Some studies on diffusion in macromolecular solutions : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy at Massey University

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SOME STUDIES ON DIFFUS ION IN MACROMOLECULAR SOLUT IONS

A thesis presented in part ial fulfilment of the requirement s for the degree of Doctor of Philosophy at Massey University

CRAIG MALCOLM TROTTER 1 9 8 0

ii ABSTRACT

The construct ion and performance of a Pulsed Field Gradient system for use with a commerc ial , high-resolut ion , Fourier-Transform NMR spectrometer

is described . The self-diffusion coefficient of benzene as measured by the calibrated system is in agreement with the current literature value , within the overall experimental error of the system ( +2% ) . The use of an external lock in conjunct ion with s ignal averaging facilitates the measurement of self-diffusion coefficients for solution components in small concentrat ions . During s ignal accumulat ions, the system exhibits freedom from the spin-echo phase and envelope instabilities mentioned as sources of error even in recent publicat ions dealing with the Pulsed Field Gradient t echnique ( e . g . von Meerwall et a l . , 1 97 9 ) . The

ability of the system to invest igate dilute solutions is demonstrated by measurements made on 0 . 5% ( w/v ) solut ions of polystyrene in carbon

tetrachloride . Homogeneity coils included in the NMR probe have allowed the self-diffusion coefficients of some single components in mult i­

component systems to be invest igated , and results for t he binary system butanol-benzene are presented .

Polymer self-diffusion coefficients have been obtained for 1 1 0 , 000 molecular weight random-coil polystyrene in the solvents carbon tetra­

chloride , deuterated-chloroform and deuterated-toluene . The Pulsed Field Gradient NMR method was used for the measurements , and the poly­

styrene concentrat ions ranged from 0 . 5% (w/v ) to 2 5% ( w/v ) . For each solvent a concentrat ion regime is found in which the de Gennes ' polymer self-diffus ion scaling law is obeyed; and the upper concentration limit at which this scaling law breaks down is defined . The self-diffusion coefficient of polystyrene in the solvent deutero-benzene has also been determined, and is shown to agree with Forced Rayle igh Scattering self­

diffus ion results for similar molecular we ight polystyrenes in normal benzene . In contrast , values of th� self-diffusion coefficient obtained for polystyrene random-coils by calculation from sedimentat ion data are shown to differ significantly from those directly determined . The mutual diffusion coefficients of the polystyrene solutions have been obtained from Quasi-E lastic Laser Light-Scattering experiments . These mutual diffusion coefficients do not approach the d irectly measured self­

diffusion coeffic ients even at concentrations where the random-coils are on average well separated . It is proposed that migrat ing polymers must suffer transient entanglement effects over the experimental time scales

employed in the diffus ion measurements .

Quasi-Elast ic Laser Light-Scattering has also been used to measure the diffusion coefficient of polystyrene latex spheres in 0 . 01M and 0 . 001M sodium chloride . Experiments were conducted over the latex sphere

concentration range 0,0 04% (w/v ) to 4 . 46% ( w/v ), and several measurements were also made for low concentrat ions of latex spheres in triply

distilled water . The diffusion coefficient was found to be ionic

iii

strength dependent over the entire concentrat ion range studied . Solut ions of polystyrene spheres at moderate concentrat ions exhibit the phenomenon of multiple scattering . The available literature on multiple scattering

is reviewed and criteria adopted for the reliable interpretation of data collected during experiments on these solutions . The diffusion

coefficients so obtained show substantial agreement with the mutual­

diffusion coefficient results of Anderson et al . , obtained by a capillary penetration technique . The conclusion reached in this sect ion of the work is that Quasi-Elast ic Laser Light-Scattering is able to provide a measure of the mutual diffusion coeffic ient in the presence of interact ions between charged macromolecules . This conclusion i s seen to be in accord with earlier laser light-scattering studies on solut ions of the protein Bovine Serum Albumin, provided that a reassessment of available mutual diffusion data on these systems is undertaken .

ACKNOWLEDGEMENTS

I wish to express my s incere appreciat ion for the guidance and careful instruct ion given by my Chief Supervisor, Dr D . N . Pinder, during the course of this research proj ect . It is also a pleasure to thank Dr P . T . Callaghan, not only for his invaluable advice during the j oint supervision of this work, but also for constructing part of the Pulsed Field Gradient apparatus .

