G. A. H. (I960). / . Chem. Soc. p. 2186. Pohl, H. A. and Pickard, W. F. (1969). "Dielectrophoretic and Electrophoretic De­

position", Electrochemical Soc. Inc. Pravdic, V. (1963). Croatica Chem. Acta 35, 233. Pravdic, V. and Mirnik, M . (1958). Croatica Chem. Acta 30, 113. Quist, J. D. and Washburn, E. R. (1940). / . Amer. Chem. Soc. 62, 3169. Robinson, McD., Pask, J. A. and Fuerstenau, D. W. (1964). J. Amer. Ceram Soc 47

(10), 516. Robinson, R. A. and Stokes, R. H. (1959). "Electrolyte Solutions" p. 91. Butterworths,

London. Rutgers, A. J., Facq, L. and van der Minne, J. L. (1950). Nature 166, 100. Seaman, G. V. F. (1975). Electrokinetic behaviour of red cells, Vol. I I , Chapter 27,

p. 1135. In "The Red Blood Cell", 2nd ed. (D. MacN. Surgenor ed.). Academic Press, New York and London.

Sears, A. R. and Groves, J. N . (1978). J. Colloid Interface Sci. 65, 479. Schenkel, J. H. and Kitchener, J. A. (1958). Experientia 14, 425. Sennett, P. and Olivier, J. P. (1965). lndust. Eng. Chem. 57 (8), 32. Shaw, D. J. (1969). "Electrophoresis." Academic Press, New York and London. Sher, L. D. and Schwan, H. P. (1965). Science 148, 229. Shergold, H. L., Mellgren, O. and Kitchener, J. A. (1966). Trans. Inst. Mining Metall.

75, C331. Smith, A. L. (1973). "Dispersions of Powders in Liquids" (G. Parfitt ed.). Chap. 3,

2nd Ed. Applied Science Publishers, Barking, England. Smith, M. E. and Lisse, M. W. (1936). J. Phys. Chem. 40, 399.

178 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

Somasundaran, P. and Kulkarni, R. D . (1973). J. Colloid Interface Sci. 45, 591. Srivastava, R. C. and Yadav, S. (1979). J. Colloid Interface Sci. 69, 280. Stigter, D . (1954). Rec. Trav. Chim. 73, 771. Stigter, D . (1964). J. Colloid Sci. 19, 252. Stigter, D. and Mysels, K . J. (1955). J. Phys. Chem. 59, 45. Stotz, S. (1978). J. Colloid Interface Sci. 65, 118. Street, N . (1964). Aust. J. Chem. 17, 828. Strickland, R. D . (1970). Analyt. Chem. 42, 32R. Sumner, C. G. and Henry, D . C. (1931). Proc. Roy. Soc. A133, 130. Tattje, P. (1941). Thesis: University of Utrecht, Netherlands. Tison, R. P. (1975a). J. Colloid Interface Sci. 52, 611. Tison, R. P. (1975b). J. Colloid Interface Sci. 53, 271. Tokiwa, F. (1972). Adv. Colloid Interface Sci. 3, 389. Uchida, S. (1956). Z. Angew. Math. Phys. VI I , 403. Ueda, S., Tsuji, F. and Watanabe, A. (1956). / . Electrochem Soc. Japan 24, 74. Ueda, S., Watanabe, A. and Tsuji, F. (1950). Reports Inst. Chem. Res. Kyoto Univ. 20,28. Ueda, S., Watanabe, A. and Tsuji, F. (1951 a and b). Memoirs of College of Agriculture,

Kyoto Univ. No. 60, p. 1 and p. 8. Usui, S. and Sasaki, H . (1978). J. Colloid Interface Sci. 65, 36. Uzgiris, E. E. (1972). Opt. Commun. 6, 55. Uzgiris, E. E. (1974). Rev. Sci. Instrum. 45, 117. Uzgiris, E. E. and Kaplan, J. H . (1974). Analyt. Biochem. 60, 455. Uzgiris, E. E. and Kaplan, J. H . (1976). / . Colloid Interface Sci. 55, 148. Van der Dr i f t , W. P. J. T. and Overbeek, J. Th. G. (1979). / . Colloid Interface Sci. 71,79. Van Gils, G. E. and Kruyt, H . R. (1936). Kolloid Chem. Beih. 45, 60. Van Oss, C. J., Fike, R. M , Good, R. J. and Reinig, J. M . (1974). Analyt. Biochem. 60,242 Visca, M . , Ardizzone, S. and Formaro, L. (1978). / . Colloid Interface Sci. 66, 95. Vorob'eva, T. A., Vlodavets, I . N . and Dukhin, S. S. (1970). Kolloid Zhurn. 32, 189

(Colloid J. (USSR) (1970)32, 152). Ware, B. R. (1974). Adv. Colloid Interface Sci. 4, 1-44. Ware, B. R. and Flygare, W. H . (1971). Chem. Phys. Letters 12, 81. Ware, B. R. and Flygare, W. H . (1972). J. Colloid Interface Sci. 39, 670. Watanabe, A . (1963). / . Electrochem. Soc. 110, 72. Watanabe, A. , Tsuji, F. and Ueda, S. (1961). J. Electrochem. Soc. Japan 29, (4), E236-40. Wilkins, D . J. and Heller, J. H . (1963). J. Chem. Phys. 39, 3401. Williams, M . (1948). Rev. Sci. Instr. 19, 640. . Wright, H . J. L . (1972). Ph.D. Thesis: University of Sydney. Zhulcov, A. N . (1974). Colloid J. (U.S.S.R.) 36, 715.

Chapter 5

Electroviscous and Viscoelectr ic Effects

It is apparent f rom the theoretical analysis of Chapter 3 that the presence of an electrical double layer exerts a profound influence on the flow behaviour of a fluid. A l l such influences are grouped together under the name of electro-viscous effects. When, for example, an electrolyte flows through electrically charged capillaries (or a porous medium) under a pressure gradient, a streaming potential is established. This potential wi l l produce a backflow of liquid by the electro-osmotic effect, and the net effect is a diminished flow in the forward direction. The liquid appears to exhibit an enhanced viscosity, if its flow rate is compared with the flow in the absence of double-layer effects (e.g. at high salt concentration or at the point of zero charge (Section 2.2)). A similar effect occurs when a dilute suspension of colloidal particles is sheared. The distortion of the double layer by the shearing process introduces an additional energy dissipation mechanism which manifests itself as an increased viscosity of the suspension. This was called the primary electro-viscous effect by Conway and Dobry-Duclaux in their review of the subject (1960).

The secondary electroviscous effect refers to the increase in apparent vis­cosity of a fluid as a result of double-layer interactions. In a porous medium it will be apparent at small values of Kr when the double layers on capillary walls cannot be properly developed. In colloidal suspension it is noticeable at quite low particle concentrations, because double-layer interaction in­fluences the details of the trajectories of approaching particles even though the average distance between particles may be much greater than the extent of the double layer.

The term tertiary electroviscous effect was applied by Conway and Dobry-Duclaux (1960) to the changes which occur in the conformation of poly-electrolytes due to changes in intramolecular double layer interactions and

180 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

the effects these have on flow behaviour. I t can be extended to include all those effects in which the geometry of the system is altered by changing double-layer interactions. Brief reviews of some of these effects have been given by Napper and Hunter (1974), Dukhin and Deryaguin (1974), Good­win (1975), and Saville (1977), but a unified treatment has not previously been attempted.

In the discussion of electroviscous effects it must be emphasized that we shall assume that the actual viscosity of the suspension medium remains unaffected by the presence of the double layers, i.e. the microscopic structure of the suspension medium is assumed to be unchanged by the local fields so that the viscosity of the liquid in the double layer is the same as that in the bulk. There are, however, situations in which the high electric fields in the neighbourhood of a surface cause a real increase in the viscosity of the sus­pension medium. This is referred to as the viscoelectric effect and wil l be considered in Section 5.3.

5 .1 . P o r o u s p l u g s a n d c a p i l l a r i e s

5 .1 .1 . The pr imary e lec t rov iscous ef fec t

For capillaries with a cross-section sufficiently large to permit complete development of double layers, the solution of the general equation (3.5.1) for the mean flow velocity would yield, for a slit-shaped capillary of height 2h (cf.eq (3.5.2)):

h2 dp s0DEz. I 3t] dz t]

(5.1.1)

so that

1 J A ^ h2 pr

where Pz = — dpjdz is the pressure gradient. Substituting for E.JPZ f rom eq (3.2.5):

v. K 3rj

1 -h2nk0_

(5.1.2)

The first term in this expression is the flow velocity per unit pressure gradient expected in the absence of double-layer effects. The second term may be taken as the expression of the primary electroviscous effect in this case (cf. Bull, 1932). The liquid would behave as though it had a viscosity,

5. E L E C T R O V I S C O U S A N D V I S C O E L E C T R I C E F F E C T S 181

r\a, given by:

1 -h2t]l0

(5.1.3)

The correction term is, in water at 25°C, given approximately by 0-848 Z 2 I ( K K ) 2 (using the fact that A 0 ( o h m ~ 1 m - 1 ) 14 x c ( m o l l - 1 ) for most salts.) This expression overestimates the primary electroviscous effect for all values of Z and xh but becomes exact as X decreases, especially for large K/Z (Table 5.1). At small values of JC/Z one must take proper account of the ion distribution in the capillaries and this is significantly modified even at quite

T A B L E 5.1

Values of the viscosity increase njti due to the primary electroviscous effect in fine pores, calculated from eq (5.1.10) with C = 0-56. Values calculated from the simplified expression (5.1.3) are shown underneath for comparison

KII

I

KII 1 2 3 4 5 6 7

8 1-013 1-046 1-085 1114 1124 1-118 1-101 1-013 1-056 1-135 1-269 1-495 1-912 2-851

10 1008 1030 1-057 1-079 1-089 1-087 1-076 1009 1-035 1083 1-157 1-269 1-439 1-711

12 1006 1-021 1-041 1-058 1-067 1-067 1-061 1-006 1-024 1-056 , 1-104 1-173 1-269 1-406

15 1-004 1-007 1-027 1-040 1-048 1-049 1-046 1004 1015 1-035 1-064 1-104 1-157 1-227

20 1-002 1-008 1-016 1-024 1-030 1-033 1-031 1-002 1-009 1-019 1-035 1-056 1-083 1-116

25 1-001 1-005 1-011 1-016 1-021 1-023 1-023 1001 1-005 1-012 1-022 1-035 1-051 1-071

30 1-001 1-004 1-007 1-012 1016 1018 1-018 1001 1-004 1-009 1-015 1-024 1-035 1-048

40 1001 1-002 1-004 1-007 1010 1011 1-012 1-001 1002 1-005 1009 1-013 1-019 1-027

50 1-000 1-001 1-003 1-005 1006 1-008 1-009 1000 1-001 1-003 1005 1-009 1-012 1-017

182 ZETA POTENTIAL IN COLLOID SCIENCE

large values of kh, especially for large C-potentials. We shall now describe a more exact expression.

In our earlier discussion of the flow in narrow slit-shaped pores we described the Hildreth (1970) approach to the problem. For the discussion of the electroviscous effects it is more convenient to use the equivalent development of Levine et al. (1975), who showed that the general equation for the stream­ing potential and electro-osmotic effects is (cf. eqs (3.6.1) and (3.6.4)):

eC_ ( 1 - G )

V o ( l + # ) (5.1.4)

In the absence of double-layer overlap the function G = 0 so we may ignore it for the moment. The value of H, on the other hand, is given by

H{t Kh) = cosh ij/{h) — 14-Ee{\ + C)

AnkTh (5.1.5)

where C = 4s0DkTn/X0r] and Ee is the electrostatic field energy per unit plate area. The value of Ee is given by Corkhill and Rosenhead (1939) in terms of elliptic integrals of the first and second kinds (see Appendix 5):

sin 2 cVc)1l2-2{E^)-E{<t>0,i)}

+ ( i - / 2 ) T O - % / ) } ] (5.1.6)

E(dj0J) = Jg° (1 - A2 sin 2 6)1/2d9 is the elliptic integral of the second kind; E(4) = E(n/2, 4) is the complete elliptic integral of the second kind. Also

A = e x p ( - t>„) and sin <£0 = d~1/2 e x p ( - lj2)

(These may be compared with eq A.5.4) In the absence of double-layer overlap, i/>„ = 0 and so A = 1 and sin 4>0 =

exp ( - 1/2). Then, f rom eq (5.1.6)

E = AnkT cos2 4>0

sin 0o - 2 { £ ( 7 t / 2 , l)-E((p0, 1)}

But E{(j)0,1) = JS° (1 - sin 2 6)ll2de = sin 4>0

and £(7i/2, 1) = £(1) = sin7t/2 = 1 so that, under these conditions:

E = AnkT COS 2 (/>c

sin 4>0

- 2(1 - sin (j)0)

Substituting for (j)0 gives us:

l6nkT . , Ee = sinh 2(r/4)

(5.1.7)

(5.1.8)

5. ELECTROVISCOUS AND VISCOELECTRIC EFFECTS 183

and thence

# = ^ s i n h 2 ( r / 4 ) [ l + C] (5.1.9)

Levine et al. (1975) arrive at eq (9) by another route which does not make clear the fact that it is an asymptotic solution valid for Kh 1 (i.e. in the absence of double-layer overlap).

The more exact expression for the apparent viscosity (replacing eq (5.1.3)) is then:

2(1 + H)(Kh)2

1 (5.1.10)

The value of C corresponding to A 0 ( o h m - 1 m ~ l ) = 14 x c (mol l - 1 ) is 0-56 in water at 298 K. (Levine et al. choose a value of C = 0-5 as typical.) Com­paring eqs (5.1.3) and (5.1.10), our more exact analysis, therefore changes the second term in the brackets in the ratio 1-5 x 0-56)/[0-848(l + H)] = 1/(1 +H).

This modified expression gives a somewhat different picture of the magni­tude of the primary electroviscous effect, especially for large values of t and small Kh. If, for example, we use X= 6 and Kh = 10 in eq (5.1.3), we obtain na = 1 -439)7—almost a 45 % increase in apparent viscosity. Using the more exact equation (5.1.10) gives r\a = I09n. I t should be noted also that the pre­dicted augmentation of the viscosity depends critically on the value chosen for C.

The magnitude of the primary electroviscous effect is shown in Table 5.1 for values of Kh 8. Lower Kh values are not calculated because the effect then becomes confused with the secondary electroviscous effect. Indeed even at Kh = 10, although double-layer overlap is minimal, the actual increase in viscosity wil l be somewhat lower than is shown in the table. Note that, at most, the increase in viscosity is about 25 % and the largest effect occurs for a Rvalue of about 6 (i.e. 150mV).

5.1 .2. The secondary e lec t rov iscous ef fect

I t is not possible to completely separate this effect from the primary effect. When double-layer overlap becomes significant we can calculate the com­bined primary and secondary effects using eq (5.1.4). The corresponding version of equation (5.1.10) for the viscosity is (Levine et al., 1975)::):

(5.1.11) ' 3CP(1 - G ) 2

2(1 +H){Kh)2

if The factor 2 in the denominator is missing from the equation given by Levine et al.

184 ZETA POTENTIAL IN COLLOID SCIENCE

The secondary effect is not, however, confined to the introduction of the function G, for we must also now use the more elaborate eqs (5.1.5 and 6) for H. Even so, at K / I = 10 when G 01 the value of r\Jr\ is very close to the values recorded in Table 5.1. Plots of rjjy are shown in Figs 5.1 and 5.2 from Levine et al. (1975). They use a slightly different C value (0-50 as against 0-56 in Table 5.1) and the slightly different values shown for r]Jn in Figs 5.1 and 5.2 are due partly to that, partly to the introduction of G, and partly to the use of the exact expression for H. Note that the maximum effect occurs for values of KII of about 1 or 2. At smaller values of Kh the double-layer overlap is very extensive, and if i/>0 is constant, the electrostatic potential over the whole of the pore is almost constant. The field strength in the pores is then quite low, and the ion distribution across the pore is uniform. Under these conditions the primary electroviscous effect is reduced because the actual streaming potential is dramatically reduced so the backflow becomes negligible. The secondary effect is reduced partly because H increases at small values oinh (Levine et al, 1975, Fig. 2) but more importantly because G approaches unity as Kh approaches zero, so the whole correction term disappears in this region.

Dukhin and Deryaguin (1974, pp. 118-123) discuss the electroviscous effects in microporous systems using the relative volume flow rate. They show that using the linearized form of the Poisson-Boltzmann equation one can predict the correct magnitude for the effect, although the Kh value at which it occurs is overestimated. For the more exact solution they use the

5. ELECTROVISCOUS AND VISCOELECTRIC EFFECTS 185

analytical approximations for the potential distribution derived by Sigal and Alekseyenko (1971) and obtain essentially the same results as those quoted above for the combined primary and secondary electroviscous effects.

5 .1 .3. The ter t iary e lec t rov iscous ef fect

The tertiary electroviscous effect should include all those processes by which the geometry of the porous medium is influenced by double-layer forces. A quantitative assessment of such processes would be extremely difficult, and no general approach has yet been attempted. The complexity of the situation can be readily appreciated: consider, for example, the question of whether double-layer repulsions would tend to reduce or increase the hydraulic permeability of a porous bed i.e. whether the effective viscosity would be in­creased or decreased. I f the particles making up the bed are not rigidly con­strained (e.g. a bed of montmorillonite or vermicuhte clay plates), double-layer repulsion would tend to separate the particles and increase the hy­draulic conductivity. On the other hand, in a sand column the hydration of the surface layers of silica may cause the formation of a gel which can swell into the pores of the column and inhibit flow. In this case, double-layer repulsion in the gel layers wil l cause a reduction in the hydraulic conductivity of the column. Some of the anomalous viscosity effects observed when water

186 ZETA POTENTIAL IN COLLOID SCIENCE

or aqueous solutions are percolated through packed beds are at least partly attributable to such processes (Grigorov and Novikova, 1955).

Dukhin and Deryaguin (1974) acknowledge the possibility of tertiary electroviscous effects of the sort referred to above but also observe that anomalous results on the flow of aqueous suspensions through porous media may be due to the existence at the solid-liquid interface of an extensive boundary layer of very much increased viscosity. This proposition wil l be discussed at greater length in Section 5.4. I t is significant that the most recent Soviet work on very fine vitreous silica capillaries (Churaev et al., 1981) clearly demonstrates the importance of swollen silica gel layers in the pro­duction of anomolous streaming potential results.

5.1.4. Exper imental data on the e lec t rov iscous ef fect in capi l lary systems

Although it is not possible to test rigorously the general equations for the (-potential developed in Chapter 3, it is possible to apply some tests to the equations for the electroviscous effects. We could, for example, determine whether the expected rise in apparent viscosity (eq (5.1.11)) occurs at small values of KII (or rcr). The comparison between theory and experiment would require a knowledge of the (-potential of the pores so the argument may be considered somewhat circular (see, for example, Anderson and Wei-hu Koh, 1977).

An alternative approach is to examine the apparent variation in (-potential with change in capillary radius. The electroviscous effects in fine pores result in a lower value of EJp being recorded for any given (-potential (see e.g. eq (3.6.13)). I f one substitutes the experimental value of EJp in the simple Smoluchowski type expression for ( (e.g. eq (3.2.5)), one obtains an arti­ficially low apparent, value of ( (( s ) , and the effect becomes more pro­nounced the smaller the radius of the capillaries. An important test of our equations can, therefore, be applied if measurements are made on a series of porous systems with decreasing pore radius but preserving the same surface properties of the pores. The varying values of ( s at different radii should be transformed into a constant value, characteristic of the surface chemistry of the pore, if the equations are correct. I t should be noted that no attempt is made in any of these tests to separate the primary and the secondary electro­viscous effects. The actual experimental tests are not easy to perform: preparation of very fine bore capillaries of constant surface properties is a task fraught with difficulties; the progress that has been made is reviewed by Dukhin and Deryaguin (1974, p. 124).

As noted in Section 3.6.1.2, Oldham et al. (1963) were able to show that their treatment of the electrokinetic behaviour of finely porous systems was able to "explain" the data of White et al. (1936) and that of Rutgers et al.

5. ELECTROVISCOUS AND VISCOELECTRIC EFFECTS 187

2 5 10 20 50 100 Radius ((im)

Fig. 5.3. Values of the uncorrected and f, the fully corrected electrokinetic potential in narrow cylindrical capillaries (from Oldham et al, (1963). f c is corrected for surface conduction but not for double-layer overlap.

0-4i 1 1 , , • ,

188 ZETA POTENTIAL IN COLLOID SCIENCE

(1959). That is to say, when the experimental data were described in terms of their equations a constant (-potential was obtained, independent of capillary size, instead of the varying value obtained for the apparent £ value (Figs. 5.3 and 5.4). A defect of this work is that Oldham et al. had to use literature data from other sources to estimate the surface conductivity in the systems they studied (see the discussion connected with eq (3.6.16) above). However, the constancy of their (-potential values for different capillary radii in water and benzene is very encouraging.

A more definitive test of the theory may be made with the data provided by Kholodnitsky and analysed by Dukhin and Deryaguin (1974, p. 126). Kholodnitsky measured streaming potentials in bundles of capillaries of varying radius: 0-5, 0-2, 0-07 and 0-025 urn (samples 1, 2, 3 and 4 in Table 5.2, respectively) with KC1 of various concentrations. He calculated ( s from the streaming potential (column 6) and ( c f rom the streaming current (column 7) and found them to be different. As noted in Section 3.2.3, the streaming current results are not affected by surface conduction corrections to the same extent as the streaming potential results. Dukhin and Deryaguin have re­calculated ( from these two sets of results, taking account of double-layer overlap (G and H in eq (5.1.11)), using the linear form of the Poisson-Boltzmann equation. Their results are shown as ( x and ( 2 in columns 8 and 9 of Table 5.2. The agreement between the two procedures is excellent except for one data item corresponding to the finest capillary at the lowest con­centration. The values of ( are also essentially independent of capillary radius except for sample 4 at the two lowest electrolyte concentrations. This is the region where the corrections are largest and the experimental difficulties are most obvious.

Much of the discrepancy for xr = 1-7 is removed by introducing the com­plete Poisson-Boltzmann expression for the potential in the pores. Sigal, using this procedure obtains d = 44mV and ( 2 = 56 mV instead of ( : = 52 mV and ( 2 = 83 mV obtained with the linearized equation (Dukhin and Deryaguin, 1974, p. 126). The remaining discrepancies could be traced to a number of possible effects:

(i) Experimental error. Despite the concordance of the streaming potential and streaming current results one must be sceptical of the indicated accuracy of ( s and ( c . Absolute accuracies better than 1 mV in these quantities are very difficult to achieve.

(ii) There are inconsistencies revealed in the tabulated values for ( l c a p — A 0 ) . The product of this quantity with the capillary radius should be a measure of the surface conduction, and although it might not be constant (see Fig. 3.13) it ought not to vary in the arbitrary manner shown in the last column of the table.

3 cs £>

Q C C .

Q E o

a CJ) C

3

c

c

T3

c

O

a VI OS co r> CM ••3- co co CM ö ö ö ó

SO CM OS t» CM r-~ Ó H Ó Ó

SO OS CN vt ,—" o\ so Ö CN Ó Ó

C O O ^ cp so CM co « ö

> B

' 2 I a ^

J x I

a

a

^ ~ !Cï ON CN a> _ 9 ? 9 r poop poop pppp

p o —«

m - 3 - o\ m m rn cn oo

ro n n >o

q\ co -- a\ <N be - r- CT\

m CN CN —(

in cp co ós -^f ( N CN H

5

• * C O -4- « C O ~

SO CM os r~-O —< OS ó ó ó ó

C O P H oo as CM t--ó ó — ™

O O O CM CM so O O O O

S S S K 2 £ 2; 2 2 2 2 2

co ob -if ó\ CM CM CN —<

SO CM OO CM CM <—f

so co t Os s ó CN TT ' — I

--3- co co 10 — I so O Os Ó Ó A - H

^- m m ^ r— CM so u-i

o - 3 - kO oo CM M

Tf SO —. •O CM Ó ö\

co so CM so >ö O —

C O CM [— O CN CN Ö Ö

so in -y O cp so CM < ÓS ob

OO CN C-~r p T* Ó « CN

OO CN c-CM —1 10 so

^ H C M C M C Ó r - o b o b ö S

0 0 — 10 so CM

t OS TT Ó A « (SI

f •* qs ^r •si- sö iö só

—< CM C O -stf- w n ^ . H ( M cn _

I

O X WO

I

O

190 ZETA POTENTIAL IN COLLOID SCIENCE

(iii) At very low electrolyte concentrations it is possible that a slight release of salts from the glass capillary wall could modify the K value and introduce some error (a possibility that was evidently envisaged by Churaev (Dukhin and Deryaguin, 1974, p. 126).

(iv) I t is possible that, in the very finest capillaries, a slight swelling of the surface layers of glass due to hydration effects could dramatically affect the value of KT. This would be in the nature of a tertiary electroviscous effect and would be most pronounced at low electrolyte concentration.

For these reasons it does not seem to be necessary to search further for an explanation of the remaining small discrepancies in the values of ( i and (2> and Table 5.2 wi l l be taken as evidence of the essential validity of the equations developed in Section 5.1 for the electroviscous effects in'capillary systems. I t should be noted, however, that Dukhin and Deryaguin draw attention to the fact that the observed values of the apparent viscosity of the liquid in the pores remain in some cases significantly higher than the calculated values (columns 10 and 11). They are inclined to attribute this to the possibility of "boundary layers" of immobile water on the pore surfaces, and this wi l l be considered further in the discussion of the viscoelectric effect (Section 5.3).

5.2. S u s p e n s i o n s o f s p h e r i c a l p a r t i c l e s

Before discussing the modifications to flow behaviour which occur as a result of electrical double layer effects the behaviour of uncharged spheres must first be examined briefly. Excellent reviews of this field have been given recently by Krieger (1972) and Goodwin (1975); aspects of the electro­viscous effects are also reviewed by Napper and Hunter (1974) and Saville (1977).

At very low particle concentrations the flow disturbance caused by one particle does not interfere with that due to other particles and so for stable colloidal suspensions the viscosity, n, is determined by the volume fraction of particles, <f>. As <£->0, the viscosity is given by the well-known Einstein equation:

^ = - 1 = 1 + ^ (5.2.1)

where nr is the relative viscosity, n0 is the viscosity of the suspension medium (assumed Newtonian) and k1 has the value 2-5 for spherical particles. The value of <p to be used in eq (1) is that corresponding to the "kinetic unit", i.e. including rigidly attached adsorbed layers of solutes or solvent. Equation (1) is the limiting form which applies to heterodisperse spherical particles,

5. ELECTROVISCOUS AND VISCOELECTRIC EFFECTS 191

but the value of kx must be modified if : (i) there is slip between the particle and the fluid (/q = I ) ; (ii) the particles are of appreciable size compared to the size of the viscometer gap; (iii) the particles are anisometric, in which case fcj becomes a function of shear rate; (iv) the disperse phase is not rigid (i.e. in emulsions with internal circulation possible); and (v) the double layer around the particles is extensive (tea < ~ 10), in which case the primary electroviscous effect must be calculated (Section 5.2.1).

At higher volume fractions the electrical and hydrodynamic interactions between the particles must be taken into account. Equation (1) can be modi­fied by taking higher terms in the power series expansion:

tit = 1 + + k2dj2 + ... (5.2.2) Considerable argument surrounds the calculation of the correct value for k2

in the absence of electrical interactions (see Goodwin, 1975). Values from about 7 to 14 have been derived depending on the model used to calculate the influence of two sphere interactions on the flow field. When electrical re­pulsion between particles is also important it can be manifest at quite small values of 0 (0 < 0-03), before the hydrodynamic forces take effect. This is the region in which the secondary electroviscous effect is studied (Section 5.2.2.).

A rather more effective equation for dealing with higher volume fractions was first suggested by Arrhenius, and a derivation was developed by Mooney (1951) using an elementary functional analysis to calculate how the addition of further solid material to an existing suspension would alter its viscosity. The resulting equation:

reduces to Einstein's equation at low volume fraction and gives a good description of the effect of dj for systems in which electrical interactions have been suppressed.

An even better representation, arrived at by a slight modification of Mooney's argument, has been given by Dougherty and Krieger (see Kreiger, 1972). Their equation is:

nr=(\-k'cby™k' (5.2.4)

where [r\] is the intrinsic viscosity, here taken as an empirical constant whose value should approach 2-5 in the absence of electrical effects. Equation (4) gives a good description of nr at high shear rate up to 0 values of ~ 0-62 (which is almost close packing) provided account is taken of the presence of emulsifier layers on the particles (Woods and Krieger, 1970).

At lower shear rates, and especially for small particles where inter-particle separations become small at quite modest volume fractions, details of the

192 ZETA POTENTIAL IN COLLOID SCIENCE

hydrodynamic encounters between two or more particles must be taken into account, and the system then exhibits a viscosity which increases with de­creasing shear rate (i.e. i t is non-Newtownian and pseudoplastic; see Ap­pendix 3).

The parameter, k', is determined by the "packing" of the sol particles and in the simplest model assumes a value which ensures that the viscosity becomes infinitely large when the particles are close packed. Using this type of description the secondary electroviscous effect would be used to calculate a modified value for k', whilst the primary electroviscous effect would modify the value of [n]. In practice, the electrical effects become obvious at very low volume fractions and they are expressed as modifications to the constants kx and k2 in eq (2).

5 .2 .1 . The pr imary e lec t rov iscous ef fec t

The problem was first examined by Smoluchowski (1916), who published without proof the following relation:

l / C « N a " nr = 1 + 2-5<j) 1 + • (5.2.5)

where X0 is the conductivity of the electrolyte. Equation (5) was expected to be valid for large values of KÜ (i.e. in the region where the effect is smallest). Krasny-Ergen (1936) subsequently published a derivation in which the second term in the parentheses was multiplied by f . The similarity of eq (5) to eq (5.1.3.) is very striking.

A more complete development has since been given by Booth (1950), who drew attention to some inconsistencies in the Krasny-Ergen analysis. (One of Krasny-Ergen's equations fails to satisfy a necessary boundary condition, and his analysis does not take account of the energy dissipation caused by the electric currents due to the diffusion gradients in the electrolyte.) Booth's equation should be valid for all values of Ka provided that (i) the average distance between particles is large compared to the thickness of the double layer (i.e. 4> is small) and (ii) the potential, or charge on the particles is not too large.

