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The effect of adsorbed poly (vinyl alcohol) on the properties of model silica suspensions

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Title:
The effect of adsorbed poly (vinyl alcohol) on the properties of model silica suspensions
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Khadilkar, Chandra, 1959-
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English
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xxiii, 436 leaves : ill. ; 28 cm.

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Subjects / Keywords:
Adsorption ( jstor )
Colloids ( jstor )
Flocculation ( jstor )
Molecular weight ( jstor )
Molecules ( jstor )
Particle interactions ( jstor )
Polymers ( jstor )
Solvents ( jstor )
Trucks ( jstor )
Viscosity ( jstor )
Polyvinyl alcohol ( lcsh )
Rheology ( lcsh )
Silica ( lcsh )
Suspensions (Chemistry) ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1988.
Bibliography:
Includes bibliographical references.
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Chandra Khadilkar.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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AA00004812_00001 ( sobekcm )

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THE EFFECT OF ADSORBED POLY (VINYL ALCOHOL)
ON THE PROPERTIES OF MODEL SILICA
SUSPENSIONS













BY

CHANDRA KHADILKAR


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1988

OF F LIBRARIES




THE EFFECT OF ADSORBED POLY (VINYL ALCOHOL)
ON THE PROPERTIES OF MODEL SILICA
SUSPENSIONS
BY
CHANDRA KHADILKAR
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
x
1988
TJ OF F LIBRARIES


ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to Dr. M.D. Sacks for
his invaluable guidance and financial support during the course of this
study. I would like to thank him for allowing me to work in the area of
colloid science and for demonstrating the importance of reproducibility
and hard work in scientific research.
I am also grateful to Dr. C.D. Batich, Dr. L.L. Hench,
Dr. D.O. Shah, and Dr. E.D. Whitney, members of supervisory committee,
for their helpful suggestions.
I would like to thank Mr. H.W. Lee, Mr. O.E. Rojas, Mr. G.W.
Scheiffele, Mr. S.D. Vora, and Mr. T.S. Yeh for their help during the
course of this work. Thanks are also due to Professor B. Moudgil for
allowing me to use spectrophotometer and Gel Permeation Chromagraphy
apparatus. I would like to thank Ms. Hazel Feagle for typing, editing,
and compilation of this dissertation.
I also wish to convey my sincere gratitude to my parents and to my
wife for their encouragement and support during the course of this work.
ii


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ii
LIST OF TABLES viii
LIST OF FIGURES ix
ABSTRACT xxii
CHAPTERS
I. GENERAL INTRODUCTION AND AIM OF THE STUDY 1
II. ADSORPTION OF POLYMERS AT SOLID/LIQUID INTERFACE 5
Introduction 5
Description of Adsorbed Polymer 9
Adsorbed Amount of Polymer (A) 9
Bound Fraction (p) 11
Direct Surface Coverage (0) 11
Segment Density Distribution (X,Y,Z) 11
Theoretical Models 12
Adsorption Energy Parameter (xs) 13
Segment-Solvent Interaction Parameter (x) 14
Polymer Adsorption Theories: General Framework ... 15
Results of Polymer Adsorption Theories 18
Effect of Adsorption Energy Parameter 19
Effect of Solvent-Segment Interaction
Parameter 19
Effect of Polymer Concentration and Molecular
Weight 19
Experimental Techniques 21
Adsorbed Amount of Polymer 21
Trains, Bound Fraction, and Direct Surface
Coverage 22
Thickness of Adsorbed Layer 23
Segment Density Distribution 26
Adsorption Energy Parameter 27
Adsorption of Polydisperse Polymers 27
Experimental Results for PVA-Water System 29
Properties of Poly (Vinyl Alcohol), PVA 29
PVA Characterization 31
Adsorbed Amount of Polymer 33
The Nature of Solid 33
iii


The Effect of Acetate Content 34
The Effect of Molecular Weight of PVA 35
The Effect of Solvency 36
Adsorbed Layer Properties and Adsorbed
Amounts 36
The Segment Density Distribution 38
The Effect of Particle Radius 38
PVA Adsorption on Silica 40
Summary 40
III. ELECTROSTATIC INTERACTIONS BETWEEN COLLOIDAL PARTICLES. 42
Introduction 42
Development of Charge at Solid-Liquid Interface 45
Dissociation of Surface Groups 45
Adsorption of Potential Determining Ions 46
Adsorption of Ionized Surfactants 46
Isomorphic Substitution 46
Electrical Double Layer 46
Double Layer Interactions 54
Interaction Between Two Flat Plates 56
Interaction Between Two Spherical
Particles 57
Van der Waal's Interaction 59
Microscopic or Van der Waal's Method 60
Flat Plates 60
Spherical Particles 61
Retardation Effect 62
Effect of Medium on the Van der Waal's Attraction .... 63
Macroscopic Approach 64
Hamaker Constants 65
The Effect of Polymer Layer on Van der Waal's
Attraction 66
Potential Energy Curves and the DLVO Theory 69
The Effect of Hamaker Constant 72
The Effect of Surface Potential 74
The Effect of Electrolyte Concentration 74
The Effect of Particle Radius 77
The Stability-Instability Approach 77
Kinetics of Coagulation 79
Summary 82
IV. EFFECT OF ADSORBED POLYMER ON DISPERSION STABILITY. ... 84
Introduction 84
Factors Influencing Steric Stabilization 85
The Adsorbed Amount of Polymer 85
The Solvent-Segment Interaction Parameter 86
The Effective Hamaker Constant and the Size
of Particles 86
Applications and Advantages of Steric Stabilization ... 87
Thermodynamic Basis of Steric Stabilization 89
iv


Polymer Solution Thermodynamics 90
The Entropy of Mixing 90
The Enthalpy of Mixing 92
The x Parameter 93
The Theta Point 95
Classification of Steric Stabilization 98
Quantitative Theories of Steric Stabilization 99
The Three Domains of Close Approach 101
The Interpenetration Domain 103
Interpenetration Without Mixing 107
The Potential Energy Diagrams 111
Time Scale of Approach of the Second Interface. ... 111
The Potential Energy of Interaction 113
Thermodynamically Limited Stability 115
Non-thermodynamically Limited Stability 119
Bridging Flocculation 122
Kinetic Aspects of Bridging Flocculation 127
Kinetics of Flocculation 130
The Potential Energy Diagrams for Bridging
Interparticle Interactions 133
V. STRUCTURE OF SUSPENSIONS 140
Introduction 140
Fractal Geometry 141
Models of Aggregate Formation 143
Eden Growth 143
Diffusion Limited Aggregation (DLA) 145
Cluster-Cluster Aggregation (CCA) 146
Hierarchial Model 148
Kinetics of Aggregation 148
Smoluchowski's Equation 150
Equilibrium Properties of Suspensions 153
The Order-Disorder Transition 154
VI. RHEOLOGICAL BEHAVIOR OF COLLOIDAL DISPERSIONS 160
Introduction 160
Viscosity Definition 161
Classification of Rheological Behavior of Colloidal
Dispersions 163
Newtonian Dispersions 165
Pseudo-plastic Dispersions 165
Dilatant and Shear Thickening Dispersions 167
Bingham Plastic Dispersions 167
Thixotropic Dispersions 167
Factors Affecting Rheological Behavior of Colloid
Dispersion 170
Interparticle Interactions 170
Brownian Motion 170
Hydrodynamic Interactions 171
Rheological Behavior of Stable Systems 171
v


The Effect of Adsorbed Layer 175
Stable Dispersions with Soft Interactions 180
Rheological Behavior of Flocculated Dispersions 181
Structure of Flocculated Dispersions 181
Flow Behavior of Flocculated Dispersions 186
Flocculated Dispersions Showing No Time Dependence . 187
Elastic Floe Model 190
VII. MATERIALS PROPERTIES AND CHARACTERIZATION/
EXPERIMENTAL PROCEDURES 196
Introduction 196
Silica as a Model Material 196
Silica Preparation and Characterization 199
Silica Washing Procedure 203
Silica Calcination Treatment 204
Effect of Calcination Treatment on the Nature of
the Silica Surface 204
Silica Size and Size Distribution 210
Silica Surface Area 215
Silica True Density 217
Poly Vinyl Alcohol 219
Synthesis and Properties 219
Solubility, Solution Behavior and Interfacial
Activity 219
Fractionation of As-received PVA 221
Acetate Content of Polymer 222
Molecular Weight and Molecular Weight
Distribution 223
Viscometry 223
Gel Permeation Chromatography 226
Conformation and Solution Parameters 239
PVA Configuration Parameters 239
The x Parameter 242
Adsorption Measurements 246
Suspension Preparation Procedure 247
Suspension Characterization 251
Electrophoresis 251
Rheological Measurements 253
Consolidation and Green Microstructure 254
Summary 256
VIII. RESULTS AND DISCUSSION 257
Electrostatically Stabilized Dispersions 257
Effect of Adsorbed PVA on the Properties of Model
Silica Dispersions 272
Effect of Amount Adsorbed 272
Effect of Silica Calcination Treatment 284
Effect of PVA Molecular Weight 318
Fractional Surface Coverage 318
Effect of Suspension pH 334
vi


Effect of pH and Molecular Eight with the Plateau
Adsorbed Amount of Polymer 346
Selection of Optimum Molecular Weight 367
Effect of Adsorbed Layer Thickness on the Maximum
Solids Loading in Suspension 372
Effect of Silica Particle Size 375
Effect of PVA Degree of Hydrolysis on Adsorption
Behavior 388
Effect of Solvent Quality on the Rheological
Behavior 400
IX. SUMMARY AND SUGGESTIONS FOR FUTURE WORK 414
Summary 414
Suggestions for Future Work 417
LIST OF REFERENCES 420
BIOGRAPHICAL SKETCH 436
vii


LIST OF TABLES
Table Page
3.1 Effect of Ionic Strength and Valency on the Electrical
Double Layer Thickness 51
3.2 Compilation of Hamaker Constants for Silica-Water-PVA
System 67
4.1 Classification of Steric Stabilization 100
6.1 Classification of Flow Behavior 164
7.1 Geometric Mean and Specific Surface Area of Various Silica
Lots Used 216
7.2 Viscometric Molecular Weight and Acetate Content of PVA
Fractions 225
7.3 Molecular Characteristics and Elution Volume of PEO
Calibration Standards 228
7.4 Gel Permeation Chromatography Data for Unfractioned PVA
Samples 229
7.5 Effect of Molecular Weight Elution Volume Calibration
Curve on the GPC Results 235
7.6 Comparison of Viscometric Molecular Weight and GPC Results
for Various PVA Fractions 238
7.7 PVA Dimensions in Solution 243
8.1 Properties of Silica and PVA Used to Investigate the Effect
of Adsorbed Amount of PVA on the Suspension Properties. . 274
8.2 Properties of Silica and PVA Used to Investigate the Effect
of Silica Calcination Temperature on the Suspension
Properties 286
8.3 Molecular Weight and Degree of Hydroxylation of PVA Samples
to Study the Effect of Degree of Hydroxylation on the
Suspension Properties 390
viii


LIST OF FIGURES
Figure Description Page
2.1 Schematic representation of an adsorbed polymer
molecule 8
2.2 Conformations of adsorbed polymer molecules (a) single
point attachment, (b) train-loop-tail adsorption,
(c) flat multiple site attachment, (d) random coil,
(e) nonuniform segment distribution, and
(f) multilayer 10
2.3 Plots of (a) degree of occupancy, 9, and (b) the
effective layer thickness, 6, as a function of the
adsorbed amount of PVA on Agl 37
2.4 Plot of segment density as a function of distance from
a surface for adsorbed PVA on PS-latex 39
3.1 Fraction of double layer potential versus distance from
a surface: (a) curves for 1:1 electrolyte at three
concentrations and (b) curves for 0.001 M symmetrical
electrolytes of three different valance types 52
3.2 Schematic illustration of the variation of potential as
a function of distance from a charged surface in the
presence of a stern layer, subscripts o at wall, 6 at
stern surface, d in diffuse layer 52
3.3 Schematic illustration of the effect of adsorbed
polymer layer on Van der Waal's attraction 68
3.4 Total potential energy of interaction V(d) = VR(d)
+ V^(d) where VR(d) is the potential energy of
repulsion due to double-layer interactions and V^(d) is
attractive potential due to Van der Waal's
interactions 71
3.5 The effect of the Hamaker constant on the total
interaction energy curves 73
3.6 The effect of zeta potential on the total interaction
energy curves 75
ix


3.7
76
The effect of concentration of 1:1 electrolyte on the
potential energy curves
3.8 The effect of particle radius on the total interaction
energy curves 78
3.9 Theoretical dependence of stability ratio on
electrolyte concentration 81
4.1 The three domains of close approach of sterically
stabilized flat plates, (i) Noninterpenetration (d >
2L); (ii) Interpenetration (L £ d £ 2L);
(iii) Interpenetration plus compression (d < L).. . 102
4.2 The distance dependence of the steric interaction
energy for two equal spheres of radius a, stabilized
by polymer layers with different segment density
distribution functions. (1) exponential; (2) constant;
(3) Gaussian; (4) radial Gaussian, d is the minimum
distance between surfaces of the spheres, 6 is the
barrier thickness, and AGS is the interaction energy.
4.3 The free energy of interaction between particles
covered by equal tails (f) and equal loops (a). For
particles covered by equal tails, (b) gives the volume
restriction effect and (c) the osmotic repulsion; (f)
is the resultant of adding (b), (c), and (e) 116
4.4 Schematic illustration of the effect of segment-solvent
interaction parameter, x, on the potential energy
diagram (A) poor solvent, x > 0.5, (B) theta solvent, x
= 0.5, and (C) good solvent, x<0.5 118
4.5 The free energy of interaction of polystyrene latex
particles stabilized by poly (vinyl alcohol) according
to Hesselink, Vrij, and Overbeek (1971); stabilizer
molecular weight 1, 8,000; 2, 17,000; 3, 28,000; 4,
43,000 120
4.6 Plots showing the effect of particle size and adsorbed
amount on the depth of the minima in the total
potential energy of interaction 120
4.7 Schematic illustration of bridging flocculation with
adsorbed polymer 123
4.8 The effect of thickness of the electrical double layer
on bridging flocculation 125
4.9 Schematic diagram showing mixing, adsorption, and
flocculation upon addition of polymeric flocculent. 128
x


132
4.10 The effect of solids loading and 6/a on collisions
Zf/Z0
4.11 Schematic representation of the approach of a second
(uncovered) particle to a covered one. It was assumed
that, at large interparticle distance H the number of
segments which adsorb on the (originally) bare particle
per unit area is equal to the number of segments per
unit area which would lie beyond Hi in the absence of
the second particle (i.e., shaded area) 134
4.12 Schematic representation of the bridging process.
(a) At large distances, a loop of i segments has its
unpurturbed configuration. (b) After adsorption of the
first segment, two bridges of i/2 segments each are
formed. (c) At shorter distances, two bridges of ifl
segments and a train of segments adsorbed on the
second surface 136
4.13 The total free energy of interaction between coated and
uncoated plates as a function of distance of
separation 137
5.1 Eden cluster produced by monomer-cluster growth.. . 144
5.2 An aggregate grown by the DLA process 144
5.3 An aggregate grown by the CCA process 147
5.4 Schematic plot of phase diagram for monosized spherical
particles. The volume fraction of solids as a function
of ionic concentration is plotted. Solid lines are
theoretical phase boundaries 156
5.5Schematic illustration showing (a) "hard" and (b)
"soft" interactions between particles. The potential
energy of interaction as a function of distance of
separation is plotted 158
6.1 Schematic illustration of the concept of viscosity
under laminar flow conditions 162
6.2 Schematic plots of (a) shear stress versus shear rate
and (b) viscosity versus shear rate for various types
of flow behaviors 166
6.3 Schematic representation of thixotropic flow behavior,
(a) shear stress versus shear rate, and (b) viscosity
versus shear rate plots 169
6.4 Schematic plot of dependence of relative viscosity on
the volume fraction solid in suspension 173
xi


6.5 Plot of relative viscosity versus volume fraction latex
particles of different sizes. Data was fitted using
Krieger equation 176
6.6 Plot of relative viscosity versus dimensionless shear
rate, Yj., for monodisperse suspensions of polystyrene
spheres at 0 = 0.50 in different fluids 178
6.7 Schematic illustration of the effect of shear on the
stability of suspensions 185
6.8 Schematic illustration of flow curve parameters for
pseudoplastic flow behavior 188
7.1 Schematic representation of various types of surface
groups present on the silica surface 206
7.2 The diffuse reflectance Fourier transform infrared
spectra of silica powders calcined at various
temperatures 207
7.3 Concentration of surface silanol groups as a function
of the temperature of calcination 209
7.4 Scanning electron micrograph of silica powder 212
7.5 A histogram (number of particles in a given diameter
class versus particle diameter) of a typical silica
batch 213
7.6 Plot of particle size distribution for silica
determined by x-ray sedimentation 214
7.7 Gas pycnometer density versus calcination temperature
for SO2 powders 218
7.8 Molecular weight calibration curves for PEO standards
and commercial PVA88. Log M is plotted as a function
of elution volume 231
7.9 GPC chromatograms, showing refractive index detector
response, h, as a function of elution volume for
several unfractioned PVA samples 233
7.10 GPC chromatograms, showing the effect of acetone
fractionation of Vinol 540 and Vinol 203 polymer
samples on distribution widths 237
7.11 Stockmayer-Fixman plot for PVA88 240
xii


7.12Plot of absorbance versus PVA concentration in solution
for various molecular weight polymers 248
8.1 Plot of zeta potential versus suspensions pH for 20
vol.% SiC>2 at ionic strength of 1 x 10^ moles/liter
NaCl 258
8.2 The effect of zeta potential on (a) shear stress versus
shear rate and (b) viscosity versus shear rate plots.
8.3 Schematic illustration showing the structural breakdown
of a floe due to applied shear 261
8.4 DLVO plots of potential energy of interaction versus
distance of separation at indicated £ potentials. . 263
8.5 Plot of relative viscosity versus zeta potential for 20
vol.% SiC>2 suspensions 265
8.6 Plot of (a) Cpp versus zeta potential and (b) Cpp
versus zeta potential square for 20 vol.% SiC>2
suspensions 266
8.7 Plot of (a) extrapolated yield stress versus zeta
potential and (b) extrapolated yield stress versus zeta
potential square for 20 vol.% SO2 suspensions. . 269
8.8 Plots of specific volume frequency versus pore radius
obtained by mercury porosimetry for sedimented samples
with indicated pH values 271
8.9 Adsorption isotherm for 20 vol.% SO2 suspensions
prepared with varying concentration of PVA with
molecular weight 24,000 at pH 3.7 273
8.10 Plots of relative viscosity versus shear rate for 20
vol.% SO2 suspensions prepared at pH 3.7 with varying
PVA concentrations and pH 7.3 (-60 mV zeta potential)
with no PVA 277
8.11 Plot of yield stress versus adsorbed amount of PVA with
molecular weight = 24,000 at pH 3.7 279
8.12 Plot of relative viscosity versus adsorbed amount of
PVA for 20 vol.% SO2 suspensions prepared with varying
PVA concentration 280
8.13 Plot of hysteresis area versus adsorbed amount of PVA
of 20 wt.% SO2 suspensions 281
8.14 Plot of (a) sediment density versus fraction plateau
coverage and (b) median pore radius versus fraction
xiii


plateau coverage for compacts prepared from 20 vol.%
SO2 suspensions with varying PVA concentrations at pH
3.7 and pH 7.3 with no polymer 283
8.15 Adsorption isotherms for silicas calcined at various
temperatures with PVA molecular weight = 215,000 at pH
3.7 285
8.16 Plot of plateau adsorbed amount versus silica
calcination temperature 288
8.17 Plots of plateau adsorbed amount versus silica
calcination temperature for (a) PEO adsorption on Cab-
O-Sil and (b) PVA adsorption on Cab-O-Sil 290
8.18 Plot of relative viscosity versus shear rate for 20
vol.% suspensions prepared with uncalcined silica and
with silica powder calcined at 500C and 700C. ... 292
8.19 Plot of yield stress versus calcination temperature for
20 vol.% silica suspensions at pH 3.7 prepared with
silica particles calcined at various temperatures. PVA
concentration in solution was sufficient to achieve
plateau coverage of the silica particles 293
8.20 Plot of relative viscosity versus calcination
temperature for 20 vol.% silica suspensions at pH 3.7
prepared with silica particles calcined at various
temperatures. PVA concentration in solution was
sufficient to achieve plateau coverage of the silica
particles 294
8.21 Plot of hysteresis area versus calcination temperature
for 20 vol.% silica suspensions at pH 3.7 prepared with
silica particles calcined at various temperatures. PVA
concentration in solution was sufficient to achieve
plateau coverage of the silica particles 295
8.22 Schematic plot of the total interaction energy versus
distance of separation between two polymer-coated
particles with varying adsorbed amounts 297
8.23 Schematic illustration (a) showing silica surface
covered with only enough PVA to make particles
hydrophobic and (b) excess of PVA adsorption prevents
cocervation 299
8.24 Schematic plot of (a) disjoining pressure as a function
of distance of separation for two parallel water films
stabilized with PVA film and (b) disjoining curve for
homopolymer. The specific attractive component in the
xiv


case of PVA derives from the hydrophobic interactions
between acetate groups 301
8.25Plot of (a) relative density for slip cast samples
versus silica calcination temperature and (b) median
pore radius versus silica calcination temperature.. 303
8.26Plot of mercury porosimetry data for slip cast samples
prepared from uncalcined and 700C calcined silica
powders. Samples were prepared from 20 vol.% silica
suspensions at pH 3.7 with the plateau coverage of SiC>2
particles with PVA of molecular weight = 215,000. . 304
8.27 Plot of (a) relative viscosity versus adsorbed amount
and (b) relative viscosity versus fractioned plateau
coverage for 20 vol.% SiC>2 suspensions prepared with
uncalcined and 700C calcined powders 306
8.28 Schematic illustration of floe structures formed with (a)
uncalcined and (b) 700C calcined silica particles at low
surface coverages with adsorbed polymer 307
8.29The depth of the free energy minimum as a function of
the total amount of polymer between the surfaces at
various solvency conditions for two molecular weights
of polymer. 0^ is the total amount of polymer between
the plates expressed as the number of equivalent
monolayers, Afmn is the interaction energy, and r is
the number of segments per chain 309
8.30 Plots of (a) yield stress versus adsorbed amount and
(b) yield stress versus fractioned plateau coverage for
20 vol.% silica suspensions prepared with 700C
calcined and uncalcined silicas with varying PVA
concentration in solution 311
8.31 Schematic illustration shows the total number of
bridges, ntotal, formed between two spherical particles
of radius a separated by distance 2h 312
8.32 Plots of (a) relative density of gravity cast samples
versus fraction plateau coverage and (b) median pore
radius versus fraction plateau coverage for compacts
prepared from 20 vol.% SO2 suspensions of uncalcined
and 700C calcined silicas with varying PVA
concentrations 315
8.33 Plots of (a) hysteresis area versus adsorbed amount and
(b) hysteresis area versus fraction plateau coverage
for 20 vol.% SO2 suspensions prepared with uncalcined
and 700C calcined silicas with varying PVA
concentrations 316
xv


8.34Plots of adsorption isotherms for two PVA samples with
different molecular weights (i.e., 24,000 and 215,000
g/mole) determined using 20 vol.% SiC>2 suspensions at
pH 3.7 319
8.35 Plots of (a) relative viscosity versus adsorbed amount
and (b) relative viscosity versus fraction plateau
coverage for two PVA samples with different molecular
weights 321
8.36 Plots of relative viscosity versus shear rate for 20
vol.% SO2 suspensions prepared using indicated
molecular weight PVA samples at fixed PVA concentration
in solution. Adsorbed amount was the same (0.15 mg
PVA/m^ SO2) for all suspensions 322
8.37 Plot of relative viscosity versus PVA molecular weight
(log scale) at fixed adsorbed amount of polymer.. . 323
8.38 Plots of (a) yield stress versus adsorbed amount and
(b) yield stress versus fraction plateau coverage for
20 vol.% silica suspensions prepared with two PVA
samples with different molecular weights 325
8.39 Plot of yield stress versus PVA molecular weight (log
scale) at fixed adsorbed amount 326
8.40 Plots of (a) hysteresis area versus adsorbed amount and
(b) hysteresis area versus fraction plateau coverage
for 20 vol.% SO2 suspensions prepared with two PVA
samples of different molecular weights 328
8.41 Plot of hysteresis area versus PVA molecular weight
(log scale) at fixed adsorbed amount of polymer.. . 329
8.42 Plots of (a) relative density of sedimented samples
versus fraction plateau coverage and (b) median pore
radius versus fraction plateau coverages for compacts
prepared using two PVA samples of different molecular
weights 330
8.43 Plot of relative density of compacts prepared using
gravity casting and slip casting versus PVA molecular
weight at fixed adsorbed amount of polymer 332
8.44 Plots of relative density versus Cpp (= 0p/p) for two
PVA samples with different molecular weights. Results
for compacts prepared at pH 3.7 and pH 7.6 with no
added PVA are also shown 333
xv i


8-45 Adsorption isotherm of PVA with molecular weight =
24,000 at two different suspensions pH's 335
8.46 Plot of plateau adsorbed amount of PVA with molecular
weight = 200,000 as a function of suspensions pH. . 336
8.47 Plots of (a) relative viscosity of 20 vol.% suspensions
versus adsorbed amount of PVA with molecular weight =
24,000 and (b) relative viscosity versus fraction
plateau coverage for suspensions prepared at pH values
3.7 and 7.6 with varying PVA concentration 338
8.48 Plots of (a) yield stress versus adsorbed amount of
polymer and (b) yield stress versus fraction plateau
coverage for suspensions prepared at pH values 3.7 and
7.6 340
8.49 Schematic illustration shows the prevention of bridging
flocculation due to diffuse electrical double layer.. 341
8.50 Plots of (a) sediment density versus fraction plateau
coverage and (b) median pore radius versus fraction
plateau coverage for compacts prepared from 20 vol.%
SO2 suspensions with varying amounts of PVA
concentrations at indicated suspension pH values. . 342
8.51 Plot of zeta potential versus adsorbed amount of PVA
with molecular weight 24,000 at suspensions pH 7.6. 344
8.52 Schematic plots of effective adsorbed layer thickness,
6, as a function of measured zeta potential at
indicated ionic strengths. The zeta potential with no
adsorbed polymer is = -65 mV 345
8.53 Plots of plateau adsorbed amounts of polymer versus PVA
molecular weight (log scale) at suspensions pHs 3.7 and
7.8. Suspensions were prepared using 20 vol.% SO2
with sufficient concentration of PVA in solution with
varying PVA molecular weights at indicated pH values. 347
8.54 Plots of relative viscosity versus shear rate for 20
vol.% SO2 suspensions at pH 7.8 with plateau adsorbed
amounts of PVAs having indicated molecular weights. 348
8.55 Plots of relative viscosity versus shear rate for 20
vol.% SO2 suspensions at pH 3.7 with plateau adsorbed
amounts of PVAs having indicated molecular weights. 349
8.56 Plots of relative viscosity versus PVA molecular weight
(log scale) for 20 vol.% SO2 suspensions with plateau
adsorbed amounts of polymer at indicated pH values. 351
xvii


8.57 Plots of hysteresis area versus PVA molecular weight
(log scale) for 20 vol.% SO2 suspensions at pH 3.7
with plateau adsorbed amounts of polymers 354
8.58 Plots of yield stress versus PVA molecular weight (log
scale) for 20 vol.% SO2 suspensions with plateau
adsorbed amounts of polymers at indicated pH values.. 355
8.59 Plots of (a) relative density of gravity cast samples
versus PVA molecular weight and (b) median pore radius
versus PVA molecular weight for compacts prepared using
20 vol.% SO2 suspensions with plateau adsorbed amounts
of polymers with different molecular weights at
indicated pH values 357
8.60 Plots of (a) relative density of slip cast samples
versus PVA molecular weight and (b) median pore radius
versus molecular weight for compacts prepared using 20
vol.% SO2 suspensions with plateau adsorbed amounts of
PVAs with different molecular weights at indicated pH
values 358
8.61 Schematic plots of relative viscosity versus volume
fraction of solids in suspensions for particles with
the indicated thicknesses of adsorbed polymer. This
thickness, 6, is indicated as a fraction of the
particle radius 360
8.62 Plot of (a) relative viscosity versus volume fraction
silica in suspensions prepared at pH 7.6 and (b)
relative viscosity versus volume fraction of latex
particles as reported by Krieger 361
8.63 Plot of relative density of gravity cast samples versus
volume fraction silica at pH 7.6 362
8.64 Plots of the adsorbed layer thickness, 6, versus PVA
molecular weight determined from the relative viscosity
values of 20 vol.% suspensions prepared at pH 7.8.
Also shown are the radius (Rg) and diameter (2 x Rg) of
gyration of the polymers in solution as determined by
intrinsic viscosity measurements 364
8.65 Plots of adsorbed PVA layer thickness, 6, on SO2
particles versus square root of PVA molecular weight.
Adsorbed layer thicknesses of PVA onto PS latex
particles is also shown 368
8.66 Schematic plots of minimum molecular weight (log scale)
required to stabilize suspensions as a function of
particle radius with different values of A /V^
ratios. The solid lines separates stable ana unstable
xviii


regions of suspensions prepared with spherical
particles of fixed size with varying molecular weight
or suspensions prepared with fixed molecular weight and
varying particle radius 370
8.67 Schematic plots of maximum true solids loading, 0 ,
achievable in suspensions prepared with spherical
monosized particles of varying size using indicated
molecular weights of polymer 374
8.68 Adsorption isotherms for PVA with molecular weight =
215,000 for 20 vol.% SO2 suspensions prepared using
0.4 urn and 0.7 um size particles with varying PVA
concentrations 376
8.69 Plots of (a) relative viscosity versus adsorbed amounts
of PVA and (b) relative viscosity versus fraction
plateau coverage for 20 vol.% SO2 suspensions prepared
using 0.4 um and 0.7 urn size particles 378
8.70 Schematic illustration showing the effect of adsorbed
layer thickness, 6, on the hydrodynamic volume of two
different size particles 379
8.71 Plots of (a) yield stress versus adsorbed amount of PVA
and (b) yield stress versus fraction plateau coverage
for 0.4 urn and 0.7 pm size particles 381
8.72 Schematic illustration showing (a) concentration
profile, 0(Z), of adsorbed layer consisting of three
regions (i) proximal (very sensitive to the details of
the interactions), (ii) central (self-similar), and
(iii) distal (controlled by a few loops and tails) and
(b) "self-similar grid" presentation of an adsorbed
polymer layer 382
8.73 Plots of hysteresis area versus fraction plateau
coverage for 0.4 urn and 0.7 urn size particles 384
8.74 Plots of (a) relative density of sedimented samples
versus fraction plateau coverage and (b) median pore
radius versus fraction plateau coverage for 0.4 um and
0.7 um size particles 386
8.75 Plots of normalized median pore radius (i.e., median
pore radius/particle radius) versus fraction plateau
coverage for 0.4 um and 0.7 um size particles 387
8.76 Adsorption isotherms for PVAs with similar molecular
weights but varying degree of hydroxylation of 20 vol.%
SO2 suspensions 389
xix


8.77Plots of relative viscosity versus shear rate of 20
vol.% silica suspensions prepared with different PVAs
with indicated degree of hydroxylation. The
suspensions are prepared at pH 3.7 with the plateau
adsorbed amounts of PVAs
393
8.78 Plots of specific volume frequency versus pore radius
for gravity cast samples prepared from 20 vol.% SiC>2
suspensions with plateau adsorbed amounts of PVAs with
indicated degree of hydroxylation 394
8.79 Plots of (a) relative viscosity versus adsorbed amounts
and (b) relative viscosity versus fraction plateau
coverage for 20 vol.% SO2 suspensions prepared using
varying PVA concentrations in solution. The degree of
hydroxylation of different PVAs used is shown in the
figure 396
8.80 Plots of hysteresis area versus fraction plateau
coverage for 20 vol.% siC>2 suspensions prepared using
PVAs with indicated degree of hydroxylation 397
8.81 Plots of (a) yield stress versus adsorbed amount and
(b) yield stress versus fraction plateau coverage for
20 vol.% SO2 suspensions prepared using PVAs with
indicated degree of hydroxylation 398
8.82 Plots of (a) relative density of gravity cast sample
versus fraction plateau coverage and (b) median pore
radius versus fraction plateau coverage of compacts
prepared from 20 vol.% SO2 suspensions with varying
PVA concentration in solution. The degree of
hydroxylation for various polymers is indicated in the
figure 399
8.83 Plots of (a) relative viscosity versus Na2S04
concentration and (b) yield stress versus Na2S04
concentration for 20 vol.% SO2 suspensions with
plateau coverages of particles with adsorbed polymer.
402
8.84Plots of relative viscosity versus shear rate for 20
vol.% SO2 suspensions with varying Na2SC>4
concentration
403
8.85 Plot of hysteresis area versus Na2SC>4 concentration in
solution suspensions were prepared at pH 7.8 with the
plateau adsorbed amount of PVA with molecular weight =
200,000 405
8.86 Plots of (a) relative density of gravity cast samples
versus Na2S04 concentration and (b) median pore radius
versus Na2SC>4 concentration 406
xx


8.87
Plots of (a) relative density of slip cast samples
versus Na2SC>4 concentration and (b) median pore radius
versus Na2SC>4 concentration in solution 407
8.88 Plots of Na2SC>4 concentration versus PVA concentration
in solution. Solid line separates single phase region
(i.e., true polymer solution) from the two phase region
(i.e., precipitated polymer and solvent) 410
8.89 Plots of relative viscosity versus Na2SC>4 concentration
for suspensions of polymer coated particles and
suspensions prepared with no added polymer at pH 7.8. 411
8.90 Plots of (a) relative density of gravity cast samples
versus Na2SC>4 concentration and (b) median pore radius
versus Na2SC>4 concentration of compacts prepared from
suspensions of polymer coated particles and suspensions
with no added polymer at pH 7.8 412
xxi


Abstract of a Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
THE EFFECT OF POLY (VINYL ALCOHOL) ON THE
PROPERTIES OF MODEL SILICA SUSPENSIONS
By
Chandra S. Khadilkar
December 1988
Chairman: Dr. Michael D. Sacks
Major Department: Materials Science and Engineering
The effect of interparticle interactions on the properties of model
suspensions of monosized, spherical silica particles was investigated.
The electrostatic interactions between particles were controlled by
changing suspension pH. The effect of adsorbed polymer was investigated
using water soluble poly (vinyl alcohol), PVA. The adsorption was
dependent on a variety of factors including the overall polymer
concentration in suspension, PVA molecular weight, PVA degree of
hydroxylation, silica surface hydroxylation, and suspension pH. The
adsorption characteristics were correlated with the state of particulate
dispersion in suspension using rheological measurements. Suspension and
green compact properties were highly dependent upon the fraction of
silica surface covered by the adsorbed polymer and the thickness of the
adsorbed polymer layer. Suggestions are made for selecting polymer
xxii


molecular weight to prepare stable suspensions with higher solids
loading.
xxiii


CHAPTER I
GENERAL INTRODUCTION AND AIM OF THE STUDY
Particle/liquid suspensions are prepared in various ceramic
processing operations such as mixing, milling, spray granulation of
powders, slip casting, tape casting, extrusion, etc. The control of
rheological behavior of suspensions is important to maximize the
processing efficiency and also to obtain the desired properties of the
final products. The rheological behavior of particle/liquid suspensions
can be modified by changing (1) particle characteristics (e.g., size and
shape distribution), (2) liquid characteristics (e.g., viscosity), (3)
particle concentration (solids loading of the suspensions), (4)
interparticle forces in suspension, and (5) additives like dispersants,
polymers, etc.
The interparticle interactions in suspensions can be broadly
classified into two categories: (1) electrostatic interactions and (2)
interactions due to adsorbed polymer. The electrostatic interactions
arise due to development of charge at the solid/liquid interface. The
electrically-charged interface leads to the formation of diffuse
"electrical double layer" surrounding each particle. The overlap of two
electrical double layers during Brownian encounter gives rise to
repulsive forces between the particles which can overcome the Van der
Waal's attraction. The effects of surface potentials (or charge) and
ionic concentrations on the stability of dispersions can be
quantitatively described using the "DLVO" theory (Chapter III).
1


2
Organic polymers are extensively used as processing aids in ceramic
forming operations. In addition to acting as binders and plasticizers,
polymers are often used to control the rheological properties of
dispersions. The interactions between particles with added polymer are
highly dependent on the adsorption behavior of polymer on particle
surfaces. Generally, at low polymer concentrations in solution,
suspensions can be destabilized (i.e., particles are aggregated) due to
"bridging flocculation" where a polymer molecule can adsorb
simultaneously onto two or more particles. At sufficiently high polymer
concentrations in solution, under certain conditions, stable suspensions
(i.e., in which primary particles are well separated) can be prepared.
Two important factors controlling the interactions between the particles
are (1) the fractional coverage of the particle by adsorbed polymer
(which affects the bridging flocculation) and (2) the thickness of the
adsorbed layer (which governs the stability at complete coverage in a
good solvent).
A model ceramic powder (i.e., well-characterized, agglomerate-free,
spherical, narrow-sized silica) was used in this investigation. The
electrostatic interactions between particles were modified by varying the
suspension pH and ionic strength. The effect of surface potentials cn
the flow curve parameters (i.e., extrapolated yield stress, relative
plastic viscosity, etc.) were analyzed using the "elastic floe" model.
Water soluble poly (vinyl alcohol), PVA, was used to investigate
the effect of adsorbed polymer on the suspension properties. PVA being a
neutral polymer, the effects of electrostatic interactions and the
effects of adsorbed polymer on the dispersion stability can be studied


3
less ambiguously. The concentration of PVA in solution can be readily
determined; hence, the adsorbed amounts of PVA could be determined
easily. The polymer adsorption behavior and rheological properties on
the same system are rarely reported in literature. The adsorption
experiments are generally carried out using dilute suspensions of high
surface area, agglomerated powders, whereas rheological measurements are
conducted on concentrated suspension where adsorption behavior and
interparticle interactions are not very well defined. In this study,
most of the rheological measurements and adsorption experiments were
carried out at the same solids loading (= 20 vol. %). The adsorption of
PVA onto the silica surface was dependent on a variety of factors
including the overall solution concentration, polymer characteristics
(PVA molecular weight and degree of hydroxylation), silica surface
nature, suspension pH, etc. The adsorption characteristics were
correlated with the state of particulates in suspension using rheological
measurements. Suspension rheological properties and green compact
properties (e.g., relative density, median pore radius, etc.) were highly
dependent on the fraction of silica covered by the adsorbed polymer and
the thickness of the adsorbed layer.
A floe structure (i.e., floe compactness, strength, etc.) consistent
with the rheological behavior and green compact characteristics is
proposed. (Experimental techniques to obtain this information are not
yet readily available). Finally, some suggestions regarding selection of
polymers to achieve stable suspensions with high solids loading are made.
In the first part of thesis, we deal with general and theoretical
concepts related to this work. In Chapter II, we describe the polymer


A
adsorption behavior at solid/liquid interface. Chapter III deals with
the electrostatic interactions between colloidal particles, and Chapter
IV deals with the effect of adsorbed polymer on dispersion stability.
The effect of interparticle interactions on the state of particulate
dispersion in suspension is described in Chapter V. The correlation
between suspension structure and rheological behavior is discussed in
Chapter VI. The silica preparation and characterization is described in
Chapter VII. PVA fractionation and characterization is also described in
Chapter VII. In Chapter VIII, we discussed experimental results obtained
in this study. First, we describe the effect of electrostatic
interactions on the rheological behavior of 20 vol.% silica suspensions.
Subsequently, PVA adsorption behavior on silica particles is correlated
with the rheological properties of suspensions and green compact
characteristics.


CHAPTER II
ADSORPTION OF POLYMER AT SOLID/LIQUID INTERFACE
Introduction
Adsorbed polymer at solid/liquid interface has a profound effect on
the stability behavior of suspensions. Typically, at low polymer
dosages, flocculation of particles in the suspension can occur, while at
the high polymer concentration, stable suspensions can be prepared.
Polymer adsorption behavior has been extensively studied because of the
present and the potential applications in industry, technology, and
medicine. The polymer adsorption behavior is important in various
processes (tape casting, slip casting, extrusion of ceramic parts,
adhesion, separation of polymers, soil improvement, etc.) and various
products (paints, cosmetics, magnetic tapes, pharmaceuticals, dyes,
foods, lubricants, etc.). Theoretical interests in the polymer
adsorption process stems from the fact that insight can be gained into
the nature of the forces acting between polymer segments and surfaces and
between particles coated with the adsorbed polymer. From the ceramic
processing point of view, by the optimization of the adsorption process,
one can control both the suspension properties as well as consolidated
microstructures obtained from these suspensions.
The adsorption behavior of the polymer is different from the
adsorption behavior of the small molecules (e.g., gas adsorption at
solid/gas interface, surfactant adsorption, etc.). This difference
5


6
arises because polymers have a large number of internal degrees of
freedom (i.e., flexibility).
Polymer molecules have a random coil type of arrangement in the
solution. A full description of the shape of a molecule requires the
description of the relative positions of each atom of the molecule (the
configuration, see Flory 69). It is common to assume that a polymer
chain consists of a number of connected chain segments for the
theoretical treatments. Then, the conformation of the adsorbed polymer
is described by specifying the relative positions of the endpoints of
these segments (Kuhn 34). To understand the adsorption behavior of
polymers, one needs to consider both the energetics and Kinetics of the
adsorption process. Polymer molecules experience short-range attractive
forces near the adsorbing interfaces. For each polymer segment adsorbed
on the surface, there is a decrease in the free energy of the system. At
the same time, upon adsorption, the random coil structure of the polymer
molecule in the solution will be distorted, which will lead to a decrease
in the number of conformations of the adsorbed polymer. This entropic
factor will oppose the adsorption process. Final conformation of the
adsorbed polymer depends on this subtle balance between the entropic and
the enthalpic terms. Also, though there is a small energy decrease due
to adsorption process per segment of the polymer molecule, large number
of segments per molecule can be adsorbed, and the total energy decrease
per molecule can be quite large. Actual bonding mechanism between the
polymer segment and the surface may involve various interactions such as
electrostatic interactions, Van der Waal's interactions, hydrogen bonding
or hydrophobic bonding.


7
The actual conformation of the adsorbed polymer near solid-liquid
interface depends upon various factors such as solid-liquid, polymer-
liquid and polymer-solid interactions, flexibility of the polymer
molecule, concentration of the polymer in the solution, etc.
Experimentally, it has been found that the adsorption isotherm
(i.e., adsorbed amount of polymer as a function of equilibrium
concentration of polymer in solution at fixed temperature and pressure)
of high molecular weight polymers display high affinity type of
character, i.e., a steep initial part of the adsorption isotherm followed
by a pseudo-plateau region. Adsorbed amounts at the plateau region are
typically on the order of approximately one to four mg/m^. This amount
is much more than that can be accommodated in a close-packed monolayer of
the polymer segments. This led early investigators to propose the
conformation of the adsorbed polymer as shown in Figure 2.1 (Jenkel and
Rumbach, 51). From Figure 2.1, it is clear that not all of the segments
of the adsorbed polymer are in direct contact with the surface. This
type of conformation is commonly called as "train-loop-tail"
conformation. "Train" is defined as a sequence of consecutive segments
in direct contact with the surface. "Loop" is defined as a sequence of
segments with the end segments in direct contact with the surface, and
remaining segments are in contact with the solution. "Tail" is defined
as the portion of the chain with end segments in contact with the
surface. From this type of conformation, it is clear that there is a
significant change in the polymer conformation upon adsorption compared
to the random coil arrangement in the solution and that the adsorbed
polymer has sufficient extension in the solution. Actual details of the


8
Train-Loop-Tail conformation of Adsorbed
Polymer
Figure 2.1
Schematic representation of an adsorbed polymer molecule.


9
conformation are dependent on the various factors mentioned previously.
Some other types of conformations of the isolated adsorbed polymer
molecules are shown in Figure 2.2 (Sato and Ruch 80).
Description of Adsorbed Polymer
The adsorption behavior of the polymers at the solid/liquid
interface is commonly characterized using the following parameters.
Adsorbed Amount of Polymer (A)
This is the most commonly measured parameter in any adsorption
experiment and is usually obtained by "solution depletion" techniques.
Adsorbed amounts are commonly expressed in the units of mg of polymer
adsorbed per unit area of the solid surface. It is also common to define
the dimensionless adsorbed amount or adsorbance, y, as the number of
segments adsorbed per surface site or the number of (equivalent) complete
monolayers than can be formed from the adsorbed amounts (Cohen Stuart
80a). This means that y is the ratio between the adsorbed weight per
unit area and the weight adsorbed per unit area in a complete monolayer,
A
mon.
Y = A/A,
mon
(2.1 )
As we have seen earlier, there is a conformational change upon the
adsorption of the polymer. Hence, for a complete description of polymer
adsorption, it is not sufficient to measure only the adsorbed amount of
polymer, as similar adsorbed amounts could be obtained for the thick
adsorbed layer of polymer with low segment concentrations or thin
adsorbed layer with high segment concentration.


10
Figure 2.2 Conformations of adsorbed polymer molecules (a) single
point attachment, (b) train-loop-tail adsorption,
(c) flat multiple site attachment, (d) random coil,
(e) nonuniform segment distribution, and (f) multilayer
(Sato and Ruch, 1980).


11
Bound Fraction (p)
This is defined as the fraction of adsorbed amount of polymer in
direct contact with the surface, i.e.,
p = A^r/A (2.2)
where A^r is the amount of polymer adsorbed per unit area in trains. The
bound fraction measures the change in the conformation upon adsorption.
The p close to one indicates the polymer lies flat on the surface having
two-dimensional structure; on the other hand, p close to 0 indicates
polymer has essentially random coil shape with no significant change in
conformation (Figure 2.c).
Direct Surface Coverage (6)
This is defined as the fraction of the available surface sites
occupied by polymer segments.
0 = Atr/A
mon
(2.3)
where A<.r and A as defined previously and from the definition of 0
mon c
and p, it follows that
Y = 0/p (2.4)
Segment Density Distribution p (X, Y, Z)
Segment density is the time-averaged volume fraction of segments per
unit volume in the vicinity of the polymer molecule. In solution, many
polymers adopt random coil conformation. For such a random coil, the
segment density averaged on all conformations is usually Gaussian in any
direction passing through the center of the molecule. (More accurately,
the distribution is better represented by a prolate ellipsoid of
revolution, see Flory 53.) Upon adsorption, there is a significant
change in the conformation of the polymer molecules. At low adsorption


12
density (i.e., isolated polymer molecules) the segment density
distribution will be dependent upon Z (i.e., distance from the adsorbing
surface) and also on X, Y (ie., position parallel to the surface). At
higher adsorption densities, there will be a significant lateral
interpenetration of the molecules (since the volume percent occupied by
the segments is approximately ten percent at typical segment densities,
i.e., there is enough empty space in a given polymer molecule) and the
segment density distribution will be dependent on Z only. This segment
density distribution is the most important feature of the adsorbed
polymer in the theory of steric stabilization. Various types of segment
density distributions have been proposed for the adsorbed polymer and
will be reviewed later (Napper 83). The segment density distribution
determines the extension of the adsorbed polymers into the solution
phase. This extension of the adsorbed polymer layer can be expressed in
terms of a thickness parameter such as the root mean square layer
thickness 6, which is defined as follows:
00
CO
(2.5)
o
o
Theoretical Models
The adsorption behavior (and hence, conformation) of polymers at
solid-liquid interface is governed by various factors including polymer
segment-surface, surface-solvent, and segment-solvent interactions. The
driving force for the adsorption is the reduction in the net free energy
due to bonding of a polymer segment to the surface. This binding process
involves the removal of the solvent molecule from the surface and
replacement with the polymer segment. Energy change associated with this


13
process is denoted by the dimensionless adsorption energy parameter, Xg-
This polymer adsorption process is opposed by the entropy changes
associated with the changes in conformation upon adsorption. Polymer
molecules have three-dimensional random coil structure in solution, and
upon adsorption, some of the segments (i.e., train) are restricted to two
dimensions.
This process is associated with considerable loss of entropy per
molecule, and the magnitude depends on the chain length and flexibility.
At equilibrium, the distribution of polymer molecules between the surface
and the solution is essentially determined by the following five
independent parameters: concentration of the polymer in solution, chain
length (molecular weight), flexibility, adsorption energy parameter, and
polymer-solvent interaction parameter. Variables, such as temperature,
time (kinetics), and polydispersity, may also play an important role.
Below, we will define some of these variables and discuss the effect of
these independent variables on the adsorption.
Adsorption Energy Parameter (Xc)
As mentioned earlier, the bonding between the surface and the
segment is the driving force for adsorption. A precise definition of Xs
is due to Silberberg (Silberberg 68). The energy change (i.e., net
enthalpy change) associated with the transfer of a polymer segment in a
pure polymer from a bulk site to a surface site, minus the corresponding
energy change for a solvent molecule in pure solvent, is denoted by
-XskT. By choosing proper reference states (i.e., pure polymer and pure
solvent), Xg is made independent of solvent-segment interactions and


14
depends only on the nature of the surface. It is clear from the
definition that for Xs > 0 indicates segments are preferred over the
solvent by the surface. Due to adsorption, polymer loses part of its
conformational entropy and will oppose the adsorption. Hence, a certain
minimum adsorption energy, denoted by Xsc i-s required for the adsorption
to take place.
Segment-Solvent Interaction Parameter (x)
Linear, flexible polymer molecules have a random coil shape in the
solution. For an "ideal" chain (i.e., polymer molecule represented as
chain consisting of volumeless, non-interacting statistical units), this
random coil conformation can be described by random walk in three
dimensions. For an ideal chain, it can be shown that the radius of
gyration is proportional to the molecular weight of the polymer and can
be represented by the following equation:
1/2 a M*0'5 (2.6)
where is the root mean square radius of gyration and is the
molecular weight of the polymer (Flory 53). However, real chain segments
have volume and interact with each other, and this effect is usually
represented by the excluded volume effect, which can be defined in terms
of either segment-segment interaction energy or in the framework of the
Flory-Huggins theory of polymer solutions as the segment-solvent
interaction parameter, x- The parameter x represents the quality of the
solvent. Theoretically, x < 0 for a very good solvent, X = 0 for an
athermal solvent, and x = 1/2 for an ideally poor or a 0 solvent, x
represents an exchange process (and in such a way that), xkT represents


15
the difference in energy of a solvent molecule immersed in a pure polymer
compared with one surrounded by pure solvent molecules. For x > 0, the
solvent is poor and segment-segment contacts are preferred over segment-
solvent contacts. The adsorption of polymer at the solid-liquid
interface leads to an increase in the concentration of the segments near
the surface, and the segment-segment or the segment-solvent interaction
parameter has a profound effect on the adsorption. In poor solvents,
adsorbed amounts of polymer are larger due to the fact that segments
prefer other segments over solvent molecules. The x parameter is one of
the basic parameters of the polymer solution thermodynamics, and hence,
plays an important role not only in the polymer adsorption behavior but
also in the theories of steric stabilization.
Polymer Adsorption Theories: General Framework
The aim of polymer adsorption theories is to try to relate the
conformations of the adsorbed polymer molecules to independent variables
(such as adsorption energy parameter, xSr solvent-segment interactions,
X, molecular weight of the polymer, M, its concentration in solution,
etc.). Various investigators have paid a lot of attention to the
theoretical development of these models (for e.g., see Eirich 77;
Hoeve 65,66,70,71; Silberberg 62,67,68; Scheutjeans and Fleer 79,80;
deGennes 80,82,87). Since a large number of polymer molecules are
involved, a statistical thermodynamic approach is usually employed.
Contributions of the entropy and the energy of many chains in a given
concentration gradient near the surface are evaluated. To evaluate these
contributions, the thermodynamic theory of polymer solutions, as


16
developed by Flory and Huggins, is often employed (Flory 53). More
recently, scaling concepts have been applied to the polymer adsorption
(deGennes 87).
To describe polymer adsorption behavior, according to the methods of
statistical mechanics, the partition function of the system is set up
(see, Cohen Stuart 80a).
-U/kT
Q = E ft e
where ft is the degeneracy, i.e., the number of different ways of
(2.7)
arranging systems having energy, U.
To evaluate ft and U, appropriate reference states are chosen
(usually unmixed pure components). To evaluate Q, all possible energy
states of the system are considered. To determine the equilibrium state
of the system, Q is maximized. For example, Q is related to the free
energy of the system by
G = -kT Ln Q (2.8)
and hence, G should be at minimal for equilibrium. To evaluate and
maximize Q, one needs to make certain assumptions, and the quality of
these assumptions leads to differences between various theories.
The earlier polymer adsorption theories (for e.g., Hoeve, Roe,
Silberberg, etc.) were formulated to describe the conformations of
isolated polymer chains. In these theories, train-loop-tail type of
polymer conformation was assumed. The chain conformation statistics,
such as average train, loop sizes, and tail, loop size distributions were
computed (but, tails were ignored). The segment-surface interaction was
taken into account using the adsorption energy parameter, Xs. (i-e.,
first layer interactions), but the solvent-segment interaction was
ignored. Since these early theories ignored solvent-segment interactions
1


17
and formulated for isolated chains (i.e., non-interacting with other
adsorbed polymer molecules), their usefulness is limited.
Later theories employed Flory-Huggins polymer solution model to
account for segment-segment and segment-solvent interactions (Hoeve,
Silberberg, Roe, Scheutjeans and Fleer). The theories of Hoeve and
Silberberg start with the train-loop model of the adsorbed polymer (i.e.,
tails were neglected) and the conformation probabilities of the adsorbed
chains were computed. To evaluate U, assumptions were made regarding the
shape of the segment density profile. Hoeve assumed the exponential
profile for the evaluation of U (i.e., segment-segment, segment-solvent
interactions) while Silberberg assumed step function for the segment
density profile (Silberberg 68). This assumption regarding segment
density profile to evaluate U has been avoided in the recently developed
theories of Roe, Scheutjeans and Fleer (Roe 74, Scheutjeans and Fleer
79,80). These theories do not assume a model for the mode of polymer
adsorption of an individual molecule. They derive the partition function
for the mixture of free and adsorbed polymer chains and solvent
molecules; a number of ways of arranging polymer chains and solvent
molecules in a given (arbitrary, but fixed) concentration gradient near
the surface was determined. Maximization of the partition function gives
the equilibrium concentration profile (Scheutjeans and Fleer 79,80). To
evaluate U, segment-segment and segment-solvent interactions were
calculated using Flory-Huggins theory. Roe neglected the tails in
evaluation of the energy term and his model predicts overall segment
density profile. Later theory (SF) gives the complete distribution of
polymer conformation near the surface and gives information about train.