I am indebted to many members of the Science Faculty for helpful

discussions, and I would particularly like to thank Mr R . C . 01Driscoll for advice concerning the e lectronic proj ects undertaken . The

financial support of Massey University during the course of this thesis is also appreciated .

I am grateful to Mr N . Foote of the Mechanical Workshop for his cheerful co-operation in performing many tasks requiring his considerable

technical expertise . The assistance of Mr R . Parsons and the Electronic Workshop staff is also appreciated .

The typists, Mesdames Maryanne Nat ion and Wendy Sigvertson are thanked for their patience and assistance . Particular thanks are due to Maryanne Nation. for concluding the typing of this thesis .

lV

ABSTRACT

ACKNOWLEDGEMENTS TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES LIST OF PLATES LIST OF SYMBOLS

TABLE OF CONTENTS

CHAPTER 1 - INTRODUCTI ON 1 . 1 . RESEARCH GOALS

1 . 2 . DIFFUSION IN MACROMOLECU�AR·SOLUTIONS

1 . 2 . a . Definition of Diffusion Coefficients 1 . 2 . b . The Direct Measurement of Macromolecular

Diffusion Coeffic ients

1 . 2 . c . Diffusion Coefficient Measurement in This Thesis

CHAPTER 2 - THE LASER LIGHT-SCATTERING EXPERIMENT 2 . 1 . HI STORICAL SKETCH AND INTRODUCTION 2 . 2 . THEORY REVIEhl

2 . 2 . a . The Laser Light -Scattering Experiment 2 . 2 . b . The Homodyne Intens ity Autocorrelat ion

Funct ion for a Very Dilute , Monodisperse Solut ion of Small Macromolecules

2 . 2 . c . The Photocount Analysis Method

2 . 2 . d . The Method of Cumulants : Application to Data Obtained from Concentrated Solutions of Macromolecules

2 . 2 . e . Implicat ions of the Angular Dependence of the Intensity Autocorrelation Funct ion

V

PAGE

ii iv

V

ix xii xiii xiv

1 1 3 3 5

7

12 12 1 5 1 5

17

1 9

2 3

Decay Rate 26

2 . 3 . INSTRUMENTATION 29

2 . 3 . a . A Digital Correlator with Blinker Facility 29 2 . 3 . b . Considerat ions for Photomultipliers as

Photodetectors

2 . 3 . c . The Laser Spectrometer

32 33

CHAPTER 3 - THE PULSED FIELD GRADIENT EXPERIMENT

3 . 1 . INTRODUCTION : THE NUCLEAR MAGNETIC RESONANCE EXPERIMENT

3.1 . a . Nuclear Magnetic Resonance 3 . 1.b . The Format ion of Spin-Echoes

3 . 2 . DIFFUSION COEFFICIENT MEASUREMENT BY NUCLEAR MAGNETIC RESONANCE

3 . 2 . a . The Attenuation of Spin-Echoes Due to Diffusion

3 . 2.b . The Pulsed Field Gradient Nuclear Magnet ic

vi

PAGE

39

39 43 45 45

Resonance Experiment 46

CHAPTER 4 - INSTRUMENTATION FOR A PULSED FIELD GRADIENT EXPERIMENT 51

4 . 1 . INTRODUCTION 51

4 . 2 . DESIGN AND CONSTRUCTION OF INSTRUMENTATION 54

4 . 2 . a . Probe and Gradient Coil Des ign 54 4 . 2 . b . A Digital Pulse Programmer for Pulsed Field

Gradient Experiment s 6 2

4,2.c . A Power Supply for Pulsed Field Gradient Generat ion

4,2 . d . The NMR Spectrometer 4 . 3 . EXPERIMENTAL CONSIDERATIONS

6 5 71 75 75 75 76 4 . 3.a . Sample Related Considerat ions

4 . 3.b . Signal Related Considerat ions

4 . 3.c. Restrictions on Gradient Pulse Parameters

4 . 3 . d . Accumulat ion and Pulse Repetition Time Effects 8 0 4 . 4. MAGNETIC FIELD GRADIENT CALIBRATION 82 4 . 4 . a . Field Gradient and RF Pulse Profiles 82 4.4 . b .. Calibrat ion and Reproducibility of the

Field Gradient 4.4 . c . Concluding Remarks

84 88

CHAPTER 5 - THE STUDY OF DIFFUSION IN RANDOM-COIL POLYSTYRENE

SOLUTIONS : EXPERIMENTAL DETAILS 93

93 5.1 .

5 . 2 .

