1

Polymer Analysis

Chapter 1. Introduction/Overview

The focus of this course is analysis and characterization of polymers and plastics. Analysis ofpolymeric systems is essentially a subtopic of the field of chemical analysis of organic materials.Because of this, spectroscopic techniques commonly used by organic chemists are at the heart ofPolymer Analysis, e.g. infra-red (IR) spectroscopy, Raman spectroscopy, nuclear magneticresonance (NMR) spectroscopy and to some extent ultra-violet/visible (UV/Vis) spectroscopy. Inaddition, since most polymeric materials are used in the solid state, traditional characterizationtechniques aimed at the solid state are often encountered, x-ray diffraction, optical and electronmicroscopy as well as thermal analysis. Unique to polymeric materials are analytic techniqueswhich focus on viscoelastic properties, specifically, dynamic mechanical testing. Additionally,techniques aimed at determination of colloidal scale structure such as chain structure and molecularweight for high molecular weight materials are somewhat unique to polymeric materials, i.e. gelpermeation chromatography, small angle scattering (SAS) and various other techniques for thedetermination of colloidal scale structure. The textbook, Polymer Characterization covers all ofthese analytic techniques and can serve as a reference for a general introduction to the analysis ofpolymeric systems. Due to time constraints we can only cover a small number of analytictechniques important to polymers in this course and these are outlined in the syllabus.

Structure/Processing/Property:

Generally people resort to analytic techniques when confronted with a problem which involvesunderstanding the relationship between properties of a processed material and the structure andchemical composition of the system. Plastics are typically complex morphological systems beingcomposed of many phases and additives, even the polymer itself being disperse in molecularweight, tacticity, crystallinity, orientation and sometimes chemical composition. Dispersion ofstructure and chemical composition means that the best tools to describe polymeric materials arealways statistical in nature. For example, a low molecular weight organic has a specific meltingpoint, while a polymer displays a range of melting with an onset, a peak and a maximum meltingtemperature. Such a melting spectrum might best be described by a Gaussian function with astandard deviation and mean. Additionally, the complication of enormous macromolecular chainsmeans that simplified descriptions are often needed to characterize polymeric materials, for examplegroup contribution methods in spectroscopy where the chain structure is decomposed into chemicalgroups which contribute to the absorption spectrum. A detailed understanding of many of theanalytic descriptions of polymeric materials is often precluded by the complexity of the situation.

Analytic Techniques:

All analytic techniques used in polymer characterization are based on specific physical principleswhich serve as a guide to understanding the basic limitations of the techniques. Often multipletechniques are available to describe the same property of a material and it is only throughunderstanding the physical basis of characterization techniques that one can pick the "right toolfor the job", the job being understanding the relationship between structure and chemicalcomposition and properties. For example, it is often found that blown films of polyethylenedisplay different tear strengths in the machine and transverse directions. This can lead to failure ofparts made from blown films such as plastic bags. It is often assumed that this directional natureof the tear strength is related to orientation induced by processing of these plastics. There areseveral analytic techniques available to describe orientation. These include construction of pole-figures from x-ray diffraction scans, calculation of orientation functions from XRD data,calculation of orientation functions from IR, NMR or Raman data, and measurement of the optical

2

birefringence for the blown films using polarized light. Each of these techniques will yield adifferent value for the orientation function!

Similarly, even the value for the degree of crystallinity in these blown films will depend on thetechnique which is used, i.e. XRD, differential scanning calorimetry, density, or IR for example.This makes a firm understanding of the physical basis of analytic techniques critical to theirapplication in polymeric systems.

Levels of Structure:

The analytic description of a complex material is strongly dependent on the size scale on which anobservation is made. For example, a semi-crystalline polymer such as polyethylene is composedof chemical units similar to olefinic waxes. These chemical units give rise to spectroscopicabsorption patterns which are largely indistinguishable from their lower molecular weightcounterparts. Similarly, the crystalline structure, which is usually of low symmetry in polymers,mimics the crystalline structure seen in lower molecular weight analogues such as waxes.

The monomer structure combined with the topological arrangement of monomers in a polymerchain give rise to helical coiling of polymer chains. This helical coiling and the weak chemicalassociations related to it give rise to some mechanical and vibrational features which can often beobserved spectroscopically. Generally, the helical coiling of monomers in a chain are evidenced bycolloidal scale structure, chain persistence (local linearity) and enhancement of the ability of longchain polymers to crystallize. NMR is a technique particularly suited for the characterization of thetopological arrangement of monomers in a chain (i.e. tacticity).

In polyethylene, local chain structure is sufficiently regular to give rise to a crystalline phase.Entanglement of chains, chain branching and the presence of endgroups prevents completecrystallization of a polymer. Polymeric materials which display crystallinity are always describedby a multi-phase model, i.e. semicrystalline, which includes an amorphous and crystalline phase incoexistence. Low transport coefficients and chain folding give rise to nano-scale crystallites whichare best observed by TEM, small-angle x-ray scattering (SAXS) and to some extent by Ramanspectroscopy (Longitudinal Acoustic Modes, LAM).

Fibrillar crystallites in polymers lead to colloidal to optical scale structures, spherulites, which aregenerally centro-symetric and which display radially oriented birefringence. These micron scalestructures are best observed using optical microscopy, SEM, small-angle light scattering (SALS)and give rise to certain features in the mechanical and transport properties of semi-crystallinepolymers.

Thus, is one in interested in a specific analytic feature of a processing operation such as filmblowing on a polymeric material such as high density polyethylene (HDPE), one is immediatelyfaced with the issue of structural level, i.e. for transport properties one might need to characterizethe orientation of chains or crystallites (lamellae), for mechanical properties, orientation ofspherulites and the amorphous component of these biphase materials. Additionally, it is expectedthat any analytic determination of these materials will be subject to a large range of statisticalvariation between samples as well as an innate distribution associated with the polydispersity of thematerial of itself.

Course Contents:

Statistics:

All properties of polymeric systems display dispersion due to 1) the limited ability of syntheticchemistry to produce monodisperse and perfect chemical structure as well as 2) the dominance of

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kinetics in processing of high molecular weight materials. Statistical analysis is critical tounderstanding and describing plastics. This course begins with a survey of the important tools todescribe statistically distributed systems which includes the major distribution functions andmathematical descriptions of the propagation of error in data sets. Analytic descriptions ofpolymers are of no use unless some description of the expected error associated with the analyticresults are presented.

Thermal and Mechanical Testing:

Thermal analysis is useful in describing solid state transitions in polymers and is of pivotalimportance to understanding mechanical properties and processing of plastics. The measurementof the glass transition and crystalline transitions using typical analytic techniques will be discussed.Polymeric systems are dominated by kinetics and this is emphasized in the use of dynamicmechanical, thermal analysis techniques to describe mechanical absorption phenomena associatedwith the glass transition and other mechanical transitions.

Absorption Spectroscopy:

The major techniques for the determination of chemical composition and molecular topologyinvolve the absorption of electro-magnetic radiation by polymers. The major techniques are IR,Raman, and NMR spectroscopy and the bulk of this course will involve these major analytictechniques. Absorption is a quantized inelastic phenomena involving the transfer of energy fromEM radiation to a material.

Diffraction and Scattering:

In addition to inelastic absorption phenomena, elastic interaction between EM radiation and amaterial is possible and this gives rise to diffraction and scattering phenomena. The smallcrystallite size and dominance of crystalline orientation in processed plastics lead to several uniqueanalytic approaches in the analysis of x-ray diffraction data in these materials. The focus in thiscourse will be on those tools used in diffraction which are specific to polymeric materials.

Small-angle x-ray scattering is a critical technique for the description of polymeric materials sincediffraction at small angles is associated with the colloidal to nano-scales which is the size range of atypical polymer chain. The colloidal scale is also associated with polymer crystallites (lamellae)and microphase separated block copolymer structures. Further, light scattering has been widelyused in polymer science to describe disordered micron scale structure as well as a primarytechnique for the determination of molecular weight from dilute solutions.

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Polymer Analysis

Chapter 2. Error Analysis/Statistical Descriptions of Data.

All Polymer Properties are Disperse:Polymeric materials are subject to dispersion in all analytic properties. For example, the meltingpoint in a low molecular weight organic or inorganic is a fixed value constant which might displaysome variability over 1 or 2 degrees. The melting point for high-density polyethylene (HDPE), forexample can vary from about 110°C to about 160°C depending on processing and generallydisplays a broad dispersion over about 10 to 20°C. Spectroscopic analysis of polymers usingtechniques such as infra red adsorption (IR) and nuclear magnetic resonance (NMR) rely mostly onlocal chemical groups, so can display fairly sharp absorption bands. In these spectroscopictechniques, dispersion is shown by the existence of peak splittings or the presence of a number ofchemical species in small amounts. Additionally, many absorption bands in polymers are difficultto describe analytically and pertain to various acoustic modes associated with the conformation oflong chain structures. These are typically broad bands due to wide dispersion in these long chainconformations. X-ray diffraction from polymer crystallites generally displays broad diffractionpeaks associated with small crystals and a high degree of disorder within crystallites. Polymersalso display dispersion in structural and chemical orientation which should be viewed withstatistics. Dispersions in mechanical properties are always seen in polymers. Statistics are alsoused to describe the dispersion of chain size, molecular weight, and topological arrangement oftacticity. All descriptions of polymer chain size are based on statistics. It is critical to realized atthe onset of any analysis of a polymeric system that every physical property will be described by adistribution.

Error Analysis in Analytic Methods:In the physical sciences each analytic measurement must be associated with an assessment of theconfidence which should be associated with the analytic description. A value for some property ofa material is of no use unless some estimate of the expected error and distribution is provided. Forexample, for a commercial sample of HDPE a technician might report the melting point of a blownfilm as 135°C. You might be involved in using this material in a packaging application where thematerial will be subjected to shipping at a maximum temperature of 95°C. From the reportedmelting point and operating temperature would you feel confident that this material would meet thespecs of the packaging company? The correct answer is that the measured value is of no usewithout a description of the statistical distribution in the samples as well as a statistical descriptionof the distribution of melting points, i.e. onset, peak and maximum melting point.

The technician should be required to measure at least 10 samples (preferably more) and calculate astandard deviation for the reported value. Additionally, similar determinations of the onset ofmelting and the maximum melting point would be needed to determine if this material meets thespecifications of the packager.

There are a number of useful texts which describe the correct handling of experimental data. Themost commonly cited reference is "Data Reduction and Error Analysis for the Physical Sciences"by P. R. Bevington, 1969 McGraw Hill. This section of the course will summarize some of themajor points of Bevington with an emphasis on applications in polymeric systems.

Types of Error:Statisticians categorize three types of error. This framework may be useful in considering aspecific experiment such as determination of the d-spacing from an XRD peak.

Illegitimate Error: These errors involve an operator error, e. g. you have placed the wrong samplein the diffractometer and are determining the d-spacing for the wrong sample. There is no

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statistical description for these errors unless you want to consider sociology. Your main protectionagainst illegitimate errors is to always consider, when faced with extremely unexpected results, thatthe results involve a human error. Always look carefully at completely unexpected results.

Systemic Error: There errors are also not subject to a statistical analysis. They result from faultycalibration of the instrument or other problems which result in a constant shift of the data. In XRDa systematic error might involve confusion of 2θ with θ leading to roughly a doubling of the d-spacings. Systemic errors can sometimes be corrected after the fact if one is careful. It isimportant to keep a good record of the analysis that was performed partly for the purpose ofcorrection of systemic errors of this type.

Random Error: Random errors are the main area which statistics can deal with (i.e. standarddeviation, mean). Random errors have two sources. 1) Random variability in the samplesthemselves. 2) Limited precision of the analytic equipment. These two sources can bedistinguished if more precise equipment is available or a standard sample is available.

In describing the error involved in an analytic measurement all types of error should be considered.

Accuracy of a measurement pertains to how close a measurement is to the actual value. Precisionrefers to the reproducibility of the measurement whether or not it is close to the actual value. Ifsystematic error is present a measurement might be extremely precise but completely inaccurate!Also, if the equipment is well calibrated a measurement could be extremely accurate but notparticularly precise. Mostly the importance of these terms is in communication of your confidencein a particular value and what the possible sources of error are.

Statistical Analysis of Measurements:

Generally a given measurement will be conducted several times in order to determine the statisticaldistribution for the measurement. The values most commonly determined are the mean, µ, and the

standard deviation, σ:

µ = ∑1

1Nxi

N

σ µ2 2

1

1

1=

−−( )∑

Nxi

N

N is the number of measurements made, the value from each measurement is xi for measurement i.The square of the standard deviation is called the variance.

Distribution Functions (see P. C. Heimenz, Polymer Chemistry, 1984, MarcelDekker, pp. 34):

If a measurement is not single valued then it is common to fit the number distribution of values to adistribution function. The simplest distribution functions will involve two parameters, the mean, µand the standard deviation, σ. More complicated continuous distribution functions will involvehigher moments of the distribution. The k'th moment of a distribution is given by:

k' th moment= −( )∑ f x xi i s

k

i

3

where fi is the fraction of all measurements which have the value xi, i.e. Ni/N, xs is basis for themoment and k is the order of the moment. For example, the mean, µ, is the first moment (k = 1)

about the origin (xs = 0). The variance, σ2, is the second moment (k = 2) about the mean (xs = µ).

The molecular weight distributions commonly used in polymer science can be described in terms ofthe more broadly used statistical description of moments.

The number average molecular weight, Mn is the same as the first moment about the origin or themean. The weight average molecular weight (mass average), Mw, is given by:

MN M

N M

f M

f Mw

i ii

i ii

i ii

i ii

= =∑∑

∑∑

2 2

or the ratio of the second to the first moment about the origin.

The polydispersity index, Mw/Mn, is the ratio of the second moment to the square of the firstmoment about the origin.

Given Mw and Mn the standard deviation of a distribution of molecular weights can be determined:

σ = −

MM

Mnw

n

1

12

Several specific distribution functions are mentioned below. A distribution is usually consideredunimodal if one of these continuous distribution functions describes the distribution of measuredvalues. If the data is best described by several of these functions it is termed bimodal, trimodal etc.A bimodal distribution in lamellar thicknesses in polyethylene might be generated if crystallizationoccurred in two distinct steps such as primary crystallization and secondary epitaxial spheruliticdecoration for instance. This distribution might be described by two Gaussian functions, so a totalof 4 parameters, 2 means and 2 standard deviations.

Binomial Distribution:

Consider a tactic polymer such as polypropylene. As discussed in class, on passing along themain chain of the polymer there is a choice in handedness for the substitutent methyl group whichcould be considered similar to a coin toss, i.e. if the substitutent occurs with the same handednessas the previous mer unit this would be similar to a coin tossed heads (R for tacticity) and if itoccurs with the opposite handedness (S for tacticity) this would be similar to a coin tossed tails. Ifthe tacticity of the polymer is determined at random (atactic) then there is equal probability for Rand S handedness. The binomial distribution describes such a situation where the "probability ofsuccess" is given by p, (consider R a success, here p = 0.5 for atactic polymers). The probabilityof observing "n" R handedness mer units in a chain of N mer units when the probability of seeingan R handiness mer unit is p is given by:

PBinomial N n pN

n N np pn N n, ,

!

! !( ) =

−( )−( ) −1

4

the mean, µ, and standard deviation, σ, for the binomial distribution are given by:

µ = Np

σ 2 1= −( )Np p

PBinomial could be used to plot a distribution function of tacticities for example. Note that there is afinite probability for a completely R polymer (0! =1).

Poisson Distribution:

When N is large and the mean, µ, is constant with sample size, the binomial distribution simplifiesto the Poisson Distribution:

P nn

ePoisson

n

,!

µ µ µ( ) = −

where the standard deviation is given by:

σ µ=

The Poisson Distribution is used to describe small samples of large populations, so is appropriatein analytic techniques which involve counting of events, such as XRD and light counters, andmass spectrometers which measure events. The standard deviation in such a countingmeasurement (counting of events) is the square root of the number of counts. For example, anintegrated diffraction peak might generate 10,000 counts on a proportional detector. The error inthis value is ± 100. If an analytic instrument does not report "counts" or "events" then thePoission distribution value for the standard deviation can not be used directly.

Gaussian Distribution:

For very large samples, N, with a finite probability of success, p, a smooth distribution is usuallyobserved. Such a distribution can be described by a Gaussian function:

P nn

Gaussian , , expµ σσ π

µσ

( ) = − −

1

2

1

2

2

The Gaussian distribution is the basis of the "bell-shaped curve". It is also used to describerandom walks in diffusion as well as the path of a polymer coil under theta-conditions. Integrationof the Gaussian distribution as a weighting factor for r2, the square of the chains end-to-enddistance yields the mean square end-to-end distance for a "Gaussian" chain, Nl2, where l is the stepsize and N is the number of randomly arranged steps in the non-interacting chain. For theGaussian chain µ = 0 and σ2 = nl2.

Other Distributions:

There are many other distribution functions commonly used in polymer science such as theLorentzian Distribution (see hand out) and the Maxwellian Distribution (see Polymer MaterialsScience by J. Schultz for instance). The definition of these distributions will rely on anunderstanding of the moments of a continuous distribution described above.

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Covariance:

In many analytic experiments several parameters are determined from a single measurementthrough the use of a fit to experimental data. For instance, a light scattering curve can be used todetermine the molecular weight (first moment about the origin) and second viral coefficient, A2,through the Zimm plot (figure 2.8 in our text). Often the two parameters which are measured arenot completely independent in terms of the fit to the data. Under such conditions it is necessary todetermine the covariance of the two parameters. The covariance reflects the degree to which twoparameters effect each other. If more than two parameters are unknown, then a covariance matrixcan be constructed to determine the extent two which any two parameters are associated. For twoparameters the covariance is defined by:

σ uv u u v v2 = −( ) −( )( )where <> indicates a mean. Covariance is included for completeness here. Determination of thecovariance is generally rare in the literature.

Propagation of Errors:

In many analytic techniques a value is measured, an error is determined, and this value is used tocalculate a parameter of interest. For example, in the determination of the modulus of a sample theextension of the sample is measured with some error and is normalized against the original lengthto determine the engineering strain. The force applied to the sample is measured with anexperimental error and is normalized by the cross sectional area to determine the stress. These twoparameters are plotted (stress versus strain) and a curve fit is used to determine the modulus at lowstrain. In order to determine the standard deviation in the modulus the experimentally determinederrors in length, force and area must be propagated to the stress and strain and the error in theseparameters must be propagated to a linear fit of the data points. Propagation of errors is arudimentary tool necessary to perform polymer analysis.

Consider a measurement of the absorption, A, and absorption coefficient, α , using a singlewavelength of light which passes through a sample and a photomultiplier tube which reportscounts. The relevant equation is the Beer-Lambert Law for linear absorption (pp. 40 and 54 inCambell):

AI

I Tcl=

=

=ln ln0 1 α

where c is the concentration of absorbing species and l is the sample thickness. T is called thetransmission. For a solid sample c = 1. Two measurements are necessary, I0 with no sample andI with a sample of thickness l. If the two measurements are I0 = 100,000 counts , and I = 20,000counts, the respective standard deviation is given by a Poission Distribution function as the squareroot of the number of events,

I0 = 100,000 ± 320 counts; I = 20,000 ± 140 counts.

To propagate the error in I0 and I to 1/T the general error propagation rule in differential form canbe used:

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σ σ ∂∂

σ ∂∂

σ ∂∂

∂∂x u v uv

x

u

x

v

x

u

x

v2 2

22

222=

+

+

Here it is safe to assume that I and I0 are uncorrelated (unrelated) so the covariance, σuv = 0. The

two standard deviations are given above. For x = 1/T, δx/δI0 = 1/I, and δx/δI = -I0/I2. The above

equation yields:

σσ σ

12

2

202 2

40

/ TI I

I

I

I= +

or

σ σ σ12

2

2

02

2

210/

/T I I

T I I( )= +

this follows the general rule for ratios given by Bevington (see handout). The propagated valueand error for 1/T is, 1/T = 5 ± 0.04.