18
loop, and tail distributions. The important difference between the Roe
and the SF theory is the contribution of the long dangling tails to the
overall segment density profile at relatively large distances from the
interface, and these long tails can play an important role in the
flocculation and stabilization of the system. Unfortunately, no simple
analytical expressions are available with these theories and substantial
computational time is required to calculate the conformation of the
adsorbed polymer.
The scaling relation applicable to semi-dilute polymer solutions has
been extended for the polymer adsorption problem by deGennes (see,
deGennes 87). This theory is limited to athermal solvent (good solvent
with x = 0) with moderate adsorption energies. Though this theory is
analytical, the results are in the form of power-laws without exact
coefficients.
In the next section, some of the predictions of these polymer
adsorption theories will be discussed. These theories are mostly
applicable for mono-dispersed homopolymers polymer on a homogeneous
substrate. Other complications arising due to inhomogeneous surface
structure and charge at the interface are not taken into account.
Results of the Polymer Adsorption Theories
In this section, the dependence of properties of the adsorbed
polymer layer (such as total surface coverage (i.e., the adsorbed amounts
A), direct surface coverage (i.e., occupancy in the first layer 0), bound
fraction (p), root mean square layer thickness (6), segment density
distribution) on various independent variables (such as solution


19
concentration, chain length, x and xs) will be reported. Only general
trends will be reported. For detailed comparison between various
theories, recent reviews are recommended (Fleer 87, Fleer and Lyklema
83).
Effect of Adsorption Energy Parameter (xJ
All theories predict that the adsorption energy parameter should
exceed a certain critical (non-zero) adsorption energy, i.e., xs > XSc
for the adsorption of polymer to take place. This critical adsorption
energy parameter xsc corresponds to the minimum energy necessary to
compensate the unfavorable entropy loss of the segment, upon adsorption,
compared to segment in solution. If the xs > Xsc then the surface
coverage increases sharply with increasing xs* At high Xs values, the
surface becomes saturated and the total surface coverage 0, becomes
independent of xs* for lattice theories, the critical value of xSc is
related to the lattice type employed (Scheutjeans and Fleer 79,80).
Effect of Solvent-Segment Interaction Energy Parameter (x)
The effect of solvent on adsorption behavior is pronounced. The
adsorbed amount of polymer increases while bound fraction p decreases
with decreasing quality of the solvent (i.e., as the solvent becomes
poorer). (Though the adsorbed amount and the direct surface coverage are
increased, the ratio, the bound fraction p is decreased.) Average loop
and tail size are increased with decreasing the quality of solvent
(Scheutjeans and Fleer 79,80).
Effect of Polymer Concentration and Molecular Weight
Adsorption isotherms for high molecular weight polymers are the high
affinity type. Adsorbed amounts are high for higher molecular weight


20
polymer. For a good solvent, adsorbed amounts tend to reach a limiting
value for high molecular weights, but from a poor solvent, various
theories predict different trends. The increased amounts of adsorbed
polymer with increasing molecular weight (and concentration in solution)
are accommodated by increases in the average size of the loop and tails.
This leads to an increase in the thickness of the adsorbed layer with
increasing polymer concentration and molecular weight. For a low polymer
concentration (typically, p < .01, where 0p is the volume fraction of
polymer in the solution), the adsorbed amount in a 0-solvent increases
linearly with log Mw. This dependence, as predicted by SF theory, is
different from the empirical power law relations, i.e., A a M, where A
is the plateau adsorbed amount of polymer (mg/m^) and a is an empirically
determined constant. For dilute concentration, the bound fraction
decreases with increasing molecular weight. Root mean square thickness
also increases with the molecular weight and the concentration.
The effect of tails, ignored in the earlier theories, is important
at finite polymer concentrations. (At extremely low polymer
concentrations, the adsorbed polymer lies in relatively flat
configuration). These tails will affect the average layer thickness and
the segment density profile at the outer region of the adsorbed layer,
and hence, will be very important for colloidal stability.
The segment density distributions predicted by various theories are
very important in various theories of steric stabilization. As mentioned
earlier, Hoeve assumed an exponential segment density distribution (Hoeve
65) whereas Silberberg assumed it to be step function (Silberberg 68).
Roe and SF theory can calculate segment density distributions without any


<
assumptions (Roe 74; Scheutjeans and Fleer 79,80). Both theories predict
an approximately exponential segment density distribution near the
surface. At larger distances, S.F. theory predicts a high density
*
compared to Roe's theory. This higher density is due to long dangling
tails. These long tails can dominate interparticle interactions and the
hydrodynamics of coated particles. The tail length increases with
increase in the molecular weight almost linearly for high molecular
weight polymer (Cohen Stuart 80a). Hence, molecular weight of the
polymer is among the most important variables for controlling rheological
and other properties of the dispersion. In the next section, various
experimental techniques used to characterize polymer adsorption will be
briefly described.
Experimental Techniques
An excellent review is available for the details of various
techniques (Cohen Stuart, et al. 86a). Adsorbed amount A, direct surface
coverage 0, bond fraction p, layer thickness 6, and segment density
distribution (Z), can be characterized experimentally.
The Adsorbed Amount of Polymer
Generally, a "solution depletion" technique is used to determine the
adsorbed amount of polymer. In this method, from the equilibrium and
initial known concentrations of polymer in solution, the adsorbed amount
of polymer is determined. Centrifugation is commonly used to separate
particles from the suspension, and the supernatant is analyzed. Various
analytical techniques are used to determine the solution such as
gravimetric, complex formation to give species which adsorb in the UV or


22
visible part of the electromagnetic spectrum (Zwick 65), etc. In certain
cases, direct determination of the adsorbed amount is possible (e.g., IR,
ellipsometry).
Trains, Bound Fraction. Direct Surface Coverage
Techniques to determine these parameters are broadly classified into
spectroscopic methods, electrochemical methods, and calorimetric methods.
(1) Spectroscopic Techniques: Spectroscopic techniques include infrared
(IR), electron spin resonance (ESR), and nuclear magnetic resonance
(NMR). Due to specific interactions between polymer segments and solid
surface, shifts in a characteristic band, either for the adsorbate (e.g.,
the carbonyl or benzene group in a polymer) or the adsorbent (e.g., the
hydroyl group on an oxide) is utilized to determine the bound fraction
using IR spectroscopy (for e.g., see Killmann 76; Takahashi et al. 80;
Fontana et al. 61,63,66; Korn et al. 80a,80b, etc.). ESR can only be
used for spin labeled polymers (Robb and Smith 74). Mobility criteria
are used to distinguish between adsorbed and non-adsorbed segments.
Segments having different mobility will give different magnetic
relaxation times and hence mobility. NMR technique is also based on the
mobility criterion to estimate p.
(2) Electrochemical Methods: Adsorbed neutral polymer affects the
electrical double layer properties (such as change in the double layer
capacitance and shift in the point of zero charge, PZC). These
properties can be utilized to determine the fraction of surface area
occupied by segments, 0 (Koopal 78).
(3) Microcalorimetric Approach: In this metliod, the heat of immersion
of adsorbent is measured at various adsorbed amounts of polymer.


23
Calibration can be achieved from the heat off immersion of a monomeric
analog compound of the adsorbed group (Korn et al. 80a,80b; Cohen Stuart
et al. 82; Killmann et al. 71; Hair 77).
There are several problems associated with these techniques, such as
distinguishing between contributions due to adsorbed and non-adsorbed
polymer, differentiation between train and loop segments, etc.
Thickness of the Adsorbed Layer
Methods to determine the thickness of the adsorbed polymer layer can
be divided into two broad categories: (1) ellipsometry and
(2) hydrodynamic methods.
(1) Ellipsometry: In this method, change in the properties of the
elliptically polarized light upon reflection, due to the adsorbed
polymer, is measured. From the measured phase shift and the amplitude of
the reflected light, under the assumption of homogeneous polymer layer,
the ellipsometric thickness and the refractive index of the film can be
calculated (e.g., see Killmann 76,77). Clearly, the assumption of
homogeneous segment density distribution leads to ambiguity in the
measured thickness. Also, this method is suitable for flat surfaces with
good reflectivity, and hence, limited mostly to bulk metal substrates and
some oxide films.
(2) Hydrodynamic Methods: These methods measure the extent of outward
shift of the slip plane due to the adsorbed polymer layer. Essentially,
they measure the drainage characteristics of the adsorbed layer, and
since the drainage characteristics of the adsorbed polymer layer (i.e.,
consisting of loops and tails) are not known, an exact definition of the
hydrodynamic thickness is not possible. Also, if the adsorbed polymer


24
layer is not homogeneous (e.g., at low adsorption densities), this method
may over estimate the thickness. To measure the hydrodynamic thickness,
several techniques have been employed.
(i) Capillarity: In this method, the inside wall of the fine capillary
is coated with the adsorbed polymer layer, and the decrease in the flow
rate due to adsorbed layer is measured (i.e., the effective decrease in
the diameter of the capillary is determined). In this method,
homogeneous coating of the capillary is critical (e.g., see Rowland and
Eirich 66; Priel and Silberberg, 78).
(ii) Viscometry: In this method, the increase in the viscosity of the
suspension of dispersed particles due to adsorbed polymer layer is
measured. Due to the adsorbed polymer layer, there is an increase in the
effective radius of the particle, hence, higher effective volume fraction
solids in the suspension. For dilute dispersions, the intrinsic
viscosity, [nl, is measured (Barsted et al. 71; Dawkins and Taylor 80)
while, for the concentrated suspensions, the high shear viscosity is
determined (Dobroszkowski and Lambourne 66). To get information about
the adsorbed layer, other parameters affecting the viscosity must be
taken into account. Factors such as electroviscous effects, aggregation
of particles, effect of shear on the thickness of the adsorbed layer,
polymer degradation, etc. can complicate the interpretation. In this
study, the adsorbed polymer thicknesses are determined from the high
shear rate viscosities. The effect of PVA molecular weight on the
adsorbed layer thickness will be reported later.
(iii) Photon Correlation Spectroscopy: In this method, the diffusion
coefficients of the particles with and without adsorbed polymer are


25
measured using "Doppler" broadening of an incident laser line (for e.g.,
see van den Boomgaard et al. 78; Kato et al. 81; Garvey et al. 76;
Killmann et al. 85,86,88, etc.). From the measured diffusion
coefficients, the particle radius, and hence, the adsorbed layer
thickness, is calculated using the Stokes-Einstein equation:
D = kT/6nn0Rh (2.9)
where R^ = hydrodynamic radius of the particle, nQ viscosity of the
suspension medium, T = absolute temperature, and k = Boltzmann constant.
This method is limited to monosized, spherical particles.
To obtain accurate values of the adsorbed layer thickness, diffusion
coefficients are determined at various solid concentrations and the plot
of diffusion coefficient vs. solid concentration is extrapolated to zero
solids concentration. This procedure then eliminates the effects of
interparticle interactions on the diffusion coefficient. As the adsorbed
polymer layer is usually a small fraction of the particle radius, it is
essential to obtain accurate values of particle radii with and without
polymer.
(iv) Electrokinetics: In this method, the decrease in the electrokinetic
potential (Zeta potential) of the charged particles, due to adsorption of
neutral polymer, is measured to estimate the hydrodynamic thickness.
Essentially, this method measures the outward displacement of the slip
plane due to adsorbed polymer. Again, several complications are present
in interpreting this data and the measured thickness is very sensitive to
the ionic strength of the solution (Koopal 78; Cohen Stuart et al.
84a,84b,85).


26
(v) Other Methods; Sedimentation rate (i.e., change in the
sedimentation coefficient due to the adsorbed polymer layer), (Garvey et
al. 74) direct force-distance measurements (i.e., half the distance at
which a certain minimum force is observed between two approaching
surfaces, coated with the adsorbed polymer), etc., have been used to
measure the thickness of the adsorbed layer (Sonntag et al. 82; Lubetkin
88; Gotze and Sonntag 87,88).
From the above discussion, it is clear that there are several
techniques available to measure the layer thickness, but due to
uncertainty about the effect of the tail-loop conformation of the
adsorbed polymer on the properties of measured layer thickness, each
technique gives some kind of average property of the adsorbed layer.
Recently, it has been shown that the hydrodynamic thickness is
essentially determined by the tails and that the loop contribution is
negligible (Cohen Stuart 86a).
Segment Density Distribution
Small angle neutron scattering (SANS) has been used to obtain the
segment density distribution for the adsorbed polymer. In this
technique, the ability of neutron to distinguish between hydrogen and
deuterium atoms, due to different coherent scattering cross sections, is
utilized. By using a suitable mixture of H2O/D2O, particles can be
contrast matched, and by measuring scattering intensity at various
angles, information about the adsorbed polymer can be obtained.
Unfortunately, the method is not sensitive enough yet to detect small
concentrations of the tail segments at larger distances (Barnett et al.
82).


27
Adsorption Energy Parameter
As mentioned earlier, Xs represents the energy change associated
with the exchange process of replacing solvent molecule from the surface
with the polymer segment. Cohen Stuart has proposed a method to measure
this parameter (Cohen Stuart 80a). In this method, a second solvent
having strong affinity for the surface is added to the solution. With
increasing concentration of the solvent (displacer), eventually polymer
can be desorbed completely from the surface, and from this critical
displacer concentration, Xs can be determined.
Adsorption of Polydisperse Polymers
Polymers used in practical application are mostly polydisperse,
and polydispersity has an important effect on the adsorption behavior.
This effect arises due to the difference in the adsorption behavior of
long and short molecules. Long molecules are preferentially adsorbed
over short molecules. As discussed earlier, the adsorption process is
controlled by the various free energy changes related to adsorption
process. The energy decrease due to adsorption of polymer segments is
opposed by the entropy loss. The decrease in entropy arises from two
contributions: (1) the loss of configurational entropy due to the
unmixing of polymer molecules and solvent molecules and (2) the decrease
in conformation entropy due to the decrease in number of different
possible arrangements of the polymer as a result of the attachment of
segments to the solid surface. In dilute solutions, which is of
practical interest in most adsorption studies, the conformational entropy
losses for one large chain compared to two shorter chains, each half the


28
length of the loner chain length, are similar. However, the decrease in
configurational entropy for the two short chains is twice that of the
longer chain. Hence, adsorption of the long chain is preferred over the
two short chains. Also, the long chain can displace the short chains
from the surface. Theory based on the above principles has been
developed, and it is successful in explaining various effects arising due
to the polydispersity of polymers (Cohen Stuart 80a,80b,84a; Koopal 81).
(1) Rounding of the adsorption isotherms: All polymer theories and
experiments using monodispersed polymers exhibit sharp adsorption
isotherms, whereas polydisperse samples lead to more rounded isotherms.
(2) Irreversibility: The observed irreversibility of adsorption
isotherms can be explained from the above theory.
(3) The effect of amount of adsorbent on the amount adsorbed and
the shape of the adsorption isotherm: Experimentally, it has been found
that the adsorption isotherms are sharper and the plateau adsorbed
amounts larger if the adsorption isotherms were determined using dilute
dispersions (i.e., small surface area/volume of the solution ratio)
(Koopal 81).
Preferential adsorption of the high molecular weight polymer over
the low molecular weight leads the fractionation of the polymer. The
molecular weight distribution in the adsorbed layer is shifted to higher
molecular weight as compared to that in the solution (Furusawa et al.
82). Also, from the practical processing point of view, the
preferentially adsorbed high molecular weight fraction may dominate the
properties of the dispersions, such as the rheology and the consolidation
behavior.


29
Additional factors, such as the charge on the solid surface and the
effect of the solid surface structure, will be discussed in the
experimental section.
Experimental Results for PVA-Water System
Properties of Poly (Vinyl Alcohol). PVA
PVA is a commonly used polymer as a steric stabilizer in aqueous
media. PVA is prepared by alcoholysis of poly (vinyl acetate), PVAc, and
is generally not hydrolysed fully. (Often the degree of hydrolyses is on
the order of 88 mole percent or greater). It is, therefore, a copolymer
of PVAc and PVA. This can be represented as follows:
CH2 = CHOAc Bolyorisation > -CH2-CH-CH2-CH-CH2-
vinyl acetate 0|c OAc
poly (vinyl acetate)
> -CH2-CH-CH2-CH-CH2-CH-
OH OAc OAc
poly (vinyl alcohol) containing acetate groups,
where symbol OAc represents acetate groups, -COOH. Copolymers have been
shown to be the best steric stabilizer for the following reasons:
Usually, they consist of two types of segments having contrasting
solubilities in a given aqueous or non-polar dispersion media. Less
soluble segments of the copolymer will preferentially attach themselves
to the solid surface, and thus, the polymer is firmly anchored to the
surface. Thus, such a copolymer then consists of two types of segments,
anchoring moieties and stabilizing moieties. Strong attachment of the
polymer with the surface will prevent desorption of polymer or lateral
movement of the polymer on the surface when two particles coated with


30
with adsorbed polymer will be encountered during Brownian collisions. In
good solvents, the stabilizing moieties will offer repulsion, due to
overlap of segments (i.e., due to osmostic effects).
This copolymeric nature of PVA makes direct comparison between
experimental results and theoretical predictions difficult and only
qualitative trends will be discussed. Also, the previously discussed
theories were mainly developed for the monosized, homopolymers, while
commercial polymers are usually polydispersed. In this section,
available literature on the PVA adsorption behavior on various substances
will be reviewed.
Two types of PVA's, i.e., partially hydrolysed, PVA88,
(approximately 88 percent hydrolysed) and completely hydrolysed, PVA98,
(greater than 98 percent hydrolysed) have been employed in various
experimental investigations. Boomgaard et al. fractionated as-received
polymer by sequential addition of acetone (i.e., a non-solvent) to
approximately five wt.% PVA solution (van den Boomgaard et al. 78).
Garvey et al. used preparative scale Gel Permeation Chromatography (GPC)
to obtain narrow molecular weight fractions (Garvey et al. 74). These
investigators did not measure the polymer molecular weight distribution
or the polydispersity index My^/M^ where Mn is number average molecular
weight and My^ is the weight average molecular weight. Acetone fractions
gave fractions of varying molecular weights, but the degree of hydrolysis
was also different for different fractions. (The degree of the
hydrolysis decreased from the approximately ninety to approximately
eighty mole percent with the decrease in the molecular weight) (van den
Boomgaard et al. 78). Other studies used as-received commercial polymer.


31
PVA Characterization
From IR and UV spectra, Koopal concluded that the commercial samples
used in his study were atactic and contained no or very little impurities
(impurities, such as 1,2 glycol units and one or two conjugated groups if
present are present as the end groups) (Koopal 78). The acetate group
distribution is "blocky" for the PVA88, and acetate groups were
distributed more or less "randomly" for the PVA98. Dunn (Dunn 80) and
Barnett et al. (Barnett et al. 82) assumed that for the PVA88, the
average block consists of three acetate groups. They assumed that the
average acetate block size increases and the width of the acetate block
size distribution increases with increasing PVA molecular weight.
The Mark-Houwink-Sakurada (MHS) equation was generally used to
determine the viscometric average molecular weight, Mv of the polymer.
[n] = kMva (MHS) (2.10)
where Cn] is the intrinsic viscosity and k and a are KKS empirical
constants. The values of the constants k and a used by various
investigators are different. This can lead to different values of Mv for
the same polymer (i.e., same [r|]). The values of k and a are dependent
on the temperature, the acetate content, and the polydispersity of the
sample. The value of constant a was in the range of 0.64 to 0.60 for
PVA98 and was in the range of 0.71 to 0.63 for PVA88 (e.g., see Koopal
78). The PVA solution properties (i.e., the segment-solvent interaction
parameter x) and polymer molecule dimensions in solution (i.e., radius of
gyration, end to end distance, etc.) were determined from the measured
intrinsic viscosity and molecular weight for a series of samples with
varying molecular weights. If these two quantities are not independently


32
available, then the MHS equation (Equaiton 2.9) was used to determine Mv
from the measured intrinsic viscosity. Hence, the values of solution
properties and polymer molecule dimensions were influenced by the values
of constants k and a used. For this reason, these values (i.e., x, *,
etc.) should be compared with caution since different values of constants
k and a are employed by various investigators.
(1) The Seqement-Solvent Interaction Parameter: x
Detailed comparison of the available values of x parameter has been
made by Koopal (Xoopal 78). At 25C in aqueous solutions, the x values
were in the range of 0.462 to 0.488 for PVA88 and in the range of 0.475
to 0.499 for PVA98. Thus, the degree of hydrolysis does not have
significant effect on the solvent quality for typical PVA polymers. It
should be noted that the aforementioned values of the x parameter
indicate that water is a relatively poor solvent for PVA. This also
suggests that intersegmental interactions occur in the polymer chain
(Koopal 78). van den Boomgaard et al. have determined the effect of
temperature on the x parameter (van den Boomgaard et al. 78). With
increase in the temperature from 25C to 50C, the x value changed from
0.464 to 0.485, indicating worsening of the solvency for PVA. Tadros and
Vincent and Barker and Garvey have determined the effect of type and
electrolyte concentration on the solvency (Tadros and Vincent 79; Barker
and Garvey 80). With increase in the electrolyte concentration, the
solvency decreases. They also found that Na2SC>4 has greater effect on
solvency compared to NaCl, i.e., lower concentration can change solvent
quality.


33
(2) The Xc Parameter:
One needs to measure the adsorption energy parameter for both
acetate and alcohol groups since the PVA adsorption mechanism may involve
adsorption of these two groups. Heat of adsorption of low molecular
weight analogues may be useful with this respect, but xs values are not
available for the various systems investigated. Typically, values in the
range 1 2 kT have been assumed for xs (Barnett et al. 82).
The Adsorbed Amount of Polymer
The Nature of Solid
PVA adsorption behavior has been studied on Agl solid particles (Fleer
71), Agl particles and sol (Koopal 78), silicas of various types (e.g.,
precipitated, Cab-o-sil, Ludox, Tadros 78), polystyrene latex particles
(made by emulsion and dispersion stabilization, Garvey, et al. 74,76),
montmorillenite clays (Greenland 62, Heath and Tadros 83). Due to
differences in the chemical nature and surface heterogeneities (e.g.,
silica powder calcined at various temperature leads to various types of
surface groups and the concentration of each group is dependent on the
thermal history; polystyrene latex particles made by dispersion
polymerization technique have more hydrophobic surface, etc.), it is not
possible to compare adsorption data on the same basis. Also, other
variables, such as solid concentration, aging time, polydispersity of the
polymer samples, and the method of sample preparation, etc., can have an
important effect. The detailed comparison with the literature results
will be made in the results and discussion section. Here, we will
briefly describe the important results.


34
The Effect of Acetate Content
The plateau adsorbed amounts (i.e., "saturation adsorbed amount" or
adsorbed amounts at "complete" surface coverage) are generally more for
PVA88 (i.e., partially hydrolysed PVA) than PVA98 (i.e., fully hydrolysed
PVA) of similar molecular weights. This effect have been found on
various substrates (e.g., Agl, Koopal 78; silica, Tadros 78; PS latex,
Barnett, et al. 82). The larger adsorbed amount for PVA88 can be due to
(i) increase in the adsorbed amount in the first layer (i.e., A ) or
mono
due to (ii) the formation of larger loops and tails. The adsorbed amount
differences in the monolayer between these two polymers cannot account
for this difference, hence, the contribution of the first layer to this
difference is small (Koopal 78). The increase in adsorption with
increasing acetate content has been attributed to greater adsorption in
loops and tails. As explained earlier, x (PVA98) and x (PVA88) are only
slightly different, and the difference in adsorption behavior cannot be
explained by the solvency effect. The preferential adsorption of acetate
groups onto Agl particles has been determined by Koopal from the
electrochemical method (i.e., from the shift in the point of zero change)
(Koopal 78). The preferential adsorption of acetate groups leads to
accumulation of acetate groups in the first layer and gives an important
contribution to the gain in the free energy of adsorption (i.e., Xs1> the
adsorption energy parameter for acetate groups is expected to be larger
than the adsorption energy parameter for alcohol groups, xs2) The
differences in flexibility of these two polymers (i.e., it can be assumed
that the flexibility of PVA88 is relatively lower due to bulky acetate
groups) will have effect on the size of the trains and loops. The train


35
size is limited by the length of the acetate blocks and lower flexibility
of PVA88 can set constraints on the minimum loop size (i.e., the average
loop size is expected to be larger for PVA 88). The preferential
adsorption of acetate groups has been confirmed by NMR studies on PS
latex particles (Barnett, et al. 82).
The Effect of Molecular Weight of PVA
As expected from the theoretical results, increased plateau adsorption
with increasing molecular weight of PVA has been observed. Generally,
this effect is represented by the following power law relation:
A = KMy,a (2.11)
where A is the plateau adsorbed amount of polymer and K and a are
empirical constants. (Please note that the Equation 2.11 is empirical in
nature. Modem adsorption theories (for e.g., SF, Roe) predict A a log
My, which is an entirely different type of functional relation). For the
adsorption of PVA98 on the PS latex (made by dispersion polymerization),
a = 0.5 has been reported (Garvey et al. 74). Weak molecular weight
dependence has been observed for adsorption on Agl particles, a = 0.1 for
PVA98 and a = 0.2 for PVA88 (Koopal 78). The hydrodynamic thickness of
the adsorbed polymer layer increases with the molecular weight (Garvey et
al 74; Killmann et al 88). The hydrodynamic thickness has been measured
using various techniques (Electrophoresis, Viscometry, PCS, ultra
centrifugation, slow speed centrifugation, direct force measurements,
etc.). Again, the relation between the hydrodynamic thickness and
molecular weight is represented by a power law. Generally, the measured
hydrodynamic thicknesses were comparable to the random coil dimensions in
solution.


36
The Effect of Solvency
Increases in temperature (van den Boomgaard et al. 78) and additions of
electrolyte (Tadros and Vincent 79; Barker and Garvey 80) increased the
plateau adsorbed amounts. This has been related to the worsening of
solvent quality with temperature and electrolyte. Decreases in the
measured hydrodynamic thickness with increasing temperature and
electrolyte concentration were observed.
Adsorbed Layer Properties and Adsorbed Amounts
It has been suggested from theoretical results (e.g., see Fleer 87)
that, although adsorbed layer properties such as 0, p, adsorbed layer
thickness, etc. are dependent on My, and solution concentration, it is
still possible to express the adsorbed layer properties as a function of
the adsorbed amount only (i.e., the properties of the adsorbed layer are
the same for the high molecular weight polymer at low concentrations and
for the low molecular weight polymer at high concentrations provided the
adsorbed amount is the same). Based on the above hypothesis, Koopal
plotted adsorbed layer properties, such as the first layer occupancy 0,
and effective layer thickness 6 (measured experimentally), as a function
of adsorbed amount (Koopal 78). His results are shown in Figure 2.3. As
expected from the theoretical and other experimental results, the root
mean square thickness increased with increasing adsorbed amounts. The
results for PVA98 and PVA88 are plotted on the same graph since no
significant difference was found in x (PVA98) and x (PVA88) values. From
the above results, he concluded that the acetate content and molecular
weight influenced the adsorbed layer properties through adsorbed amounts
only.


EFFECTIVE LAYER THICKNESS (5) (nm) DEGREE OF OCCUPANCY
37
Figure 2.3 Plots of (a) degree of occupancy, 9, and the effective
layer thickness, 6, as a function of the adsorbed amount
of PVA on Agl and (b) the fraction of segments adsorbed
in trains as a function of the total adsorbed amount
(Koopal, 1978).


38
The Segment Density Distribution
The segment density distributions have been determined (Figure 2.4)
using SANS for adsorbed PVA on PS latex particles (Barnett et al. 82).
Typically, an exponential segment density distribution was observed near
the surface. (This would be expected by the polymer adsorption theory of
Hesselink in which the conformation consists of loops and trains, but no
tailsHesselink 71a.) However, higher segment density at the
intermediate distances were related to longer "slightly folded" tails
(Barnett et al. 82). This type of segment density distribution was
different from the homopolymer PEO adsorbed on PS latex, where more or
less exponential decay in the segment density was observed. In the study
of PVA adsorbed on PS, the root mean square thickness calculated from the
segment density distribution was smaller than results obtained by PCS.
It was concluded that the SANS results were not sensitive enough to
detect tails (which are present in low concentration) and the tails are
responsible for the higher measured hydrodynamic thickness determined by
PCS (Barnett et al. 82).
The Effect of Particle Radius
The effect of particle radius on adsorption behavior and the
hydrodynamic thickness has been studied by Garvey et al. (Garvey et al.
76) and Ahmed et al. (Ahmed et al. 84). The former investigators related
the increasing hydrodynamic thickness with decreasing particle radius to
a geometric factor (Garvey et al. 76). Other groups correlated this
observation to change in the conformation of the adsorbed polymer due to
the change in the particle radius (Ahmed et al. 84).


5
£
a
£
C/D
£
t
Q
Q
W
SI
HH
S
Pi
O
£
3
2
1
LO
VO
0
12
18
24
DISTANCE FROM SURFACE (nm)
Figure 2.4 Plot of segment density as a function of distance from a surface for adsorbed
PVA on PS-latex (Barnett et al., 1982).


40
PVA Adsorption on Silica
Tadros has investigated the PVA adsorption behavior on various types
of silicas (Tadros 78). Here, we will state the results from his
investigation. A detailed comparison of his results and the results in
this study are made in Chapter VIII.
(1) The Effect of Silica Calcination Treatment:
It was observed that the plateau adsorbed amount is a strong
function of silica surface characteristics. The maximum in the plateau
adsorbed amounts was observed for approximately 700C calcined silica.
This observation was correlated with the optimum density of isolated
silanol groups on 700C calcined silica.
(2) The Effect of Surface Charge:
The maximum adsorption occurs at the point of zero change, p.z.c.,
of the oxide and progressive decrease in the adsorption was observed
above p.z.c (Tadros 78). (This effect was not observed for Agl particles
and solKoopal 78, Fleer 71).
Summary
From the above discussion, following general trends have been
established regarding adsorption behavior of PVA.
- Water is a relatively poor solvent for PVA (x is close to 0.5).
- Partially hydrolysed PVA is blocky, while for fully hydrolysed PVA,
acetate groups are randomly distributed.
- Partially hydrolysed polymer adsorbs more than fully hydrolysed PVA.
- Adsorption density increases with increase in the molecular weight.


41
- NMR and electrochemical methods suggest that the acetate segments are
preferentially adsorbed.
- Adsorbed layer properties such as bound fraction (p) and effective
thickness are functions of amount adsorbed only for PVA adsorption on
Agl.
- Segment density distribution is exponential near the surface and
relatively higher density (compared to an exponential distrituion)
observed at the intermediate distances is related to the presence of
slightly folded tails for PVA adsorption on PS.
Hydrodynamic layer thickness is substantial (five to fifty nm) and, it
is primarily the tails which are responsible for these large measured
thicknesses.
- Calcination temperature and pH are among the most important variables
controlling PVA adsorption behavior onto silica.
In the next Chapter, we will review the electrostatic interactions
between colloidal particles.


CHAPTER III
ELECTROSTATIC INTERACTIONS BETWEEN COLLOIDAL PARTICLES
Introduction
Properties of the colloidal dispersion are directly influenced by
the interparticle interactions. A colloidal dispersion is a two-phase
mixture consisting of dispersed particles (solid) in a continuous
dispersion medium (liquid). Particles are said to be colloidal in
character if at least one of its dimensions is in the size range 1 nm to
4
10 nm (1 um). In this size range, specific surface area is large
(usually few m^/gram up to 1000 m^/gram), and hence, the interparticle
interactions are dominated by the solid-liquid interface characteristics.
Also, in this particle size range, the gravitational force is not
important, and particles are moving randomly in the dispersion media due
to thermal energy, i.e., Brownian motion. Particle encounters due to
Brownian motion either leads to either formation of doublets (or higher
order multiplates) or particles remain as individual units depending on
the interparticle interactions. In the absence of any repulsive
interactions, these random collisions lead to permanent contacts between
particles and this reduces the free energy of the system. (The free
energy is lowest when the particles are all clumped together). The
origin of attractive interactions between particles is in the Van der
Waal's attraction between the atoms of the colloidal particles. The
characteristics of the aggregates formed also depend on the interparticle
42


43
forces. To prevent such aggregation of particles during collisions,
there are two mechanisms available to overcome attraction.
(1) Electrostatic Interactions: If the colloidal particles can be given
an electric charge (either positive or negative) and if all particles
have the same sign of charge, particles will repel one another during
approach.
(2) Interactions of Adsorbed Polymer: Under certain conditions (i.e.,
depending on the coverage of particle surfaces with adsorbed polymer,
thickness of the coating, solvency for polymer, etc.), adsorbed polymer
layer can prevent close approach of the particles.
These two repulsive interactions impart stability to the colloidal
dispersion. The dispersion is said to be stable if the dispersed phase
(colloidal particles) remains essentially as distinct single particles on
a long time scale (e.g., days, months, years). Such a dispersion may be
stable either due to kinetic (e.g., in the case of electrostatic
interactions) or thermodynamic reasons (e.g., stabilization with adsorbed
polymer). It is clear from the above definition that the stability
criteria is essentially based on the state of particulates in the
dispersion. The time scale is employed as a reference because, in the
absence of repulsive interactions, the number of particles (kinetic
units) in moderately concentrated suspensions can be reduced to half in a
matter of seconds due to encounters arising from Brownian motion (von
Smoluchowski 16a,16b,17).
The stability criterion based on the time scale (for e.g., time it
takes to reduce the particle concentration to half, etc.) may not be
useful to evaluate the stability of ceramic dispersions. The reasons are


44
two fold: (i) typically particles in the size range of 0.05 urn to 5-10 pm
are often employed in various ceramic processing operations, and hence,
suspensions are not strictly colloidal in nature. Also, the density of
particles is usually greater than the density of suspending media leading
to sedimentation of particles although particles are well dispersed.
(ii) the particle concentration is also quite high, and usually, it is
not possible to determine the change in particle concentration. Other
techniques, such as rheological behavior, sedimentation behavior,
properties of the sediment (porosity, average pore size, etc.) can be
used to evaluate the stability of these dispersions.
/ -6
Although interatomic attractive interactions are short range (a r
where r is the distance between the atoms), their summation over
colloidal particle sizes leads to long range attraction. To overcome
this attraction, the repulsion must also be long range.
In this chapter, we will review two components of interactions
(1) Van der Waal's attraction between colloidal particles, and
(2) electrostatic interactions.
Summation of these two components, under the assumption of
additivity, leads to well known theory developed by Deryagin and London
and independently by Verwey and Overbeek (commonly known as "DLVO"
theory) to explain the stability behavior of the electrostatically
stabilized dispersions (e.g., see Verwey and Overbeek 48). Excellent
monographs are available to discuss various aspects of this theory, hence
only the basic principles will be outlined here (e.g., see, Hiemenz 77,
Hunter 87, Overbeek 82a,82b, Lyklema 68, etc.).


45
Development of Charge at Solid-liquid Interface
There are basically four different methods by which the charge can
be developed at the solid/liquid interface (Hunter 87).
Dissociation of Surface Groups
This method is the charge determination mechanism for several oxides
(e.g., alumina, silica, etc.). The surface of these oxides is
hydroxylated to various extents. (For example, for precipitated silica
used in this investigation, the surface is almost completely
hydroxylated, i.e., the surface is nearly fully covered with silanol
groups, Si-OH). Dissociation of surface silanol groups leads to surface
charge development, as described by the following reactions:
-SiOH, + H+, ... > SiOH+2, ,
(surfdcs) (liquid) (surfdC6) |^ ^ j
(surface) (liquid) (surface) 2 (liquid)
From the above reactions, it is clear that the silica surface can develop
a positive or a negative surface charge.
A zero point of charge, (p.z.c.), is defined as a pH at which the
surface charge is zero. Another important characteristic of the oxide
material is its isoelectric point, i.e.p., which is defined as the pH at
which the electrophoretic mobility is zero. H+ and OH ions are called
the potential determining ions. In the absence of specific adsorption of
ions, the p.z.c. and the i.e.p. are the same. Below the i.e.p., the
silica surface is positively charged, and above the i.e.p., negative
charge is developed and its magnitude can be increased by increasing pH
of the solution.


46
Adsorption of Potential Determining Ions
A familiar example is silver iodide particle/water suspensions in
which the particles can preferentially adsorb an Ag+ or I- ions,
rendering them positive or negative charge, respectively.
Adsorption of Ionized Surfactants
In this case, the charge is produced by the preferential adsorption
of the ionic surfactants on the surface, for example, the preferential
adsorption of C-|2H25S04- ions from sodium dodecyl sulfate, c-| 2H25S4_Na+
surfactant.
Isomorphic Substitution
This charge development mechanism is important in the case of clay
minerals (e.g., sodium montmorillenite). Inside the solid lattice, lower
valent ions may replace higher valent ions (e.g., Al+^ replaced a Si+4
ion in the "tetrahedral silica layer," resulting in a deficit positive
charge on the particle surface.
For the present investigation, dissociation of surface silanol
groups on the silica particles is the important mechanism of charge
development. The silica used in this study has an i.e.p. near pH = 3.7.
Hence, at high pH's, the silica surface develops negative surface charge,
and at pH near 3.7, no net charge is present on the silica particles.
(Hence, there is no net electrostatic repulsion between two approaching
particles at pH = 3.7.)
Electrical Double Layer
The development of surface charge is not yet a sufficient condition
for stability because electroneutrality requires that the particle and


47
its immediate surroundings should have no net charge. In other words,
the surface charge must be balanced by an equal but opposite counter
charge in the solution. The rigid alignment of counter ions in the
solution is implausible because of thermal agitation, which causes the
counter ions to diffuse throughout the solution. To understand the
stability, it is of crucial importance to understand the distribution of
counter ions in the solution.
The stability of the charged particles can be understood
qualitatively as follows: If the counter charge is very diffusely
distributed and extends far from the particle surface, then when two
particles having the same sign of charge (and hence, same sign of the
counter ions in the diffuse layer) start approaching each other due to
Brownian motion (or due to an applied shear field), the diffuse layer
starts to overlap (even though the particles are far apart), thus giving
rise to an electrostatic repulsive force. On the other hand, when the
double layer is compressed (i.e., the counter ions are crowded close to
the particle surface), particles can approach closer before they feel the
electrostatic repulsion, and at that distance, the strong Van der Waal's
attraction leads to flocculation of the particles.
To explain the stability behavior quantitatively, a description of
the potential (or charge) distribution around the colloidal particles is
required. To describe the variation of the potential with the distance
from the charged surface, the Poisson equation is used as shown below:
V2 W = =2 (3.2)
o r
where is the potential, is the Laplace operator, Gr is the relative
permittivity, eQ is the permittivity of the free space, and p is the


48
local charge density (i.e., number of charges per unit volume). To solve
this equation, one needs to know the charge density as a function of
potential.
The work required to bring an ion to a position where the potential
V is given by Ze¥. The probability of finding an ion at that position
is given by the Boltzmann factor:
n. -Z.eW
it- *=* 1 KT 1 <3-3
io
where T is the temperature, k is the Boltzmann constant, n is the number
of ions of type i per unit volume, and n0 is the concentration far from
the surface (i.e., the bulk concentration). The valance number Z^ is
either a positive or negative integer and e is the charge on the
electron. The charge density is related to the ion concentrations, as
follows:
Z.eW
p = E n. eZ. = E Z. en. exp () (3.4)
i i .1 io ^ kT
i i
Substituting for the charge density, one obtains the Poisson-Boltzmann
equation, as follows:
Z eW
v2 ? = 4 E n. Z.e exp (£=-) (3.5)
G 6 io l kT
or i
The following assumptions were made in solving the above equation:
(1) The surface charge on the particle and the space charge in the
solution are considered as smeared out. (2) The ions are considered as a
point charges, their distribution in the solution being determined by
their valancy and not by their volume, shape or polarizability. This
assumption makes the theory non-specific (e.g., the difference between


49
Li+ and Na+ ions cannot be distinguished). (3) The solvent is considered
as homogeneous and continuous, and the solvent affects the charge
distribution through its dielectric constant Gr.
It is clear from the above equation that the potential distribution
depends in a complex way on the ionic composition of the solution. This
equation does not have an explicit general solution and has been solved
for certain limiting cases (e.g., low surface potentials) and for simple
geometries (i.e., flat plate, spherical particle, etc.).
At room temperature, the exponent Z^eV/kT = Z^V/25.4 if ¥ is
expressed in millivolts.
Debye-Huckel solved the above equation for flat plate geometry and
for low surface potentials, (i.e., ZÂ¥ < 25.4 mV). Under these
conditions, they showed that the potential decays exponentially (Hunter
87) :
V = *P exp ( -hx) (3.6)
o
where x is the distance from the interface and the h, the Debye-Huckel
parameter, is defined as follows:
H
2
e En. Z.
IQ 1
kT
2 1/2
)
(3.7)
r o
h has the units of reciprocal length, i.e., h-1 has the units of length.
The exponential rate of the potential decay is controlled by h (Equation
3.6). If the double layer thickness is defined as the distance over
which the potential drops to (1/e) of its value at the surface, then k-1
becomes the measure of the "double layer thickness." At 25C in water,
the value of k is given by:
h = 3.288 yj I (run )
(3.8)


50
where I is the ionic strength (= 1/2 E where is the ionic
concentration in mole/liter). Table 3.1 shows calculated values of
K-1 (i.e., the double layer thickness) for several different electrolyte
concentrations and valences for aqueous solutions at 25C.
From the table, it is clear that the valency of the counter ions and
the electrolyte concentration are important parameters to the control
double layer thickness, and hence, the electrostatic interactions between
colloidal particles.
Figure 3.1 shows the variation of the potential with the distance
from the surface. The drop is dramatic for the higher electrolyte
concentrations or valances.
The Poisson-Boltzmann equation has been solved for flat plates
without the Debye Huckel approximation (i.e., < 25 mV) by Gouy-
Chapman. The results are applicable only to symmetrical electrolyte,
i.e., Z+ = Z-. According to the Gouy-Chapman, the variation of potential
within the double layer can be described by the following equations
(Hunter 87):
Y = Y0 exp <-hx) (3.9)
where y is defined by the relationship
_ exp (Zeg/2kT) 1 (3 1Q)
Y exp (ZeW/2kT) + 1
and Yo is calculated from the above equation when ¥ = Vq. From Equations
3.9 and 3.10, it is clear that it is the complex ratio y that varies
exponentially with x in the Gouy-Chapman theory. For the low potentials,
as expected, the above equation reduces to the Debye-Huckel approximation
(i.e., Equation 3.6).


51
TABLE 3.1
Effect of Ionic Strength and Valency on the
Electrical Double Layer Thickness, h~1
Ionic Strength
(moles/liter)
Symmetrical
Electrolyte
Z+ : Z_
H-1
(run)
X
o
1
*>
1:1
30.41
2:2
7.60
3:3
3.36
1 x 10-3
1 :1
9.61
1 x icr2
1:1
3.04
1 x 10~1
1 :1
0.96


52
Figure 3.1
Fraction of double
layer potential
versus distance
from a surface:
(a) curves for 1:1
electrolyte at
three concentra
tions and (b)
curves for 0.001 M
symmetrical
electrolytes of
three different
valance types.
Figure 3.2
Schematic
illustration of
the variation of
potential as a
function of
distance from a
charged surface in
the presence of a
stern layer,
subscripts o at
wall, 6 at stem
surface, d in
diffuse layer.


53
To describe the potential variation as a function of distance from
the interface for spherical particles, the Poisson-Boltzmann equation
should be solved in spherical coordinates. For the case of low surface
potentials, analytical solution is available and given by the following
equation:
(r) -2- e-Kla-cl
(3.11)
o r
where (r) is the potential at a distance r from the particle center and
a is the particle radius.
In the theoretical development of the above equations, Debye-Huckel
and Gouy-Chapman treated ions as point charges, i.e., the effect of ion
size was ignored. To account for the finite volume of the ions, Stern
divided the aqueous part of the double layer by a hypothetical boundary
known as the Stern surface (Hunter 87). The Stern surface is situated at
a distance 6 from the actual surface as shown in Figure 3.2.
The Stem theory is difficult to apply quantitatively because it
introduces several parameters into the picture of double layer which
cannot be evaluated experimentally. There are several other models
available to describe the charge and the potential variation at the
charged interface, but they suffer the same problem as the Stern's model
(Hunter 87).
It is important to note that the existence of the Stern layer does
not invalidate the expressions for the diffuse part of the double layer,
but one needs to use a potential at the Stern layer, ¥5. This Stern
layer potential is usually equated with the zeta potential, which is the
potential measured using an electrokinetic method. The exact location of


54
at which the zeta potential is determined is not known, but it is assumed
to be close to the Stem layer.
In summary, in this section, we looked at the various charge
development mechanisms at the solid-liquid interface. The charge on the
solid surface leads to the distribution of the counter ions in the
solution. This model of the charged interface is often called the
electrical double layer model. The potential variation as a function of
distance from the interface can be obtained by solving the Poisson-
Boltzmann equation. Under the assumptions of low potentials (Debye-
Huckel approximation) and simple geometries, analytical solutions can be
obtained. The parameter of great importance is k (the Debye-Huckel
parameter), which can be used to describe the effect of the concentration
and valence of the counter ions on the potential distribution near the
charged surface. To describe the stability behavior of the
electrostatically stabilized dispersions, calculations are often made of
the interaction energy as a function of distance of separation between
particles. In the next section, expressions for the interaction energy
due to overlap of the electrical double layers will be developed.
Double Layer Interactions
When two particles approach each other, overlap of the double layer
occurs. The rate of approach of the two particles compared with the
relaxation time of the diffuse double layer to adjust to the new
situation is an important parameter. But, generally, two cases can be
distinguished. They are denoted as the "constant potential" and the
"constant charge" interactions.