INTRODUCTION

THE PULSED FIELD GRADIENT NUCLEAR MAGNETIC

RESONANCE EXPERIMENTS 94

5.2 . a . Preparation of Samples for the PFGNMR Study 94

5,2 . b . Analysis of PFGNMR Data 94

vii PAGE

5 . 2 . c . The Polystyrene Self-Diffus ion Coefficients 95 5 . 2 . d . Self-Diffusion of Chloroform in Polystyrene 98 5 . 3 . THE QUASI-ELASTIC LASER LIGHT-SCATTERING EXPERIMENTS 103

5 . � . a . Applicat ion of the Blinker to Concentrated

Polymer Solutions 103

5 . 3 . b . Preparation of Samples for the QELS Study 104 5 . 3 . c . Spectrometer Related Considerat ions 106 5 . 3 . d . Spurious Contributions to the Intensity

Autocorrelation Funct ion 5 . 3 . e . Analysis of QELS Data

5 . 3 . f . The Polystyrene Solution Mutual Diffusion Coefficients

CHAPTER 6 - D I FFUSION IN RANDOM-COIL POLYSTYRENE SOLUTIONS :

107 108 114

DISCUSSI ON OF RESULTS 118

6 . 1 . INTRODUCTION 118

6 . 2 . THE dE GENNES ' SCALING LAW FOR POLYMER SELF-DIFFUSION 119 6 . 3 . DIFFUSION IN RANDOM-COIL POLYSTYRENE SOLUTIONS :

DISCUSSION OF RESULTS 122

6 . 3 . a . Experimental Result s and the de Gennes ' Scaling Law

6 . 3 . b . Direct and Indirect Measurements of Polystyrene Self-Diffusion Coefficients 6 . 3 . c . A Comparison of Macroscopic and QELS

Determinations of the Mutual Diffusion Coefficient

6 . 3 . d . A Comparison of Self and Mutual.Diffusion Coefficients in Polymer Solut ions

CHAPTER 7 - POLYSTYRENE LATEX SPHERE DIFFUSION : A LASER LIGHT-

122 124

126 128

SCATTERING STUDY 135

7 . 1 . INTRODUCTION

7 . 2 . THE EFFECT OF MULTIPLE SCATTERING 7 . 3 . THE EXPERIMENTAL STUDY

7 . 3 . a . Sample Preparation 7 . 3 . b . Use of the BlinKer 7 . 3 . c . Data Analysis

7 . 3 . d . The Concentrated Solut ion Result s 7 . 3 . e . The Very Dilute Solution Results

135 137 139 139 140 141 147 151

7 . 4. COMPARISON OF RESULTS

7 . 4 . a . Theories of Macro-ion Transport in the Presence of a Concentration Gradient 7 . 4 . b . Applicat ion of Macro-ion Transport

Theories to Latex Sphere Solutions 7 . 4 . c . Interpretation of Data Obtained at High

Ionic Strength

7 . 4 . d . Interpretation of Data Obtained at Low Ionic Strength

7 . 4 . e . Comparison with Other Work CHAPTER 8 - CONCLUSIONS

APPENDICES

BIBLIOGRAPHY

8 . 1 . a . Concluding Remarks

8 . 1 . b . Towards an Improved PFGNMR Experiment

Appendix 1 - The Magnetic Field Due to an Opposed Helmholtz Pair in the Presence of Magnet Pole-p ices

Appendix 2 - The Digital Pulse Programmer Appendix 3 - Auxilary Tables

Appendix 4 - Auxilary ^· Figures

Appendix 5 - A 2 0MHz Digital-Analogue Ratemeter Appendix 6 - A Pulsed Magnetic Field Gradient

Sequence for Eddy Current Minimisat ion Appendix 7 - Published Work

PAGE 158

158 161 163

165 167

169 169 172

176 179 181 189 192

195 199 2 2 0

v iii

Figure ^{1 . 1 .}

Figure 2 . 1 .

Figure 2 . 2 .

Figure ^{2 . 3}

LI ST OF FIGURES

A comparison of QELS and diffus ion cell measurements .

A schemat ic representation of the laser light scattering experiment .

The relat ionship between the full and clipped photocount .

A schematic circuit di agram of the digital autocorre lator

Figure 2 . 4 A schemat ic diagram of the las er spectrometer Figure 2 . 5 . • The photomult iplier tube and hous ing .

Figure ^{3 . 1 .} The spin-echo experiment .