The absorption, A, is the natural log of 1/T and the propagated error is given by:

σ σ ∂∂

σA T

TA

T T2

12

2

12

21 1=

=( )/

/

( / ) /

So, A = 1.61 ± 0.01.

The sample thickness, l, is 0.1 ± 0.02 cm as calculated from a series of 10 measurements using themean and number of samples equation given above. The absorption coefficient, α , is calculated

from α = A/cl, where c is 1 for a solid sample. Replacing the variables in the equation above for

the error in 1/T, α = 16 ± 3 . Notice the reduction in the number of significantfigures associated with the large error.

It should also be noted that often the largest source of error is related to factors which are of minorsignificance to the measurement, e.g. here error in the sample thickness dominates the error in theabsorption coefficient.

Purpose of Error Analysis:

Although it is assumed that you can determine the error value for any quantity you measure inscience, the most important part of error analysis involves interpretation of the significance of theanalysis in view of the error and in view of logic and the reasonableness of the results. Erroranalysis is a critical factor in both demonstrating the scientific reasonableness of a result, as well asforming a basis of a scientific critique of work performed for you by a technician or outside lab.Error analysis is always at the heart of a scientific argument. A measured value has no meaningwithout an analysis of the associated error. Such an analysis can be quantitative, as above, orqualitative, based on your scientific judgment. The value in either case is based on thereasonableness and logic of the approach. In the remainder of this course special emphasis will begiven to qualitative and quantitative assessments of the error in the analytic techniques covered.

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Least-Squares Fits:

In order to determine analytic parameters it is often necessary to perform a curve fit to a raw dataset. For example the determination of modulus from a stress-strain measurement requires a linearfit of the type σ = Eε where s is the stress, σ = F/A, and e is the strain, ε = ∆l/l, and E is theYoungs modulus. The error in the stress measurement could be propagated from the estimatederror in the Force and area measurements. These errors can be propagated to the modulus in alinear fit through a least squares minimization of χ2. χ2 is a measure of the difference between theactual data points and the projected points associated with the fit parameter. It bears resemblance tothe variance,

χσ

2

2

21

=−( )∑ y f xi i

i

N ( )

The propagated uncertainty in the coefficients for the least squares fit can be obtained in a computerprogram by calculation of the second derivative of χ2 with respect to variation in the parameter:

σ∂ χ ∂a

jj a

22 2 2 2

2=

A full discussion of least-squares fitting routines and propagation of error is given in Bevington.For linear functions a relatively simple algorithm for propagation of error is given in the handout.

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Chapter 3. Thermal Analysis (Chapter 12 Campbell & White).

Polymers typically display broad melting endotherms and glass transitions as major analyticfeatures associated with their properties. Both the glass and melting transitions are stronglydependent on processing conditions and dispersion in structural and chemical properties ofplastics. Characterization of polymers requires a detailed analysis of these characteristic thermaltransitions using either differential scanning calorimeter (DSC) or differential thermal analysis(DTA). Additionally, polymers are viscoelastic materials with strong time and temperaturedependencies to their mechanical properties. Temperature scans across the dynamic spectrum ofmechanical absorptions are commonly required for characterization of polymers, especially forelastomers. These thermal/mechanical properties are characterized in dynamic mechanical/thermalanalysis (DMTA). Additionally, weight loss with heating is a common phenomena for polymersdue to degradation and loss of residual solvents and monomers. Weight loss on heating is studiedusing thermal gravimetric analysis (TGA).

A complete thermal analysis of a plastic sample yields inferential information concerning thechemical composition and structure of the material. Examples:

1. The Hoffman-Lauritzen description of the crystalline melting point associates shits in themelting transition temperature with the thickness of lamellar crystallites in polymers. Suchstructural based shifts would suggest further study using the Scherrer approach for diffraction peakbroadening and small-angle x-ray and TEM analysis.

2. Dramatic weight loss in a TGA analysis of nylon at temperatures above 100°C indicate someassociation of water with the nylon chemical structure. Such an observation would suggest furtherstudy using spectroscopic techniques.

3. In a polymer alloy (blend), the observation of two glass transition temperatures indicates abiphasic system, a single glass transition, a miscible system following the Flory-Fox equation.Further support for miscibility would come from microscopy and scattering (neutron, x-ray andlight can all be used to characterize miscibility).

Generally, thermal analysis is the easiest and most available of techniques to apply to a sample andfor this reason thermal analysis is often the first technique used to analytically describe a plasticmaterial.

Error analysis in thermal techniques usually is conducted by repetition of the measurement for atleast 3 to 10 identical samples in order to determine the standard deviation in the measurement. Inall thermal analysis techniques the instruments must be calibrated with standard samples displayingsharp and constant transition temperatures and enthalpies of transition.

Calorimetry (Differential Scanning Calorimetry, DSC; Differential ThermalAnalysis, DTA):

Calorimetry involves the measurement of relative changes in temperature and heat or energy eitherunder isothermal or adiabatic conditions. Chemical calorimetry where the heats of reaction aremeasured, usually involve isothermal conditions. Bomb or flame calorimeters involve adiabaticsystems where the change in temperature can be translated, using the heat capacity of the system,into the enthalpy or energy content of a material such as in determination of the calorie content offood. In materials characterization calorimetry usually involves an adiabatic measurement. Acalormetric measurement in materials science is carried out on a closed system where determinationof the heat, Q, associated with a change in temperature, ∆T, yields the heat capacity of the material,C:

2

CQ

T=

∆

At constant pressure:

dQ

dT

H

TC

P NP=

=∂∂ ,

The enthalpy can be calculated from CP through,

H T H T C dTP

T

T

( ) ( )= + ∫0

0

Instrumentation for Thermal Analysis, DTA and DSC:

Figure 12.1 of Campbell and White shows a schematic of a differential thermal analysis (DTA)instrument. The instrument is composed of two identical cells in which the sample and a reference(often an empty pan) are placed. Both cells are heated with a constant heat flux, Q, using a singleheater, and the temperatures of the two cells are measured as a function of time. If the sampleundergoes a thermal transition such as melting or glass transition a difference in temperature isobserved, ∆T = Tsample - Treference. Negative ∆T indicates an endotherm for a heating cycle.Quantitative analysis of DTA data is complicated and the instrument is usually viewed as a fairlycrude sibling of a differential scanning calorimeter (DSC) discussed below. Recent instrumentaladvancements have improved the quantitative use of DTA instruments. A DTA instrument isgenerally less expensive than a DSC. Determination of transition temperatures are accurate in aDTA. Estimates of enthalpies of transition are generally not accurate. In the DTA heat is providedat a constant rate and temperature is a dependent parameter. In the equation above for CP, thenormal order of dependent and independent parameters in the differential is reversed, so dT/dQ isactually measured rather than dQ/dT. This distinction is critically important in transitions wherekinetics become important such as in polymer melting and glass transition.

Figure 12.2 of Campbell and White shows a schematic of a differential scanning calorimeter(DSC). The arrangement is similar to the DTA except that the sample and reference are providedwith separate heaters. The independent parameter is the temperature which is ramped at acontrolled rate. Feedback loops control the feed of heat to the sample and reference so thetemperature program is closely followed. The raw data from a DSC is heat flux per time or poweras a function of temperature at a fixed rate of change of temperature (typically 10C°/min). Since theheat flux will increase with temperature ramp rate, higher heating rates lead to more sensitivethermal spectra. On the other hand, high heating rates lead to lower resolution of the temperatureof transition and can have consequences for transitions which display kinetic features.

Campbell and White go through a useful 2 page comparison of the DSC and DTA techniqueswhich should be reviewed.

Data Interpretation:

The output of a DSC is a plot of heat flux (rate) versus temperature at a specified temperature ramprate. The heat flux can be converted to CP by dividing by the constant rate of temperature change.The output from a DTA is temperature difference (∆T) between the reference and sample cellsversus sample temperature at a specified heat flux. Qualitatively the two plots appear similar.

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Both DSC's and DTA's must be calibrated, essentially, for each use since small changes in thesample cells (oil from fingers etc.) can significantly shift the instrumental calibration. For polymersamples these instruments are typically calibrated with low melting metal crystals that display asharp melting transition such as indium (Tm = 155.8°C) and low molecular weight organic crystalssuch as naphthalene. The volatility of low molecular weight organic crystals requires the use ofspecial sealed sample holders.

An instrumental time lag is always associated with scanning thermal analysis. The observedtransitions may be "smeared" by this instrumental time lag (typically close to 1°C at 10°/min heatingrate). Some account can be made for this time lag by comparison of results from different heatingrates. Often this time lag is accounted for by taking the onset of melting as the melting point ratherthan the peak value for sharp melting standard samples used in calibration. For polymer samples,significant broadening of the melting peak (up to 25 to 50C°) is the norm and this is associated withthe structural and kinetic features of polymer melting. Typically the peak value is reported forpolymer melting points. The instrumental error in temperature for a DSC is typically ±0.5 to 1.0C°.

Typical DSC trace for a Semi-Crystalline Polymer:

Endo

ther

mic

Temperature (°K)

Scan Rate 10°K/min

Glass Transition

Hysterisis

Cold Crystallization

Melting

LiquidSemi-CrystallineGlassy/Semi-Crystalline

The Figure above is a typical schematic for a heating run on a quenched sample of semi-crystallinepolymer such as polyethylene, polyester (such as PETE) or isotactic polystyrene (typical atacticpolystyrene does not display a crystalline endotherm). The left axis is (dH/dT), Cp, or heat fluxdepending on the normalization of the heat flux. Also, the left axis is often plotted with theendotherm pointing down rather than up, flipping the curve. The curve can change dramaticallywith heating rate especially with respect to the hysteresis of the glass transition (residual enthalpy)and cold crystallization phenomena. The mechanical properties of the sample change from a brittle

4

solid such as polystyrene at room temperature at the left, to a typical semi-crystalline material suchas polyethylene in a milk jug in the middle to a viscous liquid like molasses to the right.

Determination of T g : To determine Tg two lines are drawn parallel to the baseline above and below the inflection at Tg.The midpoint of the departure from the left and the intersection with the baseline nearest Tg on theright is determined by the instrument. This midpoint is taken as the glass transition temperature.The residual enthalpy associated with Tg is sometimes also recorded from the high temperaturebaseline. This residual enthalpy will decrease with heating rate and has a strong dependence on theprocessing conditions of the sample and the length of time the sample is allowed to anneal near Tg.Tg is a second order transition since it displays a discontinuity in the derivative of the enthalpy(heat capacity). It is often termed a pseudo-second order transition because it displays a finiterange of temperatures over which it occurs at finite heating rates and often displays a residualenthalpy.

Determination of T m : The melting point, Tm is determined, for broad melting polymers, by the temperature at themaximum in the (dH/dT) plot near this transition. The enthalpy of melting is determined byconstructing a baseline above the melting and extending it to below any cold crystallizationphenomena (exothermic peak below Tm and above Tg). Generally, the enthalpy of crystallization istaken as the area above this baseline. Sometimes the enthalpy of crystallization, which hasoccurred in the DSC measurement, is subtracted from this value to adjust the enthalpy as anestimate of the enthalpy associated with the original sample. The onset of melting is taken as theinitial rise of the curve above baseline. The maximum melting temperature at this heating rate istaken as the final point which deviates from the baseline.

Melting Endotherm:

Gibbs-Thompson Relationship for Polymers (Hoffman-Lauritzen Theory): (Appendix 1) In addition to the broad melting endotherms and the presence of cold crystallization as a dominantfeature in semi-crystalline polymers a number of unique complications exist in the interpretation ofcalometric data. A variant of the Gibbs-Thompson equation, known as the Hoffman-Lauritzenequation governs the relationship between structure and melting point for lamellar crystalliteswhich are present in essentially all semi-crystalline polymers. The Hoffman equation notes thatthere is a direct, inverse relationship between the undercooling at which the polymer crystallitesmelt and the thickness of the lamellar crystallites through:

LT

H TLe m

c

= +( )2σ ∂∆ ∆

where ∆T is the difference between the melting point of an infinite thickness and perfect crystallite

and the observed melting point, σe is the free energy associated with the lamellar fold surfaces and

δL is a term which accounts for the necessary deviation from equilibrium in the crystallization

process. The latter term is usually neglected, i.e. δL ≈ 0. The Hoffman approach gives rise to apossible description of the broad endotherm associated with melting of polymer crystallites, that ispolydispersity in lamellar thickness. In many plastics, particularly in branched polyethylene,multiple melting endotherms are observed and the Hoffman relationship has been used to supportthe presence of discrete populations of crystallites associated, perhaps, with initial crystallizationand formation of spherulitic structures, followed by decoration of spherulites by lower melting(i.e. thinner) lamellae.

5

A problem with the above analysis based on the Hoffman equation is that the degree of crystallinity(DOC) measured calorimetrically rarely agrees with that measured by XRD. XRD's DOC isusually taken as the best value since the myriad of complications which can be associated with athermal measurement are mostly avoided. The value of the DOC obtained from thermal analysis, ifall endotherms are included is usually higher than that of XRD which indicates that at least some ofthe endotherm measured calormetrically is associated with enthalpies of association present in theamorphous phase, such as tethered chains at the lamellar interface or interlamellar amorphouspolymer. In some highly branched polyethylenes, it has been proposed that the bulk of the meltingendotherm can be associated with such associated amorphous materials.

Melting Point Depression for Polymers: (Appendix 1) The presence of miscible, non-crystallizing components are known to depress the melting point ofall crystalline materials, i.e. adding salt to ice:

1 10T T

R

Ha

m m m

− = − ( )∆

ln

where a is the activity of the crystallizable component. Often the mole fraction of crystallizablecomponent is substituted for the activity as an ideal approximation. In polymers, the non-crystallizable component can be a low-molecular weight impurity, but is often a non-crystallizablecomonomer, an atactic segement, end-groups or a branch site. Substitution of the crystallizablefraction for "a" in the above equation yields the Sanchez-Eby Equation for copolymers. For lowfractions non-crystallizable component, such as when considering end-group effects the log termcan be approximated using, ln(a) ≈ ln(Xcry) = ln(1-Xnoncry) ≈ Xnoncry. For end-groups Xnoncry =2M0/Mn, where M0 is the molecular weight of the end-group and Mn is the number average chainmolecular weight. Thus, T0 for a polymer reflects the melting point of an infinite molecular weightsample.

For a thermodynamically miscible polymer or solvent at low concentrations the melting pointdepression can be expressed through a viral expansion where -ln(a) is replaced by a second orderviral expansion involving the interaction parameter, χ:

1 10

2

T T

R

H

V

Vm m m

mer

solventsolvent solvent− = −( )

∆φ χφ

where V represents molar volumes and φ is the volume fraction of solvent at low volume fractions.This equation is appropriate for melting point depression in the presence of a plasticizer forinstance.

The melting point of a semi-crystalline plastic sample is strongly effected by distributions inmorphology (lamellar thickness), impurity concentration and distribution in the sample, andprocessing history (thermal history) as well as the residual strain and orientation in the samples.All of these effects lead to a characteristically broad melting endotherm for plastics.

Glass Transition:

The melting point is a first order transition, in the Ehrenfest sense, since it involves a discontinuityin the first derivative of the Gibbs free energy with respect to temperature (-entropy) or pressure(volume). That is, there is a discontinuity in volume, for instance, at the melting point. The glass-transition, in the most ideal case, is a second order transition since the first derivatives of the Gibbsfree energy are not discontinuous but the second derivatives with respect to temperature (-heat

6

capacity/T), pressure (-volume * compressibility) and temperature/pressure (volume * thermalexpansion) show a discontinuity. Polymers are unique in the dominance of the glass transition asthe decisive factor in their mechanical properties. Polymers are the only material for which theequilibrium ground state is often glassy rather than crystalline. This is because topological andstereochemical constraints prevent the formation of crystals in many cases. The glass transition isoften called a pseudo-second order transition because of the dominance of kinetics. Slowercooling rates in the DSC, for instance, lead to lower measured values of the glass transition. Thistime-temperature superposition is described by the Williams Landel Ferry (WLF) equation forexample. This rate dependence hints at the basis of the glass transition in molecular motion. Theglass transition is effected by orientation, rate, molecular weight, crosslink density and impuritycontent.

Molecular Weight (Flory-Fox Approach):Through consideration of the free volume at the glass transition an expression can be obtained forthe molecular weight dependence of the glass transition temperature which is based on the idea thatchain ends lead to more free volume than mer units in the middle of chains:

T TK

Mg gn

= −∞,

where K is a constant (K = 2.1 x 105 for polystyrene, with Tg,∞ = 106°C) and Mn is the numberaverage molecular weight. The term K/Mn is proportional to the number of end groups as in themelting point equation above.

Polymer Blends (Fox Approach):The Fox equation describes the glass transition temperature of a miscible blend of two polymers, acopolymer or a plasticized polymer:

1 1

1

2

2T

m

T

m

Tg g g

= +, ,

where mi is the mass fraction of polymer "i". The Fox equation leads to a lower value of Tg thanwould be given by a simple linear rule of mixtures and reflects the effective higher free volume orrandomness due to the presence of two components in a mixture.

Systems which obey the Fox equation are considered to display intimate and uniform mixing whilethose which deviate from it, especially those that display two glass transition temperatures areconsidered to be poorly mixed.

Tacticity and T m /T g :Disubstituted vinyl polymers show dramatically different glass transition temperatures for differenttactic forms (e.g. polymethylmethacrylate, PMMA: isotactic 43°C, atactic 105°C, syndiotactic160°C). Monosubstituted vinyl polymers show a single glass transition temperature for thedifferent tactic forms (e.g. polystyrene, PS, 105°C for all tactic forms). Melting points aregenerally dramatically different for different tactic forms since the different tacticities, i.e. isotacticversus syndiotactic, show different crystalline structures.

Thermal Gravimetric Analysis (TGA):

Many DTA instruments include the ability to measure mass as a function of temperature as well asthe DTA output. In some cases such a thermal gravimetric analysis instrument is coupled with amass spectrometer or an infrared absorption instrument for analysis of decomposition gasses.

7

Typically, a superimposed plot of ∆Tsam-ref and weight can yield critical information concerning thechanges which occur on processing a polymer. For instance, the thermal cycling of a processingoperation can sometimes be mimicked in a DTA/TGA instrument to understand degradation andthermal transitions which effect the viscosity and other properties of a plastic. Often, a DTA/TGAanalysis is used to define the processing limits for a polymer, at the lower temperature associatedwith the glass transition or melting point and at the upper temperature associated with degradationof the polymer. Polymers which absorb water, such as nylon, have been studied in depth usingTGA instruments and an example of such a study is shown in figure 12.9 of Campbell and White.

Since the TGA instrument is fairly simple and self-explanatory it will not be extensively discussed.

Dynamic Mechanical Thermal Analysis (DMTA):

When a polymer is subjected to a forced mechanical vibration at a fixed frequency, temperature andelongation a fraction of the energy is absorbed and a fraction is returned elastically. This is evidentif a rubber ball is dropped on the floor. The ball bounces back to a height, L', lower than theheight from which it was dropped, L. The loss associated with the ball bouncing can be quantifiedas L" = L - L', for instance. If one were to vary temperature or speed of impact the reboundfraction would change. This will be demonstrated using a rubber ball cooled with liquid nitrogenin class.

In systems subjected to cyclic deformations or distortions, such as alternating current circuits, theloss term, L" is often associated with an imaginary component to a complex parameter, L*. So wewould write, L* = L' + i L" where i is √(-1). In complex space, a 2-D plot with the x-axis real andthe y-axis complex, the complex parameter L* is represented as a point and the angle, δ, from the

real or x-axis to a line from the origin to the point L* is given by: tanδ = L"/L'. tanδ is a"normalized" measure or the mechanical loss associated with the bouncing ball at a fixedtemperature and frequency (speed of impact).