55
In the first case, it is assumed that during the encounter of two
colloidal particles, the surface potential W0 remains constant. Under
these conditions, analysis shows that, to keep the surface potential
constant, the surface charge density, oQ, should decrease. In the second
case, it is assumed that during the encounter the surface charge density
remains constant, and in this case, the overlap of the diffuse double
layer leads to an increase in ¥0. The condition in which both >P0 and oQ
are not constant is also possible and has been called a charge regulation
(Hunter 87).
The repulsive interactions due to the overlap of the double layer
can be analyzed using two approaches (Lyklema 68).
(a) the free energy change involved when the overlap occurs, or
(b) the increase in the osmotic pressure due to accumulation of the ions
between the particles.
Following the free energy change approach, there is an increase in
the free energy of the double layer upon interaction, hence, work must be
performed to bring the particles closer. In other words, the overlap of
the double layer leads to repulsion between particles.
The repulsion energy VR(d) represents the work necessary to bring
the particle surfaces from infinity to a distance d. To calculate VR(d),
the free energy of the system as a function of the distance of separation
should be known. It is clear that VR can be represented by the following
equation:
VR(d) = 2 [G(d) G(<*>)] (3.12)
where G(d) represents the free energy at the distance d and G(<*>)
represents the free energy at the infinite separation, i.e., for the


56
isolated double layer. The factor 2 results from the fact that two
double layers are involved. On the basis of the above scheme, DLVO
theory formulates the repulsive interaction energy. The exact solution
is available only for simple shapes (e.g., flat plate) and under certain
approximations (e.g., low potentials). For two flat plates, VR as a
function of distance d has been tabulated by Overbeek for any given value
of the surface potential ?0 and concentration of ions in solution
(Overbeek. 52).
The following approximate analytical equations are often used to
represent VR.
Interaction Between the Two Flat Plates
Under the assumption of the linear superposition1 principle, and
under the condition of the constant surface potential during the double
layer overlap, the potential energy is given by the following equation
64n kTy2
V (d) = ^ exp (-Kd) (3.13)
R H
where VR is the repulsive interaction energy per unit area (J/m2) and nQ
is the ion concentration (total number of ions/m2), and the other symbols
have been previously defined.
The above expression is also valid for the constant charge case.
Comparison of the above expression with the exact results (Overbeek 52)
show that over a considerable range of overlap, the above equation is a
1 The assumption is that the potential between two interacting
particles is equal to the sum of potentials of individual double layers
at the same distances from the surface. This is valid in the case of low
surface potentials, i.e., for d > 1/k, the diffuse layer thickness, k
-1


57
good approximation, though the approximation tends to overestimate the
value of VR (Hunter 87).
A more elaborate expression, valid for higher surface potentials
under the conditions of the constant charge, is also available (Gregory
73).
Interaction Between Two Spherical Particles
This case is more important in the case of colloidal dispersions.
(1) For large values of ua: Large values of na means that the particle
radius, a, is relatively large compared to the thickness of the diffuse
layer, k-^. Under the conditions of low potentials and thin diffuse
double layer, the repulsion energy can be calculated by Deryaguin
procedure (see Hunter 87). Following this procedure, it is possible to
calculate the interaction energy between two spherical particles if the
interaction energy as a function of distance of separation is available
for the flat plate case under similar conditions. For the case of two
identical spherical particles, the energy of interaction can be
calculated from the following equation:
OO
VR(d) (sphere) = na J VR(d) (flat plate) dD (3.14)
d
where a is the radius of the particle.
Substituting for Vr (flat plate), the approximation valid for low
potentials is given by:
V_,(d) (sphere) = 2n e^aT In ( 1 + exp (-hH) ) (3.15)
R roo
Note that Equation 3.15 gives the total repulsive energy between two
spherical particles (in Joules) whereas flat plate expression (Equation


58
3.13) gives the energy per unit surface area (in Joules/m^). For the
case of somewhat higher surface potential, the interaction energy is
given by the following equation:
2
64nan kTy
V (d) (sphere) = exp (-nd) (3.16)
K 4
H
For the spherical particles at high potentials, no analytical formula is
available, but a graphical solution is available (Overbeek 52).
(2) For the case of low na, i.e., when the thickness of the diffuse
layer, h_1 is large compared to particle radius, a, the interaction
energy is given by the following equation:
4ixG aW 2
VR 2a+d 3 SXP (_Kd) (3-17>
where 3 is a factor which allows for the loss of spherical symmetry in
the double layer and has been defined by Verwey and Overbeek (Overbeek
52).
From the above equations, it is clear that the repulsion is
determined by the ionic strength (through Debye-Huckel parameter k) and
the surface potential, of the particle (i.e., radius a) is also important.
In the above equations, the finite size of counter ions (i.e.,
presence of the Stem layer) was ignored, and hence, these expressions
are not valid for the distances comparable to atomic dimensions.
Additional interactions due to solvation and hydration of ions and
hydrophobic interactions have been reported (Israelachvili and Pashley
82,83; Pashley and Quirk 84; Pashley et al. 82). Theses short range


59
interactions have been related to repeptization phenomenon (Overbeek
82b).
As mentioned earlier, the other term in DLVO calculations is the
attraction energy term arising due to the Van der Waal's interaction
between the colloidal particles leading to the flocculation of particles.
This attraction energy term will be discussed in the next section.
Van der Waal's Interactions
The Van der Waal1s interactions between neutral molecules may
originate from three possible sources: permanent dipole-permanent dipole
(Keesom), permanent dipole-induced dipole (Debye), or induced dipole-
induced dipole (London) interactions. The distance dependence of these
interactions can be represented as a power law, i.e., potential energy of
interaction a r-x where x = 6 and all interactions lead to attraction
between molecules. In the case of non-polar molecules as the dipole
moment is zero, Debye and Keesom interactions are absent. On the other
hand, London interactions, also known as dispersion forces, are always
present. The London dispersion force is attributed to correlated
electronic motion in the atoms under consideration. This correlated
motion of electrons leads to a decrease in the potential energy of the
system (i.e., attraction). This attraction energy is short range since
it is inversely proportional to the sixth power of the separation, but
the total interaction energy between two colloidal particles (i.e., a
collection of a large number of atoms) is quite large and of long-range
order and comparable to electrostatic repulsion energy under certain
conditions. The contributions to total interaction energy from the Debye


60
and Keesom interaction is usually small since dipolar contributions tend
to average out when large number of atoms are considered.
There are two methods of calculating the magnitude of Van der Waal's
attraction energy between two colloidal particles, (i) microscopic and
(ii) macroscopic procedure.
Microscopic or Van der Waal's Method
The classical procedure adopted by Hamaker was based on two
principles
(1) the additivity of London dispersion forces: The total interaction
energy between two colloidal particles was calculated using summation of
pairwise interactions between all the atoms or molecules of the two
macroscopic bodies.
(2) The summation of these interaction energies between atoms or
molecules can be replaced by an integration provided that the distance
between the particle surface is large compared with the atomic distances.
Under these assumptions, the Van der Waal's attraction energy between the
two colloidal particles can be represented as follows:
(3.13)
VA A(A) H(G)
Hence, A(A) is a function of so-called Hamaker-Van der Waal's constant of
the material and H(G) is determined by the geometry of the system. For
example,
Flat Plates
For two flat plates of substance 1, the Van der Waal's attraction
energy is given by (Hamaker 37):
A
II
12nd'
(3.19)


61
where A-] 1 is the Hamaker constant for the substance 1, d is the distance
between two flat plates, and the negative sign indicates that the energy
is attractive. The Hamaker constant A-| -j for substance 1 is defined by
Aii n qiBii ,3-20
where qi is the density of atoms in the colloidal particle and &ii is the
constant in the London equation:
V
11
(3.21)
r
describing interaction energy V-|-| between atoms or molecules. -| is
proportional to the polarizability of the atoms, and hence, increases
with the size of the atom. From the above equation, it is clear that the
Van der Waal's attraction energy VA increases with square of the density
of the substance and the interaction is of long range. (Note VA a d c
for "bulk" materials, compared to Vn a r-^ for molecular interactions.)
Spherical Particles
For two unequal spheres of radius a-) and a2 separated by a distance
d in vacuum (d is the distance between two surfaces), the Van der Waal's
attraction energy is given by the following equation (Hamaker 37):
-A
Vft (sphere)
11
12
+ 2 In
x +xy+x x +xy+x+y
, x i-xy+x
' 2
x +xy+x+y
)
(3.22)
where x = d/2a-| and y = d/2a2
For two spherical particles, having same radius a, it can be shown
that for large separations, VA decreases with the sixth power of d as in
the London expression, but for short distances (i.e., d a) VA
decreases slowly with the distance:


62
This distance dependence of the attractive (as well as the repulsive)
energy of interaction is of great importance. As it will be shown later,
at short and at long distances, the Van der Waal's attraction energy
dominates the total interaction energy while at the intermediate
distances of separation, electrostatic repulsion, which decay
exponentially with the distance, may dominate the total energy of
interaction. The above equation indicates that, for a small distance, VA
tends to assume very large negative value. However, the above equations
are no longer valid for very short distances (i.e., dimensions comparable
to the atomic dimensions) because strong Bom repulsive interactions
(usually represented as V a d-1^) which arise from the overlap of
Bom
electronic orbitals of approaching molecules, are operative.
Retardation Effect
The equations given above for VA were derived under the assumption
of the additivity of interactions between atoms. The origin of the
London-Van der Waal's interaction is the electromagnetic interactions
between atoms and molecules. The above equations for VA do not allow for
the finite time of propagation of electromagnetic waves from one atom to
the other, and the induced-dipole becomes retarded against the inducing
one when the distance between the atoms becomes comparable to the wave
length of the London frequency. This leads to reduction in the London
dispersion force between atoms. Casimir and Polder have shown that due
to retardation the inverse sixth power law [Equation (3.21)] gradually
changes into an inverse seventh power law with increasing distance


63
(Casimir and Polder 48, Gregory 69, and Visser 72). The retardation
correction is negligible when the distances between the atoms are
comparable to the atomic dimensions. However, for the interactions
between the colloidal particles at the distances of the order of ten to
one hundred nm, the retardation effect can be significant. Equations are
available to correct for the retardation effect for the flat plates and
for the spherical particles (Gregory 67). The retardation effect leads
to reduction in the attraction energy and makes the dispersion force much
longer range.
Effect of Medium on the Van der Waal1s Attraction
The equations derived above for the Van der Waal's attraction energy
can be used for the case of two colloidal particles interacting in
vacuum.
In most practical cases, colloidal particles are embedded in a
medium (e.g., water). To get the interaction energy VA in this case, the
Hamaker content An is replaced by the effective Hamaker constant. The
effective Hamaker constant, A-j3-), now depends not only on An (particle-
particle attraction) but also on A-| 3 (particle-medium attraction) and A33
(medium-medium attraction) where subscript 1 refers to particle and 3
refers to medium. The effective Hamaker constant is defined as:
A
131
+ A
33
2A
13
(3.24)
where A11 = ixq^81 A33 = nq3 833 and A13 = and it is generally
1 /2
assumed that P13 = (B11&33) and hence,
A
A
33
1/2,2
(3.25)


64
Thus, to account for the medium, A-|-j should be replaced by A131 in all
the previous equations for VA. From the above equation, it is clear that
the effective Hamaker constant is always positive (i.e., there is always
net attraction between two particles) and the magnitude of the Van der
Waal's attraction is reduced due to the presence of the medium (i.e., the
medium imparts a certain measure of stability to the dispersed
particles). By choosing A33 close to An, the Van der Waal's attraction
can be substantially reduced and at An = A33, there is no net attraction
between particles. Thus, to evaluate Van der Waal's attraction energy,
the Hamaker constants should be available for the materials under
consideration.
Macroscopic Approach
One of the main drawbacks of the microscopic or the Hamaker theory
is the assumption regarding additivity of interactions. The Van der
Waal's attraction VA has been calculated using a different approach by
Lifshitz and collaborators using the "macroscopic" approach. In this
theory, the interacting bodies are treated macroscopically, i.e.,
interacting bodies are considered as two semi-infinite phases separated
by the distance d. The bodies are characterized by their complex
dielectric constant. The interaction is evaluated using fluctuation
theory. The spontaneous electromagnetic fluctuation in one body induces
a fluctuation polarization in the other body. The correlation between
the fluctuating fields in the two objects decreases the free energy of
the system, and hence, leads to attraction. The Lifshitz theory has some
advantages over the classical microscopic theory. The assumption


65
regarding additivity is avoided and contributions of bonding between the
atoms and the molecules to the interactions are taken into account. The
Lifshitz approach is more accurate to calculate V^, but mathematics
involved is quite complicated and requires dielectric constant data over
a wide frequency range of the materials of interest. Due to these
difficulties, the Hamaker microscopic approach is generally used in
practice.
Hamaker Constants
To calculate the Van der Waal's attraction energy for practical
systems, the values of the Hamaker constants should be available. In
principle, there are two ways to estimate the Hamaker constants (see for
e.g., Visser 72, Gregory 69, Lyklema 68):
(1) Direct Calculations: In this case, the Hamaker constant can be
calculated by the microscopic approach (e.g., see equation 3.20) using
molecular properties such as polarizability or by the macroscopic
approach using dielectric constant data.
(2) Indirect Evaluation From the Experimental Data on Colloidal
Stability: In this method, the Hamaker constant will depend on the
experimental tool used to determine stability and the stability criteria
used. The Hamaker constant can be experimentally determined from various
techniques such as (i) flocculation experiments on dispersion of
colloidal particles, (ii) from the measurements of direct force of
interactions between crossed wires, (iii) from equilibrium film thickness
measurements, (iv) surface tension measurements, and (v) from rheological
data, etc. (Visser 72).


66
Table 3.2 shows the list Hamaker constants for the present system
under investigation (i.e., silica-water-PVA). Data for other materials
can be found in the above references.
The effective Hamaker constant will be also influenced by the
electrolyte concentration, surface contamination, and adsorbed polymer
layers. The effect of the adsorbed polymer layer will be discussed next.
The Effect of Adsorbed Polymer Layer on Van der Waal1s Attraction
The effect of adsorbed polymer layer of thickness 6 on the
attractive force can be compared at (i) constant center-to-center
distance of spheres, h (see Figure 3.3) or (ii) at constant separation of
outer surfaces, d.
If one makes comparison at the constant center-to-cetner distance,
then the adsorbed layer usually leads to an increase in attraction
between particles. This is due to an effective increase in the particle
size and decrease in the distance of separation between particles. If
comparison is made at the constant distance of separation d, the adsorbed
layer usually reduces the attraction between particles. In this case,
the original spheres are separated by a greater distance and generally
the adsorbed material has lower Hamaker constant. The attraction energy
between two equal spheres of radius a, separated by distance d with the
adsorbed layer thickness 6, is given by the following equation:
v_ (d)
A
-I[Hn
(AV2- AV2)2
l*11 *22 '
+ H
22
(AV2- AV2)2
1 22 33
2 H
12
1/2_ 1/22
l*11 *22 '
(AV2- A1/2)]
'*22 *33 U
(3.26)


67
TABLE 3.2
Compilation of Hamaker Constants for Silica-Water-PVA System
Material
Hamaker Constant
x 10-2 J
Technique
Reference
Silica
6.6
Macroscopic
Hunter 87
(An)
14.8
II
Visser 72
50.0
Microscopic
tl
16.4
Surface Tension
II
Water
3.7
Macroscopic
Hunter 87
(A33)
4.4
Lifshits
Visser 72
3.3-6.4
Microscopic
II
3.0-6.1
Macroscopic
II
4.8-jLO
Colloid Chemistry
II
5.5-6.4
Surface Tension
II
PVA
6.8-8.8
Microscopic
Visser 72
(Pi22)
Hamaker Constant
Material
in Water x 10-20 J
Technique
Reference
Silica
0.85
Macroscopic
Hunter 87
(A131>
PVA
0.50
Microscopic
Visser 72
(a232>


68
Figure 3.3
Schematic illustration of the effect of adsorbed polymer
layer on Van der Waal's attraction.


69
where An, A22, A33 are Hamaker constants in vacuum for solid, polymer
and liquid, respectively and H-j i, etc. are geometric functions defined
as:
(3.27)
x +xy+x x +xy+x+y x +xy+x+y
where for Hi 3, x = (d+26)/2a and y = 1, for H22, x = d/2(a+6) and y =1,
and for H12 x = (d+6)/2a and y = (a+6)/a. It can be shown from
Equations 3.26 and 3.27 that if the adsorbed layer is sufficiently thick
and when A22 = A33, the attraction between particles virtually
disappears. To evaluate the Hamaker constant A22 for the adsorbed layer,
two additional features need to be considered: (i) the adsorbed layer
will be a composite, i.e., it will consist of polymer segments plus
solvent molecules and (ii) the segment density will not be uniform. Both
of these effects on the attraction energy have been discussed by Vincent
(Vincent 74).
Potential Energy Curves and the DLVO Theory
Since the electrostatic repulsion and the Van der Waal's attraction
are assumed to operate independently and since botn are being scalar
quantities, they can be added to give the total interaction energy,
V This is the basis of DLVO theory. The total interaction energy
total
can be represented as:
V(d)
total
= VR(d) + VA(d)
(3.28)
where VR is the electrostatic repulsion energy and VA is the Van der
Waal's attraction energy term. Substituting for VR and VA from the
previously developed equations, the total interaction energy as a


70
function of distance of separation between the particles can be plotted.
In this type of plot, the attraction energy term is represented as a
negative term and the repulsive energy by a positive term. Figure 3.4
shows schematically the total interaction energy as a function of the
distance of separation between two surfaces. The shape of the total
interaction energy curve is important in determining the stability of
colloidal dispersions and can be used pictorially to show the influence
of various relevant parameters on the stability. The type of curve shown
in Figure 3.4 results because of the different distance dependences of
the interaction energy terms VR and VA. The Van der Waal's attraction
energy term is important at close approach since at close approach VA a
1/d and also at large distances when VA a d-^ due to the retardation
effect. At the intermediate distance of separation, the electrostatic
repulsion (which decreases exponentially with the distance) is more
important. The summation of these two interaction energy terms having
different distance dependences leads to a total attraction energy curve
having a maxima in a potential energy separating two minimas. The
primary minima results from the strong Van der Waal's attraction at the
short distances and the Bom repulsion due to overlap of electron clouds.
At such close distances of approach, recent experiments indicate that an
additional energy term, Vs, arising due to solvent structural effects,
should be included (e.g., see Israelachvili and Pashley 82,83). Due to
these additional complications, the exact location and depth of the
primary minima cannot be determined quantitatively. The secondary minima
results from the long-range Van der Waal's attraction and the rapid decay
of the electrostatic repulsion. The depth of the secondary minima is


potential energy
71
\Vr
\
Total potential energy of interaction V(d) = VR(d) +
VA(d) where VR(d) is the potential energy of repulsion
due to double-layer interactions and VA(d) is attractive
potential due to Van der Waal's interactions (Overbeek,
1952).
Figure 3.4


72
usually not significant compared to the thermal energy of the particles.
Hence, flocculation of particles in the secondary minima tends be weak.
The other important characteristic of the total interaction energy curve
is the presence of the potential energy maxima. If the height of the
potential barrier, vmax, is greater than the thermal energy of the
particles (i.e., vmax kT) then the potential barrier can prevent
flocculation of the colloidal particles and the dispersion is stable in a
colloid chemical sense. The fraction of particles that can surmount such
a potential barrier is given by Boltzmann's law. The fraction decreases
exponentially with increasing height of the potential barrier. vmax of
the order of 5-15 kT has been considered sufficient to achieve long-term
stability (Overbeek 82b). It should be noted that, in the case of flat
plates, the potential energy per unit area is plotted. In these cases,
the total interaction energy is obtained by multiplying by the
appropriate cross sectional area of the particles. In the case of
spherical particles, the net interaction energy is plotted. To
understand the stability behavior, factors influencing the repulsion
energy term VR, and the attraction energy term VA should be considered.
The Effect of Hamaker Constant
The range of values of Hamaker constant for substances (ignoring
retardation effects) immersed in water can be given as follows (Hunter
87):
A.^ 30 10 for metal particles
3-1 for oxides and halides, and
-20
s 0.3 x 10 J for hydrocarbons.


POTENTIAL ENERGY OF INTERACTION (J)
73
DISTANCE OF SEPARATION (nm)
50 kT
25 kT
0 kT
25 kT
Figure 3.5
The effect of the Hamaker constant on the total
interaction energy curves.


74
As discussed earlier, this difference essentially arises due to
differences in the polarizability of these materials. Figure 3.5 shows
the effect of the Hamaker constant on the total interaction energy curve.
All other factors (i.e., the surface potential ¥0 and the Debye-Huckel
parameter k) were kept constant. As expected with increasing values of
A^i/ the height of the potential barrier decreases and the depth of the
secondary minima increases.
The Effect of Surface Potential, Vn
The electrical double layer repulsion term VR is usually dominated
by two parameters (a) the near surface potential and (b) the thickness
of the double layer, 1/k. Figure 3.6 shows that the height of the
potential barrier increases as >P0 increases. (For oxide materials, such
as silica, exact value of VQ is difficult to determine experimentally due to
complications, such as specific ion adsorption and presence of the Stern
layer, etc., so usually the near-surface potential, i.e., the zeta
potential, £ is used. This experimentally determined zeta potential does
establish a lower limit of V0.
The Effect of Electrolyte Concentration
The Debye-Huckel parameter, k, which represents the thickness of the
electrical double layer, depends on both the concentration and the
valance of the indifferent electrolyte (See Equation 3.7). Figure 3.7
shows the effect of concentration of a 1:1 electrolyte on the total
potential energy curve. The potential energy barrier decreases with
increasing electrolyte concentration, and above a certain concentration,
the barrier vanishes. Thus, the addition of an indifferent electrolyte


POTENTIAL ENERGY OF INTERACTION (J)
75
DISTANCE OF SEPARATION (nm)
The effect of zeta potential on the total interaction
energy curves.
Figure 3.6


POTENTIAL ENERGY OF INTERACTION (J)
76
4
3
2
1
0
1
Figure 3.7
100 WT
DISTANCE OF SEPARATION (nm)
The effect of concentration of 1:1 electrolyte on the
potential energy curves.


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r1QcQen"6,7/") )/25,'$ 2 2 $ mP OLOLO


THE EFFECT OF ADSORBED POLY (VINYL ALCOHOL)
ON THE PROPERTIES OF MODEL SILICA
SUSPENSIONS
BY
CHANDRA KHADILKAR
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
x
1988
tJ OF F LIBRARIES

ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to Dr. M.D. Sacks for
his invaluable guidance and financial support during the course of this
study. I would like to thank him for allowing me to work in the area of
colloid science and for demonstrating the importance of reproducibility
and hard work in scientific research.
I am also grateful to Dr. C.D. Batich, Dr. L.L. Hench,
Dr. D.O. Shah, and Dr. E.D. Whitney, members of supervisory committee,
for their helpful suggestions.
I would like to thank Mr. H.W. Lee, Mr. O.E. Rojas, Mr. G.W.
Scheiffele, Mr. S.D. Vora, and Mr. T.S. Yeh for their help during the
course of this work. Thanks are also due to Professor B. Moudgil for
allowing me to use spectrophotometer and Gel Permeation Chromagraphy
apparatus. I would like to thank Ms. Hazel Feagle for typing, editing,
and compilation of this dissertation.
I also wish to convey my sincere gratitude to my parents and to my
wife for their encouragement and support during the course of this work.
ii

TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ii
LIST OF TABLES viii
LIST OF FIGURES ix
ABSTRACT xxii
CHAPTERS
I. GENERAL INTRODUCTION AND AIM OF THE STUDY 1
II. ADSORPTION OF POLYMERS AT SOLID/LIQUID INTERFACE 5
Introduction 5
Description of Adsorbed Polymer 9
Adsorbed Amount of Polymer (A) 9
Bound Fraction (p) 11
Direct Surface Coverage (0) 11
Segment Density Distribution 0(X,Y,Z) 11
Theoretical Models 12
Adsorption Energy Parameter (xs) 13
Segment-Solvent Interaction Parameter (x) 14
Polymer Adsorption Theories: General Framework ... 15
Results of Polymer Adsorption Theories 18
Effect of Adsorption Energy Parameter 19
Effect of Solvent-Segment Interaction
Parameter 19
Effect of Polymer Concentration and Molecular
Weight 19
Experimental Techniques 21
Adsorbed Amount of Polymer 21
Trains, Bound Fraction, and Direct Surface
Coverage 22
Thickness of Adsorbed Layer 23
Segment Density Distribution 26
Adsorption Energy Parameter 27
Adsorption of Polydisperse Polymers 27
Experimental Results for PVA-Water System 29
Properties of Poly (Vinyl Alcohol), PVA 29
PVA Characterization 31
Adsorbed Amount of Polymer 33
The Nature of Solid 33
iii

The Effect of Acetate Content 34
The Effect of Molecular Weight of PVA 35
The Effect of Solvency 36
Adsorbed Layer Properties and Adsorbed
Amounts 36
The Segment Density Distribution 38
The Effect of Particle Radius 38
PVA Adsorption on Silica 40
Summary 40
III. ELECTROSTATIC INTERACTIONS BETWEEN COLLOIDAL PARTICLES. . 42
Introduction 42
Development of Charge at Solid-Liquid Interface 45
Dissociation of Surface Groups 45
Adsorption of Potential Determining Ions 46
Adsorption of Ionized Surfactants 46
Isomorphic Substitution 46
Electrical Double Layer 46
Double Layer Interactions 54
Interaction Between Two Flat Plates 56
Interaction Between Two Spherical
Particles 57
Van der Waal's Interaction 59
Microscopic or Van der Waal's Method 60
Flat Plates 60
Spherical Particles 61
Retardation Effect 62
Effect of Medium on the Van der Waal's Attraction .... 63
Macroscopic Approach 64
Hamaker Constants 65
The Effect of Polymer Layer on Van der Waal's
Attraction 66
Potential Energy Curves and the DLVO Theory 69
The Effect of Hamaker Constant 72
The Effect of Surface Potential 74
The Effect of Electrolyte Concentration 74
The Effect of Particle Radius 77
The Stability-Instability Approach 77
Kinetics of Coagulation 79
Summary 82
IV. EFFECT OF ADSORBED POLYMER ON DISPERSION STABILITY. ... 84
Introduction 84
Factors Influencing Steric Stabilization 85
The Adsorbed Amount of Polymer 85
The Solvent-Segment Interaction Parameter 86
The Effective Hamaker Constant and the Size
of Particles 86
Applications and Advantages of Steric Stabilization ... 87
Thermodynamic Basis of Steric Stabilization 89
iv

Polymer Solution Thermodynamics 90
The Entropy of Mixing 90
The Enthalpy of Mixing 92
The x Parameter 93
The Theta Point 95
Classification of Steric Stabilization 98
Quantitative Theories of Steric Stabilization 99
The Three Domains of Close Approach 101
The Interpenetration Domain 103
Interpenetration Without Mixing 107
The Potential Energy Diagrams 111
Time Scale of Approach of the Second Interface. ... 111
The Potential Energy of Interaction 113
Thermodynamically Limited Stability 115
Non-thermodynamically Limited Stability 119
Bridging Flocculation 122
Kinetic Aspects of Bridging Flocculation 127
Kinetics of Flocculation 130
The Potential Energy Diagrams for Bridging
Interparticle Interactions 133
V. STRUCTURE OF SUSPENSIONS 140
Introduction 140
Fractal Geometry 141
Models of Aggregate Formation 143
Eden Growth 143
Diffusion Limited Aggregation (DLA) 145
Cluster-Cluster Aggregation (CCA) 146
Hierarchial Model 148
Kinetics of Aggregation 148
Smoluchowski's Equation 150
Equilibrium Properties of Suspensions 153
The Order-Disorder Transition 154
VI. RHEOLOGICAL BEHAVIOR OF COLLOIDAL DISPERSIONS 160
Introduction 160
Viscosity Definition 161
Classification of Rheological Behavior of Colloidal
Dispersions 163
Newtonian Dispersions 165
Pseudo-plastic Dispersions 165
Dilatant and Shear Thickening Dispersions 167
Bingham Plastic Dispersions 167
Thixotropic Dispersions 167
Factors Affecting Rheological Behavior of Colloid
Dispersion 170
Interparticle Interactions 170
Brownian Motion 170
Hydrodynamic Interactions 171
Rheological Behavior of Stable Systems 171
v

The Effect of Adsorbed Layer 175
Stable Dispersions with Soft Interactions 180
Rheological Behavior of Flocculated Dispersions 181
Structure of Flocculated Dispersions 181
Flow Behavior of Flocculated Dispersions 186
Flocculated Dispersions Showing No Time Dependence . • 187
Elastic Floe Model 190
VII. MATERIALS PROPERTIES AND CHARACTERIZATION/
EXPERIMENTAL PROCEDURES 196
Introduction 196
Silica as a Model Material 196
Silica Preparation and Characterization 199
Silica Washing Procedure 203
Silica Calcination Treatment 204
Effect of Calcination Treatment on the Nature of
the Silica Surface 204
Silica Size and Size Distribution 210
Silica Surface Area 215
Silica True Density 217
Poly Vinyl Alcohol 219
Synthesis and Properties 219
Solubility, Solution Behavior and Interfacial
Activity 219
Fractionation of As-received PVA 221
Acetate Content of Polymer 222
Molecular Weight and Molecular Weight
Distribution 223
Viscometry 223
Gel Permeation Chromatography 226
Conformation and Solution Parameters 239
PVA Configuration Parameters 239
The x Parameter 242
Adsorption Measurements 246
Suspension Preparation Procedure 247
Suspension Characterization 251
Electrophoresis 251
Rheological Measurements 253
Consolidation and Green Microstructure 254
Summary 256
VIII. RESULTS AND DISCUSSION 257
Electrostatically Stabilized Dispersions 257
Effect of Adsorbed PVA on the Properties of Model
Silica Dispersions 272
Effect of Amount Adsorbed 272
Effect of Silica Calcination Treatment 284
Effect of PVA Molecular Weight 318
Fractional Surface Coverage 318
Effect of Suspension pH 334
vi

Effect of pH and Molecular Eight with the Plateau
Adsorbed Amount of Polymer 346
Selection of Optimum Molecular Weight 367
Effect of Adsorbed Layer Thickness on the Maximum
Solids Loading in Suspension 372
Effect of Silica Particle Size 375
Effect of PVA Degree of Hydrolysis on Adsorption
Behavior 388
Effect of Solvent Quality on the Rheological
Behavior 400
IX. SUMMARY AND SUGGESTIONS FOR FUTURE WORK 414
Summary 414
Suggestions for Future Work 417
LIST OF REFERENCES 420
BIOGRAPHICAL SKETCH 436
vii

LIST OF TABLES
Table Page
3.1 Effect of Ionic Strength and Valency on the Electrical
Double Layer Thickness 51
3.2 Compilation of Hamaker Constants for Silica-Water-PVA
System 67
4.1 Classification of Steric Stabilization 100
6.1 Classification of Flow Behavior 164
7.1 Geometric Mean and Specific Surface Area of Various Silica
Lots Used 216
7.2 Viscometric Molecular Weight and Acetate Content of PVA
Fractions . 225
7.3 Molecular Characteristics and Elution Volume of PEO
Calibration Standards 228
7.4 Gel Permeation Chromatography Data for Unfractioned PVA
Samples 229
7.5 Effect of Molecular Weight - Elution Volume Calibration
Curve on the GPC Results 235
7.6 Comparison of Viscometric Molecular Weight and GPC Results
for Various PVA Fractions 238
7.7 PVA Dimensions in Solution 243
8.1 Properties of Silica and PVA Used to Investigate the Effect
of Adsorbed Amount of PVA on the Suspension Properties. . . 274
8.2 Properties of Silica and PVA Used to Investigate the Effect
of Silica Calcination Temperature on the Suspension
Properties 286
8.3 Molecular Weight and Degree of Hydroxylation of PVA Samples
to Study the Effect of Degree of Hydroxylation on the
Suspension Properties 390
viii

LIST OF FIGURES
Figure Description Page
2.1 Schematic representation of an adsorbed polymer
molecule 8
2.2 Conformations of adsorbed polymer molecules (a) single
point attachment, (b) train-loop-tail adsorption,
(c) flat multiple site attachment, (d) random coil,
(e) nonuniform segment distribution, and
(f) multilayer 10
2.3 Plots of (a) degree of occupancy, 9, and (b) the
effective layer thickness, 6, as a function of the
adsorbed amount of PVA on Agl 37
2.4 Plot of segment density as a function of distance from
a surface for adsorbed PVA on PS-latex 39
3.1 Fraction of double layer potential versus distance from
a surface: (a) curves for 1:1 electrolyte at three
concentrations and (b) curves for 0.001 M symmetrical
electrolytes of three different valance types 52
3.2 Schematic illustration of the variation of potential as
a function of distance from a charged surface in the
presence of a stern layer, subscripts o at wall, 6 at
stern surface, d in diffuse layer 52
3.3 Schematic illustration of the effect of adsorbed
polymer layer on Van der Waal's attraction 68
3.4 Total potential energy of interaction V(d) = VR(d)
+ V^(d) where VR(d) is the potential energy of
repulsion due to double-layer interactions and V^(d) is
attractive potential due to Van der Waal's
interactions 71
3.5 The effect of the Hamaker constant on the total
interaction energy curves 73
3.6 The effect of zeta potential on the total interaction
energy curves 75
ix

3.7
76
The effect of concentration of 1:1 electrolyte on the
potential energy curves
3.8 The effect of particle radius on the total interaction
energy curves 78
3.9 Theoretical dependence of stability ratio on
electrolyte concentration 81
4.1 The three domains of close approach of sterically
stabilized flat plates, (i) Noninterpenetration (d >
2L); (ii) Interpenetration (L £ d £ 2L);
(iii) Interpenetration plus compression (d < L).. . . 102
4.2 The distance dependence of the steric interaction
energy for two equal spheres of radius a, stabilized
by polymer layers with different segment density
distribution functions. (1) exponential; (2) constant;
(3) Gaussian; (4) radial Gaussian, d is the minimum
distance between surfaces of the spheres, 6 is the
barrier thickness, and AGS is the interaction energy.
4.3 The free energy of interaction between particles
covered by equal tails (f) and equal loops (a). For
particles covered by equal tails, (b) gives the volume
restriction effect and (c) the osmotic repulsion; (f)
is the resultant of adding (b), (c), and (e) 116
4.4 Schematic illustration of the effect of segment-solvent
interaction parameter, x, on the potential energy
diagram (A) poor solvent, x > 0.5, (B) theta solvent, x
= 0.5, and (C) good solvent, x<0.5 118
4.5 The free energy of interaction of polystyrene latex
particles stabilized by poly (vinyl alcohol) according
to Hesselink, Vrij, and Overbeek (1971); stabilizer
molecular weight 1, 8,000; 2, 17,000; 3, 28,000; 4,
43,000 120
4.6 Plots showing the effect of particle size and adsorbed
amount on the depth of the minima in the total
potential energy of interaction 120
4.7 Schematic illustration of bridging flocculation with
adsorbed polymer 123
4.8 The effect of thickness of the electrical double layer
on bridging flocculation 125
4.9 Schematic diagram showing mixing, adsorption, and
flocculation upon addition of polymeric flocculent. . 128
x

132
4.10 The effect of solids loading and 6/a on collisions
Zf/Z0
4.11 Schematic representation of the approach of a second
(uncovered) particle to a covered one. It was assumed
that, at large interparticle distance H the number of
segments which adsorb on the (originally) bare particle
per unit area is equal to the number of segments per
unit area which would lie beyond Hi in the absence of
the second particle (i.e., shaded area) 134
4.12 Schematic representation of the bridging process.
(a) At large distances, a loop of i segments has its
unpurturbed configuration. (b) After adsorption of the
first segment, two bridges of i/2 segments each are
formed. (c) At shorter distances, two bridges of ifl
segments and a train of segments adsorbed on the
second surface 136
4.13 The total free energy of interaction between coated and
uncoated plates as a function of distance of
separation 137
5.1 Eden cluster produced by monomer-cluster growth.. . . 144
5.2 An aggregate grown by the DLA process 144
5.3 An aggregate grown by the CCA process 147
5.4 Schematic plot of phase diagram for monosized spherical
particles. The volume fraction of solids as a function
of ionic concentration is plotted. Solid lines are
theoretical phase boundaries 156
5.5Schematic illustration showing (a) "hard" and (b)
"soft" interactions between particles. The potential
energy of interaction as a function of distance of
separation is plotted 158
6.1 Schematic illustration of the concept of viscosity
under laminar flow conditions 162
6.2 Schematic plots of (a) shear stress versus shear rate
and (b) viscosity versus shear rate for various types
of flow behaviors 166
6.3 Schematic representation of thixotropic flow behavior,
(a) shear stress versus shear rate, and (b) viscosity
versus shear rate plots 169
6.4 Schematic plot of dependence of relative viscosity on
the volume fraction solid in suspension 173
xi

6.5 Plot of relative viscosity versus volume fraction latex
particles of different sizes. Data was fitted using
Krieger equation 176
6.6 Plot of relative viscosity versus dimensionless shear
rate, "rr, for monodisperse suspensions of polystyrene
spheres at 0 = 0.50 in different fluids 178
6.7 Schematic illustration of the effect of shear on the
stability of suspensions 185
6.8 Schematic illustration of flow curve parameters for
pseudoplastic flow behavior 188
7.1 Schematic representation of various types of surface
groups present on the silica surface 206
7.2 The diffuse reflectance Fourier transform infrared
spectra of silica powders calcined at various
temperatures 207
7.3 Concentration of surface silanol groups as a function
of the temperature of calcination 209
7.4 Scanning electron micrograph of silica powder 212
7.5 A histogram (number of particles in a given diameter
class versus particle diameter) of a typical silica
batch 213
7.6 Plot of particle size distribution for silica
determined by x-ray sedimentation 214
7.7 Gas pycnometer density versus calcination temperature
for SÍO2 powders 218
7.8 Molecular weight calibration curves for PEO standards
and commercial PVA88. Log M is plotted as a function
of elution volume 231
7.9 GPC chromatograms, showing refractive index detector
response, h, as a function of elution volume for
several unfractioned PVA samples 233
7.10 GPC chromatograms, showing the effect of acetone
fractionation of Vinol 540 and Vinol 203 polymer
samples on distribution widths 237
7.11 Stockmayer-Fixman plot for PVA88 240
xii

7.12Plot of absorbance versus PVA concentration in solution
for various molecular weight polymers 248
8.1 Plot of zeta potential versus suspensions pH for 20
vol.% SÍO2 at ionic strength of 1 x 10“^ moles/liter
NaCl 258
8.2 The effect of zeta potential on (a) shear stress versus
shear rate and (b) viscosity versus shear rate plots.
8.3 Schematic illustration showing the structural breakdown
of a floe due to applied shear 261
8.4 DLVO plots of potential energy of interaction versus
distance of separation at indicated £ potentials. . . 263
8.5 Plot of relative viscosity versus zeta potential for 20
vol.% SiC>2 suspensions 265
8.6 Plot of (a) Cpp versus zeta potential and (b) Cpp
versus zeta potential square for 20 vol.% SÍO2
suspensions 266
8.7 Plot of (a) extrapolated yield stress versus zeta
potential and (b) extrapolated yield stress versus zeta
potential square for 20 vol.% SÍO2 suspensions. . . . 269
8.8 Plots of specific volume frequency versus pore radius
obtained by mercury porosimetry for sedimented samples
with indicated pH values 271
8.9 Adsorption isotherm for 20 vol.% SÍO2 suspensions
prepared with varying concentration of PVA with
molecular weight » 24,000 at pH 3.7 273
8.10 Plots of relative viscosity versus shear rate for 20
vol.% SÍO2 suspensions prepared at pH 3.7 with varying
PVA concentrations and pH 7.3 (-60 mV zeta potential)
with no PVA 277
8.11 Plot of yield stress versus adsorbed amount of PVA with
molecular weight = 24,000 at pH 3.7 279
8.12 Plot of relative viscosity versus adsorbed amount of
PVA for 20 vol.% SÍO2 suspensions prepared with varying
PVA concentration 280
8.13 Plot of hysteresis area versus adsorbed amount of PVA
of 20 wt.% SÍO2 suspensions 281
8.14 Plot of (a) sediment density versus fraction plateau
coverage and (b) median pore radius versus fraction
xiii

plateau coverage for compacts prepared from 20 vol.%
SÍO2 suspensions with varying PVA concentrations at pH
3.7 and pH 7.3 with no polymer 283
8.15 Adsorption isotherms for silicas calcined at various
temperatures with PVA molecular weight = 215,000 at pH
3.7 285
8.16 Plot of plateau adsorbed amount versus silica
calcination temperature 288
8.17 Plots of plateau adsorbed amount versus silica
calcination temperature for (a) PEO adsorption on Cab-
O-Sil and (b) PVA adsorption on Cab-O-Sil 290
8.18 Plot of relative viscosity versus shear rate for 20
vol.% suspensions prepared with uncalcined silica and
with silica powder calcined at 500°C and 700°C. ... 292
8.19 Plot of yield stress versus calcination temperature for
20 vol.% silica suspensions at pH 3.7 prepared with
silica particles calcined at various temperatures. PVA
concentration in solution was sufficient to achieve
plateau coverage of the silica particles 293
8.20 Plot of relative viscosity versus calcination
temperature for 20 vol.% silica suspensions at pH 3.7
prepared with silica particles calcined at various
temperatures. PVA concentration in solution was
sufficient to achieve plateau coverage of the silica
particles 294
8.21 Plot of hysteresis area versus calcination temperature
for 20 vol.% silica suspensions at pH 3.7 prepared with
silica particles calcined at various temperatures. PVA
concentration in solution was sufficient to achieve
plateau coverage of the silica particles 295
8.22 Schematic plot of the total interaction energy versus
distance of separation between two polymer-coated
particles with varying adsorbed amounts 297
8.23 Schematic illustration (a) showing silica surface
covered with only enough PVA to make particles
hydrophobic and (b) excess of PVA adsorption prevents
cocervation 299
8.24 Schematic plot of (a) disjoining pressure as a function
of distance of separation for two parallel water films
stabilized with PVA film and (b) disjoining curve for
homopolymer. The specific attractive component in the
xiv

case of PVA derives from the hydrophobic interactions
between acetate groups 301
8.25Plot of (a) relative density for slip cast samples
versus silica calcination temperature and (b) median
pore radius versus silica calcination temperature.. . 303
8.26Plot of mercury porosimetry data for slip cast samples
prepared from uncalcined and 700°C calcined silica
powders. Samples were prepared from 20 vol.% silica
suspensions at pH 3.7 with the plateau coverage of SiC>2
particles with PVA of molecular weight = 215,000. . . 304
8.27 Plot of (a) relative viscosity versus adsorbed amount
and (b) relative viscosity versus fractioned plateau
coverage for 20 vol.% SiC>2 suspensions prepared with
uncalcined and 700°C calcined powders 306
8.28 Schematic illustration of floe structures formed with (a)
uncalcined and (b) 700°C calcined silica particles at low
surface coverages with adsorbed polymer 307
8.29The depth of the free energy minimum as a function of
the total amount of polymer between the surfaces at
various solvency conditions for two molecular weights
of polymer. 0^ is the total amount of polymer between
the plates expressed as the number of equivalent
monolayers, Afm¿n is the interaction energy, and r is
the number of segments per chain 309
8.30 Plots of (a) yield stress versus adsorbed amount and
(b) yield stress versus fractioned plateau coverage for
20 vol.% silica suspensions prepared with 700°C
calcined and uncalcined silicas with varying PVA
concentration in solution 311
8.31 Schematic illustration shows the total number of
bridges, ntotal, formed between two spherical particles
of radius a separated by distance 2h 312
8.32 Plots of (a) relative density of gravity cast samples
versus fraction plateau coverage and (b) median pore
radius versus fraction plateau coverage for compacts
prepared from 20 vol.% SÍO2 suspensions of uncalcined
and 700°C calcined silicas with varying PVA
concentrations 315
8.33 Plots of (a) hysteresis area versus adsorbed amount and
(b) hysteresis area versus fraction plateau coverage
for 20 vol.% SÍO2 suspensions prepared with uncalcined
and 700°C calcined silicas with varying PVA
concentrations 316
xv

8.34Plots of adsorption isotherms for two PVA samples with
different molecular weights (i.e., 24,000 and 215,000
g/mole) determined using 20 vol.% SiC>2 suspensions at
pH 3.7 319
8.35 Plots of (a) relative viscosity versus adsorbed amount
and (b) relative viscosity versus fraction plateau
coverage for two PVA samples with different molecular
weights 321
8.36 Plots of relative viscosity versus shear rate for 20
vol.% SÍO2 suspensions prepared using indicated
molecular weight PVA samples at fixed PVA concentration
in solution. Adsorbed amount was the same (0.15 mg
PVA/m^ SÍO2) for all suspensions 322
8.37 Plot of relative viscosity versus PVA molecular weight
(log scale) at fixed adsorbed amount of polymer.. . . 323
8.38 Plots of (a) yield stress versus adsorbed amount and
(b) yield stress versus fraction plateau coverage for
20 vol.% silica suspensions prepared with two PVA
samples with different molecular weights 325
8.39 Plot of yield stress versus PVA molecular weight (log
scale) at fixed adsorbed amount 326
8.40 Plots of (a) hysteresis area versus adsorbed amount and
(b) hysteresis area versus fraction plateau coverage
for 20 vol.% SÍO2 suspensions prepared with two PVA
samples of different molecular weights 328
8.41 Plot of hysteresis area versus PVA molecular weight
(log scale) at fixed adsorbed amount of polymer.. . . 329
8.42 Plots of (a) relative density of sedimented samples
versus fraction plateau coverage and (b) median pore
radius versus fraction plateau coverages for compacts
prepared using two PVA samples of different molecular
weights 330
8.43 Plot of relative density of compacts prepared using
gravity casting and slip casting versus PVA molecular
weight at fixed adsorbed amount of polymer 332
8.44 Plots of relative density versus Cpp (= 0p/®p) for two
PVA samples with different molecular weights. Results
for compacts prepared at pH 3.7 and pH 7.6 with no
added PVA are also shown 333
xv i

8-45 Adsorption isotherm of PVA with molecular weight =
24,000 at two different suspensions pH's 335
8.46 Plot of plateau adsorbed amount of PVA with molecular
weight = 200,000 as a function of suspensions pH. . . 336
8.47 Plots of (a) relative viscosity of 20 vol.% suspensions
versus adsorbed amount of PVA with molecular weight =
24,000 and (b) relative viscosity versus fraction
plateau coverage for suspensions prepared at pH values
3.7 and 7.6 with varying PVA concentration 338
8.48 Plots of (a) yield stress versus adsorbed amount of
polymer and (b) yield stress versus fraction plateau
coverage for suspensions prepared at pH values 3.7 and
7.6 340
8.49 Schematic illustration shows the prevention of bridging
flocculation due to diffuse electrical double layer.. 341
8.50 Plots of (a) sediment density versus fraction plateau
coverage and (b) median pore radius versus fraction
plateau coverage for compacts prepared from 20 vol.%
SÍO2 suspensions with varying amounts of PVA
concentrations at indicated suspension pH values. . . 342
8.51 Plot of zeta potential versus adsorbed amount of PVA
with molecular weight 24,000 at suspensions pH 7.6. . 344
8.52 Schematic plots of effective adsorbed layer thickness,
6, as a function of measured zeta potential at
indicated ionic strengths. The zeta potential with no
adsorbed polymer is = -65 mV 345
8.53 Plots of plateau adsorbed amounts of polymer versus PVA
molecular weight (log scale) at suspensions pHs 3.7 and
7.8. Suspensions were prepared using 20 vol.% SÍO2
with sufficient concentration of PVA in solution with
varying PVA molecular weights at indicated pH values. 347
8.54 Plots of relative viscosity versus shear rate for 20
vol.% SÍO2 suspensions at pH 7.8 with plateau adsorbed
amounts of PVAs having indicated molecular weights. . 348
8.55 Plots of relative viscosity versus shear rate for 20
vol.% SÍO2 suspensions at pH 3.7 with plateau adsorbed
amounts of PVAs having indicated molecular weights. . 349
8.56 Plots of relative viscosity versus PVA molecular weight
(log scale) for 20 vol.% SÍO2 suspensions with plateau
adsorbed amounts of polymer at indicated pH values. . 351
xvii

354
8.57 Plots of hysteresis area versus PVA molecular weight
(log scale) for 20 vol.% SÍO2 suspensions at pH 3.7
with plateau adsorbed amounts of polymers
8.58 Plots of yield stress versus PVA molecular weight (log
scale) for 20 vol.% SÍO2 suspensions with plateau
adsorbed amounts of polymers at indicated pH values.. 355
8.59 Plots of (a) relative density of gravity cast samples
versus PVA molecular weight and (b) median pore radius
versus PVA molecular weight for compacts prepared using
20 vol.% SÍO2 suspensions with plateau adsorbed amounts
of polymers with different molecular weights at
indicated pH values 357
8.60 Plots of (a) relative density of slip cast samples
versus PVA molecular weight and (b) median pore radius
versus molecular weight for compacts prepared using 20
vol.% SÍO2 suspensions with plateau adsorbed amounts of
PVAs with different molecular weights at indicated pH
values 358
8.61 Schematic plots of relative viscosity versus volume
fraction of solids in suspensions for particles with
the indicated thicknesses of adsorbed polymer. This
thickness, 6, is indicated as a fraction of the
particle radius 360
8.62 Plot of (a) relative viscosity versus volume fraction
silica in suspensions prepared at pH 7.6 and (b)
relative viscosity versus volume fraction of latex
particles as reported by Krieger 361
8.63 Plot of relative density of gravity cast samples versus
volume fraction silica at pH 7.6 362
8.64 Plots of the adsorbed layer thickness, 6, versus PVA
molecular weight determined from the relative viscosity
values of 20 vol.% suspensions prepared at pH 7.8.
Also shown are the radius (Rg) and diameter (2 x Rg) of
gyration of the polymers in solution as determined by
intrinsic viscosity measurements 364
8.65 Plots of adsorbed PVA layer thickness, 6, on SÍO2
particles versus square root of PVA molecular weight.
Adsorbed layer thicknesses of PVA onto PS latex
particles is also shown 368
8.66 Schematic plots of minimum molecular weight (log scale)
required to stabilize suspensions as a function of
particle radius with different values of A /V^
ratios. The solid lines separates stable ana unstable
xviii

regions of suspensions prepared with spherical
particles of fixed size with varying molecular weight
or suspensions prepared with fixed molecular weight and
varying particle radius 370
8.67 Schematic plots of maximum true solids loading, 0 ,
achievable in suspensions prepared with spherical
monosized particles of varying size using indicated
molecular weights of polymer 374
8.68 Adsorption isotherms for PVA with molecular weight =
215,000 for 20 vol.% SÍO2 suspensions prepared using
0.4 urn and 0.7 um size particles with varying PVA
concentrations 376
8.69 Plots of (a) relative viscosity versus adsorbed amounts
of PVA and (b) relative viscosity versus fraction
plateau coverage for 20 vol.% SÍO2 suspensions prepared
using 0.4 um and 0.7 urn size particles 378
8.70 Schematic illustration showing the effect of adsorbed
layer thickness, 6, on the hydrodynamic volume of two
different size particles 379
8.71 Plots of (a) yield stress versus adsorbed amount of PVA
and (b) yield stress versus fraction plateau coverage
for 0.4 urn and 0.7 pm size particles 381
8.72 Schematic illustration showing (a) concentration
profile, 0(Z), of adsorbed layer consisting of three
regions (i) proximal (very sensitive to the details of
the interactions), (ii) central (self-similar), and
(iii) distal (controlled by a few loops and tails) and
(b) "self-similar grid" presentation of an adsorbed
polymer layer 382
8.73 Plots of hysteresis area versus fraction plateau
coverage for 0.4 urn and 0.7 urn size particles 384
8.74 Plots of (a) relative density of sedimented samples
versus fraction plateau coverage and (b) median pore
radius versus fraction plateau coverage for 0.4 um and
0.7 um size particles 386
8.75 Plots of normalized median pore radius (i.e., median
pore radius/particle radius) versus fraction plateau
coverage for 0.4 um and 0.7 um size particles 387
8.76 Adsorption isotherms for PVAs with similar molecular
weights but varying degree of hydroxylation of 20 vol.%
SÍO2 suspensions 389
xix