Figure ^{3 . 2 .} The pulsed magnet ic field gradient experiment .

Figure ^{4 . 1 .} The NMR probe circuitry.

Figure ^{4 . 2 .} The coil shape for the generat ion of uniform magnet ic field gradients .

PAGE

9

1 6 2 2

3 0 34 36

41 47 5 6 5 9

lX

Figure ^{4 . 3 .} The sample space geometry , and sample holder.

Figures ^{4 . 4 .} The TTL pulse t imer schematic circuit diagram .

6 0 6 3 , 64

Figure 4 . 5 .

Figure 4 . 6 .

A schemat ic circuit diagram of the Kepco power supply .

Programming the output current of the Kepco power supply by stepping the reference voltage .

Figures ^{4 . 7 .} The circuitry used in this work for the generat ion Figure ^{4 . 8 .}

Figure ^{4 . 9 .}

of magnet ic field gradient pulse s .

A block diagram of the JEOL JNM-FX ^{6 0} NMR spectrometer .

Invest igat ion of the effect of eddy current s on the PFGNMR experiment.

Figure 4.1 0 . Magnet ic field gradient and rf field magnitude- profiles along the sample y-axi s .

Figure 4 . 1 1 . The magnet ic field gradient calibrat ion experiment:

the spin-echo attenuat ion plot for water.

6 7 6 8 7 0 72 7 8

8 3

85 Figure 4 . 1 2 . The spin- echo attenuat ion plot for Analar benzene . ⁸⁷ Figure 4 . 13 . The spin-echo attenuat ion plot for Analar glycerol . ^{8 9}

Figure 4 . 14 . Figure 4 . 15 .

Figure 5 . 1 . Figure 5 . 2 . Figure 5 . 3 .

Figure 5 . 4 . Figure 5 . 5 . Figure 5 . 6 .

Figure 5 . 7 .

Figure 5 . 8 .

Figure 5 . 9 . Figure 5 . 10 .

Figure 6 . 1 . Figure 6 . 2 .

Figure 6 . 3 .

The 1H frequency domain NMR spectrum of an equi-molar mixture of benzene and butanol . . Spin-echo attenuation data for pure benzene , pure butanol and an equi-molar mixture.

. 1

The H frequency doma in NMR spectrum of 0 . 7 5gm%

polystyrene in D -toluene .

The spin-echo attenuat ion plot for 1 . 5gm% , 110 , 000M polystyrene in D-toluene. w

The self-diffusion coefficients of 110 , 000M w polystyrene , as a function of concentration , in the solvents CC14, CDC13 and D-toluene.

The spin-echo attenuation plot for 11doped" water.

The self-diffusion coefficients of chloroform in polystyrene solut ions.

The diffusion coeffic ients of 390 , 000M w polystyrene in butanone , measured by QELS in the very dilute concentrat ion regime.

The signal-t o -noise rat io obtainable in QELS

experiments at the lowest polystyrene concentration PAGE

91 92

96 97

99 100 102

109

studied . 111

Plots showing the mult i-exponential nature of the IAFs encountered in the study of the sample 2 . 0gm% 110 , 000Mw polystyrene in CC14.

Cumulant plots for the sample 2,0gm% polystyrene in CC14 .

Mutual diffusion coeffic ients obtained by QELS for solut ions of 110 , 000M polystyrene in the w

113 115

solvents CDC13 and CC14. 116

Entanglement phenomena in the polymer gel regime:

the d� Gennes ' model .

The self-diffusion coefficient s of �10,000M w polystyrene in various solvent s , normalised to solvent viscosity .

A comparison of directly and indirectly measured self-diffusion coeffic ient s of random-coil

polystyrene .

120

123

125

X

Figure 6 . 4 .

Figure 6 . 5 . Figure 6 . 6 .

Figure 6 . 7 .

Figure 7 . 1 .

Figure 7 . 2 .

Figure 7 . 3 .

Figure 7 . 4 .

Figure 7 . 5 . Figure 7 . 6 . Figure 7 . 7 .

Figure 7 . 8 .

Figure 8.1 .

A comparison of n0 values obtained by wedge interferometry with result s from QELS studies.

A comparison of mutual- and self-diffusion

PAGE

127

coeffic ients for 11 0 , 000Mw polystyrene in CC14. 1 29 A comparison of mutual- and self-diffusion

coefficients for 1 1 0 , 000M polystyrene in w CDC13 .