Any mechanical or rheological measurement can be conducted in a dynamic mode, i.e. with asinusoidally oscillating force. For example, a tensile stress experiment involving an oscillatingforce, F(ω) = F0 sin(2πωt) where F0 is the amplitude of the force, ω is the frequency and t is the

time in the inverse units of ω, gives rise to a dynamic stress, σ∗ (ω) = F(ω)/A = σ0 sin(2πωt),

where A is the sample's cross sectional area. The resulting strain, ε∗ = L(ω)/L0, for a viscoelastic

material will display an in-phase component and an out of phase component, ε∗ = ε ' sin(2πωt) +

ε" cos(2πωt). More typically, the strain is the independent parameter, ε∗ = ε0 sin(2πωt) and thestress is a complex response, since the material response is often a function of the maximum strain,ε0.

A complex Young's modulus can then be constructed, E* = E' + i E", with an associated reducedloss, E"/E' = tanδ. Similarly, a complex viscosity can be constructed for a shear experiment.Figure 12.10 in Campbell and White shows several arrangements for dynamic mechanicalexperiments.

Figure 12.11 in Campbell and White shows a typical measurement involving a glass and acrystalline transition for a semi-crystalline polymer. Below the glass transition the polymer isglassy and the elastic response is similar to a glass marble, i.e. E" is small and E' is large so tanδ =E"/E' is small. Above the glass transition the material is elastic and E" is small and E' is smaller.tanδ is slightly higher but lower than at Tg. At Tg, molecular motion leads to significant absorption

8

at the frequency of the dynamic strain leading to a large E". E' is continuous through the transitionso a peak in tanδ at the glass transition results. In class we showed that a rubber ball behaves likea piece of lead at the glass transition, i.e. a highly lossy material.

The real modulus, E', of the semicrystalline material in Figure 12.11 decays with temperatureabove the glass transition until the crystalline melting point where the material becomes a liquid.tanδ shows a monotonic increase until it peaks at the crystalline melting point due to enhancedmolecular motion when crystals begin to melt.

In many polymers a number of absorption peaks are observed below the glass transition which canbe associated with different types of molecular motion. The glass transition in this scheme istermed the primary transition temperature or α-transition, Tα, and the secondary transitions are

labeled in sequence of decreasing temperature the β, γ, δ and ε transitions with associated

temperatures Tβ for instance.

One of the uses for DMTA is in understanding the mechanical response of elastomers such as thoseused in tires. In a tire high frequency (low-temperature) loss is often considered good since itenhances the grip of a tire at high speeds (high frequency). Low frequency (high-temperature) lossis sometimes considered bad since it increases wear and reduces gas mileage. Elastomers can betuned to broaden the mechanical absorption peak in certain temperature ranges by controllingchemical composition, chemical structure, and the structure of fillers such as carbon black whichcan make up more than half of the weight of a tire. The mechanical absorption spectrum is alsocritical to a wide range of plastics since the mechanical behavior of polymers varies with bothtemperature as well as speed of deformation through the time-temperature superposition principle.

Time-Temperature Superposition:

In class a sample of silly-putty was used to demonstrate time-temperature superposition. If sillyputty is left for a long period of time (minutes) it flows (liquid at long times, low frequency orhigh-temperatures). If a ball is made it will bounce (seconds, elastic at intermediate times,frequencies and temperatures). If silly putty is rapidly ripped apart it fails like a glass (glassy atvery short times, high-frequency or low temperatures). From this observation we can considerthat raising the temperature is similar to dropping the frequency or allowing more time fordeformation.

Williams-Landel-Ferry (WLF) considered the equivalency of time and temperature in the context offree volume theory for an activated flow process in viscoelastic materials. The WLF equationyields an equivalent frequency for a given temperature relative to a ground state temperature andexperimental frequency:

ln aC T T

C T TaT T=

− −( )+ −( ) =1 1 0

2 1 00 ω ω

where C1 and C2 are constants for a given polymer and T0 is a reference temperature for a givenpolymer close to Tg. The constants in the WLF equation can be theoretically predicted from free-volume theory (C1 = 17.44 and C2 = 51.6, using T0 = Tg) or can be experimentally determined.Using the WLF equation a master curve can be constructed in temperature that corresponds to astandard frequency or a master curve in frequency can be constructed that corresponds to astandard temperature.

9

Ideal Models in the DMTA Measurement:

For an applied dynamic strain, ε∗ = ε0 sin(2πωt), an ideal Hookean elastic will respond with acomplex stress completely in phase with the applied strain:

σ* = σ0 sin(2πωt) = E ε0 sin(2πωt).

For a Hookean elastic E" = 0

For a Newtonian Fluid, an ideal viscous fluid, the stress is given by:

σ = η(dε/dt).

dε/dt for the dynamic strain, ε∗ = ε0 sin(2πωt), is given by:

dε/dt = 2πω ε0 cos(2πωt),

so, for an ideal lossy material,

σ* = 2πω ε0 η cos(2πωt),

and E' = 0.

DMTA experiments can be performed on non-Newtonian fluids such as a polymer melt todetermine the elastic component of a visco-elastic fluid or on a solid plastic or rubber to determinethe viscous component of a visco-elastic solid.

Work of Dynamic Deformation:

The work performed in a DMTA measurement per unit sample volume per cycle, W, is given by:

W d E= = ( ) = ( )∫σ ε πσ ε δ π επ

* * sin "0 0

0

2

0

2

The power consumed per unit volume is given by,

P E= ( )πω ε0

2"

DMTA Measurement of Complex Shear Modulus:

One of the most common applications of DMTA measurements is the determination of the shearloss and storage modulus, G" and G' for a polymer melt. This measurement is critical to theunderstanding of polymer processing since a polymer melt is subjected to a variety of shear rates ina typical process. For example, in an extruder at early stages low shear melts polymer pellets andpressurizes the polymer fluid in the feed and melting zones of the extruder barrel. In the extruderdie this pressurized melt is subjected to extremely high rates of shear as it is formed into the finalextruded part. The energy consumed in the process is related to the complex shear modulus andthe variation of the loss shear modulus with rate. For a dynamic shear strain,

10

γ∗ = γ0ei2πωt = γ0 sin(2πωt)

where the shear strain is defined as the change in length of a fluid with respect to a normaldirection. The complex shear stress, force per normal area dragging the fluid is given byNewton's viscosity law by,

τ η γ πωη γ* **

* *= =d

dti2

The complex viscosity is written as,

η η η* ' "= −i

(compare with complex modulus which is written E* = E' + iE"). The difference between thisequation and that for the complex Young's Modulus occurs because the complex viscosity involvesthe measurement of the compliance of the fluid to an applied strain rate rather than the strain rate ofa material due to an applied stress. The compliance, D, is the inverse of the modulus, E. Forcomplex numbers the inverse is obtained by the ratio of the complex conjugate to the magnitude ofthe complex number:

DE

conj E

E E

E

E Ei

E

E ED iD*

*

( *)

' "

'

' "

"

' "' "= =

( ) + ( )=

( ) + ( )−

( ) + ( )= −1

2 2 2 2 2 2

The shear loss modulus, G", and shear storage modulus, G', are related to the dynamic viscosityη* = η ' - iη" by,

G' " '= 2 2πωη πωη G"=

The Cox-Merz Rule gives the zero shear rate viscosity, η 0, as the magnitude of the complexviscosity,

η η η η02 2

12= = ( ) + ( )[ ]* ' "

2 Qualitative Examples of the Use of DMTA in the Plastics Industry:

1) In the processing of high density polyethylene a dramatic increase in the processing cost isassociated with small amounts of long chain branching. This can be at levels below 5 branches per1000 carbons in the main chain so can not be characterized using spectroscopic techniques.Rheological DMTA has been used to estimate the amount of long chain branching since the high-frequency shear loss modulus, G", displays a characteristic increase relative to the intermediatefrequency loss modulus for branched materials. Presumably this effect is associated withhindrance of molecular motion at high frequencies due to the presence of long chain branches.There is good correlation between this measurement and the processing costs as well as somequantitative correlation between branch content and this mechanical response. DMTA is the onlyanalytic technique which has been demonstrated to be sensitive to such weak chain branching and ithas proven critical to control of synthesis conditions aimed at reduction of long chain branching inHDPE.

2) In the elastomer industry DMTA is a standard instrument for determination of the expectedperformance of rubber products. The most common example is a automotive tire which is subject

11

to a wide range of dynamic strains in use. The frequency dependence of loss in elastomercompounds can be directly related to the performance of an elastomeric material in an automotivetire. The critical design criterion, in terms of materials, for a tire are "grip", "wear" and "fueleconomy". The "grip" of a tire is usually associated with the high-frequency (low temperature)loss. The low-frequency (high temperature) loss is associated with the wear and loss of fueleconomy. The addition of carbon black to an elastomer enhances the high frequency (lowtemperature) loss, leading to higher grip. The weight fraction of carbon in a tire is typicallybetween 40 and 60%! In carbon reinforced elastomers, this higher grip is always associated withan increase in the low frequency response leading to an increase in energy absorption or reducedfuel economy for a tire as well as an associated increase in wear. The increase in loss with additionof carbon has been related to the association of elastomer molecules with filler, so called "bound-rubber content". Loss is also associated with breakup and re-formation of the loosely associatedaggregates of carbon. In the design of elastomers for tire applications the goal is often to broadenthe absorption peaks in the DMTA spectrum and to shift absorption's to higher frequencies throughmodification of the chemical structure of the elastomer and tuning of the filler structure.

Appendix 1: Gibbs-Thompson Equation and Melting Point Depression Equations

At the equilibrium melting point a crystal and its melt have equal Gibbs free energy, Gcrystal = Gmelt.For example, in a glass of water with ice cubes at close to 0°C the ice will fuse into a larger clusterdue to epitaxial recrystallization (crystal surface nucleated crystallization). This occurs becausethere is little or no difference in free energy between water molecules in the liquid and crystallinestates. An equilibrium melting point is defined as:

∆ ∆ ∆ ∆∆

G H T S or TH

S= = − =∞ ∞0

where ∆ is the difference between the crystal and the melt and T∞ reflects ideal conditions.

There are many ways to deviate from this ideal situation. For example, the ideal crystal above is ofinfinite size. For real crystals, where surface area becomes important at small crystallite sizes thecrystal will display an equilibrium with its melt at a different temperature due to the inclusion of asurface energy term:

V G V H T V S A or TV H A

V ST

AT

V Hm s ms s∆ ∆ ∆ ∆

∆ ∆= = − − = − = −∞

∞0 σ σ σ

A melting point depression is predicted for a finite sized crystal depending on the surface to volumeratio and the surface energy to bulk enthalpy ratio. For low surface energy crystals or for largecrystals the effect is smaller. For polymers, crystallites are highly asymmetric with roughly theaspect ratio of a sheet of paper. This means that the A to V ratio is very large. Additionally, theplanar surface of polymer crystals contain chain folds which have a high tortional energyassociated with them. This means that the surface energy to enthalpy ratio is also large. Largeunder coolings are expected in polymer crystals and a direct relationship between the lamellarthickness and the melting point is predicted by the Gibbs Thompson Equation given above.

For a system such as a collection of red and blue marbles which are shaken up the entropy of thesystem can be calculated from combinatorial (or counting) statistics. The difference in entropybetween a segregated state (all blue at the top and all red at the bottom) and a totally mixed state(blue and red randomly dispersed) is given by the total number of ways to arrange the marbles inthe dispersed state, Ω:

12

∆ ΩS R= − ln( )

If an impurity is added to a crystal and it has a totally uniform dispersion then a change in entropybetween the pure crystal and the crystal with the dispersant can be calculated form a combinatorialapproach which parallels the red and blue marble law above assuming that the number of statesincreases with the molar concentration of the impurity, x:

∆S R x= − ln( )

Using an approach similar to the Gibbs Thompson discussion above, a melting point depressioncan be predicted:

∆ ∆ ∆ ∆∆

G H T S RT x or TH

S R xm m m= = − − ( ) =+ ( )

0 lnln

rearranging:

1 1

T T

R x

Hm pure

= + ( )ln

∆

1

Chapter 4. Electromagnetic Radiation in Analysis (Chapter 3 Campbell & White).

Electromagnetic radiation is a disturbance in electro-magnetic space which follows Maxwell'sdifferential equations for conversion of energy from an electrical field to a magnetic field. Thedisturbance is sinusoidal in nature for propagating EM radiation.

Electromagnetic waves display a wave nature in that the oscillating electric and magnetic fieldspropagate with a wavelength and frequency. Additionally, a special property of EM radiation isthat it displays a fixed speed in vacuum, c, and fixed velocities in other uniform media soconversion from frequency to wavelength is direct:

ν =c/λ

We can specify a type of EM radiation by specifying it's wavelength, frequency, or wavenumber,k:

k = 1/λ

EM radiation also displays a particle characteristic through the concept of the photon which isa particle of no mass. This allows a means to describe features of EM radiation which are usuallyassociated with particles such as momentum.

The energy per photon of EM radiation is related to the frequency, ν, of the Maxwellian sinusoidaloscillation through Planck's constant, h:

Energy = hν

That is, higher frequency is associated with higher energy. Then the Energy is given by hc/λ anddifferent wavelength radiations contain different amounts of energy per photon. A photon is aquantum of EM radiation that displays momentum. The momentum is expressed as:

p = h/λ =hν/c

The brilliance, brightness, flux or intensity of a particular EM radiation is related to how manyphotons are delivered in a unit area per unit time. The energy of each photon is related directly tothe wavelength. It is important to be able to distinguish in your mind betweenintensity, energy and power delivered by EM radiation.

The energy of EM radiation determines the effect of the radiation on materials and is thebasis for understanding analytic techniques which use EM radiation. The intensity determinesthe amount or number of these effects which occur. For example, x-rays are of shortwavelength and have very high energy per photon. Because of this x-rays are a type of ionizingradiation which can cause cancer in humans through formation of free radicals in your DNA forinstance. Radio waves are of very large wavelength and low energy. Radio waves pass throughour bodies constantly with no effect. High intensity radio waves can cause changes in the spinpolarization of nuclei when the nuclei are under a very strong magnetic field and this is the basis ofnuclear magnetic resonance absorption which will be discussed in class.

2

Analytic Techniques Using EM Radiation: (Low Energy to High Energy)

NAME Wavelength Common Name Effect µm

Nuclear Magnetic Resonance 101 0 Radio Nuclear(NMR) Spin

Electron Spin Resonance 107 TV Electron Spin(ESR)

Microwave Absorption 2x105 Radar Rotate PolarMolecules

Infrared Absorption IR Far IR 15.4-830 IR BondIR 2.5 -15 .4 VibrationsNear IR 0 .7 -2 .5

Raman Spectroscopy IR

Elastic Light Scattering 0.4-0.7 Visible Changes inInelastic Light Scattering Light Polarization

of Molecules

UV absorption 0.01-0.4 UV-A, UV-B Move electronsin Orbitals

Elastic X-ray Scattering 10-4-10-2 X-Rays IonizingXRD, SAXS

Inelastic X-rayESCA

Ionization (SIMS) 10-6-10-4 Gamma-Rays Ionizing10-9-10-6 Cosmic Rays Ionizing

Features of Electromagnetic Waves:

1) EM waves are described using a sin function in time and space. That is, at a fixed time thewave varies in Amplitude, A, with spatial position and for a given position in space the wavevaries in amplitude with time. This is true of both the electric field vector, E, and the magneticfield vector, B.

E = A sin[2πx/λ -2πνt]B = B0 cos[2πx/λ -2πνt] Out of phase with E by 90°

E and A have direction. λ is the wavelength, ν is the frequency.

2) The frequency and wavelength are related by the velocity of the EM wave which in vacuum isc, "the speed of light".

λ = |c|/νc = 3 x 108 m/s

3) The amplitude is related to the "intensity" of the EM radiation:Intensity = |A|2

3

Intensity doesn't have direction, amplitude does.

4) The energy of EM radiation is directly related to the energy associated wi thphenomena by which it is made and is not associated with the intensity. Planck's law states:

Energy/photon = hν = hc/λh = 6.63x10-34 J sThe idea that EM radiation has an energy is tied to the particle view of EM radiation, i.e. there is aparticle called a photon which has no mass which carries the EM energy and has momentum, p.

p = hν/c = h/λ (c/c)

5) Light, X-rays and other EM radiations are generally composed of a number of photons so onecan consider the relationship between different waves in a beam.

i.) Coherent: If all waves in a beam are in-phase, that is have the same phase angle(peaks of waves coincide in space) they are called coherent. Waves which are notcoherent can interfere with each other leading to a reduction of the intensity. Forexample, a laser beam is coherent while a flash light beam is incoherent. This is one reasonwhy a laser beam can propagate over great distances while a flash light beam quicklydissipates.

When considering interference of two waves one adds or subtracts amplitudes of theelectric field vector E. The intensity which is measured is the square of the resultingamplitude, #3 above.

ii.) Collimated: Beams with waves which are all progressing in the same direction aretermed a well collimated beam. Collimation refers to the divergence of the waves in abeam. A light bulb produces uncollimated light which spreads in all directions. The sun'srays, when they reach earth are well collimated since the angular divergence is low. A laserbeam or a synchrotron x-ray beam are well collimated due to the mechanism by which theEM radiation is produced.

iii.) Monochromatic: If all waves have the same frequency (or wavelength by #2above) they are called monochromatic (one color). A source like a light bulb, the sun, oran x-ray tube generates polychromatic (white light) radiation (many wavelengths) and asource like a laser or an x-ray synchrotron yields monochromatic radiation.

The polychromacity of EM radiation is tied to the mechanism of formation. If theformation event is specific (quantum) in terms of the energy transfer associated with theformation event, monochromatic radiation results. If the formation event is statistical(distributed in energy) polychromatic radiation results.

iv.) Polarization: The electric field vector, E, for and EM wave has a direction in aplane normal to the propagation direction. If the direction is fixed relative to thepropagation direction and this direction is the same for all waves in a beam, the EMradiation is said to be linearly polarized. Polarization can be produced by a number ofmeans: Reflection off a surface leads to linear polarization in the plane of the surface,highly birefringent materials can lead to polarization of a beam by absorption ofcomponents not with a certain polarization, some processes for formation of EM radiationproduce polarized radiation laser and x-ray synchrotrons, in some cases a grating can beused to polarize radiation (Soller slits for x-rays), diffraction leads to polarized radiationdepending on the geometry of diffraction.

4

In addition to linear polarization, waves can be elliptically and circularly polarized. Incircularly polarized beams the vector E rotates in direction along the propagation direction.Elliptically polarized radiation is a mixture of circularly and linearly polarized radiations,i.e. there is some rotation of the vector E but it is not symmetric. In this class we will onlydiscuss unpolarized and linearly polarized radiation.

7) EM radiation can always be assumed to travel in a straight line.

8) EM radiation interacts with matter in different ways depending on the energy associated with aphoton. That is, energy decides what happens while intensity decides how muchhappens. Radio waves are low energy/high wavelength (see table above) and can pass throughmost materials with no effect. IR vibrates bonds and can generate heat. Light changes thepolarization of molecules which is a minor effect. UV can dissociate weak bonds and causedegradation. X-rays are a type of ionizing radiation that can ionize atoms and molecules.Typically, x-rays have wavelengths on the Ångstrom scale. Generally, the lower thewavelength of radiation the higher the danger due to the higher energy associated with shortwavelengths see #4 above. It is also more difficult to produce and use high-energy photons sincethey must result from an associated high energy event and they are absorbed by most materialsthrough the interactions mentioned above.

9) For visible light the index of refraction, n, is used to describe the speed of the radiation in amedium.

Velocity = v = c/n.

Since the velocity is always smaller than the velocity in vacuum, n is always greater than 1. In air,n is close to 1, in silica glass n = 1.52, some materials have a very high index of refraction,Titania, TiO2, is close to 5.