8.77Plots of relative viscosity versus shear rate of 20
vol.% silica suspensions prepared with different PVAs
with indicated degree of hydroxylation. The
suspensions are prepared at pH 3.7 with the plateau
adsorbed amounts of PVAs
393
8.78 Plots of specific volume frequency versus pore radius
for gravity cast samples prepared from 20 vol.% SiC>2
suspensions with plateau adsorbed amounts of PVAs with
indicated degree of hydroxylation 394
8.79 Plots of (a) relative viscosity versus adsorbed amounts
and (b) relative viscosity versus fraction plateau
coverage for 20 vol.% SÍO2 suspensions prepared using
varying PVA concentrations in solution. The degree of
hydroxylation of different PVAs used is shown in the
figure 396
8.80 Plots of hysteresis area versus fraction plateau
coverage for 20 vol.% SÍO2 suspensions prepared using
PVAs with indicated degree of hydroxylation 397
8.81 Plots of (a) yield stress versus adsorbed amount and
(b) yield stress versus fraction plateau coverage for
20 vol.% SÍO2 suspensions prepared using PVAs with
indicated degree of hydroxylation 398
8.82 Plots of (a) relative density of gravity cast sample
versus fraction plateau coverage and (b) median pore
radius versus fraction plateau coverage of compacts
prepared from 20 vol.% SÍO2 suspensions with varying
PVA concentration in solution. The degree of
hydroxylation for various polymers is indicated in the
figure 399
8.83 Plots of (a) relative viscosity versus Na2S04
concentration and (b) yield stress versus Na2S04
concentration for 20 vol.% SÍO2 suspensions with
plateau coverages of particles with adsorbed polymer.
402
8.84Plots of relative viscosity versus shear rate for 20
vol.% SÍO2 suspensions with varying Na2SC>4
concentration
403
8.85 Plot of hysteresis area versus Na2SC>4 concentration in
solution suspensions were prepared at pH 7.8 with the
plateau adsorbed amount of PVA with molecular weight =
200,000 405
8.86 Plots of (a) relative density of gravity cast samples
versus Na2S04 concentration and (b) median pore radius
versus Na2SC>4 concentration 406
xx

8.87
Plots of (a) relative density of slip cast samples
versus Na2SC>4 concentration and (b) median pore radius
versus Na2SC>4 concentration in solution 407
8.88 Plots of Na2SC>4 concentration versus PVA concentration
in solution. Solid line separates single phase region
(i.e., true polymer solution) from the two phase region
(i.e., precipitated polymer and solvent) 410
8.89 Plots of relative viscosity versus Na2SC>4 concentration
for suspensions of polymer coated particles and
suspensions prepared with no added polymer at pH 7.8. 411
8.90 Plots of (a) relative density of gravity cast samples
versus Na2SC>4 concentration and (b) median pore radius
versus Na2SC>4 concentration of compacts prepared from
suspensions of polymer coated particles and suspensions
with no added polymer at pH 7.8 412
xxi

Abstract of a Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
THE EFFECT OF POLY (VINYL ALCOHOL) ON THE
PROPERTIES OF MODEL SILICA SUSPENSIONS
By
Chandra S. Khadilkar
December 1988
Chairman: Dr. Michael D. Sacks
Major Department: Materials Science and Engineering
The effect of interparticle interactions on the properties of model
suspensions of monosized, spherical silica particles was investigated.
The electrostatic interactions between particles were controlled by
changing suspension pH. The effect of adsorbed polymer was investigated
using water soluble poly (vinyl alcohol), PVA. The adsorption was
dependent on a variety of factors including the overall polymer
concentration in suspension, PVA molecular weight, PVA degree of
hydroxylation, silica surface hydroxylation, and suspension pH. The
adsorption characteristics were correlated with the state of particulate
dispersion in suspension using rheological measurements. Suspension and
green compact properties were highly dependent upon the fraction of
silica surface covered by the adsorbed polymer and the thickness of the
adsorbed polymer layer. Suggestions are made for selecting polymer
xxii

molecular weight to prepare stable suspensions with higher solids
loading.
xxiii

CHAPTER I
GENERAL INTRODUCTION AND AIM OF THE STUDY
Particle/liquid suspensions are prepared in various ceramic
processing operations such as mixing, milling, spray granulation of
powders, slip casting, tape casting, extrusion, etc. The control of
rheological behavior of suspensions is important to maximize the
processing efficiency and also to obtain the desired properties of the
final products. The rheological behavior of particle/liquid suspensions
can be modified by changing (1) particle characteristics (e.g., size and
shape distribution), (2) liquid characteristics (e.g., viscosity), (3)
particle concentration (solids loading of the suspensions), (4)
interparticle forces in suspension, and (5) additives like dispersants,
polymers, etc.
The interparticle interactions in suspensions can be broadly
classified into two categories: (1) electrostatic interactions and (2)
interactions due to adsorbed polymer. The electrostatic interactions
arise due to development of charge at the solid/liquid interface. The
electrically-charged interface leads to the formation of diffuse
"electrical double layer" surrounding each particle. The overlap of two
electrical double layers during Brownian encounter gives rise to
repulsive forces between the particles which can overcome the Van der
Waal's attraction. The effects of surface potentials (or charge) and
ionic concentrations on the stability of dispersions can be
quantitatively described using the "DLVO" theory (Chapter III).
1

2
Organic polymers are extensively used as processing aids in ceramic
forming operations. In addition to acting as binders and plasticizers,
polymers are often used to control the rheological properties of
dispersions. The interactions between particles with added polymer are
highly dependent on the adsorption behavior of polymer on particle
surfaces. Generally, at low polymer concentrations in solution,
suspensions can be destabilized (i.e., particles are aggregated) due to
"bridging flocculation" where a polymer molecule can adsorb
simultaneously onto two or more particles. At sufficiently high polymer
concentrations in solution, under certain conditions, stable suspensions
(i.e., in which primary particles are well separated) can be prepared.
Two important factors controlling the interactions between the particles
are (1) the fractional coverage of the particle by adsorbed polymer
(which affects the bridging flocculation) and (2) the thickness of the
adsorbed layer (which governs the stability at complete coverage in a
good solvent).
A model ceramic powder (i.e., well-characterized, agglomerate-free,
spherical, narrow-sized silica) was used in this investigation. The
electrostatic interactions between particles were modified by varying the
suspension pH and ionic strength. The effect of surface potentials cn
the flow curve parameters (i.e., extrapolated yield stress, relative
plastic viscosity, etc.) were analyzed using the "elastic floe" model.
Water soluble poly (vinyl alcohol), PVA, was used to investigate
the effect of adsorbed polymer on the suspension properties. PVA being a
neutral polymer, the effects of electrostatic interactions and the
effects of adsorbed polymer on the dispersion stability can be studied

3
less ambiguously. The concentration of PVA in solution can be readily
determined; hence, the adsorbed amounts of PVA could be determined
easily. The polymer adsorption behavior and rheological properties on
the same system are rarely reported in literature. The adsorption
experiments are generally carried out using dilute suspensions of high
surface area, agglomerated powders, whereas rheological measurements are
conducted on concentrated suspension where adsorption behavior and
interparticle interactions are not very well defined. In this study,
most of the rheological measurements and adsorption experiments were
carried out at the same solids loading (= 20 vol. %). The adsorption of
PVA onto the silica surface was dependent on a variety of factors
including the overall solution concentration, polymer characteristics
(PVA molecular weight and degree of hydroxylation), silica surface
nature, suspension pH, etc. The adsorption characteristics were
correlated with the state of particulates in suspension using rheological
measurements. Suspension rheological properties and green compact
properties (e.g., relative density, median pore radius, etc.) were highly
dependent on the fraction of silica covered by the adsorbed polymer and
the thickness of the adsorbed layer.
A floe structure (i.e., floe compactness, strength, etc.) consistent
with the rheological behavior and green compact characteristics is
proposed. (Experimental techniques to obtain this information are not
yet readily available). Finally, some suggestions regarding selection of
polymers to achieve stable suspensions with high solids loading are made.
In the first part of thesis, we deal with general and theoretical
concepts related to this work. In Chapter II, we describe the polymer

A
adsorption behavior at solid/liquid interface. Chapter III deals with
the electrostatic interactions between colloidal particles, and Chapter
IV deals with the effect of adsorbed polymer on dispersion stability.
The effect of interparticle interactions on the state of particulate
dispersion in suspension is described in Chapter V. The correlation
between suspension structure and rheological behavior is discussed in
Chapter VI. The silica preparation and characterization is described in
Chapter VII. PVA fractionation and characterization is also described in
Chapter VII. In Chapter VIII, we discussed experimental results obtained
in this study. First, we describe the effect of electrostatic
interactions on the rheological behavior of 20 vol.% silica suspensions.
Subsequently, PVA adsorption behavior on silica particles is correlated
with the rheological properties of suspensions and green compact
characteristics.

CHAPTER II
ADSORPTION OF POLYMER AT SOLID/LIQUID INTERFACE
Introduction
Adsorbed polymer at solid/liquid interface has a profound effect on
the stability behavior of suspensions. Typically, at low polymer
dosages, flocculation of particles in the suspension can occur, while at
the high polymer concentration, stable suspensions can be prepared.
Polymer adsorption behavior has been extensively studied because of the
present and the potential applications in industry, technology, and
medicine. The polymer adsorption behavior is important in various
processes (tape casting, slip casting, extrusion of ceramic parts,
adhesion, separation of polymers, soil improvement, etc.) and various
products (paints, cosmetics, magnetic tapes, pharmaceuticals, dyes,
foods, lubricants, etc.). Theoretical interests in the polymer
adsorption process stems from the fact that insight can be gained into
the nature of the forces acting between polymer segments and surfaces and
between particles coated with the adsorbed polymer. From the ceramic
processing point of view, by the optimization of the adsorption process,
one can control both the suspension properties as well as consolidated
microstructures obtained from these suspensions.
The adsorption behavior of the polymer is different from the
adsorption behavior of the small molecules (e.g., gas adsorption at
solid/gas interface, surfactant adsorption, etc.). This difference
5

6
arises because polymers have a large number of internal degrees of
freedom (i.e., flexibility).
Polymer molecules have a random coil type of arrangement in the
solution. A full description of the shape of a molecule requires the
description of the relative positions of each atom of the molecule (the
configuration, see Flory 69). It is common to assume that a polymer
chain consists of a number of connected chain segments for the
theoretical treatments. Then, the conformation of the adsorbed polymer
is described by specifying the relative positions of the endpoints of
these segments (Kuhn 34). To understand the adsorption behavior of
polymers, one needs to consider both the energetics and Kinetics of the
adsorption process. Polymer molecules experience short-range attractive
forces near the adsorbing interfaces. For each polymer segment adsorbed
on the surface, there is a decrease in the free energy of the system. At
the same time, upon adsorption, the random coil structure of the polymer
molecule in the solution will be distorted, which will lead to a decrease
in the number of conformations of the adsorbed polymer. This entropic
factor will oppose the adsorption process. Final conformation of the
adsorbed polymer depends on this subtle balance between the entropic and
the enthalpic terms. Also, though there is a small energy decrease due
to adsorption process per segment of the polymer molecule, large number
of segments per molecule can be adsorbed, and the total energy decrease
per molecule can be quite large. Actual bonding mechanism between the
polymer segment and the surface may involve various interactions such as
electrostatic interactions, Van der Waal's interactions, hydrogen bonding
or hydrophobic bonding.

7
The actual conformation of the adsorbed polymer near solid-liquid
interface depends upon various factors such as solid-liquid, polymer-
liquid and polymer-solid interactions, flexibility of the polymer
molecule, concentration of the polymer in the solution, etc.
Experimentally, it has been found that the adsorption isotherm
(i.e., adsorbed amount of polymer as a function of equilibrium
concentration of polymer in solution at fixed temperature and pressure)
of high molecular weight polymers display high affinity type of
character, i.e., a steep initial part of the adsorption isotherm followed
by a pseudo-plateau region. Adsorbed amounts at the plateau region are
typically on the order of approximately one to four mg/m^. This amount
is much more than that can be accommodated in a close-packed monolayer of
the polymer segments. This led early investigators to propose the
conformation of the adsorbed polymer as shown in Figure 2.1 (Jenkel and
Rumbach, 51). From Figure 2.1, it is clear that not all of the segments
of the adsorbed polymer are in direct contact with the surface. This
type of conformation is commonly called as "train-loop-tail"
conformation. "Train" is defined as a sequence of consecutive segments
in direct contact with the surface. "Loop" is defined as a sequence of
segments with the end segments in direct contact with the surface, and
remaining segments are in contact with the solution. "Tail" is defined
as the portion of the chain with end segments in contact with the
surface. From this type of conformation, it is clear that there is a
significant change in the polymer conformation upon adsorption compared
to the random coil arrangement in the solution and that the adsorbed
polymer has sufficient extension in the solution. Actual details of the

8
Train-Loop-Tail conformation of Adsorbed
Polymer
Figure 2.1
Schematic representation of an adsorbed polymer molecule.

9
conformation are dependent on the various factors mentioned previously.
Some other types of conformations of the isolated adsorbed polymer
molecules are shown in Figure 2.2 (Sato and Ruch 80).
Description of Adsorbed Polymer
The adsorption behavior of the polymers at the solid/liquid
interface is commonly characterized using the following parameters.
Adsorbed Amount of Polymer (A)
This is the most commonly measured parameter in any adsorption
experiment and is usually obtained by "solution depletion" techniques.
Adsorbed amounts are commonly expressed in the units of mg of polymer
adsorbed per unit area of the solid surface. It is also common to define
the dimensionless adsorbed amount or adsorbance, y, as the number of
segments adsorbed per surface site or the number of (equivalent) complete
monolayers than can be formed from the adsorbed amounts (Cohen Stuart
80a). This means that y is the ratio between the adsorbed weight per
unit area and the weight adsorbed per unit area in a complete monolayer,
A
mon.
Y = A/A,
mon
(2.1 )
As we have seen earlier, there is a conformational change upon the
adsorption of the polymer. Hence, for a complete description of polymer
adsorption, it is not sufficient to measure only the adsorbed amount of
polymer, as similar adsorbed amounts could be obtained for the thick
adsorbed layer of polymer with low segment concentrations or thin
adsorbed layer with high segment concentration.

10
Figure 2.2 Conformations of adsorbed polymer molecules (a) single
point attachment, (b) train-loop-tail adsorption,
(c) flat multiple site attachment, (d) random coil,
(e) nonuniform segment distribution, and (f) multilayer
(Sato and Ruch, 1980).

11
Bound Fraction (p)
This is defined as the fraction of adsorbed amount of polymer in
direct contact with the surface, i.e.,
p = A^r/A (2.2)
where A^r is the amount of polymer adsorbed per unit area in trains. The
bound fraction measures the change in the conformation upon adsorption.
The p close to one indicates the polymer lies flat on the surface having
two-dimensional structure; on the other hand, p close to 0 indicates
polymer has essentially random coil shape with no significant change in
conformation (Figure 2.c).
Direct Surface Coverage (6)
This is defined as the fraction of the available surface sites
occupied by polymer segments.
0 = Atr/A
mon
(2.3)
where A<.r and A as defined previously and from the definition of 0
mon c
and p, it follows that
Y = 0/p (2.4)
Segment Density Distribution p (X, Y, Z)
Segment density is the time-averaged volume fraction of segments per
unit volume in the vicinity of the polymer molecule. In solution, many
polymers adopt random coil conformation. For such a random coil, the
segment density averaged on all conformations is usually Gaussian in any
direction passing through the center of the molecule. (More accurately,
the distribution is better represented by a prolate ellipsoid of
revolution, see Flory 53.) Upon adsorption, there is a significant
change in the conformation of the polymer molecules. At low adsorption

12
density (i.e., isolated polymer molecules) the segment density
distribution will be dependent upon Z (i.e., distance from the adsorbing
surface) and also on X, Y (ie., position parallel to the surface). At
higher adsorption densities, there will be a significant lateral
interpenetration of the molecules (since the volume percent occupied by
the segments is approximately ten percent at typical segment densities,
i.e., there is enough empty space in a given polymer molecule) and the
segment density distribution will be dependent on Z only. This segment
density distribution is the most important feature of the adsorbed
polymer in the theory of steric stabilization. Various types of segment
density distributions have been proposed for the adsorbed polymer and
will be reviewed later (Napper 83). The segment density distribution
determines the extension of the adsorbed polymers into the solution
phase. This extension of the adsorbed polymer layer can be expressed in
terms of a thickness parameter such as the root mean square layer
thickness 6, which is defined as follows:
00
CO
(2.5)
o
o
Theoretical Models
The adsorption behavior (and hence, conformation) of polymers at
solid-liquid interface is governed by various factors including polymer
segment-surface, surface-solvent, and segment-solvent interactions. The
driving force for the adsorption is the reduction in the net free energy
due to bonding of a polymer segment to the surface. This binding process
involves the removal of the solvent molecule from the surface and
replacement with the polymer segment. Energy change associated with this

13
process is denoted by the dimensionless adsorption energy parameter, Xs-
This polymer adsorption process is opposed by the entropy changes
associated with the changes in conformation upon adsorption. Polymer
molecules have three-dimensional random coil structure in solution, and
upon adsorption, some of the segments (i.e., train) are restricted to two
dimensions.
This process is associated with considerable loss of entropy per
molecule, and the magnitude depends on the chain length and flexibility.
At equilibrium, the distribution of polymer molecules between the surface
and the solution is essentially determined by the following five
independent parameters: concentration of the polymer in solution, chain
length (molecular weight), flexibility, adsorption energy parameter, and
polymer-solvent interaction parameter. Variables, such as temperature,
time (kinetics), and polydispersity, may also play an important role.
Below, we will define some of these variables and discuss the effect of
these independent variables on the adsorption.
Adsorption Energy Parameter (Xc)
As mentioned earlier, the bonding between the surface and the
segment is the driving force for adsorption. A precise definition of Xs
is due to Silberberg (Silberberg 68). The energy change (i.e., net
enthalpy change) associated with the transfer of a polymer segment in a
pure polymer from a bulk site to a surface site, minus the corresponding
energy change for a solvent molecule in pure solvent, is denoted by
-XskT. By choosing proper reference states (i.e., pure polymer and pure
solvent), Xg is made independent of solvent-segment interactions and

14
depends only on the nature of the surface. It is clear from the
definition that for Xs > 0 indicates segments are preferred over the
solvent by the surface. Due to adsorption, polymer loses part of its
conformational entropy and will oppose the adsorption. Hence, a certain
minimum adsorption energy, denoted by Xsc» i-s required for the adsorption
to take place.
Segment-Solvent Interaction Parameter (x)
Linear, flexible polymer molecules have a random coil shape in the
solution. For an "ideal" chain (i.e., polymer molecule represented as
chain consisting of volumeless, non-interacting statistical units), this
random coil conformation can be described by random walk in three
dimensions. For an ideal chain, it can be shown that the radius of
gyration is proportional to the molecular weight of the polymer and can
be represented by the following equation:
1/2 a M*0'5 (2.6)
where is the root mean square radius of gyration and is the
molecular weight of the polymer (Flory 53). However, real chain segments
have volume and interact with each other, and this effect is usually
represented by the excluded volume effect, which can be defined in terms
of either segment-segment interaction energy or in the framework of the
Flory-Huggins theory of polymer solutions as the segment-solvent
interaction parameter, x- The parameter x represents the quality of the
solvent. Theoretically, x < 0 for a very good solvent, X = 0 for an
athermal solvent, and x = 1/2 for an ideally poor or a 0 solvent, x
represents an exchange process (and in such a way that), xkT represents

15
the difference in energy of a solvent molecule immersed in a pure polymer
compared with one surrounded by pure solvent molecules. For x > 0, the
solvent is poor and segment-segment contacts are preferred over segment-
solvent contacts. The adsorption of polymer at the solid-liquid
interface leads to an increase in the concentration of the segments near
the surface, and the segment-segment or the segment-solvent interaction
parameter has a profound effect on the adsorption. In poor solvents,
adsorbed amounts of polymer are larger due to the fact that segments
prefer other segments over solvent molecules. The x parameter is one of
the basic parameters of the polymer solution thermodynamics, and hence,
plays an important role not only in the polymer adsorption behavior but
also in the theories of steric stabilization.
Polymer Adsorption Theories: General Framework
The aim of polymer adsorption theories is to try to relate the
conformations of the adsorbed polymer molecules to independent variables
(such as adsorption energy parameter, xSr solvent-segment interactions,
X, molecular weight of the polymer, M, its concentration in solution,
etc.). Various investigators have paid a lot of attention to the
theoretical development of these models (for e.g., see Eirich 77;
Hoeve 65,66,70,71; Silberberg 62,67,68; Scheutjeans and Fleer 79,80;
deGennes 80,82,87). Since a large number of polymer molecules are
involved, a statistical thermodynamic approach is usually employed.
Contributions of the entropy and the energy of many chains in a given
concentration gradient near the surface are evaluated. To evaluate these
contributions, the thermodynamic theory of polymer solutions, as

16
developed by Flory and Huggins, is often employed (Flory 53). More
recently, scaling concepts have been applied to the polymer adsorption
(deGennes 87).
To describe polymer adsorption behavior, according to the methods of
statistical mechanics, the partition function of the system is set up
(see, Cohen Stuart 80a).
„ „ „ -U/kT
Q = E ft e
where ft is the degeneracy, i.e., the number of different ways of
(2.7)
arranging systems having energy, U.
To evaluate ft and U, appropriate reference states are chosen
(usually unmixed pure components). To evaluate Q, all possible energy
states of the system are considered. To determine the equilibrium state
of the system, Q is maximized. For example, Q is related to the free
energy of the system by
G = -kT Ln Q (2.8)
and hence, G should be at minimal for equilibrium. To evaluate and
maximize Q, one needs to make certain assumptions, and the quality of
these assumptions leads to differences between various theories.
The earlier polymer adsorption theories (for e.g., Hoeve, Roe,
Silberberg, etc.) were formulated to describe the conformations of
isolated polymer chains. In these theories, train-loop-tail type of
polymer conformation was assumed. The chain conformation statistics,
such as average train, loop sizes, and tail, loop size distributions were
computed (but, tails were ignored). The segment-surface interaction was
taken into account using the adsorption energy parameter, Xs. (i-e.,
first layer interactions), but the solvent-segment interaction was
ignored. Since these early theories ignored solvent-segment interactions
1

17
and formulated for isolated chains (i.e., non-interacting with other
adsorbed polymer molecules), their usefulness is limited.
Later theories employed Flory-Huggins polymer solution model to
account for segment-segment and segment-solvent interactions (Hoeve,
Silberberg, Roe, Scheutjeans and Fleer). The theories of Hoeve and
Silberberg start with the train-loop model of the adsorbed polymer (i.e.,
tails were neglected) and the conformation probabilities of the adsorbed
chains were computed. To evaluate U, assumptions were made regarding the
shape of the segment density profile. Hoeve assumed the exponential
profile for the evaluation of U (i.e., segment-segment, segment-solvent
interactions) while Silberberg assumed step function for the segment
density profile (Silberberg 68). This assumption regarding segment
density profile to evaluate U has been avoided in the recently developed
theories of Roe, Scheutjeans and Fleer (Roe 74, Scheutjeans and Fleer
79,80). These theories do not assume a model for the mode of polymer
adsorption of an individual molecule. They derive the partition function
for the mixture of free and adsorbed polymer chains and solvent
molecules; a number of ways of arranging polymer chains and solvent
molecules in a given (arbitrary, but fixed) concentration gradient near
the surface was determined. Maximization of the partition function gives
the equilibrium concentration profile (Scheutjeans and Fleer 79,80). To
evaluate U, segment-segment and segment-solvent interactions were
calculated using Flory-Huggins theory. Roe neglected the tails in
evaluation of the energy term and his model predicts overall segment
density profile. Later theory (SF) gives the complete distribution of
polymer conformation near the surface and gives information about train,

18
loop, and tail distributions. The important difference between the Roe
and the SF theory is the contribution of the long dangling tails to the
overall segment density profile at relatively large distances from the
interface, and these long tails can play an important role in the
flocculation and stabilization of the system. Unfortunately, no simple
analytical expressions are available with these theories and substantial
computational time is required to calculate the conformation of the
adsorbed polymer.
The scaling relation applicable to semi-dilute polymer solutions has
been extended for the polymer adsorption problem by deGennes (see,
deGennes 87). This theory is limited to athermal solvent (good solvent
with x = 0) with moderate adsorption energies. Though this theory is
analytical, the results are in the form of power-laws without exact
coefficients.
In the next section, some of the predictions of these polymer
adsorption theories will be discussed. These theories are mostly
applicable for mono-dispersed homopolymers polymer on a homogeneous
substrate. Other complications arising due to inhomogeneous surface
structure and charge at the interface are not taken into account.
Results of the Polymer Adsorption Theories
In this section, the dependence of properties of the adsorbed
polymer layer (such as total surface coverage (i.e., the adsorbed amounts
A), direct surface coverage (i.e., occupancy in the first layer 0), bound
fraction (p), root mean square layer thickness (6), segment density
distribution) on various independent variables (such as solution

19
concentration, chain length, x and xs) will be reported. Only general
trends will be reported. For detailed comparison between various
theories, recent reviews are recommended (Fleer 87, Fleer and Lyklema
83).
Effect of Adsorption Energy Parameter (xJ
All theories predict that the adsorption energy parameter should
exceed a certain critical (non-zero) adsorption energy, i.e., xs > XSc
for the adsorption of polymer to take place. This critical adsorption
energy parameter xsc corresponds to the minimum energy necessary to
compensate the unfavorable entropy loss of the segment, upon adsorption,
compared to segment in solution. If the xs > Xsc» then the surface
coverage increases sharply with increasing xs* At high Xs values, the
surface becomes saturated and the total surface coverage 0, becomes
independent of xs* for lattice theories, the critical value of xSc is
related to the lattice type employed (Scheutjeans and Fleer 79,80).
Effect of Solvent-Segment Interaction Energy Parameter (x)
The effect of solvent on adsorption behavior is pronounced. The
adsorbed amount of polymer increases while bound fraction p decreases
with decreasing quality of the solvent (i.e., as the solvent becomes
poorer). (Though the adsorbed amount and the direct surface coverage are
increased, the ratio, the bound fraction p is decreased.) Average loop
and tail size are increased with decreasing the quality of solvent
(Scheutjeans and Fleer 79,80).
Effect of Polymer Concentration and Molecular Weight
Adsorption isotherms for high molecular weight polymers are the high
affinity type. Adsorbed amounts are high for higher molecular weight

20
polymer. For a good solvent, adsorbed amounts tend to reach a limiting
value for high molecular weights, but from a poor solvent, various
theories predict different trends. The increased amounts of adsorbed
polymer with increasing molecular weight (and concentration in solution)
are accommodated by increases in the average size of the loop and tails.
This leads to an increase in the thickness of the adsorbed layer with
increasing polymer concentration and molecular weight. For a low polymer
concentration (typically, ®p < .01, where 0p is the volume fraction of
polymer in the solution), the adsorbed amount in a 0-solvent increases
linearly with log Mw. This dependence, as predicted by SF theory, is
different from the empirical power law relations, i.e., A a M„, where A
is the plateau adsorbed amount of polymer (mg/m^) and a is an empirically
determined constant. For dilute concentration, the bound fraction
decreases with increasing molecular weight. Root mean square thickness
also increases with the molecular weight and the concentration.
The effect of tails, ignored in the earlier theories, is important
at finite polymer concentrations. (At extremely low polymer
concentrations, the adsorbed polymer lies in relatively flat
configuration). These tails will affect the average layer thickness and
the segment density profile at the outer region of the adsorbed layer,
and hence, will be very important for colloidal stability.
The segment density distributions predicted by various theories are
very important in various theories of steric stabilization. As mentioned
earlier, Hoeve assumed an exponential segment density distribution (Hoeve
65) whereas Silberberg assumed it to be step function (Silberberg 68).
Roe and SF theory can calculate segment density distributions without any

I
assumptions (Roe 74; Scheutjeans and Fleer 79,80). Both theories predict
an approximately exponential segment density distribution near the
surface. At larger distances, S.F. theory predicts a high density
compared to Roe's theory. This higher density is due to long dangling
tails. These long tails can dominate interparticle interactions and the
hydrodynamics of coated particles. The tail length increases with
increase in the molecular weight almost linearly for high molecular
weight polymer (Cohen Stuart 80a). Hence, molecular weight of the
polymer is among the most important variables for controlling rheological
and other properties of the dispersion. In the next section, various
experimental techniques used to characterize polymer adsorption will be
briefly described.
Experimental Techniques
An excellent review is available for the details of various
techniques (Cohen Stuart, et al. 86a). Adsorbed amount A, direct surface
coverage 0, bond fraction p, layer thickness 6, and segment density
distribution (Z), can be characterized experimentally.
The Adsorbed Amount of Polymer
Generally, a "solution depletion" technique is used to determine the
adsorbed amount of polymer. In this method, from the equilibrium and
initial known concentrations of polymer in solution, the adsorbed amount
of polymer is determined. Centrifugation is commonly used to separate
particles from the suspension, and the supernatant is analyzed. Various
analytical techniques are used to determine the solution such as
gravimetric, complex formation to give species which adsorb in the UV or

22
visible part of the electromagnetic spectrum (Zwick 65), etc. In certain
cases, direct determination of the adsorbed amount is possible (e.g., IR,
ellipsometry).
Trains, Bound Fraction. Direct Surface Coverage
Techniques to determine these parameters are broadly classified into
spectroscopic methods, electrochemical methods, and calorimetric methods.
(1) Spectroscopic Techniques: Spectroscopic techniques include infrared
(IR), electron spin resonance (ESR), and nuclear magnetic resonance
(NMR). Due to specific interactions between polymer segments and solid
surface, shifts in a characteristic band, either for the adsorbate (e.g.,
the carbonyl or benzene group in a polymer) or the adsorbent (e.g., the
hydroyl group on an oxide) is utilized to determine the bound fraction
using IR spectroscopy (for e.g., see Killmann 76; Takahashi et al. 80;
Fontana et al. 61,63,66; Korn et al. 80a,80b, etc.). ESR can only be
used for spin labeled polymers (Robb and Smith 74). Mobility criteria
are used to distinguish between adsorbed and non-adsorbed segments.
Segments having different mobility will give different magnetic
relaxation times and hence mobility. NMR technique is also based on the
mobility criterion to estimate p.
(2) Electrochemical Methods: Adsorbed neutral polymer affects the
electrical double layer properties (such as change in the double layer
capacitance and shift in the point of zero charge, PZC). These
properties can be utilized to determine the fraction of surface area
occupied by segments, 0 (Koopal 78).
(3) Microcalorimetric Approach: In this metliod, the heat of immersion
of adsorbent is measured at various adsorbed amounts of polymer.

23
Calibration can be achieved from the heat off immersion of a monomeric
analog compound of the adsorbed group (Korn et al. 80a,80b; Cohen Stuart
et al. 82; Killmann et al. 71; Hair 77).
There are several problems associated with these techniques, such as
distinguishing between contributions due to adsorbed and non-adsorbed
polymer, differentiation between train and loop segments, etc.
Thickness of the Adsorbed Layer
Methods to determine the thickness of the adsorbed polymer layer can
be divided into two broad categories: (1) ellipsometry and
(2) hydrodynamic methods.
(1) Ellipsometry: In this method, change in the properties of the
elliptically polarized light upon reflection, due to the adsorbed
polymer, is measured. From the measured phase shift and the amplitude of
the reflected light, under the assumption of homogeneous polymer layer,
the ellipsometric thickness and the refractive index of the film can be
calculated (e.g., see Killmann 76,77). Clearly, the assumption of
homogeneous segment density distribution leads to ambiguity in the
measured thickness. Also, this method is suitable for flat surfaces with
good reflectivity, and hence, limited mostly to bulk metal substrates and
some oxide films.
(2) Hydrodynamic Methods: These methods measure the extent of outward
shift of the slip plane due to the adsorbed polymer layer. Essentially,
they measure the drainage characteristics of the adsorbed layer, and
since the drainage characteristics of the adsorbed polymer layer (i.e.,
consisting of loops and tails) are not known, an exact definition of the
hydrodynamic thickness is not possible. Also, if the adsorbed polymer

24
layer is not homogeneous (e.g., at low adsorption densities), this method
may over estimate the thickness. To measure the hydrodynamic thickness,
several techniques have been employed.
(i) Capillarity: In this method, the inside wall of the fine capillary
is coated with the adsorbed polymer layer, and the decrease in the flow
rate due to adsorbed layer is measured (i.e., the effective decrease in
the diameter of the capillary is determined). In this method,
homogeneous coating of the capillary is critical (e.g., see Rowland and
Eirich 66; Priel and Silberberg, 78).
(ii) Viscometry: In this method, the increase in the viscosity of the
suspension of dispersed particles due to adsorbed polymer layer is
measured. Due to the adsorbed polymer layer, there is an increase in the
effective radius of the particle, hence, higher effective volume fraction
solids in the suspension. For dilute dispersions, the intrinsic
viscosity, [nl, is measured (Barsted et al. 71; Dawkins and Taylor 80)
while, for the concentrated suspensions, the high shear viscosity is
determined (Dobroszkowski and Lambourne 66). To get information about
the adsorbed layer, other parameters affecting the viscosity must be
taken into account. Factors such as electroviscous effects, aggregation
of particles, effect of shear on the thickness of the adsorbed layer,
polymer degradation, etc. can complicate the interpretation. In this
study, the adsorbed polymer thicknesses are determined from the high
shear rate viscosities. The effect of PVA molecular weight on the
adsorbed layer thickness will be reported later.
(iii) Photon Correlation Spectroscopy: In this method, the diffusion
coefficients of the particles with and without adsorbed polymer are

25
measured using "Doppler" broadening of an incident laser line (for e.g.,
see van den Boomgaard et al. 78; Kato et al. 81; Garvey et al. 76;
Killmann et al. 85,86,88, etc.). From the measured diffusion
coefficients, the particle radius, and hence, the adsorbed layer
thickness, is calculated using the Stokes-Einstein equation:
D = kT/6nn0Rh (2.9)
where R^ = hydrodynamic radius of the particle, nQ = viscosity of the
suspension medium, T = absolute temperature, and k = Boltzmann constant.
This method is limited to monosized, spherical particles.
To obtain accurate values of the adsorbed layer thickness, diffusion
coefficients are determined at various solid concentrations and the plot
of diffusion coefficient vs. solid concentration is extrapolated to zero
solids concentration. This procedure then eliminates the effects of
interparticle interactions on the diffusion coefficient. As the adsorbed
polymer layer is usually a small fraction of the particle radius, it is
essential to obtain accurate values of particle radii with and without
polymer.
(iv) Electrokinetics: In this method, the decrease in the electrokinetic
potential (Zeta potential) of the charged particles, due to adsorption of
neutral polymer, is measured to estimate the hydrodynamic thickness.
Essentially, this method measures the outward displacement of the slip
plane due to adsorbed polymer. Again, several complications are present
in interpreting this data and the measured thickness is very sensitive to
the ionic strength of the solution (Koopal 78; Cohen Stuart et al.
84a,84b,85).

26
(v) Other Methods; Sedimentation rate (i.e., change in the
sedimentation coefficient due to the adsorbed polymer layer), (Garvey et
al. 74) direct force-distance measurements (i.e., half the distance at
which a certain minimum force is observed between two approaching
surfaces, coated with the adsorbed polymer), etc., have been used to
measure the thickness of the adsorbed layer (Sonntag et al. 82; Lubetkin
88; Gotze and Sonntag 87,88).
From the above discussion, it is clear that there are several
techniques available to measure the layer thickness, but due to
uncertainty about the effect of the tail-loop conformation of the
adsorbed polymer on the properties of measured layer thickness, each
technique gives some kind of average property of the adsorbed layer.
Recently, it has been shown that the hydrodynamic thickness is
essentially determined by the tails and that the loop contribution is
negligible (Cohen Stuart 86a).
Segment Density Distribution
Small angle neutron scattering (SANS) has been used to obtain the
segment density distribution for the adsorbed polymer. In this
technique, the ability of neutron to distinguish between hydrogen and
deuterium atoms, due to different coherent scattering cross sections, is
utilized. By using a suitable mixture of H2O/D2O, particles can be
contrast matched, and by measuring scattering intensity at various
angles, information about the adsorbed polymer can be obtained.
Unfortunately, the method is not sensitive enough yet to detect small
concentrations of the tail segments at larger distances (Barnett et al.
82).

27
Adsorption Energy Parameter
As mentioned earlier, Xs represents the energy change associated
with the exchange process of replacing solvent molecule from the surface
with the polymer segment. Cohen Stuart has proposed a method to measure
this parameter (Cohen Stuart 80a). In this method, a second solvent
having strong affinity for the surface is added to the solution. With
increasing concentration of the solvent (displacer), eventually polymer
can be desorbed completely from the surface, and from this critical
displacer concentration, Xs can determined.
Adsorption of Polydisperse Polymers
Polymers used in practical application are mostly polydisperse,
and polydispersity has an important effect on the adsorption behavior.
This effect arises due to the difference in the adsorption behavior of
long and short molecules. Long molecules are preferentially adsorbed
over short molecules. As discussed earlier, the adsorption process is
controlled by the various free energy changes related to adsorption
process. The energy decrease due to adsorption of polymer segments is
opposed by the entropy loss. The decrease in entropy arises from two
contributions: (1) the loss of configurational entropy due to the
unmixing of polymer molecules and solvent molecules and (2) the decrease
in conformation entropy due to the decrease in number of different
possible arrangements of the polymer as a result of the attachment of
segments to the solid surface. In dilute solutions, which is of
practical interest in most adsorption studies, the conformational entropy
losses for one large chain compared to two shorter chains, each half the

28
length of the loner chain length, are similar. However, the decrease in
configurational entropy for the two short chains is twice that of the
longer chain. Hence, adsorption of the long chain is preferred over the
two short chains. Also, the long chain can displace the short chains
from the surface. Theory based on the above principles has been
developed, and it is successful in explaining various effects arising due
to the polydispersity of polymers (Cohen Stuart 80a,80b,84a; Koopal 81).
(1) Rounding of the adsorption isotherms: All polymer theories and
experiments using monodispersed polymers exhibit sharp adsorption
isotherms, whereas polydisperse samples lead to more rounded isotherms.
(2) Irreversibility: The observed irreversibility of adsorption
isotherms can be explained from the above theory.
(3) The effect of amount of adsorbent on the amount adsorbed and
the shape of the adsorption isotherm: Experimentally, it has been found
that the adsorption isotherms are sharper and the plateau adsorbed
amounts larger if the adsorption isotherms were determined using dilute
dispersions (i.e., small surface area/volume of the solution ratio)
(Koopal 81).
Preferential adsorption of the high molecular weight polymer over
the low molecular weight leads the fractionation of the polymer. The
molecular weight distribution in the adsorbed layer is shifted to higher
molecular weight as compared to that in the solution (Furusawa et al.
82). Also, from the practical processing point of view, the
preferentially adsorbed high molecular weight fraction may dominate the
properties of the dispersions, such as the rheology and the consolidation
behavior.

29
Additional factors, such as the charge on the solid surface and the
effect of the solid surface structure, will be discussed in the
experimental section.
Experimental Results for PVA-Water System
Properties of Poly (Vinyl Alcohol). PVA
PVA is a commonly used polymer as a steric stabilizer in aqueous
media. PVA is prepared by alcoholysis of poly (vinyl acetate), PVAc, and
is generally not hydrolysed fully. (Often the degree of hydrolyses is on
the order of 88 mole percent or greater). It is, therefore, a copolymer
of PVAc and PVA. This can be represented as follows:
CH2 = CHOAc Bolyorisation > -CH2-CH-CH2-CH-CH2-
vinyl acetate 0|c OAc
poly (vinyl acetate)
> -CH2-CH-CH2-CH-CH2-CH-
OH OAc OAc
poly (vinyl alcohol) containing acetate groups,
where symbol OAc represents acetate groups, -COOH. Copolymers have been
shown to be the best steric stabilizer for the following reasons:
Usually, they consist of two types of segments having contrasting
solubilities in a given aqueous or non-polar dispersion media. Less
soluble segments of the copolymer will preferentially attach themselves
to the solid surface, and thus, the polymer is firmly anchored to the
surface. Thus, such a copolymer then consists of two types of segments,
anchoring moieties and stabilizing moieties. Strong attachment of the
polymer with the surface will prevent desorption of polymer or lateral
movement of the polymer on the surface when two particles coated with

30
with adsorbed polymer will be encountered during Brownian collisions. In
good solvents, the stabilizing moieties will offer repulsion, due to
overlap of segments (i.e., due to osmostic effects).
This copolymeric nature of PVA makes direct comparison between
experimental results and theoretical predictions difficult and only
qualitative trends will be discussed. Also, the previously discussed
theories were mainly developed for the monosized, homopolymers, while
commercial polymers are usually polydispersed. In this section,
available literature on the PVA adsorption behavior on various substances
will be reviewed.
Two types of PVA's, i.e., partially hydrolysed, PVA88,
(approximately 88 percent hydrolysed) and completely hydrolysed, PVA98,
(greater than 98 percent hydrolysed) have been employed in various
experimental investigations. Boomgaard et al. fractionated as-received
polymer by sequential addition of acetone (i.e., a non-solvent) to
approximately five wt.% PVA solution (van den Boomgaard et al. 78).
Garvey et al. used preparative scale Gel Permeation Chromatography (GPC)
to obtain narrow molecular weight fractions (Garvey et al. 74). These
investigators did not measure the polymer molecular weight distribution
or the polydispersity index My^/M^ where Mn is number average molecular
weight and My^ is the weight average molecular weight. Acetone fractions
gave fractions of varying molecular weights, but the degree of hydrolysis
was also different for different fractions. (The degree of the
hydrolysis decreased from the approximately ninety to approximately
eighty mole percent with the decrease in the molecular weight) (van den
Boomgaard et al. 78). Other studies used as-received commercial polymer.

31
PVA Characterization
From IR and UV spectra, Koopal concluded that the commercial samples
used in his study were atactic and contained no or very little impurities
(impurities, such as 1,2 glycol units and one or two conjugated groups if
present are present as the end groups) (Koopal 78). The acetate group
distribution is "blocky" for the PVA88, and acetate groups were
distributed more or less "randomly" for the PVA98. Dunn (Dunn 80) and
Barnett et al. (Barnett et al. 82) assumed that for the PVA88, the
average block consists of three acetate groups. They assumed that the
average acetate block size increases and the width of the acetate block
size distribution increases with increasing PVA molecular weight.
The Mark-Houwink-Sakurada (MHS) equation was generally used to
determine the viscometric average molecular weight, Mv of the polymer.
[n] = kMva (MHS) (2.10)
where Cn] is the intrinsic viscosity and k and a are KKS empirical
constants. The values of the constants k and a used by various
investigators are different. This can lead to different values of Mv for
the same polymer (i.e., same [r|]). The values of k and a are dependent
on the temperature, the acetate content, and the polydispersity of the
sample. The value of constant a was in the range of 0.64 to 0.60 for
PVA98 and was in the range of 0.71 to 0.63 for PVA88 (e.g., see Koopal
78). The PVA solution properties (i.e., the segment-solvent interaction
parameter x) and polymer molecule dimensions in solution (i.e., radius of
gyration, end to end distance, etc.) were determined from the measured
intrinsic viscosity and molecular weight for a series of samples with
varying molecular weights. If these two quantities are not independently

32
available, then the MHS equation (Equaiton 2.9) was used to determine Mv
from the measured intrinsic viscosity. Hence, the values of solution
properties and polymer molecule dimensions were influenced by the values
of constants k and a used. For this reason, these values (i.e., x, ,
etc.) should be compared with caution since different values of constants
k and a are employed by various investigators.
(1) The Seqement-Solvent Interaction Parameter: x
Detailed comparison of the available values of x parameter has been
made by Koopal (Xoopal 78). At 25°C in aqueous solutions, the x values
were in the range of 0.462 to 0.488 for PVA88 and in the range of 0.475
to 0.499 for PVA98. Thus, the degree of hydrolysis does not have
significant effect on the solvent quality for typical PVA polymers. It
should be noted that the aforementioned values of the x parameter
indicate that water is a relatively poor solvent for PVA. This also
suggests that intersegmental interactions occur in the polymer chain
(Koopal 78). van den Boomgaard et al. have determined the effect of
temperature on the x parameter (van den Boomgaard et al. 78). With
increase in the temperature from 25°C to 50°C, the x value changed from
0.464 to 0.485, indicating worsening of the solvency for PVA. Tadros and
Vincent and Barker and Garvey have determined the effect of type and
electrolyte concentration on the solvency (Tadros and Vincent 79; Barker
and Garvey 80). With increase in the electrolyte concentration, the
solvency decreases. They also found that Na2SC>4 has greater effect on
solvency compared to NaCl, i.e., lower concentration can change solvent
quality.

33
(2) The Xc Parameter:
One needs to measure the adsorption energy parameter for both
acetate and alcohol groups since the PVA adsorption mechanism may involve
adsorption of these two groups. Heat of adsorption of low molecular
weight analogues may be useful with this respect, but xs values are not
available for the various systems investigated. Typically, values in the
range 1 - 2 kT have been assumed for xs (Barnett et al. 82).
The Adsorbed Amount of Polymer
The Nature of Solid
PVA adsorption behavior has been studied on Agl solid particles (Fleer
71), Agl particles and sol (Koopal 78), silicas of various types (e.g.,
precipitated, Cab-o-sil, Ludox, Tadros 78), polystyrene latex particles
(made by emulsion and dispersion stabilization, Garvey, et al. 74,76),
montmorillenite clays (Greenland 62, Heath and Tadros 83). Due to
differences in the chemical nature and surface heterogeneities (e.g.,
silica powder calcined at various temperature leads to various types of
surface groups and the concentration of each group is dependent on the
thermal history; polystyrene latex particles made by dispersion
polymerization technique have more hydrophobic surface, etc.), it is not
possible to compare adsorption data on the same basis. Also, other
variables, such as solid concentration, aging time, polydispersity of the
polymer samples, and the method of sample preparation, etc., can have an
important effect. The detailed comparison with the literature results
will be made in the results and discussion section. Here, we will
briefly describe the important results.

34
The Effect of Acetate Content
The plateau adsorbed amounts (i.e., "saturation adsorbed amount" or
adsorbed amounts at "complete" surface coverage) are generally more for
PVA88 (i.e., partially hydrolysed PVA) than PVA98 (i.e., fully hydrolysed
PVA) of similar molecular weights. This effect have been found on
various substrates (e.g., Agl, Koopal 78; silica, Tadros 78; PS latex,
Barnett, et al. 82). The larger adsorbed amount for PVA88 can be due to
(i) increase in the adsorbed amount in the first layer (i.e., A ) or
mono
due to (ii) the formation of larger loops and tails. The adsorbed amount
differences in the monolayer between these two polymers cannot account
for this difference, hence, the contribution of the first layer to this
difference is small (Koopal 78). The increase in adsorption with
increasing acetate content has been attributed to greater adsorption in
loops and tails. As explained earlier, x (PVA98) and x (PVA88) are only
slightly different, and the difference in adsorption behavior cannot be
explained by the solvency effect. The preferential adsorption of acetate
groups onto Agl particles has been determined by Koopal from the
electrochemical method (i.e., from the shift in the point of zero change)
(Koopal 78). The preferential adsorption of acetate groups leads to
accumulation of acetate groups in the first layer and gives an important
contribution to the gain in the free energy of adsorption (i.e., Xsl> the
adsorption energy parameter for acetate groups is expected to be larger
than the adsorption energy parameter for alcohol groups, xs2)• The
differences in flexibility of these two polymers (i.e., it can be assumed
that the flexibility of PVA88 is relatively lower due to bulky acetate
groups) will have effect on the size of the trains and loops. The train

35
size is limited by the length of the acetate blocks and lower flexibility
of PVA88 can set constraints on the minimum loop size (i.e., the average
loop size is expected to be larger for PVA 88). The preferential
adsorption of acetate groups has been confirmed by NMR studies on PS
latex particles (Barnett, et al. 82).
The Effect of Molecular Weight of PVA
As expected from the theoretical results, increased plateau adsorption
with increasing molecular weight of PVA has been observed. Generally,
this effect is represented by the following power law relation:
A = KMy,a (2.11)
where A is the plateau adsorbed amount of polymer and K and a are
empirical constants. (Please note that the Equation 2.11 is empirical in
nature. Modem adsorption theories (for e.g., SF, Roe) predict A a log
My, which is an entirely different type of functional relation). For the
adsorption of PVA98 on the PS latex (made by dispersion polymerization),
a = 0.5 has been reported (Garvey et al. 74). Weak molecular weight
dependence has been observed for adsorption on Agl particles, a = 0.1 for
PVA98 and a = 0.2 for PVA88 (Koopal 78). The hydrodynamic thickness of
the adsorbed polymer layer increases with the molecular weight (Garvey et
al 74; Killmann et al 88). The hydrodynamic thickness has been measured
using various techniques (Electrophoresis, Viscometry, PCS, ultra
centrifugation, slow speed centrifugation, direct force measurements,
etc.). Again, the relation between the hydrodynamic thickness and
molecular weight is represented by a power law. Generally, the measured
hydrodynamic thicknesses were comparable to the random coil dimensions in
solution.