A comparison of mutual- and self-diffusion coefficients for 1 1 0 , 000M polystyrene in the w solvents toluene and D-toluene,

Plots showing the presence of both long and short decays in the NIAFs obtained from a QELS study of a concentrated solution of polystyrene latex spheres .

A plot showing the linear dependence of the long­

time decay rate on sin2 (8/ 2 ) , for a concentrated solution of polystyrene latex spheres .

The concentrat ion dependence o f the quantity with the dimens ions of a diffusion coefficient

which was obtained from QELS studie s of concentrated polystyrene latex sphere solut ions.

Plots showing the linear dependence of the long­

time decay rate on sin2 (8/2 ) , for representat ive samples .

As for Figure 7 . 4.

The very dilute regime latex sphere diffusion coeffic ient s .

The diameter distribut ion of the nominally 0 . 08 5�m diameter latex spheres as determined by transmission electron microscopy.

A comparison of QELS measurement s from the concentrated latex sphere solution study with DM values obtained by the technique of capillary penetrat ion.

A ramped-pulse , field gradient pulse sequence for·

eddy-current minimisat ion .

1 3 0

1 3 1

142

145

148

149

1 5 2

1 5 6

166

1 74 xi

Table 2 . 1 .

Table 4 . 1 . Table 4 . 2 .

Table 4 . 3 . Table 5 . 1 .

Table 6 . 1 .

Table 7 . 1 .

Table 7 . 2 .

Table 7 . 3 .

Table 7 . 4.

LIST OF TABLES

Specificat ions of the laser light-scattering spectrometer .

The effect of eddy.currents on the measured diffusion coefficient .

Checks for the consistency of the attenuation ratio under various rf and magnetic field gradient pulse conditions .

Gradient coil calibration experimental data .

Details of the data analysis from the samples 0 . 26gm% and 2 . 0gm% polystyrene in CC14 .

A comparison of intermolecular spacings , radii of gyration , and experimental molecular

displacements encountered at two different

PAGE

3 5

79

81 86

1 1 2

concentrat ions in 1 1 0 , 000M polystyrene solut ions . w 1 3 3

Examples of correlator off-to-on times

encountered in a study of concentrated latex sphere solutions .

Details of the data analysis for the sample 2 . 1 5gm% polystyrene latex spheres in 0.001M sodium chloride .

Diffusion coefficients determined by QELS in the very dilute latex sphere concentrat ion regime .

Theoretical and experimental values of k .

143

146

1 54 164

xii

LIST OF PLATES

PAGE

Plate 1 - The low resolution NMR probe used in this work . 61 Plate 2 ^- The JEOL JNM-FX60 spectrometer , and instrumentation

for the PFGNMR experiment . ⁷³

xi ii

J c

R

7f

no M , M w n

�

n+ _s

V

s p 8

g k . ,k .

-l. -s

E8( t ) I ( O ) I ( t )

*

LIST OF SYMBOLS

macromolecular self-diffus ion coefficient.

solvent self-diffusion coefficient.

macromolecular solution mutual-diffusion coefficient.

mean-squared displacement.

the flow of solute molecules , per unit t ime , across a unit area perpendicular to that flow.

macromolecular concentration.

infinite dilution macromolecular diffusion coefficient.

-23 -1

Boltzmann constant ( 1.38x1 0 j oule.deg ).

absolute temperature , or correlator sample t ime.

frictional coefficient per macromolecule at infinite solute dilution.

non-hydrated hard sphere radius.

the constant 3.141 59.

the solvent viscosity.

the weight and number average molecular weights.

the hydrodynamic sphere radius.

indirectly determined self-diffusion coefficient.

solute part ial specific volume.

sedimentation coefficient.

solution density.

xiv

the scattering angle ( except in Chapter 3 where 8 is the magnetisation rotation angle).

the scattering vector.

init ial and scattered wave vectors.

magnitude of the scattering vector.

vacuum wavelength of the incident laser radiation.

Planck ' s constant ( 6 . 6 2 5 6x1o-34 j oule.deg-1 ) divided by 27r.

magnitude of the total scattered electric field at the photodetector at t ime t'.

ES at the later time t'+t.

the scattered intensity, defined similarly to E8 ( o ).

the scattered intensity , defined similarly to ES ( t ).

indicates that the complex conjugate has been taken ( Chapter 2 only ).

g(1l (t)

WO

I (t ,T)

XV

the normalised intensity autocorrelation function

(second order electric field·autocorrelation funct ion ) . the normalised first order electric field autocorrelation funct ion .

incident electric field angular frequency .

the average intensity during the short t ime interval T which is centred on the time t�+t .

n(t , T) number of photodetections during· T .