Beer-Lambert Law:

As a first approximation all EM radiation can be considered to follow a linear absorption behaviordescribed by an exponential decay of intensity with thickness of a sample and with concentration ofthe absorbing species. This is critical to quantitative analysis using EM absorption which is thebasis for IR, NMR, and UV spectroscopies. The Beer Lambert Law states that the value of theabsorption is proportional to the amount of the absorbing species.

In a typical absorption experiment, a beam of EM radiation passes through a sample of thickness"t". For a small thickness, dx, the incident intensity at wavelength λ and position x, I(x), isreduced by dI due to absorption by the sample. The change in intensity is proportional to thethickness, dx, the incident intensity I(x) and the linear absorption coefficient, A:

− = =dI AI x dx cI x dx( ) ( )α

For a thicker sample this equation can be rearranged and integrated:

− =

= − = =

=

=

=

=

−

∫ ∫dI

Ic dx

c t T e

I x

I x t

x

x t

c t

( )

( )

0 0

α

α αlnI

I or

I

I0 0

5

"T" is the transmission ratio for the sample at wavelength l, and A = cα is the Absorption. TheBeer-Lambert Law given above gives t A = ln(1/T). Absorption spectra are either plotted astransmission or as absorption on the y-axis. In a transmission plot the absorption peaks pointdown. In an absorption plot the absorption peaks point up and are the ln of 1/T as above.

Scattered Intensity:

For a scattering experiment such as XRD, SAXS, or Raman scattering the Intensity is plottedversus either and energy or angular term. The integrated intensity for a scattering peak is directlyproportional to the amount of the scattering material. A Raman scattering experiment results in datawhich looks very similar to an IR absorption experiment with both plotted versus wavenumber = k= 2π/λ. IR will be a plot of Transmittance (peaks down) or Absorption (peaks up) and a Ramanpattern will be a plot of Intensity (peaks up).

Instrumentation for Spectroscopy:

Dispersive Spectrometer:

Figure 4.1 of Campbell and White, pp. 44, shows a typical dispersive optics spectrometer (UV inthis case). Even in this simple instrument the optical paths are quite complicated. The firstrequirement of a spectrometer is resolution of the spectrum of an source through a tunable devicewhich can isolate a single wavelength radiation. In a dispersive instrument this can be done with aprism or with an optical grating. The prism disperses the incident radiation due to differences inthe index of refraction of the glass prism with wavelength, n(λ). Snell's Law can be used todetermine the angle of refraction of different wavelengths:

nAir sin(θAir) = nGlass sin(θGlass)

where θAir is the angle of the incident beam with respect to a normal to the surface of the glass, and

θGlass is the angle of the beam in the prism with respect to the glass surface normal. The doublerefraction of the prism serves to disperse the incident beam into angularly diverging beams ofdifferent wavelength. The prism can be rotated to select certain wavelengths using a slit. Anoptical grating can also be used to disperse wavelengths using Bragg's Law:

sin(θ)=λ/2d

where 2θ is the angle of divergence from the incident beam.

The main problem with dispersive instruments is that almost all of the incident radiation i slost in the slit arrangement for selecting a diverging beam of the desired wavelength. Themeasurement of a spectrum involves rotation of the prism or grating.

Most spectrometers involve double beam optics (shown in Figure 4.1) because,1) the source has a spectrum of its own,2) imperfections in the prism or grating might selectively effect certain wavelengths,3) the detector usually has a spectral sensitivity which varies with wavelength and4) the instrumental sensitivity might vary in time due to atmospheric interference and fluctuations inthe electronics.Double beam optics are achieved by a partially silvered mirror which effectively splits the incidentbeam into two beams, a Reference beam and the Sample beam. A reference cell is used which

6

duplicates the sample except that no sample is present. A rotating mirror is used to alternativelysample the reference and sample beams and the transmission is measured as the ratio of these twobeams using a photomultiplier (PM) tube which results in a number of counts or events. The rawcounting error for such a spectrometer is related to the square root of the number of countsmeasured by the PM tube. Usually, this counting error is much smaller than the error introducedby the instrumental setup such as the spectral width attainable with the slit/prism optics.

Fourier Transform Instruments:

The loss of information due to the dispersion/slit optics in a dispersive spectrometer was longknown to be a major hindrance to the determination of EM absorption spectra. With thedevelopment of the Michelson Interferometer a route to alleviate this problem became clear. Figure5.2 of Campbell and White, pp. 57, shows a schematic of a Fourier Transform IR absorptionspectrometer. The Fourier transform instrument results in an interference pattern between two EMbeams which is the Fourier equivalent of an absorption spectrum. Fourier transformmathematics are critical to IR and Raman spectrometers, NMR spectroscopy, andis the basis for scattering techniques such as XRD, SAXS and SALS we wi l lspend some time discussing FTIR instruments as a simple introduction to Fouriertechniques in Polymer Analysis.

Sample

FixedMirror

MovableMirror

Detector

x

I

x

Consider the instrument shown schematically above. Polychromatic radiation (see definitionabove) is incident on a sample and the intensity versus wavelength spectrum is modified byabsorption of radiation at certain characteristic wavelengths by the sample (see below). Apolychromatic spectrum is composed of a series of EM waves of different wavelengths, withdifferent amplitudes. This continuous distribution of wavelengths can be decomposed into the plot

7

shown below, i.e. Intensity = |A2| versus wavelength. That is, the intensity versus wavelengthplot is a population distribution plot for these sine waves. A basic premise of FourierTheory is that the components of this distribution can be treated independent o fthe other components. This approach allows us to consider a single sine wave from thepolydisperse distribution, and recombine the separate sine waves into the polychromatic spectrumafter considering interference and absorption for each wavelength independently.

Inte

nsit

y

Wavelength, λ

Sample Inte

nsit

y

Wavelength, λ

Consider a single monochromatic sine wave which enters the double mirror at the center of theFTIR schematic above. Identical sine waves propagate to the movable mirror and to the fixedmirror. If the distance traveled to the fixed mirror is L then the path length of the Fixed Mirrorbeam is 2L. The movable mirror is positioned at location L+x from the central double mirror. Thepath length for the movable mirror on return to the double mirror is 2L + 2x, and the path lengthdifference for the two beams at the double mirror is 2x. The two beams, one from the FixedMirror and the other from the Movable Mirror, are recombined as they progress directly upward inthe schematic. The combination of these two beams allows for interference to occur between thetwo beams, i.e. the phase difference 2π (2x/λ) will lead to some conditions of destructiveinterference between the two beams (Amplitude of one will be positive and the Amplitude of theother will be negative). Summing the amplitude will lead to a smaller total amplitude for mostcases, and this reduction in amplitude will be a cosine wave in 4πx/λ, with a maximum intensity =|A2| at x = 0.

1.0

0.5

0.0

-0.5

-1.0

Am

plit

ude

-40 -20 0 20 40

xThis function is the Fourier transform of a single wavelength, monochromatic, distribution. Thatis, if we decompose this function into a series of sine waves we find that the function isrepresented as a single value in the amplitude versus wavelength plot.

8

Fourier transformation results in inversion of the units of the independent axis .This means that the transformation of x will result in a distribution in wavenumber, i.e. 1/x =k.

For a monochromatic wave, the interferometer results in a cosine wave centered on x=0. Thiswould result if a laser were incident on the double mirrors, for instance. For a polychromatic wavea series of amplitude versus x plots would result with different wavelengths, across the spectrumof Intensity versus Wavelength shown above. The amplitudes of these polychromatic waves sumafter passing through the optics of the interferometer. The resulting sum is a damped cosine wavein x with a central peak at x=0. The decay of this cosine wave is not monotonic, i.e. If theamplitudes at the peaks are plotted against x there is no simple function which describes the decay.The decay contains all of the information concerning the spectra incident on theinterferometer. The information, however, can not be directly interpreted from the decaypattern since it is in "inverse space". Inverse space refers to the x-dimension here. This isinverse in units to the wavenumber, k. Real space in this experiment is wavenumber. These twospaces are related by a Fourier transform.

Am

plit

ude

x0

Inte

nsi

ty

Wavenumber, k=1/ λ

FourierTransform

The biggest advantage of a Fourier transform IR instrument over the dispersive instrumentdiscussed above is that all of the incident wavelengths are used in determining the spectrum, i.e.none of the incident radiation is disposed of. In the dispersive instrument almost all of the incidentradiation is removed by the slit, see above. The narrowest resolvable feature in wavenumber spaceis much smaller in a FT instrument. There are a number of other more complicated advantages tothe Fourier Transform instrument which will not be discussed here. Dispersive instruments arecheaper since they do not require a computer, a sub-micron translation stage, i.e. x has to be on thescale of the wavelength of light, and a number of other optical components needed in the FTIRinstrument. There is also some advantage to a dispersive instrument when the interest is on asingle or very narrow range of wavelengths. Pages 57-58 of Campbell and White discuss theFTIR instrument. The governing equation for the Fourier transform in FTIR's is given there as,

G I x x dxν πν( ) = ( ) ( )−∞

∞

∫ cos 2

(note error in the equation in Campbell). G is the intensity in frequency or wavenumber space andI is the intensity measured as a function of x. This integral states that the Intensity as a function ofν is a sum of cosine waves of frequency ν in x space. Fourier transforms are invertable so we canalso say that this equation implies that the sum of intensities in frequency space at a position x yieldthe intensity as a function of x. The basic idea is that intensity can be decomposed intoindependent sine or cosine components of different amplitudes in either x or frequency space.

9

Fourier Transforms in Other Analytic Techniques:

Several other analytic techniques rely on either Fourier Transform instruments or Fourier Theory tomeasure analytic features. The most common are NMR, XRD and SAXS.

In NMR an oscillating decay of radiowave emission in time results from a radiowave pulse in time.This is similar to the decay in tone of a guitar string after being plucked. Fourier transform of thistime signal results directly in a frequency space spectrum. This will be discussed in detail whenwe discuss NMR.

In Scattering techniques such as XRD and SAXS, measurements are made in angular space whichcan be converted to wavevector or "q" by q=4π/λ sin(θ) where θ is half the scattering angle. Theunits of q are inverse size and a Fourier transform of the intensity in q-space results in the radialdistribution function in real space (size space). We will discuss this when we discuss small anglescattering. This topic is also covered in Cullity in his Appendix on inverse space. (note Cullityuses "s" rather than "q". These two quantities are related by a constant factor of 2π.)

PDF File: (Click to Down Load): Chapter5.pdf

Polymer Analysis

= Back to TOC

= To Syllabus

Chapter 5. IR Spectroscopy and Raman Scattering

(Chapter 5 Campbell & White).

Bristol University IR Spectroscopy

Whitworth College IR/NMR problems

Scimedia on spectroscopy

CSU on IR

http://chipo.chem.uic.edu/web1/ocol/spec/IR.htm (Application/Chemistry)

http://www-wilson.ucsd.edu/education/spectroscopy/spectroscopy.html (Physics)

http://www.columbia.edu/cu/chemistry/edison/IRTutor.html (Spectra) Good Movies

http://avogadro.chem.iastate.edu/chem572/ (Chemistry)

Introduction:

The energy associated with electro-magnetic radiation in the infrared range (just above visible in wavelength) is sufficient to excite vibrationsof chemical bonds. IR spectroscopy and Raman scattering both involve IR wavelength radiation and both characterize vibrations of chemicalbonds. For this reason they are usually considered as a group although the instrumental details for the two techniques are significantlydifferent.

The vibration of any structure is analyzed in terms of the degrees of freedom which the structure possesses. For example, a sphere has 3degrees of transitional freedom and 0 degrees of rotational freedom since rotation does not result in a perceptibly different state. A singlesphere or a single atom does not have vibrational states. When two spheres are bonded the group has 3*2 degrees of translational freedom.

The grouping of 2 spheres, as a unit, possesses 3 degrees of translational freedom and 2 degrees of rotational freedom, since rotation about theaxis of the two spheres does not result in a perceptible change. When considering vibrational states we fix the reference frame on the groupingof objects, so the degrees of freedom for the grouping are subtracted from the total number of translational degrees of freedom for theindividual spheres. For two spheres there are 3*2 - (3+2) = 1 degree of vibrational freedom. This means that the IR/Raman spectra for adiatomic molecule such as CO will have one absorption band. This vibration would involve stretching and compressing of the CO bond. Forgroupings of spheres with more than 2 members the number of vibrational states is 3n-6 so a molecule such as water has 3 vibrational stateswhich results in three absorption bands in IR and Raman. These are symmetric stretching of the H-0 bonds, asymmetric stretching of the H-Obonds and a scissors bending of the HOH structure. The number of stretching vibrations is n-1 and the number of bending vibrations is 2n-5.

Carbon dioxide, O=C=O, is a linear molecule so the number of degrees of freedom are 3n-5 rather than 3n-6, i.e. one of the molecularrotational degrees of freedom, rotation about the molecular axis, does not result in a perceptible change.

The number of IR and Raman absorption bands is calculated from the number of degrees of translational freedom for the collection of atomsin a molecule, 3n, minus the number of degrees of translational and rotational freedom for the molecule as a whole, usually 6, but 5 for a linearmolecule.

These normal modes of vibration are useful for consideration of relatively small molecules, i.e. Benzene (C6H6) has 30 absorption bands in IRand Raman, each of which can be described in detail.

Consider a polymer molecule such as a 100,000 gm/mole sample of polystyrene. This molecule contains about 1,000 mer units or 16,000 atoms!The number of vibrational states for this molecule are 3*16,000 - 6 or about 50,000 different vibrations. It is impossible to identify all of thevibrational states for such a molecule. In polymer analysis we can greatly simplify the characteristic spectra for a chain by considering therepeating chemical groups which occur in the chain as independently contributing to the IR and Raman spectra. This approach is called thegroup contribution approach, and for polystyrene, would involve consideration of the major bands due to aromatic ring bands, C-Hvibrations, C-C and C=C vibrations. In the group contribution method many of the weaker bands are simply ignored and we concentrate onthe few high absorption bands which serve as a finger print for a particular polymer.

IR Active Bands:

The possible vibrations of a molecule are sensitive to IR absorption if the vibration results in a change in the dipole moment, u, of the molecule.The dipole moment is the product of the charge times distance and is similar to the moment of inertia in mechanics except that charge is theweighting factor rather than mass. When an EM wave in the IR wavelengths irradiates a molecule the electric field acts on the chargedistribution in the molecule, i.e. the more polar the molecule the larger the effect. The oscillation of the EM electric field, if of the quantizedfrequency for absorption by a particular bond, will set the bond in motion, vibrating at the specific frequency needed for that vibrationalexcitation. The IR absorption experiment involves the oscillating electric field changing the charge distribution so that a dipole is enhanced ordiminished. Strong IR absorption bands occur for polar groups such as OH, Cl, and the C=O bond.

In determining if a vibration is IR active consider if there is a change in the sum of the charge*distance vectors, i.e. for a symmetric stretch ofO=C=O (linear molecule) the movement of the left O is offset symmetrically by the movement of the right O so there is not net change in the charge* distance vector, not change in the dipole moment so this is not IR active. For a non-linear molecule like HOH (shaped like a V) there is a changein the dipole moment for a symmetric stretch so the vibration is IR active.

Raman Active Bands and The Raman Scattering Experiment:

The Raman scattering experiment involves shifts in the wavelength of an incident monochromatic beam. Raman scattering uses a laser as alight source (IR uses a mercury lamp or other broad spectrum source). The laser is usually in the optical wavelengths. The incident lightcauses motion of electrons in bonds. These moving electrons reemit light of the same wavelength for elastic scattering. Light scattering issensitive to the mobility of electrons in bonds. The mobility of electrons in a bond is called the Polarizability of the bond, i.e. a measure of howeasy it is to move electrons and polarize a bond. For bonds with a strong dipole moment (which are IR active) the mobility or polarizability isusually low. For bonds which have a weak dipole moment (which are IR inactive) the polarizability is usually high and the vibrational states ofthe bond are Raman active. IR and Raman activity are complimentary and the two techniques are used to fully characterize the vibrationalstates of molecules.

Raman scattering is based on a scattering event as described above. Figure 5.4 of Campbell and White shows a schematic of a Ramanspectrometer. A laser (usually an argon laser) is incident on a sample. Scattered light is collected usually at 90deg. to the incident beam. Thespectrum of the scattered light is measured using either a dispersive spectrometer or a Fourier transform spectrometer. Small shifts in thewavelength from the incident wavelength due to inelastic scattering are measured in the Raman spectrometer.

In the Raman measurement, an incident EM wave induces polarization of a bond through the EM wave's electric field. The energy of thisexcitation of the atom is h[nu]0, where [nu]0 is a larger frequency (and energy) than the IR range, [nu]vibration. There is a possibility that some ofthe energy of this excited state can be transferred to the atom in terms of a vibration of the bond, h[nu]vibration. Since the source of this energy isa scattering event, the absorption will be stronger for more polarizable bonds, i.e. bonds with more freedom of movement for the electrons.The loss of energy for the scattered EM wave due to transfer to a vibrational state for the bond is called a Stokes event and the resultingscattered wave is of higher wavelength and lower energy, EStokes = h[nu]0 - h[nu]vibration. The elastically scattered beam of energy ERayleigh = h[nu]0

is 10,000 times more intense than the Stokes line. It is also possible for an incident EM wave to interact with a bond which is alreadyvibrationally excited. In this case an Anti-Stokes line of higher energy results, EAnti-Stokes = h[nu]0 + h[nu]vibration. The anti-Stokes line is muchweaker than even the Stokes line. If a number of absorptions occur for a material, then the spectral distribution of scattered light can bemeasured and the shifts from the Rayleigh (elastic scattering) line converted to wavenumber. A spectrum very similar to an IR absorptionspectrum results.

In determining if a vibration is Raman active consider if there is a change in the volume of the electron cloud, i.e. for a symmetric stretch ofO=C=O (linear molecule) the movement of the left O is in the opposite direction of the movement of the right O so there is a net change in thevolume of the electron cloud within the molecule, this vibration is Raman active. For the asymmetric stretch the movements of the two O's are inthe same direction so the volume increase on the left is offset by a volume decrease on the right and the asymmetric stretch is not Raman active.

Since the source of the Raman spectrum is a scattering event, the Scattered Intensity is directly proportional to the concentration of speciesgiving rise to the Raman lines. This is different than in an IR absorption experiment which follows Beer's Law discussed in the previoussection.

The following two spectra compare the IR and Raman absorption from nylon-6,6: -(CO)-(CH2)6-(NH)-(CO)-(CH2)4-(CO)-n. NH stretch is thehighest wave number absorption. This is a polar bond so strongly absorbs in IR and weakly in Raman. CH stretch is a doublet below 3000cm-1

(asymmetric and symmetric) which is less polar than NH and has a strong absorption in Raman but a weaker absorption in IR. Carbonylstretch is a signature band (about 1750cm-1) in IR which is weak in Raman (highly polar and volumetrically inflexible bond).

From "Polymer Characterization" by B. J. Hunt and M. I. James, Blackie Press 1993.

Intensity and wavenumber of absorptions in IR and Raman

The wavenumber (energy or frequency) of an IR/Raman Absorption depends on the mass of the atoms connected to a chemical bond, thestrength of the chemical bond, and the geometry of the molecule. There are two basic types of vibrations, Stretches and Bends. Bends requireless energy so occur at lower frequencies for the same or similar bonds. There are generally two types of stretching, symmetric and asymmetric.There are many types of bends, Twisting, Rocking, Scissoring, Tortional, Breathing (For ring molecules) and other specialized bends.Symmetric stretches require lower energy than asymmetric stretches.