36
The Effect of Solvency
Increases in temperature (van den Boomgaard et al. 78) and additions of
electrolyte (Tadros and Vincent 79; Barker and Garvey 80) increased the
plateau adsorbed amounts. This has been related to the worsening of
solvent quality with temperature and electrolyte. Decreases in the
measured hydrodynamic thickness with increasing temperature and
electrolyte concentration were observed.
Adsorbed Layer Properties and Adsorbed Amounts
It has been suggested from theoretical results (e.g., see Fleer 87)
that, although adsorbed layer properties such as 0, p, adsorbed layer
thickness, etc. are dependent on My, and solution concentration, it is
still possible to express the adsorbed layer properties as a function of
the adsorbed amount only (i.e., the properties of the adsorbed layer are
the same for the high molecular weight polymer at low concentrations and
for the low molecular weight polymer at high concentrations provided the
adsorbed amount is the same). Based on the above hypothesis, Koopal
plotted adsorbed layer properties, such as the first layer occupancy 0,
and effective layer thickness 6 (measured experimentally), as a function
of adsorbed amount (Koopal 78). His results are shown in Figure 2.3. As
expected from the theoretical and other experimental results, the root
mean square thickness increased with increasing adsorbed amounts. The
results for PVA98 and PVA88 are plotted on the same graph since no
significant difference was found in x (PVA98) and x (PVA88) values. From
the above results, he concluded that the acetate content and molecular
weight influenced the adsorbed layer properties through adsorbed amounts
only.

EFFECTIVE LAYER THICKNESS (5) (nm) DEGREE OF OCCUPANCY
37
Figure 2.3 Plots of (a) degree of occupancy, 9, and the effective
layer thickness, 6, as a function of the adsorbed amount
of PVA on Agl and (b) the fraction of segments adsorbed
in trains as a function of the total adsorbed amount
(Koopal, 1978).

38
The Segment Density Distribution
The segment density distributions have been determined (Figure 2.4)
using SANS for adsorbed PVA on PS latex particles (Barnett et al. 82).
Typically, an exponential segment density distribution was observed near
the surface. (This would be expected by the polymer adsorption theory of
Hesselink in which the conformation consists of loops and trains, but no
tails—Hesselink 71a.) However, higher segment density at the
intermediate distances were related to longer "slightly folded" tails
(Barnett et al. 82). This type of segment density distribution was
different from the homopolymer PEO adsorbed on PS latex, where more or
less exponential decay in the segment density was observed. In the study
of PVA adsorbed on PS, the root mean square thickness calculated from the
segment density distribution was smaller than results obtained by PCS.
It was concluded that the SANS results were not sensitive enough to
detect tails (which are present in low concentration) and the tails are
responsible for the higher measured hydrodynamic thickness determined by
PCS (Barnett et al. 82).
The Effect of Particle Radius
The effect of particle radius on adsorption behavior and the
hydrodynamic thickness has been studied by Garvey et al. (Garvey et al.
76) and Ahmed et al. (Ahmed et al. 84). The former investigators related
the increasing hydrodynamic thickness with decreasing particle radius to
a geometric factor (Garvey et al. 76). Other groups correlated this
observation to change in the conformation of the adsorbed polymer due to
the change in the particle radius (Ahmed et al. 84).

5
£
a
I/)
£
tí
Q
Q
W
SI
HH
s
o
£
3
2
I
LO
VO
0
12
18
24
DISTANCE FROM SURFACE (nm)
Figure 2.4 Plot of segment density as a function of distance from a surface for adsorbed
PVA on PS-latex (Barnett et al., 1982).

40
PVA Adsorption on Silica
Tadros has investigated the PVA adsorption behavior on various types
of silicas (Tadros 78). Here, we will state the results from his
investigation. A detailed comparison of his results and the results in
this study are made in Chapter VIII.
(1) The Effect of Silica Calcination Treatment:
It was observed that the plateau adsorbed amount is a strong
function of silica surface characteristics. The maximum in the plateau
adsorbed amounts was observed for approximately 700°C calcined silica.
This observation was correlated with the optimum density of isolated
silanol groups on 700°C calcined silica.
(2) The Effect of Surface Charge:
The maximum adsorption occurs at the point of zero change, p.z.c.,
of the oxide and progressive decrease in the adsorption was observed
above p.z.c (Tadros 78). (This effect was not observed for Agl particles
and sol—Koopal 78, Fleer 71).
Summary
From the above discussion, following general trends have been
established regarding adsorption behavior of PVA.
- Water is a relatively poor solvent for PVA (x is close to 0.5).
- Partially hydrolysed PVA is blocky, while for fully hydrolysed PVA,
acetate groups are randomly distributed.
- Partially hydrolysed polymer adsorbs more than fully hydrolysed PVA.
- Adsorption density increases with increase in the molecular weight.

41
- NMR and electrochemical methods suggest that the acetate segments are
preferentially adsorbed.
- Adsorbed layer properties such as bound fraction (p) and effective
thickness are functions of amount adsorbed only for PVA adsorption on
Agl.
- Segment density distribution is exponential near the surface and
relatively higher density (compared to an exponential distrituion)
observed at the intermediate distances is related to the presence of
slightly folded tails for PVA adsorption on PS.
Hydrodynamic layer thickness is substantial (five to fifty nm) and, it
is primarily the tails which are responsible for these large measured
thicknesses.
- Calcination temperature and pH are among the most important variables
controlling PVA adsorption behavior onto silica.
In the next Chapter, we will review the electrostatic interactions
between colloidal particles.

CHAPTER III
ELECTROSTATIC INTERACTIONS BETWEEN COLLOIDAL PARTICLES
Introduction
Properties of the colloidal dispersion are directly influenced by
the interparticle interactions. A colloidal dispersion is a two-phase
mixture consisting of dispersed particles (solid) in a continuous
dispersion medium (liquid). Particles are said to be colloidal in
character if at least one of its dimensions is in the size range 1 nm to
4
10 nm (1 um). In this size range, specific surface area is large
(usually few m^/gram up to 1000 m^/gram), and hence, the interparticle
interactions are dominated by the solid-liquid interface characteristics.
Also, in this particle size range, the gravitational force is not
important, and particles are moving randomly in the dispersion media due
to thermal energy, i.e., Brownian motion. Particle encounters due to
Brownian motion either leads to either formation of doublets (or higher
order multiplates) or particles remain as individual units depending on
the interparticle interactions. In the absence of any repulsive
interactions, these random collisions lead to permanent contacts between
particles and this reduces the free energy of the system. (The free
energy is lowest when the particles are all clumped together). The
origin of attractive interactions between particles is in the Van der
Waal's attraction between the atoms of the colloidal particles. The
characteristics of the aggregates formed also depend on the interparticle
42

43
forces. To prevent such aggregation of particles during collisions,
there are two mechanisms available to overcome attraction.
(1) Electrostatic Interactions: If the colloidal particles can be given
an electric charge (either positive or negative) and if all particles
have the same sign of charge, particles will repel one another during
approach.
(2) Interactions of Adsorbed Polymer: Under certain conditions (i.e.,
depending on the coverage of particle surfaces with adsorbed polymer,
thickness of the coating, solvency for polymer, etc.), adsorbed polymer
layer can prevent close approach of the particles.
These two repulsive interactions impart stability to the colloidal
dispersion. The dispersion is said to be stable if the dispersed phase
(colloidal particles) remains essentially as distinct single particles on
a long time scale (e.g., days, months, years). Such a dispersion may be
stable either due to kinetic (e.g., in the case of electrostatic
interactions) or thermodynamic reasons (e.g., stabilization with adsorbed
polymer). It is clear from the above definition that the stability
criteria is essentially based on the state of particulates in the
dispersion. The time scale is employed as a reference because, in the
absence of repulsive interactions, the number of particles (kinetic
units) in moderately concentrated suspensions can be reduced to half in a
matter of seconds due to encounters arising from Brownian motion (von
Smoluchowski 16a,16b,17).
The stability criterion based on the time scale (for e.g., time it
takes to reduce the particle concentration to half, etc.) may not be
useful to evaluate the stability of ceramic dispersions. The reasons are

44
two fold: (i) typically particles in the size range of 0.05 urn to 5-10 pm
are often employed in various ceramic processing operations, and hence,
suspensions are not strictly colloidal in nature. Also, the density of
particles is usually greater than the density of suspending media leading
to sedimentation of particles although particles are well dispersed.
(ii) the particle concentration is also quite high, and usually, it is
not possible to determine the change in particle concentration. Other
techniques, such as rheological behavior, sedimentation behavior,
properties of the sediment (porosity, average pore size, etc.) can be
used to evaluate the stability of these dispersions.
/ -6
Although interatomic attractive interactions are short range (a r
where r is the distance between the atoms), their summation over
colloidal particle sizes leads to long range attraction. To overcome
this attraction, the repulsion must also be long range.
In this chapter, we will review two components of interactions
(1) Van der Waal's attraction between colloidal particles, and
(2) electrostatic interactions.
Summation of these two components, under the assumption of
additivity, leads to well known theory developed by Deryagin and London
and independently by Verwey and Overbeek (commonly known as "DLVO"
theory) to explain the stability behavior of the electrostatically
stabilized dispersions (e.g., see Verwey and Overbeek 48). Excellent
monographs are available to discuss various aspects of this theory, hence
only the basic principles will be outlined here (e.g., see, Hiemenz 77,
Hunter 87, Overbeek 82a,82b, Lyklema 68, etc.).

45
Development of Charge at Solid-liquid Interface
There are basically four different methods by which the charge can
be developed at the solid/liquid interface (Hunter 87).
Dissociation of Surface Groups
This method is the charge determination mechanism for several oxides
(e.g., alumina, silica, etc.). The surface of these oxides is
hydroxylated to various extents. (For example, for precipitated silica
used in this investigation, the surface is almost completely
hydroxylated, i.e., the surface is nearly fully covered with silanol
groups, Si-OH). Dissociation of surface silanol groups leads to surface
charge development, as described by the following reactions:
-SiOH, _ , + H+, . ... > SiOH+2, , *
(surfdcs) (liquid) (surfdC6) |^ ^ j
(surface) (liquid) (surface) 2 (liquid)
From the above reactions, it is clear that the silica surface can develop
a positive or a negative surface charge.
A zero point of charge, (p.z.c.), is defined as a pH at which the
surface charge is zero. Another important characteristic of the oxide
material is its isoelectric point, i.e.p., which is defined as the pH at
which the electrophoretic mobility is zero. H+ and OH ions are called
the potential determining ions. In the absence of specific adsorption of
ions, the p.z.c. and the i.e.p. are the same. Below the i.e.p., the
silica surface is positively charged, and above the i.e.p., negative
charge is developed and its magnitude can be increased by increasing pH
of the solution.

46
Adsorption of Potential Determining Ions
A familiar example is silver iodide particle/water suspensions in
which the particles can preferentially adsorb an Ag+ or I- ions,
rendering them positive or negative charge, respectively.
Adsorption of Ionized Surfactants
In this case, the charge is produced by the preferential adsorption
of the ionic surfactants on the surface, for example, the preferential
adsorption of C-|2H25S04- ions from sodium dodecyl sulfate, c-| 2H25S°4_Na+
surfactant.
Isomorphic Substitution
This charge development mechanism is important in the case of clay
minerals (e.g., sodium montmorillenite). Inside the solid lattice, lower
valent ions may replace higher valent ions (e.g., Al+^ replaced a Si+4
ion in the "tetrahedral silica layer," resulting in a deficit positive
charge on the particle surface.
For the present investigation, dissociation of surface silanol
groups on the silica particles is the important mechanism of charge
development. The silica used in this study has an i.e.p. near pH = 3.7.
Hence, at high pH's, the silica surface develops negative surface charge,
and at pH near 3.7, no net charge is present on the silica particles.
(Hence, there is no net electrostatic repulsion between two approaching
particles at pH = 3.7.)
Electrical Double Layer
The development of surface charge is not yet a sufficient condition
for stability because electroneutrality requires that the particle and

47
its immediate surroundings should have no net charge. In other words,
the surface charge must be balanced by an equal but opposite counter
charge in the solution. The rigid alignment of counter ions in the
solution is implausible because of thermal agitation, which causes the
counter ions to diffuse throughout the solution. To understand the
stability, it is of crucial importance to understand the distribution of
counter ions in the solution.
The stability of the charged particles can be understood
qualitatively as follows: If the counter charge is very diffusely
distributed and extends far from the particle surface, then when two
particles having the same sign of charge (and hence, same sign of the
counter ions in the diffuse layer) start approaching each other due to
Brownian motion (or due to an applied shear field), the diffuse layer
starts to overlap (even though the particles are far apart), thus giving
rise to an electrostatic repulsive force. On the other hand, when the
double layer is compressed (i.e., the counter ions are crowded close to
the particle surface), particles can approach closer before they feel the
electrostatic repulsion, and at that distance, the strong Van der Waal's
attraction leads to flocculation of the particles.
To explain the stability behavior quantitatively, a description of
the potential (or charge) distribution around the colloidal particles is
required. To describe the variation of the potential with the distance
from the charged surface, the Poisson equation is used as shown below:
V2 W = —=2— (3.2)
o r
where is the potential, is the Laplace operator, Gr is the relative
permittivity, eQ is the permittivity of the free space, and p is the

48
local charge density (i.e., number of charges per unit volume). To solve
this equation, one needs to know the charge density as a function of
potential.
The work required to bring an ion to a position where the potential
V is given by Z¿e¥. The probability of finding an ion at that position
is given by the Boltzmann factor:
n. -Z.eW
it- ■ «P 1 KT 1 <3-3’
io
where T is the temperature, k is the Boltzmann constant, n¿ is the number
of ions of type i per unit volume, and n¿0 is the concentration far from
the surface (i.e., the bulk concentration). The valance number Z¿ is
either a positive or negative integer and e is the charge on the
electron. The charge density is related to the ion concentrations, as
follows:
Z.eW
p = E n. eZ. = E Z. en. exp (———) (3.4)
i i .1 io ^ kT
i i
Substituting for the charge density, one obtains the Poisson-Boltzmann
equation, as follows:
—Z eW
V2 ? = —4— E n. Z.e exp (—^—) (3.5)
G 6 io i kT
or i
The following assumptions were made in solving the above equation:
(1) The surface charge on the particle and the space charge in the
solution are considered as smeared out. (2) The ions are considered as a
point charges, their distribution in the solution being determined by
their valancy and not by their volume, shape or polarizability. This
assumption makes the theory non-specific (e.g., the difference between

49
Li+ and Na+ ions cannot be distinguished). (3) The solvent is considered
as homogeneous and continuous, and the solvent affects the charge
distribution through its dielectric constant Gr.
It is clear from the above equation that the potential distribution
depends in a complex way on the ionic composition of the solution. This
equation does not have an explicit general solution and has been solved
for certain limiting cases (e.g., low surface potentials) and for simple
geometries (i.e., flat plate, spherical particle, etc.).
At room temperature, the exponent Z^eV/kT = Z^V/25.4 if ¥ is
expressed in millivolts.
Debye-Huckel solved the above equation for flat plate geometry and
for low surface potentials, (i.e., Z¿¥ < 25.4 mV). Under these
conditions, they showed that the potential decays exponentially (Hunter
87) :
V = ¥ exp (-hx) (3.6)
o
where x is the distance from the interface and the h, the Debye-Huckel
parameter, is defined as follows:
H
2
e En. Z.
IQ 1
€ € kT
2 1/2
)
(3.7)
r o
h has the units of reciprocal length, i.e., h-1 has the units of length.
The exponential rate of the potential decay is controlled by h (Equation
3.6). If the double layer thickness is defined as the distance over
which the potential drops to (1/e) of its value at the surface, then k-1
becomes the measure of the "double layer thickness." At 25°C in water,
the value of k is given by:
h = 3.288 ->J I (run )
(3.8)

50
where I is the ionic strength (= 1/2 E where is the ionic
concentration in mole/liter). Table 3.1 shows calculated values of
K-1 (i.e., the double layer thickness) for several different electrolyte
concentrations and valences for aqueous solutions at 25°C.
From the table, it is clear that the valency of the counter ions and
the electrolyte concentration are important parameters to the control
double layer thickness, and hence, the electrostatic interactions between
colloidal particles.
Figure 3.1 shows the variation of the potential with the distance
from the surface. The drop is dramatic for the higher electrolyte
concentrations or valances.
The Poisson-Boltzmann equation has been solved for flat plates
without the Debye Huckel approximation (i.e., < 25 mV) by Gouy-
Chapman. The results are applicable only to symmetrical electrolyte,
i.e., Z+ = Z-. According to the Gouy-Chapman, the variation of potential
within the double layer can be described by the following equations
(Hunter 87):
Y = Y0 exp <-hx) (3.9)
where y is defined by the relationship
_ exp (Zeg/2kT) - 1 (3 1Q)
Y exp (ZeW/2kT) + 1
and Yo is calculated from the above equation when ¥ = Vq. From Equations
3.9 and 3.10, it is clear that it is the complex ratio y that varies
exponentially with x in the Gouy-Chapman theory. For the low potentials,
as expected, the above equation reduces to the Debye-Huckel approximation
(i.e., Equation 3.6).

51
TABLE 3.1
Effect of Ionic Strength and Valency on the
Electrical Double Layer Thickness, h~1
Ionic Strength
(moles/liter)
Symmetrical
Electrolyte
Z+ : Z_
H-1
(run)
X
o
1
1:1
30.41
2:2
7.60
3:3
3.36
1 x 10-3
1 :1
9.61
1 x icr2
1:1
3.04
1 x 10~1
1 :1
0.96

52
Figure 3.1
Fraction of double
layer potential
versus distance
from a surface:
(a) curves for 1:1
electrolyte at
three concentra¬
tions and (b)
curves for 0.001 M
symmetrical
electrolytes of
three different
valance types.
Figure 3.2
Schematic
illustration of
the variation of
potential as a
function of
distance from a
charged surface in
the presence of a
stem layer,
subscripts o at
wall, 6 at stem
surface, d in
diffuse layer.

53
To describe the potential variation as a function of distance from
the interface for spherical particles, the Poisson-Boltzmann equation
should be solved in spherical coordinates. For the case of low surface
potentials, analytical solution is available and given by the following
equation:
» (r) . » -2- e-Kla-cl
(3.11)
o r
where *P (r) is the potential at a distance r from the particle center and
a is the particle radius.
In the theoretical development of the above equations, Debye-Huckel
and Gouy-Chapman treated ions as point charges, i.e., the effect of ion
size was ignored. To account for the finite volume of the ions, Stern
divided the aqueous part of the double layer by a hypothetical boundary
known as the Stern surface (Hunter 87). The Stern surface is situated at
a distance 6 from the actual surface as shown in Figure 3.2.
The Stem theory is difficult to apply quantitatively because it
introduces several parameters into the picture of double layer which
cannot be evaluated experimentally. There are several other models
available to describe the charge and the potential variation at the
charged interface, but they suffer the same problem as the Stern's model
(Hunter 87).
It is important to note that the existence of the Stern layer does
not invalidate the expressions for the diffuse part of the double layer,
but one needs to use a potential at the Stern layer, ¥5. This Stern
layer potential is usually equated with the zeta potential, which is the
potential measured using an electrokinetic method. The exact location of

54
at which the zeta potential is determined is not known, but it is assumed
to be close to the Stem layer.
In summary, in this section, we looked at the various charge
development mechanisms at the solid-liquid interface. The charge on the
solid surface leads to the distribution of the counter ions in the
solution. This model of the charged interface is often called the
electrical double layer model. The potential variation as a function of
distance from the interface can be obtained by solving the Poisson-
Boltzmann equation. Under the assumptions of low potentials (Debye-
Huckel approximation) and simple geometries, analytical solutions can be
obtained. The parameter of great importance is k (the Debye-Huckel
parameter), which can be used to describe the effect of the concentration
and valence of the counter ions on the potential distribution near the
charged surface. To describe the stability behavior of the
electrostatically stabilized dispersions, calculations are often made of
the interaction energy as a function of distance of separation between
particles. In the next section, expressions for the interaction energy
due to overlap of the electrical double layers will be developed.
Double Layer Interactions
When two particles approach each other, overlap of the double layer
occurs. The rate of approach of the two particles compared with the
relaxation time of the diffuse double layer to adjust to the new
situation is an important parameter. But, generally, two cases can be
distinguished. They are denoted as the "constant potential" and the
"constant charge" interactions.

55
In the first case, it is assumed that during the encounter of two
colloidal particles, the surface potential W0 remains constant. Under
these conditions, analysis shows that, to keep the surface potential
constant, the surface charge density, oQ, should decrease. In the second
case, it is assumed that during the encounter the surface charge density
remains constant, and in this case, the overlap of the diffuse double
layer leads to an increase in ¥0. The condition in which both >P0 and oQ
are not constant is also possible and has been called a charge regulation
(Hunter 87).
The repulsive interactions due to the overlap of the double layer
can be analyzed using two approaches (Lyklema 68).
(a) the free energy change involved when the overlap occurs, or
(b) the increase in the osmotic pressure due to accumulation of the ions
between the particles.
Following the free energy change approach, there is an increase in
the free energy of the double layer upon interaction, hence, work must be
performed to bring the particles closer. In other words, the overlap of
the double layer leads to repulsion between particles.
The repulsion energy VR(d) represents the work necessary to bring
the particle surfaces from infinity to a distance d. To calculate VR(d),
the free energy of the system as a function of the distance of separation
should be known. It is clear that VR can be represented by the following
equation:
VR(d) = 2 [G(d) - G(<*>)] (3.12)
where G(d) represents the free energy at the distance d and G(<*>)
represents the free energy at the infinite separation, i.e., for the

56
isolated double layer. The factor 2 results from the fact that two
double layers are involved. On the basis of the above scheme, DLVO
theory formulates the repulsive interaction energy. The exact solution
is available only for simple shapes (e.g., flat plate) and under certain
approximations (e.g., low potentials). For two flat plates, VR as a
function of distance d has been tabulated by Overbeek for any given value
of the surface potential ?0 and concentration of ions in solution
(Overbeek. 52).
The following approximate analytical equations are often used to
represent VR.
Interaction Between the Two Flat Plates
Under the assumption of the linear superposition1 principle, and
under the condition of the constant surface potential during the double
layer overlap, the potential energy is given by the following equation
64n kTy2
V (d) = exp (-Kd) (3.13)
R H
where VR is the repulsive interaction energy per unit area (J/m2) and nQ
is the ion concentration (total number of ions/m2), and the other symbols
have been previously defined.
The above expression is also valid for the constant charge case.
Comparison of the above expression with the exact results (Overbeek 52)
show that over a considerable range of overlap, the above equation is a
1 The assumption is that the potential between two interacting
particles is equal to the sum of potentials of individual double layers
at the same distances from the surface. This is valid in the case of low
surface potentials, i.e., for d > 1/k, the diffuse layer thickness, k
-1

57
good approximation, though the approximation tends to overestimate the
value of VR (Hunter 87).
A more elaborate expression, valid for higher surface potentials
under the conditions of the constant charge, is also available (Gregory
73).
Interaction Between Two Spherical Particles
This case is more important in the case of colloidal dispersions.
(1) For large values of ua: Large values of na means that the particle
radius, a, is relatively large compared to the thickness of the diffuse
layer, k-^. Under the conditions of low potentials and thin diffuse
double layer, the repulsion energy can be calculated by Deryaguin
procedure (see Hunter 87). Following this procedure, it is possible to
calculate the interaction energy between two spherical particles if the
interaction energy as a function of distance of separation is available
for the flat plate case under similar conditions. For the case of two
identical spherical particles, the energy of interaction can be
calculated from the following equation:
OO
VR(d) (sphere) = na J VR(d) (flat plate) dD (3.14)
d
where a is the radius of the particle.
Substituting for Vr (flat plate), the approximation valid for low
potentials is given by:
V_,(d) (sphere) = 2n€ e^aV In ( 1 + exp (-hH) ) (3.15)
R roo
Note that Equation 3.15 gives the total repulsive energy between two
spherical particles (in Joules) whereas flat plate expression (Equation

58
3.13) gives the energy per unit surface area (in Joules/m2). For the
case of somewhat higher surface potential, the interaction energy is
given by the following equation:
2
64nan kTy
V (d) (sphere) = —— exp (-nd) (3.16)
K Z
H
For the spherical particles at high potentials, no analytical formula is
available, but a graphical solution is available (Overbeek 52).
(2) For the case of low na, i.e., when the thickness of the diffuse
layer, h_1 , is large compared to particle radius, a, the interaction
energy is given by the following equation:
4ixG € aW 2
VR 2a+d ° 3 ' SXP (_Kd) (3-17>
where 3 is a factor which allows for the loss of spherical symmetry in
the double layer and has been defined by Verwey and Overbeek (Overbeek
52).
From the above equations, it is clear that the repulsion is
determined by the ionic strength (through Debye-Huckel parameter k) and
the surface potential, of the particle (i.e., radius a) is also important.
In the above equations, the finite size of counter ions (i.e.,
presence of the Stem layer) was ignored, and hence, these expressions
are not valid for the distances comparable to atomic dimensions.
Additional interactions due to solvation and hydration of ions and
hydrophobic interactions have been reported (Israelachvili and Pashley
82,83; Pashley and Quirk 84; Pashley et al. 82). Theses short range

59
interactions have been related to repeptization phenomenon (Overbeek
82b).
As mentioned earlier, the other term in DLVO calculations is the
attraction energy term arising due to the Van der Waal's interaction
between the colloidal particles leading to the flocculation of particles.
This attraction energy term will be discussed in the next section.
Van der Waal's Interactions
The Van der Waal1s interactions between neutral molecules may
originate from three possible sources: permanent dipole-permanent dipole
(Keesom), permanent dipole-induced dipole (Debye), or induced dipole-
induced dipole (London) interactions. The distance dependence of these
interactions can be represented as a power law, i.e., potential energy of
interaction a r-x where x = 6 and all interactions lead to attraction
between molecules. In the case of non-polar molecules as the dipole
moment is zero, Debye and Keesom interactions are absent. On the other
hand, London interactions, also known as dispersion forces, are always
present. The London dispersion force is attributed to correlated
electronic motion in the atoms under consideration. This correlated
motion of electrons leads to a decrease in the potential energy of the
system (i.e., attraction). This attraction energy is short range since
it is inversely proportional to the sixth power of the separation, but
the total interaction energy between two colloidal particles (i.e., a
collection of a large number of atoms) is quite large and of long-range
order and comparable to electrostatic repulsion energy under certain
conditions. The contributions to total interaction energy from the Debye

60
and Keesom interaction is usually small since dipolar contributions tend
to average out when large number of atoms are considered.
There are two methods of calculating the magnitude of Van der Waal's
attraction energy between two colloidal particles, (i) microscopic and
(ii) macroscopic procedure.
Microscopic or Van der Waal's Method
The classical procedure adopted by Hamaker was based on two
principles
(1) the additivity of London dispersion forces: The total interaction
energy between two colloidal particles was calculated using summation of
pairwise interactions between all the atoms or molecules of the two
macroscopic bodies.
(2) The summation of these interaction energies between atoms or
molecules can be replaced by an integration provided that the distance
between the particle surface is large compared with the atomic distances.
Under these assumptions, the Van der Waal's attraction energy between the
two colloidal particles can be represented as follows:
(3.13)
VA • - A(A) • H(G)
Hence, A(A) is a function of so-called Hamaker-Van der Waal's constant of
the material and H(G) is determined by the geometry of the system. For
example,
Flat Plates
For two flat plates of substance 1, the Van der Waal's attraction
energy is given by (Hamaker 37):
A
II
12nd'
(3.19)

61
where A-| 1 is the Hamaker constant for the substance 1, d is the distance
between two flat plates, and the negative sign indicates that the energy
is attractive. The Hamaker constant A-| -j for substance 1 is defined by
An â– " qiBii ,3-20
where qi is the density of atoms in the colloidal particle and i is the
constant in the London equation:
V
11
(3.21)
r
describing interaction energy V-|-| between atoms or molecules. &-|-| is
proportional to the polarizability of the atoms, and hence, increases
with the size of the atom. From the above equation, it is clear that the
Van der Waal's attraction energy VA increases with square of the density
of the substance and the interaction is of long range. (Note VA a d z
for "bulk" materials, compared to Vn a r-^ for molecular interactions.)
Spherical Particles
For two unequal spheres of radius a-) and a2 separated by a distance
d in vacuum (d is the distance between two surfaces), the Van der Waal's
attraction energy is given by the following equation (Hamaker 37):
-A
Vft (sphere)
11
12
+ 2 In
x +xy+x x +xy+x+y
, x -t-xy+x
' 2
x +xy+x+y
)
(3.22)
where x = d/2a-| and y = d/2a2
For two spherical particles, having same radius a, it can be shown
that for large separations, VA decreases with the sixth power of d as in
the London expression, but for short distances (i.e., d « a) VA
decreases slowly with the distance:

62
This distance dependence of the attractive (as well as the repulsive)
energy of interaction is of great importance. As it will be shown later,
at short and at long distances, the Van der Waal's attraction energy
dominates the total interaction energy while at the intermediate
distances of separation, electrostatic repulsion, which decay
exponentially with the distance, may dominate the total energy of
interaction. The above equation indicates that, for a small distance, VA
tends to assume very large negative value. However, the above equations
are no longer valid for very short distances (i.e., dimensions comparable
to the atomic dimensions) because strong Bom repulsive interactions
(usually represented as V„ a d-1^) which arise from the overlap of
Bom
electronic orbitals of approaching molecules, are operative.
Retardation Effect
The equations given above for VA were derived under the assumption
of the additivity of interactions between atoms. The origin of the
London-Van der Waal's interaction is the electromagnetic interactions
between atoms and molecules. The above equations for VA do not allow for
the finite time of propagation of electromagnetic waves from one atom to
the other, and the induced-dipole becomes retarded against the inducing
one when the distance between the atoms becomes comparable to the wave
length of the London frequency. This leads to reduction in the London
dispersion force between atoms. Casimir and Polder have shown that due
to retardation the inverse sixth power law [Equation (3.21)] gradually
changes into an inverse seventh power law with increasing distance

63
(Casimir and Polder 48, Gregory 69, and Visser 72). The retardation
correction is negligible when the distances between the atoms are
comparable to the atomic dimensions. However, for the interactions
between the colloidal particles at the distances of the order of ten to
one hundred nm, the retardation effect can be significant. Equations are
available to correct for the retardation effect for the flat plates and
for the spherical particles (Gregory 67). The retardation effect leads
to reduction in the attraction energy and makes the dispersion force much
longer range.
Effect of Medium on the Van der Waal1s Attraction
The equations derived above for the Van der Waal's attraction energy
can be used for the case of two colloidal particles interacting in
vacuum.
In most practical cases, colloidal particles are embedded in a
medium (e.g., water). To get the interaction energy VA in this case, the
Hamaker content An is replaced by the effective Hamaker constant. The
effective Hamaker constant, A-j3-), now depends not only on An (particle-
particle attraction) but also on A-| 3 (particle-medium attraction) and A33
(medium-medium attraction) where subscript 1 refers to particle and 3
refers to medium. The effective Hamaker constant is defined as:
A
131
+ A
33
2A
13
(3.24)
where A11 = ixq^81 , A33 = nq3 833 and A13 = and it is generally
1 /2
assumed that P13 = O11&33) , and hence,
A
A
33
1/2,2
(3.25)

64
Thus, to account for the medium, A-|-j should be replaced by A131 in all
the previous equations for VA. From the above equation, it is clear that
the effective Hamaker constant is always positive (i.e., there is always
net attraction between two particles) and the magnitude of the Van der
Waal's attraction is reduced due to the presence of the medium (i.e., the
medium imparts a certain measure of stability to the dispersed
particles). By choosing A33 close to An, the Van der Waal's attraction
can be substantially reduced and at An = A33, there is no net attraction
between particles. Thus, to evaluate Van der Waal's attraction energy,
the Hamaker constants should be available for the materials under
consideration.
Macroscopic Approach
One of the main drawbacks of the microscopic or the Hamaker theory
is the assumption regarding additivity of interactions. The Van der
Waal's attraction VA has been calculated using a different approach by
Lifshitz and collaborators using the "macroscopic" approach. In this
theory, the interacting bodies are treated macroscopically, i.e.,
interacting bodies are considered as two semi-infinite phases separated
by the distance d. The bodies are characterized by their complex
dielectric constant. The interaction is evaluated using fluctuation
theory. The spontaneous electromagnetic fluctuation in one body induces
a fluctuation polarization in the other body. The correlation between
the fluctuating fields in the two objects decreases the free energy of
the system, and hence, leads to attraction. The Lifshitz theory has some
advantages over the classical microscopic theory. The assumption

65
regarding additivity is avoided and contributions of bonding between the
atoms and the molecules to the interactions are taken into account. The
Lifshitz approach is more accurate to calculate V^, but mathematics
involved is quite complicated and requires dielectric constant data over
a wide frequency range of the materials of interest. Due to these
difficulties, the Hamaker microscopic approach is generally used in
practice.
Hamaker Constants
To calculate the Van der Waal's attraction energy for practical
systems, the values of the Hamaker constants should be available. In
principle, there are two ways to estimate the Hamaker constants (see for
e.g., Visser 72, Gregory 69, Lyklema 68):
(1) Direct Calculations: In this case, the Hamaker constant can be
calculated by the microscopic approach (e.g., see equation 3.20) using
molecular properties such as polarizability or by the macroscopic
approach using dielectric constant data.
(2) Indirect Evaluation From the Experimental Data on Colloidal
Stability: In this method, the Hamaker constant will depend on the
experimental tool used to determine stability and the stability criteria
used. The Hamaker constant can be experimentally determined from various
techniques such as (i) flocculation experiments on dispersion of
colloidal particles, (ii) from the measurements of direct force of
interactions between crossed wires, (iii) from equilibrium film thickness
measurements, (iv) surface tension measurements, and (v) from rheological
data, etc. (Visser 72).

66
Table 3.2 shows the list Hamaker constants for the present system
under investigation (i.e., silica-water-PVA). Data for other materials
can be found in the above references.
The effective Hamaker constant will be also influenced by the
electrolyte concentration, surface contamination, and adsorbed polymer
layers. The effect of the adsorbed polymer layer will be discussed next.
The Effect of Adsorbed Polymer Layer on Van der Waal1s Attraction
The effect of adsorbed polymer layer of thickness 6 on the
attractive force can be compared at (i) constant center-to-center
distance of spheres, h (see Figure 3.3) or (ii) at constant separation of
outer surfaces, d.
If one makes comparison at the constant center-to-cetner distance,
then the adsorbed layer usually leads to an increase in attraction
between particles. This is due to an effective increase in the particle
size and decrease in the distance of separation between particles. If
comparison is made at the constant distance of separation d, the adsorbed
layer usually reduces the attraction between particles. In this case,
the original spheres are separated by a greater distance and generally
the adsorbed material has lower Hamaker constant. The attraction energy
between two equal spheres of radius a, separated by distance d with the
adsorbed layer thickness 6, is given by the following equation:
v_ (d)
A
-iCHn
(AV2- AV2)2
l*11 *22 '
+ H
22
(A1/2- AV2)2
1 22 33
2 H
12
1/2_ 1/22
l*11 *22 '
(AV2- A1/2)]
'*22 *33 U
(3.26)

67
TABLE 3.2
Compilation of Hamaker Constants for Silica-Water-PVA System
Material
Hamaker Constant
x 10-2® J
Technique
Reference
Silica
6.6
Macroscopic
Hunter 87
(An)
14.8
II
Visser 72
50.0
Microscopic
tl
16.4
Surface Tension
II
Water
3.7
Macroscopic
Hunter 87

4.4
Lifshits
Visser 72
3.3-6.4
Microscopic
II
3.0-6.1
Macroscopic
II
4.8-jLO
Colloid Chemistry
II
5.5-6.4
Surface Tension
II
PVA
6.8-8.8
Microscopic
Visser 72
(Pi22)
Hamaker Constant
Material
in Water x 10-20 J
Technique
Reference
Silica
0.85
Macroscopic
Hunter 87

PVA
0.50
Microscopic
Visser 72
(a232>

68
Figure 3.3
Schematic illustration of the effect of adsorbed polymer
layer on Van der Waal's attraction.

69
where An, A22, A33 are Hamaker constants in vacuum for solid, polymer
and liquid, respectively and Hn, etc. are geometric functions defined
as:
(3.27)
x +xy+x x +xy+x+y x +xy+x+y
where for Hn, x = (d+26)/2a and y = 1, for H22» x = d/2(a+6) and y =1,
and for H12» x = (d+6)/2a and y = (a+6)/a. It can be shown from
Equations 3.26 and 3.27 that if the adsorbed layer is sufficiently thick
and when A22 = A33, the attraction between particles virtually
disappears. To evaluate the Hamaker constant A22 for the adsorbed layer,
two additional features need to be considered: (i) the adsorbed layer
will be a composite, i.e., it will consist of polymer segments plus
solvent molecules and (ii) the segment density will not be uniform. Both
of these effects on the attraction energy have been discussed by Vincent
(Vincent 74).
Potential Energy Curves and the DLVO Theory
Since the electrostatic repulsion and the Van der Waal's attraction
are assumed to operate independently and since botn are being scalar
quantities, they can be added to give the total interaction energy,
V . This is the basis of DLVO theory. The total interaction energy
total
can be represented as:
V(d)
total
= VR(d) + VA(d)
(3.28)
where VR is the electrostatic repulsion energy and VA is the Van der
Waal's attraction energy term. Substituting for VR and VA from the
previously developed equations, the total interaction energy as a

70
function of distance of separation between the particles can be plotted.
In this type of plot, the attraction energy term is represented as a
negative term and the repulsive energy by a positive term. Figure 3.4
shows schematically the total interaction energy as a function of the
distance of separation between two surfaces. The shape of the total
interaction energy curve is important in determining the stability of
colloidal dispersions and can be used pictorially to show the influence
of various relevant parameters on the stability. The type of curve shown
in Figure 3.4 results because of the different distance dependences of
the interaction energy terms VR and VA. The Van der Waal's attraction
energy term is important at close approach since at close approach VA a
1/d and also at large distances when VA a d-^ due to the retardation
effect. At the intermediate distance of separation, the electrostatic
repulsion (which decreases exponentially with the distance) is more
important. The summation of these two interaction energy terms having
different distance dependences leads to a total attraction energy curve
having a maxima in a potential energy separating two minimas. The
primary minima results from the strong Van der Waal's attraction at the
short distances and the Bom repulsion due to overlap of electron clouds.
At such close distances of approach, recent experiments indicate that an
additional energy term, Vs, arising due to solvent structural effects,
should be included (e.g., see Israelachvili and Pashley 82,83). Due to
these additional complications, the exact location and depth of the
primary minima cannot be determined quantitatively. The secondary minima
results from the long-range Van der Waal's attraction and the rapid decay
of the electrostatic repulsion. The depth of the secondary minima is

potential energy
71
\Vr
\
Total potential energy of interaction V(d) = VR(d) +
VA(d) where VR(d) is the potential energy of repulsion
due to double-layer interactions and VA(d) is attractive
potential due to Van der Waal's interactions (Overbeek,
1952).
Figure 3.4

72
usually not significant compared to the thermal energy of the particles.
Hence, flocculation of particles in the secondary minima tends be weak.
The other important characteristic of the total interaction energy curve
is the presence of the potential energy maxima. If the height of the
potential barrier, vmax, is greater than the thermal energy of the
particles (i.e., vmax » kT) then the potential barrier can prevent
flocculation of the colloidal particles and the dispersion is stable in a
colloid chemical sense. The fraction of particles that can surmount such
a potential barrier is given by Boltzmann's law. The fraction decreases
exponentially with increasing height of the potential barrier. vmax of
the order of 5-15 kT has been considered sufficient to achieve long-term
stability (Overbeek 82b). It should be noted that, in the case of flat
plates, the potential energy per unit area is plotted. In these cases,
the total interaction energy is obtained by multiplying by the
appropriate cross sectional area of the particles. In the case of
spherical particles, the net interaction energy is plotted. To
understand the stability behavior, factors influencing the repulsion
energy term VR, and the attraction energy term VA should be considered.
The Effect of Hamaker Constant
The range of values of Hamaker constant for substances (ignoring
retardation effects) immersed in water can be given as follows (Hunter
87):
A.^ 30 - 10 for metal particles
3-1 for oxides and halides, and
-20
= 0.3 x 10 J for hydrocarbons.

POTENTIAL ENERGY OF INTERACTION (J)
73
DISTANCE OF SEPARATION (nm)
50 kT
25 kT
0 kT
25 kT
Figure 3.5
The effect of the Hamaker constant on the total
interaction energy curves.

74
As discussed earlier, this difference essentially arises due to
differences in the polarizability of these materials. Figure 3.5 shows
the effect of the Hamaker constant on the total interaction energy curve.
All other factors (i.e., the surface potential ¥0 and the Debye-Huckel
parameter k) were kept constant. As expected with increasing values of
A^i/ the height of the potential barrier decreases and the depth of the
secondary minima increases.
The Effect of Surface Potential, Vn
The electrical double layer repulsion term VR is usually dominated
by two parameters (a) the near surface potential and (b) the thickness
of the double layer, 1/k. Figure 3.6 shows that the height of the
potential barrier increases as Vq increases. (For oxide materials, such
as silica, exact value of VQ is difficult to determine experimentally due to
complications, such as specific ion adsorption and presence of the Stern
layer, etc., so usually the near-surface potential, i.e., the zeta
potential, £ is used. This experimentally determined zeta potential does
establish a lower limit of V0.
The Effect of Electrolyte Concentration
The Debye-Huckel parameter, k, which represents the thickness of the
electrical double layer, depends on both the concentration and the
valance of the indifferent electrolyte (See Equation 3.7). Figure 3.7
shows the effect of concentration of a 1:1 electrolyte on the total
potential energy curve. The potential energy barrier decreases with
increasing electrolyte concentration, and above a certain concentration,
the barrier vanishes. Thus, the addition of an indifferent electrolyte

POTENTIAL ENERGY OF INTERACTION (J)
75
DISTANCE OF SEPARATION (nm)
The effect of zeta potential on the total interaction
energy curves.
Figure 3.6

POTENTIAL ENERGY OF INTERACTION (J)
76
4
3
2
1
0
1
Figure 3.7
100 WT
DISTANCE OF SEPARATION (nm)
The effect of concentration of 1:1 electrolyte on the
potential energy curves.

77
can cause a hydrophobic colloid to undergo flocculation. For a given
salt, the critical concentration needed to induce aggregation can be
predicted quantitatively by DLVO theory.
Effect of Particle Radius
The total interaction energy is influenced by the particle radius as
follows. The Van der Waal's attraction energy is proportional to radius,
a, and the electrical double layer repulsion is proportional to a ,
hence, the potential energy barrier increases with a. Thus, the bigger
particles are more likely to be stabilized by the electrostatic repulsion
(Figure 3.8).
In summary, for preparing electrostatically stabilized dispersions,
one should operate under the conditions of high zeta potentials and low
ionic strengths. We will examine the effect of electrolyte concentration
in more detail as it proves the validity of the DLVO theory.
The Stability - Instability Approach
From the above discussion, it is clear that the form of the total
potential energy curve gives an explanation for the stability behavior of
lyophobic dispersions. When the potential energy barrier has a large
positive value (usually greater than = 10kT), the system is kinetically
stable due to large activation energy opposing transition from the
secondary minima to the primary minima. With decreasing height of the
potential barrier, V , the transition from stable to unstable
max
dispersion is facilitated. Theoretically, the onset of instability can
be defined by the following condition (Overbeek 52):
V{d)
total
total
= 0 and dV(d)
/dd = 0
(3.29)

POTENTIAL ENERGY OF INTERACTION (J)
78
DISTANCE OF SEPARATION (nm)
The effect of particle radius on the total interaction
energy curves.
Figure 3.8

79
and the electrolyte concentration at which these conditions are fulfilled
gives a theoretical critical coagulation concentration. The
experimentally observed dependence of the critical coagulation
concentration on the sixth power of the valence of the counter ions for
high surface potentials and on the second power of the valence of the
counter ions at low surface potentials can be explained using the above
criteria. The successful explanation of the dependence of the critical
coagulation concentration on the valence is one of the important
contributions of the DLVO theory. The DLVO theory explains the stability
criterion with the help of interaction energy plots while experimental
determination of the stability involves some additional approximations.
The effects of electrolyte concentration and the valence can be
controlled experimentally, and the change in the particle concentration
is measured using various techniques (i.e., direct particle counting,
light scattering, etc.). To correlate the stability to instability
transition to the critical coagulation concentration, the kinetics of
coagulation should be considered.
Kinetics of Coagulation
The rate of coagulation in the absence of any potential barrier was
first examined by von Smoluchowski (Smoluchowski 16a,16b,17). The rate
of disappearance of primary particles in the initial stages of the
coagulation could be written as:
- = K N 2 (3.30)
dt o o
where N0 = the number of particles per unit volume present at time t = 0
and Kq is a rate constant. It was shown that, for the case of rapid

80
coagulation (i.e., coagulation with no potential barrier), K0 = 8naD,
where D = the diffusion coefficient of a single particle and a = the
collision radius. In the presence of an energy barrier, the flocculation
rate is slower and characterized by a rate constant, K, where K = K0/W
and W is called the stability ratio. The above equation can be written
as:
= KN 2 = (K /W)N 2 (3.31)
dt o o o
and W = 1 corresponds to rapid coagulation and W > 1 indicates a slow
rate of coagulation. Experimentally, the rapid coagulation rate, Kq, is
determined using either direct particle counting or light scattering
technique. From the experimentally determined values of K0 and K, the
stability ratio W under various electrolyte concentrations can be
obtained. Typically, log W versus log C is plotted, and the sharp change
of gradient at a particular electrolyte concentration establishes the
critical coagulation concentration (see Figure 3.9).
According to DLVO theory, the stability ratio W is related to the
potential energy maxima Vmax by the following equation
°° 2
W = 2a f exp (V /kT) dh/ (d + 2a) (3.32)
J max
o
where a is the particle radius and d is the distance between particle
surfaces. Reerink and Overbeek showed that the slope of the log W versus
log C curve is given by the following equation (Reerink and Overbeek 54):
d log W/ d log C = - 2.06 x 107 (ay^ / Z^) (3.33)
where Z is the valance of electrolyte, and y was previously defined
(Equation 3.10). The above equation has been confirmed experimentally
for various systems, but the dependence of slope on the particle radius

81
Figure 3.9 Theoretical dependence of stability ratio of electrolyte
concentration (Overbeek, 1952).

82
has not confirmed. The above theory has been modified to account for a
hydrodynamic correction (e.g., see Honig et al. 71). The hydrodynamic
correction attempts to take into account the fact that the particles are
slowed down in the final stages of approach due to difficulty in removing
the remaining liquid film between the particles. This treatment shows
that the diffusion coefficient D is also a function of distance of
separation between the particles and D approaches zero as 2d/a approaches
zero, and that the rate of rapid coagulation can be reduced to a little
less than half of the Kq value predicted by von Smoluchowski's equation.
Summary
The following comments can be made regarding the DLVO theory and the
electrostatic stabilization:
- The origin of the charge on colloidal particles is well understood.
- The effect of valence of the electrolyte on the critical coagulation
concentration can be understood from the Debye-Huckel parameter k.
- The slopes of d log W/d log C have correct order of magnitude.
But, still there are certain weak points in this theory. They are
as follows:
- The effect of the finite volume of the ions have been taken into
account by the Stern's theory, but the pertinent parameters are difficult
to measure experimentally.
- The surface potential cannot be measured experimentally due to the
presence of the Stem layer. Often, experimentally measured zeta
potential is substituted for WQ.

83
- The expected dependence of d log W/d log C on particle radius is not
confirmed, and whether this discrepancy is due to experimental problems
or to a serious flaw in the theory is not yet known (Overbeek 82b).
- The minimum distance of closest approach of two particles is not known.
- The time scale of various relaxation processes occurring in the double
layer are not incorporated in the theory (i.e., whether to use the
constant charge or constant potential case).
- The recent direct force measurements between two macroscopic surfaces
have confirmed the validity of the DLVO theory at the distances larger
than 5 to 10 nm. But, the additional short range (1 to 10 nm) repulsive
interactions due to changes in the liquid structure near the charged
surface and attraction due to hydrophobic forces need to be considered.
Quantitative treatments of these interactions are not yet available.