Cp(t ,T) the un-normalised , full photocount autocorrelation function .

nk( T) the clipped count .

k clipping level.

Ck( t) , Ck(mT) - s ingle-clipped , un-normalised photocount autocorrelation function .

( 2) (t T) - gk '

r r G(r) llr /rr DML

A coh I

y

i , j ,k

normalised , single-clipped photocount autocorrelat ion function .

the decay rate of a photocount autocorrelation function . average decay rate of a photocount autocorrelation function . normalised distribution of the decay rates .

rth normalised moment of G(f).

the quantity extracted by a cumulants analysis of QELS data obtained from concentrated , non-random-coil , macromolecular solutions.

the variance of the decay rate distribution .

the calculated baseline: the value of the normalised , single-clipped autocorrelation funct ion at infinite t ime . the detector coherence area .

the solid angle subtended by the source , at the detector (Chapter 2 only ; otherwise an Ohm symbol) .

nuclear spin quantum number ( Chapter 3 only ; otherwise symbolises current) .

magnitude of the steady magnetic field . nuclear gyromagnet ic ratio.

laboratory frame unit vectors rotating frame unit vectors .

magnitude of the nuclear magnetic moment ( Chapter 3 only ; otherwise the multiplier "micro" i . e . x10 ^-

6

^).

M

t p T

G

A(O) A(

^{G )}

T s 6 0

d P(�D) QT L1,L2,L3

Rin'R1,R2,R3 c

1,

^c

2

^,c

3 D1-D4 21 QA,QB

Rs Rcc

go

Aco

R

F

£ M

m

V

the magnitude of the macroscopic magnetisation.

Larmor frequency.

the magnitude of a magnetic field applied along the

� .

X -aXlS,

the time for which �1 is applied.

the 90°-180° rf pulse spacing.

the magnitude of the magnetic field gradient.

xvi

the magnitude of the spin-echo at t=2T in the absence of an applied magnetic field gradient.

the magnitude of the spin-echo when the field gradient is applied.

the time for which a nucleus remains at a position

^z.

the rms jump displacement of a nucleus.

the magnetic field gradient pulse separation.

the magnetic field gradient pulse width.

the separation between the 90 rf pulse and the first

0

gradient pulse.

the probability of obtaining a phase difference �n·

quality factor of the transmitter circuit.

inductances in the NMR probe circuitry.

resistances in the

NMR probe circuitry.

capacitances in the NMR probe circuitry.

diodes in the NMR probe circuitry.

separation of the magnet pole-pieces.

sets of Darlington connected transistors.

the Kepco sensing resistor.

Kepco current control potentiometer.

residual field gradients due to inhomogeneities in the steady field �o·

the eddy current induced gradient.

the off-resonant frequency.

the ratio of the correlator off-to-on times.

the slope of a ln(NIAF-1) vs. time plot; or the initial slope of such a plot if the plot is multi-exponential.

the radius of random-coils in the very dilute regime.

the Kuhn statistical segment length.

the polymer molecular weight.

molecular weight of

^{£ .}

Flory number (except in Chapter 3 where

^AV

denotes the

off-resonant frequency).

c *

£

a

E (r)

ljJO

R

K -1 e f�?:

xvii

the polymer concentrat ion such that the average distance between two neighbouring coil centres is equal to 2RF.

the average distance between polymer entanglements in the gel regime .

the chain disentanglement t ime . the long t ime decay rate .

the quantity with the dimensions of a diffusion coefficient�

calculated from the long-time part of a ln ( NIAF-1) vs.

t ime plot .

the calculated mean latex sphere separat ion . latex sphere surface charge density .

permitt ivity of the latex sphere . latex sphere radius .

latex sphere bulk modulus . the permitt ivity of free space . defined through the equation

DM ⁼ D0 (l + k . c + --- )

the pair potential energy for two charged particles separated by a centre-to-centre distance r,

the surface potential of a charged sphere .

the normalised intersphere separation ( Chapter 7 only) . the Debye-Huckel shielding length.

-1 9 electronic charge (1 . 602x1 0 C) .

parameters used to evaluate equation 7 . 6 . isothermal osmotic compres s ibility .

volume fraction .

the rise-t ime of a ramped magnet ic field gradient pulse .