For a simple stretching vibration,

see: http://chipo.chem.uic.edu/web1/ocol/spec/IR1.htm p

where k is a spring constant for the tensile deformation of a bond (bond strength) and u is the geometric mean mass of the two atoms, u = m1

m2/(m1 + m2) at the ends of the bond. This is similar to Campbell and White's description of LAM modes in Raman for polymer crystals whichwe will discuss near the end of this chapter,

Campbell and White pp. 77 and R. G. Snyder, S. J. Krause, J. R. Scherer, J. Pollym. Sci. Polym. Phys. ED. 16 1593 (1978, where n= 1, 3, 5, 7, 9...

where L is the length of a sequence of C-C bonds, n is the order of the vibration (like the modes of a guitar string), [rho] is the density and E isthe Young's modulus for deformation of a series of C-C bonds. These equations quantify that higher mass leads to smaller frequencies andstronger bonds lead to higher frequencies of vibration. These equations also indicate that IR absorption is a critical tool for the quantitativedetermination of certain molecular features such as bond strength or bond modulus.

In most IR books one considers simple molecules first to gain a feel for the position of absorption peaks in the IR spectrum. Usually water andcarbon dioxide are discussed first. Often both water and CO2 are present as impurities in polymers so it is important to be able to identifythese bands which are not related to the material. Both of these molecules have 3 atoms. Water is not a linear molecule so 3n - 6 = 3 vibrationsare expected in the IR and Raman spectra. The three vibrations are shown below (movies of these vibrations are available from the internetsite mentioned).

Symmetric Stretch Asymmetric Stretch Symmetric Bend

3652 cm-1 3756 cm-1 712 cm-1

IR active IR active IR active

From: http://chipo.chem.uic.edu/web1/ocol/spec/IR1.htm (see main site above).

The three vibrations are symmetric and asymmetric stretching of the H (white ball) - C (red ball) bonds, and a symmetric bending (scissorsbend) of the H-C bonds. There is no asymmetric bend since such a vibration would result in a rotation of the molecule and no relative changeof the position of the atoms. These vibrations follow the general rule that there are N-1 stretches (3-1=2) and 2N-5 bends (6-5=1). Thesymmetric stretch is an easier deformation than the asymmetric stretch so the asymmetric stretch occurs at a higher wavenumber. Thebending vibration is much easier than stretching so this occurs at a much lower wavenumber.

Carbon dioxide also displays, 2 stretching and 1 bend vibration.

Symmetric Stretch Asymmetric Stretch Symmetric Bend

1340 cm-1 2350 cm-1 666 cm-1

Raman active IR active IR active

Not IR active

Chloroform and deutero-chloroform are a common example of the effect of mass on absorption bands.

Chloroform deutero-Chloroform

H-C or D-C stretch 3035 cm-1 2250 cm-1

Other Band (Deutero Bend) 1224 cm-1 910 cm-1

Cl Umbrella Bend 765 cm-1 740 cm-1

Group Contribution Method:

As noted above, the large number of atoms in a polymer chain makes the possible number of IR/Raman bands enormous, 3n-6, where n is onthe order of 5,000 to 10,000. Synthetic polymer chains are composed of repeated chemical groups, mer units, which are arranged about thechain axis in a similar fashion for all of these groups. The simplest approach to considering the IR/Raman absorption patterns from syntheticpolymers is to identify characteristic chemical groups which give rise to absorptions. This approach, of identifying chemical groups asindependent contributions to a complex IR/Raman pattern, is called the group contribution method. The basic assumption of the groupcontribution method is that vibrations from most chemical species are little effected by their bonding to the polymer chain. This approach isaccurate in the sense that absorptions for most chemical groups will fall in a limited range which can be distinguished from other absorptionsdue to the strength of the absorption, for example polar bonds have strong absorptions in IR, combined with the range of wavenumber wherethe absorptions occur.

Below are three "cheat sheets" for IR group contributions which you should be familiar with.

http://chipo.chem.uic.edu/web1/ocol/spec/IRTable.htm (See main site above).

This chart and the following table are from: From: http://chipo.chem.uic.edu/web1/ocol/spec/IRTable.htm (see main site above).

* 3700 - 2500 cm-1: X-H stretching (X = C, N, O, S)

* 2300 - 2000 cm-1: CX stretching (X = C or N)

* 1900 - 1500 cm-1: CX stretching (X = C, N, O)

* 1300 - 800 cm-1: C-X stretching (X = C, N, O)

http://chipo.chem.uic.edu/web1/ocol/spec/IRTable.htm (See main site above).

This chart and the following table are From a course on polymer chemistry by D. Tirrell and T. J. McCarthy.

C-H Stretch

Most polymers contain C-H bonds and this is a fairly polar bond so that a strong IR absorption band is usually observed for the C-H stretchnear 3000cm-1.

The Following two figures and text are from Paul R. Young's web page, The 4 examples serve to demonstrate the importance of the C-H stretch inidentification of organic materials such as polymers.

http://chipo.chem.uic.edu/web1/ocol/spec/IR1.htm

"The infrared spectrum of benzyl alcohol displays a broad, hydrogen-bonded OH stretching band in the region 3400 cm, a sharp unsaturated(sp) CH stretch at about 3010 cm and a saturated (sp) CH stretch at about 2900 cm; these bands are typical for alcohols and for aromaticcompounds containing some saturated carbon. Acetylene (ethyne) displays a typical terminal alkyne CH stretch, as shown in the second

panel."

Methylene, -(CH2)-: The spectra of nylon 6-6 above has an example of methylene CH stretch. This is a doublet to the right of 3000cm-1 (lowerwavenumber) 2926 (a) 2853(s). A scissors bend is also seen at about 1465cm-1.

Methyl, -(CH3): Methyl groups (such as in polypropylene, -(CH2)-(CH(CH3))-) also display a doublet in the CH stretch region just below3000cm-1, 2962 (a), 2872 (s). The bend vibration for methyl groups is a doublet (methylene is a singlet), 1450 (a) and 1375 (s). The symmetricbend for a methyl group is called an umbrella bend vibration for obvious reasons. Compare PE, PP and polyisobutylene IR spectra below.You should be able to distinguish these three spectra. Notice the broadening of the CH stretch region due to both methyl and methylenegroups in PP an PIB.

Near and between 1400 and 1500 PE has a single peak for the CH bend. For PP and PIB this methylene bend is present also. Below 1400 adoublet appears for PP and PIB indicating methyl bends in asymmetric and symmetric modes.

Other features in the fingerprint region (below 1500) are distinctive for the 3 polymers.

Alkenes, C=CHR: Unsaturation has a signature effect on the CH stretch shifting it to higher wavenumbers, 3020-3080 (to the left of 3000). It iseasy to identify unsaturated hydrocarbons in IR by the CH stretch region.

The figure below is an IR transmission spectra for polystyrene -(CH2)-(CH(C6H5))-. The aromatic ring, (C6H5), gives rise to the group of bandsabove 3010. The bands below 3010 are from the saturated main chain CH groups. Aromatics display a distinctive C=C stretch at about1600cm-1 which in combination with unsaturated CH stretch above 3000 identifies polymers containing aromatics. The ring breathingvibration at 1600 is always very sharp and strong.

Polyisoprene displays a broad CH stretch region associated with a mixture of saturated and unsaturated CH stretches, the broad band at

about 960 and the sharp band at 750 are associated with out of plane bends for the CH bond attached to the C=C bond. In simple alkenes thesetwo bands are used to distinguish between trans and cis stereoisomers, 960 for trans (H on opposite sides of the C=C bond) and 750 for cis (H

on same side of C=C bond).

Alkynes, C---C-H (triple bond): As the CC bond becomes stronger the CH stretch vibration goes to higher wave number (3300cm-1 foralkynes). A weak resonance at 2100 to 2200 for the triple bond occurs. The latter bond vibration has a low dipole change so is weak in IR butstrong in Raman.

Carbonyl, C=O: The C=O stretch is the most distinctive absorption in IR due to the high change in dipole moment on vibration and the uniquerange of wavenumber where this vibration occurs, 1700 to 1780cm-1. The carbonyl stretch occurs in many commodity polymers such aspolycarbonate, polyvinylacetate, polymethylacrylate and in nylon (see above, IR Raman comparison).

For acetone the C=O stretch occurs at 1724 (CH2O), for aldehydes (R-CH=O) at 1730 and for methyl acetates at 1745cm-1 (R-(C=O)-OCH3).For methyl acetates the O-CH3 stretch occurs at 1100 to 1280cm-1.

You should be able to identify the carbonyl contribution to the following spectra.

Alcohols, (-OH): The OH stretch occurs at 3500 to 3650 cm-1 (higher than CH). A C-O stretch occurs at 1100 to 1200 cm-1. These can beidentified in the spectra of polyvinyl alcohol below.

Amines, (-NH): The N-H stretch occurs between that of CH and OH. It is also intermediate in the strength of the IR absorption, see nylonRaman and IR patterns above.

Nitriles, (-C---N) triple bond CN: The CN is a strong absorption band which is seen in polyacrylonitrile (PAN) and copolymers with styrene(styrene acrylonitrile copolymers SAN). The absorption occurs in the 2200 to 2300 cm-1 range.

Halides, (C-Cl, C-F): The C-Cl stretch is a strong absorption at low wavenumber, 760-540 cm-1. Two examples of halide stretch from PVC -(CH2-CHCl)- and teflon -(CF2-CF2)- are shown below.

Other Examples from Web:

"Saturated and unsaturated CH bands also shown clearly in the spectrum of vinyl acetate (ethenyl ethanoate). This compound also shows atypical ester carbonyl at 1700 cm and a nice example of a carbon-carbon double bond stretch at about 1500 cm. Both of these bands are shifted

to slightly lower wave numbers than are typically observed (by about 50 cm) by conjugation involving the vinyl ester group."

Selected Applications of IR/Raman:

Tacticity and Crystallinity:

In some cases it is possible to assign certain absorption bands with tacticity in polymer chains. In most cases only a qualitative measure oftacticity is gained from IR and Raman spectroscopy. Figures 5.13 and 5.18 of Campbell and White and the following figure show thequalitative differences observed in IR/Raman spectra for tactic forms of polypropylene. In some cases specific bands are associated withspecific conformations which are possible in tactic forms. Many of these conformational differences disappear when samples are run in themelt. There is a significant overlap between tacticity and crystallinity determinations in IR analysis of polymers.

From "Polymer Characterization" by B. J. Hunt and M. I. James, Blackie Press 1993.

Raman spectra form polypropylene particles. Atactic: weak, diffuse Isotactic: sharp and strong. CH3 umbrella bend is to the left.

The presence of crystallinity also leads to predictable changes in IR patterns and has been used to qualitatively determine the degree ofcrystallinity for instance.

From "Polymer Characterization" by B. J. Hunt and M. I. James, Blackie Press 1993.

LAM Modes in Raman (Crystallite Thickness):

Low frequency regions (< 30cm-1 in polyethylene) have been associated with longitudinal acoustical modes (accordion modes) for the planarzigzag chain conformation in a lamellar crystallite. From the frequency of absorption, the length of such a planar zigzag chain can bedetermined using a simple model:

From R. G. Snyder, S. J. Krause, J. R. Scherer, J. Pollym. Sci. Polym. Phys. ED. 16 1593 (1978, where n = 1, 3, 5, 7, 9...

From R. G. Snyder, S. J. Krause, J. R. Scherer, J. Pollym. Sci. Polym. Phys. ED. 16 1593 (1978).

From R. G. Snyder, S. J. Krause, J. R. Scherer, J. Polym. Sci.; Polym. Phys. Ed. 16 1593 (1978).

The figure above shows a comparision between the LAM method and the use of small angle x-ray scattering (SAXS) to determing the lamellarthickness. From such a comparison the constants in the LAM equation can be determined. For PE the lamellar thickness, L ~ 6000/[nu].

Orientation:

If polarized radiation is used in IR/Raman it is possible to determine the relative orientation of specific absorbing groups in a processedsample. The usual way to do this is to determine the Herman's Orientation function, f, for the group of interest. f has a value of 0 forunoriented samples, 1 for samples oriented in the machine direction and -1/2 for samples oriented perpendicular to the machine direction butin the plane of observation. The absorption ratio for a given bond is measured by rotating the sample parallel and perpendicular to theincident direction of polarization, R = Aparallel/Aperpendicular. The angle between the bond axis and the polymer chain axis, a, needs to bedetermined from molecular models if chain orientation is of interest. The orientation function is then given by:

The first function is a generic description of the uniaxial Herman's orientation function. The second function is calculated for directionalabsorption in IR. The following figure from Hunt and James shows the complexity involved in calculation of IR absorption orientation. Wilkes

has written a good review article on this subject, G. Wilkes (1971), Adv. Polym. Sci., 8, 91.

From "Polymer Characterization" by B. J. Hunt and M. I. James, Blackie Press 1993.

The IR and Raman (lower) patterns below show the type of changes in IR and Raman patterns which are observed for oriented samples ofPET.

From "Polymer Characterization" by B. J. Hunt and M. I. James, Blackie Press 1993.

The orientation function can be calculated through a number of techniques some of which are shown in the following figure for PET as afunction of draw ratio.

From "Polymer Characterization" by B. J. Hunt and M. I. James, Blackie Press 1993.

Real time studies of deformation can yield information as to which chemical groups are involved in mechanical manipulation of samples. Thefollowing spectra are from continuous deformation of a polyether-polyurethane sample. These show the wealth of information which is

available in a rheo-optical study using IR/Raman.

From "Polymer Characterization" by B. J. Hunt and M. I. James, Blackie Press 1993.

= Back to TOC

= To Syllabus

1

Chapter 6. NMR Spectroscopy (Chapter 6 Campbell & White).

http://www.shu.ac.uk/schools/sci/chem/tutorials/molspec/nmr1.htmhttp://www.informatik.uni-frankfurt.de/~garrit/biowelt/nmr.htmlhttp://www-wilson.ucsd.edu/education/spectroscopy/nmr.html (Physics)

Introduction:

Nuclear magnetic resonance is an absorption spectroscopy involving the absorption of radiofrequency EM waves. Since we have already covered IR absorption spectroscopy it is appropriateto compare these two techniques, building on what we already know. The energy associated witha photon in the radio frequencies is extremely small compared to IR frequencies. In fact, we areconstantly being irradiated by radiowaves with no effect. Absorption spectroscopies rely on thetransfer of energy (hν) from an electro-magnetic wave to a quantized transition in a material. ForIR the quantized transition (a transition with a fixed energy) is the vibration of a chemical bond.

NMR involves changes in the spin state of the nucleus of an atom. Not all nuclei display spin.In order to display spin a nucleus must have an odd number of protons or neutrons. Hydrogen hasone proton, so displays spin (1H or proton NMR). Deuterium has one proton and one neutron andalso displays spin. Other atoms that display spin are isotopes of common elements (deuterium isan isotope of hydrogen). The most common are 13C, 19F, 15N and 29Si. An atom with spin has anon-zero spin quantum number, I. For most nuclei of interest to polymer scientists the spinquantum number is 1/2. Deuterium and Nitrogen 15 have spin quantum numbers of 1. Thenumber of spin states possible for a nucleus is given by 2I+1. For most nuclei ofinterest to polymer scientists there are two spin states. IR absorption also involves two states, i.e.vibrating and not vibrating. In NMR the two states correspond with two orientations of magneticmoment vector for the nucleus with respect to an external magnetic field as discussed below.

For a quantized transition to occur a system must be constrained, i.e. a guitar string must be undertension for the production of a note, atoms must be bonded for an IR absorption. In NMR theconstraint which leads to quantized transitions is applied to the sample externally in a large staticmagnetic field. If a nucleus spins and is composed of charged particles it possesses a magneticmoment associated with the angular velocity of the charged particles. The magnitude of themagnetic moment, µ, is proportional to the spin quantum number,

µ β= g IN N

gN is a constant and βN is the nuclear magneton, given by,

βπN

eh

mc=

4

where all the parameters are constant and related to a proton. The magnetic moment is a vector, µ,and in vector form it is defined as,

µ γ= I

where I is the angular momentum vector, given by.

Ih

I I= +( )2

1π

2

and γ is the magnetogyric ratio for the nucleus. The frequency of absorption in IR is directlyproportional to the magnetiogyric ratio,

υ γ πµ= =BB

Ih002

where B0 is a strong magnetic field which is applied to the sample.

The nucleus can then be thought of as a magnet, and in the absence of a magneticfield these tiny magnets are randomly arranged in a sample with no preferreddirection for the magnetic moment vectors. Application of a radiofrequency EMwave to such a sample has no effect, i.e. there is no absorption. There is notabsorption because the system can no tell the difference between magnetic vectorspointing up or down or any other direction since these directions have notreference base. This is analogous to a guitar string which is not constrained ortwo atoms which are not bonded in IR. In the absence of constraints there is noperceptible absorption.

If a strong magnetic field is applied to the nuclei, they can tell the difference between alignment inthe direction of the applied magnetic field and opposed to the applied magnetic field. In NMR theconstraint which leads to quantized transitions is applied by the spectrometer. Because of this thefrequency of absorption varies with the applied magnetic field and there is no absolute frequency orwavelength for a given absorption. NMR spectra are not plotted as absorption versus wavenumberas IR spectra are, they are plotted as absorption versus chemical shift, δ. The chemical shift forproton NMR is the difference between the frequency of absorption of the sample and a standard,tetramethylsilane (TMS) normalized by the frequency of absorption of TMS,

δυ υ

ν=

−Sample TMS

TMS

*106

δ is expressed in parts per million (ppm) so the above equation is multiplied by 106.

In IR we consider two states for a bond, vibrating and non-vibrating. The transition associatedwith the change from non-vibrating to vibrating leads to the absorption at fixed wavenumbers. InNMR several states are potentially possible depending on the spin quantum number. The permittedstates are given by the allowable values for the magnetic quantum number, mI = I, I-1,...-I. For I= 1/2 there are two states possible, mI = 1/2 and mI = -1/2. For I = 1 three states are possible, mI =1, 0, -1. For protons the two allowable states are generally spoken of as parallel and anti-parallelto the applied field.

Intensity of Absorption:

The strength of an IR absorption band depends on the change in dipole moment for the bond onvibration, i.e. how polar a bond is. The strength of a NMR absorption band depends onthe magnitude of the magnetogyric ration, γ, i.e. how large the magnetic dipolemoment is. The absorption in IR is also proportional to the concentration of the absorbing bond.NMR depends on the presence of specific isotopes. In considering the strength of a NMRabsorption band we consider the "natural abundance" of these isotopes. For example, 99.98percent of hydrogen atoms are 1H, and 0.0156 percent are deuterium, 2H. The magnetogyric ratiofor hydrogen is 26,700 while for deuterium is 4,100. This means that proton NMR (1H) results in

3

100 times the signal as deuterium if a sample contains natural abundance hydrogen. A similarcomparison shows that proton NMR absorption is about 50 times stronger than 13C NMR (naturalabundance of 13C is about 1%).

An NMR absorption peak for a given nucleus is directly proportional to thenumber of these atoms in a sample.

A major difference between analysis of NMR spectroscopy in polymers and IR spectra is that allabsorption bands in NMR are uniquely identifiable.

The NMR Experiment, Pulsed NMR:

The simplest NMR experiment is the observation of "free induction decay" from a radio frequencypulse. This is analogous to the plucking of a guitar string. The free induction decay looks similarto the raw data obtained form a FTIR instrument except that the x-axis is time rather than space.Fourier transform of the spectra results in inverse units, i.e. frequency, which is converted to δ.

N

S

NS

(1 )

N

S

NS

(2 )

RF Pulse90° to

B0

Prec

ess

ion

N

S

NS

(3 )

ObserveFree

InductionDecay

π/2Pulse

Free Induction Decay

Time

RF

Sig

nal

t 0 td

In the NMR experiment many pulses are applied and the FID patterns summed to enhance thesignal. The delay time between pulses, td, determines the smallest frequency which is observed.The width of the initial π/2 pulse, t0, determines the largest frequency which is observed. Thesummed decay pattern is Fourier transformed to obtain the spectrum in frequency.

4

The NMR instrument is capable of many other more complicated experiments involvingsequencing of pulses and observation of kinetic phenomena in spin relaxation which will bediscussed later in this chapter.