CHAPTER IV
EFFECT OF ADSORBED POLYMER ON DISPERSION STABILITY
Introduction
There are essentially two methods to impart stability to the
dispersion of colloidal particles. In the last chapter, we reviewed the
DLVO theory of electrostatic interactions between two charged particles,
and potential energy versus distance diagrams were used to explain the
stability behavior. In the case of electrostatic stabilization, the
overlap of the electrical double layers around the charged particles give
rise to repulsive interactions.
Absorbed polymer can affect two particle interactions in the
following ways. It can increase the stability by increasing
electrostatic repulsion between particles and/or decreasing the Van der
Waal's attraction. The adsorbed polymer can also impart stability due to
the additional "steric" component of repulsion. In this case, the
overlap of the adsorbed polymer layers give rise to repulsive
interactions. The protective action of the adsorbed polymer is often
called "steric stabilization," (especially for the effects of non-ionic
polymers on colloid stability) (Napper 83). The term 'steric
stabilization' was introduced by Heller and Pugh (Heller and Pugh 54).
As pointed out by Napper, the term 'steric' used in this context is a
misnomer, and it has little in common with steric effects present in
organic chemistry (Napper 83). The phenomenon of steric stabilization
has a general thermodynamic basis. Steric interactions are of two major
84

85
kinds namely "entropic" and "mixing" interactions. Entropic interactions
result from the loss of configurational freedom of the adsorbed polymer
on approach of a second particle. The configuration entropy is decreased
because less total volume is available for each chain due to
"compression" of the chains. This effect is also called the "volume
restriction" or the "elastic effect." The mixing interaction arises due
to interpenetration of the adsorbed layers. This leads to a build up in
the segment concentration in the interaction zone, and hence, an increase
in the osmotic pressure. Hence, this mixing term is also known as the
"Osmotic Repulsion" effect. The microscopic origin of these interactions
will be considered later. As in the case of polymer adsorption behavior
(Chapter II), interactions with adsorbed polymer will involve various
interactions such as polymer-solvent, solvent-surface, and surface-
polymer and as such is a very complicated problem.
Factors Influencing Steric Stabilization
Excellent reviews are available in this area, and hence, only a
brief discussion will be attempted (Vincent 74; Tadros 82; Sato and Ruch
82; Napper 83; Lyklema 68; Vincent and Whittington 81). The principle
factors governing the stability of particles with adsorbed layer are:
The Adsorbed Amount of Polymer
The adsorbed amount of polymer is related to the average thickness
of the adsorbed layer (or, more importantly, to the segment density
distribution). The adsorbed polymer should completely cover the particle
surface to prevent "bridging flocculation." Generally speaking, in the
absence of electrostatic interactions, at low polymer concentrations

86
(usually 0-1000 ppm), the dispersion is destabilized due to bridging
flocculation, and at higher polymer dosages, the dispersion can be
stabilized under certain conditions due to the steric interactions. To
prepare stable dispersions, the adsorbed layer should have sufficient
thickness to overcome the Van der Waal's attraction.
The Solvent-Segment Interaction Parameter, x
The origin of the steric stabilization is in the polymer solution
thermodynamics. For the case of thick adsorbed polymer layers, the
stability behavior can be related to the solvent-segment interaction
parameter, x (i.e., it is independent of particle size, Hamaker constant,
etc.) (Napper 83). The interactions between adsorbed polymer molecules
are very similar to the interactions between the molecules in solution
and directly related to the stability of long chains in solution.
The Effective Hamaker Constant and the Size of the Particles
These parameters are important when the thickness of the adsorbed
polymer layer is small compared with the particle radius. For the case
of thin adsorbed layers on the large particles, the Van der Waal's
interactions are important.
The above factors can be controlled to influence the stability of
polymer coated particles. For example, the adsorbed amount of polymer
can be varied by changing the molecular weight of the polymer, chemical
nature of the polymer, polymer concentration, etc. The solvency for the
polymer molecules can be changed, for example, by changing electrolyte
concentration or change in the temperature. As in the case of
electrostatic stabilization, the effect of various parameters can be
explained using the total interaction energy curves.

87
Applications and Advantages of Steric Stabilization
Steric stabilization has been used in a variety of technological
applications and products (for example, paints, inks, glues, dispersion
of latex particles, ceramic particulates, etc.) because it can offer
several distinct advantages over electrostatic stabilization. These
advantages will be discussed below:
(1) Relative Insensitivity to the Presence of Electrolytes
The thickness of the adsorbed layer (in the case of non-ionic
polymer) is relatively insensitive to the presence of electrolyte, and
hence, can overcome the Van der Waal's attraction. This is in sharp
contrast with the effect of electrolyte concentration on the thickness of
the electrical double layer. As discussed in the last chapter, the
thickness of the electrical double layer, k-1, decreases with increasing
electrolyte concentration. This effect can lead to the coagulation of
electrostatically stabilized dispersion on the addition of high
concentrations of electrolytes.
With a suitable steric stabilizer, coagulation can be prevented at
high salt concentrations and dispersions of high solid volume fractions
can be prepared. At low electrolyte concentrations, the thickness of the
electrical double layer can be quite substantial. Concentrated
dispersions of charged particles at low ionic strengths can have very
high viscosities because of electroviscous effects (for e.g., see, von
Smoluchowski 16a, Booth 48, Chan and Goring 66a, Chan et al. 66b, Stone-
Masui and Watillon 68). In certain cases, this effect cannot be overcome
by the addition of salt due to coagulation.

88
(2) Equally Effective in Aqueous and Non-aqueous Dispersion Medias
From the practical experience, it can be concluded that the
electrostatic stabilization is less effective in non-aqueous dispersion
media than it is in aqueous media (Napper 83, Vincent 74). This is
primarily due to the low relative dielectric constant of most non-aqueous
media. Steric stabilization is equally effective in both aqueous and
non-aqueous media, and for this reason, steric stabilization is often
employed in non-aqueous dispersion media.
(3) Reversibility of Flocculation
A sterically stabilized dispersion can be flocculated by adding non¬
solvent to the dispersion (or by changing the temperature). This
behavior can be reversed by adding more solvent to the system (or by
reversing the temperature change). In contrast to this behavior, the
dispersion stabilized solely by an electrostatic mechanism can be
coagulated by the addition of electrolyte, but that coagulation is
usually irreversible to subsequent dilution. The difference in behavior
between electrostatically and sterically stabilized dispersions may be
due to the fact that the latter dispersions are thermodynamically stable,
while the former dispersions are kinetically stable (Napper 83). The
difference between these two cases will become clear when we consider the
difference between the total interaction energy diagrams under these two
stabilization mechanisms. In the latter case, the coagulation state is
the lowest energy state, and it can be reversed only if additional work
is done on the system (Overbeek 82b). One important consequence of the
thermodynamic stability of sterically stabilized dispersion is that

89
particles redispersed spontaneously after drying (Napper 83). Sterically
stabilized dispersions also often display good freeze-thaw stability.
Thermodynamic Basis of Steric Stabilization
The stability of sterically stabilized dispersion is
thermodynamically limited provided that the following three requirements
are fulfilled: (a) strong anchoring of polymer, (b) complete coverage
of the surface, and (c) sufficiently large thickness of the adsorbed
polymer layers. Under these condition, the thermodynamic limit of
stability is largely dependent on the chemical nature of both the
stabilizing moieties and the dispersion medium (i.e., solvent-polymer
interactions). The stability is usually independent of particle nature
and size and is independent of the chemical nature of the anchor polymer.
The stability to instability transition can be induced in at least three
different ways: by changing the temperature, by increasing the pressure,
and by adding to the dispersion a miscible non-solvent medium for the
stabilizing moieties (Napper 83). All of the above methods decrease the
solvency of the dispersion medium for the stabilizing moieties. The
transition from long-term stability to catastrophic flocculation occurs
at the critical flocculation point (UFPT). This may be a critical
flocculation temperature (CFT), a critical flocculation pressure (CFP),
or a critical flocculation volume (CFV) of non solvent. It is possible
to induce instability by increasing the temperature or cooling the
dispersion. When the flocculation is induced by heating, CFT is usually
called an upper critical flocculation temperature (UCFT), and when
flocculation is induced by cooling, it is said to occur at a lower

90
critical flocculation temperature (LCFT). In principle, all sterically
stabilized dispersions show a LCFT and a UCFT although both are not
always experimentally accessible.
The thermodynamic limit is set by the polymer-solvent interactions,
and hence, there is a strong correlation between the critical
flocculation point and the theta point (0 point) of the stabilizing
moieties. The theta point represents a transition point with respect to
segment-solvent interactions. At this point, the interaction between two
polymer segments changes from net repulsive to attractive. To understand
this correlation between 0 point and the stability, we will review
briefly polymer solution thermodynamics.
Polymer Solution Thermodynamics
The best known theory of polymer solution was as developed by Flory
and Huggins (Flory 53). The Flory-Huggins theory is based on statistical
thermodynamics and describes the free energy changes associated with
mixing of pure solvent and pure polymer. This is accomplished the
separate calculations of the entropy of mixing and the enthalpy of
mixing. Then, these two terms are combined to calculate the free energy
change associated with the mixing processes as follows:
AGm = AH,,, - TAS,,, (4.1)
where m denotes mixing.
The Entropy of Mixing
To calculate the configurational entropy, the lattice model was
employed. In the model, a polymer molecule was assumed to consist of x
segments and the size of each segment was assumed to be equal to the size

91
of solvent molecules. The total configurational entropy of the polymer
solution arises from the variety of ways of arranging the polymer and
solvent molecules. This was accomplished by placing a chain molecule
segment by segment on an empty or partially filled lattice. This
sequence of steps constitutes a three-dimensional random walk. A step in
such a walk is possible if the lattice site towards which it was directed
is unoccupied. To calculate the probability of finding a vacant lattice
site, the mean field approximation was made. It was assumed that all the
previously placed segments are uniformly distributed so that the solution
is a completely random mixture and local concentration was replaced by an
average segment concentration. Due to this assumption, the Flory-Huggins
theory is valid for relatively high polymer concentrations when the
polymer coils overlap extensively. With the above assumptions, the
configuration in entropy of mixing n-j solvent molecules and n2 polymer
molecules is given by the following expression1:
ASm * -k (n-j In v-j + n2 In V2) (4.2)
where v-| and V2 are the volume fractions of solvent and solute,
respectively, and are defined as:
v-] = n-|/(n-] + xn2> and V2 = (n2*x)/(n-j + xn2> (4.3)
and x is defined as:
Molar volume of solute ,A , s
x = (4.4)
Molar volume of solvent
The above expression for the configurational entropy of mixing is
analogous to the expression for mixing of small molecules:
1 It should be noted that the expression for the configurational
entropy of mixing can be arrived at following free volume theories (Flory
53).

92
ASm = -k (n-j In x-| + n2 In X2) (4.5)
where x¿ = mole fraction of species i.
The primary difference between the entropy of mixing expressions for
small molecules (Equation 4.5) and for polymer solutions (Flory-Huggins
Equation 4.2) is that mole fraction statistics is replaced by volume
fraction statistics. This difference arises due to the dissimilarity in
the sizes of solvent and polymer molecules. Since the values of volume
fractions v-| and V2 are smaller than the unity, the entropy of mixing is
always a positive quantity and mixing is spontaneous unless opposed by
the enthalpy of mixing, AHm. We will examine the enthalpy of mixing in
the next section.
The Enthalpy of Mixing
Mixing can be considered as a quasi-chemical reaction between
solvent contacts and segment contacts. Following Flory's lattice model,
it can be represented as:
(1-1) + (2-2) > 2(1-1) (4.6)
where 1 and 2 refer to solvent molecules and polymer segments,
respectively, and the change in the enthalpy of formation of 1-2 contacts
is given by:
Ad)
12
d>
12
“221
(4.7)
where d>-| -j, <022» and 0012 are the enthalpies associated with the respective
pair contacts. For the case of random mixing of n] solvent molecules and
n2 polymer molecules, it can be shown that (Flory 53):
AH = ZAd),_n,v_ (4.8)
m 12 12
where Z is the coordination number of the lattice site and V2 is the
volume fraction of polymer. Now, if we write:

93
AH = kTxn,v_ (4.9)
m i ¿
AH ZA Then, we have x = , — m = ——— (4.10)
kTn1v2 kT
The parameter x is called the Flory-Huggins interaction parameter. It is
defined in such a way that x kT represents the difference in energy of a
solvent molecule immersed in a pure polymer compared with that in pure
solvent.
The expression for the total free energy of mixing can be obtained
by combining equations for the enthalpy and entropy changes and is given
as follows:
- TASm =
AG = kT {n. In v, + n. In d, + n, v, x) (4.11)
m 1 i ¿ ¿ i ¿
As mentioned earlier, AS,,, is always positive, hence, it contributes to a
negative value of AGm. The AHm contribution can be either positive or
negative depending on the sign of x- For non-aqueous solvents, x is
often positive (i.e., mixing is endothermic), and hence, contact
dissimilarity often opposes mixing. Note that xkT was originally
introduced as a change in internal energy, but the above treatment will
be essentially the same if xkT is considered as a free energy change for
the process (Flory 53). In this way, x determined experimentally
incorporates both enthalpy and entropy change associated with the
dissimilar contact formation.
The x Parameter
From the above expression for the free energy of mixing, the
chemical potential of the solvent and solute can be determined. The

94
chemical potential difference is related to the osmotic pressure as
follows:
n = - (ui - ui°) / Vi (4.12)
where n is the osmotic pressure, (ui - 4i°) represents the change in the
chemical potential of the solvent and V-| is the solvent molar volume. It
can be shown that the osmotic pressure can be represented as (see Flory
53) :
rt/C2 = RT {B1 + B2C2 + B^2 + . . . } (4.13)
where C2 is the polymer concentration. B-) is the first viral coefficient
and B-| is related to the number average molecular weight (i.e., 11/C2 =
RT/ according to van't Hoff osmotic pressure equation).
B2 and B3 are second and third viral coefficients. According to the
Flory-Huggins theory, B2 is given by the following equation:
b2 = ~ X) V22/ v,2 M2 (4.14)
where are the molar volume of solute and solvent and M is the
molecular weight of the polymer.
Thus, the x parameter is related to the osmotic pressure of the
solvent. As mentioned earlier, the interaction parameter x can be
considered as a contact dissimilarity free energy, i.e., it contains both
enthalpic and entropic contributions. Flory has defined enthalpy (h-¡)
and entropy (W-j ) of dilution parameters as follows (Flory 53):
AH1 = RTh1v22 and A§1 = W^2 (4.15)
Here AH-j and AS-j are partial molar enthalpy and entropy. It can be shown
that x is related to hi and W-] as follows:
<5 - x) - (h, - V
(4.16)

95
and this important result allows the interaction free energy to be
related to its enthalpic and entropic contributions.
The Theta Point
The equation for the second viral coefficient (B2) shows that its
value becomes zero when x = 1/2 (Equation 4.14). The point at which the
second viral coefficient B2 vanishes is known as the theta (0) point. It
has been also called the Flory point. In polymer/solvent systems, the
second viral coefficient becomes equal to zero at some temperature,
called the theta temperature. Then, the theta temperature is defined as
follows:
H T
0 = so that W1 - H1 = 4>1 (1-0/T) = (1 - x) (4.17)
Clearly, if T = 0, then, x = 1/2. The parameter x can be used to compare
different solvents at a single temperature, while the 0 temperature can
be used to describe changes in solvent goodness arising due to
temperature changes. At the theta conditions (i.e., x = 1/2 or T = 0),
polymer solutions behave ideally. Under 0 solvency conditions, high
molecular weight polymer molecules behave as if they were ideal small
molecules. At 0 conditions, the excluded volume effects and
intermolecular attractions are canceled out. Thus, the 0 point
represents the transitional point with respect to solvent-segment
interactions. At 0 = T or x = 1/2, the free energy of interaction is
zero, i.e., there is no preference for the solvent or segment. When 0 <
T or x < 1/2, this case corresponds to the position derivation from the
ideality. Under these conditions, dissimilar contacts are favored. For

96
a given system of solvent and polymer, the value of x < 1/2 indicates
that the solvent is a good solvent, x = 0 is called an athermal solvent
for which the enthalpy of mixing is zero. When 0 > T (or x > 1/2), there
is a negative derivation from the ideality, and under these conditions,
there is a mutual attraction between polymer segments.
The value of x is not predicted by the Flory-Huggins theory and must
be measured experimentally. According to this theory, experimentally
determined values should be independent of the concentration of the
polymer. Measured values of x parameter are usually concentration
dependent, and for many systems, the free energy change associated with
the exchange process is predominantly entropic in origin, and enthalpic
contribution is relatively small. Also, the observation of phase
separation upon heating non-aqueous polymer solutions cannot be explained
by the original Flory-Huggins theory. To account for phase separation
upon heating, free volume theories have been proposed (see Flory 70,
Napper 83).
These theories account for the differences in the free volumes for
the solvent and solute molecules at higher temperatures. Flory has
developed an elaborate equation-of-state theory for non-aqueous polymer
solutions that incorporates the changes in free volume of mixing as well
as the combinational and contact dissimilarity contributions (Flory 70).
The concepts underlying the theory can be used to describe microscopic
processes that generate steric stabilization in both non-aqueous and
aqueous dispersions. The free volume of the solvent near its critical
point is much larger than the free volume of the polymer molecules. Near
the critical point, solvent molecules essentially behave like a gas,

97
while due to the covalent bond constraints, there is no substantial
increase in the free volume of the polymer molecules. When polymer
molecules are placed in contact with solvent vapor, these segments can
act as condensation sites. This condensation of solvent molecules
provides a negative contribution to mixing entropy and, if sufficiently
large, can promote phase separation. The enthalpy change associated with
condensation is positive and opposes phase separation. The free volume
dissimilarity provides the rationale for the following observations: (1)
the observed phase separations on heating, (2) the strong entropic
contribution to x that opposes mixing, and (3) the concentration
dependence of x as an increase in the polymer concentration provides more
condensation sites for the solvent molecules.
To explain the microscopic origin of the steric stabilization, the
free energy changes associated with the combinational (as described
earlier), contact dissimilarity and free volume dissimilarity should be
considered (Napper 83). At normal temperatures, combinational entropy is
responsible for the stability of non-aqueous dispersion. During the
Brownian encounter of two colloidal particles, the adsorbed polymer
layers will interpenetrate. This leads to an increase in the
concentration of polymer segments in the interaction zone and transfer of
solvent molecules into the dispersion media. Thus, there is a decrease
in the entropy of mixing between polymer segments and solvent molecules
on close approach of particles. The loss of entropy generates the
observed steric stabilization. At high temperatures, the contribution
due to the differences in free volume can become more important, and

98
thus, flocculation can occur at the critical temperature of the
dispersion medium.
At present, there is no satisfactory theory for the thermodynamics
of aqueous polymer solutions. The strong interactions between polar
water molecules and water soluble polymers makes description speculative
(Napper 83). The contributions of combinational and free volume
dissimilarity terms to free energy are essentially the same as in non-
aqueous dispersions. The main contribution to stability arises from the
contact dissimilarity term due to the specific interactions (such as
hydrogen bonding) between water molecules and polar stabilizing moieties
(for example, hydroxyl groups of PVA). During the interpenetration of
the adsorbed polymer layers, the bound water is released from the
interaction zone into the bulk dispersion media. This process of removal
of bound water is associated with the positive enthalpy change that
opposes flocculation. This enthalpic stabilization is more common in
aqueous dispersions (Napper 83).
Classification of Steric Stabilization
Three different types of steric stabilization can be classified as
enthalpic, entropic, or combined enthalpic-entropic. This classification
is based on the overall free energy change during the encounter of two
sterically stabilized particles and can be represented as follows:
AGp = AHp - TASp (4.18)
where AGp represents the free energy of flocculation of a pair of
particles, and AHp and ASp are the enthalpic and entropic contributions,
respectively. A positive value of AGp indicates the particles are

99
sterically stabilized. The sign and the magnitude determines the type of
steric stabilization as shown in the Table 4.1.
In enthalpic stabilization, the enthalpy changes associated with the
overlap promotes stabilization whereas the corresponding entropy change
promotes flocculation. At room temperature, the enthalpic contribution
is dominant, and hence, promotes stabilization. In entropic
stabilization, the entropy change promotes stability, whereas it is
disfavored by the enthalpy change. In this case, the overall
contribution of the entropy change prevails. In the case of combined
enthalpic-entropic stabilization, both the enthalpic and the entropic
changes contribute to stability.
At room temperature and pressure, entropic stabilization is more
common in non-aqueous dispersion media, and enthalpic stabilization is
more common in aqueous media. However, there are several examples
contradicting the above statement (Napper 83).
Quantitative Theories of Steric Stabilization
In order to discuss the stability of colloidal particles, it is
essential to know the pairwise interactions between two particles. In
the case of two charged particles, the stability behavior can be
explained from the well-known DLVO theory (See Chapter II). In the case
of steric stabilization, no such general quantitative theory is
available. The reasons for this are two-fold. One is the lack of
quantitative theory of polymer solution thermodynamics which is the
foundation of any theory of steric stabilization. Second, in order to
calculate the distance dependence of steric interactions, it is necessary

100
TABLE 4.1
Classification of Steric Stabilization Depending on the Sign and
the Magnitude of Enthalpic and Entropic Contributions
Stabilization
AGp
ahf
ASp
|AHp|/|TASp|
Type
Flocculation
£0
+
+
>1
Enthalpic
Heating to UCFT
*0
-
-
£1
Entropic
Cooling to LCFT
£0
+
-
Combined
Enthalpic-
Entropic
Not accessible

101
to predict the conformation of the adsorbed polymer sandwiched between
two flat plates. It is not possible to predict quantitatively the
conformation of free polymer molecules in solution except for the
limiting case such as 0 solvent. Hence, a general quantitative
description is not available. Despite these difficulties, various models
have been developed to predict the distance dependence of steric
interactions (Mackor 51; Fisher 53; Clayfield and Lumb 66,68; Meier 67;
Dolan and Edwards 74; Ottewill and Walker 68; Napper 77; Scheutjeans and
Fleer 85; Hesselink 77; Hesselink et al. 71b; Bagchi and Void 72; Bagchi
74a,74b; deGennes 82,87).
The Three Domains of Close Approach
Figure 4.1 shows two parallel flat plates separated by a distance d
and coated by the adsorbed polymer layer of thickness L. The three
domains at close approach are classified as follows:
(a) d > 2L: This is called non-interpenetrational domain. In this
case, the plates are too far apart for the adsorbed layers to overlap,
and there is no steric interaction.
(b) L <, d ú 2L: This is called the interpenetrational domain. Once the
distance of separation is less than twice the thickness of the adsorbed
layer, the polymers attached to opposing surfaces undergo
interpenetration. The increased segment concentration in the overlap
region forces solvent molecules into the bulk dispersion media, and thus,
reduces mixing between the polymer segments and the solvent molecules.
In a good solvent (x < 0.5), the demixing of segments and solvents raises
the free energy of the system leading to steric stabilization. In worse

102
L
L
The three domains of close approach of sterically
stabilized flat plates, (i) Noninterpenetration (d > 2L);
(ii) Interpenetration (L £ d £ 2L);
(iii) Interpenetration plus compression (d < L) (Napper,
1983).
Figure 4.1

103
than 0 solvent, overlap leads to attraction. The repulsion arising due
to interpenetration is known as the "mixing" or "osmotic" effect.
(c) d < L: This is called interpenetrational-plus-compression domain.
The free energy change in this case is derived from two components: a
solvent-segment mixing term (as above) and an elastic component. The
elastic effect arises due to the compression of the adsorbed layer by the
opposing surfaces. This would reduce the volume available for the
adsorbed molecules, and hence, restrict the number of possible
configurations. This reduces the configurational entropy and,
irrespective of the solvent quality, leads to repulsion between two
surfaces. This effect is also called the "volume restriction" or
"elastic" effect.
The classical approach for calculating steric interactions has been
to consider two limiting cases: (1) interpenetration without compression
and (2) compression without any interpenetration. These two terms are
added to calculate total steric interactions. We will consider these two
limiting cases in more detail.
The Interpenetration Domain
The free energy of mixing of polymer segments with solvent molecules
in a small volume dV is given by the Flory-Krigbaum theory as follows:
¿(AC^) = kT (6n-| In v-| + Sn-j^x) (4.19)
where 6n-| = number of solvent molecules in the volume element, and vi and
V2 are the volume fractions of solvent and solute, v-j « 1 - V2 ■ 1 -
P2vs» where 02 = segment density distribution function of the polymer, V3
is the volume of the polymer segment. On expanding the logarithmic

104
function and ignoring terms higher than second order, the above equation
becomes:
6(AGM) = (kT/V^ { - (1 -x) P2VS + (1 /2-x-] ) P22VS2} dV (4.20)
where V-] is the volume of the solvent molecule.
This equation is applicable to all domains of approach.
The mixing free energy change for the approach of two sterically
stabilized particles, AGM, is given by the following equation:
AG*1 = AG(d) - AG (to) (4.21)
where AG(d) = the mixing free energy at distance = d, and AG(®) = the
mixing free energy when the particles are separated by infinite distance.
After substituting the Equation 4.19 into Equation 4.20, the following
result is obtained:
AGM = kT (Vs2/V1)(l - x) tí P22(d)dV - J p22(»)dV] (4.22)
v v
Thus, to evaluate the mixing free energy change, the following
quantitives should be known:
(i) The distance dependence of the segment density distribution, i.e.,
when the particles are separated by a distance d and when they are
infinitely separated, and
(ii) Geometry of the interacting bodies: Usually, two flat plates or
spherical particles are considered. Various treatments of steric
stabilization can be classified based on the assumptions regarding the
segment density distribution.
(1) Theoretically Derived Segment Density Distributions: Various
theoretical models are available to predict the segment density
distribution of the adsorbed polymer with or without the presence of
other interfaces. Meier (Meier 67) calculated the segment density

105
distribution for the isolated adsorbed polymer, and Hesselink (Hesselink
71) derived the segment density distributions for the case of isolated
loop and tail conformation. Segment density distributions also have been
derived from lattice models (SF) (Scheutjeans and Fleer 85) and from
scaling concepts (deGennes 80). Napper has called these theories ab
initio theories (Napper 83).
(2) Experimentally Determined Segment Density Distributions: Recently,
it has become possible to obtain segment density distribution from the
small angle neutron scattering, (SANS) technique (Barnett et al. 82).
(The segment density distribution is exponential near the surface of the
physically adsorbed polymer, but this technique is not sensitive enough
to pick up the signal from the dilute tails far away from the interface.)
It is then possible to use the experimentally determined segment density
distribution to carry out necessary integration. Also, from
experimentally determined force vs. distance curves, it may be possible
to deduce the information about the segment density distribution.
(3) Assumption Regarding the Segment Density Distribution: This class
of theories has been called the pragmatic theories of steric
stabilization by Napper (Napper 83). Various types of segment density
distributions can be assumed, such as constant segment density,
exponential, the radial Gaussian, etc.
Only two equations for interactions between polymer-coated spherical
particles will be reported here. According to the HVO theory (Hesselink
et al. 71b), the mixing free energy for spheres is given by the following
equation:
AGM = [2n5/2/27] v2 kT a 3/2 (a2-1) SM
(4.23)

106
where a = particle radius, d = the minimum distance of surface
separations of the spheres, v = the number of tails (loops) per unit
area, and a is the expansion factor defined as a = 1/2 / Q1/^2.
Where 0V2 refers to unpurturbed dimensions (i.e., under 0
conditions), V2 is the r.m.s. end to end length of the polymer
molecule in solution.
The distance dependence of SM is given by the following equation for
equal tails S*^ = 1.2nd^ / V2 exp {- 1.2d^ / }.
And for equal loops, SlM = 3nd^ / 1/2 exp {_ 3^2 / }> a is
related to (1/2 - x) by the following equation (Flory 53):
(9/2ti) 3/2 V 2
1 o
{l - x)
3/2
(4.24)
For the case of constant segment density distribution with the
adsorbed layer thickness 6, Fisher gave the following expression (Fisher
58):
AGM = 4naw2N^ (v^/V^H^ - x) kT (1 - d/26)2 (4.25)
where w = weight of stabilizing moieties attached per unit area, =
Avogadro's number, and are the molar volumes of the solvent and
polymer, respectively, and = V^/M is the partial specific volume of
polymer.
From the above two equations, it is clear that the free energy of
mixing (hence, steric stabilization) is controlled by the following
factors: (i) particle size a, (ii) amount adsorbed (w or v), (iii)
solvent quality as determined by x or a, (iv) molecular weight of the

107
polymer (through r.m.s. end to end distance or 6, the adsorbed layer
thickness), and (v) segment density distribution.
The effect of the above parameters on the interparticle particle
interactions will be reviewed in detail later. From Equations 4.23 and
4.25, it is clear that at 0 conditions (x = 0.5 or a = 1), the net free
energy of mixing is zero, and in a good solvent (x < 0.5 and a > 1),
repulsion prevails, while in a poor solvent, attraction between particles
leads to catastrophic flocculation.
Interpenetration Without Mixing (Elastic Repulsion)
This is the second term arising due to the compression of the
adsorbed layer upon closer approach of two coated particles during the
Brownian collision. This term becomes particularly important when the
separation distance is less than the adsorbed layer thickness, i.e., (L <
d). This leads to the reduction in the available volume to adsorbed
molecules, and hence, restricts the number of possible configurations for
adsorbed polymer chains. Thus, the loss in number of configurations on
the approach of the second particle can be related to the decrease in the
configurational entropy, AS, using the Boltzmann equation:
AS = k In (4.26)
Q(co)
where £1 is the total number of possible configurations when the second
interface is present at a distance d, and ft(®) is the total number of
possible configurations when the two particles are infinite distance
apart. Evaluation of Q(d), and £}(«>) is model dependent. Usually, the
effect of solvent is ignored. Mackor first attempted to evaluate these
terms assuming a simple model based on inflexible rods terminally

108
attached, but freely jointed to a flat interface (Mackor 51). Clayfield
and Lumb used Monte Carlo computations to calculate steric repulsion for
flexible, terminally attached macromolecules (Clayfield and Lumb 66).
Meier (Meier 67), and subsequently Hesselink (Hesselink 71a), applied
random flight statistics to evaluate loss in configurational entropy for
equal size loops and tails. A free energy change from the elastic
effects for two flat plates is given by:
AGvr = 2vkTV(d) (4.27)
where v is the adsorbed amount of tails (loops) per unit area, and V(d)
determines the distance dependence of the elastic free energy
contribution. The value of V(d) needs to be evaluated numerically for
the case of equal loops and equal tails. Approximate analytical
expressions are also available (Hesselink et al. 71b). Subsequently, the
effect of tail size distribution on the elastic free energy change has
been considered, and it was shown that, with the wider tail size
distribution, the repulsion curves becomes less steep (Hesselink 77).
Bagchi (Bagchi 74a,74b) has proposed that, at high surface coverage, the
adsorption layers will be dense and mixing will be improbable. On close
approach of the two particles, the polymer layers will be compressed with
the exclusion of solvent from the adsorption layer. This mechanism is
called the "denting mechanism." The denting hypothesis has been
criticized by Napper on the grounds that volume fraction of segments in
the interactional zone rarely exceeds 0.1, and denting puts severe
constraint on the conformation of stabilizing polymer, and hence,
interpenetration is strongly favored over denting (Napper 83).

109
To evaluate the distance dependence of the elastic free energy
contribution to steric stabilization, Hesselink assumed a theoretically
derived segment density distribution, while Bagchi assumed a constant
segment density. Evans et al. have shown that the elastic free eriergy
change for spherical particles is given by (Evans et al. 77):
AG^ = 2navkTSE (4.28)
where v = wNA/M, w = weight of stabilizing chains per unit area of
surface, M = molecular weight of the polymer, and SE is the distance
dependent function for elastic repulsion. The SE is a function of
adsorbed layer thickness 6 and the minimum distance between surfaces of
the spheres, and it is a very sensitive function of the form of segment
density distribution. The specific case of polystyrene latex spheres
carrying adsorbed poly (vinyl alcohol) and dispersed in water has been
considered by Smitham and Napper (Smitham and Napper 79). The effect of
the assumed form of the segment density distribution on the elastic free
energy repulsion is shown in Figure 4.2. It can be seen that, in the
region, 6 < d < 26, G(uniform) > G(Gaussion) = G(radial Gaussian) >»
Gg(exponential), and at d < = 6/3 G(exponential) > G(uniform).
Flory's classical lattice theory of polymer solutions has been used
to evaluate interpenetration and elastic interactions, and hence, has a
number of limitations discussed previously. The possible use of free
volume theories (Flory 69) to evaluate mixing interactions have been
considered by Evans and Napper (Evans et al. 77), but the mathematics is
quite involved. The separation of steric interactions into elastic and
mixing terms is open to criticism. To avoid this artifact, Dolan and
Edwards (Dolan and Edwards 74) proposed a theory based on a self-

no
10
at
% 5
0
Figure 4.2
1
The distance dependence of the steric
interaction energy for two equal spheres of
radius a, stabilized by polymer layers with
different segment density distribution
functions. (1) exponential; (2) constant;
(3) Gaussian; (4) radial Gaussian, d is
the minimum distance between surfaces of
the spheres, 6 is the barrier thickness,
and AG^ is the interaction energy (Smitham
and Napper, 1979).

Ill
consistent field approach. They evaluated the free energy term as a
configurational term by treating the interactions between all polymer
segments (i.e., intermolecular as well as intramolecular interactions).
Then, calculations are limited to the good solvent regime, and as
expected, predict strong repulsive interactions as predicted. Levine et
al. have applied the self-consistent field approach to treat polymer
adsorption process (Levine et al. 78). Scheutjeans and Fleer have
extended this approach to polymer adsorption (Scheutjeans and Fleer
79,80) and to calculate the potential energy diagrams for steric
interactions under variety of conditions (Fleer and Scheutjeans 86).
That is the topic of the next section.
The Potential Energy Diagrams
The two factors contributing to steric stabilization are the loss of
configurational entropy of two adsorbed polymer layers and the free
energy of mixing. To describe the interactions, it was assumed that the
particles were uniformly coated with the adsorbed polymer layer (i.e.,
complete surface coverages). This situation is valid at high polymer
concentrations.
Time Scale of Approach of the Second Interface
Generally, two cases are considered (a) full equilibrium and (b)
restricted equilibrium. (This situation is similar to constant potential
and constant change case with the electrostatically stabilized
dispersions.) In full equilibrium, the polymer is allowed to leave the
gap between two particles during the 3rownian encounter. Increased

112
polymer segment concentration in the overlap region puts severe
entropical restriction on the adsorbed polymer if it cannot leave the
gap. Polymer desorption, being a very slow process, is questionable to
what extent the full equilibrium condition is applicable. The time, xen,
it takes for two spherical colloidal particles of radius a to travel to
distance 6 corresponding to the thickness of an adsorbed layer during
Brownian collision is given by (Hesselink et al. 71b):
Ten = 3nna6^ / 2kT (4.29)
where the symbols have their usual meanings.
For example, in the case of 0.5 pm particles with the adsorbed layer
thickness of 50 nm dispersed in water (i.e., viscosity n = 1 cp), the
time of encounter is approximately 3 milliseconds. We can introduce the
ratio (Lyklema 85):
x
D = --en (4.30)
Xdes
where tdes equals the time of desorption. In practice, tdes is usually
very high, and depending upon conditions, it may be of the order of hours
(Cosgrove and Fergie-Woods 87). Hence, D « 1 means that the polymers
have no time to desorb. From the scaling theory, deGennes has proved
that in full equilibrium (i.e., D » 1), only attraction is found between
two particles, and hence, steric stabilization cannot be explained
(deGennes 82). This has also been proved from the self-consistent field
lattice theory (Fleer and Scheutjeans 86). In this case, desorption of
the polymer develops empty sites on the surface which can be bridged by
the polymer molecules adsorbed on the other particles. Therefore, the
case in which the amount of polymer is assumed to be constant during

113
particle encounter is more important (i.e.., D « 1). In this case,
"local" or "restricted" equilibrium is assumed, and the segment density
distribution and the distribution of trains, loops, and bridges adjust to
a variable particle separation. This assumption has been used by all
theories described earlier (e.g., Hesselink et al., Bagchi, Fisher,
etc.).
The Potential Energy of Interaction
The total interaction between two polymer-coated particles is given
by:
AGT = AGE + AGA + A3S (4.31)
where AGE is the electrical double layer repulsion, AGA is the Van der
Waal's attraction, and AGS is the steric interaction energy which
consists of two additive contributions, i.e.,
AGS = AG^ + AG14 (4.32)
where AGVR is the elastic or volume restriction on entropic contribution,
and AG*4 is the mixing energy term. Both AGE and AG^ are modified due to
the presence of the adsorbed polymer layer (Void 61, Vincent 74). First,
we will consider a case where AGE is negligible, and the adsorbed polymer
layers have no effect on AG*^. (This approximation is usually valid as
the concentration of segments in the adsorbed layer is generally quite
low, and the Hamaker constant of the adsorbed polymer layer is similar to
that of the dispersion media—e.g., see Bagchi 74a,74b). Under these
assumptions, the total interaction energy as a function of distance
between two flat plates is given by the following equations:

114
. „T . „VR _M 131
AG = AG + AG —
12nd
(4.33)
and substituting for AGVR and AG** from the HVO theory, we obtain
(Hesselink 77):
A
AGT = 2vkTV(d) + 2(2n/9)i/2 v2kT(a2-1) M(d) ^ (4.34)
12nd
Thus, the main parameters determining the total free energy change of
interaction, AGT, on the approach of two flat plates, each carrying an
adsorbed polymer layer are:
1. The adsorbed amount of polymer through v, where v is the number
of tails (loops) per unit area and w = vM/Nav, where w is the
adsorbed amount of polymer per unit area, and M is the
molecular weight and Nav is Avogadro's number.
2. The dimensions of the adsorbed tails: where is the mean
square end to end distance of the tail in solution.
3. The solvent quality x as expressed in terms of the expansion
factor a.
4. The mode of attachment of the polymer, i.e., tails or loops.
V(d) and M(d) are the volume restriction and osmotic repulsion
functions that give the dependence of AGT on the distance d
between the surfaces. M(d) and V(d) are extremely dependent on
the mode of adsorption (i.e., conformation of the adsorbed
polymer). Generally, the values of M(d) and V(d) are tabulated
(Hesselink et al. 71b), and simple analytical approximate
expressions are also available (Hesselink 77) and dependent on
the form of the segment density distribution assumed.

115
Figure 4.3 shows the schematic of the total interaction energy as a
function of distance of separation between two plates with the adsorbed
polymer layer. Various contributions to the total interaction energy are
also shown. At large values of d > 2L (where L is the adsorbed layer
thickness), the Van der Waal's attraction predominates, but at close
approach, very steep repulsion prevents further approach. The mixing
interactions, AG**, starts at higher values of d (usually L <, d £ 2L) than
elastic interaction, AG^ (usually L £ d). This is expected since the
mixing interactions start as soon as the polymer layers overlap, whereas
the elastic repulsion starts after further closer approach of the two
surfaces. Unlike the total interaction energy plots for the
electrostatically stabilized system, the sterically stabilized system
does not display the characteristic maxima and primary minima.
Sterically stabilized systems display only one minima, AGmin, over the
whole range of d and the stability/flocculation behavior is controlled by
the magnitude of AGmj^ Depending on the shape of the total interaction
energy curve, the dispersion stability can be classified into various
categories. The sign and magnitude of various contributions to the total
interaction energy diagram are determined by the parameters, such as
solvent quality, molecular weight, polymer concentration in solution,
etc. Sterically stabilized dispersions can be classified into the
following general categories:
Thermodynamically Limited Stability
The stability/instability transition of the sterically stabilized
dispersions provides an insight about the thermodynamic factors
controlling the stability. This type of steric stabilization is

116
The free energy of interaction between particles covered
by equal tails (f) and equal loops (a). For particles
covered by equal tails, (b) gives the volume restriction
effect and (c) the osmotic repulsion; (f) is the
resultant of adding (b), (c), and (e) (Hesselink. et al.
1971).
Figure 4.3

117
exhibited by colloidal particles with a firmly anchored polymer layer of
sufficient thickness. In this case, the Van der Waal's interactions
between the particles are unimportant, hence, the size of the particles
has no significant effect on the stability. The stability behavior is
essentially determined by the polymer solution thermodynamics, i.e.,
entropic and enthalpic interactions between the adsorbed polymer layer
during the Brownian encounter. Hence, it is not surprising that the
strong correlation exists between the CFT (i.e., critical flocculation
temperature), CFV (i.e., critical flocculation volume of the non¬
solvent), and the theta conditions for the polymer chains in solution.
As discussed earlier, the theta condition represents the thermodynamic
limit of stability for an infinite molecular weight polymer. Above 0
temperature (or in a poor solvent with x > 0.5), there is a net segment-
segment attraction, and below 0 temperature (or in a good solvent with x
< 0.5), net segment-segment repulsion exists. At the 0 temperature,
there is no net free energy change associated with the overlap. The
effect of the solvent-segment interaction parameter x on the potential
energy diagram is shown in Figure 4.4.
As shown in Figure 4.4A, in a poor solvent, the depth of the
potential energy minima AGm^n is sufficient to cause the flocculation of
the particles. In this case (note that the depth of AG . is much larger
min
than the Van der Waal's attraction at a distance of separation of the
order of 2L), the sign of mixing term is negative since there is a net
segment-segment attraction. (This can be also seen from the Equations
4.23 and 4.25, when for the poor solvent, a < 1 or x > 0.5 and AG1* is
negative.) At the 0 conditions, Figure 4.4B, AGM = 0 (since a = 1 at x =

POTENTIAL ENERGY OF INTERACTION (J)
POOR SOLVENT THETA SOLVENT GOOD SOLVENT
agvr
Schematic illustration of the effect of segment-solvent interaction parameter,
X, on the potential energy diagram (A) poor solvent, x > 0.5, (B) theta
solvent, x = 0.5, and (C) good solvent, x < 0.5.
Figure 4.4
118

119
0.5), and the elastic contributions impart stability to the dispersion.
The elastic contribution, which is related to the decrease in the
conformational entropy of the adsorbed polymer molecules upon approach of
the second particle, is always repulsive. In a good solvent, Figure
4.4C, the mixing and the elastic interaction energy contribute to steric
stabilization.
It has been experimentally observed that the CFT is independent of
the molecular weight of the polymer (if sufficiently thick adsorbed layer
is present) and the chemical nature of the polymer (as CFT is related to
segment-solvent interactions) (e.g., see Napper 83). Exceptions to the
above obeservations have also been reported in the literature (Napper
83). In this case, the stability in worse than 0 solvent has been
observed and referred to as "enhanced steric stabilization." The exact
mechanism is not clearly understood but has been related to the loop-tail
adsorption of the polymer.
Non-thermodynamically Limited
In this case, the flocculation is observed in better than 0
solvents. The molecular weight of the polymer and the Van der Waal's
attraction play an important role in determining the flocculation
behavior. Usually, large particles coated with the low molecular weight
polymers are less stable, i.e., the relative thickness of the adsorbed
layer (6/a) is an important parameter. The effect of molecular weight on
the total interaction energy plots is shown in Figure 4.5.
If the adsorbed layer thickness is relatively small, then, an
appreciable attraction exists between two particles. But with increase
in the adsorbed layer thickness, the depth cf
is reduced

Q mln/kT
120
Figure 4.5 The free energy of interaction of polystyrene latex
particles stabilized by poly (vinyl alcohol) according to
Hesselink., Vri j, and Overbeek (1971); stabilizer
molecular weight 1, 8,000; 2, 17,000; 3, 28,000;
4, 43,000 (Tadros, 1982).
Figure 4.6 Plots showing the effect of particle size and adsorbed
amount on the depth of the minima in the total potential
energy of interaction.

121
considerably. Hence, above certain critical molecular weight (or the
absorbed layer thickness), the dispersion stability will be
thermodynamically limited (i.e., determined by the solvent quality). The
effect of particle size and the adsorbed amount (or the thickness) on the
depth of the minima in the total potential energy of interaction is shown
in Figure 4.6. At a given distance of separation, the depth of AG .
increases with increase in the particle size since the Van der Waal's
attraction increases with the size of the particles. Similar trends can
be expected with the increase in the effective Hamaker constant of the
materials involved. The critical molecular weight of the polymer
required to stabilize the dispersion will increase with the Hamaker
constant of the material (e.g., silica < alumina < zirconia < metal
powders).
It should be noted that the origin of the minima in the total
interaction energy curves is different in the above two cases. In the
case of a thin adsorbed polymer layer, the depth of is controlled
by the Van der Waal's interactions (through thickness and particle size),
while in the case where flocculation is induced due to thermodynamic
reasons (i.e., poor solvent), the magnitude of AGm^n is usually large.
If the flocculation is induced by thermodynamic factors, then, the
flocculation behavior can be reversed, for example, by decreasing the
temperature or adding more solvent to the dispersion, and volume fraction
of solids, 0, does not play any role. If the depth of AGmin is
determined by the Van der Waal's attraction, then, the resulting
flocculation is usually weak since the magnitude of AG is usually
small. In this case, the existence of a critical volume fraction solids

122
0C has been observed below which flocculation cannot be observed (Long et
al. 73). The existence of a critical flocculation volume fraction solids
0C has been related to the configurational entropy loss of the particles
due to flocculation (Long et al. 73). The change in the configurational
entropy due to flocculation (i.e., AS = S^, - SJ. J decreases
' floe dispersed
with increasing volume fraction of solids. At the critical volume
fraction 0C, the decrease in the configurational entropy is balanced by
the gain in energy due to flocculation.
Bridging Flocculation
During a Brownian collision, an adsorbed polymer chain from one
particle and can attach itself to the empty sites on the other particle
as shown in Figure 4.7. This process of destabilization of dispersions
at low polymer concentrations has been called "bridging flocculation."
If the dimensions of the flocculent polymer molecules are comparable to
the dimensions of colloidal particles, then, a polymer molecule can
attach itself to several particles. This phenomenon was first pointed
out by Ruehrwein and Ward (Ruehrwein and Ward 54). The process of
bridging flocculation is employed in various operations, such as water
purification, selective flocculation in the mineral industry, etc. It
has been experimentally found that, for a given polymer, there is an
optimum concentration beyond which poorer flocculation is observed. This
observation is quite consistent with the bridging hypothesis, which
requires that the particle surface should be partially covered with the
adsorbed polymer, so that attachments with other polymer segments can be
formed. Thus, the fractional surface coverage is one of the important

123
Schematic illustration of bridging flocculation with
adsorbed polymer.
Figure 4.7

124
parameters controlling the phenomenon of bridging flocculation.
Typically, high molecular weight polyelectrolytes of the same sign as the
dispersed particles or polyelectrolytes of opposite sign of the dispersed
particles are used in application. This area has been reviewed
extensively by Vincent (Vincent 74 and Vincent and Whittington 81). Non¬
ionic PVA was used in this study, hence, we will focus on the effect of
non-ionic polymers on bridging flocculation.
The important parameters controlling the stability behavior are:
(i) polymer concentration, (ii) molecular weight and structure, (iii)
electrolyte concentration, and (iv) particle concentration.
The following general trends are observed (Vincent 74). The optimum
polymer concentration decreases with increase in the molecular weight of
the polymer, and the concentration range over which the increased
flocculation efficiency is observed decreases with increase in the
molecular weight (Vincent 74). The chemical structure of the polymer
does not play an important role with regard to their flocculating
capacity. (It was observed that homopolymer PSO or co-polymer PEO-
pclypropelene oxide had no first order effect on flocculating capacity.
Also, PVA88 and PVA98 were equally effective in flocculating
dispersions.) But the chemical nature of the polymer has a pronounced
effect on the stability at high polymer concentrations. (This effect may
be due to the stronger anchoring and improved adsorption in the case of
co-polymers.) (Vincent 74).
In the case of charged particles, it has been observed that the salt
additions favor flocculation. The effect of ionic strength on
flocculation behavior is shown in Figure 4.8. With increasing ionic

125
Low Ionic Concentration
High Ionic Concentration
Figure 4.8
The effect of thickness of the electrical double layer on
bridging flocculation.

126
strength, the electrical double layer thickness decreases (i.e., the
Debye-Huckel parameter, h-1 decreases, as discussed in Chapter II).
Hence, the distance of closest approach between two spherical particles
(approximately 2/h) will also be decreased. To form polymeric bridges
between two approaching particles, the extension of the adsorbed polymer
molecules (tails or loops) should be greater than the thickness of the
electrical double layer. At higher ionic strengths, the electrical
double layer is less extensive and bridging is more likely.
The above trends with regard to the efficiency of flocculation are
measured using various experimental techniques, such as light scattering
(turbidity), refilteration rates, sedimentation behavior, etc., and
experiments are usually carried out at low particle concentrations.
Additional effects due to possible differences in the method of mixing,
rates of polymer adsorption and particle collision in the system studied
makes quantitative comparison impossible. Fleer has compared the
relative merits of two methods of mixing, i.e., the so-called "one
portion" and "two portion" methods (Fleer 71). In the former method,
dispersion is added to the polymer solution all at one time. In the
latter method, a given portion of the dispersion is added to the polymer
solution (the polymer concentration and the volume of the dispersion
chosen so as to give complete coverage of the particles) and mixed
together, and the remainder of the dispersion (containing bare particles)
is added subsequently. Thus, in the two portion method, bridging
flocculation takes place between sterically stabilized particles and bare
particles. It has been speculated that this method may improve the
reproducibility of the flocculation process, and hence, process control

127
at the same added polymer concentration (Lyklema 85). These two ways of
mixing can lead to a dramatic difference in the final flocculation
results. This shows that the kinetics play an important role in the
flocculation experiment.
Kinetic Aspects of Bridging Flocculation
The rate of coagulation in the case of electrostatically stabilized
dispersions was considered in Chapter II. At the moment of addition of
polymers to a dispersion, several rate processes are initiated (Figure
4.9) (Akers 75, Gregory 88).
(i) Mixing of polymer molecules among the particles, (ii) Diffusion of
polymers and attachment to the particles, i.e., adsorption.
(iii) Rearrangement of the adsorbed polymer chains towards equilibrium
arrangement (reconfiguration). (iv) Collisions between coated particles
resulting in aggregate formation.
In the case of polymer flocculation, the rate of polymer adsorption
should be considered. Generally, two limiting cases are considered: (i)
the time required (or rate) for adsorption and reconformation is longer
than the collision between two particles and (ii) the adsorption and
reconformation processes is much faster than the collision between
approaching particles (Gregory 88).
It is usually assumed that the added polymer distributes itself
instantly and uniformly throughout the dispersion.
To estimate the rate of polymer adsorption, Gregory treated the
adsorption of polymer molecules as a process of heteroflocculation and
assumed that each polymer particle contact leads to permanent attachments

128
O
O
O
rS\j
2as-
'vA-
*2
oS*
O ^ ¡¿y
h o-i»
I oV 4 ^
/
/
y v
Figure 4.9
Schematic diagram showing mixing, adsorption, and
flocculation upon addition of polymeric flocculent
(Gregory, 1988).