The equations of motion that must be solved for the particles and the electrolyte are essentially those involved in the electrophoresis problem (Section 3.7) but with appropriate modifications to the boundary conditions. Booth's solution was expressed in the form:

^ = 1 + 2 - 5 ^ 1 + f>„r^

Only the first two coefficients were evaluated, and since b1 = 0, the final

5. ELECTROVISCOUS AND VISCOELECTR1C EFFECTS 193

expression for the viscosity could be written:

n, m 1 + 2-50 [1 + q*Z2Z(Ka) (1 + Ka)2] (5.2.6) where

^akT-EU^zfcor" q 4nr,0e2ZUiniZÏ

and Z(JOZ) is given as a power series in KÜ ; co; is the velocity per unit force of an ion of type i, and if all ions are assumed to have the same mobility the expression for q* can be simplified and eq (.6) becomes:

(5.2.7)

where £ is the permittivity of the medium. The limiting form of Z(KO) for large Ka was given by Booth (1950) as Z(KÜ) = 3/2n{Ka)4 so that under those conditions eq (7) reduced to the Krasny-Ergen relation.

A similar relation, with the second term in parentheses in eq (5) multiplied by 3-08 rather than f was derived by Finkelstein and Cursin (1942) for values of Ka 17. An attempt has also been made by Street (1958) to apply a similar procedure to that used in deriving eq (5.1.3). A simple, but significant, typographical error in the statement of Street's equation was corrected by Whitehead (1969), who noted that the equation should read:

^ = ( 1 + 2 ^ K 1 + 2 ^ ( l ) 2 ( 1 + Kf l )V) (5-18)

Note that this equation is quadratic in (h so that it purports to represent both the primary and secondary electroviscous effects. For the primary electro­viscous effect it reproduces the Smoluchowski equation with the second term in parentheses multiplied by (1 + KO)2/5. For values of Ka> 1-23 i t therefore predicts a larger effect than the Smoluchowski equation as Stone-Masui and Watillon (1970) have pointed out. Since the experimental evidence suggests that even the Smoluchowski equation is an overestimate, we shall not discuss eq (8) further.

The most elaborate of the above treatments is that of Booth, and although it should apply for all Ka values it is restricted to small values of the (-potential (Z < ~ 1) and to small Peclet numbers. (The Peclet number Pe = au/cokT where u is the particle velocity; it measures the extent to which the movement of the fluid, relative to the particle, disturbs the ionic atmosphere. For small values of Pe the double layer is only slightly distorted from its equilibrium shape.) Russel (1978a) extended the analysis to larger values of the Peclet number {Pe -4 Ka and Ka 1) but still only for small potentials. His result agrees with that of Krasny-Ergen (i.e. the large Ka value arrived at by Booth)

^ = 1 + 2 - 5 0 1 + ( f iQ 2

4nA0ri0a (Ka)2(l + Ka)2Z{Ka)

I

194 ZETA POTENTIAL IN COLLOID SCIENCE

in the limit of zero shear rate, as would be expected. For higher shear rates eq (7) is modified, at high KO, by the inclusion of an additional term:

r 3 f&Y 1 (5.2.9)

The net effect is to make rir decrease with increase in particle velocity (i.e. with increase in shear rate). Equation (9) thus suggests that systems exhibiting the primary electroviscous effect should be to some extent pseudoplastic (see Appendix 3).

Finally on the theory of the primary effect, Sherwood (1978) has extended the Booth analysis for low shear rates {Pe < 1) to higher values of £. The results of his numerical calculations are shown in Fig. 5.5 as a comparison with the original Booth estimates. They show that Booth's analysis is excellent even for high values of £ provided Ka is very small. At higher values of Ka, the effect of introducing the more exact form of the Poisson-Boltzmann equation is to lower the primary electroviscous effect at intermediate values of m ( ~ 1-10) and raise it at higher values (m > 100).

5.2.2. The secondary e lec t rov iscous ef fec t

This effect is ascribed to the overlap of double layers on two approaching particles. Since the frequency of such interactions is proportional to 0

5. ELECTROVISCOUS AND VISCOELECTRIC EFFECTS 195

(Manley and Mason, 1954), i t appears as a correction to k2 in eq (5.2.2). Fortunately, the effect is so large that it can be studied at concentrations below those where particle self-crowding is important {(f) < 0-03) so that the uncertainy in the value of k2 for uncharged spheres is unimportant.

The approach to a theoretical analysis was suggested by Harmsen et al. (1953), and a rather simple derivation was undertaken by Chan et al. (1966) based on the following model: as two particles approach, the repulsion between them causes an increase in the centre to centre distance of the re­sulting doublet, and as the doublet rotates in the shear field, the energy dissipated is correspondingly larger. The separation distance was calculated by balancing the hydrodynamic forces with the repulsive force, since the van der Waals attraction is negligible at the distances involved. Their result was:

fc2 = M 2 5 n + ^ J (5.2.10)

where 2SC is the distance of closest approach of the particles (which depends on the (-potential (see Appendix 5)). A rather more elaborate expression was subsequently derived by Blachford et al. (1969) in terms of the Huggins coefficient, which in our terminology is the value of k2jk\.

More recently, Russel in two papers (1976, 1978b) has reanalysed the problem from a more fundamental standpoint and has provided, in the first paper, estimates of k2 for the condition £ < 1 and KÜ X 1. In the second paper, he establishes an analytical solution which applies to arbitrary values of £ and Ka; in both cases it is assumed that the electrostatic repulsion forces are dominant. This last restriction is well justified since it is certainly true in the situations in which the secondary electroviscous effect can be examined experimentally.

Russel's analysis is given in terms of a dimensionless parameter, a, which, for small values of Ka, is given by:

oc = 47T£0^>5 a K a exp(2Ka) (5.2.11) kT

a measures the ratio of the electrostatic forces (see Appendix 5) to the thermal diffusion (Brownian motion) forces. The complete expression for the relative viscosity in the limit of low shear rates {Pe <g 1) is:

nr=\+ 2-5R0(h + 2-5O3o0)2 + & In ( I n " ) (5.2.12) In a \ ln(a/ lna) / (Ka)

The term in (j) is identical to the result obtained by Booth for the primary electroviscous effect eq (5.2.7) and might be expected to apply only for small values of X- As we have seen already, however, in the work of Sherwood (1978),

196 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

the linearized Poisson-Boltzmann theory works quite well for modest values of X, except at some intermediate values of Ka. Russel's approach uses, for the electrostatic potential, analytical approximations which are appropriate for small and for large Ka respectively. His analysis is therefore most likely to break down at intermediate values of K A . As wi l l be seen in Section 5.3.2 it is precisely in this region that the results can be compared with the experi­mental values. I t should be noted that the term containing a in eq (12) is usually very much larger than 2-5/J2,, so that the limitations in the Booth analysis of the primary electroviscous effect (Sherwood, 1978) do not inter­fere much with the experimental testing of eq (12).

For large values of Ka, the electrostatic forces are better represented by (Bell and Peterson, 1972):

Fel = (47ce„)8D^0 m tanh2(^j e x p [ - K(r - 2a)] (5.2.13)

and the corresponding value of a is:

a = (4ms0) • l 6 D k I a • ^ exp(2Ka) tanh2( P) (5.2.14) (ze) V 4 /

The further restrictions on eq (12) imposed by the method of analysis, are that (i) the distance over which the electrostatic force can exert itself is large compared to the particle size, (ii) the particles are far enough apart to prevent hydrodynamic interactions and (iii) the electrostatic forces are not so long-ranged that multiparticle effects need to be taken into account.

5.2.3. Exper imental ev idence on the pr imary and secondary e lec t ro­v iscous ef fects in suspens ions

Experimental tests of the equations developed in Section 5.2.1. for the primary electroviscous effect are not easy to mount, since the effect is rather small and is easily confused by other effects which can contribute to departures of the constant fct from its Einstein value of 2-5. Apart f rom the factors mentioned under eq (5.2.1), the presence of a small degree of aggregation due to in­complete dispersion can cause significant departures from kx = 2-5 even in the absence of charge effects (Gillespie, 1963; Stone-Masui and Watillon, 1968). Goodwin (1975) suggests that the presence of a small degree of aggregation may explain why Chan and Goring's (1966) measurements of the primary electroviscous effect in the region l < K a < 4, gave results sig­nificantly higher than those predicted by the Booth equation (5.2.7) and approaching those of the Smoluchowski equation at the upper l imit (KÜ = 4).

More recent studies by Stone-Masui and Watillon (1968, 1970) are con­fined to the region Ka = 0-81-9-28 and at the upper limit the Smoluchowski

5. E L E C T R O V I S C O U S A N D V I S C O E L E C T R I C E F F E C T S 197

equation is reasonably accurate, but for all other cases only the Booth equation gives good agreement with the experimental data. A l l the other equations greatly exaggerate the significance of the primary effect, and in any case, in the region for which they were derived (m large), the effect is so small that they are unlikely to be of any practical significance. In the low Ka region the Booth equation shows that kx is raised from 2-5 to at most about 4, even for ( potential values of 100 mV. Although there is some slight un­certainty in the Ka values appropriate to some of the experimental systems (see Russel 1978b), this argument may be regarded as satisfactory.

Stigter (1967) has applied the Booth equation to the primary electro­viscous effect in micellar solutions. Experimental data obtained by Kushner et a!. (1952) could be explained very well, provided that a Stern layer of about 4-5 A thickness was postulated around the micelle. The concentration range studied was from 0-008-0-1 mole l " 1 which would correspond to m values in the range 0-5-2-0. Sherwood's (1978) analysis (Fig. 5.5) would predict an error of the order of 10-20% in the Booth expression for these Ka values and the measured (-potential (which varied from X= 5 to 3-5). The analysis is, however, not sensitive enough to pick up such a small correction since the micelles may also depart slightly from spherical.

The Stone-Masui and Watillon (1968) data were collected at very low particle concentrations (0 < 0-03) and at moderate to high shear rates (y = 50-1200 s" 1). More recently Wang (1970) obtained some data on more concentrated sols (0 = 0-13-0-40) and analysed them using the Mooney equation (3) with the constant 2-5 replaced by the parameter, a'. A plot of 0/ln nr against 0 should be linear (Saunders, 1961) with intercept 1/a' and slope - k'/a.'. Wang found «.' = 2-65 at high shear rate. The departure from the Einstein value could be explained by the presence of an emulsifier layer of thickness 16 A—which seems reasonable. This a' value was independent of electrolyte concentration over the range 0 < c < 10" 1 M . The calculation of Ka values in these systems is complicated by the high volume fractions in­volved. Even at the lowest volume fraction (0 = 0-139) and in the absence of added salt, the ionic strength due to the particles themselves and their associated double layers is in excess of 10" 2 M . The corresponding Ka values are all greater than 20, so the primary electroviscous effect should be negli­gibly small; Wang's high shear result does not therefore contradict the theory. At lower shear rates he found that a.' increased with decrease in the shear rate and suggested that this was direct evidence for a shear rate de­pendence of the primary electroviscous effect. The low shear data could be very approximately accounted for by assuming that under these conditions the individual particles have their true volumes augmented by a layer of thickness about 1 / K (rather more at higher shear rates). A similar result had been obtained earlier by Brodnyan and Kelley (1965). The earlier experi-

198 ZETA POTENTIAL IN COLLOID SCIENCE

merits of Schalier and Humphrey (1966), using a similar technique of analysis, also revealed a shear rate dependence for r\r at high volume fractions, but only for small particle sizes [a = 88 nm). For a = 557 nm and 1305 nm the systems were Newtonian. Wang's experiments were all conducted on systems with a = 166 nm so the shear rate dependence is critically dependent on particle size. The values of a! obtained by Schalier and Humphrey (1966) were all less than 3-03, and one system even showed a value significantly less than the Einstein prediction of 2-5. This result is very strange indeed, for it was for the shear-thinning system measured at low shear rate ( 1 0 s _ 1 ) where r\r is significantly higher than it is at high shear rate.

A more detailed analysis of the shear dependence of nr is given by Krieger (1972) who showed that at high volume fractions and small particle sizes (a < 500 nm) shear thinning occurs even when the electrical effects have been completely suppressed and that it can be attributed to the formation, by Brownian motion, of temporary doublets which are destroyed by the shear field. For larger particles the formation of more permanent aggregates dominates the behaviour, which is therefore quantitatively somewhat dif­ferent. To establish whether there is any influence of shear rate on the primary electroviscous effect, which on the face of it seems doubtful, one would first have to separate out the shear rate effects which occur even in suspensions of rigid uncharged spheres at high volume fractions.

Although Russel's (1978a) analysis provides a theoretical reason for the primary electroviscous effect to decrease with increase in shear-rate (eq (5.2.9)), i t cannot be expected to apply quantitatively to the very high volume fraction data involved in these experiments. Russel has tested his equation against data obtained by Chan and Goring (1966) for lower values of 4>, and it is clear from his examination that better experimental data is needed on the shear dependence of the coefficient Such data wil l , of course, be very difficult to obtain with the necessary accuracy.

A final note of caution is appropriate here. The Mooney equation (3) is a semiempirical relation and there is no a priori reason why the parameter, a', obtained from it at high values of (p should be a true reflection of the primary electroviscous effect as defined above. When higher-order interactions be­come significant the particles in the suspension spend a considerable part of their time in interactions which modify both their own double layers and the shear field around them. To separate out the primary effect it is therefore necessary to extrapolate back to 4> = 0 from the lowest accessible particle concentration. The discordance between Wang's (1970) results and those of Stone-Masui and Watillon (1968) suggests that the long extrapolation in the former case (from <j> = 0-14) masks the true behaviour for small volume fractions.

The best data for testing the secondary electroviscous effect is that pro-

5. ELECTROVISCOUS AND VISCOELECTRIC EFFECTS 199

vided by Stone-Masui and Watillon (1968). They compared their experi­mental data with the theoretical model of Chan et al. (1966) and found that the theory predicted the general qualitative features of the effect but that it was not very satisfactory for quantitative analysis. Russel (1976) used the same data to test his more elaborate analysis and found it necessary to postulate that multiparticle interactions become important at relatively low volume fractions. In a more recent paper, however, (Russel 1978b) it is suggested that the previous discrepancy was due to the neglect of the counter-ions in the calculation of the K values. The dependence of k2 on (JOZ) 5 (see eq (5.2.12)) makes the result very sensitive to the value of K and when K is adjusted for this effect the agreement between theory and experiment is

0 ' 0-10 0-20 Ratio (counterions !added electrolyte)

Fig. 5.6. Comparison of theoretical predictions with experimental data for the secondary electroviscous effect. O: K value calculated from added electrolyte (Stone-Masui and Watillon (1968)); • : K value interpolated at constant ionic strength (Stone-Masui and Watillon, 1976, priv. comm.) (From Russel, 1978b.)

200 ZETA POTENTIAL IN COLLOID SCIENCE

vastly improved. The results shown in Fig. 5.6 suggest that, at least for low volume fractions, the present theory is in a satisfactory state.

At higher volume fractions the situation becomes very complicated indeed and multiparticle and cooperative effects are certainly significant. Experi­mental evidence on this point is provided by the work of Brodnyan and Kelley (1965) and Krieger and Eguiluz (1976), who show that in well-dialysed latices (ch = 0-4) the multiparticle effects become so important that they modify the behaviour considerably. In these systems most of the ions present are those contributed by dissolution from the latex, and the behaviour of added ions is not at all like that to be expected from the usual D L V O theory of colloid interaction. Valency effects are non-existent and the added ions seem to act almost entirely by modifying the particle charge. Such systems can form ordered arrays at fairly low volume fractions (see Krieger, 1972) where they behave as liquid crystals stabilized by Coulomb forces and separated by distances of the order of the wavelength of light. Optical, rheo-optical and elasticoviscous effects can be observed in such systems. They exhibit yield stresses which decrease with decreasing volume fraction and with increasing electrolyte concentration. The second electroviscous effect in such systems is a function of the equivalent concentration of added electro­lyte and does not depend on the type of ion or its valency. Unfortunately, the theoretical treatments cannot at this stage be applied to such systems because they combine high potentials with high volume fractions.

5.2.4. The ter t iary e lec t rov iscous ef fect

The viscosity of a polyelectrolyte solution depends upon the conformation adopted by the polymer molecule and this is determined by the intramolecular interactions between the various segments. At high ionic strengths the re­pulsive interactions are minimized and the chains often tend to adopt a more or less compact configuration. Reducing the ionic strength wil l in­crease the repulsions and the chain wi l l tend to open out. Thus a polymer solution which behaves like a collection of spheres at high ionic strength may be better represented as a number of rigid or kinked cylinders at lower ionic strength. Although the difference in the behaviour at the extremes of salt concentration may be explained in this qualitative way it is very difficult to set up a quantitative theoretical description of the process. The changes in conformation can be expected to influence the value of both kx and k2 in eq (5.2.2).

The early work in this area is described by Conway and Dobry-Duclaux (1960), and more recent developments are reviewed by Overbeek and Wiersema (1967). Since these analyses are all conducted in terms of the

5. ELECTROVISCOUS AND VISCOELECTRIC EFFECTS 201

charge on the segments rather than the potential, we would need to introduce further assumptions i f we wished to use them to gain further insight into the interpretation of (-potential. The more recent analyses of Stigter (1975, 1979) suggest that the electrophoretic mobility of poly-(acrylic acid)-and D N A can be adequately described over a range of electrolyte concentrations by treating them as long cylinders (see Sections 3.7.3 and 4). The tertiary electroviscous effect is evidently not very great in these cases. We wil l there­fore, turn now to a quite different area for evidence of a tertiary electro­viscous effect of a quite markedly different kind.

Coagulated colloidal suspensions also exhibit what may be called a tertiary electroviscous effect, in the sense that their viscous behaviour de­pends upon the attractive and repulsive forces between the particles (Hunter and Nicol, 1968) and the way these forces modify the geometry of the system. The actual flow behaviour obtained depends to some extent on the shear history of the sample but reproducible behaviour (at least with respect to simple shear) can be obtained i f the sample is first subjected to a very high shear rate. Measurements can then be taken of the shear stress-shear rate relation at lower shear rates and a typical result would be as shown in Fig. 5.7. The parameters which describe the high shear rate behaviour are then: (i) the intercept of the linear part of the curve on the shear stress axis, xB, called the Bingham Yield Value; (ii) the critical shear rate, y0, at which the curve appears to become linear; (iii) the slope of the line at high shear rate rjPL, called the plastic viscosity. A l l three quantities can be shown to depend on the (-potential of the particles in the coagulum. A model to explain this behaviour has been proposed by Firth and Hunter (1976), who suggest that the flow units in the system consist of more or less spherical floes containing many hundreds of primary particles linked in a three-dimensional array and trapping a significant volume of the suspension medium. The openness or compactness of the floe is measured by the floe volume ratio CFP = <pF/(j)P

where 4>F is the volume fraction of floes and 4>P is the volume fraction of particles. Open, loosely packed floes (large CFP) are expected to occur in systems with very strong attractive forces between the particles; as the attractive forces are diminished (by raising the (-potential (see Appendix 5.2)) the floe must become more compact to withstand the initial very high shear rate to which it has been subjected f j c „ ) . Using a very elementary stress analysis on the floe Firth and Hunter (1976) show that:

CFp — 20r]oryc \2d\

(5.2.15)

where dx is the distance at which the force between particles is a maximum and r is the particle radius. The bracketed term measures the maximum force which can be withstood by the bond between two particles: i t is calculated

202 ZETA POTENTIAL IN COLLOID SCIENCE

Fig 5 7 Variation in the basic flow diagram (see Appendix 3) as £ potential is varied in a co­agulated poly (methyl methacrylate) suspension, r = 0-44 urn; ionic strength = 2 x 1 0 mol 1 at <j> = 0-07. £ (in mV): (1) 14-6; (2) 21-6; (3) 29-3; (4) 35-9; (5) 55; (6) supernatant. Suspension 5 is disperse and follows the Einstein equation (from Friend and Hunter, 1971.)

from the D L V O theory of colloid stability (Appendix 5). The plastic viscosity is then obtained by assuming that the total energy dissipation, E, can be broken into two terms: (i) a viscous energy, E„ due to the flow of the sus­pension medium around the floes and (ii) an additional dissipation within the floes {Ef). This sort of breakdown had been suggested earlier by Michaels

and Bolger (1962). . The viscous energy dissipation is assumed to be given by the Einstein

5. VISCOELECTRIC AND ELECTROVISCOUS EFFECTS 203

equation (5.2.1) but with <f> = <f>F, and is calculated from the plastic viscosity:

EV = E - E f = ( x - xB)y = r,PLy2 (5.2.16)

whilst the plastic viscosity, tjPL, is related to the volume fraction of floes, (f)F

by an empirical equation like (5.2.3) or (5.2.4). (The simple Einstein relation, (5.2.1), breaks down at the volume fractions considered here.)

Measuring T as a function of y for various systems of known qbP and dif­ferent £ potential enables CFP to be studied as a function of £. Firth and Hunter have applied the model to a number of systems, including kaolinite dispersions in water and water emulsions in benzene. The most accurately controlled systems are, however, monodisperse latices of poly- (methyl-methacrylate) or poly(styrene) of particle size about lOOnm. The relation between CFP and £ for such a system is shown in Fig. 5.8, which conforms with eq (15). I t can also be shown experimentally (Firth, 1976) that CFP is proportional to r _ 1 in some systems, but in some cases this relation breaks down, perhaps because yCrr varies with r.

The critical shear rate, y0, is assumed to correspond to the point at which two approaching floes can just be torn apart by the shear field. At lower shear rates the shearing stress is insufficient to break the floc-floc bonds, and aggregates of floes are built up. Using the same simple stress analysis Firth and Hunter (1976) find, for yQ, the value:

(5.2.17)

r,2(mv)2

Fig. 5.8. Plot of the floe volume ratio CFP = cj>F/<pF as a function of f 2 for poly (methylmeth-acrylate) suspensions in glycerol/water mixtures of different viscosities. • : 0%; v : 12-5%; O: 31-2% glycerol. r= 120nm. (Hunter and Frayne, 1979.)

204 ZETA POTENTIAL IN COLLOID SCIENCE

where nF is the number of bonds formed between two floes during a collision. Again the dependence of y0 on £ 2 can be demonstrated experimentally (Fig. 5.9).

Finally, the value of xB is estimated by examining the additional energy, E f , dissipated within the floes during a collision (van de Ven and Hunter, 1977), and this leads to:

T B = f = ( ^ ) ^ i C „ (5.2.18)

The term in brackets has been found to be independent of y (Hunter and Frayne, 1980) and the proportionality between t B and CFP implies that xB

should also depend on ( 2 , which it clearly does (Fig. 5.10). The internal consistency of the model is also indicated by the fact that the plots of CFP, of y0 and of xB against ( 2 all allow an estimate of the parameter i t to be made (dx is the particle separation at which the force is a maximum) and all these estimates give dl = 6 + 1 A.

5.3. T h e v i s c o e l e c t r i c e f f e c t

One of the most pervasive problems in the interpretation of (-potential is that of defining the position of the plane of shear (Section 1.2). So far we have assumed that it is near to, if not coincident with, the OHP, so that ( » \j/d. In this and subsequent chapters, a variety of evidence is presented for the belief that ( is a useful indicator of the magnitude of the diffuse double-layer potential, especially for situations involving interparticle interaction. We

5. ELECTROVISCOUS AND VISCOELECTRIC EFFECTS 205

wish now to examine, however, the extent to which the concept of a shear plane can be modified without destroying the theoretical analysis developed in Chapter 3, in which i t was assumed that the suspension medium retained its bulk values of viscosity and permittivity right up to the shear plane, at which point the viscosity became infinitely large. The electric field in the neighbourhood of the interface is expected to exert some influence on the structure of the surrounding fluid and the extent to which the viscosity is modified is called the viscoelectric effect.

5 . 3 . 1 . M o d i f i e d v iscos i ty and permi t t i v i t y in the doub le layer

In the derivation of Smoluchowski's equation, eq (3.1.1) may be written:

„ , d f dv.\ E " p d X + d-x{r,-dt)dx = 0 ^

and the more general form of Poisson's equation (2.3.1) used to substitute for p:

„ d ( # \ d ( dv.\

Integrating this expression from the bulk solution, where both dé/dx and dvjdx are zero (Fig. 3.1) and up to a point near the surface gives (Overbeek 1952):

dv„ sE. dé

r,2(mv)2

Fig. 5.10. Dependence of the Bingham Yield Value, T b , of coagulated sols on £ 2 .

206 ZETA POTENTIAL IN COLLOID SCIENCE

and a second integration from the bulk solution where v = ve0 and é = 0 yields:

• = u„ o n

= - s 0

o >7 (5.3.4)

In the usual derivation (eq (3.1.3)) both D and r\ are assumed to be constant, but Davies and Rideal (1961) attempted to rectify this situation by esti­mating the influence of the electric field in the double layer on D and r\. Using the data available at the time they concluded that D/r\ would begin to fall in value if the field strength exceeded about 105 Ve in 1 and would be reduced essentially to zero for fields greater than 106 V cm l . Such fields are not at all impossible in the near neighbourhood of the surface. Whereas Davies and Rideal analysed the situation graphically, an analytical solution was provided by Lyklema and Overbeek (1961), who considered the situation where D is maintained constant and r\ is considered to vary in accordance wi th :

ri(x) = no (5.3.5)

as suggested by the work of Andrade and Dodd (1951). n0 is the viscosity of the bulk liquid where the field is zero and ƒ is the viscoelectric constant which is assumed to be a characteristic for each liquid and independent of con­centration. Substituting for dé/dx from eq (2.3.15) gives, for the mobili ty:

Um = £0D dé

,1 +2/Csinh 2(ziA/2) (5.3.6)

Here C = 4n°kT/e0D ( = 14-1 x 1 0 1 5 c v o l t 2 m - 2 at 25°C i f c is the electrolyte concentration in mol 1"1).

Lyklema and Overbeek (1961) gave three analytical expressions for this integral depending on whether 2fC was equal to, greater than, or less than unity. From what follows it wi l l be apparent that only the latter condition is likely to be of much interest in which case the mobility is given by (Hunter, 1966):

»7o where

ze • 2 / C ) - 1 / 2 arctanh

z D (1 - f C ) 1 1 2 t a n h y

(5.3.7)

(5.3.8)

£ is the apparent value of £ which would be calculated from the Smolu-l~„oU n a t i o n from any given mobility. As 2fC approaches unity, £ s can

5. ELECTROVISCOUS AND VISCOELECTRIC EFFECTS 207

be simplified to: 2kT , zl

tanh— ze 2 [Cshfc= i = tanh— (5.3.9)

as suggested by Lyklema and Overbeek (1961). For small values of £ and/or C and especially for the small values of ƒ that we shall concentrate on, the approximate solution:

kT Cs = C-fC— (sinh z£"-z£) (5.3.10)

was given by Lyklema and Overbeek. Their estimate of the value o f / ( = 10-2 x 10 1 6 volt 2 m 2 ) was based on a comparison with the known behaviour of several organic liquids, and the expected corrections to the Smoluchowski equation were very significant, especially at high concentrations (e.g. 20% correction in 0-01 M solution for £ = 75 mV).

In a later analysis, Stigter (1964a) used eq (6) to calculate the shift, A, m the apparent position of the plane of shear due to the enhanced viscosity near the surface of a soap micelle. He found that, for 2fC < 1:

A = séK

where arctanl sé cosh- j - arctanh

2fC U ~ 2/C)_

1/2

(5.3.11)

The experimental value of A, based on a concerted analysis of various ex­periments on micelles of sodium dodecyl sulphate appears to be equal to or less than 1 A, which would correspond to a value of about 1 0 " 1 6 m 2 v o l t - 2

for the constant ƒ With his detailed model of the micelle-solution interface, Stigter (1964b) shows that a significant part of the discrepancy between é i and £ arises from the fact that £ measures the average potential in the plane of shear. The additional contribution to é i due to the discrete nature of the ions in the head group amounts to as much as 50-75 mV for the dodecyl sulphate and dodecyl ammonium ions. (See Fig. 6.11 for i/r;.)

A very low estimate for ƒ was also arrived at by Hunter (1966), who exam­ined the limiting behaviour of eq (8) as £ becomes infinitely large. I t may readily be shown from eq (8) that:

l im £ s = ^ ( 1 - y c f * in ( I ± £ z ^ > ( 5 3 1 2 ) zeK J ' \ \ - ( 1 -2fC)1/2J

and i f one can find a system in which £ is expected to increase but £ s is found to reach a limiting value, this should provide a means of estimating an upper limit for ƒ

208 ZETA POTENTIAL IN COLLOID SCIENCE

Haydon's (1960) data on paraffin oil drops stabilized by anionic and cationic soaps appear to provide such a system, since he determined the (Smoluchowski) (-potential and also estimated the surface potential as the density of surface charge was increased at constant ionic strength. Hunter (1966) showed that Haydon's data indicated an upper l imit for ƒ of 1-2 x 1 0 " 1 7 m 2 v o l t - 2 for water. Watillon and Joseph-Petit (1966) also concluded from measurements on latex systems that the limiting value of ( 5 calculated from eq (8) was too low unless one reduced significantly the value o f f used by Lyklema and Overbeek (1961). A value of 1 0 " 1 7 m 2 vo l t" 2 in eq (11) corresponds to a A value of 0-22 A for ( = 100 mV and c = 0-01 M . (Note that very small shifts in the position of the shear plane correspond to large changes in the value of ƒ.)

More recently, Hunter and Leyendekkers (1978) estimated ƒ for water in the neighbourhood of clay mineral surfaces and found, f rom a variety of viscosity data, a value o f f = 10 " 1 5 m 2 vo l t" 2 , very close to the initial Lyklema and Overbeek (1961) estimate. They suggested that the surface may in this case be able to introduce some unique structuring effect on the contiguous water phase, although a much smaller effect than is often assumed. Since that analysis appeared Stigter (1980) has shown that the apparently augmented viscosity between two clay plates can be quite adequately explained by the wall effect so that ƒ values of 1 0 " 1 7 V " 2 m 2 are valid even in this case. (The wall effect is the effect produced on the motion of a sphere confined between two walls by the fact that the fluid velocity is zero at the walls and not at infinity as is required for application of the Stokes equation linking the diffusion of the sphere to the liquid viscosity.)

Returning to the - lower estimates of f which seem to be appropriate in most if not all systems, the correction to the Smoluchowski equation is very much reduced and the effect of the field strength on the viscosity becomes comparable to its effect on the permittivity and both should then be taken into account simultaneously. The Davies and Rideal (1961) procedure can be used, or the problem can be treated analytically as follows.