The nuclear dipole tilts at an angle θ with respect to the static magnetic field when the RF pulse is

applied. The magnetic dipole then rotates about the static field at this precession angle, θ. Theprecession angle is determined by the RF field strength, H1, the pulse time, t0, and themagnetogyric ration of the nucleus, γ.

θ γ= H t1 0

Position of Absorption Peaks in NMR Spectra:

Consider an isolated nucleus in a static magnetic field of B0 = 14,000 Gauss. The frequency ofabsorption for different nuclei varies according to the magnetogyric ratio,

Frequency in Mega Hertz (MHz)

1H19F31P13C14N 2H

60581382

This would be called a 60 megahertz NMR instrument because the proton NMR resonance is closeto 60 megahertz. A typical proton NMR spectra will go from 0 to 10 ppm meaning that the entirespectra will be 60MHz ± 0.0006 MHz. The absorption from other nuclei will not effect the protonspectra at all since their absorptions are at far different frequencies.

NMR is extremely sensitive to the "chemical environment" of a nucleus. "Chemical environment"means the local magnetic environment. The local magnetic environment is changed by "shielding"or "deshielding" depending on how the chemical bonds which are attached to the nucleus withdrawelectrons from the electron cloud of a bare atom. Electron withdrawing groups such as thearomatic ring, deshield a proton for instance and give rise to a deshielded proton with a large δ.Tetramethyl silane (TMS) has highly shielded protons so is used as a standard for the 0 point of aNMR spectra. The only absolute point for a given nucleus would be a nucleus stripped of allelectrons. Since it is not possible to obtain such a completely deshielded nucleus TMS is used as astandard.

5

0-10Shielded

Deshielded

δ ppm

TMS

I

H

H

HH

H

H

H

H

H

Phenet hyl Iodide

The chemical environment of protons changes the frequency of absorption as shown in the cartoonof the absorption spectra for phenethyl iodide above. TMS is highly shielded so is a good 0 pointfor frequency.

The "Chemical Environment" reflects the bonding geometry of a molecule. Two protons withidentical chemical environments "see" the same view of the molecule from their position in thebonding structure. The same view includes the stereochemical arrangement of the molecule, i.e.the handedness of the structure. The "view" which effects proton absorption is three bonds indistance, that is, structural differences more than three bonds away don't matter. Additionally, thepresence of resonance structures such as the aromatic ring in phenethyl iodide makes all of thearomatic protons see the same chemical environment.

The aromatic protons in phenethyl iodide are identical. The methyl iodide group at the other end ofphenethyl iodide has two protons with identical chemical environments, so give rise to a singlepeak at the far right of the spectrum. The two methylene protons have different chemicalenvironments because of their stereochemical relationship to the methyl iodide group.

Protons with the same chemical environment give rise to a single peak, so simple or symmetricmolecules have single peaks in proton NMR spectra. Examples are water, benzene andcyclohexane. The aromatic ring in benzene is strongly electron withdrawing, so the protons arestrongly deshielded. The oxygen in water is also electron withdrawing and deshielding. Thecyclohexane ring is only weakly electron withdrawing so weakly deshielding in proton NMR.

6

0-1.4-5.2-7.3

HH

H

H

H

H

O

H H

H

H

H

HH

H

H

H

H

H

H

H

Benzene Water Cyclohexane

Si

C C

CC

HH

H

H

H

HH

H

HH

H

H

TMS

The intensity in proton NMR is directly proportional to the number of protons in the structure withidentical chemical environments. Toluene has 5 aromatic protons (-7.3ppm) and 3 methyl protons(-2.5 ppm) with an absorption intensity ratio of 5:3.

CH

H

H

H

H

H

H

H

Toluene

0-2.5-7.3

5

3

7

Multiplet Splittings:

Nuclei are magnetically coupled through 3 or fewer bonds. Since each proton can exist in one oftwo states, I = 1/2 or I = -1/2, neighboring protons (3 or fewer bonds away) will create 2 magneticenvironments for a given proton. A single absorption peak will be split into two peaks for a singleneighboring proton. The number of splittings is given by n+1 where n is the numberof neighboring (within 3 bonds) magnetically equivalent protons. If there aretwo types of neighboring (within 3 bonds) protons the number of splittingsmultiplies. (The general rule is 2nI + 1 splittings.) These multiplet splittings will be centered onthe normal absorption frequency or δ of the proton. For example, in isopropyl benzene the 6methyl proton band will be split into two bands of equal intensity by the single methylene proton.The 6 methylene protons will split the single methylene proton into 2nI+1 absorption bands or 7bands (I = 1/2).

Definition of Magnetic Equivalence:

Protons are equivalent if:1) Same chemical shift, δ.2) Same coupling constant to all nuclei in the molecule, JHa.

Examples:(CH3)2CHBr 2 types CH3 and CHPropene CH3CHCH2

H3C

C

H

C

Hcis

Htrans 4 types Hcis, Htrans, CH3, CH

Magnetic Equivalence Means Same Magnetic EnvironmentThere is not way to magnetically distinguish the nucleiThe Nuclei's view of the molecule is the same both in terms of bonds and in terms ofstereochemistry. Nuclei can only sense 3 bonds away, with no conjugated double or triple bondsbetween them and can only sense other nuclei with similar nuclear magnetic transitions.

8

CH

H

H

H

H

H

C

C

H

H

H

H

H

H

Iso-Propyl Benzene

0-1-3-7.3The frequency of the distance between splittings of an absorption band i sindependent of the applied field. This is called the J coupling constant. The J spin couplingconstant is an absolute value for a given pair of nuclei, i.e. JHMethyl is a fixed value on a frequencyscale (but not on a δ scale). The intensity of the split bands follows the binomial distribution (n isthe number of chemically identical protons splitting a given band):

n Relative Intensity of split bands.1 1 12 1 2 13 1 3 3 14 1 4 6 4 15 1 5 10 10 5 16 1 6 15 20 15 6 1

For coupling by non-chemically identical protons the binomial distributions multiply, i.e. forsplitting by two non-chemically identical protons 4 bands of equal intensity would result.

Unlike IR, all absorption bands in an NMR spectrum can be completelyidentified. This makes NMR an extremely powerful tool for chemical identification. NMR isalso extremely flexible since you have complete control over the constraint which leads to theabsorption. The drawback to NMR is the cost of the instrument, $250,000 to $1e6, and theexpertise needed for more than routine identification. The cost and expertise are roughly both anorder of magnitude higher than IR.

Example: Ethanol

9

Chemical Shift Series:

Shielding is directly related to the electron density around the nucleus. There are a number ofhomologous series (series varying a chemical group to observe the chemical shift) whichdemonstrate this.

C

H

H

H

C

H

H

X

e-

δ-

δ+

δ+

Chemical Shifts/Electronegativity for CH3CH 2X X Electronegativity( χ ) Chemical Shift δ for CH 2

-SiEt3 1.9 0.6-H 2.2 0.75-CEt3 2.5 1.3-NEt2 3.0 2.4-OEt 3.5 3.3-F 4.0 4.0

Double bonds and Rings with conjugated bonds (high electron orbitals) deshield associatedprotons:

Compound Structure Chemi cal Shift δ for CH 2

ethane CH3CH3 0.9ethylene CH2=CH2 5.0 (Current)ethyne HC = CH 2.3 (No Current)

10

Benzene φ-H 7.3 (Current)

Parts of the NMR Spectra:

In summary the NMR spectra is composed of absorption peaks at a value of chemical shift, δ,relative to a reference material, usually TMS. The amplitude is proportional to the number ofmagnetically equivalent protons of that type in the molecule. When neighboring protons (3 bondsor less away) are present the peak amplitude is split into several peaks following the binomialdistribution and separated by the J coupling constant which is fixed in value, in terms offrequency, of the J coupling (not δ) according to the type of proton which is causing the splitting(see figure below):

From Hunt and James, "Polymer Characterization"

Tacticity:

The tetrahedral bond of carbons can be thought of as a tripod with a bond sticking straight up. If apolymer chain is attached to the bond sticking straight up and the chain is attached to one of thelegs of the tripod then substitutent groups attached to the other two legs have a choice of beingplaced to the right or left. This can be depicted in a Neuman projection along the chain back bonewhere the circle and three line apex are two carbons along the main chain connected by a bond.

11

PolymerChain

PolymerChain

Ha Hb

H Cl

The presence of the Cl substitutent group allows for the distinction of Ha and Hb. H a is gaucheto Cl and Hb is anti to the Cl group. For each mer unit in this polyvinylchloride moleculethere will be an anti and a gauche methylene proton. The relationship between these mer unitstereo chemistries gives rise to tacticity. Because of this the smallest grouping for tacticity is adiad, two mer units.

Diad Tacticity:

For two substitutent groups in a three carbon sequence the substitutents can be located with thesame handedness (Meso) or with opposite handedness (Racemic).

C

Cl H

C

Ha

Hb

C

Cl

H

Gaucheto Cl's

Anti to Cl's

Mesoa and b not equivalent

C

Cl H

C

Ha

Hb

C

H

Cl

Racemica and b equivalent

Both are Gauche to 1and Anti to other

12

From an NMR perspective, Racemic diads give rise to two magnetically equivalent protons on themethylene group while Meso diads give rise to two magnetically different protons on the methylenegroup.

NMR can not sense diad tacticity because the proton on the substituted carbon cansense two diads (on either side). The smallest unit of tacticity which NMR candetect in polymers is a triad (3 mer units). Because of this triad tacticity is theusual way to refer to polymer stereochemistry. NMR can also sense higher oddnumber groupings of tactic mer units, with diminishing resolution, pentads ( 5 ) ,heptads (7) etc.

C C

R

H

H

H

C C

R

H

H

H

C C

R

H

H

H If H "sees"

this merH also "sees"

this mer

Note that the handedness, anti or gauche, is independent of rotation about the C-C bonds whichoccurs in all single bond chains. The particular arrangement shown above is merely forconvenience of comparison between meso and racemic diads and does not reflect the actualconformation of the chains in a polymer which would be reflected by a distribution of rotationalorientations reflecting the potential energy diagram for C-C bond rotation as show below forpolyethylene (PE does not display tacticity),

C

C

H H

H H

Pot

enti

al

Ener

gy

Rot ati on

Triad Tacticity:

A triad is composed of two diads which share a central mer unit. There are three possibilities fortriad tacticity based on diad tacticity:

Isotactic, meso+meso mm 1Syndiotactic, racemic+ racemic rr 1Heterotactic, racemic + meso or meso + racemic rm or mr 2

13

A polymer with no preferred tacticity, an atactic polymer, has a random statistical distribution ofdiad tacticities so it would have 25% isotactic triads, 25% syndiotactic triads and 50% heterotactictriads. A polymer with 50% meso and 50% racemic diads does not necessarily have an atactic triaddistribution, just as an atactic triad distribution does not necessarily have an atactic (random) pentaddistribution. An atactic (random) pentad distribution does imply atactic triad and diad distributions.

Since syndiotactic is composed of two racemic units, and because the protons in a racemic diad aremagnetically equivalent (see above), then syndiotactic triads will have the fewest number ofmagnetic types of protons and the fewest peaks and splittings. This is shown for PMMA in figure6.6 of Campbell and White shown below (isotactic top, syndiotactic bottom):

C CPolymer Polymer

H

H

C

C H

H H

OO

C

H

H

H

(α)(β) PMMA

α-CH3

β-CH2

α-COOCH3

β-CH2

α-CH3α-COOCH3

From Campbell and White "Polymer Characterization"

PMMA was one of the first polymers studied in depth for tacticity using proton NMR (see texts byBovey from the 1970's). For syndiotactic PMMA (bottom) 3 main absorptions are observed, α-

CH3 at 0.91, β-CH2 at 1.9 and α-COOCH3 at 3.6. For isotactic polymer (top curve) the α-CH3 is

more deshielded 1.20, the β-CH2 becomes a quartet centered at 1.9 and the α-COOCH3 remains a

singlet at 3.6. The splittings of the β-CH2 in what should be a sequence of 1:1:1:1 is due to twotypes of methylene groups, termed erythro, e, (more deshielded) and threo, t, (less deshielded)corresponding to the bottom and top protons in the molecular sketch above. Each of these peaksare split into two peaks by the other leading to and expected splitting of 4 equal peaks, with areported J coupling constant of about 0.2 ppm. The e and t protons are separated by 0.7 ppmwhich can be verified by molecular modeling.

A higher resolution NMR can resolve higher order stereosequences as shown below for isotacticand atactic PMMA. You should compare the information content of the 60 MHz spectrum above tothe 500MHz spectra below. (Again, 60 MHz refers to the natural resonance frequency of a protonfor a given magnetic field of the instrument, ν α B0.)

14

From Hunt and James "Polymer Characterization"

15

From Hunt and James "Polymer Characterization"

A similar comparison of signal can be made for the 100 and 500 MHz spectra of polyvinyl chloridegiven below. β-methylene protons occur at 1.5-2.5 range and the single α-methine proton isobserved at about 4.6. In many polymers resolution of splittings for individual stereochemicalpeaks is not possible and this is illustrated by the PVC spectra. In such cases the tactic sequencescan be identified with complicated NMR techniques such as 2-D NMR (see Campbell and White).Generally you will find a reference which has identified the peaks associated with certain tacticgroupings for vinyl polymers and use the integrated areas of these peaks to determine the triadtacticity of a given polymer. (Note that the PMMA example given above is the best resolvedspectra of commodity polymers, i.e. sharpest peaks, and this is related to the structure of PMMA).

C C

H

H

H

Cl

PolymerPolymer(α)(β)

Methine

Methylene

16

From Campbell and White, "Polymer Characterization"

From Hunt and James "Polymer Characterization"

The splittings of the tactic peaks in the proton NMR spectrum of PVC, shown above, are notresolvable on typical NMR spectrometers. Use of a different nucleus, 13C, can overcomeproblems with resolution of this type. The 125 MHz, 13C spectra for PVC is shown below.Notice the higher resolution even compared to the 500 MHz proton spectra shown above. (Notethat spectrometer magnetic field strength is in reference to the proton resonance frequency even if adifferent nuclei is probed.)

From Hunt and James "Polymer Characterization"

13C NMR:

17

1) Low Natural Abundance: Since most polymers are composed of hydrogen and carbon, thenatural alternative nucleus for NMR is 13C. There are a number of major differences betweenproton and carbon 13 NMR. First, the natural abundance of 13C is much lower than 1H (12C doesnot display spin since the number of protons and neutrons are both even). The natural abundanceof 13C is about 1.1 % while that of 1H is close to 100%. Since only nuclei of similar magneticresonance can lead to coupling and splitting of the absorption peaks, the low natural abundance of13C leads to no splittings of the absorption peaks. The sensitivity of absorption of a RF pulse andthe associated decay are also much lower for 13C.

2) Large Chemical Shifts: The range of proton absorptions are on the order of 10ppm relative toTMS. For 13C the range of absorptions are on the order of 200ppm relative to TMS. The 13Cspectrum has more than an order higher resolution when compared to 1H spectra as can be seen inthe PVC spectra above.

3) The large abundance of 1H nuclei compared with 13C leads to loss of 13C resolution and signaldue to weak coupling of 13C and 1H resonances. This problem is amplified in solid samples, socalled solid state 13C NMR.

Cross Polarization: The low abundance of 13C leads to poor absorption of the RF pulse in a FT-NMR experiment.This limitation can be over come by exciting the protons in a sample followed by a sequence of twoseries of long-time pulses which make the 13C and 1H nuclei resonate at the same frequency. Thelatter is called the "Hartman-Hahn" condition and the process is called "cross-polarization" and thetime of cross polarization is called the "contact time" or "spin-lock time". This cross polarizationacts as a strong pulse for the carbon 13 nuclei (see figure below).

From Hunt and James "Polymer Characterization"

Proton Decoupling:

Cross-polarization leads to a large enhancement of the excitation of 13C nuclei. The large numberof 1H in the sample, however, interfere with the decay of the isolated 13C nuclei due to weakinteraction of the spins. For example, this would be like trying to play a guitar under water, that is

18

despite the difference in resonance frequency for a swimming pool full of water and the guitarstring, there is transfer of energy to the pool from the guitar. This dampening of the 13C signal canbe removed by a strong radio frequency signal which essentially holds the protons in a highlyresonating state so they are not capable of absorbing resonance from 13C nuclei. Cross polarizationand spin decoupling were critical developments for the wide use of 13C NMR. The figure belowshows the effect of proton decoupling on a carbon 13 NMR signal.

From Hunt and James "Polymer Characterization"

Magic Angle Spinning and Solid State 13 C NMR:

All of the previous discussion was based on "solution NMR" where a polymer sample is dissolvedin a solvent at 1 to 20% concentration. One reason for studying polymers in solution is that theanisotropy of the magnetic moment with respect to the macromolecule (Chemical ShiftAnisotropy, CSA) is averaged out due to thermal motion of the molecule in solution. Chemicalshift anisotropy has the effect of smearing out the NMR signal as can be seen by comparison of thebottom and second to the bottom spectra in the previous figure. In the solid state, i.e. a semi-crystalline or glassy polymer, CSA has a severe effect on the spectra in broadening the absorptionpeaks and the effect becomes worse the higher the restriction of mobility of the chains ormolecules. Through a tensoral analysis of the magnetic moments in a molecule it is possible todemonstrate that a "Magic Angle" exists with respect to the applied magnetic field at which rapid

19

spinning of the solid sample leads to minimization of absorption line broadening due to chemicalshift anisotropy.

Examples of 13 C NMR spectra:

Several examples of 13C NMR spectra from Campbell and White, and Hunt and James are givenbelow. The PMMA spectra should be compared with the proton NMR spectra given above.

From Campbell and White "Polymer Characterization"

From Campbell and White "Polymer Characterization"

From Campbell and White "Polymer Characterization"

20

From Hunt and James "Polymer Characterization"

Other Nuclei:

A number of plastics and elastomers are based on nuclei other than 13C or 1H. Two examples aregiven below, 29Si (natural abundance 4.7%) and 19F (natural abundance 100.0%). Silicon 29NMR is conducted similar to carbon 13 NMR while Fluorine 19 parallels closely proton NMR.The advantage of using these alternative nuclei is that the degree of C substitution on Si can bedirectly determined and a much higher resolution of fluorinated tacticity can be determined.

21

From Hunt and James "Polymer Characterization"

From Hunt and James "Polymer Characterization"

1

Chapter 7. XRD (Chapter 8 Campbell & White, Alexander "X-ray Diffraction Methods in Polymer Science").

The general principles of diffraction are covered in Cullity, "Elements of X-ray Diffraction". Ifyou are unfamiliar with XRD you will need to review or read Cullity Chapters 1-7 and theappendices. Alexander's text referenced above is also useful as an introduction to XRD but is lessgeneral and at a slightly more advanced level.

There are a number of differences between x-ray diffraction in polymers and metallurgical (Cullity)or ceramic diffraction.

1) Polymers are not highly absorbing to x-rays. The dominant experiment is a transmissionexperiment where the x-ray beam passes through the sample. This greatly simplifies analysis ofdiffraction spectra for polymers but requires somewhat specialized diffractometer from thosecommonly used for metallurgy (usually a reflection experiment).

From Alexander, "X-Ray Diffraction Methods in Polymer Science"

For transmission geometry the optimal sample thickness is 1/µ where µ is the linear absorptioncoefficient. Typically the optimal thickness for a hydrocarbon polymer is 2 mm. (See Cullity forcalculation of optimal thickness for a diffraction sample in transmission).

2) DOC: Polymers are never 100% crystalline. XRD is a primary technique to determine thedegree of crystallinity in polymers.

3) Synthetic polymers almost never occur as single crystals. The diffraction pattern frompolymers is almost always either a "powder" pattern (polycrystalline) or a fiber pattern (orientedpolycrystalline). (Electron diffraction in a TEM is an exception to this rule in some cases.)