129
(Gregory 88). Assuming that the particles and polymer molecules as
spherical particles with radius a-| and a2 and the rate constants for
particle-polymer molecule and particle-particle collisions are given by
the Smoluchowski' s theory. The ratio between the adsorption time, t^,
and collision time, tf, is given by (Gregory 88):
tft/tf = -4 [In (1 —f) ] a.^/ (a1 + a2>2 (4.35)
where f is the fraction of the added polymer adsorbed at time t^. It can
be shown from the above equation that, for very different values of a-|
and a2, the adsorption step is faster (i.e., tf » t^). Also, if t^ »
tf, there might be an observable time lag between the addition of polymer
and the onset of flocculation.
If one assumes that, at the moment of polymer adsorption, the
conformation of the adsorbed polymer is similar to its random coil
solution conformation, then, the time required for a polymer to achieve
its equilibrium configuration is largely a matter of guesswork. Times of
the order of hours or days may be involved (Cosgrove and Fergie-Woods
87). The reconformation rate will a have significant effect on bridging
flocculation. As discussed in Chapter II, at extremely low polymer
concentrations at equilibrium, the adsorbed polymer has flatter
conformation (i.e., short sizes of loops and no tails), and hence, will
be less effective than random coil adsorption (Scheutjeans and Fleer
79,80). This non-equilibrium random coil conformation of the adsorbed
polymer will extend farther from the surface and will also cover less
surface area and the particle may enhance the bridging flocculation. It
can be expected that, at high volume fraction of solids in suspension,
the rate of collisions between particles will be high, i.e., there may be

130
a considerable number of collisions between particles in the time period
of the reconformation step. At low particle concentrations, the
reconformation step may be faster and the extent of flocculation will be
less. Although there is very little experimental evidence on this point,
it can be concluded that "non-equilibrium flocculation" may be occurring
at high particle concentrations, where the time of flocculation is very
short (usually a fraction of a second) (Gregory 88).
Kinetics of Flocculation
We have already discussed the aggregation kinetics of
electrostatically stabilized dispersions in Chapter III. Smoluchowski's
second order kinetic equation for irreversible aggregation in the absence
of long-range particle interactions is given as:
dN
1 = -KN 2 (4.36)
dt 1
where K is rate constant for the process, and dN-j/dt is the rate of
disappearance of primary particles N-j. Bridging flocculation is also an
example of irreversible flocculation, and the above equation is modified
and given as follows (Smellie and LaMer 58, Healy and LaMer 62):
dN
â– â–  1 = -0 (1-0) KN 2 (4.37)
dt 1
where 0 is the fraction of surface covered by the polymer. LaMer called
E = 0 (1-0) an efficiency factor. It represents the probability that the
surface of one of the particles is covered by polymer (0) and that the
other particle has bare patch (1-0) during collision. In this
derivation, it has been assumed that the polymer adsorption reaches
steady state very rapidly compared to the rate of particle collision.

131
From the Equation 4.37, it is clear that at extremely low fractional
coverates (i.e., 9-*0), and near complete coverages (i.e., 0->1), dN-|/dt
will approach zero. This can be represented as follows:
dN1
—— 0 as 9 ■* 0 as 9 -* 1 (4.38)
and that dN-|/dt is maximum at 0 = 0.5 (i.e., half surface coverage).
Usually, 0 is defined as 0 = (adsorbed amount)/(plateau adsorbed amount),
i.e., fraction plateau coverage. By defining 0 this way, the
conformation change of the adsorbed polymer with the surface coverage (or
polymer concentration) is ignored. Hogg has modified the collision
efficiency factor to account for the reorientation of particles during
collision (Hogg 84). This model predicts that the collision efficiencies
are very high (close to approximately 100%) over a broad range of surface
coverages (0 = 0.1 to 0.9, typically) for the flocculation of large
particles with low molecule weight polymers (i.e., « a, where
is the root mean square radius of gyration of the polymer molecule in
solution and a is the particle radius.)
The observed initial rate of flocculation, K, is often higher than
the rate predicted by Smoluchcwski for diffusion-controlled aggregation
(i.e., K = 8kT/3n, where n is the viscosity of the dispersion media).
This has been attributed to (1) reduced hydrodynamic interactions between
the particles and (2) an increase in the effective collision radius of
the particles when they have an adsorbed polymer layer. Walles has shown
that the ratio of number of collisions per unit time 2, between the bare
and covered particles, to the number of collisions, ZQ, between bare
particles is given by (Walles 68):

5/a
Figure 4.10 The effect of solids loading 0 and 6/a on collisions Zf/ZQ (Walles, 1968).

133
Z (2 + 6/a)2 (1.69q - 2)4 , _-1/3
—— = — and q = 0 (4.39)
o 4 (1,69q - 2 - 6/a)4
This ratio increases with the adsorbed layer thickness, 6, for a given
particle radius, a and solid volume fraction, 0 (Figure 4.10).
The Potential Energy Diagrams for Bridging Interuarticle Interactions
At incomplete surface coverage, the free energy of interaction is a
compromise between an attractive term due to polymer bridging and a
repulsion which is caused by segment overlap and conformational entropy
loss. There are essentially two ways to calculate these interactions
(Vincent and Whittington 81). In the first case, polymer is assumed to
be adsorbed in its equilibrium conformation on one surface, and the
change in free energy is calculated when a second surface is brought to a
distance d < L where L is the adsorbed layer thickness. In the second
approach, a polymer molecule is generated between two surfaces separated
by a distance d. Fleer has calculated the total interaction energy
between covered and bare surfaces as follows (Fleer 71). He assumed the
polymer is adsorbed in train-loop conformation (i.e., tails were
ignored). The total interaction energy is divided into three
contributions: (a) the adsorption attraction term, G^, (b) the
configuration repulsion free energy, G1^3, and (c) the free energy change
due to mixing, G1*. To calculate G^, he assumed that all segments in
loops in the region (L-d) are considered to be adsorbed on the bare
surface in trains (Figure 4.11). Then, G^ is given by:
G^ = -N€
(4.40)

134
Figure 4.11 Schematic representation of the approach of a second
(uncovered) particle to a covered one. It was assumed
that, at large interparticle distance H, the number of
segments which adsorb on the (originally) bare particle
per unit area is equal to the number of segments per unit
area which would lie beyond Hi in the absence of the
second particle (i.e., shaded area) (Fleer, 1971).

135
where N is the number of segments per unit area in the region (L-d), and
G is the adsorption energy of the segment on surface. To evaluate, N, he
assumed the exponential segment density distribution, p(x), as given by
Hoeve (Hoeve 65,70):
OO
N = J p(x) dx (4.41)
b
for homopolymer adsorption. At very short distances, the maximum
adsorbed amount on the second surface was assumed to be a fraction of the
adsorbed amount on the first surface.
The repulsive contribution to which results from the reduction
in the configurational entropy of the chains for d < L was calculated
using Hesselink's equations (Hesselink 71). Hesselink et al. have
derived the expression for the free energy of a loop and bridge on flat
plate as a function of distance of separation between two flat plates.
Schematic processes of bridging by a single loop consisting of i segments
is shown in Figure 4.12.
The free energy associated with the above process is given by:
OO
= J [2vq (iH,d) - (i,“)] di (4.42)
i=0
where Hi is the number of loops of size i per unit area and vj-, (ÍH»d) is
the configurational free energy of a bridge of in elements at an
interparticle distance d. (i,°°) is the configurational entropy of an
unconfined loop of i elements. The total free energy of interaction
between coated and uncoated plates as a function of distance of
separation in the absence of electrostatic interactions is shown in
Figure 4.13, where

136
Figure 4.12 Schematic representation of the bridging process, (a) At
large distances, a loop of i segments has its unpurturbed
configuration. (b) After adsorption of the first
segment, two bridges of i/2 segments each are formed.
(c) At shorter distances, two bridges of in segments and
a train of segments adsorbed on the second surface
(Fleer, 1971).

137
Figure 4.13 The total free energy of interaction between coated and
uncoated plates as a function of distance of separation
(Fleer, 1971).

138
Gbr(d) = + G^h + Ga (4.43)
and 0a is the Van der Waal's attraction energy. It can be seen that
there is a deep minima in the Gbr(d) plot, indicating strong
flocculation. The magnitude, Gm¿n and the position d ^ of this
potential energy minima will have an important effect on structure and
the properties of the floes formed. A perturbatory technique of
monitoring suspension structure, such as rheological measurements, will
be influenced strongly by these parameters.
In the second approach to calculate the free energy of interaction,
the polymer molecule is generated between two confining surfaces at
separation d. In the simple case of non-adsorbing polymer between two
plates, the decrease in the configurational entropy of constrained
polymer molecule compared to an equivalent unconstrained polymer leads to
repulsion. The distance dependence of such repulsive interaction energy
is given as G1^3 approximately d~^ from random walk and self-avoiding walk
analysis between two plates (Richmond and Lai 74, Hesselink 69). If the
polymer segments can adsorb on the approaching surface, the bridging of
two surfaces by the polymer chain leads to attraction. Properties, such
as the probability of bridging by the chain, the mean bridge length,
etc., of the adsorbed chain have been determined as functions of segment-
surface interaction energy and separation distance from the Monte Carlo
simulation (Clark and Lai 82). Scheutjeans and Fleer have extended their
polymer mean field adsorption theory to calculate the free energy of
interaction for the case of bridging flocculation (Scheutjeans and Fleer
85; Fleer and Scheutjeans 86). They have considered two cases (a)
restricted equilibrium—when the adsorbed amount remains constant and (b)

139
full equilibrium—when the polymer can freely enter or leave the gap
between the surfaces. First, the segment density distribution in the gap
was computed. From the known segment density distribution, the total
free energy of interaction as a function of distance of separation was
determined. deGennes also has employed scaling concept to calculate the
energy of interactions at partial coverage (deGennes 82,87). The effect
of various parameters (molecular weight, adsorbed amount, etc.) on
bridging flocculation will be considered in the results and discussion
section. The effect of these parameters on the rheological behavior of
suspensions will be discussed with the help of total potential energy
diagrams.

CHAPTER V
STRUCTURE OF SUSPENSIONS
Introduction
The total potential energy of interaction between two colloidal
particles was reviewed in the last chapter. In the case of
electrostatically stabilized dispersions (with no added polymer) under
the conditions of high zeta potential and low ionic strength, the
electrostatic repulsion leads to ordering of the particles in suspension.
Ordering has been observed in suspensions of spherical monosized,
particles (e.g., lattices, silica, etc.) and results in bright iridescent
colors due to Bragg diffraction when the interparticle spacing is in the
range of wavelength of visible light (e.g., Forsyth et al. 78, Hachisu
and Takano 82, Hiltner and Krieger 69, Krieger and O'Neill 68, Ohtsuki,
et al. 81, Takano and Hachisu 78, Tomita et al. 78, Yoshimura and Hachisu
83). This ordering of colloidal particles resembles liquid to solid
transition of atomic systems and has been analyzed using statistical
mechanics and Monte Carlo approach (Snook and Van Megan 82, Van Megan and
Snook 84). It is important to note that the above techniques are valid
under the equilibrium conditions. The order-disorder transition also has
been observed in the case of sterically stabilized dispersions where the
repulsion arises due to overlap of the adsorbed polymer layers.
For suspensions in which the potential between particles is strongly
attractive (e.g., Van der Waal's attraction with no electrostatic
repulsion, polymer induced bridging flocculation at low polymer
140

141
concentrations, particles coated with adsorbed polymer in a poor solvent,
etc.) and the depth of the potential energy minima > = 5 kT, encounters
due to Brownian motion lead to aggregation of particles. This process of
aggregation is random process, and the aggregate structure is not in
thermodynamic equilibrium. The aggregation process is relevant to many
scientific phenomena, e.g., dendritic growth, gel formation and
flocculation of particles, kinetics of polymerization, tumor growth, etc.
The resulting non-equilibrium structures (for example, dust, soot, cell
colonies, electrolytic deposition, etc.) are not merely amorphous blobs
but exhibit symmetry despite their random growth process (Sanders 86).
These structures produced by the short-range forces are statistically
well-defined on larger scales, and the long-range structure is
independent of the details of the interparticle potential. These
structures can be described using Mandelbrot's formulation of fractal
geometry (Mandelbrot 82).
Fractal Geometry
The fractal dimension, df, of an object may be defined by the
following equation:
M a ldf (5.1)
where M is the mass of the object and 1 is a length which specifies the
characteristic size of the object. For Euclidian objects, df, equals the
dimension of the space, d (for example, in three dimensions, the mass of
a sphere M a , where R is the radius of the sphere and d = 3).
Fractals are objects for which df is less than the dimensions of the
space (i.e., df £ d). For example, the mass of the polymer molecule

142
generated by random walk in three dimension is proportional to the square
of the r.m.s. radius, i.e., M a , hence, it is a fractal with df =
2.0. Fractal objects are scale invariant or self-similar (i.e., objects
which look the same under different magnification) and do not exhibit any
characteristic length scale. Since df has many properties of a dimension
but is often a fractional, it is called the fractal (broken) dimension.
The fractal dimension of objects can be determined in several ways
(Meakin 86). One way is to measure the radius of gyration, Rg, of the
object as function of mass, M, as it grows. Alternatively, M and Rg
might be measured for an ensemble of objects of different sizes, and df
is determined by the relation M a Rgdf. The above mass-length
relationship is observed over finite length scales in real systems. For
example, in the case of colloidal aggregates, the scaling relation is
observed between the lower length scale corresponding to the particle
diameter and the upper length scale corresponding to the diameter of
aggregate. Knowledge about the fractal geometry of colloidal aggregates
is important since the physical behavior (e.g., rheology, sedimentation
behavior, etc.) of dispersions will be directly related to its fractal
geometry.
The fractal nature of the colloidal aggregates has been confirmed
from x-ray scattering (e.g., see Schaefer et al. 84, Dimon et al. 86),
small angle neutron scattering (Weitz et al. 85), direct observation
under transmission electron microscope (Weitz and Olivera 84), and
dynamic light scattering experiments. The fractal dimensionality is also
important in studying the kinetics of aggregation since knowledge of df
is essential in formulating the appropriate Smoluchowski equation for the

143
aggregation kinetics (Jullien and Kolb 84a, Jullien et al. 84b).
Kinetics give the cluster-size distribution as a function of time and
will be discussed later.
Models of Aggregate Formation
Computer simulation technique has been extensively used to study
aggregation and growth processes. This technique to generate ramified
structures can be broadly classified into two categories: (a)
simulations describing non-equilibrium irreversible growth processes
(such as flocculation) and (b) equilibrium models such as percolation to
describe critical phenomenon (for example, phase transition, sol-gel
transition, magnetization (M) near the curie point, etc.). In this
section, we will briefly describe various models proposed to described
the random growth processes.
Eden Growth
In Eden growth process, one starts with a seed (one lattice site on
a lattice) and randomly picks and occupies one of the neighboring sites
(Eden, 61). This process is repeated over and over, and the resulting
structure is shown in Figure 5.1. The structure grown in this fashion
has uniform interior and smooth surfaces relative to their radius, and
hence, is not a fractal object. This fact is important since it
demonstrates that not all random growth processes lead to ramified
fractal objects. The Eden growth is realized when growth occurs from
monomers with large functionality. This growth model has been proposed
by Keefer and Schaefer for the base-catalyzed polymerization of silicic
acid. They have modified the Eden's model (called Poisoned Eden growth

144
Figure 5.1
Eden cluster produced by monomer-cluster growth (Schaefer
et al., 1984).
Figure 5.2 An aggregate grown by the DLA process (Sanders, 1986).

145
model) to account for the change in the mean functionality and the
variance of the monomers (Keefer and Schaefer 86).
Diffusion Limited Aggregation (DLA)
Witten and Sanders developed a model for diffusion-limited
aggregation process which produces complex random dendritic structures
(Witten and Sander 81, Meakin 83a). This model provides the basis for
obtaining a better understanding of a variety of diffusion limited
processes such as random dendritic growth and flocculation of colloidal
particles, etc. The growth starts from a single immobile growth site at
the origin of a lattice. A single particle is then added at some random
site at large distance from the origin and allowed to random walk (i.e.,
diffuse) until it comes into contact with the seed and sticks. Then,
another random walker is released and allowed to walk until it sticks to
either of the previous particles, and this process is continued. Figure
5.2 shows an example of an aggregate grown by this model. This model has
been extensively investigated and aggregates grown show scale-independent
correlations over an arbitrarily large range of distances with the
fractal dimension of df = 5/3 for two-dimensional and df = 2.5 for three-
dimensional clusters. The clusters grown using the DLA technique
resemble objects such as snow flakes, coral, etc. In the DLA model, only
one particle is allowed in the vicinity of growing cluster which is
unrealistic for many colloidal systems. A variation of DLA model has
been proposed by Meakin and Kolb et al. and will be discussed next
(Meakin 83a,85 and Kolb et al. 83).

146
Cluster-Cluster Aggregation (CCA)
This model is more relevant to the study of flocculation of
colloidal particles, hence, it will be discussed here in more detail. In
this model, nG, identical particles of radius Rq and mass m=1 are
initially distributed randomly in a volume VQ. The initial concentration
P0 â–  no/vo is considered sufficiently low so that the average distance
between the particles is large. The particles are allowed to move
randomly (i.e., by Brownian motion) and independently of one another.
During the course of diffusion, if two particles come into contact, they
will stick together due to strong short-range attractive interactions.
In this way, two-particle clusters are generated, and these will diffuse
randomly in a similar manner. The possibility of rotation of clusters is
neglected. Also, some kind of mobility criterion can be introduced (for
example, the mobility of cluster may vary with their mass) to mimic
experimental physical conditions. If a cluster contacts another cluster,
then, the contacting clusters can merge to form a single cluster; that
is, a cluster of mass m-| colliding with a cluster of mass m2 forms a
single new cluster of mass (mi + m2). In this process, the clusters
formed never change their internal structure and all contacts are rigid.
This process (if continued) results in a single large cluster (see Figure
5.3). The clusters grown with this model are also fractal objects with
fractal dimension of df = 1.45 - 1.5 for two-dimensional simulations and
df = 1.75 for three-dimensional clusters (Meakin 83a,83b, Kolb et al.
83). A number of modifications of the basic CCA model have been already
carried out. The effect of sticking probability on the fractal
dimensionality has been investigated by Jullien and Kolb and have shown

147
Figure 5.
An aggregate grown by the CCA process (Jullien et
al. 1984).

148
that a very small sticking probability raises the fractal dimensionality
from about 1.45 to 1.55 in two-dimensional simulations and from about
1.75 to 2.0 in three-dimensional simulations (Jullien and Kolb 84a).
Cluster formation process with a finite sticking probability is known as
a Reaction Limited Cluster Aggregation, RLCA, process. Reorganization of
the aggregate has also been considered, but no significant change in the
effective fractal dimensionality was observed (Meakin 86a).
Hierarchial Model
Another simpler model used to investigate the structural properties
of clusters is known as the Hierarchial Model (Jullien et al. 84b). In
this model, one starts with N0 = 2^° particles. In step 1, N-| = N0/2
clusters are formed and each cluster consisting of two particles is
formed by a process similar to DLA, i.e., one particle is kept fixed at
the origin and the other particle is introduced at a random position far
away from the particle and random walk of the second particle continues
until it meets the immobile particle. In the next step, N'2 = N-¡/2
clusters are formed and each cluster has four particles. Four-particle
clusters are formed by contact between two two-particle clusters during
random walk. Eventually, a single cluster with N0 particles is formed
having effective fractal dimensionality df approximately 1.42 for two-
dimensional simulations.
Kinetics of Aggregation
The above described models, such as DLA and CCA deals with the
structural (or morphology) properties of aggregates. The dynamics of
cluster formation is also a very important problem and intensive research

149
efforts have been directed towards that goal (e.g., see Family and Landau
84, Pietronero and Tosatti 87). Essentially, there are two different
ways to study the growth process. The first approach is a kinetic
approach based one the Smoluchowski's equation. With this approach,
under certain assumptions, the change in the cluster mass (size)
distribution as a function of time can be described. The second approach
is based on dynamic computer simulations to describe the time dependent
cluster size distribution (e.g., DLA, CCA models). It is important to
distinguish between two regimes of aggregation: (1) flocculation regime,
i.e., where growing clusters stay far apart and (2) kinetically induced
gelation phenomenon—where large network structures are formed (Jullien
et al. 84b). This difference between these two cases can be seen more
easily from a quantity known as the effective average cluster density,
P
eff
N (R)d
c
V
m
d-d
‘V
(5.2)
o
where pQ = n0/V0 is the initial particle concentration, i.e., the total
number of particles, nQ, contained in a volume VQ. Nc is the total
number of clusters at a given time with average radius R and mass m.
For fractal objects, in = Rdf where df is the fractal dimension and d is
the dimension of the space (df < d). From the above definition, it is
clear that the effective cluster density increases with increasing
average cluster mass. Thus, no matter how low the initial concentration
pQ, eventually the cluster covers all space, i.e., Peff » 1. This case
is known as the kinetically induced gelation phenomenon. The gelation
regime also exhibits scaling properties different from the scaling

150
properties of the flocculation regime. These two limiting scaling
regimes are separated by a crossover for intermediate concentration.
Smoluchowski1s Equation
Kinetics of aggregation can be studied from the Smoluchowski's
equation as follows. These are two processes contributing to the kinetic
evolution of the cluster size distribution (a) the collisions of cluster
i and cluster j leading to the formation of cluster (i + j) and (b)
dissociation of a large cluster into two smaller ones. The
Smoluchowski's coagulation equation is given by:
E
i+j=k
k. .C.C. -
i 3
C„ .E k^.C.
k j=1 kj 3
(5.3)
where Cj^ = Cfc(t) is the time dependent concentration of clusters of
discrete mass k (k * 1, 2, . . .), and ¿^(t) represents the time rate of
change of concentration Cfc(t) and kjj is concentration-independent
collision kernel which determines the aggregation mechanism (Brownian
motion, gravitational settling, etc.). The above equation represents
that the population of k-clusters is increased by the collisions between
i clusters and j clusters where i + j » k and decreased when k cluster
combines with any other cluster. One needs to solve these coupled
differential equations to calculate C^it). Equation 5.3 is based on the
assumption that all particles are randomly distributed and are
uncorrelated at all times. The kernel K^j represents the probability of
meeting cluster i and j. The Equation 5.3 has been solved for some forms
of kjj to arrive at complete time dependent solutions of Cfc(t) in closed
form (e.g., Bowen et al. 85, Ernst 86, Ziff 84). In some of these cases,

151
the solution is found to show a gelation transition after a finite period
of time tc at which an infinite cluster (the gel) is formed which
manifests itself through a violation of the conservation law for
the total number of finite size clusters, i.e., = E kC^ / 0 where M-|
is the first moment of the cluster size distribution M-] ^E.jkC^ and
represents the total mass of the system (Ziff 84).
The coagulation equation has been solved for the cases (a) where k^j
is assumed to be independent of the size of the reacting cluster, i.e.,
kij = A = constant and (b) kjj is proportional to the size of clusters,
i.e., k¿j = (i + j)B or k^j = (ij)C where B and C are constants (e.g.,
see Bowen et al. 85). Recently, the following forms of coagulation
kernel have been considered: the product form kjj = (ij)w and the sum
form kjj = (i + j)w. For w = 1, the exact solution is available (e.g.,
see Ziff 84, Ernst 86). It has been shown that the product kernel
describes a gelation transition provided w > 1/2 while product kernel
with 0 < w (e.g., see Ernst 86).
In conclusion, it can be said that the aggregation process is
controlled by the mobility criterion (e.g., kjj in Smoluchowski's
equation) used. The information about structural properties (i.e., the
fractal dimension) of the cluster or cluster aggregates have been
obtained from computer simulation or experimentally from static or
dynamic scattering (x-ray, light, neutrons) techniques. With regard to
the dynamics of aggregation, the computer simulations are far ahead than
the experimental observations. Techniques, such as flow cell have been
applied to determine cluster size distribution and temporal evolution of

152
cluster size distribution during Brownian coagulation (e.g., see Bowen et
al. 84b,85). Also, the importance of short-range forces (i.e.,
hydrophilic-hydrophobic interactions, steric effects, etc.) have been
neglected in these treatments which may be important in real systems.
To control the properties of dispersion (such as rheological,
sedimental behavior, etc.) adequate control over size, morphology, and
strength of various aggregates (clusters) is essential. For full control
of the properties of suspension, it is essential to consider the
following points (Tadros 86):
(1) The aggregation mechanism (flocculation under Brownian motion,
shear flocculation, bridging flocculation, etc.) should be
known.
(2) The magnitude of the binding forces in aggregated structure
(i.e., Van der Waal's coagulation, bridging flocculation,
magnitude of the depth of minima in the total potential energy
diagram, etc.) should be known.
(3) The time evolution of aggregation process (aging effects)
should be understood.
(4) The mechanical properties of individual cluster or network
(gel) and mechanism of structural breakdown under applied shear
should be known.
(5) The reversibility of the aggregated structure (i.e., related to
thixotropic behavior in rheology) and the properties of the
subunit formed from the breaking process should be understood.

153
Very little is known regarding point 4 and 5, and hence, it is
impossible at present to describe the rheological properties of
flocculated concentrated suspensions quantitatively.
Equilibrium Properties of Suspension
It is well known that charged monodisperse particles, (lattices,
silica, TÍO2, etc.) form ordered structures under certain conditions.
In ordered state, particles form a three-dimensional crystalline
structure and has a bright iridescent color due to Bragg diffraction of
visible light. The order formation is related to the long-range
electrostatic repulsion between particles. The secondary minima in the
potential energy diagram (i.e., attraction, see Figure 3.6) cannot
explain the long-range reversible order formation. It has been observed
that additions of electrolyte results in order-disorder transition,
opposite to behavior expected from the secondary minima explanations (as
additions of electrolyte lead to increase in the depth of secondary
minima, and hence, should promote ordering).
Since a colloidal dispersion consists of large number of particles
(108-1019 / cm8), correct formulation to describe equilibrium structure
and ordering is based on the statistical mechanics principles. The
importance of statistical mechanics is summarized as follows from the
article by Israelachvili and Ninham:
"a knowledge of the interaction free energy between two atoms
or particles, taken in isolation, may tell us little of the
properties of an ensemble of such particles . . . Thus, the
first moral to be learned from statistical mechanics is that
the existence of a minima in the two-particle interaction free
energy in the associated or ordered state does not guarantee
the formation of this state. Conversely, the existence of an
associated or ordered state does not necessarily imply that the

154
particles are sitting at a separation where there is a minimum
in two particle interaction free energy." (Israelachvili and
Ninham 77)
From a known potential energy function (e.g., potential energy of
interaction as a function of distance of separation given by DLVO
theory), statistical mechanics can be applied to determine the excess
thermodynamic properties of dispersion (e.g., radial distribution
function, osmotic pressure, elastic constants, and phase equilibria,
etc.—Snook and Van Megan 82). Within the framework of statistical
mechanics, various approximate theories, such as integral equation
methods, hard-sphere perturbation theory, and lattice models are
available. Theoretically, exact Monte Carlo and molecular dynamics
computer solutions are also available (Barker and Henderson 76).
The Order-Disorder Transition
The dispersion of monosized colloidal particles is modelled as a
collection of hard spheres. From computer experiments, Alder et al. have
shown that hard spheres will undergo a liquid-solid phase transition
between volume fraction 0.5 - 0.55 (Alder and Weinright 62, Alder et al.
68). The volume fractions of co-existing disordered, 0h¿l, anc* ordered,
Qq1, phases have been determined for the hard sphere model and are given
as follows: ®h<3 = 0.494 and 0ho = 0.540 (e.g., see Hachisu and Takano
82).
In the case of electrostatically stabilized dispersions, the
effective double layer thickness, Aa, is added to the particle radius, a,
to calculate the effective hard sphere volume fraction of solids. It is

155
possible to construct real volume fraction hard spheres versus salt
concentration phase diagram from the following equations:
0ho = 0.55 = 0 ^ (1 + Aa/a)3, and
° o.true
0hd = 0.50 = 0 (1 + Aa/a)3
d,true
: 5.4)
As described in Chapter II, Aa is a strong function of electrolyte
concentration and valency of ions (i.e., Aa a h“1). Schematic phase
diagram for electrostatically stabilized dispersions is shown in Figure
5.4. The volume fractions of coexisting ordered and disordered phases
are plotted as a function of electrolyte concentration in solution. From
Figure 5.4, it is clear that (i) , and 0 t increases with the
d,true o,true
increasing concentration of the electrolyte (as Aa decreases with
increases in electrolyte concentration). (ii) At high electrolyte
concentration 0 ^ and 0O . should approach the hard sphere limits
o,true d,true
(i.e., 0 -» 0.55 and 0. . -* 0.50). (iii) At low electrolyte
o,true d,true
concentrations 0. ^ and 0 . becomes independent of the electrolyte
d,true o,true
concentration, and (iv) the different between 0J . and 0
d,true o,true
decreases as the electrolyte concentration decreases. The detail
comparison between various statistical model in predicting order-disorder
transition is done by Van Megan and Snook. (Van Megan and Snook 84). As
expected, hard sphere model gives semi-quantitative agreement with the
experimental observations regarding phase stability and more elaborate
model for the electrostatic potential energy at low ionic concentrations
(= £ 10“5 moles/liter) is required. It has been reported that by
softening the interparticle repulsion (i.e., by decreasing the
electrolyte concentration), the transition in the ordered phase from face
centered cubic to body centered cubic structure has been observed (Megan

.6
.5
. 4
.3
. 2
. 1
O
10 10 10 10 10 10
IONIC CONCENTRATION (moles/liter)
Figure 5-4 Schematic plot of phase diagram for monosized spherical particles. The volume
fraction of solids as a function of ionic concentration is plotted. Solid
lines are theoretical phase boundaries.
156

157
and Snook 84). The difference between the hard interactions and soft
interactions is shown in Figure 5.5.
Other properties of dispersions, such as radial distribution
function, coordination number (number of nearest neighbors), osmotic
pressure, elastic constants, etc. show dramatic change near order-
disorder transition. For example, the coordination number changes
abruptly near order-disorder transition from * 2 to 12 (Van Megan and
Snook 84). The osmotic pressure versus volume fraction solids plot shows
a discontinuity near order-disorder transition.
Another important feature of the ordered dispersion is its ability
to sustain shear stress like a molecular crystal. Properties, such as
viscoelastic behavior (shear modulus) or steady flow behavior
(viscosity), show abrupt changes near order-disorder transition in the
case of electrostatically stabilized latex dispersion (Van Megan and
Snook 84). Order-disorder phenomenon has also been observed with the
anisotropic particles. For example, dilute solution of rod shaped
tobacco mosaic virus separates into two phases where the top layer is
isotropic and the bottom is ordered phase (Forsyth et al. 78). Disk
shaped clay particles under low electrolyte concentrations also separate
into two phases. The effect of poly dispersity on order-disorder
transition has been theoretically studied. From the molecular dynamics
calculations, it has been found that ordered and disordered phases are
distinguishable up to certain distribution width and is unaffected,
but 0oh decreased with increasing distribution width (Van Megan and Snook
84).

ENERGY OF INTERACTION (J)
"hard" interactions
"soft" interactions
Figure 5.5 Schematic illustration showing (a) "hard" and (b) "soft" interactions between
particles. The potential energy of interaction as a function of distance of
separation is plotted (Tadros, 1986).
158

159
In summary, we considered two distinct mechanisms of structure
formation in colloidal dispersions. The short-range attractive
interactions lead to irreversible flocculation into primary minima. The
structural properties of aggregates can be determined from fractal
geometry principles, while kinetics of aggregation can be investigated
from computer simulation experiments or from Smoluchowski's equation.
The long-range ordering of the monosized colloidal particles, when
repulsive forces are present, can be explained from the equilibrium
statistical mechanics or from Monte Carlo computer calculations.

CHAPTER VI
RHEOLOGICAL BEHAVIOR OF COLLOIDAL DISPERSION
Introduction
Rheology is defined as the science of flow and deformation. The
rheological properties of a colloidal dispersion are among its most
important characteristics. Here, we will focus on the rheological
properties of the concentrated suspensions. The distinction between
"concentration" and "dilute" dispersion can be made as follows (Tadros
86). In a dilute dispersion, the average particle separation distance is
much greater than the range of interparticle forces. In this case,
colloidal particles are undergoing Brownian motion, and the particle
interaction can be represented by two-body collisions. As the particle
concentration in the suspension increased, the volume fraction occupied
by the solid is increased and the forces of interactions (i.e.,
attraction or repulsion) plays an important role in determining the
properties of the dispersion. As discussed in the last chapter,
electrostatic repulsion under certain conditions (i.e., high surface
potentials, low electrolyte concentrations) leads to ordering of the
spherical particles. These long-range ordered suspensions are referred
to as "solid" suspensions (Tadros 86). In between these two extreme
cases of "dilute" and "solid" suspensions are systems defined as
"concentrated" suspensions. With these suspensions, many body
interparticle interactions are important, and the particle translational
motions are restricted. In this case, the challenge is to correlate the
160

161
observed rheological properties to the structure of the dispersed phase
(Chapter IV) and to the interparticle forces (Chapters II and III). It
will be shown that the rheological techniques can provide information
about the stability of the dispersions. In the next section, we will
define various types of flow behavior encountered in rheological
experiments.
Viscosity Definition
The viscosity of a liquid is a property which defines the resistance
of liquid to flow. The viscous nature of a liquid is due to the
molecular attraction that offers resistance to shear, and consequently,
to the resulting flow induced by shear. To understand the concept of
viscosity more clearly, consider a model situation as shown in Figure
6.1. As shown in the figure, two parallel flat plates are separated by a
distance x, and the space between the plates is filled by a viscous
liquid. The bottom plate is stationary and force F is applied on top
plate of area A in a tangential direction, so that the top plate moves
with a constant velocity V in the given direction y, parallel to the
bottom plate. A thin layer of liquid adjacent to each plate will move at
the same velocity as the plate. (This is the "no slip" assumption and
holds true for most liquids.) Thus, the liquid molecules near the top
plate will be moving with the velocity V while velocity at the bottom
plate is zero. Molecules in liquid layers between these two extremes
will move at intermediate velocities. Under steady state conditions, the
force F required to produce the motion becomes constant and will be

Figure 6.1
Schematic illustration of the concept of viscosity under laminar flow
conditions.
162

163
related to the velocity gradient, dV/dx. The tangential force acting per
unit area i.e., the shear stress, x, is given by the following equation:
x (shear stress) =
F (Force)
(6.1 )
A (Area)
and units of shear stress are dynes/cm^ or N/m^. Viscosity of the liquid
can be defined in terms of two measurable quantities, i.e., shear stress,
x, and shear rate, y, as follows:
, . .. v x (Shear Stress) , <â– 
n (viscosity) - . (shear Rate, ,6-2)
Y
A liquid exhibiting a linear relationship between shear stress and shear
rate is called a Newtonian liquid. Newtonian flow behavior is exhibited
by single phase liquids with simple molecules, solutions of low
molecular-weight (non-polymeric) materials, and dilute suspensions of
spherical particles in simple liquids.
Classification of Rheological Behavior of Colloidal Dispersion
Colloidal dispersion is a two-phase mixture of solid particles
dispersed in a continuous liquid phase, and hence, the rheological
properties of the dispersion is dependent on the nature of the components
involved (i.e., solid and liquid) and also on the interactions between
these phases (solid-solid and solid-liquid). Based on the rheological
behavior, dispersions can be classified into two broad categories (a)
Newtonian and (b) non-Newtonian. Those dispersions that strictly follow
Newton's law of viscosity are called Newtonian dispersions. All other
types of rheological behavior are classified as non-Newtonian
dispersions. For non-Newtonian dispersions, it is convenient to define
an apparent viscosity na

TABLE 6.1
Classification of Flow Behavior
Suspensions
Viscoelastic
t—*
cr>
-P>
No-Yield Stress Yield Stress Thixotropic
Rheopectic
Purely Viscous
Time-Independent Time-Dependent
Newtonian Bingham
Pseudoplastic Yield-pseudoplastic
Dilatant
Yield-dilatant

165
na - t/y (6.3)
The apparent viscosity is a function of y for non-Newtonian dispersions.
(For Newtonian dispersions, n is independent of shear rate.) A
colloidal dispersion can behave as a pure viscous substance (i.e., a
substance which does not have the ability to recover its deformation
imposed by the action of shear stress after the shear stress is removed)
or show characteristics of viscoelastic materials (i.e., the ability of a
substance to partially recover the deformation on removal of shear
stress). The behavior of non-Newtonian pure viscous dispersions can also
show time-dependent flow properties. The classification of flow behavior
is shown in Table 6.1. We will discuss various types of flow behaviors
with Figure 6.2, which shows the shear stress-shear rate and
corresponding viscosity-shear rate plots (known as "rheograms" or "flow
curves"). In this section, we will define the terms in Table 6.1.
Various processes leading to flow behavior will be discussed later.
Newtonian Dispersions: As discussed earlier, Newtonian dispersions can
be characterized by a linear relation between shear stress and shear rate
and the ratio of shear stress to shear rate is independent of the shear
rate. This type of behavior is exhibited by dilute dispersion of
spherical particles where the interparticle interactions can be ignored.
The viscosity of the dispersion is higher than the viscosity of the
continuous liquid media since the perturbation of streamlines due to
presence of particles leads to higher rates of energy dissipation during
laminar flow of the dispersion.
Pseudoplastic Dispersions: Dispersions showing a decrease in viscosity
with increased shear rate (shear thinning) are described as

166
Figure 6.2
Schematic plots of (a) shear stress versus shear rate and
(b) viscosity versus shear rate for various types of flow
behaviors.

167
pseudoplastic. The shear thinning behavior is generally produced by the
reversible breakdown of a three-dimensional network structures (formed by
flocculation of the particles), alignment of anisotropic particles, etc.
Dilatant and Shear Thickening Dispersions: Dispersions showing
increasing viscosity with increasing shear rate (shear thickening) are
described as dilatant. This type of flow behavior is usually exhibited
by dispersions of rigid particles at high concentration. Since this type
of flow behavior was not exhibited by dispersions investigated in this
study, it will not be discussed further.
Bingham Plastic Dispersions: Ideally, Bingham plastic dispersions
exhibit a yield stress, xg, which is a certain minimum value of shear
stress that must be exceeded before flow takes place. Once the critical
shear stress is reached, flow starts and then the dispersion behaves as a
Newtonian dispersion, i.e., shear stress is proportional to the shear
rate. The slope of this straight line is called the plastic viscosity.
The yield stress is indicative of the strong attractive interactions
between particles. After the yield point, dispersions may show yield-
pseudoplastic or yield-dilatant type of flow behavior. Pseudoplastic (or
shear thinning) flow curves can be modeled as ideal Bingham dispersions
and flow curve properties such as apparent yield stress and plastic
viscosity can be related to the structural properties of dispersion.
Thixotropic Dispersions: For the dispersions discussed above, it was
assumed that, for any given shear stress, there is only one associated
shear rate (and viscosity) and, hence, the flow curves were independent
of additional factors such as shear history, time scale, etc. In this
type of dispersion, there is an instantaneous response to sudden changes

168
in shear. However, there exists a class of dispersions for which the
response time is considerable. Pseudoplastic dispersions exhibiting
time-dependent shear stress-shear rate are referred to as thixotropic.
Thixotropy in a dispersion is generally due to breakdown of some loose
network structure under applied shear stress. (This type of network
structure is formed during flocculation stage.) Hence, the rate of
rebuilding the structure after being destroyed by the shear stress is
important in determining the thixotropic behavior. This structural
rebuilding process is governed by the Smoluchowski's kinetic equation
(Chapter IV). The presence of thixotropy can be detected either by
measuring shear stress under constant shear rate conditions as a function
of time or by studying the ascending and descending shear stress-shear
rate curves under certain programmed conditions (i.e., shear rate is
increased from zero to a certain peak value in a given time, and then
decreased from a maximum shear rate to zero in the same time and the
corresponding shear stress is measured). A thixotropic dispersion under
such a program would produce a hysteresis loop for x versus i as shown in
Figure 6.3. If the program time is kept constant, then, the hysteresis
area can be related to the degree of thixotropy.
From the above discussion, it is clear that the interparticle forces
(and, hence, stability) and the structure of the dispersion can be
assessed from the rheological measurements. Since this study was aimed
at correlating stability behavior to rheological properties of the
dispersion, we will discuss this correlation in the next section. The
main factors that affect the flow behavior of colloidal suspensions are
Brownian motion, hydrodynamic interactions, and interparticle forces of
attraction and repulsion.

VISCOSITY SHEAR STRESS
169
Schematic representation of thixotropic flow behavior,
(a) shear stress versus shear rate, and (b) viscosity
versus shear rate plots.
Figure 6.3

170
Factors Affecting Rheological Behavior of Colloidal Dispersions
Interparticle Interactions: Interparticle interactions are determined by
the physicochemical bulk and surface properties of the components
(Chapter II and Chapter III). We have already reviewed electrostatic and
steric interactions (with adsorbed polymer) between two particles. We
have shown that the repulsive interactions can be represented by hard
sphere model or by soft interactions (Figure 5.9). The rheological
behavior of colloidal dispersions can be classified into two categories:
(1) Rheological properties of stable dispersions when (i) a net repulsion
exists between two colloidal particles (such as electrostatically
stabilized particles) or (ii) the depth of the pseudosecondary minima is
much smaller than the thermal energy of particles (in the case of
sterically stabilized dispersions). (2) Rheological behavior of
flocculated dispersion, in this case, net attraction between particles,
leads to structure formation (i.e., non-equilibrium aggregation) in
dispersion. The origin of attraction can be due to Van der Waal's
attraction, bridging attraction, flocculation under the condition of
change in solvency, or flocculation in a good solvent in the
pseudosecondary minima due to Van der Waal's attraction when the
thickness of the adsorbed polymer is not sufficient. The effect of all
of the above interactions on the rheological properties have been
investigated in this study.
Brownian Motion: Colloidal particles distributed in a liquid undergo
random motion which effectively acts as a dispersive mechanism. Brownian
motion tends to oppose buildup of structure in the dispersion. This
effect is more important in the rheology of dispersion of smaller

171
particles (usually £ 0.5 urn) when Brownian motion and shear field are
superimposed during flow.
Hydrodynamic Interactions: Hydrodynamic effects result from the mere
presence of solid particles in the flowing media. The presence of
particles leads to perturbations of the flow field and a corresponding
increase in the energy dissipation. The hydrodynamic interactions are
determined by the geometrical and mechanical properties of the
components. The continuous phase affects the flow through its viscosity,
and rigid particles through their concentration, size distribution, and
shape. The effect of these three factors on the rheological properties
will be discussed in the next section.
Rheological Behavior of Stable Systems
First, we will consider the rheological properties of dispersions
with hard sphere interactions, often called 'neutrally' stable
dispersions. In this case, the effect of attraction and repulsion is
minimized, and the main forces responsible for the rheological behavior
are hydrodynamic and Brownian forces. This type of hard sphere system
shows either Newtonian or non-Newtonian flow behavior depending on the
particle concentration and the relative importance of Brownian motion and
hydrodynamic interactions.
Einstein first described the dependence of the viscosity of a
suspension on the volume fraction solid at low particle concentration by
the following equation (Einstein 86):
nrel = n/nQ = 1 + 2.50 (6.4)

172
where n , is the relative viscosity, n is the suspension viscosity, and
rel
r¡0 is the viscosity of the suspending media, 0 is the volume fraction of
particles. The above equation is valid for rigid, uncharged, spherical
particles at very low particle number concentrations (i.e., hydrodynamic
interactions between particles were ignored). At higher particle
concentrations, the dependence of relative viscosity hrel on the volume
fraction solid is shown in Figure 6.4. The slope of nrel - 0 curve
reaches a limiting value of 2.5 only in the very dilute region. At
intermediate concentrations, the n , increases more rapidly with 0 than
rel
predicted by Einstein's equation (Equation 6.4). Above a certain volume
fraction, the packing fraction 0p the dispersed particles locks into a
rigid structure and flow ceases (i.e., n ,-»«■). The fractional
rel
dependence of nrel = h(0) has been a subject of many studies both
experimentally and theoretically.
For a mor.odisperse system, the dependence of Newtonian viscosity on 0
can be represented by a power series expression (e.g., see Goodwin 75):
Hrel = n/n0 = 1 + k-¡0 + k202 + k30^ + . . . (6.5)
In this expression, k-¡ = 2.5 is similar to the Einstein's equation for
rigid, uncharged, spherical particles. The coefficient k2 is calculated
from the perturbation of streamlines by collision doublets, and k3, k4,
etc. are used to describe higher-order collisions. The range of values
for k2 is 4 - 14.1 depending on the assumptions involved and experimental
estimates of the value of k3 varying = 16-50 (Goodwin 75). The power
series expression is empirical in nature, and there is no theoretical
justification for Equation 6.5. Thomas has suggested the following form

RELATIVE VISCOSITY
173
Figure 6.4
Schematic plot of dependence of relative viscosity on the
volume fraction solid in suspension.

174
based on an analysis of extensive data on the relative viscosity of
suspensions of spherical particles (Thomas 65):
nrel = n/n0 * 1 + 2.50 + 10.050^ + a exp (B0) (6.6)
where A and B are two adjustable parameters.
Mooney has derived the following equation based on purely geometric
packing considerations (Mooney 51):
nrel = n/n° = 6X9 (6*7)
The above equation is reduced to Einstein's equation when 0 -* 0. The
value of parameter k is chosen such that the viscosity will become
infinite when the packing of particles leads to complete mechanical
interlocking, i.e., when:
k0max = 1 <6*8)
Mooney argued that a simple cubic packed system will flow and a face-
centered cubic packed system will not, and hence:
1.35 < k < 1.91 (6.9)
Various values of k have been suggested for different packing
considerations. For example, for particles with net repulsive
interparticle forces, k = 1.65 has been suggested (Goodwin 75). On the
other hand, for the particles with net attraction, if one assumes that
the highest volume fraction at which slip can occur is random packing of
spheres, then, k = 1.56 (since 0max = 0*64 for random packing of
spheres).
Krieger and Dougherty have suggested the following equations from the
nrel - 0 studies of polymer lattices (Krieger and Dougherty 59):
nrel = [1 ’ (®/®p)r[n]0p
(6.10)

175
where 0p is the so-called packing fraction and [r|] is the intrinsic
viscosity defined as follows:
Cn]
Lim
0-0
0
1)
(6.11)
and Cn] = 2.5 for rigid, uncharged spheres. This equation fits the nrel
- 0 data of lattices of different sizes stabilized by non-ionic surface
active agent as shown in Figure 6.5.
The Effect of Adsorbed Layer
The hard sphere approximation can be applied to sterically stabilized
colloidal dispersions if proper account is taken of the adsorbed layer
volume. The adsorbed polymer layer thickness, 6, is usually added to the
particle radius a to account for the increase in the effective volume
fraction of solid. The effective volume fraction of solid, 0eff, is
related to the true solid volume fraction 0 and the adsorbed layer
thickness, 6, by the following relation:
®eff * ®true ^1 + 6/a)3 (6.12)
The value of 0eff should be used in Equations 6.4, 6.5, 6.6, 6.10, etc.
for 0 in the case of sterically stabilized dispersion. This equation can
also be used to evaluate the hydrodynamic thickness, 6, of the adsorbed
polymer layer if 0true and ®rei i-s known (i.e., if 0eff can ^ determined
from the experimentally established nrel~® relation for a given system).
Using this approach, the dependence of the effective hydrodynamic
thickness on the molecular weight of the polymer has been determined in
this study and the results will be discussed in Chapter VIII.

176
4>
Figure 6.5 Plot of relative viscosity versus volume fraction latex
particles of different sizes. Data was fitted using
Krieger equation (Krieger, 1972).

177
Equations 6.5, 6.6, 6.7, and 6.10 are generally used to relate
relative viscosity to the volume fraction of solids at high shear rates
where the rheological behavior is essentially determined by the
hydrodynamic interactions. The case where Brownian and hydrodynamic
interactions are important have been analyzed by Krieger using the
principle of corresponding rheological states (Krieger 72). The
appropriate dimensionless group characterizing the process is given by
the product of shear rate y, and the time scale of Brownian diffusion, tr
= 6nn0a^ / kT. This gives the dimensionless shear rate Yr = 6n.n0a^Y/kT.
According to the principle of corresponding state, the relative viscosity
(under steady state and laminar flow conditions), of neutrally buoyant
(i.e., density of particles = density of medium) particles is only a
function of volume fraction of solid and dimensionless shear rate, i.e.,
Hrel = f(0,Yr). Thus, at given volume fraction solids, nrel is the
unique function of Yr* Now, the same value of Yr can be obtained by
changing particle radius a or viscosity of the medium or shear rate y*
The validity of this approach has been proved by measuring the relative
viscosity of latex dispersions at a fixed solids loading. The
dimensionless shear rate Yr was varied by either (i) changing the
viscosity of medium (i.e., n0) (Figure 6.6) or (ii) by changing the
particle size (i.e., a), (Krieger 72). The superimposition of the
rheological curves, where hre^ is plotted as a function of dimensionless
parameter Yr# conclusively proves that the rheological behavior is
affected by Brownian motion and hydrodynamic interactions in this case.
The flow curve shown is shear thinning (i.e., nrel
decreases with

178
kT
Figure 6.6 Plot of relative viscosity versus dimensionless shear
rate, Yr, for monodisperse suspensions of polystyrene
spheres at 0 = 0.50 in different fluids (Krieger, 1972).