Booth (1951) derived for the dielectric constant:

D = D0[l — B(dyj/dx)2] (5.3.13)

where B is expected to have a value of about 4 x 1 0 " 1 8 m 2 v o l t " 2 at 25°C in water. The field strength in the diffuse layer must be calculated using the more general form of the Poisson equation (2.3.1) and is found to be (Hunter, 1966):

5. ELECTROVISCOUS AND VISCOELECTR1C EFFECTS 209

Substituting eqs (5), (13) and (14) into (4) gives:

uE = -1o

x (2 + pV2)dè

0 5 + f ( \ - p ^ ) I B ( 5 3 J 5 )

where p = l - \2BC sinh2(i/;/2). Provided ( and c are sufficiently small ( C < 0 - 0 1 M for ICI == 100 mV), one can simplify eq (15) and show that the dielectric and viscosity effects are then simply additive (Hunter, 1966). For larger ( and/or c values the combined effect is rather larger than the sum of the two separately. This is clearly shown in Fig. 5.11 where a comparison is made with Haydon's (1960) data.

An exact analytical solution of the integral in eq (15) can be given in terms of elliptic integrals of the third kind, but it is easier and more accurate to integrate the function numerically; the magnitude of the correction function for various values of B, f c and \ is given graphically in Figs 5.12 and 5.13 (from Hunter (1966)). These figures may be used to estimate the regions in which one might confidently seek to equate the measured ( potential with the total diffuse double-layer potential.

-150

- 5 0

-10 -20 -30 Surface charge density ( i tC cm 2 )

-40

Fig. 5.11. Comparison between Haydon's (1960) data and the calculated curves for various values of /and B (see text). Full curves: Haydon's data. Curve ( a ) , / = 1-25 x 1 0 - I 7 m 2 v o l r 2

B = 0; (b)f = 2-50 x 1 0 - 1 7 m 2 volt" 2, B = 0; (C) /= B = 1-25 x 10" 1 7 m 2 volt" 2. (From Hunter (1966)).

210 ZETA POTENTIAL IN COLLOID SCIENCE

i 1 1 r

J I i i L 4 6 8 10 12

4

Fig. 5.12. Correction curves for converting "Smoluchowski" -potential (£s) to true C-potential, for various values of/and B. All points along a curve show a constant value for £/£s as indicated. Full lines: f/B= 1; Broken lines: f/B= 10 (from Hunter (1966)).

5.4. P o s i t i o n o f t h e p l a n e o f s h e a r

Early estimates of the position of the plane of shear, with respect to the inter­face, were based on a simple Gouy-Chapman picture of the double layer. Thus, Eversole and Boardman (1941) considered the dependence of C on the concentration of indifferent electrolyte using eq (2.3.16) in the form:

Fig. 5.13. As for Fig. 5.12. Full lines: f/B = 100; Broken linesf/B = 1000. (From Hunter, 1966.)

5. ELECTROVISCOUS AND VISCOELECTRIC EFFECTS 211

where \\it is the potential at a distance t from the interface. The data tabulated by Eversole and Boardman (1941) are plotted in Fig. 5.14 in accordance with eq (5.4.1). Values of / range from 16-49 A. The very large value (63 A), criticized by Overbeek (1952, p. 229) as being undoubtedly too large, pre­sumably comes from data like that shown by the open circles to which the model clearly does not apply. The value of 49 A attributed to an unspecified ceramic surface is obviously at variance with the value obtained by Hunter and Alexander (1963) on the kaolinite surface and shown in Fig. 5.14 for comparison. The lower values obtained in these plots might bear some semblance to reality, especially if one modified the Gouy-Chapman model to take account of the compact layer (Section 2.4), but the upper values are generally considered to be excessive (Overbeek, 1952, p. 229). The apparent confirmation of these high values o f f , obtained by Eversole and Lahr (1941) in experiments with water adsorbed on quartz plates, has been criticized by Elton (1946), who showed that the experimental errors involved in that work were too great to reach a conclusion which was statistically valid.

In more recent work an attempt is made to take account of the detailed structure of the interface in order to compare ( with the most appropriate double layer potential. Thus Haydon (1960) compared his (-potential measurements on soap-stabilized oil drops with a calculated value of the potential in the plane of the head-groups using a theoretical analysis of Haydon and Taylor (1960). He found that at low values of the surface charge

o-

ffig. 5.14. Plots of In tanh (zf/4) as a function of K (eq 5.4.1) from the data collected by Eversole and Boardman (1941). Data from Hunter and Alexander (1963) on kaolinite are included for comparison.

212 ZETA POTENTIAL IN COLLOID SCIENCE

density the two potentials were identical (see Fig. 5.11), but they diverged as the stabilizing head groups came closer together. Although some criticism has been levelled at the theoretical calculations of Haydon and Taylor by Levine et al. (1964) this particular conclusion is not likely to be challenged (Carroll, 1970).

I t was noted in Section 5.3 that Stigter's (1964) estimate of the distance between the plane of the headgroups and the shearing surface in micellar solutions was less than 1 A. More recently Carroll and Haydon (1975) have extended Haydon's examination of emulsion systems, stabilized by soaps, and concluded that the shear plane coincides with the plane of centres of the headgroups up to a surface density of charge of about 0-1 C m - 2 . Above that value it moves more or less quickly out a distance of a few angstroms towards the outer edge of the headgroups as the counterions of the diffuse double layer become squeezed out by the increasingly densely packed head-groups. A similar model was developed by Hunter (1966) to account for Haydon's earlier (1960) data. Carroll (1970) also made an attempt to take account of the influence of the local ionic concentrations on the permittivity and viscosity as Dukhin and Deryaguin (1974, p. 184) suggest is essential. The difficulties involved in assessing these effects in combination are rather formidable, and it is doubtful whether this more extensive analysis is justified until the behaviour of ionic solutions in high fields is better understood. Nevertheless Carroll's conclusions offer a plausible extension of the model proposed by Stigter (1964) who placed the shear plane within 1 A of the outer sheath of the headgroups (either inside or outside).

Another estimate of the position of the shear plane is provided by the work of Mysels et al. (Mysels and Cox, 1962; Lyklema et al., 1965), who studied the drainage of films drawn from detergent solutions. The thickness, <5, of the f i lm depends on the velocity, v, of withdrawing and is given by Frankel's equation:

1 , k ^ (542) v213 yll6{dg)112

where y is the surface tension, d is the density, k is a constant (1 < k < 2) and g is the acceleration due to gravity. The equation is derived on the as­sumption that r\ is constant over the entire f i lm cross-section, and the fact that the plot of S against v2/3 is linear down to very small thicknesses suggests that the thickness of the rigid layer is, at each face of the fi lm, 16 + 8 A. Since the detergent itself takes up 16 A it is concluded that the rigid water layer has thickness 0 ± 8 A.

The above remarks refer to systems involving surfactants and it is im­portant that we determine whether the conclusions can be carried over into other systems. One such system for which a plausible determination has been

5. ELECTROVISCOUS AND VISCOELECTR1C EFFECTS 213

made of the position of the plane of shear is the silver iodide surface in the neighbourhood of the p.z.c. Smith (1973) considers the function:

S = r dc # o _d(pAg)_ # 0 d(pAg)_

(5.4.3)

At low potentials we have (c.f. equation (2.3.11)):

C = i A d e x p ( - K A ) (5.4.4)

where A is the distance from the OHP to the plane of shear. The factor dé0jd(pAg) = N is the Nernst factor, which may be taken as constant and experimentally determinable and the quantity déd/dé0 in the absence of specific adsorption is obtained from the relation

K ^ j ^ - r - (5.4.5)

where Kt is the integral capacitance of the inner layer.

# 0 1 da0 a0 dK< Q IT = 1 + ~ F ' T T _ ~ Ï ^ - T 7 - = l + — as<7 0->0 (5.4.6) # d &i déd Kf déd Kt

where Cd is the diffuse layer capacitance ( = % D K ) . Substituting in eq (3) then yields:

S - ^ N - ^ l + ^ e x p O c A ) (5.4.7)

The experimental plot of S""1 against Cd is shown in Fig. 5.15. The intercept at Cd = 0 (and hence K = 0) is very close to 1/59 ( m V - 1 ) and the linearity suggests that A is small. The slope gives a value of K, = 3 0 u F c m " 2 which is in good agreement with the results from direct adsorption studies and again implies that A is small. Assuming a maximum error in S~l of 10%, Smith infers that A must be less than 1 A at all concentrations i f the potential is low. At higher potentials there is no reason to suppose that the shear plane moves far f rom the OHP. The observed behaviour can usually be fitted if account is taken of adsorption in the compact layer (see Chapters 6,7 and 8). We might recall here that Ottewill and Woodbridge (1964) used low values oldrjdpAg (i.e. values less than the Nernst prediction) to infer that the surface regions of their silver halide sols contained many defects (see Section 2.6). As Smith (1973) pointed out, the lower values of dQdpAg are expected on the basis of a simple Gouy-Chapman-Stern model of the double layer. The values obtained by Ottewill and Woodbridge (1964) were highly vari­able and depended on the method of preparation of the sol. Perhaps both effects are involved to some extent. Certainly if the Smith effect were de-

214 ZETA POTENTIAL IN COLLOID SCIENCE

T

I i i I 1 I I I I 0 20 40 60

Fig. 5.15. Experimental plot of 5 " 1 against C j ( = KS) for Agl in K N 0 3 (aq)at 25°C. At C j = 0, S = 59 mV; slope yields Kt = 30 u F c m - 2 . (From Smith (1973) with permission.)

ducted from the experimental results of Ottewill and Woodbridge, their estimates of the numbers of defects involved would be reduced to more reasonable values.

Lyklema (1977) has also examined in some detail the relation between \j/d

obtained from coagulation data on A g l sols and ( measured on the same system. He also concluded that up to quite high values of yjd (i.e. up to values of ApAg = 6, corresponding to i ^ o > 3 0 0 m V ) the two are identical (see Section 7.2). Chelidze et al. (1973) reached the same conclusion from their dielectric dispersion studies on chloroprene and poly- (vinyl chloride).

Another promising method of comparing ( wi th the \j/d (or é;) potential is that using the electrical conductivity of colloidal suspensions. A theoretical description has been developed by Dukhin and Semenikhin (1970) and an attempt made by Sonntag and Pilgrimm (1976) to apply it to a dispersion of small silica particles. The results were not an unqualified success, for various reasons described in the latter paper, but the method clearly warrants more work. I t rests, of course, on the development of an adequate theory of the contribution of the inner-layer ions to the overall conductivity of the suspension. Measurements at different frequencies (Sidorova et al, 1976) may provide a means of distinguishing the contributions of the compact layer from that of the diffuse layer. (See also Drobosyuk and Talmud, 1977.)

In an extensive analysis of the problem, Haydon (1964) concluded that measured (-potentials are limited by the low ratios oïD/n which occur in the

5. ELECTROVISCOUS AND VISCOELECTRIC EFFECTS 215

double layer and cited his own (1960) data to support this belief. Since then Carroll and Haydon (1975) have shown that that data can be explained by steric effects (cf. Hunter (1966)). There remains other data, however, especi­ally on oxides and clay mineral surfaces, which have been brought forward to support claims for a more or less extensive region of structured water at certain interfaces. Much of the early work was based on the permeability of packed beds where geometrical factors were obscure and electrical effects in very fine pores were not taken properly into account. Thus Low (1961) and Drost-Hansen (1977) presented a considerable volume of experimental data which could be explained on the basis of extensive water structure at inter­faces, and Deryaguin (1966) described a series of ingenious experiments to demonstrate its existence. Unfortunately, much of that evidence is now re­garded as equivocal for one reason or another. The presence of impurities, (Deryaguin et al, 1974), the solubility of oxides in water to form polymeric species which could influence flow, the possibility of microbial contamination in long-term flow investigations, the effects of swelling of porous oxide surfaces (Perram et al, 1973, 1974), the difficulty of ensuring a particular model geometry—all these and other objections have been raised by the sceptics who feel that such "extended water structure" could all too easily be used to "explain" almost any experimental outcome. As noted in Section 5.1.3 the most recent and careful Soviet studies on the streaming potential behaviour of fine vitreous silica capillaries (Churaev et al., 1981) show clearly that gel layers grow slowly on such surfaces and can be swept away by application of high pressure gradients. On quartz, however, even shear stresses as high as 10 4 dyne c m " 2 fail to produce any shift in the plane of shear (Zorin et al, 1977).

One "oxide" system which has been examined in great detail is that of exfoliated mica, and the work of Israelachvili and Adams (1978) on the forces between mica sheets in electrolyte solution suggests that there is no extensive water structure at the interface. They were able to measure the forces down to separations of the order of 10 A and found general agreement with double layer theory; the refractive index of the liquid also maintained its bulk value at these very small separations. Of course, these were equilibrium rather than flow measurements and the compressive forces rather than the resistance to a shearing stress were being measured, but it seems reasonable to assume that an extended water structure, i f i t were present, would be reflected in some way in their measurements.

Israelachvili and Adams' (1978) results are indeed consistent with the notion that only the first few water layers (perhaps 5-6 A) are held immobile by the mica surface. This would be consistent with the general notion that r ~ é d . A similar conclusion was reached by Smith (1976) for the T i 0 2

surface using a modification of the method described above (Fig. 5.15;

216 ZETA POTENTIAL IN COLLOID SCIENCE

see Section 7.4). A similar conclusion is also reached by Deryaguin et al. (1972) in a study of capillary osmosis through glass membranes.

Although there remains a good deal of data, especially in the Soviet litera­ture, which would suggest that extended water structures occur on some sur­faces, much of it is in relatively inaccessible journals and at least some of i t lacks the necessary experimental detail to allow a ful l evaluation. Dukhin and Deryaguin (1974, p. 188) analysed some of this data, and Deryaguin (1966) has argued for such extended structures on (at least some) oxide sur­faces over many years. (See, for example, the criticisms of the "blow-off" method of measuring surface viscosity by Bascom and Singleterry (1978) and the reply by Deryaguin et al. (1978).) Such advocacy cannot be lightly dis­missed but in subsequent chapters we shall find that a good deal of data on various surfaces, including oxides, can be rationalized on the assumption that £ ~ é d , and to avoid additional arbitrariness in the models we use, we shall often make the equation £ = èd. Although this procedure may lead to some error in some situations, the advantages that accrue from having a well-defined value of the potential in the diffuse double layer greatly outweigh any disadvantages.

R e f e r e n c e s

Anderson, J. L . and Wei-hu Koh (1977). J. Colloid Interface Sci. 59, 149. Andrade, E. N . Da C. and Dodd, C. (1951). Proc. Roy. Soc. (London) A204, 449. Bascom, W. D . and Singleterry, C. R. (1978). J. Colloid Interface Sci. 66, 559. Bell, G. M . and Peterson, G. C. (1972). J. Colloid Interface Sci. 41, 542. Blachford, J., Chan, F. S. and Goring, D . A . I . (1969). J. Phys. Chem. 73, 1062. Booth, F. (1950). Proc. Roy. Soc. (London) 203A, 533. Booth, F. (1951). J. Chem. Phys. 19, 391, 1327, 1615. Brodnyan, J. G. and Kelley, E. L . (1965). / . Colloid Sci. 20, 7. Bull, H . B. (1932). Kolloid-Z. 60, 130. Carroll, B. J. (1970). Ph.D. Thesis, Univ. of Cambridge. Carroll, B. J. and Haydon, D . A. (1975). J. Chem. Soc. Faraday I. 71, 361. Chan, F. S., Blachford, J. and Goring, D . A. I . (1966). J. Colloid Interface Sci. 22, 378. Chan, F. S. and Goring, D . A. I . (1966). J. Colloid Interface Sci. 22, 371. Chelidze, T. L . , Shilov, V. N . and Baran, A . A . (1973). Colloid J. (U.S.S.R.) 34, 699. Churaev, N . V. and Gorokhov, M . M . (1975). In "Research in Surface Forces", Vol . 4,

pp. 176-181 (B. V. Deryaguin, ed.). Consultants Bureau, New York. Churaev, N . V., Sergeeva, I . P., Sobolev, V. D . and Deryaguin, B. V. (1981). J. Colloid

Interface Sci. I n press. Conway, B. E. and Dobry-Duclaux, A. (1960). In "Rheology: Theory and Applica­

tions", Vol . 3, p. 83 (F. R. Eirich, ed.). Academic Press, New York and London. Corkill , A. J. and Rosenhead, L . (1939). Proc. Roy. Soc. 172A, 410. Davies, J. T. and Rideal, E. K . (1961). In "Interfacial Phenomena". Academic Press,

London and New York. Deryaguin, B. V. (1966). Disc. Faraday Soc. 42, 109.

5. ELECTROVISCOUS AND VISCOELECTRIC EFFECTS 217

Deryaguin, B. V.; Dukhin, S. S. and Koptelova, M . M . (1972). J. Colloid Interface Sci 38, 584.

Deryaguin, B. V., Zorin, Z. M . , Rabinovitch, Ya. I . and Churaev, N . G. (1974). J. Colloid Interface Sci. 46, 437.

Deryaguin, B. V., Karasev, V. and Chromova, E. (1978). J. Colloid Interface Sci. 66,573. Drobosyuk, V. M . and Talmud, S. G. (1977). Colloid J. (U.S.S.R) 39, 111. Drost-Hansen, W. (1977). J. Colloid Interface Sci. 58, 251. Dukhin, S. S. and Deryaguin, B. V. (1974). Electrokinetic phenomena, In "Surface and

Colloid Science," Vol. 7 (E. Matijevic ed.). John Wiley, New York. Dukhin, S. S. and Semenikhin, N . M . (1970). Kolloid Zhur. 32, 360. Elton, G. A . H . (1946). / . Chem. Phys. 14, 463. Eversole, W. G. and Boardman, W. W. (1941). J. Chem. Phys. 9, 789. Eversole, W. G. and Lahr, P. H . (1941). J. Chem. Phys. 9, 686. Finkelstein, B. N . and Cursin, M . P. (1942). Acta Physiocochim,, S.S.S.R. 17, 1. Firth, B. A . (1976). J. Colloid Interface Sci. 57, 257. Firth, B. A , and Hunter, R. J. (1976). J. Colloid Interface Sci. 57, 266. Friend, J. P. and Hunter, R. J. (1971). J. Colloid Interface Sci. 37, 548. Gillespie, T. (1963). / . Colloid Sci. 18, 32. Goodwin, J. W. (1975). In "Colloid Science", Vol. 2, pp. 246-293. Specialist Periodical

Reports. Chemical Society, London. Grigorov, O. N . and Novikova, N. -A. (1955). Kolloidnyi Zhur. 17, 278. Harmsen, G. J., van Schooten, J. and Overbeek, J. Th. G. (1953). J. Colloid Sci. 8, 72. Haydon, D . A . (1960). Proc. Roy. Soc. (London) 258A, 319. Haydon, D . A . (1964). In "Recent Progress in Surface Science" p. 140 et seq (J. F.

Danielli, K . G. A. Pankhurst and A . C. Riddiford, eds.). Academic Press, London and New York.

Haydon, D . A. and Taylor, F. H . (1960). Phil. Trans. A252, 225; A253, 255. Hildreth, D . (1970). J. Phys. Chem. 74, 2006. Hunter, R. J. (1966). J. Colloid Interface Sci. 22, 231. Hunter, R. J. and Alexander, A. E. (1963). J. Colloid Sci. 18, 820. Hunter, R. J. and Frayne, J. (1979). / . Colloid Interface Sci. 71, 30. Hunter, R. J. and Frayne, J. (1980). J. Colloid Interface Sci. 76, 107. Hunter, R. J. and Leyendekkers, J. V. (1978). J. Chem. Soc. Faraday I. 74, 450. Hunter, R. J. and Nicol, S. K . (1968). / . Colloid Interface Sci. 28, 250. Israelachvili, J., and Adams, G. (1978). / . Chem. Soc. Faraday I. 74, 975. Krasny-Ergen, W. (1936). Kolloidzeit. 74, 172. Krieger, I . M . (1972). Adv. Colloid Sci. 3, 111. Krieger, I . M . and Eguiluz, M . (1976). Trans. Soc. Rheology 20, 29. Kushner, L . M . , Duncan, B. C. and Hoffman, J. I . (1952). J. Res. Natl Bur. Std. 49, 85. Levine, S., Bell, G. M . and Pethica, B. A . (1964). J. Chem. Phys. 40, 2304. Levine, S., Marriott, J. R. and Robinson, K . (1975). J. Chem. Soc. Faraday II, 71, 1. Low, P. F. (1961). Adv. in Agronomy 13, 269. Lyklema, J. (1977). J. Colloid Interface Sci. 58, 242. Lyklema, J. and Overbeek, J. Th. G. (1961). J. Colloid Sci. 15, 501. Lyklema, J., Scholten, P. C. and Mysels, K . J. (1965). J. Phys. Chem. 69, 116. Manley, R. and Mason, S. G. (1954). Canad. J. Chem. 32, 763. Michaels, A. S. and Bolger, J. C. (1962). Ind. Eng. Chem. (Fundamentals) 1, 153. Mooney, M . (1951). J. Colloid Interface Sci. 6, 162. Mysels, K . J. and Cox, M . C. (1962). / . Colloid Sci. 17, 136. Napper, D . H . and Hunter, R. J. (1974). In " M T P International Review of Science",

218 ZETA POTENTIAL IN COLLOID SCIENCE

Physical Chemistry Series 2, Vol. 7, pp. 161-213. (M. Kerker ed) Butterworths, London.

Oldham, L B., Young, F. J. and Osterle, J. F. (1963). J. Colloid Sci. 18, 328. Ottewill, R. H. and Woodbridge, R. F. (1964). J. Colloid Sci. 19, 606. Overbeek, J. Th. G. (1952). In "Colloid Science" Vol. I , p. 200. (H. R. Kruyt ed.),

Elsevier, Amsterdam. Overbeek, J. Th. G. and Wiersema, P. H. (1967). In "Electrophoresis" Vol. 2 (Milan

Bier ed.), Academic Press, New York and London. Perram, J., Hunter, R. J. and Wright, H. J. L. (1973). Chem. Phys. Letters 23, 265. . Perram, J., Hunter, R. J. and Wright, H. J. L. (1974). Aust. J. Chem. 27,461. Russel, W. B. (1976). J. Colloid Sci. 55, 590. Russel, W. B. (1978a). J. Fluid Mech. 85, 673. Russel, W. B. (1978b). J. Fluid Mech. 85, 209. Rutgers, A. J., de Smet, M. and Rigole, W. (1959). J. Colloid Sci. 14, 330. Saunders, F. L. (1961). J. Colloid Sci. 16,13. Saville, D. A. (1977). Ann. Rev. Fluid Mech. 9, 321. Schalier, E. J. and Humphrey, A. E. (1966). J. Colloid Interface Sci. 22, 573. Sherwood, J. (1978). Ph.D. Thesis, Cambridge University. Sidorova, M. P., Tasev, D. K. and Fazilova, M . (1976). Colloid J. (U.S.S.R) 38, 723. Sigal, V. L. and Alekseyenko, A. M. (1971). Kolloidnyi Zhur. 33, 737. Smith, A. L. (1973). In "Dispersions of Powders in Liquids" (G. D. Parfitt ed) p. 113.

Applied Science, London. Smith, A. L. (1976). J. Colloid Interface Sci. 55, 525. Smoluchowski, M. von (1916). Kolloidzeit. 18,190. Sonntag, H. and Pilgrimm, H. (1976). Progr. Colloid Polymer Sci. 61, 87. Stigter, D. (1964a). J. Phys. Chem. 68, 3600. Stigter, D. (1964b). J. Phys. Chem. 68, 3603. Stigter, D. (1967). J. Colloid Interface Sci. 23, 379. Stigter, D. (1975). J. Colloid Interface Sci. 53, 296. Stigter, D. (1979). J. Phys. Chem. 83, 1663, 1670. Stigter, D. (1980). Soil Soc. Amer. J. (submitted). Stone-Masui, J. and Watillon, A. (1968). / . Colloid Interface Sci. 28, 187. Stone-Masui, J. and Watillon, A. (1970). / . Colloid Interface Sci. 34, 327. Street, N . (1958). J. Colloid Sci. 13, 288. Van de Ven, T. G. M . and Hunter, R. J. (1977). Rheol. Acta 16, 534. Wang, Y. L. (1970). J. Colloid Interface Sci. 32, 633. Watillon, A. and Joseph-Petit, A. M . (1966). Disc Faraday Soc. 42, 143. Webb, J. T., Bhatnagar, P. D. and Williams, D. G. (1974). J. Colloid Interface Sci. 49,

346. White, H. L., Monaghan, B. and Urban, F. (1936). J. Phys. Chem. 40, 207. Whitehead, J. R. (1969). J. Colloid Interface Sci. 30, 424. Woods, M . E. and Krieger, I . M. (1970). J. Colloid Interface Sci. 34, 91. Zorin, Z. M., Lashnev, V. I . , Sidorova, M . P., Sobolev, V. D. and Churaev, N. V. (1977).

Colloid J. (U.S.S.R.) 39,1012.

Chapter 6

Applications of the Zeta Potential

In the preceding chapters the groundwork has been laid for the interpretation of the zeta potential and the relation between electrokinetic potentials and flow behaviour of colloidal and capillary systems has been examined. In this and the following chapters we shall consider the applications of the (-potential to other areas of colloid science, particularly the extent to which the (-potential can be used as a tool to study the detailed features of charge and potential distribution at interfaces, in the presence of simple ions and in more complicated situations involving surfactants, multivalent ions, polymers and even proteins.

In this chapter we shall introduce a few preliminary concepts relating to the adsorption behaviour of ions and then go on to outline the application of the (-potential in some practical situations like colloid stability behaviour and flotation. No attempt wil l be made to review the literature exhaustively; it is far too large, too diffuse and in many respects simply misleading, and a catalogue of experimental data provides little or no insight into the signifi­cance and relevance of the (-potential measurement. The reasons for this confusion are not hard to find. The ready availability of simple, yet sophisti­cated, electronic equipment has meant an influx into the scientific literature of electrokinetic numbers for a vast array of adsorbent-adsorbate systems. Many of these systems are far from equilibrium, dominated by unknown impurities and controlled by a suite of interacting equilibria; often several parameters are varied at once, and in many cases the adsorbent and/or the adsorbate are undergoing chemical reactions with the solvent. In such systems, even if individual measurements are found to be reproducible, a detailed description of the interface may not be possible, although general qualitative insights may be obtained (see e.g. James et al., 1977).

This does not mean that complex systems cannot or should not be exam-

220 ZETA POTENTIAL IN COLLOID SCIENCE

* A* Frenuentlv it is possible to establish a useful but yet difficult to measure,

empirical correlation ^ W t i c ^ S of ^ e system (see e.g. Cserfalvi parameter and the i g g ^ a n d control of coagulation or e , 1974, 1976). Typcal y, the i o n ( t h i c k e n i n g or floeculation steps » large-scale^ solid 1 iq m e a s u r e m e n t of zeta clarification) operations can be achieved oy f w a t e r t u r b i d i t y potentials and so-called "zeta-potented control o removal has been demonstrated by ^ * ^ s e p a r a t i o n operations by.

Extension of this sort of control o f s o h d l U q u ^ > ^ ^

routine p o t e n t i a l ^ ^ ^ ^ m V ^ S ^ m 0 re t rad i t iona lengmeermtes t couMöe f a s t e r or more tions. Its rationale is simply that «i many• z e t a -po ten t i a l precise to control a thickener or clanfie^perauon y ^ measurement and that a strong ^ f ^ S e ^ s t e m . In such corn-measured number and the 7^fé^L, no claim need be made

— S co'agulation or Hocculation

^ ^ ^ ^ ^

with the risk of passmg adverse J ^ 8 ™ » « ° ° J ° w e s b a l l ^ „ t r a t e a t ^

6 1 I o n i c a d s o r p t i o n a t i n t e r f a c e s

I f we survey the literature in ^ ^ ^ ^ ^ ^ have been reported and concentrate on t h o s e n u m b e r of solute surface or colloidal particle was used andwhere a sm ^ | species was present and « ^ ^ J ^ J or solute species, dispersed phases and an even greater a m y ot « « o ^ | I t would be possible tc.devote ^ i ^ b k o r g U c oxides, silver data on each of the foU^^ a ^ ^ c a t e s ^ p h i d e minerals, hahde sols, glass, clays and c 0 ^ c ^ ° b I c o l l o i k s , metal oxide films

^SSt^Jt^^ « * ~ n a t u r a l i ^ t n ^

- - c s a n d t h e . p h c r ^ ^ ^ ï ^

6. APPLICATIONS OF THE ZETA POTENTIAL 221

JS: of adsron °f — °f

can draw on well-doaunented Ï S S Ï Ï ^ a n d d e v e l o P ™ n t In general the emphasi' ^ d a t a "

species at interfaces- thus e l e c t r o n . L 7 ^ adsorption of solute

interface and the mechanism of a S ^ > Z ^ o f t h e

achieve this end, e m p h a s i s ^ b e t S ^ f ' t h 3 t T o

quantities with various variloljTrathefth ° n T f ^ ° f e l e c t r o k i ^ t i c electrokinetic potential measurement ^ " " b y a s i n S I e

stenau. Illustrated are t n r e f v a r k £ ? f T 5 figUre d u e t o

terms of a simple S t o ^ S S ^ I ^ ^ F " same zeta-potential, i.e. = f , = r „ H Ü t i . « , V a n a t l o n s ^ the particulartheinnerpartofthedo-ibl 3; .! Y ° f t h e i n t e r f a c e a » d *» Thus the same z e t a ' S ^

^ ^ r ^ ^ : ^ ~ ».óhWith a low to moderate

p o r t i n g electrolyte c o n c e S L ^

I HP Shear plane

222 ZETA POTENTIAL IN COLLOID SCIENCE

Case 2. Lower surface potential but still positive (-><,,), little Stern layer adsorption and low concentration of electrolyte so that there is considerable extension of the diffuse layer.

Case 3. Negative but small surface potential (i> 0 3), strong, "super-equiva­lent" adsorption in the Stern plane and moderate extension of the diffuse layer, i.e. moderate concentration of supporting electrolyte.

These are not hypothetical cases and, in fact, approximate closely to the

following systems: Casel. F e 2 0 3 1 0 " 2 M K C 1 p H 4 0 1 ( T 4 M K 2 S 0 4

Case 2. F e 2 0 3 1 ( T 4 M K C 1 p H 7-0 Case 3. F e 2 0 3 1 ( T 3 M K C 1 p H 8-5 1 0 " 4 M long-chain ammonium salt. +

An understanding of the mechanism of adsorption of H + , K + , Cl , R N ( C H 3 ) 3

and SOl' can only be obtained by systematic measurement of the change in zeta potential with each variable.

To simplify the presentation and to maximize the chance of isolating distinct patterns of zeta-potential data that should be valid for any system, the following chapters are divided on the basis of classes of solutes or ad-sorbates. The particular differences between the various kinds of substrates or adsorbents are noted but not stressed, and no attempt is made to survey all examples in the literature of well-documented electrokinetic potentials. The emphasis is almost solely on aqueous systems since our understanding of non-aqueous systems is meagre. The dominant effect of trace water and the need for very high voltage power supplies are just two reasons why non­aqueous systems have received only minimal study (see e.g. Jackson and Parfitt, 1971).

In what follows, two important assumptions wi l l be made:

(i) Where actual zeta-potential data are reported, i t is possible, for that system, to calculate unambiguously the (-potential from the measured electrokinetic quantity. (ii) Unless explicitly stated otherwise, the shear plane is located at the outer edge of the inner part of the double layer, i.e. either at the Outer Helmholtz Plane (OHP) or the Stern layer, depending on the model used to describe the interface. A corollary of this assumption is that the shear plane does not "move" during changes in occupancy of the inner part of the double layer.

The classification of solutes and adsorbates is based on the distinctions introduced in Chapter 2. I t is first necessary to identify the ionic species that exert a fundamental control on the surface charge and the potential at the surface of the dispersed phase (the potential determining ions). We must then

6. APPLICATIONS OF THE ZETA POTENTIAL 223

identify the ionic species that control in general the extension of the double layer out into the solution but which are not involved in any specific inter­action per se with the surface (indifferent electrolyte ions). Finally, we con­sider ionic species which, f rom electrokinetic potential measurements, appear to enter the inner part of the double layer and undergo a specific interaction with the surface.

Having identified the effects of the simple electrolyte species, we can then examine the electrokinetic behaviour of the surfaces in the presence of more complex solutes such as the transition or hydrolysable metal ions, surfactants, simple organic ions and molecules, polymers and polyelectrolytes, etc.

6.2. S i m p l e i n o r g a n i c i o n s a s s o l u t e s

6.2 . 1 . Po ten t i a l - de te rm in ing ions and ind i f fe rent ions

We discussed, in Section 2.2, the behaviour of A g + and I ~ ions in the A g l -water system and showed how they determined the surface potential, i j / 0 , in that system (eq (2.2.4)). One would expect that, by the same token, the potential determining ions for the BaS0 4 surface would be B a 2 + and SO 2.", but only an experimental examination can establish that this is in fact so (Buchanan and Heymann, 1948). In these cases, the potential-determining ions (p.d.i.) are the ions that actually generate the surface charge (see Appendix 6).

G-o = XziFTi (6.2.1) i

where the r ( s are the surface excesses (in moles per unit area) of the potential determining ions. In other cases the p .d i . control the surface charge in a fundamental way. Consider, for example, a colloidal latex particle (consisting of a polymer which is insoluble in water). In many cases (e.g. polystyrene latices) the polymerization process leaves behind a number of — C O O H groups which, because of their polar character, tend to accumulate at the polymer solution interface. The surface charge is then governed by the ioniza­tion of these groups:

— C O O H - * — C O O " + H +

for which

v_H+

K« = ~Z— (6-2.2) vo

where Ka is the dissociation constant for the surface groups, and v_, v 0 are the numbers of negative and neutral groups per unit area, respectively, at a

224 ZETA POTENTIAL IN COLLOID SCIENCE

particular interfacial concentration, H+, of protons. I f we identify the poten­tial I/JQ as the potential at the plane of carboxylate groups, which plane has a surface charge <J0, then from the Boltzmann expression:

where [ H + ] is the bulk concentration of protons. Thus hydrogen ions exert a fundamental control (eq (6.2.3)) on the total double-layer potential and p H is the fundamental p.d.i. concentration scale.

A final example concerns the simple inorganic oxides (e.g. S i 0 2 , A 1 2 0 3 , T i 0 2 ) or biosurfaces which owe their surface charge to zwitterionic centres such as amino acids. The surface hydroxyl groups on oxides can be con­sidered to be amphoteric (able to gain or lose a proton). For example

H + + — A l — O H ^ — A l — O H J (6.2.4)

— A l — O H -> A l — O " + H + (6.2.5)

The surface charge of such an interface is given by:

a0 = e(v+ - v _ ) (6.2.6)

where e is the electronic charge and v + and v_ are the numbers of positive and negative sites per unit area. Alternatively, we may imagine the surface to be made up of neutral sites that adsorb H + and O H - ions. The net surface charge can then be expressed in terms of the adsorption densities, T H + and r 0 H _ in mol per unit area as before:

o-o = F ( r H + - r O H _ ) (6.2.7)

We shall discuss the potential-determining ion concept further in the next chapter, but at this stage it is important to note that for solids such as Agl or A 1 2 0 3 and the other oxides, there must exist a particular bulk concentra­tion of p.di . for which the surface charge is zero (Section 2.2). This condition is termed the point of zero charge (p.z.c.) and is determined by a direct measurement of the surface charge as a function of p.d.i. concentration. There is another point which may or may not be coincident with the p.z.c. and this is the point at which the p.d.i. concentration has been adjusted to make the zeta potential zero; this is called the isoelectric point of i.e.p. (see Section 6.2.3.).

6.2.2. The po in t .o f zero charge

The actual establishment of the point of zero charge is not an easy matter, so it is worth while outlining the usual technique for finding it by titration with

each of the p.d.i.s in turn (Overbeek, 1952, p. 161; Freyberger and de Bruyn, 1957; Parks and de Bruyn, 1962).

The method relies on the assumption that i f the solid (say an oxide) is prepared at its p.z.c. and then dialysed and dried, when it is placed in a solution of the same p H as the p.z.c. i t wil l cause no change in that pH, irrespective of the electrolyte concentration provided that the electrolyte is not specifically adsorbed. Since one does not know the p.z.c. at the beginning of the exercise, the following procedure is adopted:

A known mass m, of the solid oxide is added to a known volume, V, of the electrolyte of known concentration (e.g. 1 0 _ 2 M K N 0 3 ) . The initial p H (called p H 0 ) is noted and the sample is then titrated with, say, 10" 2 M N a O H and the volume required to achieve each solution p H might be as shown in Fig. 6.2. (A suitable time must elapse after each addition to establish equil­ibrium with the surface.) I f on the same diagram we superimpose the titration curve for F ml of 1 0 - 2 M K N 0 3 alone, then the volume v t corresponds to the amount of base taken up by the oxide surface in order to establish equil­ibrium with a solution of p H = p H ^ The net increase in (negative) surface charge, per unit area is therefore

- ( r + - r _ ) = i ^ i (6-2.8) mA

224 ZETA POTENTIAL IN COLLOID SCIENCE

particular interfacial concentration, i / s

+ , of protons. I f we identify the poten­tial r j / 0 as the potential at the plane of carboxylate groups, which plane has a surface charge c r 0 , then from the Boltzmann expression:

K s = ^ r H + ] e x p ( ^ ) (6.2.3)

where [H + ] is the bulk concentration of protons. Thus hydrogen ions exert a fundamental control (eq (6.2.3)) on the total double-layer potential and p H is the fundamental p.d.i. concentration scale.

A final example concerns the simple inorganic oxides (e.g. S i 0 2 , A 1 2 0 3 , T i 0 2 ) or biosurfaces which owe their surface charge to zwitterionic centres such as amino acids. The surface hydroxyl groups on oxides can be con­sidered to be amphoteric (able to gain or lose a proton). For example

H + + — A l — O H -> — A l — O U t (6-2.4)

— A l — O H A l — O ~ + H + (6.2.5)

The surface charge of such an interface is given by:

cr0 = e ( v + - v _ ) (6.2.6)

where e is the electronic charge and v + and v_ are the numbers of positive and negative sites per unit area. Alternatively, we may imagine the surface to be made up of neutral sites that adsorb H + and O H - ions. The net surface charge can then be expressed in terms of the adsorption densities, T H + and r O H _ in mol per unit area as before:

cr0 = F ( T H + - r O H _ ) (6.2.7)

We shall discuss the potential-determining ion concept further in the next chapter, but at this stage it is important to note that for solids such as A g l or A 1 2 0 3 and the other oxides, there must exist a particular bulk concentra­tion of p.d.i. for which the surface charge is zero (Section 2.2). This condition is termed the point of zero charge (p.z.c.) and is determined by a direct measurement of the surface charge as a function of p.d.i. concentration. There is another point which may or may not be coincident with the p.z.c. and this is the point at which the p.d.i. concentration has been adjusted to make the zeta potential zero; this is called the isoelectric point or i.e.p. (see Section 6.2.3.).

6.2.2. The po in t .o f zero charge

The actual establishment of the point of zero charge is not an easy matter, so it is worth while outlining the usual technique for finding it by titration with

each of the p.d.i.s in turn (Overbeek, 1952, p. 161; Freyberger and de Bruyn, 1957; Parks and de Bruyn, 1962).

The method relies on the assumption that i f the solid (say an oxide) is prepared at its p.z.c. and then dialysed and dried, when it is placed in a solution of the same p H as the p.z.c. i t wi l l cause no change in that pH, irrespective of the electrolyte concentration provided that the electrolyte is not specifically adsorbed. Since one does not know the p.z.c. at the beginning of the exercise, the following procedure is adopted:

A known mass m, of the solid oxide is added to a known volume, V, of the electrolyte of known concentration (e.g. 1 0 _ 2 M K N 0 3 ) . The initial p H (called p H 0 ) is noted and the sample is then titrated with, say, 1 0 - 2 M N a O H and the volume required to achieve each solution p H might be as shown in Fig. 6.2. (A suitable time must elapse after each addition to establish equil­ibrium with the surface.) I f on the same diagram we superimpose the titration curve for Kml of 10" 2 M K N 0 3 alone, then the volume v x corresponds to the amount of base taken up by the oxide surface in order to establish equil­ibrium with a solution of p H = p H x . The net increase in (negative) surface charge, per unit area is therefore

- ( r + - r _ ) = ^ — ( 6 . 2 . 8 ) mA

226 ZETA POTENTIAL IN COLLOID SCIENCE

Fig. 6.3. Relative surface charge (a) can be converted into absolute surface charge (b) provided that all curves pass through a common point since this can be identified as the p.z.c.

6. APPLICATIONS OF THE ZETA POTENTIAL 227

for Vj in cm 3 , i f A is the area per unit massf. Very accurate measurements of the volume added are clearly necessary to establish CT0 with the necessary accuracy. The quantity obtained from eq (8) is the relative amount of O H " adsorbed at p H x . I f this calculation is repeated, for all p H > p H 0 and the comparable data for acid titration are also obtained, a plot of relative charge against p H for 1 0 - 2 M K N 0 3 can be constructed. The same procedure can then be repeated in 1 0 " 3 M K N 0 3 and in 1 0 " 4 M K N 0 3 . When all this data is plotted together, one hopes to obtain a result like that shown in Fig. 6.3a. The point where all three isotherms cross one another can be identified as the p.z.c. because only at that point is the surface charge independent of the concentration of the supporting electrolyte (assuming there is no specific adsorption). Figure 6.3a can then be redrawn as 6.3b to give the absolute value of a0. As Lyklema (1977a) points out, this procedure amounts to finding the point at which the Esin-Markov coefficient (/?) is zero (see Section 2.4.3). At the mercury-solution interface, /? is defined as:

where fis is the chemical potential of the supporting electrolyte (s); measures the effect of the supporting electrolyte on the surface charge at constant polarizing potential, E. The analogue of eq (9) in colloidal systems is (Lyklema, 1972):

where a{ is the activity of the potential-determining ion. The only point at which CTQ is independent of p.s at a particular p.d.i. concentration is the p.z.c. and then only if there is no specific adsorption. I f no point of coincidence is observed, it must be assumed that K N 0 3 is not an indifferent electrolyte in this case and the experiment must be repeated with an alternative electrolyte. The experiment, therefore, also serves to define what is meant by the term "indifferent electrolyte".

Accurate assessment of the surface charge density of course also requires an accurate measurement of the surface area of the solid, and this is by no means a trivial problem. As Lyklema (1977a) points out, the particles are usually irregular in shape and may "age" on standing (Ostwald ripening). Direct (electron microscopic) measurements of average particle sizes may underestimate the area because of surface roughness and possible collapse of the surface on a hydrous oxide or a "spongy" organic colloid. Indeed, any

fThis procedure is of course, only valid if there are no other processes competing for the O H " ion (e.g. dissolution of the solid).

228 ZETA POTENTIAL I N COLLOID SCIENCE

procedure which requires a dry sample (e.g. BET adsorption or Knudsen flow of a gas at low pressure) may give an area which for all its accuracy has little relation to the effective area in solution.

Measurement by adsorption (e.g. of a dye) from solution requires a know­ledge of the effective cross-sectional area of the adsorbing molecule, and this may vary from one substrate to another (Adamson, 1976). Van den Hul and Lyklema (1967) argue that negative adsorption is an appropriate procedure for determining specific surface areas. This procedure has been used for some time by soil scientists working with clay minerals where the high surface charge and high areas make the negative adsorption (of negative ions) quite significant (see e.g. Schofield and Samson, 1954). The main feature of this method is that it cannot adequately estimate areas on which double layer development is impaired (e.g. in narrow cracks) but in some cases this may be quite appropriate. A more detailed discussion of surface area determina­tion is given in the recent review by James and Parks (1980).

The point of zero charge is itself an important characteristic of the solid. Typical values for some solids are given in Tables 6.1 and 6.2 (from Fuer­stenau, 1971) from which the activities of other potential determining ions at the p.z.c. can be calculated if the hydrolysis equilibria and solubility are known. The values quoted, especially for the oxides, are meant as guides only and should be used with caution. The actual value for a particular material can be influenced by its source, or preparation method, pretreatment and the presence of trace impurities. For example, Yopps and Fuerstenau (1964) quote values for various forms of alumina ranging f rom 6-6 to 9-4, while Roy and Fuerstenau (1972) showed that the "normal" value of 9T for pure a-alurnina can be shifted down to less than 8 by doping with T i 0 2 and up to 10 by doping with MgO. Materials with p.z.c.s at high p H cannot be in­vestigated in glass or silica apparatus because of the effects of dissolved silica at high p H (see Section 6.2.3).

TABLE 6.1 The point of zero charge of some ionic solids (from Fuerstenau, 1971)

Material p.z.c.

Fluorapatite, Ca 5(P0 4) 3(F, OH) pH 6 Hydroxyapatite, Ca5 (P0 4 ) 3 (OH) pH 7 Calcite, CaC0 3 pH 9-5 Fluorite, CaF2 pCa 3 Barite (synthetic), BaSO 4 pBa 6-7 SiYvei iodide, \ g l P^ê 5 6

6. APPLICATIONS OF THE ZETA POTENTIAL 229

TABLE 6.2

The point of zero charge of some oxides (from Fuerstenau, 1971)

Material p-zx. = pH.

Quartz, Si0 2 2-3-7 Cassiterite, Sn0 2 4-5 Rutile, T i 0 2 6-0 Hematite (natural), Fe 2 0 3 4-8" Hematite (synthetic) 8-6

Corundum, A1 2 0 3 9-0 Magnesia, MgO 12

" Probably contaminated with S i 0 2 (see Section 6.2.3 and Fig. 6.5).

The point of zero charge can also be correlated with surface thermo­dynamic properties such as the heat of neutralization (de Bussetti et al, 1972) and, more particularly, the heat of immersion (Healy and Fuerstenau, 1965) and through this with the field strength generated at the crystal surface (Chessick and Zettlemoyer, 1959). This latter quantity is significantly af­fected by the details of the charge distribution in the solid and so too is the point of zero charge (Levine et al, 1970). A striking correlation is presented by Healy et al (1966) for the series of manganese oxides stretching from the almost non-polar d-Mn02 (pH, = 1-5) to the polar /?-Mn0 2 (pH_. = 7-3). The p.z.c. increases regularly as the atomic packing in the crystal lattice increases and, hence, the electric field strength just outside the lattice also increases.

6 .2.3. The isoelectric point

We may now ask "What sort of electrokinetic behaviour wi l l be expected of say, an oxide surface?" Figure 6.4 shows some typical data for the (-potential of T i 0 2 as a function of p H at various concentrations of K N 0 3 . The point at which ( = 0 is called the isoelectric point (i.e.p.), and the fact that it is inde­pendent of the ionic strength of the K N 0 3 suggests that (i) K N 0 3 is an indifferent electrolyte in this system and (ii) the i.e.p. is in this case the same as the p.z.c. The dramatic effect that p H exerts on the (-potential (and, by inference, the surface charge) also indicates that H+and OH'are potential-determining ions in this system. It should be emphasized, however that a

230 ZETA POTENTIAL IN COLLOID SCIENCE

40

C(mV) 2 0

0

- 2 0

- 4 0

- 6 0

- 8 0

4 5 6 7 8 9 10 pH

Fig. 6.4. The variation of the C-potential of 0-05 g l " 1 colloidal T i 0 2 as a function of pH in aqueous solutions of K N 0 3 at 25°C (after Wiese and Healy, 1975). Note that the most steeply sloping curve is now the one of lowest indifferent electrolyte concentration (cf. Fig. 6.3).

for aged a-Si0 2- An extensive compilation of p.z.c. and i.e.p. data is given by Parks (1965) in his review.

When an i.e.p. and p.z.c. value do not coincide in a given system, it is often not clear that the difference is fundamental. Thus a charge determination (i.e. titration) is conducted on a dispersion at a high volume fraction of solids (i.e. a high surface area) where the effect of a trace impurity, introduced perhaps with the supporting electrolyte material or with the water, is mini­mized simply because the trace impurity is spread over such a large area. However, since a low percentage solids dispersion is required in a micro­electrophoresis experiment (Section 4.3), the same amount of impurity is distributed over a very much lower area. Its effect on the i.e.p. wi l l be much greater than its effect on the p.z.c.

An impurity often ignored is soluble silicate derived by prolonged storage of electrolyte solutions in glassware prior to use, and this problem is par­ticularly severe in alkaline solution. In a recent careful study of the problem Furlong et al. (1980) show that even at p H 6 a shift of 0-2 p H units occurs in the i.e.p. of T i 0 2 (from 5-95 to 5-75) over a period of 2 months. At p H 10, however, a shift of 0-8 p H units occurred in the i.e.p. of A 1 2 0 3 (from 9-6 to 8-8) in a mere 16 hours when stored in a glass vessel. (Fig. 6.5). Because of the low i.e.p. of S i 0 2 the effect is always to shift the i.e.p. of the substrate to lower

T — i i i i i i i i i i i r

J I i il i 1 i 1 i I i I L

6. APPLICATIONS OF THE ZETA POTENTIAL 231

p H values. When a measured i.e.p. is less than the corresponding p.z.c, silica contamination must be a prime suspect.

Contamination problems of this sort can be minimized by taking some elementary precautions in the (-potential measurement such as (i) storing samples either dry or at high volume fraction in aged hard glass (at neutral or slightly acid pH) or in well-washed polythene containers (ii) using the dispersion as a wash liquid for all surfaces with which the sample is to come in contact so that any readily adsorbable impurities wil l be removed and (in) preparing low volume fraction samples for microelectrophoresis not by dilution but by centrifuging down a more concentrated suspension and re­turning some of the solid to the supernatant.

stored a , nH , n o ' t - ° f A!2°VPr4°dUCed b y d e P ° s i t i o n ° f «>luble silica on the surface when AI O a d f d f0 5 £ f T - ? t 1 0 M K N O equilibrated overnight under N 2 at P H 5, then zUntl , , • 5 6 r I d d l s P e r s l o n equilibrated for 2 hrs before pH increased stepwise allowing 15 m.ns. equilibration at each pH. Closed triangles as before but after standing over­k o m ^ H ' S ^ S T ^ " '° ^ " P " '° "** ^ 3 5 A B ° V £ -

232 ZETA POTENTIAL I N COLLOID SCIENCE

Although changes in indifferent electrolyte concentration do not change the i.e.p. they do affect the (-potential at other p.d.i. concentrations, as is obvious from Fig. 6.4. A simple Gouy-Chapman model of the double layer would predict a decrease in ( with increase in concentration of indifferent electrolyte if the shear plane was assumed to be a fixed distance f rom the surface (Fig. 6.6). We have already noted, however (Section 5.4), that this model gives a poor quantitative description of the behaviour. We wil l shortly investigate more realistic models which predict the same general effect but involve physically more reasonable positions for the plane of shear.

I t should be noted that one important effect of the indifferent electrolyte is that it strongly influences the surface charge when \ j / 0 0. Figure 6.6 shows that as K increases the potential falls off more rapidly at the surface. Since, (see 2.3):

(2.3.21)

we see that increasing the indifferent electrolyte concentration forces more potential determining ions to be adsorbed at the surface (eqs 6.2.1 and 7) in order to maintain a constant surface potential.

6. APPLICATIONS OF THE ZETA POTENTIAL 233

6.2.4. Spec i f i ca l l y adsorbed ions

Any ion whose adsorption at the surface is influenced by forces other than simply by the electrical potential there can be regarded as being specifically adsorbed. The additional forces may be "chemical" in nature (i.e. involving some degree of cövalent bonding with surface atoms) or they may be more "physical" (e.g. van der Waals forces between the ion and the surface or between clusters of specifically adsorbed ions, as occurs in the adsorption of surfactant ions). For "chemical" forces to be involved it is clear that ad­sorption must occur into the inner or compact part of the double layer, as i t does at the mercury-solution interface (Grahame, 1947). There are situations, however, where "specific" ion effects could be involved but the ions are adsorbed only at the Outer Helmholtz Plane (see Fig. 2.9) and retain their hydration sheaths (see e.g. Lyklema, 1977b).

Specifically adsorbed ions can be recognized by their ability to reverse the sign of the (-potential, whereas indifferent ions can only reduce ( asymp­totically to zero. Figure 6.7 from Modi and Fuerstenau, 1957 shows clearly the difference in the behaviour of a positively charged alumina surface towards (i) p.d.i.: H + and O H - (ii) the indifferent ions C I - and N O J , N a +

and B a 2 + and (iii) the specifically adsorbed ions SO 2." and S 2 0^" . Note

T

t o " 7 i o " 6 10 5 i o 4 i o ~ 3 i d " 2 i o " '

Concentration of electrolyte (equivalents per litre)

Fig. 6.7. Zeta potential of A 1 2 0 3 (corundum) in solutions of various electrolytes. The concen­tration unit is equiv. I " 1 which corresponds to m o l l - 1 except for the last three salts for which the abscissa would need to be divided by 2 to obtain mol l" 1 .

234 ZETA POTENTIAL I N COLLOID SCIENCE

Fig. 6.8. Zeta potential of an oxide at various pH values in the presence of different types of electrolyte: I : indifferent electrolyte; I I : physically adsorbed cation; I I I : physically adsorbed anion; I V : chemically adsorbed anion. (After Fuerstenau.)

that adding more acid at first causes an increase in (-potential, but at higher acid concentrations the acid itself begins to contribute to the ionic strength; it wi l l therefore affect the extension of the double layer, and ( is diminished in magnitude.

A useful distinction can also be made between specifically adsorbed ions and those which are chemically adsorbed. Figure 6.8 (after Fuerstenau) shows that a physically adsorbed ion does not affect the p.z.c. ( = i.e.p.) but can reverse the sign of the (-potential. Its effects are apparent only when the underlying surface is of opposite sign. A chemisorbed ion, on the other hand,

Fig. 6.9. Reversal of the sign of the (-potential by the specific adsorption of an anion, using a simple Stern model of the interface.

6. APPLICATIONS OF THE ZETA POTENTIAL 235

shifts the p.z.c. and can remain adsorbed even when the underlying surface has the same sign as itself. This situation wil l be examined in more detail in Section 8.2 when the adsorption of surfactants, especially onto oxide sur­faces, is studied.

The mechanism by which reversal of the (-potential occurs is illustrated in Fig. 6.9 for a simple Stern model of the interface. As the concentration of the specifically adsorbed anion, in this example, increases, not only is ( reversed but at the same time the positive surface charge given by eq (2.3.21) increases. An increase in positive charge at fixed concentration of p.d.i. with increase in concentration of specifically adsorbed ions, is equivalent to a shift in the p.z.c. to lower concentrations of positive potential determining ions, i.e. to high p H or pAg for oxides and silver halides respectively.

By the same token, specific adsorption of an anion causes a shift, in the opposite direction, of the point at which ( changes sign. (Following Fuer­stenau we wi l l call this the point of zeta potential reversal (p.z.r.) when it is caused by a specific adsorption process. We shall reserve the term Le.p. for a point where ( = 0 because the p.d.i. concentration has been suitably ad­justed.) The specifically adsorbed anion tends to make ( more negative, and a more positive surface potential is required to offset the effect. The potential profiles at the p.z.r. and at the p.z.c. in the presence of anion specific adsorp­tion are illustrated in Fig. 6.10. Note that in case (b) i j / 0 = \ j / a because, by definition cr0 = 0 and so

by eq (2.3.21).

r r = 0

x = 0

Stern layer (a) Stern layer (b)

Fig. 6.10. Potential profile for the Stern Model of the double layer (a) at the p.z.r. and (b) at the p.z.c. in the presence of a specifically adsorbed anion. The shear plane is here assumed to co­incide with the Stern plane but this is not an essential feature of the model.

236 ZETA POTENTIAL I N COLLOID SCIENCE

6.3 . C h a r g e a n d p o t e n t i a l d i s t r i b u t i o n f o r t h e G o u y - C h a p m a n -S t e r n - G r a h a m e ( G C S G ) m o d e l o f t h e i n t e r f a c e

In what follows we shall adopt, unless otherwise stated, the Gouy-Chapman-Stern-Grahame (GCSG) model of the double layer as illustrated in Fig. 2.9. For the Agl-solution interface we shall also assume that the surface potential is given by the Nernst equation (2.2.4). For other colloidal interfaces, such as the oxides and organic polymer latices, more elaborate models are re­quired to obtain a satisfactory relationship between \j/0 and the concentra­tion of potential determining ions. The assumptions about the compact and diffuse double layers, however, are essentially the same in all models, al­though some authors prefer to reduce the number of arbitrary model parameters by, for example, allowing the I H P and OHP to become coincident and calling this the Stern Plane. For this simpler model fa = t/^ = i / ^ and agreement between theory and experiment for (-potential data can usually only be achieved by placing the shear plane some little distance into the diffuse layer (or by the adoption of rather unlikely values for the ion adsorp­tion potentials).

The basic equations for the GCSG model are outlined by Smith (1973). Electrical neutrality requires that:

ff0 + <T£ + <T<, = 0 (2.4.10)

and the integral capacity of the whole of the inner region per unit area K, is given by:

- = — + — (2.4.14) K K, K2

I f we assume for simplicity that the dielectric permittivity, eh is constant throughout the compact region, we can write (see Appendix 2 and note also the discussion in Section 2.4.2):

K=^ (6.3.1) xz

so that ffnX

t o - ^ - f 1 (6-3.2)

fa - fa = £ — (6.3.3.)

which may be compared with eqs (2.4.5) and (24.6). Combining eqs (2) and (3) we have:

Ob o ^ - x j

6. APPLICATIONS OF THE ZETA POTENTIAL 237

At the p.z.c, we have a0 = 0 and, hence, f rom eq (2)

<Ao = fa as is shown in Fig. 6.1 lb . From eq (4):

ff,(*2-*i) ^o~fa = (6.3.5)

and the potential profile on this model would be as shown in Fig. 6.1 lb . Only in the absence of specific adsorption (at = 0) do we have

ij/0 = ^. = x j j i = 0 at the p.z.c.

At the p.z.r., ( = 0 = fa and hence from eq (2.4.9) ad = 0, so from eq (2.4.10) 0-0 = - and from eq (6.3.3) fa = fa = 0, as shown in Fig. 6.1 la. Then, from (6.3.2):

>Ao = KXn

(6.3.6)

Equations (2.4.10) (6.3.2 and 3) and (2.4.9) are not sufficient to determine the six unknowns: \j/0, \j/b \j/d, a0, a-, and aA, even i f arbitrary assignments are made to xu x2 and K. In addition we need two further equations: one for the surface potential and one linking fa and o> We discussed in Section 2.4.1 the Stern model for developing this last equation:

cr; = z+eNyX8 e x p [ - (z+e^ + 6±)/kT] (2.4.4)

and noted that the chemical adsorption potential 6±, at least for simple

***

(a)

Fig. 6.11. Potential profile for the G C S G model of the double layer (a) at the p.z.r. and (b) at the p.z.c. in the presence of a specifically adsorbed anion.

238 ZETA POTENTIAL I N COLLOID SCIENCE

inorganic ions on mercury, was essentially independent of the charge on the surface if the electrostatic potential fa in eq (2.4.4) was replaced by the micropotential i / ^ = i / ^ + cba where the self-atmosphere potential, <pa, was given by:

for systems with high values of the diffuse layer capacitance. Smith (1973) points out that this equation is strictly valid only when the surface and the diffuse layers may be regarded as conductors. For the case where the permit­tivity of the solid is similar to that in the compact part of the double layer (which is the more usual situation in colloid chemistry) the corresponding equation is:

(6-3-7)

l i K 1 = K 2 = 2K, then cba f rom eq (2.4.16) is - a JAK, while eq (6.3.7) gives - aJ2K so the self-atmosphere potential in colloid systems may be expected to be even more significant than it is on mercury. I f i t is not taken into ac­count we wil l expect the "chemical" adsorption potential 6± in eq (2.4.4) to depend linearly on the surface charge. Equations (2.4.16) and (6.3.7) can be combined to yield

0 . = ^ (6-3-8)

where g has a value between 0-25 and 0-5, and eq (2.4.4) can then be re­arranged to give as an explicit function of CTt:

z . ^ . = M _ 0 ; _ / c T \n(ajsf) (6.3.9) K.

'• where 8'h the "true" adsorption potential, should now be independent of surface charge and si is equal to z&NyX*.

The final equation needed for solution of this sort of model is a relation linking the concentration of p.d.i. with the surface potential. For the .silver halide-solution interface we shall assume the validity of the Nernst equation:

, kT. [ A g + ]

= 2 - 3 - ( p A g p . z . c . - p A g ) (2.2.4) ze

6. APPLICATIONS OF THE ZETA POTENTIAL 239

and a similar equation should hold for any system in which the potential-determining ions are themselves constituents of the crystal lattice (e.g. BaS0 4 ) so that the assumptions involved in deriving eq (2.2.4) can be as­sumed to hold (see Section 2.2). In all other cases it wi l l be necessary to set up a more elaborate expression for i j / 0 , and indeed this wil l prove to be one of the most difficult aspects of the problem.

For polymer latices, the best current models postulate that one or more dissociation reactions occur at the interface (the site-dissociation models reviewed by Healy and White (1978)). For oxide and clay mineral surfaces the most sophisticated models (see e.g. James and Parks, 1980) involve site dissociation and also site-binding (i.e. complex formation or chemical bind­ing of the electrolyte ion, e.g. N a + , C I " , to the surface charge groups). Whilst this latest approach achieves a significant reconciliation of the data on charge and electrokinetic potential i t does so at the expense of blurring the distinc­tion between p.di. , specifically adsorbed and indifferent electrolytes.

In all of the site-dissociation models i t is the surface charge on which attention is focused. The surface potential is then calculated as a dependent variable using rather simple Boltzmann equations to link the surface and bulk concentrations of the ions. As we study the available data we shall find that the discreteness-of-charge effect has hardly ever been introduced into colloid adsorption theories and the general level of sophistication of the theoretical analysis has, perforce, been rather low.

6.4. Z e t a p o t e n t i a l a n d c o l l o i d s t a b i l i t y

The stability of a lyophobic colloidal sol, according to the D L V O theory (Deryaguin and Landau, 1941; Verwey and Overbeek, 1948) is determined by the balance between the repulsive and attractive forces which the particles experience as they approach. I f stability is caused by the particle charge (electrostatic stabilization), the repulsion force depends on the degree of double layer overlap, as described in Appendix 5. The attractive force is provided by the van der Waals interaction which depends on d~" where 2d is the separation between the particles and n ta 3-4. The total potential energy of interaction, VT, can readily be altered by altering the magnitude of the repulsion, either by increasing the ionic strength of the solution (adding indifferent electrolyte) or changing the surface potential on the particles.

I t has long been recognized (see e.g. Tuorilla, 1928) that the (-potential is a very good index of the magnitude of the repulsive interaction between colloidal particles and measurements of (-potential are commonly used to assess the stability of a colloidal sol (see e.g. Riddick, 1968). Figure 6.12 shows some typical plots of the total potential energy of interaction between

240 ZETA POTENTIAL I N COLLOID SCIENCE

x10'

Fig. 6.12. Total potential energy of interaction between flat plates of kaolinite as function of separation at pH 10-4 for various electrolyte concentrations assuming C = i ^ a n d ^ = 2 x 10 J. Curve (1): K = 5 x 1 0 6 c m _ 1 , £ = - 52mV; (2): K = 4 x 10 s c m - 1 , f = - 53 mV; ( 3 ) K = 3-5 X l O ' c m - 1 , £ = -54 -5mV; (4): K = 3 x 1 0 6 c m - 1 , f = -57 -3mV; (5): K = 2 X lO'cm" 1 , f = - 6 5 m V ; (6): JC = 1 0 6 c m _ 1 , J = - 8 0 m V ; (7): K = 5 x 10 s c m - 1 , f = - 96mV. The predicted c.c.c. value between curves 2 and 3 is observed experimentally but the value of A=2x 10" 1 9 J is rather high. (From Hunter and Alexander, 1963).

kaolinite particles at different concentrations of an indifferent electrolyte. Note that as the indifferent electrolyte concentration increases, the (-potential decreases and a quite sharply defined coagulation concentration can be identified. This is the point at which the potential energy barrier opposing coagulation just disappears; i t is called the critical coagulation concentration (c.c.c). An estimate of the c.c.c. can be obtained by using an approximate expression for the potential energy of attraction (VA = — A/A&nd2) and the potential energy of repulsion:

(\AnkT VR = °Z^Lyi e x p ( - 2Kd) (6.4.1)

/c

where y = [exp (i£ 0/2) - l]/[exp(i// 0/2) + 1]. When the potential energy barrier just disappears we have:

d{VA + V R ) = 0 and (VA + V R ) = 0 (6.4.2)

and it is not difficult to show (Overbeek, 1952, p. 306) that at that point Kd= 1. Substituting this value into eqs (1) and (2) and recalling the definition of K we find that:

(47t £ o ) 3 .0-107D 3 (/c7;)V ccc. (mole 1 ' ) = N ^ 2 { z e ) e ^

where NA is the Avogadro number and the quantities on the r.h.s. are in SI units. (For CGS units the constant is 107 and 47t£0 = 1 statfaradcm - 1 ) . At

6. APPLICATIONS OF THE ZETA POTENTIAL 241

25°C in water (D = 80), for particles of high potential {y = 1) we have

c.c.c. (mole 1 _ 1 ) = 874 x 10 ~ *°/z6A 2 (6.4.4)

where the Hamaker constant A, is in joules. This very strong dependence of the critical coagulation concentration on the valency, z, was a well-known empirical rule (known as the Schultze-Hardy rule) and its derivation in this way was an early triumph of the D L V O theory. Values of the c.c.c. for 1:1 electrolytes are normally around 150 m i l l i m o l e r 1 for highly charged Agl sols which requires a value for A of about 24 x 1 0 " 1 9 J (or ~ 58kT), which is rather larger than the theoretical value. Considering the approximations involved in deriving eq (4) the agreement is quite remarkable.

Rather better estimates of the value of A are obtained from coagulation data if, instead of using as the potential characterizing the double layer, we use (.

An empirical relation between (-potential and coagulation behaviour was first provided by Eilers and Kor f f (1940) using published data on a variety of systems. They showed that the onset of instability is associated with a rapid decrease in the value of the function (2//c. This "Eilers and Korf f Rule" does not seem to have attracted much attention in recent years, although a theoretical justification for it was provided by Deryaguin (1940a, b) on the basis of the Debye-Hiickel approximation. The expression for the total energy of interaction between two approaching spheres (corresponding to the force equation (A5.2.3) for flat plates) can be written:

V (H) = 2ne0Da£2 l n [ l + exp( - KH)] - —|- (6.4.5)

where H is the distance of closest approach of the spheres whose radius is a. The first term is the repulsion energy generated by double-layer overlap and the second term is the van der Waals attraction.

Equation (5) can be rewritten :

V (H) = kTb{\n{\ + e x p ( - KH)) - m/tcH} (6.4.6) where

27re0Da(2

AK yn =

(247te0/)(2) Deryaguin (1940b) shows that eq (6) wil l exhibit a minimum value and also a maximum (at smaller H values) provided m is less than about 0-5. I t can be seen from Fig. 6.12 that rapid coagulation occurs near to the point at which

242 ZETA POTENTIAL I N COLLOID SCIENCE

the maximum in V begins to disappear (cf. curves 3 and 2), and Deryaguin shows that a rapid decrease in stability can be expected when m < \ , which occurs when

47re0DC2 , < A.

K

For a given system the quantities A and D are fixed, so that varying the electrolyte (in type and concentration) should produce instability at a particular value of ( 2 / K . Eilers and Korf f (1940) find that instability occurs at values of ( 2 / K of about 1 0 _ 2 - 1 0 - 3 (mV) 2 cm. This corresponds to A values (in water) of about 1 0 _ 1 8 - 1 0 - 1 9 J (i.e. 200-20/cr), which again are rather too large, but the relation remains a useful one for establishing semi-empirical correlations.

To the extent that ( measures the potential characterizing the diffuse part of the double layer it is hardly surprising that i t should provide a good description of the coagulation process, since it is the diffuse layer potentials that are involved in double-layer overlap. Typical data on this approach are provided by Choudhury (1960), Ottewill and Wilkins (1962), Parfitt and Picton (1968) and Dumont et al. (1976) among many others; the utility of the (-potential concept in more complex situations is demonstrated by the work of Brooks and Seaman (1972) and Brooks (1973) on the aggregation of red blood cells in the presence of dextran.

A much more exacting test of the use of the (-potential to characterize coagulation behaviour would require not merely a correlation with the c.c.c where rapid (i.e. particle diffusion rate controlled) coagulation begins, but rather an understanding of the region of slow coagulation. When the (-potential is high enough to produce a significant potential barrier opposing coagulation the rate of coagulation is slowed by a factor, W, called the stability ratio (Overbeek, 1952, p. 285).

_ Total number of particle collisions Number of collisions forming permanent doublets

To generate a very stable sol requires W values f rom 106 to 109 say, for moderately concentrated sols. The behaviour of marginally stable sols {W= 1-20) has been studied by Wiese and Healy (1975) (Fig. 6.13), and the relation between the theoretical values of W (calculated from the (-potential) and the experimental values is shown for T i 0 2 and A 1 2 0 3 sols in Fig. 6.14. The correlation as the p.d.i. concentration is varied is very good indeed and rapid coagulation occurs in all cases at a value of | ( | = 14 + 4mV, corre­sponding to ( 2 / K = 6 x 10" 4 (mV) 2 cm (or A at MKT).

By contrast, the earlier work of Ottewill and Rastogi (1960), in which

6. APPLICATIONS OF THE ZETA POTENTIAL 243

7-5 8-0 8-5 9 0 9-! pH

Fig. 6.13. (a) (-potential as a function of pH for A 1 2 0 3 (0-15gl _ 1 ) in indifferent electrolyte solution. The i.e.p. = p.z.c. at pH 9 is clearly demonstrated. (Wiese and Healy, 1975.) (b) Stability ratio for the same A 1 2 0 3 sol as a function of pH. The minimum value correlates well with the (-potential. (From Weise and Healy, 1975.)

244 ZETA POTENTIAL I N COLLOID SCIENCE

20

10

I I 1 1 1 1

Theoretical

1 1 1

Experimental

—

1 0 ~ 4 M K N 0 3

11 —

1 1 j 1

1 1 1 1

— \\ j •

-\\ l \ \ \

{ 1 J Ti 0 2

| | 1 1 1 1

A l 2 0 ;

1 1 I 5 6 7 8 9

pH

Fig. 6.14. Comparison of theoretical (full curves calculated from measured (-potentials) and experimental (broken curves) stability ratios for T i 0 2 (0-05 g l _ 1 ) and A 1 2 0 3 (<H5gI - 1 ) colloids in 1 0 - 4 M K N 0 3 . (From Wiese and Healy, 1975).

coagulation was induced by varying the double-layer potential using a positive surfactant ion, shows a very different behaviour pattern (Fig: 6.15). No very satisfactory explanation has been provided for the fact that the stability minimum does not correspond in each case with the minimum in (-potential. In these experiments the value of ^ a is being varied whilst i/rQ is constant (since the p.d.i. concentration is fixed) and some change is possibly occurring in the degree of hydrophobicity of the surface (see Section 8.2), but the result is still puzzling; the same Agl sol when positively charged and coagulated with a negative surfactant does not show these anomalous effects (Ottewill and Watanabe, 1960). I t must be noted, however, that the maximum

6. APPLICATIONS OF THE ZETA POTENTIAL 245

level of coagulation does not always occur at the minimum in the (-potential. Where there are polymeric hydroxylated ions (Healy and Jellet, 1967) or organic polymers present (Black et al, 1966; Cooper and Marsden, 1978), other effects such as steric stabilization (Napper and Hunter, 1972) may dominate the situation.

The (-potential has also proved to be a valuable guide to the behaviour of mixed systems undergoing heterocoagulation (Hogg et al, 1966; Usui, 1972). Such systems are often difficult to study because of the dissolution and redeposition of constituent ions. Thus Healy et al (1973) found that in mixed S n 0 2 / A l 2 0 3 and T i 0 2 / A l 2 0 3 systems, the dissolution of A l 3 + and its redeposition as an hydroxylated species eventually converted all surfaces into A 1 2 0 3 though the system initially behaved as a mixed one (see also McAtee and Wells, 1967). Hydrolysable metal ions have a profound effect on colloid stability due to their ready adsorption on oxide and silver halide surfaces (see e.g. Kratohvil and Matijevic, 1967; Matijevic et al, 1968). Recognition of the possibility of hydrolysable metal ion deposition, and its effect on (-potential, is often the key to understanding of these more complex systems (James et al, 1977b). Heterocoagulation is much more easily studied

246 ZETA POTENTIAL I N COLLOID SCIENCE

when there is no such possibility (see e.g. Cooper et al, 1973; Kitahara and Ushiyama, 1973; James et al, 1977a) and in such cases the (-potential, especially for aqueous systems, is a valuable guide to stability behaviour.

6.5. S e d i m e n t a t i o n v o l u m e a n d s e t t l i n g t i m e

I t has long been recognized that there is a close correlation between sedi­mentation volume and colloid stability (Overbeek, 1952, p. 355). When a well-dispersed sol settles, i t does so slowly and tends to form a very dense deposit. A coagulated sol, on the other hand, settles rapidly because of the formation of aggregates of particles, and the final sediment volume is large because the aggregates form loose open structures as they come together and stick at the point of first contact.

Michaels and Bolger (1962) showed that the final sediment height, Z f , was determined by the volume fraction of floes in the suspension {<bF) and the initial sediment height, Z 0 according to:

Z f = ^ + Constant (6.5.1)

using the data of Gaudin and Fuerstenau (1958) on CaO suspensions, and Firth (1976) showed that the same relation held for his T i 0 2 dispersions. The floe volume fraction is related to the particle volume fraction by the floe volume ratio: CFP = cbF/(l>P, and we have already discussed (Section 5.2.4) the dependence of CFP on the (-potential. According to the model of Firth and Hunter (1976) which so successfully describes the rheological behaviour of coagulated sols:

1 C f p — 20rj0ryc \2di

(5.2.15)

For systems which have been strongly sheared after coagulation (at shear rate yCn), these equations suggest that the relative sediment height after a fixed time interval (i.e. Z f / Z 0 ) would decrease as ( 2 . The more usual situa­tion, in which the sol is only gently sheared after addition of the coagulating reagents, has not yet been subjected to detailed analysis. There is no doubt, however, that the (-potential wi l l find a significant role in the understanding of such systems, and in the effective dewatering of slurries (Wright and Kitchener, 1976).

Closely related to this phenomenon is the settling velocity which clearly depends on the degree of aggregation in the system. Fuerstenau and his co­workers use this as a technique for locating the p.z.c. or ie.p. of then systems. The correlation between i.e.p. and settling time for an alumina sol is well AfmnnitmteA in the work of YOÜDS and Fuerstenau (1964) .

6. APPLICATIONS OF THE ZETA POTENTIAL 247

More recently Webb et al (1974) used sedimentation velocities in a test of D L V O theory for anatase ( T i 0 2 ) sols and found that the values of the Hamaker constant, A, calculated from the (-potential were dependent on the electrolyte concentration, indicating a failure of the simple theory. Their mobility measurements were made in a mass-transport type cell which can in some cases yield erroneous data and they worked in the region 4 < Ka < 20 where corrections to the (-potential are at a maximum. Though they used the complete Wiersema et al treatment (see Section 3.7) this is not a solution when the particles are more cylindrical than spherical. Their conclusion should, therefore, be reassessed using the Stigter analysis of the mobility (Section 3.7.3). They also used the Nernst equation to calculate the surface potential and we now recognize that procedure to be inadequate (Section 2.2).

6.6. E l e c t r o p h o r e t i c d e p o s i t i o n

This is a technique which is widely used in industry to produce a thin coherent coating on a conducting base. Sometimes the final product retains both the conductor and the fi lm, as in the production of activated cathodes for cathode ray guns, the deposition of phosphors on T.V. screens, or the de­position of rust prevention coatings on automobile chassis. In other cases, such as the production of some rubber products, a f i lm is deposited from a latex mixture onto a metal former, f rom which it can subsequently be removed.

We have already discussed (Section 4.3.4) the theory of mass transport of materials to an electrode by electrophoretic migration, and the way this depends on the (-potential. The actual deposition process is also governed by the (-potential since the applied electric field in the neighbourhood of the electrode must be sufficiently high to overcome the double layer repulsion. Overbeek (1952, p. 234) gives some references to the early literature, in­cluding the theoretical analysis in Verwey and Overbeek (1948, p. 178) and the work of H i l l et al (1947) on the deposition process. Most of the more recent work is confined to the patents literature but some fundamental work was described by Brown and Salt (1965) and, more recently, Furuno et al (1976).

Since many of the industrial processes are conducted in non-aqueous media, most of the ideas we are developing in this treatment are of only marginal utility. The deposition of polymer coatings from aqueous solution is, how­ever, an exception and there has appeared in recent years a spate of publica­tions in the Soviet colloid literature. Tikhonov et al (1976) gave a brief discussion of conditions favouring effective deposition from aqueous dis-

248 ZETA POTENTIAL I N COLLOID SCIENCE

persions onto dielectrics. Ankundinova-Preikshaite et al. (1974) studied deposition of a polyamide on to metal whilst other latices were studied by Eliseeva etal. (1977). Ul'berg etal. (1977) studied the formation of composites of polymer and electrochemically deposited metals, having previously discussed the theory of deposition from aqueous polymer dispersions (Deinega et al., 1976). Deposition from dipolar solvents was examined by Kolesov et al. (1977) (tungsten on to graphite from butanol) and by Panov and Petrov (1975), who paid particular attention to the influence of the (-potential of their alkaline-earth metal oxides on deposition from amyl alcohol suspensions. Malov et al. (1974) applied a correction to the deposi­tion equations to take account of the non-uniformity of the electric field in the neighbourhood of the deposit. Their equations are, however, couched in terms of the electrokinetic charge rather than the (-potential.

6.7. F l o t a t i o n

Undoubtedly the most widespread and effective method of separating es­sentially pure minerals from their ores is that of flotation. Every year, about a billion tons of ore are treated and the economic value of the product runs into many billions of dollars.

Fuerstenau and his collaborators have made extensive used of the (-potential in developing their understanding of the processes involved in mineral flotation. That work has been reviewed by Apian and Fuerstenau (1962), Fuerstenau and Healy (1972) and Fuerstenau and Raghavan (1976), and the particular significance of surfactant adsorption on surfaces in general is discussed by Fuerstenau (1971). The data wi l l be examined in some detail in Chapter 8; here only the general correlations need to be noted.

6 . 7 . 1 . Col lec tor adsorp t ion

Figure 6.16 shows the close correlation that exists between the measured (-potential of the surface and the ability of an anionic or a cationic surfactant to induce flotation. These collectors are able to adsorb only when the charge on the particle is opposite in sign to that of the collector (Section 8.2). The function of the collector is to convert the hydrophilic oxide surface into a hydrophobic one so that an air bubble can attach itself to the surface. The collector does this by adsorbing with its head group towards the surface and its long hydrocarbon chain then projects out into the solution (Fig. 6.17). The effect on the hydrophobicity is best measured by measuring the contact angle between the surface, the solution and an air bubble pressed against the solid. Figure 6.18 (from Fuerstenau, 1957) shows how close the correlation

6. APPLICATIONS OF THE ZETA POTENTIAL 249

Fig. 6.16. The dependence of the flotation properties of goethite (FeO(OH)) on surface charge Lower curves show the flotation recovery in 10" 3 M solutions of dodecylammonium chloride" sodium dodecyl sulphate and sodium dodecyl sulphonate. (From Iwasaki, Cooke and Colombo' (1960), redrawn from Apian and Fuerstenau (1962).)

la)

ST

2E

(b)

- m

Fig. 6.17. Schematic picture of the adsorption of surfactants on goethite (FeO(OH)) (a) Lone-chain sulphate at pH < 6-7. (b) Long-chain ammonium salt at pH > 6-7.

250 ZETA POTENTIAL I N COLLOID SCIENCE

T — l — i — I — r — T — i — [ — r - f i i l r

0 2 4 6 8 10 12 14

PH

Fig. 6.18. Correlation of adsorption density, contact angles, and zeta potentials with flotation of quartz with 4 x I O " 5 M dodecylammonium acetate additions (From Fuerstenau, 1957).

is between (-potential, contact angle, surfactant adsorption and flotation recovery.

6.7.2. A c t i v a t i o n

Sometimes it is not the surface charge that determines the collector adsorption but rather the charge in the Stern layer, as measured by the (-potential. Positively charged alumina (pH 6) can, for example, be floated with a positive collector (dodecylammonium chloride) provided sufficient sulphate ion is present (Modi and Fuerstenau, 1960). As noted above (Fig. 6.7), the sulphate ion is able to reverse the sign of the (-potential of alumina from positive to negative and is referred to in this context as an activator.

6. APPLICATIONS OF THE ZETA POTENTIAL 251 Metal hydroxide precipitation pH values

F e l l A l PbColl MnE Mg

Fig. 6.19. The effect of cation hydrolysis in the activation of quartz for flotation with a sul-phonate (6 x 10"5 M) as collector. Metal ion used at a concentration of 2 x 10"* M. (From Fuerstenau, et al., 1963.)

Activation can also occur by adsorption of a hydrolysable metal cation (Fig. 6.19) if the p H is suitably adjustedf. In 8.3 we wil l see how studies of the (-potential have aided understanding of this adsorption process.

6.7.3. The s l ime coa t ing p rob lem

When an ore material is crushed to a size that allows the desired mineral to be released there is sometimes produced at the same time a significant quantity of another (gangue) material in highly divided form. I f this impurity should coat the surface of the valuable mineral, i t could significantly impair the recovery—and certainly affect the puri ty—of the product.

The importance of electrical interactions between the slime and the under­lying mineral has long been recognized (Ince, 1930; Sun, 1943). Fuerstenau et al. (1958) correlated the slime coating density with the flotation recovery and the properties of the electrical double layer. More recently, Hunter and Neville (1980) showed that the (-potential can be used as a valuable guide to the interaction energies involved in removing slime coatings. Using the ideas that have evolved from a study of the rheology of colloidal dispersions (Section 5.2.4) they were able to show that the efficiency of slime coating removal is directly related to the interparticle attractive force (as determined by the square of the (-potential) (Fig. 6.20).

f i n Fig. 6.19 the decreasing side of the flotation recovery curve for P b 2 + and A l 3 + have been omitted for clarity. Flotation ceases at pH 12 and 7 respectively (Fuerstenau et al., 1963).

252 ZETA POTENTIAL I N COLLOID SCIENCE

Fig 6 20 Removal of a goethite slime coating from an underlying resin particle (C = — 29 mV). The (-potential of the goethite was varied by changing the solution pH whilst the resin particle remained unaffected. The figure shows the amount of goethite remaining after a fixed time ot shearing in a bottom drive macerator at 275 r.p.m.

6.8. C o r r e l a t i o n of z e t a w i t h o t h e r p r o p e r t i e s

Chibowski and Waksmundski (1978a, b) gave a striking demonstration of the use of the (-potential in investigating wettability and spreading pressure, 7te, of a hydrocarbon f i lm on a hydrophobic surface. They began (1978a) by measuring the streaming potential of a sulphur surface which had been ex­posed to various volumes of a normal paraffin and subsequently washed with water. They observed a strong correlation between the (-potential and the volume of hydrocarbon with which the surface had been in contact and also a strong dependence of ( on the number of carbon atoms in the chain.

The equilibrium spreading pressure of the hydrocarbon film is given by the Bangham-Razouk (1937) equation:

vdlnp (6.8.1) RT

6. APPLICATIONS OF THE ZETA POTENTIAL 253

RT

which is derived from the Gibbs adsorption isotherm. (The application of this equation to the study of wettability and other interfacial phenomena is discussed by Zettlemoyer, by Razouk and by Fowkes in the symposium volume edited by Fowkes (1969)). Chibowski and Waksmundski (1978b) make the rather bold assertion that the pressure,/?, in eq (1) can be replaced by the (-potential to give:

*J vdlnC (6.8.2)

where V is the molar volume of the hydrocarbon, and S is the surface area of the solid. The volume, v, is assumed to be the volume to which the surface was initially exposed. Thus, it is assumed that the subsequent washing does not remove any hydrocarbon. In fact the evidence for this assertion is very weak, especially for volumes above about three monolayers but this has little effect on the main argument. No real justification for the substitution of pressure by the (-potential is given save that a correlation exists and the results appear to be quite reasonable.

I t must be remarked that the reproducibility and quoted accuracy of the data is surprising considering how easy it is to contaminate hydrophobic surfaces (Israelachvili, priv. comm.), but the data do appear to be consistent with the author's hypothesis. The original work on sulphur and Teflon has recently been extended by Chibowski (1979) to the quartz-w-heptane-water system. One surprising feature of the results is that they appear to support a vertical orientation for the adsorbed film. (The (-potential is markedly dependent on the chain length (Fig. 6.21) and is uniformly more negative for odd-numbered chains than for even-numbered chains. Such an effect would hardly be expected i f the chains were horizontal. Also, the cross-sections agree with a vertical rather than a horizontal orientation.) This result is in contrast to the assumed orientation of the hydrocarbon chains of surfactants on a by Jrophobic surface (see Section 8.2), but in that case the adsorption occurs f rom water and the main driving force is the reduction in interfacial energy between hydrocarbon chains and water.

One significant outcome of this work is the calculation of reasonable values for the dispersion contribution to the surface energy of the sulphur, Teflon and quartz surfaces using the theoretical ideas of Fowkes (1968) and Zettle­moyer (1969).

A rather more direct correlation of (-potential with other electrochemical properties is provided by Hoffman and Billings (1977). They show that the reduction overpotential of the silver bromide surface varies with bromide ion concentration through the influence of the latter on the diffuse double-layer potential and this is, of course, reflected in a concomitant change in (-potential. Though obvious to a colloid chemist, the value of the electrokinetic potential

254 ZETA POTENTIAL I N COLLOID SCIENCE

-120

_ -115 > E ~Z -no

£ - 1 0 5 O

n -100

-95

-90 J _ _L 6 7 8 9 10 11 12 13 14 15 16

No. of carbon atoms C„

Fig. 6.21. Relationship between the (-potential values of sulphur in double distilled water and ' the number of C atoms in the M-alkane deposited on the sulphur surface. Note that £ was not measured for C n and C 1 2 . (From Chibowski and Waksmundski, 1978a.)

has not often been recognized by electrochemists in the past. The growing importance of semiconductor devices wil l no doubt foster a greater interest in this area of common ground between colloid and electrochemistry. The relation between electron affinity, flat band potential and (-potential demon­strated by Butler and Ginley (1978) and, more particularly, the work of Schenck (1977) on MOSFET (Metal Oxide Semiconductor Field Effect Transistor) devices with aqueous electrolyte as conductor element show how frui tful such an approach can be.

R e f e r e n c e s

Adamson, A. W. (1976). "Physical Chemistry of Surfaces" 3rd Ed. Wiley Interscience, New York.

Ankundinova-Preikshaite, R. K., Bal'tenas, R. A. and Iozenas, A. L. (1974). Colloid J. (U.S.S.R.) 36, 337.

Apian, F. F. and Fuerstenau, D. W. (1962). Principles of nonmetallic mineral flotation, chapter 7. In "Froth Flotation—50th Anniversary volume." Amer. Inst. Min. Eng., New York.

Bangham, D. H. and Razouk, R. I . (1937). Trans. Faraday Soc. 33, 1459. Black, A. P., Birkner, F. B. and Morgan, J. J. (1966). J. Colloid Interface Sci. 21, 626. Brooks, D. E. (1973). J. Colloid Interface Sci. 43, 714. Brooks, D. E. and Seaman, G. V. F. (1972). Nature New Biology 238, 251. Brown, D. R. and Salt, F. W. (1965). J. Applied Chem. 15,40. Buchanan, A. S. and Heymann, E. (1948). Proc. Roy. Soc. (London) A195, 150. De Bussetti, S. G., Tschapek, M . and Heliny, A. K. (1972). J. Electroanal. Chem. 36,507. Butler, M. A. and Ginley, D. S. (1978). J. Electrochem. Soc. 125, 228. Chessick, J. J. and Zettlemoyer, A. C. (1959). Advan. Catalysis 11, 263. Chibowski, E. (1979). J. Colloid Interface Sci. 69, 326.

6. APPLICATIONS OF THE ZETA POTENTIAL 255

Chibowski, E. and Waksmundski, A. (1978a). J. Colloid Interface Sci. 64, 380. Chibowski, E. and Waksmundski, A. (1978b). / . Colloid Interface Sci. 66, 213 Cserfalvi, T., Meisel, T. and Pungor, E. (1974). J. Electroanal. Chem. 53, 365. Cserfalvi, T., Meisel, T. and Pungor, E. (1976). / . Electroanal. Chem. 74, 377 Choudhury, B. K. (1960). Nature 185, 308. Cooper, W. D., Fearon, P. and Parfitt, G. D. (1973). J. Chem. Soc. Faraday 169, [71

1261. 1 J ' Cooper, W. D. and Marsden, R. S. (1978). J. Chem. Soc. Faraday 174, 2271. Deinega, Yu. F., Ul'berg, Z. R., Estrela-Lopez, V. R. and Nizhnik, Yu. V (1976)

Colloid J. (U.S.S.R.) 38,781. Deryaguin, B. V. (1940a). Trans. Faraday Soc. 36, 730. Deryaguin, B. V. (1940b). Trans. Faraday Soc. 36, 203. Deryaguin, B. V. and Landau, L. D. (1941). Acta Physicochim. SSSR 14, 633; (see Deryaguin, B. V. (1940). Trans. Faraday Soc. 36, 203). Dumont, F., Dang Van Tan and Watillon, A. (1976). J. Colloid Interface Sci. 55, 678. Eilers, H. and Korff, J. (1940). Trans. Faraday Soc. 36, 229. Eliseeva, V. I . , Moore, A. P., Ryabinina, T. I . and Zuikov, A. V. (1977). Colloid J

(U.S.S.R.) 39,479. Firth, B. A. (1976). / . Colloid Interface Sci. 57, 257. Firth, B. A. and Hunter, R. J. (1976). J. Colloid Interface Sci. 57, 266. Fowkes, F. M . (1968). / . Colloid Interface Sci. 28, 493. Fowkes, F. M . (1969). "Hydrophobic Surfaces." The Kendall Award Symposium,

ACS 1968. Academic Press, New York and London. Freyberger, W. L. and de Bruyn, P. L. (1957). / . Phys. Chem. 63, 1475. Fuerstenau, D. W. (1957). Amer. Inst. Min. Eng. Trans. 208, 1365. Fuerstenau, D. W. (1971). In "The Chemistry of Biosurfaces" Vol. 1, p. 143 (M. L.

Hair ed.). Marcel Dekker, New York. Fuerstenau, D. W., Gaudin, A. M . and Miaw, H. L. (1958). Amer. Inst. Min. Eng Trans

211, 792. Fuerstenau, D. W. and Healy, T. W. (1972), Principles of mineral flotation. Chap. 6.

In "Adsorptive Bubble Separation Techniques". Academic Press, New York and London.

Fuerstenau, D. W. and Raghavan, S. (1976). Some aspects of the thermodynamics of flotation. Chap. 3 in "Flotation, A. M. Gaudin Memorial Volume" (M. C. Fuer- • stenav ed.). Amer. Inst. Min. Eng., New York.

Fuerstenau, M . C, Martin, C. C. and' Bhappu, R. B. (1963). Amer. Inst. Min. Eng Trans. 226, 449.

Furlong, D. N., Freeman, P. A. and Lau, A. C. M. (1981). J. Colloid Interface Sci. 80,20. Furono, N. , Kawai, H. and Qyabu, Y. (1976). J. Colloid Interface Sci. 55, 297. Gaudin, A. M . and Fuerstenau, D. W. (1958). Eng. Min. J. 159, 110. Grahame, D. C. (1947). Chem. Revs. 41, 441. Healy, T. W. and Fuerstenau, D. W. (1965). J. Colloid Interface Sci. 20, 376. Healy, T. W., Herring, A. P. and Fuerstenau, D. W. (1966). J. Colloid Interface Sci. 21,

435. Healy, T. W. and Jellett, V. R. (1967). / . Colloid Interface Sci. 24, 41. Healy, T. W. and White, L. R. (1978). Adv. Colloid Interface Sci. 9, 303

H f y ' I : T?e' a R ' Y a t e s ' D - R ^Kavanagh, B. V. (1973). J. Co/bidMer-joce oc/. 42, 647,

256 ZETA POTENTIAL I N COLLOID SCIENCE

Hogg, R., Healy, T. W. and Fuerstenau, D. W. (1966). Trans. Faraday Soc. 62, 1638. Hunter, R. J. and Alexander, A. E. (1963). J. Colloid Sci. 18, 820. Hunter, R. J. and Neville, P. C. (1980). Colloids and Surfaces 1, 257. Ince, C. R. (1930). Amer. Inst. Min. Eng. Trans. 87, 261. Iwasaki, I . , Cooke, S. R. B. and Colombo, A. F. (1960). Rep. Investigation No. 5593,

U.S. Bureau of Mines. Jackson, P. and Parfitt, G. D. (1971). KolloidZ. 244, 240. James, R. O., Homola, A. and Healy, T. W. (1977a). / . Chem. Soc. Faraday I. 73,1437. James, R. O., Wiese, G. R. and Healy, T. W. (1977b). J. Colloid Interface Sci. 59, 381. James, R. O. and Parkes, G. A. (1980). In "Surface and Colloid Science" Vol. 11 (E.

Matijevic ed.) John Wiley, New York. Kitahara, A. and Ushiyama, H. (1973). J. Colloid Interface Sci. 43, 73. Kolesov, I . K , Lunina, M. A. and Khachaturyan, M . A. (1977). Colloid J. (U.S.S.R.)

39, 850. Kratohvil, J. and Matijevic, E. (1967). J. Colloid Interface Sci. 24, 47. Levine, S. and Bell, G. M . (1962). J. Colloid Sci. 17, 838. Levine, P. L., Levine, S. and Smith, A. L. (1970). J. Colloid Interface Sci. 34, 549. Lyklema, J. (1972). J. Electroanal. Chem. 37, 53. Lyklema, J. (1977a). In "Trends in Electrochemistry." (Bockris, J. O'M., Rand, D. A. J.

and Welch, B. J. eds.) p. 159. Plenum Publishing Corp, New York. Lyklema, J. (1977b). / . Colloid Interface Sci. 58, 242. McAtee, J. L., Jr and Wells, L. M. (1967). J. Colloid Interface Sci. 24, 203. Malov, V. A., Kalminskaya, I . A., Bezruk, V. I . , Lazarev, A. N. and Lavrov, I . S. (1974).

Colloid J. (U.S.S.R.) 36, 348. Matijevic, E., Levit, A. B. and Janauer, G. E. (1968). J. Colloid Interface Sci. 28, 10. Michaels, A. S. and Bolger, J. C. (1962). Ind. Eng. Chem. Fund. 1, 24. Modi, H. J. and Fuerstenau, D. W. (1957). J. Phys. Chem. 61, 640. Modi, H. J. and Fuerstenau, D. W. (I960). Amer. Inst. Min. Eng. Trans. 217, 381. Napper, D. H. and Hunter, R. J. (1972). Hydrosols In MTP International Review of

Science. Phys. Chem. Series 1 (M. Kerker ed.). Vol. 7, p. 241. Butterworths, London. Ottewill, R. H. and Rastogi, M . C. (1960). Trans. Faraday Soc. 56, 880. Ottewill, R. H. and Watanabe, A. (1960). KolloidZ. 170, 132; 173, 7. Ottewill, R. H. and Wilkins, D. J. (1962). Trans Faraday Soc. 58, 608. Overbeek, J. Th. G. (1952). In "Colloid Science" (H. R. Kruyt ed.). Vol. 1. Elsevier,

Amsterdam. Panov, V. I . and Petrov, V. A. (1975). Colloid J. (U.S.S.R.) 37, 359. Parfitt, G. D. and Picton, N. H. (1968). Trans. Faraday Soc. 64, 1955. Parks, G. A. (1965). Chem. Rev. 65, 177. Parks, G. A. and de Bruyn, P. L. (1962). J. Phys. Chem. 66, 967. Riddick, T. M. (1968). "Control of Colloid Stability through Zeta Potential." Zeta

Meter Corp, New York. Roy, P. and Fuerstenau, D. W. (1972). Surface Sci. 30, 487. Schenck, J. F. (1977). / . Colloid Interface Sci. 61, 569. Schofield, R. K. and Samson, H. R. (1954). Disc. Faraday Soc. 18, 135. Smith, A. L. (1973). In "Dispersions of Powders in Liquids" (G. D. Parfitt ed.). Chapter

3. Applied Science, London. Sun, S. C. (1943). Amer. Inst. Min. Eng. Trans. 153, 479. Tikhonov, A. P., Senapova, O. V. and Krivoshchepov, A. F. (1976). Colloid J. (U.S.S.R.)

36, 926. Tuorilla, P. (1928). Kolloid-Beih. 27, 44.

6. APPLICATIONS OF THE ZETA POTENTIAL 257

Ul'berg, Z. R., Deinega, Yu.F. and Estrela-Lopez, V. R. (1977). Colloid J. (U.S.S.R.) 39,105.

Usui, S. (1972). In "Progress in Surface and Membrane Science" (Danielli, J. F., Rosen­berg, M . D. and Cadenhead, D. A. eds) Vol. 5, p. 223. Academic Press, New York and London.

Ven den Hul, H. J. and Lyklema, J. (1967). J. Colloid Sci. 23, 500. Verwey, E. J. W. and Overbeek, J. Th. G. (1948). "Theory of Stability of Lyophobic

Colloids." Nth Holland, Amsterdam. Webb, J. T., Bhatnagar, P. D. and Williams, D. G. (1974). J. Colloid Interface Sci. 49,

346. Wiese, G. R. and Healy, T. W. (1975). / . Colloid Interface Sci. 51, 427. Wright, H. J. L. and Kitchener, J. A. (1976). J. Colloid Interface Sci. 56, 57. Yates, D. E. and Healy, T. W. (1975). J. Colloid Interface Sci. 52, 222. Yopps, J. A. and Fuerstenau, D. W. (1964). J. Colloid Sci. 19, 61. Zettlemoyer, A. C. (1969). In "Hydrophobic Surfaces" p. 1. (F. M. Fowkes ed.).

Academic Press, New York and London.

Chapter 7

Influence of Simple Inorganic Ions on Zeta Potential

7.1. Introduction

Electrokinetic studies of aqueous interfaces have most often been undertaken to elucidate the mechanism of adsorption of a particular inorganic or organic solute at the interface. Where such studies have not included a preliminary survey of simple variables, such as potential-determining ions (p.d.i.) and indifferent electrolyte concentrations, the usefulness of the 1 electrokinetic results is limited. In this chapter we wish to consider the effect of the p.d.i. and simple electrolytes (usually 1:1) on the electrokinetic potentials of several classes of substrate and to remove as much as possible the influence of more complex species such as surfactants, polymers, hydrolysable metal ions, etc. I t wi l l not be possible to consider every substrate which has been studied, but rather we shall consider three groups of systems: (i) silver halide-water, (ii) polymer latex-water and (iii) inorganic oxide-water interfaces, all in simple 1:1 electrolyte. For silver halide systems A g + and I ~ ( o r ' C l - or Br~) are p.d.i., whereas for almost all other systems that we shall consider, H + and O H - exert a direct or indirect potential-determining role. The overwhelming importance of the role of p H and the fundamental role played by H + activity in the electrokinetic properties of many interfaces cannot be stressed too often.

In each of the sections that follow, the measured zeta potential behaviour wil l be discussed on the basis of various models. The silver halide-water interface is considered in terms of the classical Gouy-Chapman-Stern-Grahame (GCSG) model of the electrified interface. The polymer latex colloids are examined in terms of a very simple site-dissociation-Gouy-Chapman model, while oxides are considered in terms of a more complex

7. S I M P L E I N O R G A N I C I O N S A N D Z E T A P O T E N T I A L 259

site-dissociation-ion-binding model as well as in terms of the classical GCSG model. In this way we can consider all the available models and stress their usefulness or otherwise in the task of understanding variations in electro­kinetic potentials of simple interfaces in the presence of p .d i . and indifferent simple electrolytes only.

7.2. Surfaces obeying the Nernst equation

7.2.1. The silver halide-water interface

Silver halide sols have long occupied a position of pre-eminence in colloid chemistry. They were, until recent years, practically the only hydrophobic inorganic sols referred to in classical textbooks. There are good reasons for studying silver halide colloids, considering their importance in the photo­graphic process and in cloud-seeding operations. One might also argue that a Ag|Ag halide electrode used in conjunction with a colloidal silver halide sol gives the experimenter a powerful tool for measuring interfacial electro­chemical parameters like surface charge and potential (see Appendix 6). And finally, by beginning with ultra-pure silver nitrate and, say, potassium iodide, one can produce a colloidal sol that is likely to have minimal im­purities.

There are, however, a few disadvantages inherent in the use of silver halide sols in fundamental colloid chemistry. To begin with, these sols can show variations in crystal structure and morphology, and in statu nascendi sols (i.e. sols studied in the process of formation) wi l l exhibit important ageing changes. A more serious, yet subtle problem associated with such an over­emphasis on the chemistry of silver halide sols, in contrast to other colloids, is that the important metal oxide, metal sulphide and alumino-silicate colloids have been largely excluded from investigation or assumed to behave like the silver halide sols. We shall see that this almost universal extrapolation of the results of silver halide research to the properties of other sols has serious limitations.

The classical theory of the electrical double layer was developed and tested against the experimental data derived from the mercury-solution interface (Section 2.4.1). The principal advantages of that system are (i) the ease with which a clean surface can be maintained, (ii) the smoothness of the surface and (iii) the fact that a direct measurement of interfacial tension can be made. Adsorption at such an interface can be studied using the Gibbs adsorption equation which, for a system containing a single salt dissolved in water, may be written (Delahay, 1965) (Appendix 6):

— dy = a0dE± + r _ dus (7.2.1)

260 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

where y is the interfacial tension, E+ is the e.m.f. read on a potentiometer connected between the (dropping) mercury electrode and a half cell which is reversible to the cation of the salt, and T_ is the surface excess of the anion of that salt, whose chemical potential in the solution is LIS. Measurement of the variation of surface tension with solution composition, at constant applied e.m.f., gives a direct measure of adsorption density:

P-) = - r _ ; ( f ) . — r + (7.2.2)

Measuring the variation of surface tension with change in "polarization" (i.e. change in E+ or E_) gives a direct measure of surface charge:

8 7 > =-a-0 (7-2.3)

Both of these measurements are done under conditions of zero current flow, i.e. the only current which flows when the potentiometer reading is changed is that which is required to establish the new double-layer conditions—no current flows through the interface. Such a system is said to be perfectly (or ideally) polarized, and it can be arranged on the mercury-solution interface because of the high overvoltage required before H 2 or 0 2 are evolved from that surface.

When a silver-silver iodide electrode is used in place of the mercury elec­trode the situation is very different. Now the e.m.f. read on the potentiometer is determined by the concentration of A g + and I - ions in the solution, and i f the potentiometer is moved from that e.m.f. reading, a current wi l l flow. Such a system is said to be completely reversible. Fortunately, i t turns out that despite the differences between the two systems there are many similarities. In particular an equation very similar to eq (1) can be written for the Agl-solution interface (see Appendix 6), so many of the techniques and ideas established on the mercury solution interface can be carried across, even though a direct measurement of interfacial tension, y, is not possible.

Although eqs (2) and (3) cannot be used directly we can take advantage of the very large surface area of a silver iodide sol to measure directly the amount of A g + or I ~ ion adsorbed on the particles by the titration technique discussed in Section 6.2 using an A g - A g I electrode to monitor the activity of, say, I - ions in solution. This technique has been ably exploited by the Dutch school of colloid scientists so that our knowledge of the cr0 versus é0

relation on the A g l surface is extensive (see e.g. Overbeek, 1952, p. 162; Lyklema, 1977a).

Unfortunately, there does not appear to be a comparable collection of data published on the C-potential of such systems. The very early work of Lange and Crane (1928) serves to establish the i.e.p. as p i = 10-6 (pAg = 5-4)

7. S I M P L E I N O R G A N I C I O N S A N D Z E T A P O T E N T I A L 261

but otherwise is of little use because they did not control the ionic strength of their solutions. Troelstra and Kruyt (1942) confirmed the i.e.p. by electro­phoresis in 0-012 M K N 0 3 solution, but most of the data presented in their paper is for multivalent cation systems and these are likely to show pro­nounced specific adsorption effects. They are, therefore, of no use for the present purpose. The data in 0-012 M K N 0 3 is presented as electrophoretic mobility against pAg or p i but the conversion to C-potential is complicated somewhat by uncertainties in the value of Ka. Assuming, as Overbeek (1952) does, that Ka is sufficiently large to allow the Smoluchowski equation (3.3.1) to be applied we obtain the results shown in Fig. 7.1 which, according to Lyklema (1977b) have been confirmed by Bijsterbosch (1965). I t must be pointed out, however, that Bijsterbosch was, himself, far from happy with the Agl system as a model for electrokinetic investigations, and the data compiled by Hough and Ottewill (priv. comm.) and displayed in Fig. 7.2 make it clear that for large values of pAg there is considerable uncertainty in the electrophoretic mobility and, therefore, also in the C-potential. Part of the difficulty here may be caused by differences in the effective Ka values of different samples but the data of Ottewill and Woodbridge (1964) clearly indicate that seeking to prepare monodisperse samples wil l not of itself overcome the problem. I t is symptomatic of the difficulties involved that when

PAg

Fig. 7.1. Comparison of experimental data (•) for C-potential of Agl in O 0 1 2 M K N 0 3 (from Troelstra and Kruyt, 1942) with the theoretical predictions from simple G C S G theory with no specific adsorption.

ZETA K y S E K H A L IN COEEOID SCIENCE

14 12 10 8 6 t i l

\ ^ \ ' 6 8 O pAg

Fig 7 2. A compilation of data from various sources on the C-potential of Agl suspensions. A Hough; • Osseo-Asare; O: Bijsterbosch; Rastogi; • Watanabe; o : Mirmk; • . Troelstra (after Hough, priv. comm.).

Lyklema (1977b) wished to make a comparison of éd (from coagulation data) with C-potential he was forced back to the original Troelstra and Kruyt data Before discussing Lyklema's approach we shall examine the very simplest type of model consistent with modern views of the double layer.

In the absence of specific adsorption we have r j ; = 0 and <r0 = - o* trom eq (2.4.10). Then from eq (6.3.4):

( 7 2 - 4 )

and a d = ^lsmh{zeédl2kT) (2.4.9)

K

whilst è0 can be obtained from eq (2.2.4) as a function of pAg. Solving these equations simultaneously allows ^ to be calculated as a function of pAg provided K is known. The value of K, the integral capacity of the inner part

1. SIMPLE INOB.Gf'LNIC I C J N S K K D ZETA POTEKTUvL 261

oï the double layer, can he obtained from the charge-potential data obtained by "Lyklema (,196V). A constant value of about 18 uF c m - 2 for the negatively charged silver halide surface gives a reasonable fit to the data and using this value in eq (4)^gives the ipd values shown in Fig. 7.1. Choosing the value K = l l -9uFcm"- 2 , given by Lyklema (1977) for the differential capacity of the Agl surface in OT M K N 0 3 apparently underestimates the C value implied by the Troelstra and Kruyt data but would be much closer to that suggested by the other data displayed in Fig. 7.2. The significance of the uncertainty in

/ca value is also shown in Fig. 7.1. Ifthe value is assumed to beica= WOrather than the infinite value required for application of the Smoluchowski equation then a significant change occurs in the anticipated behaviour. The broken line shows the expected C-potential, calculated from the Smoluchowski equation, if the true C-potential is equal to \j/d as given by the upper curve in Fig. 7.1.

Lyklema (1977b) has used this same data to test a slightly more elaborate model involving a Stern plane at which the potassium ions are adsorbed with a slight specific adsorption potential (0+ = 2-5 - AkT) which he attri­butes to an entropie effect of the potassium ions on the surface water (Fig. 7.3). His value of 0-54nm for the Stern layer thickness corresponds to K = 1 8 u F c m - 2 if the relative permittivity of the Stern layer water is 11, which is by no means an unreasonable figure.

pAg Fig. 7.3. Comparison of the experimental data (•) for C-potential of Agl in 0012M K N 0 3

(from Troelstra and Kruyt, 1942) with estimates of based on stability behaviour. (After Lyklema, 1977b.)

264 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

Values of the (-potential at lower concentration (0-001M K N 0 3 ) are provided by Osseo-Asare (1972) for both positively and negatively charged A g l at 20°C and these are shown in Fig. 7.4. In this case the KU value is given as unity, and the (-potentials were calculated by Osseo-Asare using the re­lation :

C (mV) = 20-4 x uE (7-2-5)

where uE is the mobility in urn s" 1 per V c m " l . This relation applied only the Henry correction (eq (3.3.5)) to the mobility and though adequate at low potentials somewhat underestimates the (-potential at high potentials. Application of the more exact O'Brien and White (1978) analysis to the

I { i i i ' pAg

ma 7 4 C-potential of Agl in 0-001 M K N 0 3 , m = 1 and T = 293 K . • are calculated using the H e n r v c o S n ( f r o m Osseo-Asare (1972)). O are obtained from the complete relaxation S oft fBrien and White (1978). The full line is the theoretically predated value of in the absence of specific adsorption and assuming K = 18 uF cm .

7. S I M P L E I N O R G A N I C I O N S A N D Z E T A P O T E N T I A L 265

extreme data points raises the absolute value of the (-potential significantly. The curve is drawn using the same model as for Fig. 7.1 with K = 18 uF cm ~ 2

again. The agreement is, to say the least, satisfactory. Note, however, that the value of K is somewhat lower than that found by Smith (see Section 5.4) for the same system.

We may conclude that the very limited amount of data on the (-potential of A g l in indifferent electrolyte solutions can be adequately explained on the basis of the Nernst equation and a simple Gouy-Chapman-Stern model with \jsd = (. This system should, however, be subjected to further electrokinetic study.

7.2.2. The calcium oxalate system

Another surface for which one might anticipate that the Nernst equation (2.2.4) would be applicable is that of calcium oxalate monohydrate, as studied by Curreri et al. (1979). This material is an important constituent of kidney stones, so its colloidal characteristics (especially coagulation behaviour) are of intrinsic medical interest, but, in addition, it has many of the features considered important in a system to be used to investigate the Gouy-Chapman-Stern model of the double layer.

"I t is sparingly soluble, is stable in water, equilibrates rapidly, does not form gels, has no hydroxylated surface groups, forms well-defined particles, and gives no evidence of solid-state conduction processes at or near room temperature. Furthermore, its solution chemistry has been investigated extensively because of the importance of crystallization phenomena in renal lithiasis."

Curreri et al. (1979) were able to match the (-potential behaviour of calcium oxalate in a variety of electrolyte solutions at various p H values, using (i) the Nernst equation for \j/0, (ii) the Stern equation (2.4.2), (iii) an inner layer capacitor equation (2.4.5) and (iv) the equation ( = é6. To do so, however, they had to postulate that the C a 2 + and C2Ol~ ions could function not only as p.d. ions but also as specifically adsorbable ions. They justified this as­sumption on the grounds that hydrated forms of these ions could enjoy a special relationship with the surface whereas only the dehydrated ions could be incorporated into the crystal lattice. The specific adsorption potentials for these ions and also for SO 2 . - were all around 4kT. This wil l be discussed further in Section 8.1.

7.3. The polymer colloid-water interface

In its broadest sense the term "polymer colloids" covers that intermediate zone between what most chemists would regard as "polymers" on the one

266 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

hand and "colloids" on the other. Exploration of this kind of definition would indeed generate new ways of thinking of polymers, colloids and poly­mer colloids. For our present purposes we shall restrict ourselves to a con­sideration of synthetic polymer latex sol particles.

The classification due to Robertson (1975) is of value in illustrating the range of polymer colloid surfaces. Thus he subdivides the various surfaces into:

(a) Non-ionogenic particle surfaces (i) with no stabilizer added

(ii) with an ionogenic stabilizer (iii) with a non-ionic stabilizer

(b) Ionogenic^ particle surfaces (i) with no stabilizer

(ii) with an ionogenic stabilizer (iii) with a non-ionic stabilizer

The non-ionogenic surfaces without obvious stabilizers may be compared to gas bubbles, emulsifier-free oil drops, pure paraffin, etc, all of which exhibit measurable electrokinetic properties. The origin of the surface charge that produces the measured zeta potential of such surfaces is by no means well understood. Similarly the electrokinetic properties of ionogenic and non-ionogenic surfaces with coatings of non-ionic stabilizer are themselves a special and difficult case which wi l l be considered in the next chapter.

Those surfaces whose charge is due to either built-in ionizable surface groups or a layer of adsorbed ionic surfactant are much simpler systems and we shall concentrate on them in this present section.

7.3.1. General electrokinetic properties of polymer colloids The important "polymer colloids" or polymer latex colloids are often stabilized electrostatically by the presence of sulphonate or carboxylate, and sometimes trialkylammonium functional groups on their surfaces. The surface charge and all other double-layer charge and potential parameters are quite clearly controlled by the ionization of such acid groups. Wi th in­creasing p H the fraction of, say, — C O O - groups on the surface increases in the same way as the fraction of OAc~ species in acetic acid solutions. Thus at a p H equal to the surface vKa of the groups, the fraction o f — C O O groups is 0-5.

The other great advantage of the use of latex colloids in electrokinetic work is that they can be prepared as perfect spheres and in an essentially $ Ionogenic: i.e. surfaces that can dissociate to yield an ion (usually a proton) to the solution. This makes the surface either neutral or negatively charged in the usual case.

7. S I M P L E I N O R G A N I C I O N S A N D Z E T A P O T E N T I A L 267

monodisperse form. This allows direct and unequivocal conversion of the electrophoretic mobility to the zeta potential using the methods of Chapter 3. The mean particle radius and the statistical spread of radii about this mean can be determined very accurately. Care has to be taken to correct for shrink­age of the particles in the electron microscope, or for coagulation and co­incidence effects in Coulter Counter determination of mean particle radius. However these corrections or precautions are minor and quite typically a latex particle size can be specified with an accuracy better than 5 % (Ottewill and Shaw, 1966).

Again with attention to minor precautions, i t is found that microscopic, gas adsorption (BET), Coulter Counter, light scattering and other methods yield results that agree closely with one another so that the particles may be regarded as hard, non-porous spheres.

While latex particles are almost ideal from the point of view of the physics of conversion of mobility to zeta potential, there are a number of more chemical aspects of their colloidal behaviour that give rise to some concern. First, i t is usually not possible to obtain a density of ionizable groups greater than about 100 A 2 per group or site. I f i t is recalled that i t is usual to assume a site density of 5 x 10 1 4 c m - 2 (approx. 20 A 2 per site) for silver halide, metal oxide, etc. aqueous interfaces, then for the purpose of comparison of theory with experiment there is a big gap between the maximum charge density of such materials and that of the most highly charged latex particle. Recent advances in polymer chemistry (see e.g. L iu and Krieger, 1978; Homola and James, 1977) may serve to narrow this gap and diminish this particular inconvenience; metal oxide sols can now also be prepared in spherical (or other regular) forms (Matijevic, 1977) although in some cases these are porous.

The other concern about polymer latex particles and the interpretation of their electrokinetic potentials relates to uncertainties about the exact nature of the surface region of such particles. Much evidence suggests that their surface structure consists of flat hydrophobic regions with well-anchored (bound-in) functional groups; the whole surface region is assumed to be impermeable to water and to have little by way of dangling or interwoven polymer chains. I t is also assumed that there is no gel layer, nor is there any entrapment of excess emulsifier or initiator impurities. I t must be admitted, however, that there exists a possibility that some of the surface charge groups reside behind the physical surface, possibly associated with some "buried" water. Such an arrangement would be thermodynamically unstable and would be expected to persist for an appreciable time only i f the overlying (more hydrophobic) polymer layer(s) were coherently cross-linked.

In some cases, particularly those where the surface charge is introduced via adsorbed surfactants, i t is possible that desorption takes place. Other iso-

268 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

lated reports indicate unexplained increases in the negative zeta potential of PTFE latex particles at low pH, hydrolysis reactions wherein surface groups are reacted off the surface: hysteresis in (-pH isotherms between upscale and downscale p H runs, and significant differences in zeta potentials between washed, dialysed, "extensively dialysed", decanted or ultrafiltered prepara­tions of the same particles.

Despite these uncertainties and unknowns, the polymer latex particles with well-anchored stable surface groups still provide the best systems for unambiguous mobility to zeta potential conversion and in most cases are excellent systems for the use of electrokinetic potentials to elucidate adsorp­tion mechanisms.

The influence that surface group ionization has on the electrokinetic potential is shown very clearly by the results of Fig. 7.5 taken from one of several studies by Ottewill and Shaw (1967). With increasing number of surface carboxyl groups per unit area the plateau or maximum value of the electrophoretic mobility increases, but the inflexion region, where the number of ionized groups increases rapidly with increase in pH, is similar for all cases.

With such a monofunctional surface there is no Le.p. and no zeta reversal. The indifferent role of simple 1:1 electrolyte is demonstrated by the results in Fig. 7.5, where it can be seen that ( approaches zero asymptotically as the

U10" 4) - 4 r

Fig. 7.5. Mobility as a function of bulk pH for three different latices: Dotted line: Latex F (low charge)

A: 5 x 10~*MNaCl. Full lines: latex C (intermediate charge).

• : 5 x 1 0 _ 4 M N a C l . O : 1 0 _ 2 M N a C l . • : 5 x 10" 2 MNaCl.

Broken line: latex I (high charge). O: 5 x 10" 2 MNaCl.

(Redrawn from Ottewill and Shaw (1967) with permission.)

7. S I M P L E I N O R G A N I C I O N S A N D Z E T A P O T E N T I A L 269

p H approaches zero and, more importantly, there is no crossover of the isotherms for latex C in the low p H region. In general terms | ( m a x | obtained from the high p H plateau region of each curve is changing in much the way one would expect from changes in 1/K with increasing indifferent electrolyte concentration. The quantitative analysis of such zeta potential data in terms of electrical double-layer theory requires a knowledge of the relationship of the total double-layer potential é0 to the pH. (Note that, as pointed out in Section 6.3, the Nernst equation can no more be applied in this case than i t can for the oxide-water interface.) Some of the solutions to this problem have been reviewed by Healy and White (1978) and we shall draw heavily on their analysis in the next section. They relate tj/0 and p H using the so-called site-dissociation model which was foreshadowed by the earlier work of Hunter and Wright (1971) and Levine and Smith (1971). The model is tested by examining simultaneously the dependence of aQ on p H (from titration measurements) and ( on pH, at different concentrations of indifferent electrolyte. Because of the complexities of the calculation and a desire to keep the model as simple as possible, Healy and White (1978) begin by assuming that there is no Stern layer and that the potential falls off from the surface in accordance with the Poisson-Boltzmann equation (2.3.7). They therefore compare (-potentials with the potential, è, at a distance of 10-20 A from the surface, using eq (2.3.16). In a brief coda to the original paper they have also introduced a "zeroth order Stern Layer correction" in which the Stern layer charge is taken as zero. This correction accounts only for the "distance of closest approach" of the counterions and, as we shall see, the evidence is that in these systems the counterions do not seem to be greatly restricted in their approach to the plane of the surface charge.

Before beginning the theoretical analysis of the latex surface we should consider some recent evidence by Bagchi et al. (1979) which at first sight appears to simplify the problem greatly. Their measurements of surface charge of po lyv iny l toluene) latices by titration (cr0) give values very close to those obtained by microelectrophoresis (<rc <jd) which suggests that in this system there is no Stern layer charge. That is not too surprising, but their further equation of \j/0 with ( is not justified for it involves the assumption that the Gouy-Chapman model applies right up to the particle surface. They have no independent assessment of \j/0 and so have simply calculated it from eq (2.3.23); i t is hardly surprising that they obtain the same value as ( since that follows immediately from the equation tr 0 = ar because ar is related to ( by the same equation. A further difficulty with their work is that the microelectrophoresis results are obtained by first adsorbing the latex particles on to larger AgCl particles. Their assertion that this gives the true mobility of the latex particles is not borne out by a comparison with their own results using the moving boundary technique.

270 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

7.3.2. Site-dissociation models of the polymer colloid-water interface

7.3.2.1. Single (acid) site dissociation Consider a surface containing Ns simple acid groups (—AH) per unit area, dissociating thus:

A H - A - + H + (73.1)

so that: [ A ~ ] . a s ( H + ) ( 7 3 2 )

[AH]

The activity of hydrogen ions at the particle surface, a s (H + ) is given by:

a , ( H + ) = a t ( H + ) e x p ( - i p o ) ( 7 - 3 - 3 )

where ab(U+) is the activity in the bulk. The surface charge (<r0) is given simply by

- c r 0 = e [ A - ] (7-3.4)

whilst the total number of surface groups {Ns) per cm 2 is:

N , = [ A - ] + [AH] (7-3.5)

or, in terms of the fractions a_ and ce0 of A - and A H groups a_ + a 0 = 1 (7-3.6)

and [A"] - o

a_ = (7.3.7)

From eqs (2), (3), (5) and (7) the fractional surface charge a_ is given by:

(7.3.8) a_ = 1 + - V - - e x p ( - ^ o )

I t is perhaps now instructive to express eq (8) in terms of <T0 rather than the fractional charge a_. Thus,

_ n m _ (7.3.9) ff° l + ( f l 6 / X J e x p ( - ^ 0 )

and from simple Gouy-Chapman theory (with at = 0 and z = 1):

c T d = - f J o = ^ s i n h l A 0 / 2 . (7.3.10)

where the important term £, is (cf. eq (2.3.23)):

*J&N* (7.3.11) 4NAc

7. S I M P L E I N O R G A N I C I O N S A N D Z E T A P O T E N T I A L 271

(Here K is the Debye-Hückel reciprocal length ( c m - 1 ) , NA is Avogadro's number; c is the bulk electrolyte concentration in mol d m - 3 ) ; t\ is a dimen-sionless quantity which combines site density and electrolyte concentration effects (see Table 7.1).

T A B L E 7 .1

Range of £ values for most polymer colloid systems

Ns (sites cm 2 )

c 5 x 10 1 4 5 x 101 2

(mol d m - 3 ) (20A2 per site) (2000A2 per site)

10"5 2000 20 10"3 200 2-0 K T 1 20 0-2

Solving eqs (9) and (10) provides an explicit relationship between the surface charge and surface potential in terms of pH, K and Ns. The easiest form of the solution for computational purposes is

^-ïè^-^'i^h-1) <7'312)

from which a direct calculation of the p H required to establish any arbitrary $ 0 may readily be calculated, given Ns, Ka and the bulk electrolyte concen­tration (or K ) .

The principal features of this simple single acid site-dissociation model are shown in Fig. 7.6 where a_, i f f Q and ij/ (x= lOA), respectively are plotted as functions of bulk p H for various values of ionic strength and Ka and for Ns = 5-08 x 10 1 3 sites c m - 2 . The values of the parameters selected follow those appropriate to the detailed study by Ottewill and Shaw (1966) on carboxylate latex colloids with maximum surface charges varying from 0-81 x 10 1 3 to 7-0 x 1 0 1 3 c m " 2 and ionic strengths of 5 x 10~ 4, 10" 2 and 5 x 1 0 - 2 M for each of the five latex preparations. The value of ip at 10 A out from the surface was obtained from eq (2.3.16).

The expected relations between charge and potential on the one hand and p H at various values of ionic strength are shown clearly; at high ionic strength the high charge and rapid increase in charge with p H is matched by a low tj/0

and low rate of increase in i//0 or i^(10A). Notice also that both \j/0 and è (10 A) approach a maximum value at high p H and that the magnitude of this maximum is in each case independent of the pKa selected.

Healy and White (1978) compared the predictions of this model with the experimental (-potential data of Ottewill and Shaw (1966). Though the gen-

272 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

Fig. 7.6. Variation with pH of the surface charge (<J0), the surface potential (i^0) and the potential 10 A out from the surface as predicted by the simple site dissociation model. (Ns is taken as 5-08 x 1 0 u c m _ 2 . ) (After Healy and White, 1978.)

eral features of the behaviour are satisfactorily explained, there remain some quantitative discrepancies, especially for the lower electrolyte concentrations (Fig. 7.7). I t should be noted, however, that this is a rather stringent test, since the value of Ns is assumed to be given and the model is used to calculate the maximum value expected for the (-potential at pHs far above the pKa

value. A more significant failure of this model is its prediction of the influence of indifferent electrolyte concentration on the maximum (-potential. I t is clear from Fig. 7.7 that this model predicts that dQd log c is of the order of - 60 mV. This is the slope that is predicted by the simple Gouy-Chapman Theory with constant diffuse layer charge (see Section 7.6). The usual value _ i _ i i „ /^r\ or\\ . . . - \ r

7. S I M P L E I N O R G A N I C I O N S A N D Z E T A P O T E N T I A L 273

140

I M

5 x 10~ 4

Latex

A

I

Silica

• '

I 1

10" 2

O O 5 x 10" 4 M

120 - 5 x 1 0 - 2 V V •

~ 100 > E

- -

« 8 0 O CNJ

- A

A • ^ 60 A

A 10 _ 2 M

4 0 -0 /

O V v

V

5 x 10~2 M

20

0 1

V

" I i 1

20

0 1 0 1 2 1 0 1 3 1 0 1 4

( sites cm" 2)

1 0 1 5

Fig. 7.7. Comparison of experimental ^-potentials for latex colloids (Ottewill and Shaw, 1966) and silica (Wiese et al., 1971) with the predictions of the simple dissociation model. (After Healy and White, 1978.)

Healy and White (1978) also tested the predictions of the model against experimental data for charge versus p H using the data of Ottewill and Yates (1975) and again found good agreement at high ionic strength ( C = 0 T M ) but less satisfactory agreement at c = 1 0 - 3 M .

Before leaving this model we must examine what it predicts about devia­tions from Nernst behaviour. Although the surface does not have a p.z.c. we can rearrange eq (12) to read (for \j/0 in mV):

* 0 = 59-8 { P K a - pH) - 59-8 l o g 1 0 ( * — - l ) (7.3.13)

Since the second term depends on \j/0, no simple explicit expression is possible but the form of the ij/0-pll relation, as shown in Fig. 7.6, clearly departs sig­nificantly from the Nernst equation (2.2.6). In particular, the value of di^ 0/<ipH depends strongly on the indifferent electrolyte concentration. Only at infinite dilution does i t appear to approach the classical value of 2-303.R77F ( = 59-8 mV per p H unit). A general analysis of the Nernst equation in this and other site-dissociation systems has been given by Healy et al. (1977) and we shall

274 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

7.3.2.2. Two-site dissociation models There are three common situations involving two-site models:

(i) Surfaces that possess two distinct (acid) dissociation sites. This is the sort of surface commonly found in latex colloids when both sul-phonic or sulphate (i.e. strong acid) groups and carboxylic (weak) acid groups may arise from the stabilizer, or by oxidation reactions induced by the initiator. Such systems cannot exhibit a p.z.c.

(ii) Amphoteric surfaces in which the same surface group may dissociate to yield a proton (eq (7.3.1)) or take up a proton:

A H + H + ^ A H J (7.3.14)

K+ = ' (7-3-15)

The species A H 2 may be regarded as a diprotic acid with dissociation con­stant K + given by

[ A H K ( H + ) [ A H 2

+ ]

and X _ given by eq (7.3.2). This model is also examined in detail by Healy and White (1978).

(iii) Zwitterionic surfaces. In this case two distinct groups are present: one capable of accepting a proton and one capable of dissociating to leave a negative surface site.

Situations (ii) and (iii) both lead to the possibility of a point of zero charge but are quite distinct, as is clear from the treatment of the i / ^ - p H relationship in the two cases (Healy et al, 1977). The type (ii) situation is more commonly encountered in oxide systems and, since the experimental data is confined to those systems, we shall postpone further discussion of this system to Section 7.4. I t might be noted at this point that the so-called "amphoteric" latices of Homola and James (1977) are actually of type (iii).

A brief analysis of the type (i) situation has been given by Healy and White (1978). They define the surface p H (pH s ) as (cf. Hartley and Roe, 1940):

p H s = - l o g i r A ( H + ) = p H + Jj!L (7.3.16)

and show that the fractional surface charge, a_, exhibits a two-step nature (as would be expected) (Fig. 7.8). The exact shape of the curve depends on the relative number of weak acid (NA) and strong acid (NB) sites. Healy and White (1978) did not calculate the expected (-potential for this model, but a similar model (with site-binding also included) is discussed by James and Parks (1980) as a basis for describing the behaviour of clay minerals. We shall therefore, postpone discussion of this point until Section 7.4.

The Zwitterionic surface has also been discussed in general terms by Healy

7. S I M P L E I N O R G A N I C I O N S A N D Z E T A P O T E N T I A L 275

I 1 i^l I I I I I 0 1 2 3 4 5 6

p H s

Fig. 7.8. Variation with surface pH (pH s) of the fractional surface charge oc_, for a two-site surface with various ratios of acid and basic groups. (After Healy and White, 1978.)

et al. (1977) and by Healy and White (1978), but we shall follow the alterna­tive treatment by Rendall and Smith (1978) since they have made a direct comparison of their model predictions with (-potential data. The two ap­proaches are very similar, and a comparison of the corresponding values of the main parameters in the two different formulations is given by Healy and White (1978) (their table 3.2). The Rendall and Smith (1978) model is a development of an earlier analysis of the oxide-solution interface by Levine and Smith (1971), a simplified version of which is described by Smith (1976). These models wil l be discussed in Section 7.4.

The Zwitterionic surface occurs commonly in biological systems where the two different surface groups are the carboxylic acid and the amino group. The same two groups occur in synthetic polyamides, and it is the nylon surface to which Rendall and Smith have applied their model.

Consider an interface which can develop positive and negative surface charges by the ionization of basic (e.g. amino) and acidic (e.g. carboxyl) groups. The density of surface basic and acidic sites is represented by Ns+

and JVS_, respectively, of which n+ and «_ are ionised at any given solution pH. Writ ing n + /Ns+ = cc+ and «_/JV s_ = cc_ then a_ is again given by eq (7.3.8) with Ka = K_. This can be rearranged to read:

p K _ = p H + 0-4343 l n { ( l - «_)/«_} + 0-4343i//0 (7.3.17)

The corresponding expression for pK+ is:

pK+ = p H + 0-4343 l n . { a + / ( l - <x+)} + 0-4343i>0 (7.3.18)

276 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

where K+ is also defined as an acid dissociation constant. Adding and sub­tracting these equations leads to two important expressions:

pK+ + pJSL = 2pH + 0-4343 In ^ — + 0-8686</>0

and

p K _ - p K + = 0-4343 In

(1 - « + ) « .

( l - « _ ) ( l - a + )

a_ . a 4

(7.3.19)

(7.3.20)

Differentiation of eq (19) with respect to p H gives:

<#0 kT kT = -2-303— + _

tipH e 2e da.

rf(pH)V«-+

1 da.+ ( 1 r — +

1

Éf(pH)\a+ 1—a+ /

(7.3.21)

The first term is the Nernst slope (N) (see eq (5.4.7)), and eq (21) shows that these systems can be expected to depart markedly from Nernstian behaviour, as Healy et al. (1977) pointed out. Rendall and Smith (1978) give a number of useful simplifications near to the p.z.c. The one of interest for their experi­mental test applies when Ns+ < N s _ and p K _ < pK + . Then:

do~r

where

and K 5 = i V s _ e 2 a _ ( l - a _ ) / * T

(7.3.22)

(7.3.23)

<70 = e(2V s + a + - / Y s _ a _ )

= e(/V s + - N s _ a _ ) (7.3.24)

At this point the function S is again introduced (as in eq (5.4.3)):

(7.3.25)

(7.3.26)

\d(s>H)Jr^ and a new function (Smith, 1976), S u is also defined:

S 1 = 4pH)

I t can then be shown (Smith, 1976) that if specific adsorption is negligible,

N S r ^ l + c Z - L + i - ) (7.3.27)

where K is the integral capacitance of the entire compact part of the double-

7. S I M P L E I N O R G A N I C I O N S A N D Z E T A P O T E N T I A L 277

layer (eq (2.4.14)). Using an argument similar to that shown in Fig. 5.15, Rendall and Smith again show that the distance, A, between the OHP and the shear plane is very small (certainly < 5 A) . Their plot of NS~1 against Cd

(the limiting value of the diffuse layer capacitance as ( approaches zero) allows the apparent integral capacitance KA to be estimated, where

K ^ ^ K - ' + Kr1 (7.3.28)

In order to interpret their electrokinetic data in detail, Rendall and Smith first assume that £ = \j/d. Then, in the absence of specific adsorption, eq (7.3.10) holds (with {j/0 replaced by {j/d). This enables <r0 to be calculated directly from the C-potential. The data at low p H then allow an estimate to be made of Ns+ ( a a0/e) and then eq (24) applied at the p.z.c. ( = i.e.p.) gives

pH

Fig. 7.9. Comparison of experimental C-potential data with theoretical curves calculated from a two-site model for nylon at 25°C. Electrolyte concentration: • : 1 0 " 4 M ; D : 1 0 _ 3 M ; A : 5 X 1 ( T 3 M ; O : 1 0 _ 2 M .

Model Parameters: Ns+e = 0 - 6 2 u C c m - 2

J V s _ e = 1-42 uC cm" 2

i.e.p. = 5-4; pK_ =5-5 KA = 8-6; Ks = 13-2; K = 25 u F o n " 2

(After Rendall and Smith, 1977.)

278 ZETA POTENTIAL IN COLLOID SCIENCE

a value for ATs_a_e\ This can be used in eq (17) to estimate p K _ which for both of the nylons studied turns out to be 5-5. Given the value of cc_ one can calculate Ks from eq (23) and hence K from eq (28). For both nylons K turns out to be about 25-26 uF c m - 2 , very similar to the values obtained by this method for the A g l surface (Fig. 5.15).

Once K is known, it is possible to calculate the entire ( - p H curve, using an iterative procedure. A trial ( value is used to calculate ad ( = <T0) and hence ij/0 using the expression (from eq (6.3.4) with er; = 0):

^o = ^ + f = C - f (73.29)

The extent of ionization of the surface groups is then calculated from eqs (17) and (18) and this gives a different estimate of a0. The process is repeated until the values from the electrokinetic data and the dissociation equilibria are in agreement. I t is evidently possible to obtain such agreement with a value of p K + = 10-0 for both nylons. The theoretical curves for the system are shown in Fig. 7.9, and it is clear that this model is able to describe, in con­siderable detail, the behaviour of this sample of nylon under a variety of conditions and with very reasonable values for the parameters. The proof would be even more compelling if there were independent direct measure­ments of the surface charge (from titration measurements).

I t would be a very useful exercise to apply this approach to the data of Homola and James (1977) on "amphoteric" (actually Zwitterionic) latices. In their systems the ratio of acidic to basic groups is controlled (so the i.e.p. varies) and at least for one system the charge and (-potential data are both available.

7.4. The oxide-water interface

Inorganic oxides such as silica (S i0 2 ) , titania ( T i 0 2 ) , zirconia ( Z r 0 2 ) and alumina (A1 2 0 3 ) , have been studied by electrokinetic techniques for many years because of their technological importance. I t has also been suggested— e.g. by Chakravarti and Talibudeen, 1961) and more recently Hall (1966)— that natural clay minerals are coated with alumina-silica colloidal material and the surface properties of clays can reflect oxide behaviour as much as that of the clay lattices themselves. As with the latex colloids, experimental surface charge and zeta potential data on oxides confirm the potential-deter­mining role of protons (Parks, 1965).

The establishment of the surface charge may occur by two distinct but essentially equivalent mechanisms :

7. SIMPLE INORGANIC IONS AND ZETA POTENTIAL 279

(1) the adsorption of protons or hydroxyls on to an amphoteric site: H+ O H -

M O H 2

+ , M O H >MO_ + H 2 0 (7.4.1)

or (2) the formation in solution of hydroxylated metal species which deposit on the surfaces:

M " + + x O H ^ M ( O H ) ^ l ) +

M ( O H ) ? _ J C ) + + O H " -> M ( O H ) ? ; r 1 ) +

M(OH)<T *>+ + H + -> M ( O H ) ? r f + » + H 2 0

Alternatively the species M(OH)„ may be considered to form directly on the solid surface. No distinction need be made between these possible mechanisms if one is concerned only with the equilibrium charge-potential relationships and the representation of the behaviour by the sorts of simple model with which we have been concerned to date. For a more detailed discussion of the relationship between the two processes, see Parks (1965). The site-dissociation models, with which we wil l deal below, concentrate attention on mechanism (i).

7.4.1. Genera! electrokinetic properties of oxide-water interfaces

A typical set of electrokinetic data for an oxide-aqueous solution system is shown in Fig. 7.10. The symmetric shape of the ( - p H plot near the i.e.p. is a general characteristic of these systems; it indicates clearly the significance of the H + and O H " ions in determining the state of surface charge. The fact that the i.e.p. at p H 6-7 is independent of electrolyte concentration suggests that it is also a p.z.c. (see Section 6.2.1), and indeed this has been established by titration studies. The titration data of Yates and Healy (1976) give a value for the p.z.c. of high purity T i 0 2 (rutile) of p H 5-8 + 0-1 in excellent agreement with the i.e.p. value found for the same system by electrokinetic measurements (Wiese and Healy, 1975). The same agreement is recorded for a -Fe 2 0 3

(haematite) by Breeuwsma and Lyklema (1971) and Parks and de Bruyn (1962) at p H = 8-5 ± 0-2. More recently Ray and Khan (1974) have obtained a p.z.c. and an i.e.p. for chromic oxide at p H 8-4 in the presence of sodium chloride and nitrate.

For the Si0 2 -water interface, an i.e.p. can be established only by extra­polation since it occurs at about p H 2-5 (Wiese et al., 1971) and the acid side of the curve is not accessible to electrokinetic measurement. This system could therefore be treated by the single acid site-dissociation model described in Section 7.3 (Healy and White, 1978). I t may be noted here that consistent and reproducible data can be obtained on vitreous and a-quartz (SiO-,1

280 ZETA POTENTIAL IN COLLOID SCIENCE

l r

Fig. 7.10. C-potential as a function of pH for goethite (FeO(OH)) from the work of Iwasaki, Cooke and Colombo (1960) Rep. Investigation No. 5593, US Bureau of Mines. (Redrawn from Fuerstenau and Healy (1972)).

provided the surfaces are well aged prior to measurement (see Section 7.4.2). One of the most significant general features of the behaviour of oxide

systems is shown in Fig. 7.11. Compared to the mercury or A g l surface, the surface densities of charge are very much higher at comparable values of the surface potential. Indeed, since the Nernst equation no longer holds in these systems (see Section 2.2) the discrepancy is even larger than is indicated in Fig. 7.11, because the |i> 0 | values for the oxides are much lower than the corresponding values for A g l (i.e. when A p H = ApAg). The capacitance of the oxides is, therefore, even higher than Fig. 7.11 indicates. The other im­portant features are (i) the r j 0 - p H curve is convex to the p H axis and (ii) the very large values of <r0 (obtained from titration data) are accompanied by quite modest values of the electrokinetic potential. These unusual features are even more pronounced in the clay mineral systems.

The first detailed experimental evidence that the oxide-solution interface

7. SIMPLE INORGANIC IONS AND ZETA POTENTIAL

- 2 5 , -

281

Glass

I ApAg (for Agl)

Fig. 7.11. Comparison of surface charge on various surfaces at comparable values of the surface potential. Values for mercury are obtained by equating the applied polarising potential E to 2-3Q3RT(ApAg)/F. (After Lyklema, 1971.)

was somewhat unusual was compiled by Hunter and Wright (1971) and, at much the same time, by Lyklema (1971). Prior to that time it had been as­sumed that the Gouy-Chapman-Stern-Grahame (GCSG) model of the double layer contained sufficient flexibility to describe the charge-potential relationships on any surface, with reasonable values of the adjustable par­ameters. Hunter and Wright showed that no set of parameters could ade­quately describe the (-potential data on silica (Gaudin and Fuerstenau, 1955) and alumina (Modi and Fuerstenau, 1957) in the presence of simple inorganic ions unless one allowed the value of \j/0 to be arbitrarily adjusted, i.e. not in accordance with the simple Nernst equation (2.2.6). The possible failure of the Nernst equation for oxides was discussed in Section 2.2. More serious was the fact that, even with an adjustable value of \j/0, i t was possible to get very good agreement with the electrokinetic data whilst the measured surface charge was grossly underestimated (Wright and Hunter, 1973).

The very high values of (titratable) surface charge on oxides were also being cited by Lyklema (1971) as possible evidence for a porous layer, at least on some oxides. Values as high as 70uCcm~ 2 were found on a -Fe 2 0 3

and even higher values occurred on some silica preparations. These values are close to or may even exceed the usually accepted maximum site density for — O H groups on an oxide surface (20 A 2 per surface group corresponds

282 ZETA POTENTIAL IN COLLOID SCIENCE

to m 8 0 u C c m - 2 ) . Considering that the oxides are supposed to be rather weak acids these results were, to say the least, unusual.

Two theoretical approaches to the problem were pursued in parallel (i) the porous interface model and (ii) the site-dissociation model and we shall now examine them in that order. Before doing so, however, we digress briefly to discuss the preparation of oxide surfaces.

7.4.2. Ageing, solubility and leaching reactions

The inorganic oxides and related materials are often slow to come to equi­librium with a particular aqueous electrolyte system. This is not unexpected, since the process of preparation of the equilibrium solid-solution interface often involves immersion of a partially or totally dehydrated or even de-hydroxylated dry solid into the electrolyte solution. Even where the interface is prepared in situ by, say, homogeneous precipitation followed by some form of washing or dialysis, i t is expected that the establishment of the final equi­librium structure of the solid and solution sides of the interface wil l take a finite time. The solid side reaction may involve rearrangement and con­densation of the nuclei or crystallites into a final crystal form or involve some penetration of water and ions into the surface structure as, for example, in the slow rehydration of the T i 0 2 surface after calcining.

A more general view is that the oxide must dissolve until the solution com­position corresponds to the solubility ( K 5 0 ) condition. Some oxides, notably preleached T i 0 2 (rutile), a - A l 2 0 3 (corundum) and a -Fe 2 0 3 (haematite) appear to dissolve so slowly that, for the purpose of interpretation of electro­kinetic potentials, they may be regarded as essentially "insoluble".

The ageing or solubility reactions of silicas, including quartz, amorphous silica, vitreous silica, Pyrex and other glasses represents a special case (see e.g. van Lier et al., 1960) and one in which electrokinetic studies have proved extremely useful. The data shown in Fig. 7.12 (from Wiese et al., 1971) represent well cleaned, aged silica whether it be vitreous or crystalline. General agreement has now been established between various laboratories in that reproducible zeta potential data on S i 0 2 can be obtained with nitric acid washed surfaces that have been subsequently aged at 25-3 5°C for long periods of time in distilled water or that have been washed and then steamed. The long-term stability of the zeta-pH isotherms for H N 0 3 washed-steamed silicas and the agreement obtained between various laboratories when this treatment is used suggest that silica so treated represents a base response. The effect of pre treatment of quartz with H F and N a O H wash cycles is to alter drastically the electrokinetic properties away from the base curve, and very long ageing times are required to regenerate the standard or base response (Fig. 7.13). (See also Kulkarni and Somasundaran (1973) on this point.)

7. SIMPLE INORGANIC IONS AND ZETA POTENTIAL 283

284 ZETA POTENTIAL IN COLLOID SCIENCE

There is room for some argument on the question of whether this "stan­dard" response surface is or is not porous. From its mode of preparation it could well be quite openly porous and able to rapidly establish equilibrium with a solution. The slower response of the H F treated surface would then be ascribed to the need to first establish an open porous ("gel") structure on the surface rather than to the slow hydrolysis reactions proposed by Kulkarni and Somasundaran (1973). Arguing against this interpretation is the data of Smit et al. (1978a), who find no evidence of gel layers thicker than ~ 3 A on H F treated vitreous silica after soaking in NaCl solution at pH 10 for periods up to 28 hours. Unfortunately, their (-potential measurements were made after soaking in the measuring solution for 20 hours so, not surprisingly, they make no mention of time effects. This silica is, of course, very different from the silica whose charge characteristics are shown in Fig. 7.11; Smit et al. quote its surface charge as only about — 10 uC c m - 2 at p H 10, which clearly does not require a gel model interpretation. In a subsequent paper, Smit et al. (1978b) showed that thin films of microporous silica on the surface of silica rods show adsorption and (-potential behaviour consistent with the gel model of Perram et al. (1973, 1974) to be described in the next section.

7.4.3. The porous gel model of the oxide-solution interface

This was suggested by Lyklema (1971) as a possible explanation for the very high values of titratable charge on some oxide surfaces. The basic idea was that H + and O H " ions could penetrate the surface layers of the oxide and react there with amphoteric—OH groups. In this way quite large amounts of "surface" charge could develop whilst retaining a reasonable separation between the charged groups. At the same time, if counterions were permitted to enter this porous layer, the net electrical potential at the outer edge of the porous layer would be reduced considerably in magnitude. Thus high "surface charge density" could be reconciled with modest values of the (-potential. (It should be noted that the rather low values of | ( | were also consistent with the stability behaviour of these systems.)

Lyklema (1968) attempted to quantify the porous gel model to some extent, as did Wright (1972), Levine et al. (1972) and Perram (1973). A complete analysis of the model in its most general form (for unsymmetric electrolytes) was given by Perram et al. (1973,1974), and only in this paper is a simultane­ous test applied to both the surface charge and the (-potential for the same system, under the same conditions, with the same parameter values, so we shall concentrate on that treatment.

The concentrations of all ions in the gel layer (c() are assumed to be related to the bulk concentrations (cf) by expressions of the form:

c^cfexpl-ieV + ixJ/kT] (7.4.2)

7. SIMPLE INORGANIC IONS AND ZETA POTENTIAL 285

where fit is the specific adsorption potential (corresponding to ö ± in eq (2.4.3)) and ¥ is the local potential in the gel relative to a zero value in the bulk of the electrolyte. Poisson's equation (2.3.1) then has a unique solution when the appropriate boundary conditions are introduced for (i) the bulk solution and (ii) the junction of gel and solid and when also continuity of the potential and the dielectric displacement is ensured at the gel-electrolyte boundary. The solution even for the case of a 1:1 electrolyte cannot be written explicitly but involves solution of two simultaneous transcendental equations to obtain the (-potential and the titratable charge (see Hunter, 1975). The anticipated potential distribution is as shown in Fig. 2.13 for the l iquid-l iquid interface. On this model the titratable charge is equal in magni­tude and opposite in sign to the sum of (i) the diffuse layer charge and (ii) all of the charge due to indifferent ions in the gel ( C T J . The (-potential is identified with the potential at the junction between the gel and the electrolyte. There is, thus, no separate Stern layer postulated. In a sense the entire gel layer may be taken to be a generalization of the idea of the Stern layer for the case where the surface is significantly disordered.

The adjustable parameters of the model are:

(i) the thickness, L , of the gel layer;

(ii) the values of for each of the indifferent ions;

(iii) the dielectric permittivity (sg) of the gel region;

(iv) the dissociation constants, K+, X _ (defined as in eq (7.3.15)).

Actually only one of these Ks is independently adjusted since they must satisfy:

1/2 = 10pH= (7.4.3)

where p H - is the p.z.c. in the absence of specific adsorption. For purposes of illustration a value of eg = 40e0 was taken, though the

results are not very sensitive to this parameter. Values of ^ = — kT for the counterion and L = 4 0 A for the gel thickness on T i 0 2 gave a reasonable fit to the charge data as is apparent f rom Fig. 7.14(a).

The (-potential data are somewhat sparse (when a plot is done against indifferent electrolyte concentration) and clearly much more evidence would be needed in order to properly assess the predictions of the model. Though it predicts values of the right order of magnitude, the experimental data, especially near to the p.z.c, do not show such a marked dependence on electrolyte concentration as the model predicts (Fig. 7.14(b)). This is partly due to one of the simplifications made in the model analysis. The specific adsorption potentials / i £ were put equal for anions and cations and the lines of best fi t are a compromise between the charge and the (-potential data.

286 ZETA POTENTIAL IN COLLOID SCIENCE

Fig. 7.14a. Titratable surface charge on T i 0 2 as a function of indifferent electrolyte concentra­tions at various pHs. Curve A: pH7; &: pH8; C: pH9. /u; = - kT, gel thickness = 40A. (From Perram, Hunter and Wright, 1974.)

Fig. 7.14b. C-potential as a function of indifferent electrolyte concentration for pH4 (curve A) and pH5 (curve B). Theoretical curves from Perram et al. (1974), with U j = — kT and gel thick­ness 40 A. Experimental data from Wright (1972.).

(Note that since T i 0 2 has a p.z.c. at p H 5-8 the charge data are for a negatively charged gel whereas the (-potential data are for a positively charged gel.)

It should be noted in concluding this section that there is no independent experimental evidence for a porous layer on the T i 0 2 surface. Indeed, tri t ium exchange studies (Yates, 1975) indicate the contrary. Titania may also not be the most appropriate oxide to use in a test of a general theory because it may well have cation (Ti" + ) sites as well as surface hydroxy Is. On the other hand, some silica systems (see Section 7.4.2) and a-haematite (Breeuwsma and Lyklema, 1971) may well be best treated by this model. What is required for an adequate test is more extensive data on both charge and (-potential as functions of indifferent electrolyte concentration at dif-ferent.pH values for well-characterized systems. I t should be reiterated at this point that the porous model described above requires that the surface layers of the solid be penetrated to some depth not just by hydrogen (and hydroxyl) ions but also by the indifferent electrolyte counterions.

7.4.4. The site-dissociation model for oxides

As noted in Section 7.4.1 the most appropriate form of the site-dissociation model for oxide systems considers the surface to be composed of amphoteric sites which can become charged either positively or negatively, depending on the pH, (Levine and Smith, 1971). The basic approach involves analysing

7. SIMPLE INORGANIC IONS AND ZETA POTENTIAL 287

how the chemical behaviour affects the charging process and thence calcu­lating the resulting surface potential, i//0, as a function of solution pH. The Nernst equation is not expected to hold for such systems (Levine and Smith, 1971; Healy et al, 1977). Because Healy and White (1978) have made a direct comparison of their version of this model with the available experimental data on oxides we shall use their approach rather than the earlier, very similar development of Levine and Smith (1971). The relationship between the two different formulations is illustrated by Healy and White (1978) in their table 3.2. Essentially it amounts to simplifying the Zwitterionic model (Section 7.3.2.2) by noting that:

N Ns^=Ns+=^ (7.4.4)

so that at the p.z.c, where n+ = the fraction of negative or positive sites, 0O is:

2Nt. ) :

Equation (7.3.20) can then be written

(7.4.5)

r2> ApX = p i c : _ - p X + = 2 1 o g 1 0 l - ) (7.4.6)

where the parameter, <5, is given by:

3 = W \ (7.4.7) ( l - 2 0 o ) 1 ;

The quantities ApX or <5 or 0O are very important in this model for they deter­mine the extent to which the system departs from Nernst-type behaviour (Fig. 7.15). The departure can also be calculated from eq (7>3.21), which in this case reduces to (Smith, 1976):

# o <ipH

= N - k T 1 d<7° 2iVs<?2 0O dpU

where N = — 2-303kT/e is the Nernst slope. The second term becomes important only for small values of 0O. As 0O approaches its maximum value of 0-5, so that 5 becomes very large ( > 100, say) and ApK becomes negative ( < 3) the system would begin to show Nernst-type behaviour. This corresponds to the A g l interface where the p.z.c. occurs at a point where there is an equal (and large) number of positive and negative sites. For oxide systems, ApK is commonly in the range + 3 ( T i 0 2 ) to + 6 (Fig. 7.16).

At the p.z.c. in these systems the majority of the amphoteric sites is un­charged ( 1 0 _ 1 > 0n > l O - 5 ) .

288 ZETA POTENTIAL IN COLLOID SCIENCE

1 0 0

r i i i

Nernst potential ( A p H = 3 )

• " - _ ^ A p / f = 0

" ^ - ^ A p A ^ ^ ^ - ^ ^

- \ > y A p / r = 6

-

I

log, 0<r

Fig. 7.15. The variation of \j)a with indifferent electrolyte concentration for a non-Nernstian surface at three pH units distance from the i.e.p. (After Healy and White, 1978.)

The p H at the p.z.c. is given by (from eqs (7.3.19) and (7.4.5)):

pH- = ( P K + + p K _ )

(7.4.8)

and Healy and White (1978) introduce an hypothetical Nernst potential defined by:

éN = 2-303—(pH. - pH) = 2-303—ApH (7.4.9) e e

The analogue of eq (7.3.9) for the surface charge can then be written:

o-Q _ <5 s inh^jy - 1 > 0 ) (7.4.10) eNs~ ~ 1 + 6 coshW^ - i£ 0)

In the absence of specific adsorption we have a0 = -ad and eqs (7.3.10) and (7.4.10) then give:

, - i • , f$o\ g j f a h Q f a - y o ) (7.4.11) \2 J l+ö cosh(i>w - <Ao)

where £ is defined in eq (7.3.11). This equation can be solved graphically by plotting the left and right hand

7. SIMPLE INORGANIC IONS AND ZETA POTENTIAL 289

sides against $ 0 for various values of <5 and ApH, to yield the equilibrium value of the surface potential as a function of pH .

The effect of ApK on the predicted a0 — ApH curve for various oxides is shown in Fig. 7.16. Note that this type of model can at least reproduce the broad features of the charge-pH data and, in particular, the difference between SiO z and the other oxides can be ascribed to a larger value of ApX for S i 0 2 than for the others. Though a value of ApK = 10 appears to be necessary to fi t the charge data for S i 0 2 , Healy and White show that such a value does not f i t the data for the electrokinetic potential (Fig. 7.17), if one equates £ to the potential at a distance of 20 A from the surface (using eq (2.3.16)). Perhaps the discrepancy could be partially removed by incorpor­

ating a more sophisticated Stern Layer model but it is clear once again that simultaneous fitting of both charge and (-potential data provide a very strong test for any model of the interface.

2 3

Fig. 7.16. Comparison of the experimental surface charge-pH isotherms for various oxides with theoretical curves based on a simple site-dissociation model. The numbers attached to the curves refer to the ApK values used in their calculation. Ns = 5 x 10 1 4 sites per cm 2 and c = 0-1 M in all cases. (After Healy and White, 1978.)

290 ZETA POTENTIAL IN COLLOID SCIENCE

pzc

Fig. 7.17. The effect of indifferent electrolyte and pH on the value of \j/ at 20 A from a negative surface. Experimental points are ^-potential values for silica. Full curves are calculated assuming ApK = 6 and broken curves assumed ApK = 10.

A somewhat better reconciliation of charge and potential data is possible for T i 0 2 (Fig. 7.18) but even here the value of \a%/d log c| predicted by the model is again too large, especially for large values of ApH. As noted above in connection with Fig. 7.7, this is a common feature of simple Gouy-Chapman models of the interface.

7.4.5. Site-dissociation—site-binding models for oxides

An alternative method of rationalizing the very large surface charges and \cw ^-rjoltnU&vS c£ oxides, invokes, the population ot a direct binding, oi

COUTtóroS lo the svxrfa.ce daai®*. i\\\non%n this procedure compromises

7. SIMPLE INORGANIC IONS AND ZETA POTENTIAL 291

1 2 3 4 5 6 ApH

Fig. 7.18. Comparison of experimental data for ^-potential and surface charge with theoretical predictions from a site-dissociation model. Experimental data are for an oxide surface T i O , Model parameters: K' = K" = 0-135;

JCj.=T40- . K 2 = 2 0 u F c m - 2

Full lines are for ApK = 3-0; broken lines ApK = 4-0. Charge date from Yates and Healy (1980); Potential data from Weise and Healy (1975).

The idea of incorporating specific chemical dissociation processes in solution and at the surface into electrical double layer theory is a natural outgrowth of the studies of Sturnm and his co-workers (see e.g. Huang and Stumm, 1973) and of Posner and Quirk (see e.g. Bar-yosef et al., 1975). The basic notion is that negative and positive sites formed by reactions like those shown in eqs (7.3.1) and (7.3.14) can, in effect, be neutralized by reactions

such as:

A " + N a + ^ ± A N a (7.4.12)

^ + C \ - ^ A H j C l (7.4.13)

292 ZETA POTENTIAL IN COLLOID SCIENCE

The surface charge density, as measured by titration, is then given by:

o-o = e { [ A H 2

+ ] + [AH 2C1] - [ A " ] - [ANa]} (7.4.14)

whereas the charge density balancing the diffuse layer charge is:

cr0 + cr; = e { [ A H 2

+ ] - [ A - ] } (7.4.15)

and the Stern layer charge is:

o-; = e { [ A N a ] - [ A H 2 C l ] }

In their original paper Yates et al. (1974) considered the association of the counterion with the surface group to form a small dipole but in working through the consequences of the model they were forced to assume that the total energy of the dipole in the electric field at the surface was simply — e(i//0 — which corresponds to assuming that the counterion resides in the I H P where the potential is (see Fig. 2.9). Apart from neglecting the discreteness of charge effect (Section 2.4.3), which is common practice, this procedure assumes that both counterions have the same approach distance when they undergo association with the surface and, furthermore, the num­ber of surface sites (N\ in eq (2.4.2) )in the Stern layer is equal to the number of positive ( A H 2 ) or negative (A~) sites at each pH. This amounts to a rather severe restriction on the normal Stern model.

Equations very similar to (7.3.19 and 20) can still be written, whilst the Nernst equation becomes modified to read:

, « \ _ T I _TT1 kT [AH «Ao = 2-303 — [pH. - pH] - — In ^ — ± ± (7.4.16)

e 2e [A J

Since the degree of binding depends upon the bulk electrolyte concentration it is clear from eq (16) that i / f 0 wil l also depend on electrolyte concentration at constant pH. The diffuse layer charge is again given by:

c r d = - ^ s i n h ( | ) (2.4.9)

= -(0-0 + 6-)= - e { [ A H 2 + ] - [ A - ] }

where Ns includes all the terms in eq (14) together with [AH] . The inner and outer layer capacitances, Kx and K2 are introduced to define the potential differences in the compact layer:

= (7A17)

and K2

h = + P (7-4.18) A . ,

7. SIMPLE INORGANIC IONS AND ZETA POTENTIAL 293

This model therefore requires for its solution estimates of Ns, Kt and K2, K+ and a knowledge of the p H at the p.z.c. (from which K_ follows from eq (7.4.8)). I t also requires estimates of K" and K', the equilibrium (dissocia­tion) constants for the reactions represented by eqs (7.4.12 and 13). I t turns out that these dissociation (or complexation) constants are very important in this model and Davis et al. (1978) describe in some detail the limitations of the method used by Yates et al. (1974) to evaluate these quantities. They propose a modified approach which leads to constants which are essentially independent of electrolyte concentration. A detailed account of the extra­polation procedures developed by James et al. (1978) and required to obtain values for K' and K" is given in the review by James and Parks (1980) and need not be repeated here. The point to be noted is that Davis et al. (1978) regard the association reaction represented by eq (7.4.12) as rather less im­portant than a simple exchange reaction between positive ions and the surface hydrogens attached to amphoteric groups:

A H + N a s

+ ^ A ~ N a + + H s

+ (7.4.19)

They therefore write in place of K" the constant

[ A - N a + ] [ H + ] S

[ A H ] [ N a + ] ,

[ A " N a + ] [ H + ]

(7.4.20)

[AHlfNa" 1 expOA,--^) (7.4.21)

where the subscript s refers to surface values. Equation (21) is derived from (20) assuming that the N a + ions occupy the IHP (Fig. 2.9) whilst the H +

ions occupy the surface plane. Figure 7.19 shows how this model describes the data on T i 0 2 in the pre­

sence of K N 0 3 solution at several concentrations. The agreement both with a0 (especially at high electrolyte concentration) and with £ data is an im­provement over that shown in Fig. 7.18 but is to be expected considering the larger amount of experimental information fed into this model. The most striking features are (i) the much larger value assigned to ApK on this model and (ii) the value assigned to the intrinsic dissociation constants of the bound NO3 and K + ions. If, as an approximation we set i/^- ~ yj0 in eq (21) then we can write:

* ^ = f r ? (7.4.22)

so that the quoted values of *K'£i and X _ correspond to K" = 10' V 9 . Yates et al. (1974) show that the site binding model can be related to the more traditional form of the Stern model by writing:

294 ZETA POTENTIAL IN COLLOID SCIENCE

Fig. 7.19. The surface charge density (Yates, 1975) and C-potential (Wiese, 1973) of a T i 0 2

dispersion as a function of pH at various concentrations of K N 0 3 and at 25°C. The theoretical curves are calculated from the site-dissociation-site-binding model with Ct = H O u P c m " 2 and pJCim = 2-7; pKf = 9-1; *pKj& 3 = 4-2 and *pK$ = 7-2.

K " = [ ^ - ] ^ j ] e x p ( - f c ) (7.4.23)

Comparing this with the usual expression for Stern layer adsorption (eq (2.4.4)) (where o+ = e [ A " K + ]) requires that the specific adsorption poten­tial be given by:

9^kTii§s) (7A24)

A value of K" = 1 0 " 1 9 thus corresponds to a specific adsorption potential of about — 8-4 kT. By the same token, the value of 0_ is about — 7-5 kT. I t must be admitted that these values are larger (in absolute magnitude) than one would expect, especially considering the fact that the p.z.c. is independent of electrolyte concentration. One is forced, on this model, to conclude that these simple inorganic ions are very considerably adsorbed at the p.z.c. but

7. SIMPLE INORGANIC IONS AND ZETA POTENTIAL 295

in equal amounts at all concentrations. That is a proposition that many would find difficult, if not impossible, to accept, but before rejecting the model out of hand it would be wise to explore the insights it offers us into the nature of surface processes on oxides. Davis et al. (1978), for example, give a rather striking illustration of its use in distinguishing possible competing models for the adsorption of M g 2 + ions on to T i 0 2 - Even if in the long run this particular approach is superseded it can, for the time being at least, be used to rationalize and codify a significant body of information. Perhaps more importantly it acts as a stimulus for the collection of more meaningful data on oxide, polymer latex and related systems.

7.4.6. The Stern model of the oxide-solution interface

The site-dissociation-site-binding model described in Section 7.4.5 is a special case of the application of a Gouy-Chapman-Stern-Grahame (GCSG) model to the oxide surface, and its relationship to the conventional Stern model has already been discussed. The preliminary studies of Hunter and Wright (1971) and of Wright and Hunter (1973) on the electrokinetic behaviour of simple inorganic ions on oxide surfaces have also been men­tioned (Section 7.4.1), though they were not described in detail. That work concerned itself solely with (-potential data and did not attempt to reconcile charge and potential data on the same system. The fact that the GCSG model could reasonably represent the dependence of £ on the concentration c, of indifferent electrolyte ought not to occasion any surprise considering the theoretical work of Anderson (1958) (see Section 7.6). He showed that a simple adsorption model, when coupled with the usual expression for the diffuse layer charge, e.g. eq (2.3.27), could produce a number of different types of (-log 1 0 c relations. Both positive and negative slopes are possible, depending on the details of the isotherm parameters and the electrolyte concentration and, what is more significant, an analytical explanation can be provided for the observation that d\Q/d log c is never as high as 59 mV. A closer examination of Anderson's data on T i 0 2 reveals, however, that his model would give far too low a value for the surface charge at each pH. This is precisely the same conclusion as that arrived at by Wright and Hunter (1973) and was the point which led them to examine the porous gel model (Section 7.4.3). The important point to note here is that the limitations of the GCSG model may not be apparent unless one conducts detailed studies of both charge and potential over wide ranges of concentration for both the p.di . and other ions.

Before leaving this point we must draw attention to the analysis by Smith (1976) of the electrokinetic data on oxides. Using the technique described in connection with Fig. 5.15 for Agl , and again for a Zwitterionic polymer

296 ZETA POTENTIAL IN COLLOID SCIENCE

latex in Section 7.3.2.2 Smith is again able to show that, at least in the neigh­bourhood of the p.z.c., the GCSG theory describes the (-potential data well and the position of the shear plane must be within at most a few Angstrom units of the plane where \f/d is measured.

Smith's analysis also provides an estimate of the apparent compact layer capacitance KA which he finds is 18 uF c m " 2 for T i 0 2 and 5-6uFcm 2 for S i 0 2 . The value of KA is determined by contributions from the region be­tween the surface and the IHP and between the I H P and the OHP. I t is therefore dominated by the smaller of these two values which is generally assumed to be the capacitance of the outer part of the compact region K2. Values of around 2 0 u F c m " 2 for this capacitance are generally found to be satisfactory for the site-dissociation-site-binding model. By contrast the capacitance of the inner part of the compact region is very high on oxide surfaces. Values well in excess of lOOuFcm" 2 are commonly assumed (see Fig. 7.19 for example). A value of 140uFcm" 2 corresponds to D/x1 = 1-6 x 1 0 n m _ 1 which for a dielectric constant of 16 (reflecting significant water dipole orientation) would require a plate separation of only 1 A. The implication is that the plane of centres of adsorbed ions is closer to the plane of centres of the surface charge than would be expected on the basis of ionic radii (especially i f the ions are to some extent hydrated). I t must be assumed then, that the adsorbed counterions can penetrate the plane of the surface charge to some extent. In the terms of the original site-binding model of Yates et al. (1974) this was accommodated by assuming that the surface ionic dipoles were oriented almost parallel to the surface (Fig. 7.20).

Fig. 7.20. Relation between surface charge ( - ) and adsorbed charge ( + ) in the site-dissociation model of Yates et al. (1974). Note the direction of the surface dipole.

7. SIMPLE INORGANIC IONS AND ZETA POTENTIAL 297

7.5. Clay mineral systems

The importance of clay minerals in the ceramic, paper and oil industry and in agriculture has led to their being studied in great depth over a long period. Unfortunately, much of the early data is vitiated by the fact that most clay minerals are unstable in aqueous solution, especially at low pH. Slow dis­solution of silica and especially alumina from the crystal edges (Eeckman and Laudalout, 1961) and redeposition on the crystal faces led to consider­able uncertainty and variability in surface properties, particularly when the clay was "purified" by the old electrodialysis technique. The early, very extensive studies of Marshall and his co-workers (see e.g. Marshall, 1949) are, therefore, best understood as mixed clay mineral-oxide systems. The more recent preparation techniques, introduced by Coleman and his as­sociates (Harward and Coleman, 1954; Coleman and Craig, 1961) involve passing the clay through a strong acid exchange column in the H +-saturated form, or rapid washing with strong mineral acid (pH 3) in the presence of, say, N a + ion to exchange with A l 3 + as it is formed. This technique, used by Posner and Quirk (1964), produces a much cleaner clay surface which dis­plays the characteristic features of a strong acid cation exchanger (probably on the crystal faces) with some weak acid sites, probably confined to the crystal edges. I t might be expected, therefore, that clays prepared by both the old and the newer methods would be described by the two-site binding model mentioned in Section 7.3.2.2. A description of the application of such a model to the clay mineral surface has recently been given by James and Parks (1980), who also give an excellent review of the earlier work in the field.

Two separate acid dissociation sites are postulated (SOH and TOH), and exchange reactions occur between them and an inorganic cation:

SOH + N a s + - S O " N a + + Hs+ ; * * £ s o t f = t S O " N a + ] [ H + L ( ?

N3.SOH- [ S O H ] [ N a + L ( '

and similarly for T O H . The normal acid dissociation constant is:

T O H ^ T O " + H+ ; K™T0H = (7.5.2)

with a similar expression f o r A * ^ . The titratable surface charge density, tT 0, is given by:

o-o = - e{[SO~] + [ S O " N a + ] + [ T O " ] + [ T O - N a + ] } (7.5.3)

whilst the Stern layer charge is:

ff, = e{[SO"Na + ] + [TO"Na + ]} (7 .S41

298 ZETA POTENTIAL IN COLLOID SCIENCE

and the diffuse layer charge is again given by eq (2.4.9). Equations (7.4.17 and 18) also hold for the potential distribution in the compact layer.

James and Parks (1980) have solved this model by matching the experi­mental titration data for Putnam clay at various clay concentrations as measured by Marshall and Krinbi l l (1942). The dissociation constants used in this comparison were:

P ^ S O H = 6-8; p * X j S L S O H = 3 2; pK^oH = 2-5

p K ' " ' T o H = H O; p * K k n a ,TOH = 74 ; p*KgTOH = 6-8 Again, eq (7.4.22) holds, so the specific adsorption potential being assumed here for N a + ions on the T O " sites is 6+ = - 12-3 kT, when measured in terms of the usual Stern theory (eq (7.4.24)). I t cannot be denied that this is a disturbingly large value (in absolute magnitude) but this is more or less to be expected, considering the very high surface charge on clay minerals and the very low values of (-potential commonly measured. Unfortunately, there were no (-potential values with which to compare the calculated values of \j/d but the results obtained were of the right magnitude and showed the kind of p H dependence commonly encountered in experiments. An example of the rather complicated (-potential behaviour which can be exhibited by these clay systems is shown in Fig. 7.21 from Friend and Hunter (1970). I t is

- 8 0 -i i

-

_ - 6 0 >

•E

- y • (2 )

\ -

- 4 0 -( 3 ) J L / V

1 1

log c 10

Fig. 7.21. C-potential as a function of indifferent electrolyte concentration for (1) lithium ver-miculite, (2) butyl ammonium vermiculite and (3) sodium bentonite. (From Friend and Hunter, 1970.)

7. SIMPLE INORGANIC IONS AND ZETA POTENTIAL 299

apparent that simultaneous measurements of charge and (-potential as functions of p H and "indifferent" electrolyte concentration, now that we recognize their significance, could lead to a much better understanding of the nature of the surface of these materials. The alternative approach using the simple GCSG model is evidently inadequate, as Williams and Williams (1978) have pointed out.

7.6. Application to other surfaces

One test that can be applied to systems for which the simple Gouy-Chapman theory might prove adequate is to examine the dependence of (-potential on indifferent electrolyte concentration, c at constant surface potential or charge. I f one assumes that the diffuse layer charge is independent of c (an unlikely situation), it is not difficult to show, from eq (2.3.16), that (Anderson, 1958):

d C ^ = - 2 - 3 0 3 ^ t a n h f ^ (7.6.1) dlogcj^ ze \2kT

This equation predicts that |( | wi l l decrease by about 60 mV per ten fold change in concentration if ( is sufficiently high. Such behaviour is rarely, i f ever, observed experimentally. An alternative and more likely hypothesis is that \j/0 is constant and that ( i s measured at some fixed distance, t, from the surface (Fig. 6.6). This corresponds to the Eversole and Boardman (1941) model, discussed in Section 5.4. Those systems which follow eq (5.4.1) with modest values of t ( < 10 A) can be quite satisfactorily modelled in this way, although one ought not to interpret this as meaning that such systems do indeed have a simple Gouy-Chapman double layer. The variation of (-potential with indifferent electrolyte concentration can then be determined by differentiation of eq (5.4.1):

* ^ = - 2 -303^ . (7.6.2) d log c j ^ t ze \2kT)

= - 2 - 3 0 3 ^ (7.6.3) 2e

These equations predict a very slow change in | ( | with concentration of in­different electrolyte at low concentration, gradually increasing as the con­centration rises. Such behaviour is commonly encountered experimentally. A rather detailed discussion of the dr\d log c relation for a Gouy-Chapman double layer was given by Anderson (1958) who incorporated a simple ad­sorption isotherm into the model but did not analyse the (i/^ 0 = constant)

300 ZETA POTENTIAL IN COLLOID SCIENCE

situation leading to eqs (2) and (3). He did, however, show that a number of different situations can be expected to give rise to a value of d£/d log c which is much less than the 60 mV per tenfold change in concentration which is sometimes claimed to be the expected value for a Nernst-type interface.

The Gouy-Chapman-Stern-Grahame model has been applied to a variety of surfaces to describe their behaviour in the presence of simple in­organic ions. The effort has been only partly successful, for reasons which should be obvious from the analysis given earlier in this chapter. The most important area remaining is that of biological systems for which a huge amount of electrokinetic data has been published. Unfortunately, it is very rarely analysed in terms of detailed double-layer models.

Overbeek (1950) drew attention to the early work of Bull and others which showed that for protein-covered surfaces the charge calculated from electro­kinetic data was always less than ( ~ f ) that calculated from titration data. This is of course readily understood if the shear plane is located some little distance from the plane of the surface charge. Rather than simply invoking an arbitrarily adjustable shear plane, however, it would be preferable to have an analysis done in terms of the site-dissociation (and, if necessary, site-binding) models described above. The most appropriate model for a protein-covered surface would probably be the two-site, Zwitterionic surface treated in Section 7.3.2.2, and certainly the data provided by Nordt et al. (1978), on the hardened red blood cell (Fig. 7.22) show similarities to that of Rendall and Smith (Fig. 7.9). The red cell membrane contains a number of con­stituents, of course, and the treatment applied by Nordt et al. is said to selectively remove just one (the sialic acid). I t would be interesting then to see. whether the Rendall and Smith (1978) analysis describes the data with an appropriate variation in the number of acid sites as the sialic acid is removed. The fact that the treatment causes the i.e.p. to move to higher p H suggests that there is more involved than the disappearance of — C O O H groups.

One biological system for which a detailed analysis has recently appeared is the D N A molecule. In a series of papers, Stigter has been able to reconcile various aspects of the transport properties of D N A in a very satisfactory way even though his calculations of the relaxation effect may be somewhat under­estimated (see Section 3.7.3 for a brief discussion of the comparable work of van der Dr i f t et al. (1979)). Stigter (1975) began by improving the description of the double layer around D N A using the cylindrical solution of the Poisson-Boltzmann equation (2.3.38) to replace the usual Debye-Hücke l approxima­tion (see Section 3.7.3). This improves considerably the agreement between the (-potential calculated from electrophoresis and the value calculated from the surface charge. In subsequent papers Stigter has improved the analysis for the electrophoresis of cylinders and shown how to take account

7. SIMPLE INORGANIC IONS AND ZETA POTENTIAL 301

pH Fig. 7.22. pH-mobility relation for red blood cells (hardened in formaldehyde). In A the sus­pending medium is 0-15 M NaCl. In B it is 0-03 M NaCl/022 M sorbitol.

O Control cells • After removal of 50 % of sialic acid A After removal of all sialic acid.

Broken curve indicates systems which did not return to their original mobility when returned to pH7, following exposure to the indicated pH. (From Nordt et al, 1978).

of orientation (Stigter, 1978a, b). He has also (1978c) described the applica­tion of the Gouy-Chapman-Stern theory to protein systems, drawing par­ticular attention to the fact that "for double layers of high potential, in equilibrium with an electrolyte solution in which all ions have the same valency, the structure of the inner part of the double layer remains nearly unchanged upon dilution of the bulk electrolyte". Finally, in a more recent

302 Z E T A P O T E N T I A L I N C O L L O I D S C I E N C E

publication (Stigter, 1979) he presented the comparison shown in Fig. 7.23 between the fractional charge on a D N A molecule as measured by electro­phoresis and by conductance and the adsorption isotherm which is required to describe the ion binding. I t should be noted that the ion-binding constant cited here is the association constant for the counter ion to the surface charge:

A " 4- N a + A - N a + ; K = [— A ~ N a + ]

[ - A - ] [ N a + ]

The values of \ down to 1 m ° l ~ 1 therefore correspond to moderate to very weak binding ( — 3-3 to + 1-5/crfor 6+, using eq (7.4.24)).

One important outcome of this analysis is that it shows that, at least for D N A , there is no contribution to the conductivity from the ions behind the shear plane.

1-0

0 -8

0-6

0-4

0-2

I I I I I I |

K = 1 / 2 5 6 •

T T

K = 1/16

• • Me 4 NCl 0 • KCl O • NaCl A A L iCl

I I I 1 I 1 I

• d I I I I I

•

I I I I I I I I I I 1 I I I

10 Molarity

10 1

Fig. 7.23. Kinetic charge of DNA (— ctj per phosphate group, in various salt solutions. Open points calculated from electrophoresis, filled points from conductance data, curves from Lang-muir adsorption model with electrostatic corrections. (From Stigter (1979) with permission.)

References Anderson, P. J. (1958). Trans. Faraday Soc. 54, 562. Bagchi, P., Gray, B. V. and Birnbaum, S. M . (1979). J. Colloid Interface Sci. 69, 502. Bar-Yosef, B., Posner, A . M . and Quirk, J. P. (1975). J. Soil Sci. 26, 1. Bérubé, Y . G. and de Bruyn, P. L . (1968). J. Colloid Interface Sci. 27, 305. Bijsterbosch, B. H . (1965). Thesis. State University of Utrecht. Breeuwsma, A. and Lyklema, J. (1971). Disc. Faraday Soc. 52, 324. Chakravarti, S. N . and Talibudeen, O. (1961). Soil Science 92, 232.

7. S I M P L E I N O R G A N I C I O N S A N D Z E T A P O T E N T I A L 303

Coleman, N . T. and Craig, D . (1961). Soil Sci. 91, 14. Curreri, P., Onoda, G. Y. Jr and Finlayson, B. (1979). J. Colloid Interface Sci. 69, 170 Davis, J. A. , James, R. O. and Leckie, J. O. (1978). J. Colloid Interface Sci. 63,480. Delahay, P. (1965)! "The Electrical Double Layer and Electrode Kinetics." Inter-

science, New York. Eeckman, J. P. and Laudalout, H . (1961). Kolloid-Zeit. 178, 99. Friend, J. P. and Hunter, R. J. (1970). Clays and Clay Minerals 18, 275. Fuerstenau, D . W. and Healy, T. W. (1972). Principles of mineral flotation. Chapter 6.

In "Adsorptive Bubble Separation Techniques". Academic Press, New York and London.

Gaudin, A. M . and Fuerstenau, D . W. (1955). Trans. Amer. Inst. Min. Eng. 202, 66. Hall , E. S. (1966). Disc. Faraday Soc. 42, 197. Harward, M . E. and Coleman, N . T. (1954). Soil Sci. 78, 181. Hartley, G. S. and Roe, J. W. (1940). Trans. Faraday Soc. 36, 101. Healy, T. W. and White, L . R. (1978). Adv. Colloid Interface Sci. 9, 303. Healy, T. W., Yates, D . E., White, L . R. and Chan, D . (1977). J. Electroanal. Chem. 80,

57. Homola, A . and James, R. O. (1977). J. Colloid Interface Sci. 59, 123. Huang, C. P. and Stumm, W. (1973). J. Colloid Interface Sci. 43, 409. Hunter, R. J. (1975). In "Modem Aspects of Electrochemistry" (J. O 'M. Bockris and

B. Conway eds.) Vol . I I , pp. 33-84. Plenum Press, New York and London. Hunter, R. J. and Wright, H . J. L . (1971). J. Colloid Interface Sci. 37, 564. James, R. O., Davis, J. A . and Leckie, J. O. (1978). / . Colloid Interface Sci. 65, 331. James, R. O. and Parks, G. A. (1980). In "Surface and Colloid Science". Vol. I I .

(E. Matijevic ed.). Wiley-Interscience, New York. Kulkarni, R. D . and Somasundaran, P. (1973). Proc. Symposium on Oxide-Electrolyte

Interfaces, Miami (R. S. Alwit t ed.) p. 31. The Electrochemical Society, New York. Lange, E. and Crane, P. W. (1928). Z.physik. Chem. 141, 225. Levine, S. and Smith, A . L . (1971). Disc. Faraday Soc. 52, 290. Levine, S., Smith, A . L . and Brett, A. C. (1972). Proc. 7th Intern. Congr. Surf Activity,

Zurich, p. 603. Liu , L.-J. and Krieger, I . M . (1978). In "Emulsions, Latices and Dispersions" p. 41.

(P. Becher and M . N . Yudenfreund eds.) Marcek Dekker, New York. Lyklema, J. (1961). Trans. Faraday Soc. 59, 418. Lyklema, J. (1968). J. Electroanal. Chem. 18, 341. Lyklema, J. (1971). Croatica Chem. Acta. 43, 249. Lyklema, J. (1977a). In "Trends in Electrochemistry" (J. O 'M. Bockris, D . A. J. Rand

and B. J. Welch eds.) p. 159. Plenum Press, New York. Lyklema, J. (1977b). J. Colloid Interface Sci. 58, 242. Marshall, C. E. (1949). "Colloid Chemistry of the Silicate Minerals" 1st ed. Academic

Press, New York. Marshall, C. E. and Krinbi l l , C. A . (1942). J. Phys. Chem. 46, 1077. Matijevic, E. (1977). J. Colloid Interface Sci. 58, 374. Modi , H . J. and Fuerstenau, D . W. (1957). J. Phys. Chem. 61, 640. Nordt, F. J., Knox, R. J. and Seaman, G. V. F. (1978). / . Cell. Physiol. 97, 209. O'Brien, R. W. and White, L . R. (1978). J. Chem. Soc. Faraday II, 74, 1607. Onoda, G. Y. and de Bruyn, P. L . (1966). Surface Sci. 4, 48. Osseo-Asare, K . A. (1972). M.Sc. Thesis, Univ. of California, Berkeley. (See also

Osseo-Asare, K . A. , Fuerstenau, D. W., and Ottewill, R. H . Amer. Chem. Soc. Symposium No. 8, p. 63. (1978).