2

4) Microstructure: Crystallite size in polymers is usually on the nano-scale in the thicknessdirection. The size of crystallites can be determined using variants of the Scherrer equation.

5) Orientation: Polymers, due to their long chain structure, are highly susceptible toorientation. XRD is a primary tool for the determination of crystalline orientation through theHermans orientation function.

6) Polymer crystals display a relatively large number of defects in some cases. This leads todiffraction peak broadening (see Campbell and White or Alexander for details).

7) Polymer crystallites are very small with a large surface to volume ratio which enhances thecontribution of interfacial disorganization on the diffraction pattern.

8) SAXS: Due to the nano-scale size of polymer crystallites, small-angle scattering isintense in semi-crystalline polymers and a separate field of analysis based on diffraction at anglesbelow 6° has developed (see Alexander and Chapter 8 of these notes for details).

Introduction:

Diffraction or scattering is a separate category of analytic techniques using electromagneticradiation where the interference of radiation arising from structural features is observed. Theinterference pattern is the Fourier transform of the pair wise correlation function. The pairwise correlation function can be constructed in a though experiment where a multiphase material isstatistically described by a line throwing experiment. If lines of length "r" are thrown in to a 2phase material there is a probability that both ends of the lines fall in the dilute phase. Thisprobability in 3-d space changes with the size of the line, "r", and a plot of this probability as afunction of "r" is a plot of the pair wise distribution function. For a crystal the two phases areatoms and voids and peaks in the pair wise correlation function occur at multiples of the latticespacing. Interference which results from correlations of different domains or atoms is usuallyassociated with the "Structure Factor" or "Interference Factor", S2(2θ). Interference can also occurif the individual domains are prefect structures such as spheres. For a sphere, there is a sharpdecay in the pair wise correlation function near the diameter of the sphere and this sharp decayresults in a peak in the Fourier transform of the correlation function. For a metal crystal thiscorresponds to the atomic form factor, f2(2θ). For larger scale domains interference associated

with the form of the scattering units is generally termed the "form factor", F2(2θ).

The scattered intensity as a function of angle is then the product of two terms, the form factor(f2(2θ) or F2(2θ)) and the structure factor (S2 (2θ)):

I(2θ) = Constant F2(2θ) S2 (2θ)

For XRD the form factor is usually obtained from tabulated values and the major interest is in theStructure factor. For small angle scattering dilute conditions are usually of interest making thestructure factor go to a constant value of 1 and the form factor for complex structures areinvestigated.

Thus, the basic principles of scattering and diffraction are the same, while the implementation ofthese principles are quite different.

Bragg's Law:

3

Cullity and Alexander derive Bragg's Law using the mirror analogy (specular analogy). It can alsobe derived from interference laws or using "inverse space" (see appendix in Cullity). The featuresof Bragg's Law is that structural size is inversely proportional to a reduced scattering angle, sohigh angle relates to smaller structure and low angle relates to large structure. Small-anglescattering measures colloidal to nano-scale sizes. There is no large scale limit to diffraction. Thesmall scale limit (i.e. the smallest measurable size) is λ/2 as is inherent in Bragg's Law:

d = λ/2 (1/sinθ)

θ is half of the scattering angle measured from the incident beam. The 1/sinθ term in Bragg's law

acts as an amplification factor. The minimum value of which is 1 for 2θ = 180° (direct backscattering). The maximum value of the amplification factor is ∞ so that theoretically no size limitexists with a given radiation of wave length λ . In reality the diffraction geometry and coherencelength of the radiation leads to a large scale limit on the micron scale.

Typically diffracted intensity if plotted as a function of 2θ. Since the d-spacing is of interest one

might wonder why diffraction data isn't plotted as a function of sinθ or 1/sinθ. This is in fact

done with the use of the "scattering vector" q or s. q = 4π/λ sin(θ) = 2π/d and s = 2/λ sin(θ) =1/d. The appendix of Cullity gives a good description of diffraction in "q" or "s" reciprocal space.

The Fourier transform of the real space vector, "r", used to determine the pair wise correlationfunction is the scattering vector "q".

Review of Crystalline Polymer Morphology:

"Molecular" scale Crystalline Structure:

Consider that we can form an all-trans oligmeric polyethylene sample an bring it below thecrystallization temperature. The molecules will be in the minimum energy state and will bein a planar zigzag form. These molecular sheets, when viewed from end will look like aline just as viewing a rigid strip from the end will appear as a line.

Crystal systems are described by lattice parameters (for review see Cullity X-ray Diffraction forinstance). A unit cell consists of three size parameters, a,b,c and three angles α, β, γ.Cells are categorized into 14 Bravis Lattices which can be categorized by symmetry forinstance. All unit cells fall into one of the Bravis Lattices. Typically, simple molecules andatoms form highly symmetric unit cells such as simple cubic (a=b=c, α=β=γ=90°) orvariants such as Face Center Cubic or Body Centered Cubic. The highest density crystal isformed equivalently by FCC and Hexagonal Closest Packed (HCP) crystal structures.These are the crystal structures chosen by extremely simple systems such as colloidalcrystals. Also, Proteins will usually crystallize into one of these closest packed forms.This is because the collapsed protein structure (the whole protein) crystallizes as a unit celllattice site. In some cases it is possible to manipulate protein molecules to crystallize inlamellar crystals but this is extremely difficult.

As the unit cell lattice site becomes more complicated and/or becomes capable of bonding indifferent ways in different directions the Bravis lattice becomes more complicated, i.e. lesssymmetric. This is true for oligomeric organic molecules. For example olefins (such asdodecane (n=12) and squalene (n=112)) crystallize into an orthorhombic unit cells which

4

have a, b and c different while α=β=γ=90°. The reason a, b and c are different is thedifferent bonding mechanisms in the different directions. This is reflected in vastlydifferent thermal expansion coefficients in the different directions. The orthorhombicstructure of olefinic crystals is shown below. Two chains make up the unit cell lattice site(shown in bold). The direction of the planar zigzag (or helix) in a polymer crystal isalways the c-axis by convention.

a

bc

PE/Olefin crystal structure.

See also, Campbell and White figure 8.4. Chain Folding:

The planar zigzag of the olefin or PE molecule crystallize as shown above into an orthogonal unitcell. This unit cell can be termed the first or primary level of structure for the olefin crystal.Consider a metal crystal such as the FCC structure of copper. The copper atoms diffuse tothe closest packed crystal planes and the crystal grows in 3-dimensions along low-indexcrystal faces until some kinetic feature interferes with growth. In a pure melt with lowthermal quench and careful control over the growth front through removal of the growingcrystal from the melt, a single crystal can be formed. Generally, for a metal crystal there isno particular limitation which would lead to asymmetric growth of the crystallite and fairlysymmetric crystals result.

This should be compared with the growth of helical structures such as linear oligomeric olefins,figure 4.1 on pp. 143 of Strobl. Here there is a natural limitation of growth in the c-axisdirection due to finite chain length. This leads to a strongly preferred c-axis thickness forthese oligomers which increases with chain length. In fact, a trace of chain length versuscrystallite thickness is a jagged curve due to the differing arrangement of odd and evenolefins, but the general progression is linear towards thicker crystals for longer chains untilabout 100 mer units where the curve plateaus out at a maximum value for a given quenchdepth. (Quench depth is the difference between the equilibrium melting point for a perfectcrystal and the temperature at which the material is crystallized.)

5

Number of mer units

Cry

stal

lite

Thi

ckn

ess Depends on

Quench Depth∆T

Schematic of olefin crystallite thickness as a function of the chain length.

The point in the curve where the crystallite thickness reaches a plateau value in molecular weight isclose to the molecular weight where chains begin to entangle with each other in the melt andthere is some association between these two phenomena. Also, the fact that this plateauthickness has a strong inverse quench depth dependence suggests that there is someentropic feature to this behavior (pp. 163 eqn. 4.20 where dc is the crystallite thickness andpp. 164 figure 4.18 Strobl).

Considering a random model for chain structure such as shown in figure 2.5 on pp. 21 as well asthe rotational isomeric state model for formation of the planar zigzag structure in PE, pp. 15figure 2.2, it should be clear that entropy favors some bending of the rigid linear structure,and that this is allowed, with some energy penalty associated with gauche conformation offigure 2.2. Put another way, for chains of a certain length (Close to the entanglementmolecular weight) there is a high-statistical probability that the chains will bend even belowthe crystallization temperature where the planar zigzag conformation is preferred for PE.When chains bend there is a local free energy penalty which must be paid and this can beincluded in a free energy balance in terms of a fold-surface energy if it is considered thatthese bends are locally confined to the crystallite surface as shown on pp. 161 figure 4.15;and pp. 185 figure 4.34.

There are many different crystalline structures which can be formed under different processingconditions for semi-crystalline polymers (Figures 4.2- 4.7 pp. 145 to 149; figure 4.13 pp.157; Figure 4.19, pp. 165; figure 4.21 pp. 170). As a class these variable crystallineforms have only two universal characteristics:

1) Unit cell structure as discussed above.2) Relationship between lamellar thickness and quench depth.

This means that understanding the relationship between quench depth and crystallite thickness isone of only two concrete features for polymer crystals. John Hoffman was the first todescribe this relationship although his derivation of a crystallite thickness law borrowedheavily on asymmetric growth models form low molecular weight, particularly ceramic anmetallurgical systems. Hoffman's law is given in equation 4.23 on pp. 166:

nT

H T Tm e f

mf

f

* ,=−( )∞

∞

2σ∆

, Hoffman Law

6

where n* is the thickness of the equilibrium crystal crystallized at T (which is below theequilibrium melting point for a crystal of infinite thickness, Tf

∞), σ is the excess surface

free energy associated with folded chains at the lateral surface of platelet crystals, and ∆His the heat of fusion associated with one monomer.

Hoffman's law can be obtained very quickly for a free energy balance following the "Gibbs-Thomson Approach" (Strobl pp. 166) if on considers that the crystals will formasymmetrically due to entropically required chain folds and that the surface energy for thefold surface is much higher than that for the c-axis sides..

σ

R

t

At the equilibrium melting point ∆G∞ = 0 = ∆H - T∞ ∆S, so ∆S = ∆H/ T∞. At some temperature, T, below the equilibrium melting point, The volumetric change in free energy

for crystallization ∆fT = ∆H - T ∆S = ∆H(1 - T/T∞) = ∆H(T∞ - T)/ T∞.

The crystallite crystallized at "T" is in equilibrium with its melt and this equilibrium state is adjustedby adjusting the thickness of the crystallite using the surface energy, that is,

∆GT = 4Rt σside+ 2R2 σ - R2t ∆fT = 0 at T. That is, At T the crystallite of thickness "t" is in equilibrium with its melt and this equilibrium is

determined by the asymmetry of the crystallite, t/R. If ∆fT = ∆H(T∞ - T)/ T∞. is use in thisexpression,

4t σside+ 2R σ = R t ∆H(T∞ - T)/ T∞.

Assuming that σside <<< σ, and "t"<<<"R" then,

t = 2 σ T∞./( ∆H(T∞ - T))which is the Hoffman law.

The deeper the quench, (T∞ - T), the thinner the crystal and for a crystal crystallized at T∞, thecrystallite is of infinite thickness. (Crystallization does not occur at T∞).

Nature of the Chain Fold Surface:

In addition to determination of T∞, the specific nature of the lamellar interface in terms of molecularconformation is of critical importance to the Hoffman analysis. There are several limitingexamples, 1) Regular Adjacent Reentry, 2) Switchboard Model (Non-Adjacent

7

Reentry), 3) Irregular Adjacent Reentry (Thickness of interfacial layer is proportionalto the temperature).

The synoptic or comprehensive model involves interconnection between neighboringlamellae through a combination of adjacent and Switchboard models.

The interzonal model involves non-adjacent reentry but considers a region at the interface wherethe chains are not randomly arranged, effectively creating a three phase system, crystalline,amorphous and interzonal.

Several distinguishing features of the lamellar interfaces are characteristic of each of these models.Adjacent Uniform and Thin Fold Surface High Surface EnergySwitchboard Random chains at interface, Broad interface, Low Surface EnergyIrregular Adjacent Temperature Dependent interfacial thickness Intermediate Surface EnergyInterzonal Extremely Broad and diffuse interfaces with non-random interfacial chainsSynoptic Interfacial properties are variable depending on state of entanglement and

speed of crystallization.

8

The Hoffman equation states that the lamellar thickness is proportional to the interfacial energy sowe can say that Adjacent reentry favors thicker lamellae since adjacent reentry has thehighest interfacial energy and the more random interfacial regions should display thinnerlamellae.

Colloidal Scale Structure in Semi-Crystalline Polymers:

Lamellae crystallized in dilute solution by precipitation can form pyramid shaped crystallites whichare essentially single lamellar crystals (figure 4.21 for example). Pyramids form due tochain tilt in the lamellae which leads to a strained crystal if growth proceeds in 2dimensions only. In some cases these lamellae (which have an aspect ratio similar to asheet of paper) can stack although this is usually a weak feature in solution crystallizedpolymers.

Lamellae crystallized from a melt show a dramatically different colloidal morphology as shown infigure 4.30 pp. 182, 4.13 on pp. 157, 4.7 on pp. 149, 4.6 on pp. 148, 4.4 and 4.5 on pp.147 and 4.2 on pp. 145. In these micrographs the lamellae tend to stack into fibrillarstructures. The stacking period is usually extremely regular and this period is called thelong period of the crystallites.

Long Period

Amorphous

Crystalline

The long period is so regular that diffraction occurs from regularly spaced lamellae at very smallangles using x-rays. Small-angle x-ray scattering is a primary technique to describe thecolloidal scale structure of such stacked lamellae. The lamellae are 2-d objects so a smallangle pattern is multiplied by q2 to remove this dimensionality (Lorentzian correction) andthe peak position in q is measured, q*. q= 4π/λ sin(θ/2), where θ is the scattering angle.

Bragg's law can be used to determine the long period, L = 2π/q*. Figure 4.8 on pp. 151shows such Lorentzian corrected data. The peak occurs at about 0.2 degrees! In somecases the x-ray data has been Fourier transformed to obtain a correlation function for thelamellae which indicate an average lamellar profile as shown in figure 4.9 pp. 152.

The degree of stacking of lamellae would appear to be a direct function of the density ofcrystallization, i.e. in lower crystallinity systems stacking is less prominent, and the extentof entanglement of the polymer chains in the melt. You can think of lamellar stacking asresulting from a reeling in of the lamellae as chains which bridge different lamellae furthercrystallize as well as a consequence of spatial constraints in densely crystallized systems.

In melt crystallized systems, many lamellar stacks tend to nucleate from a single nucleation site andgrow radially out until they impinge on other lamellar stacks growing from other nucleationsites. The lamellar stacks have a dominant direction of growth, that is, they are laterallyconstrained in extent, so that they form ribbon like fibers. The lateral constraint in meltcrystallized polymers is primarily a consequence of exclusion of impurities from thegrowing crystallites.

9

Fibrillar Growth Front

Excluded Impuriti es

Excluded Impuriti esNucleat ion

Sit e

"Impurities" include a number of things such as dirt, dust, chain segments of improper tacticity,branched segments, end-groups and other chain features which can not crystallize at thetemperature of crystallization. Some of these "Impurities" will crystallize at a lowertemperature so it is possible to have secondary crystallization occur in the interfibrillarregion. Despite the complexity of the "impurities" it can be postulated that the impuritiesdisplay an average diffusion constant, D. The Fibrillar growth front displays a lineargrowth rate, G. Fick's first law states that the flux of a material, J, is equal to the negativeof the diffusion constant times the concentration gradient ∆c/∆x. If we make an associationbetween the flux of impurities and the growth rate of the fibril then Fick's first law can beused to associate a size scale, ∆x with the ratio of D/G. This approach can be used to

define a parameter δ, which is known as the Keith and Padden δ-parameter, δ = D/G. Thisrule implies that faster growth rate will lead to narrower fibrils. Also, the inclusion of highmolecular weight impurities, which have a high diffusion constant, D, leads to widerfibrils. There is extensive, albeit qualitative, data supporting the Keith and Padden delparameter approach to describe the coarseness of spherulitic growth in this respect.

Branching of Fibrils: Dendrites versus Spherulites.

Low molecular weight materials such as water can grow in dendrite crystalline habits which insome ways resemble polymer spherulites (collections of fibrillar crystallites which emergefrom a nucleation site). One major qualitative difference is that dendritic crystalline habitsare very loose structures while spherulitic structures, such as shown in Strobl, fill space indense branching. At first this difference might seem to be qualitative.

10

120°

In low-molecular weight materials such as snowflakes or ice crystallites branching always occursalong low index crystallographic planes (low Miller indices). In spherulitic growth there isno relationship between the crystallographic planes and the direction of branching. It hasbeen proposed that this may be related to twinning phenomena or to epitaxial nucleation ofa new lamellar crystallite on the surface of an existing lamellae. A definitive reason fornon-crystallographic branching in polymer spherulites has not been determined but itremains a distinguishing feature between spherulites and dendrites.

(Incidentally, the growth of dendrites can occur due to similar impurity transport issues as thegrowth of fibrillar habits in polymers. In some cases a similar mechanism has beenproposed where rather than impurity diffusion, the asymmetric growth is caused by thermaltransport as heat is built up following the arrows in the diagram on the previous page.)

Non-crystallographic branching leads to the extremely dense fibrillar growth seen in figures 4.4 to4.7 of Strobl. In the absence of non-crystallographic branching, many of the mechanicalproperties of semi-crystalline polymers would not be possible. As was mentioned above,non-crystallographic branching may be related to the high asymmetry and the associatedhigh surface area of the chain fold surface which serves as a likely site for nucleation ofnew lamellae as will be discussed in detail below in the context of Hoffman/Lauritzentheory.

The formation of polymer spherulites requires two essential features as detailedby Keith and Padden in 1964 from a wide range of micrographic studies:

1) Fibrillar growth habits.2) Low angle, Non-crystallographic branching.

Polymer Spherulites.

Figure 4.2 pp. 145 shows a typical melt crystallize spherulitic structure which forms in most semi-crystalline polymeric systems. The micrographs in figure 4.2 are taken between crossedpolars and the characteristic Maltese Cross is observed and described on the followingpage. The Maltese cross is an indication of radial symmetry to the lamellae in thespherulite, supporting fibrillar growth, low angle branching and nucleation at the center ofthe spherulite. In some systems, especially blends of non-crystallizable and crystallizablepolymers, extremely repetitive banding is observed in spherulites as a strong feature, figure

11

4.7 pp. 149. Banding is especially prominent in tactic/atactic blends of polyesters and it isin these systems in which it has been most studied. It has been proposed by Keith thatbanding is related to regular twisting of lamellar bundles in the spherulite (circa 1980).Keith has proposed that this twisting is induced by surface tension in the fold surfacecaused by chain tilt in the lamellae (circa 1989). Since most spherulites crystallize in anextremely dense manner it has been difficult to support Keith's hypothesis withexperimental data. Regular banding has, apparently, no consequences for the mechanicalproperties of semi-crystalline polymers so has been essentially ignored in recent literature.

XRD of Polymers:Four main features of XRD are of importance to Polymer Analysis:

1) Indexing of Crystal Structures2) Microstructure3) Degree of Crystallinity4) Orientation

1) Indexing of Crystal Structures: Indexing of crystal structures is similar to thedescriptions in Cullity and other metallurgical texts. The main difference is that polymer crystalscan not be formed in perfect crystals, so single crystal or Laue patterns are not possible. Also,polymer crystals tend to be of low symmetry, orthorhombic or lower symmetry, due to theasymmetry in bonding of the crystalline lattice, i.e. the c-axis is bonded by covalent bonds and thea and b axis are bonded by van der Waals interactions or hydrogen bonds. Additionally, the unitcell form factor tends to be fairly complicated in polymer crystals. Several unit cells for polymersare shown below:

Nylon 66, from Alexander, "X-Ray Diffraction Methods in Polymer Science"

12

Polybutadiene (PBD), from Alexander, "X-Ray Diffraction Methods in Polymer Science"

Poly(ethylene adipate), a polyester, from Alexander, "X-Ray Diffraction Methods in PolymerScience"

Lattice parameters in polymer crystals are strongly temperature dependent as shown in thefollowing diagram:

13

From Balta-Calleja and Vonk, "X-ray Scattering of Synthetic Polymers"

Polymer lattice parameters are also dependent on strain as shown in the following diagram:

From Balta-Calleja and Vonk, "X-ray Scattering of Synthetic Polymers"

Notice that the c-axis (covalent main chain bonds) is much less dependent on thermal or mechanicalstrain.