179
increasing Yr) which may be related to the structure of the suspension.
At low shear rates, Brownian motion dominates and this tends to
distribute particles randomly throughout the dispersion. The particles
arranged in such a random way tend to jam the flow when forced to move in
a shear field. Hydrodynamic forces tend to form ordered layered
structural coinciding with the planes of constant shear. These layers
can glide over one another in the flow direction, and hence, offers less
resistance to flow, i.e., lower viscosity. A theory has been developed
to explain non-Newtonian flow in hard spheres dispersions by Krieger.
The relative viscosity as a function of dimensional shear stress xr is
given by the following equation (Krieger 72):
n , = tv +
rel 1r
n2r " nlr
1 + b.T
(6.13)
r
Hence, H2r and hlr are the low and high shear limiting viscosities.
Dimensionless shear stress xr = a^-x/kT where x is the shear stress, a is
the particle radius, and kT has the usual meaning. The parameter b is a
concentration (®) independent parameter.
For electrostatically stabilized dispersions, particle sizes up to
approximately 0.5 urn follows the superposition as predicted by the
rheological equation of state. For larger particles and at higher shear
rates, the hydrodynamic interactions dominate the flow behavior and
relative viscosity is essentially independent of particle size.
Important conclusions from the _bove discussion are: (1) Neutrally
stable dispersions of colloidal particles at high solid concentrations
may exhibit non-Newtonian flow behavior. (2) The suspension viscosity of
concentrated dispersions of smaller particles will be higher at fixed low

180
shear rate and volume fraction solids than the dispersions of larger
particles. (3) Sterically stabilized dispersions can be represented as
hard spheres if the adsorbed polymer layer thickness is added to the
particle radius to calculate effective volume fraction solids (0eff'*
Stable Dispersions with Soft Interactions
This represents the case where the particle interactions are
dominated by long-range interactions (either electrostatic or steric
repulsion). The appropriate dimensionless group characterizing these
types of suspensions is given by n0a^Y/€902 where € is the dielectric
constant of the medium and V0 is the surface potential (e.g., see Russel
80, Tadros 86). Here again, repulsive force plays an important role at
low shear, while hydrodynamic forces play a dominate role at high shear.
The effect of electrostatic forces on the viscosity of aqueous suspension
of charged particles is classified into three categories (Krieger 72).
(1) The primary electroviscous effect arises due to distortion under
shear of the electrical double layer. (2) The secondary electroviscous
effect arises due to electrostatic interactions between the double layers
of different particles at finite concentrations. (3) The third
electroviscous effect is due to the distortion of the particle itself due
to the electrostatic forces. Clearly, this effect is not important in
the case of rigid spherical particles.
As discussed in the last chapter, the thickness of the double layer
is controlled by the ionic strength of the solution, and hence,
electroviscous effects are more important under high charge, low ionic
strength conditions. Long-range electrostatic interactions can lead to
ordering of particles under certain conditions (see Chapter V), and

181
hence, higher viscosities and viscoelastic behavior can be observed. The
amount of electrolyte added to suspension (ionic strength 1 x 10“^ M NaCl
moles/liter) in the present investigation was sufficient for
electroviscous effects to be neglected.
Rheological Behavior of Flocculated Dispersions
The rheological behavior of unstable dispersions (i.e., when net
attraction exists between particles) is a more difficult problem from the
theoretical as well as practical point of view. This is due to the
formation of non-equilibrium structure at rest resulting from the weak
Brownian motion. The quantitative description of the flow behavior of
the flocculated dispersions is difficult because the role of
interparticle forces on the properties of the non-equilibrium structures
(e.g., structural properties such as average size, size distribution of
particle-clusters, effect of shear on these structures, etc.) is not yet
well understood, and the description of the structure itself is not a
trivial problem.
Structure of Flocculated Suspensions
As mentioned in the last chapter, various types of non-equilibrium
structures can be formed. Two extreme cases are the formation of chain¬
like structures or the formation of more spherically-shaped clusters of
particles. These two shapes are the extreme simplifications of the real
structures and are often used as a structural model for flocculated
dispersions. In real systems, various intermediate structures are
expected. At high particle concentrations, the difference between the

182
two cases must vanish. The structure of the dispersion has a profound
effect on the properties of the dispersion such as its rheological
behavior, sediment density and porosity, filterability, compressibility,
etc. The properties of the flocculated suspension (and hence, its
structure) are influenced by various factors, such as particle size and
shape, solid surface characteristics, particle concentration, mixing
conditions, shear history, interparticle forces, etc. In the case of
electrostatically stabilized dispersions factors such as pH, electrolyte
concentration, and ion type control the dispersion properties. If
polymer is present, then additional parameters such as polymer
concentration in solution (i.e., adsorbed amount of polymer or fractional
coverage), polymer molecular weight, and polymer characteristics are
important variables controlling the dispersion properties.
Various types of structural models have been used to describe the
properties of flocculated dispersions. Here, only a brief summary is
given. The recent developments in describing non-equilibrium structures
have been discussed in Chapter V.
Michaels and Bolgers assumed that the basic flow units in flocculated
dispersion are small particle clusters consisting of randomly packed
spherical particles called floes (Michaels and Bolgers 62). At low shear
(or at rest), the floes group into clusters of floes called aggregates.
The aggregates may form a network which can fill the entire volume of the
dispersion and which can give the dispersion its plastic and structural
properties. A similar model was also used by Void (Void 63) to explain
the large sediment volume of silica sols in organic solvents. From
computer simulation, she showed that floes grown from successive random

183
additions of individual spherical particles have a roughly isometric
"core" of slowly diminishing density and "tentacles" giving the floe a
rough surface. Also, the mean density of the floe decreases with
increasing size and is given by the following equation:
p = 69 N (6.14)
where p is the mean density of the core containing N particles. This
functional dependence is very similar to the one used to describe a
fractal object (Chapter V).
Firth and Hunter proposed an elastic floe model to describe the flow
behavior of flocculated dispersions (Firth and Hunter 76a, Hunter 82, van
de Ven and Hunter 79). In their model, a typical floe is assumed to
consist of a string of particles linked together in a more or less
regular three-dimensional network. A parameter known as the floe volume
ratio Cpp = 0p / 0p measures the compactness of the floe, and Cpp has
been related to interparticle forces and shear history. Later, we will
discuss the elastic floe model in more detail. From the above
discussion, it is clear that the internal arrangement of particles in a
given floe is either assumed to be uniform (random packing of spheres or
some uniform low packing structure) or similar to a fractal object (mean
density decreasing with floe size).
The information regarding the temporal evolution of the floe size
distribution can be obtained from Smoluchowski's kinetic equations.
Applied shear will have an effect on the collision frequency (i.e.,
kinetics of flocculation) (Zeichner and Schowalter 77). It will induce
breakdown of aggregate network and floes, and it can change stability
conditions (van de Ven and Mason 76). It has been shown in the case of

184
dilute dispersions that the hydrodynamic interactions (which are ignored
in Smoluchowski's equation) will retard the flocculation rate (Honig et
al. 71). The effect of shear on the flow stability of dilute and
electrostatically stabilized dispersions have been investigated (Zeichner
and Schowalter 77, van de Ven and Mason 76). The stability diagram as
shown in Figure 6.7 illustrates that, with increasing shear rate, weakly
coagulated (in secondary minima) doublets can be separated into single
particles. With further increases in shear rate (i.e., kinetic energy of
particles), particles can surmount the potential energy barrier and
particles can be coagulated into primary minima. Very little information
is available regarding the rupture susceptibility of flocculated
structures. Usually, the exponential type of relation between average
stable floe radius and shear rate is assumed as shown below:
R = Y 'a (6.15)
where R is the average floe radius, y is the shear rate and a is an
empirical constant. Sonr.tag and Russel have modified Alder and Mills'
treatment of uniform porous floe breakup to accounts for the breakup of
floes with fractal characteristics (Sonntag and Russel 87, Alder and
Mills 79). They showed that the location of rupture approaches the
surface as R becomes large due to the decreased strength of the network
(i.e., floes are assumed to be fractal objects, and hence, the local
density is a decreasing function of distance from the center of gravity
of the floe. Due to lower density, floes are expected to be weaker near
surface compared to core.)
Various methods have been used to characterize flocculated
structures. The most simple is the direct observation of structure using

ZETA POTENTIAL (mV)
185
SHEAR RATE (s'1)
Figure 6.7
Schematic illustration of the effect of shear on the
stability of suspensions (Zeichner and Schowalter, 1977).

186
optical microscopy or using scanning or transmission electron microscopy.
Freeze-fracture technique has been developed to observed concentrated
dispersions (for e.g., see, Menold et al. 76). It is obvious that these
techniques can be applied to obtain structural information by application
of quantitative microscopy if flat sections through the freeze-dried
dispersions can be prepared. Changes in structure can be detected and
analyzed by physical techniques such as electrical conductivity,
dielectric measurements, optical measurements, and scattering techniques
(Mewis and Spaull 76). For fractal clusters with fractal dimensionality,
df < 2, the two-dimensional projections can be used to determine df of
three-dimensional structures (Weitz et al. 84). Another method of
obtaining information on random structures is computer simulation.
Various cluster growth models, such as diffusion limited growth (DLA) and
cluster-cluster aggregation model (CCA), have been discussed in detail in
the last chapter. From the above discussion, it can be concluded that
detailed information regarding the properties of the structure is still
lacking. The recent applications of fractal geometry for irreversible
processes and the solutions of Smoluchowski's equation under different
kernel seems to be a step in the right direction.
Flow Behavior of Flocculated Dispersions
Flocculated dispersions show pseudoplastic (or shear thinning)
behavior at low volume fractions of solids, while more concentrated
dispersions display plastic behavior. Suspensions flocculated with
polymeric additives often show time dependent flow or thixotropic flow
behavior. There are a few equations which are commonly used to describe

187
shear thinning flow behavior (Goodwin 75, Tadros 80). Often these
equations are derived based on the assumption that the shear thinning
flow behavior results from the breakdown of three-dimensional network
structure under shear, but the analysis is rarely carried out in detail
to determine all the constants of the model. Hence, these equations are
empirical in nature.
Flocculated Dispersion Showing No Time Dependence
As shown in Figure 6.8, three parameters are often used to
characterize the pseudo plastic flow curve: (1) D0, the critical shear
rate at which the flow curve becomes linear, (2) np^, the plastic
viscosity which is the slope of the linear portion of the shear stress-
shear rate curve, and (3) Tv,, an apparent yield stress obtained by
extrapolation of the linear portion of the shear stress-shear rate curve
to D = 0.
We will briefly review some of the models put forward to relate the
dependence of the flow curve parameters to the interparticle interactions
and floe structure. To obtain a satisfactory model of the flow process,
it is necessary to identify, at the microscopic level, various energy
dissipation processes. From the characteristic fluid motion, it should
be possible to calculate the forces necessary to produce postulated
structural changes or deformations of flocculated dispersion.
In a flocculated system, it is assumed that dynamic equilibrium
exists between aggregate growth and destruction at any shear rate. High
shear rate shifts the equilibrium in the direction of better dispersion,
whereas low shear favors aggregation. It is generally assumed that above

SHEAR STRESS, T
SHEAR RATE, D
Figure 6.8
Schematic illustration of flow curve parameters for pseudoplastic flow
behavior.
188

189
a critical shear rate, aggregates are broken down to individual particles
or at least to single floe.
Albers and Overbeek equated the maximum hydrodynamic force, FH,
exerted on a pair of flow units (in their case, particle-doublet) with
the maximum interaction force, Fmax (Albers and Overbeek 60). To
evaluate FH, the expression derived by Goren for two "touching" spheres
in couette flow is used (Goren 71). The maximum hydrodynamic force is
given by the following equation:
Fjj = 6.12 n n0 D (6.16)
where a is the particle radius, n0 is the solvent viscosity, and D is the
shear rate.
The energy dissipation in the flow process can be divided into two
parts: (i) the part due to the flow of fluid around the flow units
(assuming the flow units are impermeable) and (ii) additional energy
dissipation due to attractive interactions within or between flow units.
This breakdown of total energy was first suggested by Goodeve and
developed by Gillespie who showed that the Bingham yield value could be
related to the interaction energy between the flow units by (Goodeve 39
and Gillespie 60):
where 0 is the volume fraction of flow units, R is the radius of the flow
unit, and Eggp is the energy required to separate a doublet of flow
units. This model has been applied by Neville and Hunter for the case of
reversible flocculation of sterically stabilized dispersions (Neville and
Hunter 74, Tadros 84). The linear relationship between tj-, and 0^

190
(Equation 6.13) has been confirmed experimentally by various
investigators (see e.g., Firth 76a, Michaels and Bolgers 62, Hunter and
Nicol 68).
Elastic Floe Model
Hunter et al. have shown that much of the flow behavior of dilute
dispersions (0 = 0.01 - 0.1) can be explained with the "elastic floe
model." In this model, the basic flow units are considered to be
"elastic" floes which can undergo extension and compression during
rotation in the shear field. The floes are formed from the association
of particles or "flocculi" (i.e., cluster of particles, or hard
agglomerates, which cannot be broken by shear). The most important
characteristic of the floe is the degree to which it can entrap liquid
(i.e., the floe packing density). This is measured by a quantity called
the floe volume ratio, Cpp, and given is by the following equation:
Cpp * 0p / 0p (6.18)
where 0p is the volume fraction of floes and 0p is the volume fraction of
particles. A higher value of Cpp indicates a more open floe structure.
The flow curve parameters, such as extrapolated yield stress, xj-,,
relative plastic viscosity, np, and critical shear rate, D0, are related
to floe structure and interparticle forces as follows.
To describe the flow behavior by the above parameters, the energy
dissipation process was separated into two types, as described earlier,
(1) energy dissipation, Ey, due to the viscous flow of medium around the
flow units and (2) energy dissipated in overcoming interactions between

191
particles. The shear stress-shear rate relation can be described by the
following equation above D0:
T = Tp + hpl-D = + xv (6.19)
and energy dissipation by:
"total
t-D
T3
D +
HplD"
and
(6.20)
Ey — Hpi•0^ — TvD
Coagulated systems often show a linear shear stress-shear rate
relationship above D0 (Figure 6.8). This means that the relative plastic
viscosity is constant (i.e., the effective volume fraction of floes 0p,
does not alter during the measurement procedure above D0). At high shear
rate, the flocculi can be transferred from one floe to another during
collisions, but the value of 0p (or Cpp) remains constant for all D > D0.
Substituting for npi in Equation 6.20 in terms of 0p (using Equations 6.4
and 6.18), we have:
Ev = npl D2 = nQ D2 (1 + 2.5 Cpp 0p) (6.21)
In using the above equation, it is implicitly assumed that the floes are
spherical and are impervious to the suspension medium. At high shear
rates, floe deformation due to rotation cannot be neglected. The energy
associated with floe deformation and associated with the transfer of
flocculi from one floe to another is estimated by considering detailed
interactions between two floes as they collide and get separated by the
shear field (van de Ven and Hunter 79). The elastic floe model predicts
how the value of Cpp (which is related to ripi—Equation 6.21) depends on
the colloidal properties of the dispersion. By equating the shear stress
exerted by the applied shear field F^, to the strength of the floe Fmax«
the following equation for Cpp is obtained:

192
CFP
1.5 +
5n D a
o CFP
12d
A - - B(€,Hd1)Q2 }
(6.22)
where DCpp is the maximum shear rate to which the system is subjected, A
is the Hamaker constant, d-| is the distance between particles at which
maximum interactive force occurs, B is a function determining dependence
of electrostatic repulsion on the dielectric constant, e, of the medium,
Debye Huckel parameter k, and zeta potential, £.
The minimum value of 1.5 for Cpp can be explained since any aggregate
must trap some liquid. The dependence of Cpp on 1/a at constant zeta
potential has been demonstrated by Firth (Firth 76b). The linear
dependence of Cpp on zeta potential squared has also been demonstrated
(Firth 76b, Hunter and Frayne 80). In addition, the calculated values of
d-j from the experimental data are reasonable (i.e., in the range of 1 to
10 °A). The decrease in the Cpp with increase in the repulsion indicates
that the floes formed under repulsive interactions are more compact.
Although the above trends have been verified, the quantitative agreement
is very poor.
In order to calculate the Bingham yield value (an apparent yield
stress), following Michaels and Bolgers, Hunter considered, the energy
dissipation during the rupture of the floe doublet. In the elastic floe
model, it is assumed that two elastic floes when collided can form a floe
doublet. The energy required to break the floe doublet is supplied by
the shear field. This energy consists of two parts: (i) the energy to
break the bonds between two floes forming a doublet and (ii) the energy
required to stretch the bonds within the floe as the tension must be
transmitted from the shear field to the floc-floc interface. Since the

193
number of bonds between floes is relatively small, the energy required to
break, them is relatively small when compared with the energy due to bond
stretching. The energy required to stretch one bond is smaller than that
required to break it, but the number of bonds involved in stretching are
much more than bond breaking. The energy needed to stretch bonds
consists of several parts. (1) Elastic energy required to overcome the
interparticle forces which are keeping the particles in a primary minima
position; (2) Viscous energy dissipated to overcome the viscous drag as
particles will be displaced during stretching; and (3) Viscous energy
dissipation due to liquid movement inside floes during collision.
Van de Ven and Hunter have derived the necessary equations to
calculate the above energy terms. From the order of magnitude
approximations, they showed that only the contribution due to fluid
movement inside the floe is important. From the above analysis, Bingham
yield is given by:
x„ =
B'*noD r2 6, »2p Cpp
6 a3
(6.23)
where 6} is the increase in distance between the spheres as a result of
stretching (a few angstrom units at the most), &' is a constant (27/5), A.
is the orthokinetic efficiency which depends weakly on shear rate
—0.18
(A. á y * ), r is the floe radius, and a is the particle radius.
The trends predicted by the above equation have been verified by the
linear relations for (i) x^ vs. 0p^ (Firth 76b, Michaels and Bolgers 62,
Hunter and Nicol 68), (ii) x^ vs. Cpp (or more correctly xg vs. 1/E2)
(Hunter and Frayne 80), and (iii) a^ x^ vs. Cpp 0p2) (Hunter and Frayne
80).

194
The critical shear rate, D0, is related to the number of floc/floc
bonds, np, by the following relation:
nFFM
5n
(6.24)
where is the maximum force of attraction between particles. The
elastic floe model also suggests following relation between Cpp and F«:
CFP= 1‘5+
M
bn a
o
(6.25)
where b is constant approximately equal to 10. Equation 6.25 is
incorrect by orders of magnitude. A reasonable value for is s 1.6 x
10-9 N (i.e., force of attraction between two spherical partices of
radius a = 0.5 urn, separated by distance of 5°A. The value of Hamaker
constant A = 1 x 10-^ J). Substituting nQ = .001 Pas and Cpp in the
range 2-7, we obtain (from Equation 6.25):
Fm = 1.25 x 10-17 N (Cpp = 2) and FM = 3.2 x 10~16 (Cpp = 7)
If we compare F^ values (1.25 x 10~^7 - 3.2 x 101^ N) predicted by
Equation 6.25 with the resonable estimates (Fm = 1.6 x 10“^ N), we
observe that Equation 6.25 is in error by orders of magnitude.
The floe radius, r, is
Ev = np]D2 = n0D2 (1 +2.5 Cpp 0p) (6.21)
by the following equation:
r = (1.2 n np) ~V2 (6.26)
Hunter claims that the above equation predicts the correct order of
magnitude of the floe radius (one to few microns) at D0, but since the
value of nF used in the above equation is erroneous (as r¡p was calculated

195
from Equation 6.25), the value of the floe radius cannot be assumed to be
valid.
In summary, it can be said that the quantitative description of the
flow behavior of the flocculate dispersion is clearly lacking. The model
described above is one of the few models which tries to relate the
colloidal parameters (i.e., particle radius, zeta potential, Hamaker
constant, 9, etc.) of the dispersion to the flow parameters (i.e., xg,
Hpi, etc.) of the dispersion and is successful in predicting certain
trends. The whole treatment is based on the network floe model and
dependence of Cpp on the interparticle forces. Since this link is not
yet firmly established, the complete solution is not yet possible.
Hunter's elastic floe model is developed for the dispersions with only
electrostatic interactions. Flocculation with the adsorbed polymer
introduces additional complications since the exact nature of the
interaction forces at partial coverage is not yet known, and the kinetics
of the polymer adsorption process also play an important role.

CHAPTER VII
MATERIALS: PROPERTIES AND CHARACTERIZATION/EXPERIMENTAL PROCEDURES
Introduction
In this chapter, we will review the characteristics of materials
used in this investigation. All dispersions were prepared using
spherical, narrow-sized silica particles in water. Poly (vinyl alcohol),
PVA, having different molecular weights and acetate contents were used in
this investigation.
Silica as a Model Material
Narrow-sized agglomerate-free spherical silica powder was used in
this investigation. There are several advantages of using homogeneous
particles of uniform size and shape (Overbeek 82b). The colloid
stability (or interaction between particles) of dispersion is usually
explained from the interactions between two particles, and if all
particles are of the same size and shape, then, the explanations are more
satisfactory. Spherical shape and narrow size distribution is important
in studying various scientific phenomena, e.g., Brownian motion,
coagulation, light scattering by colloidal particles, liquid-solid phase
transition in concentrated dispersions, sintering, etc. Monosized
spherical particles can be used as a calibration standard (for example,
spherical latex particles are used to calibrate scanning electron
microscope, SEM, magnification).
196

197
Polymer lattices with very narrow-size distribution can be made
reproducible and have been used as a "model" material in testing various
theories of colloidal stability (Hearn et al. 81). For such studies, the
surface characteristics of the particles are very important, and hence,
usually emulsifier-free systems are used (e.g., see Krieger 72,
Vanderhoff et al. 70, Hearn et al. 81). The detailed surface
characterization of latex particles is not yet possible (initiator use
and efficiency of cleaning have influence on the surface
characteristics). The stability of polymer lattices is usually explained
from the electrostatic interactions, but the absence of steric
contribution to stability is not yet conclusively proved (Hearn et al.
81, Killmann et al. 88).
The silica used in this investigation has certain advantages with
respect to studying the factors affecting processing of ceramic powders.
It is expected that the material characteristics, such as density,
surface properties, Hamaker constant, etc., will have a significant
effect on the properties of dispersion. The surface characteristics of
silica are modified by calcination treatment and have been characterized
using Fourier Transform Infrared Spectroscopy, (FTIR) (Sacks et al. 87).
In ceramic processing operations, dispersions are usually made by mixing
powders and additives in appropriate solvent. In contrast, latex
dispersions are prepared by emulsion or suspension polymerization
technique (i.e., latex particles are grown in-situ) and these dispersions
are used to investigation the stability behavior under various
conditions. Thus, additional complications due to the sequence of
additions of various components and mixing conditions cannot be studied.

198
Also, the stability mechanism can be studied unambiguously (as initially,
no polymer or surfactant is present on the silica surface). The surface
of the precipitated silica is well-covered with hydroxyl groups, -OH.
The surfaces of many oxides, such as alumina, zirconia, etc., and other
technically important ceramics such as silicon carbide, silicon nitride,
etc., are also covered with surface hydroxyl groups. Therefore, the
effect of surface characteristics on the dispersion properties,
investigated for silica in this study, may have applications in colloidal
processing of other ceramics. Recently, there has been substantial
interest in the processing of monosized powders using the colloidal
processing route. Bowen and Barringer have stated two postulates to
improve manufacturability of value added ceramics (Bowen, 84a):
(1) Powders with narrow-sized distributions are easier to process into
uniform green microstructures (i.e., uniformity with respect to pore size
and size distribution), which results in easier control of the
microstructure during sintering. (2) To achieve uniform green
microstructure, interparticle forces (electrostatic, solvation, or
steric) between colloidal particles should be controlled by controlling
surface chemistry. (Postulate 1 has been recently challenged by Sacks
and coworkers (Sacks 88—unpublished work).
The effect of the green density (i.e., green microstructure) on the
sintering behavior has been studied in the case of monosized titania,
TiC>2, by Barringer (Barringer and Bowen 82), and for spherical monosized
silica, SÍO2, by Sacks and Tseng (Sacks and Tseng 84, Shinohira et al.
78). They showed that the densely packed samples can be sintered at much
lower temperatures (clear glass can be obtained at 1100°C, much below its

199
melting point). Also, TiC>2 can be sintered at 1050°C to greater than 99
percent of the theoretical density (Barringer and Bowen 82). In this
work, the effect of various factors on the stability of narrow-sized
silica dispersions of moderate silica concentration (twenty volume
percent) and on the green microstructure is investigated. The effect of
starting green density (packing) on the evolution of the microstructure
during sintering is under investigation (Vora and Sacks 88—unpublished
work).
Silica Preparation and Characterization
The silica used in this investigation was prepared using the Stober
et al. method (Stober et al. 68). Spherical silica particles of uniform
size are prepared by hydrolysis of tetraalkylsilicate and subsequent
condensation of silicic acid in alcoholic solutions containing water and
ammonia at room temperature. The hydrolysis of tetraethylorthosilicate
(Si(OC2H5)4), TEOS, and subsequent condensation of resulting silanol
groups, (= SiOH), can be represented by the following reaction:
= SiOEt + H20 -♦ = SiOH + EtOH (Hydrolysis) (7.1)
2 = SiOH -» = Si-O-Si = + H2O (Condensation) (7.2)
The rates of hydrolysis and condensation reactions are important in
determining the final form of the silica. If the above reaction is
carried out at low pH (acid catalyst) then, silica gel is obtained
(Keefer 84). In basic solution, the hydrolysis reaction proceeds by
nucleophilic substitution, i.e., negatively charged hydroxide ion attacks
the positively charged silicon atom (Keefer 84). In this reaction, TEOS
is hydrolysed at a slower rate to form monomeric hydrolysis product, OH-

200
Si(OC2H5)3> and subsequent hydrolysis reaction proceeds at a faster rate
to produce orthosilicic acid, Si(OH)4. Subsequent condensation and
cross-linking of orthosilicic acid polymers gives silica particles of
which the interior consists essentially of silicon and oxygen atoms with
hydroxyl groups attached to silicon only around the outside.
In the Stober method, ammonia acts as a catalyst for hydrolysis
reaction and as a morphological modifier to make particles spherical
(Stober 68). The final particle size depends mainly on the initial water
and ammonia concentration, the particular silicon alkoxide (methyl,
ethyl, pentyl esters, etc.) and alcohol (methyl, ethyl, butyl, pentyl
alcohol, etc.) mixture that is used (Stober 68), and the temperature of
the reactants (Tan et al. 87). To obtain monosized colloidal
dispersions, nucleation and growth processes should be separated in time
(Overbeek 82b, Sugimoto 87). In this method, all nucleation takes place
in a very short period and additional material is supplied in such a way
to growing nuclei that no supersaturation, and hence, no further
nucleation takes place. Sometimes, early nucleation can be replaced by
addition of seed material. Overbeek has shown that, if the nucleation
takes place in a very short time, then, the growth process will decrease
the relative width of the particle size distribution (compared to the
relative width of the nuclei size distribution) or at worst will remain
constant (Overbeek 82b). This controlled nucleation and growth technique
has been applied for synthesizing monodispersed oxide powders besides
silica, such as Cr(OH)3 (Demchak and Matijevic 69), AlOOH (Brace and
Matijevic 73), and TÍO2 (Matijevic et al. 77). Silica powder used in
this investigation was prepared as follows:

201
(i) Materials: Analytical grade chemicals were used. For initial
silica batches, ethyl alcohol* was distilled. Tetraethylorthosilicate**,
TEOS, was distilled under vacuum at approximately 55°C (lower than its
boiling point at one atmosphere pressure 88°C). Concentrated ammonia
solution*** with approximately 30 wt.% ammonia, (NH3) was used as-
received. From the trial experiments, it was found that the distillation
of ethyl alcohol and TEOS is not necessary to obtain spherical powder,
hence, distillation procedures were discontinued (but distillation of
TEOS may have an effect on the average silica size obtained at the
particular concentrations of TEOS, alcohol, and ammonia).
(ii) Cleaning: All glassware was first cleaned with soap solution. Then
rinsed thoroughly with tap water. Then, it was cleaned with two percent
hydrogen fluoride, HF, solution and then rinsed with distilled water.
All glassware was rinsed with ethyl alcohol just before use. Since
controlled nucleation produces monosized silica particles, thorough
cleaning of the glassware is essential.
(iii) Mixing: An appropriate volume of alcohol was measured using a two
liter measuring cylinder and added to a six-liter flask. Then,
concentrated ammonia solution was added to the alcohol under the hood,
and the mixture was stirred for approximately five minutes. TEOS was
then added to the alcohol-ammonia mixture, and then, the flask was
covered with a wax film.**** After an (invisible) hydrolysis reaction in
*Fisher Scientific Co., Fairlawn, NJ.
**Fisher Scientific Co.
***Fisher Scientific Co.
****Fisher Scientific Co.

202
which silicic acid is formed, the condensation reaction of the
supersaturated silicic acid was indicated by the increase in the
opalescence of the mixture starting approximately two minutes after the
addition of TEOS. The time at which opalescence first appears is related
to the final particle size. Generally, for the smaller sizes, the time
is longer. Stober has indicated that the above reaction is completed in
approximately fifteen minutes. To assure that the reaction was complete,
the mixture was kept stirring for at least one hour after the
precipitation of silica particles. Most of the silica powder used in
this was study was prepared using the following concentrations of the
chemicals:
TEOS: .063 liter TEOS/liter alcohol corresponding to .28 moles
TEOS/liter alcohol. (TEOS molecular weight: 208.34 and density at room
temperature: .9358 g/cm3).
Ammonia: 0.189 liter concentrated ammonia/liter alcohol corresponding to
approximately 3 moles NH3/liter alcohol and 6.5 moles I^O/liter alcohol.
(Density of the 30 wt.% ammonium hydroxide solution is .89 grams/cm3.)
Typically, four liters of alcohol was used per silica batch giving
approximately sixty grams of powder. After one hour, the silica powder
was separated using a teflon-coated filtration apparatus.* Millipore
GVHP filter paper* with .22 micron pore size was used. To increase the
filtration rate, the pressure was applied using compressed commercial
nitrogen gas. The filter cake was dried under the hood at room
temperature.
*Millipore Corp., Bedford, MA.

203
Silica Washing Procedures
Room temperature dried silica cake was ground using morter and
pestle. To remove adsorbed ammonia and alcohol, silica powder was
calcinated at 90°C for = 24 hours. To remove soluble impurities and to
remove remaining ammonia, silica was washed with deionized water* as
follows.
An approximately five volume percent silica dispersion using
deionized water was prepared. To facilitate dispersion, samples were
sonicated** for at least 15 minutes. The dispersion was filtered using
filtration apparatus and 0.22 urn filter paper. Fresh deionized water was
added to the filtration apparatus and filtration was continued. The
electrical conductivity of the filtrate was measured using a conductivity
meter*** (The measured conductivity gives a rough estimate of the
concentration of the soluble impurities in water; higher conductivity
corresponds to higher concentration of soluble impurities). When the
conductivity of the filtrate was smaller than twice the conductivity of
the fresh deionized water, filtration was discontinued. (Typically,
conductivity values in the range of 0.5 x 10"6 - 1 x 10~6 1/ohm cm were
measured for the deionized water. Filtration was discontinued when the
filtrate conductivity was < 1.5 x 10“® 1/ohm cm. For comparison, the
conductivity of a 1 x 10“2 M KC1 solution at 25°C is 1.41 x 10~3 1/ohm
cm.) The above washing procedure is satisfactory since the ionic
strength was kept constant at 1 x 10~2 moles/liter NaCl for most of the
*Continental water supply.
**Model W-375, Heat Systems, Ultrasonics, NJ.
***Model-70 CB, The Barnstead Company, Boston, MA.

204
dispersions prepared in this study. The filter cake was dried in the
oven at 60°C and then it was crushed, and the powder was stored in
plastic bottles.
Silica Calcination Treatment
To investigate the effect of silica-surface properties on the
polymer adsorption (and hence, on the stability of dispersions), silica
powders were calcined at various temperatures. Ground silica powder was
calcined in a high purity (> 99%) dense alumina crucible* at the desired
temperature for six hours. The sample was heated to calcination
temperature in three hours and was cooled in three hours (i.e., total
cycle time is three hours of heating, six hours of calcination, and three
hours cooling, totalling twelve hours). Powder was loosely packed to
avoid aggregate formation during the calcination treatment. After the
calcination treatment was over, powder was transferred (while it was warm
= 50°C) to a cleaned and dried polypropylene bottle, and the cap was
sealed using electrical tape.
Effect of Calcination Treatment on the Nature of the Silica Surface
Surface of the precipitated (and calcined at temperatures < 100°C) silica
powder is covered with the densely packed surface hydroxyl groups called
surface silanol groups, = Si-OH, and physically adsorbed water.
Physically adsorbed water can be removed by degassing silica under vacuum
at room temperature (Hair 67). At high temperatures, two adjacent
*Coors, alumina crucible.

205
silanol groups undergo a reversible condensation reaction to form a
siloxane bond, = Si-O-Si = , with evolution of water:
2 = SiOH -* = Si-0-Si = + H20 (7.3)
This reaction is completely "reversible" up to 400°C (Hair 67).
Hence, a heat treatment of silica reduces the concentration of the
surface hydroxyl groups. There are essentially two types of silanol
groups present on the silica surface (i) 'Free' (or isolated) silanol
groups which do not interact with other OH groups and (ii) 'bound'
silanols which are close enough to other OH groups to form hydrogen
bonds. The various types of surface groups present are shown in Figure
7.1. Heating silica above 400°C causes drastic irreversible elimination
of bound hydroxyl groups, and at temperatures around 800°C, only isolated
silanol groups remain on the silica surface (Hair 67). In the infrared
spectrum, these two types of silanol groups appear as quite different
species. Figure 7.2 shows the diffuse reflectance Fourier transform
infrared (FTIR)* spectra of silica powders calcined at various
temperatures (Sacks et al. 87). The uncalcined and 200°C calcined
powders are extensively hydroxylated (and also have physically adsorbed
water). The broad asymmetric absorption bond observed in the range =
3000 to 3600 cm-1 in these materials is associated with OH stretching
vibrations in hydrogen-bonded molecular water and to silanol groups
bonded to molecular water (Hair and Hertl 69, Kiselev and Lygin 75). The
broad absorption around = 3660-3680 cm-1 is associated with weakly
hydrogen bonded SiOH groups and adsorbed water molecules. A strong
absorption peak develops at = 3750 cm-1 for silica calcined at
*Model MX-1, Nicolet Analytical Co., Madison, WI.

206
SILOXANE
ISOLATED HYDROXYL GROUP
HYDROGEN BONDED HYDROXYL GROUP
Figure 7.1
Schematic representation of various types of surface
groups present on the silica surface.

REFLECTANCE
207
WAVENUMBERS (cm1)
Figure 7.2
The diffuse reflectance Fourier transform infrared
spectra of silica powders calcined at various
temperatures (Sacks et al., 1987).

208
temperatures above 500°C. This peak has been associated with free
(isolated) SiOH groups on silica (Kiselev and Lygin 75). The
concentration of surface hydroxyl groups as a function of calcination
temperature has been studied by many investigators using techniques, such
as chemical and isotropic substitution, spectral methods, etc. (Hertl and
Hair 68). Figure 7.3 shows the compilation of results by Kiselev. From
the figures, it is clear that the hydroxyl group density decreases from
about 4.6 nm~2 at 180°C to values as low as 0.5 nm-^ at 900°C. The
importance of isolated silanol groups in polymer adsorption behavior has
been studied by various investigators (e.g., see Rubio and Kitchner 76,
Tadros 78). Although the heat treatment reduces the total concentration
of the surface silanol groups, the concentration of isolated silanol
groups is maximum in a certain range of calcination temperatures (usually
700-800°C) (see Figure 7.2). The effect of increasing the concentration
of isolated silanol groups on the polymer adsorption densities will be
discussed in detail later. However, it can be stated that the maximum
amount of adsorbed polymer on silica occurs after calcination at = 700°C.
This result is consistent with observations by other investigators (Rubio
and Kitchner 76, Tadros 78, Moudgil and Cheng 86). The effect of surface
hydroxylation on dispersion stability will be described in the next
chapter. It is important to note that the silica surface is
heterogeneous with respect to adsorption and the degree of heterogeneity
depends on the calcination treatment. In the case of uncalcined silica
(i.e., the surface is well-covered with hydroxyl groups), hydrogen
bonding between surface hydroxyl groups and OH groups of poly (vinyl
alcohol), PVA, will be an important adsorption mechanism. For silica

209
i
E
Z
o
>—I
E-
<
a
H
Z
H
U
z
o
u
p
CALCINATION TEMPERATURE (°C)
CALCINATION TEMPERATURE (°C)
Figure 7.3
Concentration of surface silanol groups as a function of
the temperature of calcination ((a) Bode et al., 1967 and
(b) Kiselev, 1975).

210
calcined at higher temperature (i.e., for example, 700°C silica), the
surface will be largely hydrophobic so that adsorption by means of
hydrophobic bonding is also likely to occur. (At high temperature,
hydrogen bonding between isolated silanol groups and -OH groups of PVA is
still likely to be an important mechanism). The increase in the
adsorption density of partially hydrolysed (88^6) PVA has been associated
with hydrophobic bonding between acetate groups and hydrophobic surface
sites (Cohen Stuart 80a, Tadros 78). The silica surface can be made
hydrophobic by chemically grafting polymers on to the silica surface by
in-situ polymerization and stable non-aqueous silica dispersions can be
prepared (Laible and Hamann 80). Since only aqueous silica dispersions
were prepared in this study, we will not further discuss other techniques
of rendering the silica surface hydrophobic.
Silica Size and Size Distribution
The average size and size distribution of the silica particles were
determined using two techniques: (i) scanning electron microscopy, SEM,
and (ii) x-ray sedimentation. Samples for SEM examination were prepared
as follows.
A dilute dispersion (approximately 300 ppm silica) of silica in
ethyl alcohol was prepared by ultrasonication treatment for fifteen
minutes. A drop of this dispersion was placed on polished and cleaned
(with alcohol) aluminum sample holder and was carefully dried. (To avoid
dust pick up, a plastic cup was placed on sample holders, or
alternatively, a glass slide cleaned with a Chormic Sulfuric acid
cleaning solution was also used). All samples were sputter coated with

211
thin layers of gold (= 100 °A) to avoid charging of the sample during SEM
examination. Several photomicrographs were analyzed for each silica
batch to determine particle size and size distribution. Particle
diameter was measured using a digitizing tablet* and a specially
developed computer program. Figure 7.4 shows a SEM photomicrograph of
silica particles used in this investigation. Spherical shape and narrow
size distribution is quite evident from the photomicrograph. Since the
particle size distribution exhibited nearly log-normal distribution
characteristics, the geometric mean and standard deviation were used to
characterize a silica batch. For perfectly monosized particles, the
geometric standard deviation would be 1. For the silica batches in this
study, the geometric standard deviation was = 1.1 or less. Typically,
two hundred particles were analyzed to obtain average size. (It was
found that the difference in the mean between 50 and 1000 particles was
statistically insignificant). To represent the particle size
distribution, a histogram (number of particles in a given diameter class
versus particle diameter) can be constructed. A typical histogram with a
geometric mean = 0.42 urn and geometric standard deviation, In Og, =1.1
is shown in Figure 7.5.
Median diameter and particle size distribution information was also
obtained from the x-ray sedigraph. A plot of cumulative mass percent
versus equivalent spherical diameter is shown in Figure 7.6. Samples for
x-ray sedigraph was prepared as follows.
Silica powder was ground using a mortar and pestle. An
approximately three volume percent silica dispersion at pH = 7.5 was
*Model 9111A, Hewlett Packard, CA.


15
W
hJ
u
HH
H
35
Oh
fe
o
Pi
w
M
9 -
6 -
3 -
SILICA
t—r
*=f
ri~l~
.3
.35
.4 .45 .5 .55
PARTICLE DIAMETER (|im)
.6
Figure 7.5 A histogram (number of particles in a given diameter class versus particle
diameter) of a typical silica batch.
213

100
80
60
40
20
0
EQUIVALENT SPHERICAL DIAMETER (jun)
.6
Plot of particle size distribution for silica determined by x-ray
sedimentation.
214

215
prepared (to ensure electrostatic stabilization) and ultrasonicated for =
two hours to breakdown agglomerates. The x-ray sedigraph result shows
(Figure 7.6) a small fraction of larger particles in the suspension
(i.e., > 0.8 pm), compared to the SEM results. This may reflect the
formation of temporary doublets, triplets (or higher order clusters),
etc. during Brownian collisions at the relatively high (= 3 vol. %) solid
concentration and/or may indicate the presence of a small concentration
of hard agglomerates that formed during processing (e.g., drying, pre¬
calcination treatment, etc.).
Silica Surface Area
The powder specific surface area (m^/gram) was measured using
nitrogen gas adsorption using the multipoint BET method*. Uncalcined
powders were outgassed at temperatures = 80°C (i.e., below the drying
temperature of 90°C) under flowing nitrogen, and calcined samples were
outgassed at the temperatures in the range 150 to 200°C for at least
three hours. Surface area measurements were repeated at least two times,
and the average surface area of a given silica lot was used to report the
adsorbed amount of polymer in the units of mg polymer adsorbed/m^ solid
surface. Typical results for various silica lots are shown in Table 7.1.
No significant change in the measured surface area with the calcination
temperature was found by Sacks and Tseng indicating an absence of
micropore formation during the dehydroxylation treatment (Sacks and Tseng
84). Also, close agreement was found between surface area calculated
(Sp) from the average particle radius a, and the powder true density, p
*Model OS-7, Quantachrome Corp., Syosett, NY.

216
TABLE 7.1
Geometric Mean and Specific Surface Area of Various Silica Lots Used
Calcination Geometric Mean BET Calculated
Silica Temperature from SEM Surface Area Surface Area
Lot # (°C) (urn) (m^/gram) (m^/gram)
700
0.70
5.16
4.08
tl
0.42
8.34
6.80
ft
0.42
8.00
6.72
• f
0.43
7.90
6.64
fl
0.44
8.00
6.50
50
0.45
8.07
6.36
700
0.45
8.03
6.36
850
0.45
7.51
6.36
8

217
(i.e., Sp = 3/p.a) and the measured value obtained from BET. Van Helden
et al. reported that uncalcined powder (dried at 25°C) is highly porous
to nitrogen, i.e., it has a high surface area compared to powder heated
to 90°C (Van Helden et al. 81). In both cases, area determined with 0H“
ions was the same. Vrij et al. suggested that, upon heating the diameter
of the surface micropores decrease and become impermeable to nitrogen.
In summary, the powder used in this investigation (heated at temperatures
> 90°C) has no significant amount of surface porosity.
Silica True Density
Powder true density of Stober silica has been measured using helium
gas pycnometry*. The increase in the true density with calcination
temperature is shown in Figure 7.7 (Sacks and Tseng 84). True density
increases as water and ethoxy groups are removed during low temperature
calcination. Density values for powders calcined at temperatures above
800°C (Figure 7.7) are slightly higher than the usual reported value for
fused silica, 2.20 g/cm^. The reasons for this observation is not yet
clear (Sacks and Tseng 84). Thermogravimetric results show that = 12
wt.% loss occurs on heating uncalcined silica samples (Sacks and Tseng
84). The endothermic peak (centered at = 160°C) observed in Differential
Thermal Analysis, DTA, shows that much of the bound water is removed at
temperatures below 200°C.
*Model PY-5, Quantachrome Corp., Syosett, NY.

DENSITY (g/cm3)
2.5
2. 4
2.3
2.2
2. 1
2
Figure 7.7
Silica
1 1 1 1 i I i I i I i
0 200 400 600 800 1000 1200
CALCINATION TEMPERATURE (°C)
Gas pycnometer density versus calcination temperature for SÍO2 powders (Sacks
and Tseng, 1984).
218

219
Poly (Vinyl) Alcohol
Synthesis and Properties
Poly (vinyl alcohol), PVA, is a water soluble polymer with simple
chemical structure. The basic unit is -(CH2-CHOH)Vinyl acetate is
polymerized to form poly vinyl acetate, PVAj., and the subsequent
hydrolysis reaction forms PVA. In industrial practice, complete
hydrolysis is seldom achieved, and the catalyst used determines the
residual acetate distribution. An alkaline catalyst favors 'blocky'
distribution whereas acidic promotes random distribution of acetate
groups, (CH3COO-) (Pritchard 70). The majority of samples used in this
study were supplied by Air Products and Chemicals*. The following
physical properties of PVA are important for adsorption studies.
Solubility. Solution Behavior and Interfacial Activity
These properties are dependent not only on the primary chemical
structure of the molecules, but secondary effects, such as branching, end
groups, irregularity in chain, stereoregularity, residual acetate
content, distribution of acetate groups, and molecular weight of the
polymer.
PVA is essentially an uncharged polymer. Structural impurities,
such as carboxyl and sulphate groups, will result in a charged polymer
(Pritchard 70, Koopal 78). Fleer has shown from electrophoresis and
intrinsic viscosity measurements at various ionic strengths that PVA is
an uncharged polymer (Fleer 71). PVAc is often a branched polymer, but
most side chains may split off during saponification, and thus, the
resulting PVA has little branching or is an unbranched polymer (Pritchard
*Air Products and Chemicals, Inc., Allentown, PA.

220
70). Stereoregularity of the polymer affects properties such as
crystallinity and solubility. Industrial samples are usually atactic and
tacticity of a polymer is not an important factor in adsorption studies
(Koopal 78).
The rate of dissolution and the solubility of PVA strongly depends
upon the degree of hydrolysis. PVA containing approximately twelve
percent acetate groups easily dissolves in cold water. More highly
hydrolysed PVA (< = 2% acetate) dissolves quickly only at elevated
temperature (e.g., = 80°C) and has a tendency to age and form aggregates
at room temperature. The main reason for this phenomenon is the intra-
and intermolecular hydrogen bonding between the hydroxyl groups which
decreases the dissolution rate and the solubility. Residual acetate
groups are hydrophobic in nature and prevent hydrogen bonding of hydroxyl
groups and improve the solubility characteristics. To break the internal
hydrogen bonds and to dissolve the PVA, a temperature above glass
transition point ( = 75°C) is required. The following procedure was used
to prepare polymer solutions with known concentration.
To prepare 100 ml of polymer solution of known concentration, PVA
was weighed using an analytical balance and polymer was added to a beaker
with =75 cm^ Qf deionized water (water was stirred using magnetic
stirrer). PVA powder was slowly added to the vortex to avoid formation
of lumps. Stirring was continued until the polymer particles were
wetted. Then, the solution was heated for approximately thirty minutes
at = 85°C using a double walled jacket apparatus and a temperature
controlled circulator. The solution was cooled to room temperature and

221
was transferred to a 100 ml volumetric flask. The beaker was thoroughly
rinsed with deionized water and the solution was transferred to a 100 ml
volumetric flask. Deionized water was then added to make 100 ml of
polymer solution of known concentration. At room temperature, solutions
prepared within completely hydrolysed PVA (> 2% acetate) are reasonably
stable (Gruber et al. 74). Also, polymer concentrations used were
relatively low (usually, at most, 3 g PVA/100 cc). Hence, polymer
solutions with higher acetate content (> 10%) were used for up to three
days before being discarded. Solution of PVA, hydrolysed > 98%, were
discarded after twenty-four hours.
Fractionation of As-received PVA
As-received polymer samples were fractionated using a sequential
precipitation technique to remove high and low molecular weight
impurities (van den Boomgaard et al. 78). To obtain polymer samples with
varying molecular weights and acetate contents, various commercial
samples were fractionated. Preparative scale Gel Permeation
Chromatography, GPC, has been used by Garvey et al. to obtain various
molecular weight samples with narrow molecular weight distribution
(Garvey et al. 74). But, the above method is not practical for our study
since larger quantities of samples are necessary to study suspension
properties. (Preparative scale GPC gives few mg of samples while
approximately ten grams of PVA samples were used to study various aspects
of dispersion stability.) The following procedure was followed to
fractionate samples.

222
One liter of polymer solution with = 5 w/v % PVA was prepared
following the procedure described before (i.e., initially wetting the
polymer beads while stirring, and then, heating the solution to = 85°C
for thirty minutes). The polymer solution was cooled to room
temperature, and then, acetone was added to this polymer solution while
stirring. When the solution became turbid, some more acetone was added
( = 100 cc) in order to produce a sufficient amount of gel phase. The
solution was stirred for one more hour and was left to sediment for three
to four days. The precipitated polymer forms a strong gel-sediment at
the bottom of the container. The top solution was transferred to another
container, and more acetone was added to precipitate the next fraction.
Each polymer was separated into four to five fractions. Each fraction
was washed with excess acetone and dissolved in water and kept in an oven
at = 40°C for one day to remove the last traces of acetone. Then, films
were cast from this viscous solution and dried at » 60°C for two days.
These films were stored in a plastic bottle and were useful for
accurately weighing the required amount of polymer. These samples were
then characterized for acetate content and molecular weight as follows.
Acetate Content of Polymer
The acetate content of the fractions was determined by a
saponification technique (Garvey et al. 74, Pritchard 70). Approximately
0.13 grams of polymer was refluxed for 1/2 hour with 20 ml of 0.1 N NaOH
solution. The unreacted amount of NaOH was determined by titration with
0.1N HC1 using a phenol red indicator. From the amount of NaOH consumed,
the acetate content was calculated using the following formula:

223
-4
Mol % hydrolysed = — ^ X 1<~* ^ x 100 (7.4)
(w - 42 x 10 y)
where w is the weight of the sample and y is the ml of 0.1 N NaOH
consumed. Since the end point of the titration can be determined
accurately within ± 0.1 ml, the acetate content can be determined within
± 1% of the true value. The results of polymer characterization are
reported in Table 7.2.
Molecular Weight and Molecular Weight Distribution
Polymer samples were characterized for their molecular weight and
molecular weight distribution using viscometry and Gel Permeation
Chromatography, GPC.
Viscometry
Viscosity measurements were carried out at 30 ± .05°C using an
Ubbelohde suspended level dilution viscometer. Polymer solutions were
prepared following the procedure described earlier. The polymer
concentration was typically = 0.6 g PVA/dL. Samples were filtered using
glass fiber filter paper. Special attention was paid to good temperature
control, vertical alignment of the viscometer, filling of the viscometer,
and cleanliness. After the experiment, the viscometer was thoroughly
rinsed with deionized water using a peristaltic pump. Then, it was
rinsed with filtered methanol and dried at 40°C. Ten ml of sample was
introduced and flow time was measured using a stop watch. Three
consecutive readings within ± .01 sec were used for calculations. Then,

224
three ml of deionized water was added and the sample was mixed and
equilibrated for fifteen minutes. The above procedure was continued
until four readings were recorded. The flow time for deionized water was
periodically checked to ensure consistency. The flow time for water was
about 159 seconds. Intrinsic viscosities, [p], for each PVA sample were
determined using the equation of Huggins:
rel
= [p]H + k
^H Cp
and Martin:
(7.5)
in jSSi , ln [„] + k’ [,] C (7.6)
P F
where nrei is the viscosity excess ratio and nrei = (t - t0)/t0 where t
is the flow time for the polymer solution with concentration, Cp, and t0
is the flow time for water, k'^ and k'^ are the Huggins and Martin
constants. Since water is only a poor solvent for PVA, ChIm was used to
calculate the viscometric average molecular weight, Mv (Koopal 78). (In
good solvents, th