Line widths are broad for polymer diffraction and a substantial amorphous peak is usually present.

14

Isotactic Polystyrene, from Alexander, "X-Ray Diffraction Methods in Polymer Science"

2) Microstructure:

Cullity deals with metallurgical crystals where crystallite sizes are typically larger than a micron.With a monochromatic incident beam the diffraction pattern from a single crystal is a sequence ofspots where the Bragg condition is met for certain orientations of crystals (see "a" in figure below).As the crystallite size becomes smaller, more crystallites meet the Bragg condition and the radialorientation of these crystallites cover a broader spectrum of angles ("b" and "c" below), eventuallyforming Debye-Scherrer powder pattern rings ("c" below). If crystallite sizes approach 0.1 micron(1000Å), the Debye-Scherrer ring begins to broaden ("d" in figure below).

15

From Cullity, "Elements of X-Ray Diffraction

Polymer crystals are on the order of 100Å in thickness. Broadening of the diffraction lines due tosmall crystallite size becomes a dominant effect and the breadth of the diffraction lines can be usedto measure the thickness of lamellar crystals using the Scherrer equation:

tB

=( )

0 9.

cos

λθ

λ is the x-ray wavelength, B is the half width at half height for the diffraction peak in radians and θis half of the diffraction angle. The Scherrer equation is derived in Cullity and other texts. Use ofthe Scherrer equation is a primary technique to determine lamellar thickness in polymer crystallites.This can be used in conjunction with the Hoffman-Lauritzen (Gibbs-Thompson) equation forstudies of crystallization.

In addition to Scherrer broadening diffraction lines can be broadened in polymers due to defects inthe structure. This will not be covered in detail in this course but is described in Campbell andWhite and in Alexander's text.

3) Degree of Crystallinity:

Polymers are never 100% crystalline since the stereochemistry is never perfect, chains containdefects such as branches, and crystallization is highly rate dependent in polymers due to the highviscosity and low transport rates in polymer melts. A primary use of XRD in polymers is

16

determination of the degree of crystallinity. The DOC is determined by integration of a 1-d XRDpattern such as that shown below for polyethylene.

From Balta-Calleja and Vonk, "X-ray Scattering of Synthetic Polymers"

The determination of the degree of crystallinity implies use of a two-phase model, i.e. the sample iscomposed of crystals and amorphous and no regions of semi-crystalline organization. Thealternative to the two-phase model is a paracrystalline model which was popular in the early daysof polymer science. There are limits to the two-phase model, particularly for fairly disorganizedpolymer crystalline systems such as polyacrylonitrile (PAN). Most polymer systems are amenableto the two-phase model but you should keep in mind that the 2-phase model ignores interfacialzones where the density may differ from that of the amorphous.

The integrated XRD intensity measures the volume fraction crystallinity, φc. Other techniques suchas density gradient columns (see Campbell and White or DSC) measure a mass fractioncrystallinity Ψc. The two fractions are related by the density ratios, where ρc is the crystalline

density, ρ is the bulk sample density and ρa is the amorphous density,

ψ φ ρρ

ψφ ρρc

c cc

c a= −( ) =−( )

and 11

17

If the density of the sample is known from a density gradient column, the weight fraction degree ofcrystallinity can be obtained using:

ψ ρρ

ρ ρρ ρc

c a

c a

= −−

Determination of φc from the XRD pattern under the 2-phase assumption involves separation thediffraction pattern into three parts, 1) Crystalline; 2) Amorphous and 3) Compton Background(Incoherent Scattering). The diffracted intensity if proportional to the amount of each of thesecontributions. Consider the 2-d diffraction pattern shown just above section 2) above. The 1-ddiffraction pattern is a line cut through this pattern as shown below:

1-d Slice of2-d Pattern

2-d Diffraction Patter n

q

I

The actual scattered intensity is related to a volume integral of the diffracted peak in the 2-d pattern,

V I q dV q I q dqc c q c∝ ( ) = ( )∞ ∞

∫ ∫0

2

0

Then the volume fraction degree of crystallinity is given by the ratio of the integral of the crystallinediffraction intensity over the total coherent scattering, i.e. after subtracting the incoherentscattering:

φc

c

Compton

q I q dq

q I q I q dq

=( )

( ) − ( )[ ]

∞

∞

∫

∫

2

0

2

0

This procedure is called the Ruland method and is valid for:1) Random crystallite orientation (Powder pattern)2) 3-d crystalline ordering3) Validity of integrals for finite angles of measure, i.e. there is a point in angle where thecrystallinity is not significant to I(q)

18

4) Crystalline peaks can be separated from the amorphous halo

The Ruland equation can be modified for crystalline defects as described in Campbell and Whiteand Alexander. Usually the simple form given above is sufficient. The Ruland method is shownin the figures below where Iq2 is plotted as a function of q (or s).

From Alexander, "X-Ray Diffraction Methods in Polymer Science"

19

From Alexander, "X-Ray Diffraction Methods in Polymer Science"

4) Orientation:

Orientation is covered in the later chapters of metallurgical diffraction texts such as Cullity (seefigure below).

20

From Cullity, "Elements of X-Ray Diffraction

Orientation is a more dominant effect in polymer samples especially processed plastics (see figurebelow). Orientation is a dominant feature in control of the mechanical and physical properties ofpolymers.

From Tadmor and Gogos, "Principles of Polymer Processing"

21

From Alexander, "X-Ray Diffraction Methods in Polymer Science"

22

From Alexander, "X-Ray Diffraction Methods in Polymer Science"

There are a number of techniques for the quantification of orientation from diffraction data. Cullitydescribes the use of stereographic projections on a Wulff Net (shown below left). The Wulff net isuseful if single crystals are studied and it is desired to determine the orientation with respect to thediffraction experiment such as in orientation of semi-conductor samples for cleavage. In mostpolymer applications it is desired to determine the distributions of orientation for a polycrystallinesample with respect to processing directions such as the direction of extrusion, (machine directionMD), the cross direction (CD) and the sample normal direction (ND). A more useful stereographicprojection for these purposes is the polar net or pole figure (shown below right).

23

From Alexander, "X-Ray Diffraction Methods in Polymer Science"

The pole figure is a slice across the equator of the sphere of projection with the MD usually definedat the top of the pole figure and either the CD or ND as the right side. Normals to planes areprojected from the south pole to the point of intersection on the sphere of projection and where theycross the equatorial plane a point is plotted on the pole figure. A typical polar figure for aprocessed polymer is shown in the figure below for the (110) and (020) normals for thepolyethylene orthorhombic crystalline structure. Notice that the plane normals appear as atopographical plot since there is a distribution in orientation. The (110) and (020) reflections arethe two dominant peaks in the 1-d diffraction pattern for PE shown above (start of section 3).

24

From Balta-Calleja and Vonk, "X-ray Scattering of Synthetic Polymers"

The following figure shows the type of qualitative analysis of orientation which can be performedusing pole figures. Generally, pole figures are constructed by computer software which is part ofa diffractometer capable of measurement of pole figures such as the Siemens D-500.

25

From Alexander, "X-Ray Diffraction Methods in Polymer Science"

The pole figure can give a qualitative picture of orientation in a polymer sample. Quantitativemeasures of orientation can be obtained by considering a radial plot of diffraction data.

2-d Diffraction Patter n

φ

I

MD

TD

0 36 0

φ

The intensity for a given diffraction line (given 2θ) has two peaks as a function of radial angle, φreflecting the two normals to the diffraction plane relative to the MD/TD plane. The Hermansorientation function can be calculated for a given plane from the φ dependence of the diffractedintensity:

26

f( ) cos11021

23 1= −( )φ

where <cos2φ> is the average cosine squared weighted by the intensity as a function of the radialangle for the (110) plane. The Hermans orientation function has the behavior that f = 1corresponds to perfect orientation in the φ = 0 direction, f = 0 for random orientation and f = -1/2

for perfect orientation normal to the φ = 0 direction. If the orientation function is calculate fororthogonal axis such as the a, b, and c unit cell directions for the PE unit cell then fa + fb + fc = 0.The orientation function for the unit cell vectors can be determined from geometry if the angularrelationship between a plane normal and the unit cell direction is known. <cos2φ> is calculated by:

cos

, sin cos

, sin

( )

( )

( )

2110

1102

0

2

110

0

2φ

φ θ φ φ φ

φ θ φ φ

π

π=( )

( )

∫

∫

I d

I d

The figure below shows the behavior of the Hermans orientation function for the three unit celldirections in PE as a function of processing conditions in a fiber spinning process.

From Tadmor and Gogos, "Principles of Polymer Processing"

The orientation function is directly related to polymer properties as shown in the example below.

27

From Tadmor and Gogos, "Principles of Polymer Processing"

Small Angle X-ray Scattering (SAXS)

We have considered that Bragg's Law, d = λ/(2 sinθ), supports a minimum size of measurementof λ/2 in a diffraction experiment (limiting sphere of inverse space) but does not predict amaximum size, i.e. the point (000) of inverse space reflects infinite size. In class we havediscussed the use of diffraction to measure crystalline and amorphous structures on the atomicscale, but clearly, many morphologies are of importance that have characteristic sizes muchlarger than the atomic scale. In metals this was first noted by Guinier in his development of thetheory of Guinier-Preston zones. Guinier was one of the fathers of an outgrowth of diffractionaimed at large-scale structures in the 1950's. Bragg's Law predicts that information pertaining tosuch nano- to colloidal-scale structures would be seen below 6° 2θ in the diffractometer trace. Itis possible to design specialized instruments to measure down to less than 1/1000 of a degree formeasurement of up to 1-micron scale structures using x-rays!

The characteristics of materials at these larger size scales are fundamentally different than atatomic scales. Atomic scale structures are characterized by high degrees of order, i.e. crystals,and relatively simple and uniform building blocks, i.e. atoms. On the nano-scale, the buildingblocks of matter are rarely well organized and are composed of rather complex and non-uniformbuilding blocks. The resulting features in x-ray scattering or diffraction are sharp diffractionpeaks in the XRD range and comparatively nondescript diffuse patterns in the SAXS range.

In XRD the scattered intensity depends on the Lorentz-Polarization factor which is essentiallyequal to 1 below 6° 2θ. For disorganized systems the multiplicity factor is 1 and the structurefactor, |F2|, generally does not reflect order so involves only a form factor for the nano-scalestructures that give rise to scattering, i.e. regions of differing electron density. In XRD theatomic scattering factor, f2, was equal to the square of the number of electrons in an atom at lowangles, n e

2(1/q), where q is 4π sin(θ)/λ. Additionally, the intensity of scattering is known to beproportional to the number of scattering elements in the irradiated volume, Np(1/q). Then, insmall-angle scattering we can consider a generalized rule that describes the behavior of scatteredintensity as a function of Bragg size "d" or "r" that is observed at a given scattering angle2θ, where r = 1/q.

I(q) = Np(1/q) ne2(1/q) (1)

From this simplified rule of thumb we can derive most of the general rules of small-anglescattering in a less than rigorous manner. This approach, however, is extremely useful for asimple understanding of small-angle scattering. Scattering laws in the small angle regimedescribe two main features that are observed in a log Intensity versus log q plot. First, typicalscattering patterns display power-law decays in intensity reflecting power-law scaling features ofmany materials. Secondly, power-law decays begin and end at exponential regimes that appearas knees in a log-log plot. These exponential knees reflect a preferred size as described by r =1/q for the knee regime.

All scattering patterns in the small-angle regime reflect a decay of intensity in q and this can beeasily described by considering that at decreasing size scales the number of electrons in aparticle is proportional to the decreasing volume, while the number of such particles increases

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with 1/volume. Then the scattered intensity by equation (1) is proportional to the decay of theparticle volume with size. This analysis implies that the definition of a particle, i.e. r, does notnecessarily reflect a real domain, but reflects the size, r, of a scattering element that could be acomponent of a physical domain.

Porod's Law:

Next consider a sharp smooth surface of a particle such as a sphere. The surface can bedecomposed into spherical scattering elements that bisect the particle/matrix interface. Thenumber of such spheres is proportional to the surface area for the particles divided by the areaper scattering element, r2 or 1/q2, while the number of electrons per particle is just proportional tor3 or 1/q3. Using equation (1) with Np = Sq 2 and ne = 1/q3, yields I(q) = S/q4, or Porod's Law forsurface scattering. Porod's Law can be used to measure the surface area of domains in the nano-scale. The rigorously derived form of Porod's Law is,

I(q) = Ie 2π ρ2 S/q4 (2)

where ρ is the electron density difference between particulate domains and the matrix materialand Ie is a constant.

High Density Polyethylene showing XRD at high-q, SAXS at intermediate q and LS (lightscattering) at low-q. Two Porod Regimes are observed.

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Light Scattering and SAXS from non-woven mat of micron scale fiber glass.

There are three other general categories of power-laws that are well defined in small-anglescattering, A) Surface-Fractal Laws, B) Diffuse interface Laws and C) Mass-Fractal orDimensional Laws . Additionally there is the possiblity of D) polydispersity of particle sizeleading to power-law decays.

A) Surface Fractal Laws.

Systems that do not have smooth surfaces are often described by a scaling law where the surfacearea, S(r), is a function of the size of measurement. For example, the coastline of an island willincrease if it is measured with smaller rulers since smaller rulers are capable of measuring thenooks and cranies of the coast line. For a smooth surface S(r) = r2, and for a rough surface S(r) =rds, where ds is the surface fractal dimension that varies from 2 to 3. Inserting such a scaling lawinto the discussion of Porod's Law above, yields I(q) proportional to qds-6. Surface fractalsdisplay power-law decays weaker than Porod's Law and are termed positive deviations fromPorod's Law.

B) Diffuse interfaces.

Diffuse interfaces are observed when a concentration gradient is observed at an interface such asin mixing of two liquids or dissolution of a particle. A diffuse interface can be modeled as apower-law distribution of particles of differing electron density. Through this type analysis it ispossible to describe the Schmidt exponent, β, where I(q) goes as q-(4 + β), and β describes theconcentration decay of the diffuse interface.

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C) Dimensional scattering laws.

Particles can be described in terms of their dimension in the sense that a rod is 1-D, a disk 2-Dand a sphere 3-D. Such a dimensional description implies that the mass of the object depends onthe size of observation, "r", raised to the dimension. Because of this it is natural to expectpower-law scattering to result form such low dimension objects. Consider that scattering from arod is observed at "r" = 1/q between the rod length, L, and the rod diameter, D. Then the numberof scattering elements in the rod of size "r" is equal to L/r or Lq. The number of electrons perscattering element is given by rD2 or D2/q. Using equation (1) we have I(q) = LD4/q, or a q-1

decay in intensity for the 1-D object.

For a disk at size scales, r = 1/q, between the disk diameter, D, and thickness, t, the number ofscattering domains is equal to the disk area, D2, divided by the scattering domain size, r2, or Np =D2/r2 = D2q2. The number of electrons per scattering domain is given by the volume of a domain,r2t= t/q2. Equation 1 yields I(q) = D2t2/q2, or I(q) proportional to 1/q2. Again the scattering isproportional to q-df where df is the dimension of the object.

For a mass-fractal object such as a polymer coil the mass is given by the size raised to thedimension, i.e. for a polymer coil the end to end distance R is given by n1/2 l, so the mass, n isproportional to R2. In this sense a Gaussian polymer coil is a 2-D object. If such an object isbroken into scattering elements of size r, the number of such elements is given by R2/r2 = R2q2

and the number of electrons in an element is ne = r2 = 1/q2. Then equation 1 yields I(q) = R2/q2,or I(q) is proportional to 1/qdf.

D) Polydisperse Particles

Many systems display dispersion in particle size in the small-angle regime. In some cases thesedispersions are broad enough that they can be observed in terms of a power-law regime inscattering, i.e. when the dispersion in size occurs over a decade or more in size.

Guinier's Law:

We have so far discussed power-law decays in scattering that are mostly defined between twosize limits, i.e. for a rod a power-law decay of q-1 is observed between the rod length and the roddiameter. Power-laws merge with a generic description of a discrete size at these limits, i.e. at q= 1/L or q = 1/D for a rod. For an isotropic system we can consider a particle in an averagesense by allowing the particle structure to be averaged with respect to position and rotation.

The scattering event involves interference from waves emanating from two points in the particleseparated by a distance r = 1/q. Then the probability of constructive interference involves theprobability that given a point lying in an average particle, one finds a second point also in theaverage particle at a distance r = 1/q. For an isotropic system one must consider first averagingany starting point in a particle and secondly averaging any direction for the vector r. This leadsto a double summation that is identical to the determination of the moment of inertia for aparticle. When the electron density is used as the weighting rather than the mass density, thismoment of inertia is called the radius of gyration of the particle. For a system of disperse shaped

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and sized particles the radius of gyration reflects a second moment of the distribution of theshape and size about the mean.

The process of obtaining Rg involves two steps, first averaging all possible positions in theparticle from which a vector "r" can start and be within the particle. Second, determining theprobability that a randomly directed vector "r" from an arbitrary starting point in the particle willfall in the particle. The meaning of this probability p(r), in the vicinity of r = the particle size,can be graphically represented by a Gaussian probability cloud created by the summation of allpossible positions of the particle where the center of the probability cloud is in the particle phase.

At very low q this corresponds to the volume fraction particles squared. At sizes, r = 1/q, closeto the average particle size or radius or gyration, this probability is reflected by a decayingexponential function. The decaying exponential function can be written in terms or r or in termsof q. (Fourier transform of a Gaussian distribution is a Gaussian distribution). Such an analysisleads to Guinier's Law, where the average size is reflected in the radius of gyration, Rg. Rg isthe moment of inertia for the particle using the electron density rather than the mass as aweighting factor.

I(q) = Np ne2 exp(-q2Rg

2/3)

Special Scattering Functions:

Sphere: radius R

I(q) = N n23(sinqR - qR cosqR)/(q3R3)2

Rg = R/1.29 = R√(3/5)

Rod: Length 2H; Diameter 2R

I(q) = N n2π exp(-q2R2/4) / (2qH)

Rg overall = R2/2 + H2/3

Disk: diameter 2R; thickness 2H

I(q) = N 2n2 exp(-q2H2/3) / (q2R2)

Rg overall = R2/2 + H2/3

Gaussian Polymer Coil: n persistence units; l persistence length

I(q) = I0 2(Q - 1+exp(-Q)/Q2

Q = q2Rg2

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Rg overall = n1/2l/√6

Unified Function: (one level)

I(q) = G exp(q2Rg2/3) + B q*-P

q* = q/(erf(qRg/√6))3

This is a generic function where G, Rg, P and B are defined according to local functions. Forexample, for a sphere, G = N n2; Rg = R/1.3; P = 4; B = 2π G S/V2 = 9 G/(2 R4). (Sphere S =4πR2; V = 4πR3/3) Other examples can be found in:

Small-Angle Scattering from Polymeric Mass Fractals of Arbitrary Mass-Fractal Dimension,Beaucage, G. , J. Appl. Crystallogr. (1996), 29, 134-146.

Approximations leading to a unified exponential/power-law approach to small-angle scattering,Beaucage, G., J. Appl. Crystallogr. (1995), 28(6), 717-28.

Correlation Function Analysis: