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Rheology of Noncolloidal Suspensions of Spheres in Oscillatory Shear Flow and the Dynamics of Suspensions of Rigid Fibers

Permanent Link: http://ufdc.ufl.edu/UFE0021331/00001

Material Information

Title: Rheology of Noncolloidal Suspensions of Spheres in Oscillatory Shear Flow and the Dynamics of Suspensions of Rigid Fibers
Physical Description: 1 online resource (142 p.)
Language: english
Creator: Bricker, Jonathan M
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The work presented in this dissertation provides a significant contribution to understanding the rheological behavior of suspensions systems. Specifically, this work provides results from a comprehensive set of experiments on the rheology of suspensions of noncolloidal spheres undergoing oscillatory shear flow. The experiments are complimented by results from the first dynamic simulation of suspensions of noncolloidal spheres in unsteady shear flows. Additionally, results from experiments on the steady shear rheology of model suspensions of rigid ellipsoids are presented. Experiments on suspensions of spheres undergoing oscillatory shear flow show that the rheology is strongly influenced by the applied strain amplitude. At each amplitude, the steady value of the complex viscosity decreases with total strain for high strain amplitudes and increases for low amplitudes. The transition point at which the qualitative behavior changes occurs at an amplitude-to-gap ratio between 0.1 and 0.5 and is independent of the particle size distribution and suspension system. The steady state value of the complex viscosity is a nonmonotonic function of the applied strain amplitude, with a minimum viscosity observed at an amplitude-to-gap ratio of 1. The experiments suggest that shear-induced migration is of no consequence, and that the observed behavior is instead due to changes in the suspension microstructure. The rheology observed in the experiments is largely confirmed by Stokesian dynamics simulations. Simulations of suspensions of noncolloidal spheres undergoing unsteady simple shear flows are used to evaluate the evolution of the stresses with time along with the corresponding microstructural development. Similar to the experiments, the shear stress drifts with total strain before attaining a steady state that depends upon the applied strain amplitude. The steady state viscosity obtained from the shear stresses exhibits a nonmonotonic dependence on the applied shear rate that agrees qualitatively with the experimental results. An analysis of the suspension microstructure at steady state reveals three distinct microstructures that correlate to the observed rheology. Hydroclusters, ordered layers, and crystalline structures can all be induced by simply altering the applied strain amplitude. Results from both simulations and experiments indicate irreversibility over the range of strain amplitudes studied. The rheology of semi-dilute suspensions of rigid polystyrene ellipsoids at rotational Peclet numbers greater than 10$^3$ was studied for two different aspect ratios. The ellipsoid suspensions exhibit shear thinning behavior for both aspect ratios. Although in agreement with previous experiments, rate dependent rheology is not predicted by theories and simulations. Direct comparison of the results with rheological measurements on suspensions of spheres with material properties identical to the ellipsoids suggest that colloidal interactions are improbable sources of the shear thinning behavior. A mechanism for the observed shear thinning behavior is proposed which involves a competition between particle drift due to shear-induced migration, and thermal diffusion. For the largest value of the shear rate, the relative viscosity scales linearly with dimensionless number density regardless of the aspect ratio. At the lowest shear rate, the relative viscosity does not scale linearly with dimensionless number density and a dependence on the ellipsoid aspect ratio is apparent.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jonathan M Bricker.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Butler, Jason E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021331:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021331/00001

Material Information

Title: Rheology of Noncolloidal Suspensions of Spheres in Oscillatory Shear Flow and the Dynamics of Suspensions of Rigid Fibers
Physical Description: 1 online resource (142 p.)
Language: english
Creator: Bricker, Jonathan M
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The work presented in this dissertation provides a significant contribution to understanding the rheological behavior of suspensions systems. Specifically, this work provides results from a comprehensive set of experiments on the rheology of suspensions of noncolloidal spheres undergoing oscillatory shear flow. The experiments are complimented by results from the first dynamic simulation of suspensions of noncolloidal spheres in unsteady shear flows. Additionally, results from experiments on the steady shear rheology of model suspensions of rigid ellipsoids are presented. Experiments on suspensions of spheres undergoing oscillatory shear flow show that the rheology is strongly influenced by the applied strain amplitude. At each amplitude, the steady value of the complex viscosity decreases with total strain for high strain amplitudes and increases for low amplitudes. The transition point at which the qualitative behavior changes occurs at an amplitude-to-gap ratio between 0.1 and 0.5 and is independent of the particle size distribution and suspension system. The steady state value of the complex viscosity is a nonmonotonic function of the applied strain amplitude, with a minimum viscosity observed at an amplitude-to-gap ratio of 1. The experiments suggest that shear-induced migration is of no consequence, and that the observed behavior is instead due to changes in the suspension microstructure. The rheology observed in the experiments is largely confirmed by Stokesian dynamics simulations. Simulations of suspensions of noncolloidal spheres undergoing unsteady simple shear flows are used to evaluate the evolution of the stresses with time along with the corresponding microstructural development. Similar to the experiments, the shear stress drifts with total strain before attaining a steady state that depends upon the applied strain amplitude. The steady state viscosity obtained from the shear stresses exhibits a nonmonotonic dependence on the applied shear rate that agrees qualitatively with the experimental results. An analysis of the suspension microstructure at steady state reveals three distinct microstructures that correlate to the observed rheology. Hydroclusters, ordered layers, and crystalline structures can all be induced by simply altering the applied strain amplitude. Results from both simulations and experiments indicate irreversibility over the range of strain amplitudes studied. The rheology of semi-dilute suspensions of rigid polystyrene ellipsoids at rotational Peclet numbers greater than 10$^3$ was studied for two different aspect ratios. The ellipsoid suspensions exhibit shear thinning behavior for both aspect ratios. Although in agreement with previous experiments, rate dependent rheology is not predicted by theories and simulations. Direct comparison of the results with rheological measurements on suspensions of spheres with material properties identical to the ellipsoids suggest that colloidal interactions are improbable sources of the shear thinning behavior. A mechanism for the observed shear thinning behavior is proposed which involves a competition between particle drift due to shear-induced migration, and thermal diffusion. For the largest value of the shear rate, the relative viscosity scales linearly with dimensionless number density regardless of the aspect ratio. At the lowest shear rate, the relative viscosity does not scale linearly with dimensionless number density and a dependence on the ellipsoid aspect ratio is apparent.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jonathan M Bricker.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Butler, Jason E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021331:00001


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RHEOLOGY OF NONCOLLOIDAL SUSPENSIONS OF SPHERES IN OSCILLATORY
SHEAR FLOW AND THE DYNAMICS OF SUSPENSIONS OF RIGID FIBERS


















By
JONATHAN MARK( BRICK(ER


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

2007


































S2007 Jonathan Mark Bricker




































To my son, Aiden









ACKENOWLED GMENTS

I am fortunate to have met many special people and friends who have contributed to

my research over the years. Without them, my success would not have been possible and I

extend a heartfelt appreciation to these individuals.

Foremost, I must thank my research advisor, Jason E. Butler. Over the years,

Jason has taught me the science of fluid mechanics through both experimental and

computational means. He has instilled in me the qualities of a good researcher, namely he

has taught me to be persistent and inquisitive concerning all things known and unknown.

His patience allowed me to discover concepts on my own and his passion and drive

inspired and challenged me. Jason allowed me to mature as a researcher, and as a person.

For these things, I am grateful.

I also benefitted greatly from the advice and direction of my committee members.

Specifically, I extend thanks to Dr. Anthony Ladd for his helpful discussions and criticism

of my work over the years and Dr. Ranga Narali- .Il .Il who has taught me to constantly

question the status quo. Although not on my committee, I thank Dr. Kirk Ziegler for his

insight on life in general.

I owe thanks to my colleagues, Philip D. Cobb, Joontack Park, and Hyun-Ok Park

for their willingness to listen and their patience in teaching me. I extend special thanks to

O. Berk Usta, who was instrumental in the progression of my work. In the process, he has

become a valuable friend whose sincerity and unselfishness I will never forget. I would also

like to thank staff members Dennis Vince and Jim Hinnant for their help in keeping things

operational in the laboratory. Their work has been invaluable over the years.

I am fortunate to have developed many friendships at the University of Florida. I

would like to thank Matt Monroe, Patrick McE~inney, Michael June, and Darren McDuff

for their friendship on and off the court. I also extend thanks to Colin Sturm for his

wisdom and humor. I look forward to conducting business with him in the future.










Finally, none of this would have been possible without the loving support of my

family. I especially thank my maternal grandparents, Martin and Mary Schwerthoffer for

their support. I also thank my mother, Alariann Bricker for instilling in me the strength

and confidence needed to complete this work. I also thank my fiancee, Alma Hernandez,

for her patience and love over the years. Most importantly, I thank my son, Aiden Bricker,

who made my doi~ bright unconditionally.











TABLE OF CONTENTS

page

ACK(NOWLEDGMENTS .......... . .. .. 4

LIST OF TABLES ......... ..... .. 8

LIST OF FIGURES ......... .... .. 9

ABSTRACT ......... ...... 13

CHAPTER

1 THE RHEOLOGY OF SUSPENSIONS: PAST EXPERIMENTS,
SIMULATIONS, AND THEORIES ...._ .. .. 15

Introduction ......... ..... .. 15
Suspensions of Spheres ......... . 17
Suspensions of Fibers ......... . . 22

2 OSCILLATORY SHEAR OF SUSPENSIONS OF NONCOLLOIDAL
PARTICLES ...... ......_ 25

Introduction ......... ..... .. 25
Experiment ......... ..... .. 27
Results ............ .............. 32
Steady Shear ......... . .. 32
Oscillatory Shear: Monodisperse Suspension .... ... .. 36
Oscillatory Shear: Comparisons Between Systems ... .. .. 41
Discussion ......... . .... .. 43
Steady Shear ......... .. .. 43
Oscillatory Shear ......... .. .. .. 45
Origin of oscillatory behavior ...... .. . 45
Strain dependence of the results .... ... .. 48
Effect of suspension characteristics .... .. .. 49
Conclusions ........ .... .. 52

3 CORRELATION BETWEEN STRESSES AND MICROSTRUCTURE IN
CONCENTRATED SUSPENSIONS OF NON-BROWNIAN SPHERES
SUBJECT TO UNSTEADY SHEAR FLOWS .... .... .. 53

Introduction ......... .... .. 53
Rheology Simulations ......... . .. .. 54
The Stokesian Dynamics Method . .... .. 55
Evaluation of Rheology ........ .. .. 58
Implementation ........ . .. 60
Results .. ......... ......... .. 61
Steady Shear ........ ... .. 62











Shear Reversal ........ ... .. 66
Oscillatory Shear ........ ... .. 70
Rheology ......... . .. 70
Suspension microstructure . ...... .. 74
Discussion .............. .. ............ 77
Strain Dependent Diffusivity ........ .. 79
Suspension Microstructure and Rheology .... .... .. 82
Effect of the Repulsive Force on the Rheology .. .. .. 86
Normal Stresses ...... .. .. .. 87
Relationship Between Oscillatory and Shear Reversal Rheology .. .. .. 89
Conclusions ........ .... .. 90

4 RHEOLOGY OF SEMI-DILUTE SUSPENSIONS OF RIGID POLYSTYRENE
ELLIPSOIDS AT HIGH PECLET NUMBERS .... .... .. 92

Introduction ........ .... .. 92

Experiment ........ .... .. 93
Results .. ......... ......... .. 97
Discussion ............ ............ 104
Rate Dependent Rheology ....... ... .. 104
Scalings of 9, at High Per ........ ... .. 109
Conclusions ........ ..... .. 110

5 CROSS-STREAM MIGRATION OF RIGID BROWNIAN FIBERS
UNDERGOING SIMPLE SHEAR FLOW NEAR A SOLID BOUNDARY .. 112

Introduction ........ .... .. 112
Model ........ .... .. 113
Velocity Expressions ........ . .. 114
Probability Distribution ........ ... .. 118
Calculation of Orientation Moments .... .. .. 120
Results and Discussion ........ .. .. 122
Center of Mass Velocity ........ ... .. 122
Center of Mass Distribution ....... ... .. 123
Conclusions ........ ..... .. 127

6 CONCLUSIONS ........ . .. 128

REFERENCES ......... . .... .. 132

BIOGRAPHICAL SK(ETCH ......... . .. 142










LIST OF TABLES


Table page

2-1 Summary of the monodisperse and polydisperse suspension systems studied .. 30

4-1 Summary of the characteristic length scales, volume per particle, and surface
area per particle of polystyrene particles ...... .. . 95










LIST OF FIGURES

Figure page

1-1 Images of woven K~evlar fabrics ......... ... 16

1-2 Electrorheologfical suspension composed of silica spheres in corn oil .. .. .. 17

1-:3 Summary of observations of the net bulk migration of particles undergoing
oscillatory pipe flow versus dimensionless strain amplitude .. .. .. 18

1-4 Phase diagram taken from the work of Ackerson [:3] for Brownian suspensions
undergoing oscillatory shear flow as a function of volume fraction and strain
amplitude ......... .... . 19

1-5 Dimensionless diffusivity in the flow (closed) and gradient (open) direction as a
function of strain amplitude for concentrated noncolloidal suspensions .. .. 21

1-6 Schematic showing the steady state spatial and orientational distribution of
fibers prior to (left) and following (right) simple shear flow .. .. 2:3

1-7 Relative viscosity of semi-concentrated suspensions of fibers as a function of
shear rate ......... .... . 24

2-1 SENT images and respective size distributions of the particles used in the
oscillatory experiments ......... . 28

2-2 Relative viscosity plotted as a function of time for monodisperse PMMA spheres
in ITCON/water/Nal (triangles), monodisperse poli--li-i. in.- spheres in
polyalkylene glycol (squares), monodisperse PMMA spheres in EG/glycerol
(crosses), and polydisperse PMMA spheres in EG/glycerol (circles) undergoing
steady shear in the Couette geometry . ..... .. 3:3

2-3 Relative viscosity plotted as a function of time for monodisperse PMMA spheres
in ITCON/water/Nal (triangles), monodisperse poli--li-i. in.- spheres in
polyalkylene glycol (squares), monodisperse PMMA spheres in EG/glycerol
(crosses), and polydisperse PMMA spheres in EG/glycerol (circles) undergoing
steady shear in the parallel-plate geometry ...... .. .. :35

2-4 Images of the ITCON oil/water suspending liquid in the parallel-plate geometry :36

2-5 Input strain waves (dashed line) and resulting output stress waves (solid line)
plotted versus time for the ~=0.40 suspension of monodisperse polili-(i. 1n.-
spheres suspended in polyalkylene glycol ...... .. :38

2-6 Relative complex viscosity as a function of time for a monodisperse suspension
in the Couette geometry .. ... . :39

2-7 Relative complex viscosity as a function of strain and amplitude-to-gap ratio
for a monodisperse suspension in the Couette geometry .. .. .. 40










2-8 Relative complex viscosity as a function of total strain and frequency for a
nionodisperse suspension in the Couette geometry .. . .. 41

2-9 Comparison of the responses observed in parallel-plate (closed symbols) and
Couette (open symbols) geometries for a nionodisperse suspension system .. 42

2-10 Comparison of the responses observed in the Couette geometry for
nionodisperse suspension systems containing polystyrene (open symbols) and
PMMA (closed symbols) ......... .. .. 4:3

2-11 Comparison of the responses observed in the Couette geometry for suspensions
of PMMA spheres with nionodisperse (closed symbols) and polydisperse (open
symbols) size distributions ......... .. .. 44

2-12 Difference between the final and initial values of the complex viscosity
normalized with respect to its initial value in the Couette geometry plotted as a
function of aniplitude-to-gap ratio ....... ... .. 46

2-1:3 Complex viscosity (evaluated at y=15000) plotted as a function of applied strain
amplitude for the Couette and parallel-plate geometries .. .. .. 47

2-14 Complex viscosity (evaluated at y=15000) plotted as a function of applied strain
amplitude and concentration for the parallel-plate geometry .. .. .. .. 50

2-15 Complex viscosity (evaluated at y=15000) plotted as a function of concentration
for the parallel-plate geometry ......... .. .. 51

:3-1 Schematic showing particle pairs exposed to simple shear flow .. .. .. .. 54

:3-2 The periodic cell used in the simulations ...... .. . 56

:3-3 Particle contribution to the shear stress, o-,,, as a function of areal fraction, 4,
for steady shear flow ......... .. .. 6:3

:3-4 Normal stress o-y, as a function of areal fraction for steady shear flow .. .. 64

:3-5 Normal stress difference NI~ as a function of areal fraction 4 for steady shear flow 64

:3-6 Pair distribution, g(r), as a function of radial distance and areal fraction
integrated over all angles . .. ... ... .. 65

:3-7 Angular pair distribution, g(0), at contact for particles located in the center of
the gap (1:3
:3-8 Particle contribution to the shear stress as a function of total strain upon
reversal of shear flow . .. ... ... .. 67

:3-9 First normal stress difference as a function of total strain upon reversal of the
flow ..... ............... ..... 68










3-10 Pair distribution, g(0), at contact for particles in the center of the gap
(13
3-11 Input strain wave (dotted line) and the resulting output stress wave (solid line)
plotted versus strain for simulations at #=0.60 for a gap of 15 .. .. .. 70

3-12 Particle contribution to the shear stress plotted versus total strain for oscillatory
shear flow at applied strain amplitudes of A= 0.1, 0.25, 0.5, 1, and 2 .. .. 71

3-13 Large strain viscosity in oscillatory shear flow normalized by the steady state
viscosity in steady shear flow as a function of strain amplitude .. .. .. .. 72

3-14 Normal stress normalized by the steady shear value plotted as a function of
strain amplitude for oscillatory shear flow for #=0.60 and a gap of 15 .. .. 74

3-15 Second moment of the particle distances (at steady state) from the centerline
as a function of strain amplitude ......... .. 75

3-16 The radial pair distribution as a function of the radial distance integrated over
all angles (0<0<180) for a simulation at #=0.60 and a gap of 15 .. .. .. .. 76

3-17 The angular pair distribution, g(0), at contact as a function of theta for a
simulation at #=0.60 and a gap of 15 . ..... .. 78

3-18 Mean squared displacements nondimensionalized by the particle radius, a,
plotted versus total strain for a strain amplitude of A=1 .. .. 79

3-19 Dimensionless hydrodynamic diffusivities plotted versus strain amplitude for
simulations at #=0.60 and a gap of 15 ...... .. . 81

3-20 Initial configuration of particles within the shear cell presented along with the
instantaneous steady state particle configurations for strain amplitudes of A=5,
1, and 0.1 .. .......... ........... 83

3-21 Schematic showing the sliding mechanism (a) observed for intermediate strain
amplitudes versus the 'locked 1.v. -r' mechanism (b) observed for low strain
amplitudes .. ... . .. 84

3-22 Steady state viscosity in oscillatory shear flow normalized by the corresponding
steady state viscosity in steady shear flow as a function of strain amplitude .. 86

4-1 Scanning electron micrographs of poli--i-1. n--particles .. .. .. 93

4-2 Normalized size distributions for ellipsoids with an average aspect ratio of m4
(solid) and =7 (dashed) ... . .. 97

4-3 Relative viscosity as a function of time and shear rate for polystyrene ellipsoids
suspended in a polyalkylene glycol/water/K(Cl mixture ... .. .. 98

4-4 Relative viscosity as a function of time and volume fraction for polystyrene










ellipsoids suspended in a p..hl- I11:ylene glycol/water/K(Cl mixture. The ellipsoids
have an average aspect ratio of 4 ........ ... .. 99

4-5 Relative viscosity as a function of time and volume fraction for polystyrene
ellipsoids suspended in a p..hl- I11:ylene glycol/water/K(Cl mixture. The ellipsoids
have an average aspect ratio of 7 ........ ... .. 100

4-6 Relative viscosity as a function of shear rate and volume fraction for polystyrene
ellipsoids suspended in a p..hl- I11:ylene glycol/water/K(Cl mixture. Each data
point represents the apparent viscosity after 360 seconds of shear at each
corresponding shear rate. The ellipsoids have an average aspect ratio of 4 .. 101

4-7 Relative viscosity as a function of shear rate and volume fraction for polystyrene
ellipsoids suspended in a p..hl- I11:ylene glycol/water/K(Cl mixture. Each data
point represents the apparent viscosity after 360 seconds of shear at each
corresponding shear rate. The ellipsoids have an average aspect ratio of 7 .. 102

4-8 Comparison of the relative viscosity measured in the cone-and-plate (dotted
line) and parallel-plate (solid line) geometry for suspensions of polystyrene
ellipsoids with L/d=4 in a polyalkylene glycol/water/K(Cl mixture .. .. .. 103

4-9 SEM images of processed polystyrene spheres before (left) and after (right)
washing .... ........ .............. 105

4-10 Relative viscosity as a function of shear rate and aspect ratio for different
suspension systems ......... . .. 106

4-11 Steady state relative viscosity at Per=106 aS a funCtiOn Of Volume fraction and
aspect ratio ......... ... .. 110

4-12 Steady state relative viscosity at Per=104 and 106 aS a funCtiOn of dimensionless
number density and aspect ratio ........ ... .. 111

5-1 Geometry and notation used to describe a fiber in solution undergoing simple
shear flow near a solid boundary ........ ... .. 113

5-2 Numerically calculated ensemble average of the orientation moments as a
function of Pe, ......... .. .. 121

5-3 Effect of fiber orientation on the center of mass drift velocity, x2 .. .. .. .. 122

5-4 Schematic summarizing the effect of fiber orientation on the drift velocity .. 123

5-5 Contribution of the shear flow (Pe(a~)) and Brownian torque (24(/))) to the
migration of a rigid fiber plotted as a function of Per. ... .. . .. 124

5-6 Center of mass distribution, n(Z2), plotted as a function of x2/L and Pe .. 125

5-7 Depletion thickness normalized by fiber length plotted as a function of Pe .. 126










Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

RHEOLOGY OF NONCOLLOIDAL SUSPENSIONS OF SPHERES IN OSCILLATORY
SHEAR FLOW AND THE DYNAMICS OF SUSPENSIONS OF RIGID FIBERS

By

Jonathan Mark Bricker

August 2007

C'I I!r: Jason E. Butler
Major: Chemical Engineering

The work presented in this dissertation provides a significant contribution to

understanding the theological behavior of suspensions systems. Specifically, this work

provides results front a comprehensive set of experiments on the theology of suspensions of

noncolloidal spheres undergoing oscillatory shear flow. The experiments are complimented

by results front the first dynamic simulation of suspensions of noncolloidal spheres in

unsteady shear flows. Additionally, results front experiments on the steady shear theology

of model suspensions of rigid ellipsoids are presented.

Experiments on suspensions of spheres undergoing oscillatory shear flow show that the

theology is strongly influenced by the applied strain amplitude. At each amplitude, the

steady value of the complex viscosity decreases with total strain for high strain aniplitudes

and increases for low aniplitudes. The transition point at which the qualitative behavior

changes occurs at an aniplitude-to-gap ratio between 0.1 and 0.5 and is independent

of the particle size distribution and suspension system. The steady state value of the

complex viscosity is a nonnionotonic function of the applied strain amplitude, with a

nmininiun viscosity observed at an aniplitude-to-gap ratio of 1. The experiments -II---- -1

that shear-induced migration is of no consequence, and that the observed behavior is

instead due to changes in the suspension microstructure. The theology observed in

the experiments is largely confirmed by Stokesian dynamics simulations. Simulations

of suspensions of noncolloidal spheres undergoing unsteady simple shear flows are










used to evaluate the evolution of the stresses with time along with the corresponding

microstructural development. Similar to the experiments, the shear stress drifts with total

strain before attaining a steady state that depends upon the applied strain amplitude.

The steady state viscosity obtained from the shear stresses exhibits a nonmonotonic

dependence on the applied shear rate that agrees qualitatively with the experimental

results. An analysis of the suspension microstructure at steady state reveals three distinct

microstructures that correlate to the observed theology. Hydroclusters, ordered 1.v. r~s, and

crystalline structures can all be induced by simply altering the applied strain amplitude.

Results from both simulations and experiments indicate irreversibility over the range of

strain amplitudes studied.

The theology of semi-dilute suspensions of rigid poli--r i-i. in-- ellipsoids at rotational

Peclet numbers greater than 103 WaS Studied for two different aspect ratios. The

ellipsoid suspensions exhibit shear thinning behavior for both aspect ratios. Although

in agreement with previous experiments, rate dependent theology is not predicted by

theories and simulations. Direct comparison of the results with theological measurements

on suspensions of spheres with material properties identical to the ellipsoids -II_ -r that

colloidal interactions are improbable sources of the shear thinning behavior. A mechanism

for the observed shear thinning behavior is proposed which involves a competition between

particle drift due to shear-induced migration, and thermal diffusion. For the largest value

of the shear rate, the relative viscosity scales linearly with dimensionless number density

regardless of the aspect ratio. At the lowest shear rate, the relative viscosity does not scale

linearly with dimensionless number density and a dependence on the ellipsoid aspect ratio

is apparent.









CHAPTER 1
THE RHEOLOGY OF SUSPENSIONS: PAST EXPERIMENTS, SIMULATIONS, AND
THEORIES

Introduction

Suspensions of particles exist in a vast array of materials and exhibit a variety of

useful properties that are utilized in many industrial and technological applications.

Examples of suspensions exist in biomaterials such as blood [33], and household goods

such as inks, paints, and cement [40, 88, 131]. As a result of the ubiquitous nature of this

particular class of complex fluids, many studies have been conducted to understand the

dynamics of suspension systems.

The theology of suspension systems varies widely depending on the nature of

the individual phases and the interactions among them. In general, the addition of

solid particles to a suspending liquid contributes to the total stress of the suspension

resulting in an enhanced viscosity, even at low volume fractions [13, 56, 57, 86]. In

certain applications, particles are specifically used to alter the viscosity of a substance.

For example, xantham gum is a rigid rod-like polysaccharide commonly used as a food

thickener. In certain suspension systems, the viscosity depends on the rate of deformation.

Examples of suspensions which typically exhibit rate dependent theology include systems

which contain Brownian particles [3, 16, 17, 62, 111] or particles which interact through

electrostatic interactions [71, 115, 134, 135]. Shear thinning phenomena occurs when the

applied shear rate become large enough such that the stresses disturb the equilibrium

particle distribution. Household paints, which thin upon application with a roller, are

shear thinning fluids composed of colloidal silica spheres in an organic solvent [40]. At

higher volume fractions and shear stresses, shear thickening occurs. This property is

particularly advantageous in body armor technology [44, 79, 130]. For example, Lee et al.

[89] show that by impregnating K~evlar woven fabrics with a shear thickening suspension of

colloidal silica in ethylene glycol, the effectiveness of body armor is significantly enhanced

as shown in Figure 1-1. Electrorheological fluids are another example in which interactions
























Figure 1-1. Images of woven K~evlar fabrics without (left) and with (right) impregnation of
a colloidal shear thickening suspension following ballistics testing. The image
is taken from the work of Lee et al. [89], who found that the addition of a
shear thickening fluid to K~evlar dramatically improves the efficiency of bullet
proof vests.


between particles affect the suspension theology. Electrorheological fluids contain

polarizable particles in a nonpolar suspending medium. When an electric field is applied to

the suspension, the particles form dipoles, allowing fibrous particle chains to form in the

direction parallel to the electric field as shown in Figure 1-2. These chains dramatically

increase the viscosity and cause the suspension to have a yield stress proportional to the

square of the electric field [22, 84, 139]. The unique properties of these fluids are utilized

in automotive applications such as shock absorbers and variable-differential transmissions,

and in industrial applications such as variable flow pumps [72]. Other fluid properties,

such as electrical and thermal conductivity, can be altered by the addition of particles

such as carbon nanotubes, which form lightweight high-strength composites with superior

material properties [15, 63, 67, 140].

Clearly, suspensions have a wide variety of uses owing to the unique properties that

develop as a result of the interactions between phases. Understanding the theological

properties of suspension systems is crucial to improving consumer products and industrial

processes. For suspensions of spherical particles, the literature is rich and much is

known about the theology, including macro- and microscopic details. On the other
















C1 II

1


Figure 1-2. Electrorheological suspension composed of silica spheres in corn oil taken
from the work of K~lingenberg et al. [84]. The left image shows the random
distribution of particles prior to application of an electric field and the right
image shows the formation of chains under the presence of an electric field. ER
fluids are used in a variety of automotive and industrial applications [72].


hand, suspensions of fibers are not fully understood and disparities currently exist in the

literature.

Suspensions of Spheres

In the limiting conditions of steady Stokes flow of rigid spherical particles suspended

in a Newtonian liquid, the theology is fairly well understood as reviewed by Stickell and

Powell [126]. The theology in time-dependent flows such as shear reversal and oscillatory

shear flow is less understood. For example, in oscillatory shear flow, the theology is

oftentimes unexpected and qualitatively differs from steady shear.

The phenomenon of shear-induced particle migration can he qualitatively different

depending on the nature of the flow. Shear-induced particle migration is a phenomenon

in which particles exposed to an inhomogeneous shear flow undergo a bulk migration

from regions of high shear rate to regions of low shear rate [5]. The role of shear-induced

particle migration on suspension theology can he profound and oftentimes leads to

anomalous results [116]. Shear-induced migration has been observed in steady shear flows

such as pressure-driven flow [7, 29, 7:3, 9:3, 125], narrow-gap Couette flow [28, :37, 65, 91],

and wide-gap Couette flow [1, 109]. There are currently two competing models which

describe shear-induced particle migration in steady flows. In the diffusivee flux' model [91,









Some: migration Migration to
to center, 0.07-0.38 center, 2.0-4.64
UTniform,
0.064 Migration to
Migration center, axial
to wall, 0.03 segregation, 2.0


10' 10-1 1 1
AIR

Figure 1-:3. Summary of observations of the net bulk migration of particles undergoing
oscillatory pipe flow versus dimensionless strain amplitude. The schematic is
taken from the work of Butler et al. [:30]. For reference, particles in steady
pipe flow undergo a net migration to the center of the pipe and reverse
migration does not occur.


109], the particle flux arises due to the self-diffusion of particles, while in the 'suspension

balance' model [10:3], the particle flux arises directly from the particle stress. Both models

successfully describe the steady state concentration profiles in a number of steady flows.

However, in oscillatory pressure-driven flow, experiments [:30] and simulations [97] show

that shear-induced particle migration is a function of the applied strain amplitude as

summarized in Figure 1-:3. For large amplitudes of oscillation, particles migrate to the

center of the pipe, as they do in steady pipe flow. At low strain amplitudes where one

might expect a uniform concentration profile, particles migrate to the walls of the pipe.

Currently it is unclear if existing models can account for the reverse migration observed

in oscillatory shear flow, or if a similar reverse migration phenomena occurs for other

oscillatory shear flows, such as narrow-gap Couette flow.

In addition to macroscopic theological differences, oscillatory and steady flow

of suspensions can yield distinct behavior at the microscopic level. In concentrated

suspensions of Brownian spheres, the suspension microstructure depends on the balance

between Brownian and hydrodynamic forces, as well as the flow type [1:35]. In steady

shear, simulations [62, 110, 111] and experiments [:3, 4] show three distinct microstructural










CAGED

0.60
GLASB CSACSA S S
(0.595)
ZIG-ZAG

c CRYSTAL CA C CS\ SC 8
<(0.553)


0 .5 COEX A AC C 8 S
S(0.497)
LIQUID A A C SA A
(0.470)


LIQUID
(0,412) A A A A A
0.10 I I ~BLOCKED
0.4 0.8 1.4 2.0 2.6 5.0
STAIAIN AMPLITUDE

Figure 1-4. Phase diagram taken from the work of Ackerson [3] for Brownian suspensions
undergoing oscillatory shear flow as a function of volume fraction and strain
amplitude. The symbols correspond to amorphous (A), FCC < ti--r I1 (C), and
sliding 1.07;- (S) structures. The shear and are not observed in the steady shear of Brownian suspensions at
volume fractions in which the equilibrium phase is liquid-like (4 < 0.47).


phases depending upon the ratio of hydrodynamic to Brownian forces. Amorphous

structures, string-like structures, and hydroclusters [17, 46, 95] can all be induced by

altering the ratio of hydrodynamic to Brownian forces. When exposed to oscillatory

shear flow, Brownian suspensions show an additional microstructural phase. Figure 1-4

shows the structures obtained from light scattering experiments on Brownian suspensions

undergoing oscillatory shear flow as a function of concentration and strain amplitude [2-4].

At certain strain amplitudes, < ti--r I11;1,-- structures are formed even when the equilibrium

structure is liquid-like. Such structures cannot he induced by steady shear, unless the

particle volume fraction is large enough such that the equilibrium structure is crystalline.

Although microstructural differences in suspensions of Brownian spheres are well-defined,

it is unclear whether or not similar differences occur in noncolloidal suspensions.










Oscillatory flow of suspensions often returns unexpected results, such as the presence

of a threshold for irreversibility. It is now accepted that hydrodynamic interactions cause

shear-induced self diffusion of noncolloidal particles in concentrated suspensions [5, 55,

126]. Eckstein et
the irregular motion of particles similar to random walks. In fact, the hydrodynamic

diffusivity of a tracer particle has been measured by tracking the motion of a single

particle in suspension [55, 90]. Several sources exist for the irreversibility. For example,

although the trajectories of two isolated smooth particles interacting in Stokes flow are

symmetric and reversible, three-body interactions can lead to .I-i-ii. !l ( 1; irreversible

particle displacements [26, 5:3, 91]. Irreversibility of particle trajectories can also occur as

a result of interactions between rough spheres [8, 4:3, 114]. In oscillatory shear flow, the

hydrodynamic diffusivity of noncolloidal particles is a function of applied strain amplitude.

Figure 1-5 shows results from experiments and simulations on noncolloidal suspensions

performed by Pine et
diffusivity decreases dramatically. As a result, Pine et
threshold strain amplitude below which particle trajectories are reversible. Above the

critical threshold, particles diffuse irreversibly with time, as most physical processes do

[121]. The results of Pine et
shear of noncolloidal suspensions [65, 69] which show changes in the suspension viscosity

over large strains even at small strain amplitudes, as well as with experiments and

simulations on the shear-induced migration of noncolloidal suspensions in oscillatory pipe

flow [:30, 97], which show a net migration of particles at low strain amplitudes. As a result,

it is unclear whether or not a threshold for irreversibility truly exists.

This work addresses some of the unanswered questions associated with the theology

and microstructure of noncolloidal suspensions of spheres undergoing oscillatory shear

flow. C'!s Ilter 2 contains a comprehensive set of results from experiments which show

that the theology is a strong function of the applied strain amplitude. Rheological









100

10-1 -*-


~s~ 1 0 0 o a
3 ** o

e o Dx (nu m)
010" "
a o Dz (nu m)

10-6 *De
O Dz (exp)

0 1 2 3 4 5 6
Strain amplitude, yo

Figure 1-5. Dimensionless diffusivity in the flow (closed) and gradient (open) direction
as a function of strain amplitude for concentrated noncolloidal suspensions.
Results from experiments (black) and simulations (gray) are taken from Pine
et al. [112].


changes occur over large total strains for all strain amplitudes studied. The transient

theology is attributed to changes in the suspension microstructure as opposed to a

shear-induced migration phenomena which occurs only for large strain amplitudes.

The experiments also show evidence of a nonmonotonic dependence of the steady state

viscosity on strain amplitude, which has not been reported previously. The theological

observations are confirmed by Stokesian dynamics simulations in CI Ilpter 3. In addition

to the macroscopic behavior, the suspension microstructure can he evaluated. This is the

first study to identify and correlate the theological behavior of noncolloidal suspensions

undergoing oscillatory shear flow to the suspension microstructure. The simulations

show that the nonmonotonic behavior of the steady state viscosity correlates to three

distinct microstructures induced by oscillatory shear flow. Depending on the applied

strain amplitude, < li- I11;1,-- structures, ordered 1.wr-is, or hydroclusters are observed.

Additionally, simulation results show that the particle diffusivity is finite for all strain










amplitudes, indicating that irreversibility occurs even for the smallest strain amplitude

studied.

Suspensions of Fibers

Suspensions of fibers can yield widely different properties when compared to

suspensions of spheres. For example, maximum random packing depends on particle

aspect ratio and is generally lower for fibers due to an enhanced excluded volume effect

[1:37]. The sedimentation of suspended particles also depends on particle shape. Studies

on the sedimentation of rigid non-Brownian fibers [:31, 76] show that as an initially

homogeneous suspension of fibers settles, clusters form as a result of instabilities from

hydrodynamic interactions between individual fibers; similar instabilities are not observed

in suspensions of spheres. As a result of the added complexity of suspensions containing

non-spherical orientable particles, there is considerable interest in understanding the

dynamics of such systems.

1\uch work has been done to study the dynamics of rigid fiber suspensions. In the

dilute concentration regime, where the motion of one fiber is not affected by the presence

of neighboring fibers, the theology is well understood [19, 82, 120], and quantitatively

agrees with theories [1:36]. In the semi-dilute concentration regime, where the rotation of

each fiber is severely restricted by neighboring fibers, the theology is not as well-described.

For Brownian fibers, predictions of the shear theology, which are based on theories of

diffusion [50, 59, 60], qualitatively predict correct trends, such as shear thinning behavior

[1:36], however the quantitative agreement with experiments is poor. In the absence of

thermal motion, where only hydrodynamic interactions between fibers are considered,

much larger disparities exist among theories, simulations, and experiments.

Theories on semi-dilute suspensions of non-Brownian fibers predict a viscosity which

depends upon the concentration and orientation distribution of fibers [12, 47, 120]. The

available theories do not predict a dependence of the theological properties on shear rate.

In simple shear flow, suspensions of fibers will display a decrease in viscosity with time,





initial distribution i > 0

Figure 1-6. Schematic showing the steady state spatial and orientational distribution
of fibers prior to (left) and following (right) simple shear flow. Fibers rotate
and align in the flow direction as a result of an applied shear flow. Alignment
occurs at steady state regardless of the magnitude of the shear rate.


regardless of the magnitude of the shear rate, as the orientation distribution transitions

from isotropic to one in which the particles are aligned in the flow direction, as shown

schentatically in Figure 1-6. The steady state alignment of fibers is independent of the

applied shear rate. Indeed, studies on the orientation distribution of non-Brownian fibers

in simple shear flows show that fibers orient with the flow direction indefinitely [128] with

the exception of periodic flipping motion of individual fibers [82]. As a result, the theology

of non-Brownian suspensions of fibers is not expected to be rate dependent. Simulations

of non-Brownian fiber suspensions also show no dependence of the steady state theology

on shear rate [113, 129, 141]. Experimental results however, qualitatively disagree with

theories and simulations. Figure 1-7 shows a compilation of experiments on non-Brownian

fiber suspensions, which consistently show shear thinning behavior. Interpretation of

experimental results are often complicated by the presence of non-ideal characteristics such

as large size distributions, boundary effects, and other nonhydrodynantic effects. Whether

or not a rate dependence is observed in a model suspension of fibers, is unknown. Clearly,

a study of the theology of well-defined fiber suspensions is needed.

This work addresses the issue of the rate dependent theology of non-Brownian fiber

suspensions. Results front experiments on suspensions of polystyrene ellipsoids in the

senli-dilute concentration regime are provided in ChI Ilpter 4. This is the first study in











J


+


no -Bow ia fies[0]

whc herelgyo upesosofrdlkeprile sealae it epctt h

rheology+ ofssesoso pee aigietclmtra rpris iia opeiu









migration and thema diffusion. rO









CHAPTER 2
OSCILLATORY SHEAR OF SUSPENSIONS OF NONCOLLOIDAL PARTICLES

Introduction

Studies on suspensions of noncolloidal particles have primarily concentrated on the

response when exposed to steady shear flow [34, 65, 132], thus the theology of these

suspensions is fairly well-defined as recently reviewed by Stickell and Powell [126].

In steady flow, noncolloidal suspensions at vanishing Reynolds numbers undergo a

diffusive-like motion which is due solely to hydrodynamic interactions. Whereas the

trajectories of particles undergoing simple pairwise interactions are symmetric and

reversible [14], inhomogeneities in the particle surface can cause permanent displacements

of particles from their original streamlines, resulting in irreversible interactions [43, 114].

Shear-induced diffusion of particles in steady shear flow has been observed in both

experiments [26, 55, 90] and simulations [52, 53]. The shear-induced diffusion explains the

apparently random trajectories that particles undergo in concentrated suspensions exposed

to steady shear and has been used to explain such phenomena as shear-induced particle

migration [91, 109]. This chaotic motion is intimately related to the time and spatial

configuration of the particles, closely linking the theology to the microstructure.

Whereas the theology of suspensions in steady shear is well-understood, the behavior

of noncolloidal suspensions in oscillatory flow is less understood. This is due in part to few

experiments investigating unsteady flows. Gadala-Maria [64] studied suspensions exposed

to oscillatory shear, finding that above a certain strain amplitude, the stress response

became nonlinear. The theology was found to be independent of the frequency and the

amplitude of the stress response drifted lower in time until a steady state was achieved

after approximately 100 cycles [65]. Gondret and Petit [69] found a similar drift for a

strain amplitude of 0.1, however a steady state was achieved after a much larger number

of cycles. Gondret and Petit [69] explained the observed decrease in viscosity with time as

being caused by microstructural changes due to a regular ordering of particles into 1 e. r~s










parallel to the flow direction, as evidenced by optical microscopy experiments. Breedveld

et
suspensions in a Couette geometry and found that the theology was both frequency

and amplitude dependent and that plateaus existed at both small and large aniplitudes

which were separated by a transient period in which the response signals were nonlinear.

In contrast to Gondret and Petit [69], Breedveld et
aniplitudes, the particle positions remained unchanged with no evidence of a shear-induced

microstructure. Similar to Breedveld et
hydrodynantic diffusivities from simulations and experiments indicating that at small

strain aniplitudes the particle diffusivity became increasingly small such that particles

returned to their initial positions after each oscillation. Pine et
transition at high strain aniplitudes above which the flow becomes irreversible. Narunli et


and explain the unusual output waveforms in terms of the transient responses observed in

shear reversal experiments [85, 100].

Significant gaps therefore exist in understanding the characteristic behavior of

noncolloidal suspensions in oscillatory shear flow. In this chapter, suspensions of

noncolloidal particles are exposed to oscillatory shear to investigate the response as a

function of the applied strain amplitude. The goal is to gain some understanding of

the dynamics within unsteady flows. In the following sections, we present results front

experiments of noncolloidal suspensions exposed to both steady and oscillatory shear flow.

The results are presented for five different systems in both Couette and parallel-plate

geometries using the same instrument. A detailed description of the experiment is

provided along with details concerning the characterization of each suspension system.

Results are presented for the cases of steady shear flow and oscillatory flow, and

conclusions of the work are presented in the last section.










Exp er ime nt

Three sets of particles were used to determine the effect of size distribution and

particle type on the suspension theology. Figure 2-1 shows SEM images of two sets of

the particles with size distributions for all three sets. The distributions were measured

using a Coulter counter and were verified using scanning electron microscopy. The

polystyrene and PMMA particles obtained from Sapidyne Industries were found to be

monodisperse with similarly shaped size distributions and mean diameters of 99 + 6 pm

and 100 + 8 pm, respectively. The PMMA particles obtained from Bangs Laboratories

had a much broader size distribution and a mean diameter of 61 + 16 pm as measured in

the laboratory, though the manufacturer reported a mean diameter of 83 pm. The surface

roughness was evaluated using scanning electron microscopy, and for all of the spheres

used, the size of the surface roughness features never exceeded 1pm. The sphericity was

also evaluated using scanning electron microscopy. A form factor was evaluated according

to the equation F = 4xrA,/P2, Where A, is the particle area and P is the perimeter. For

each set of particles, a similar form factor of F e 0.91 was measured, whereas a factor of 1

corresponds to a perfect sphere.

Three different suspending liquids were investigated and are listed in Table 2-1. For

the polystyrene spheres, a pl~uki I11:ylene glycol oil (Aldrich) was used and found to be

Newtonian with a viscosity of 1.87 Pa s and a measured density of 1.052 g/cm3. TWO

different suspending liquids were prepared for the PMMA spheres. The first consisted of

a mixture of equal amounts by volume water and 75-H-90000 UCON oil (Dow C'I. ..... I1).

To effectively match the densities of the PMMA and suspending liquid, sodium iodide

(25' by weight) was added to the water prior to mixing. The composition of the

suspending liquid is comparable to previous experiments [29, 30, 125]. Although the

UJCON oil/water suspending liquid was found to have a constant short-time viscosity of

0.97 Pa s, a significant drift in the viscosity with time was observed in the parallel-plate

geometry due to a slow evaporation of the suspending liquid. This drift was found to











































Diameter (ptm)


Diameter (ptm)


SENT images and respective size distributions of the particles used in the
oscillatory experiments. The left image shows the monodisperse PS particles
(also representative of the monodisperse PMMA particles). The distributions
shown are for monodisperse PS (solid line) and PMMA (dashed line). The
particles shown on the right are the polydisperse PMMA particles and their
respective distribution. The distributions were determined using a Coulter
counter and verified using SENT imaging. The number distributions are
normalized by the peak values and the scale bar on the accompanying images
is 200pm.


Figure 2-1.










be minimized by using the Couette geometry since the fluid in a Couette gap is not

directly exposed to the environment. Due to the presence of evaporation in this system, an

additional suspending liquid identical to the one used in the experiments of Han et al. [74]

was used. In this system, a mixture of approximately equal amounts by mass of ethylene

glycol and glycerol were mixed based on the measured density of the particles. The

short-time viscosity of the suspending liquid was found to be 0.0924 Pa s and although

the suspending liquid has been reported to be Newtonian in previous studies [74], a t:I' .

decrease in the viscosity was observed over 24 hours when sheared in the parallel-plate

geometry at a constant shear rate of 24s-1. A smaller decrease of 5' was observed in the

Couette geometry over the same time period and shear rate. The decrease probably results

from absorption of moisture from the surrounding air, which is minimized in the Couette

geometry since the exposed surface area to volume ratio is smaller.

A summary of the suspension systems studied is provided in Table 2-1. To confirm

close matching of densities between the suspending liquid and particulate phases,

suspensions at a volume fraction of 4 = 0.10 were set out for a period of 24 hours. In all

systems, no apparent sedimentation or buonnely was observed during this period. Unless

noted otherwise, the volume fraction of particles henceforth was set at 4 = 0.40 for all

experiments. All suspensions were prepared by gently hand mixing the particles in small

increments until a homogeneous state was reached. The suspensions were placed under

vacuum prior to testing to eliminate any air bubbles entrained within the suspension.

The maximum particle-based Reynolds number for the systems used is 10-7, so the

effects of inertia are expected to be minimal. Likewise, thermal diffusion is negligible

for the relatively large particle sizes since the minimum particle-based Peclet number is

10'. Other colloidal interactions, such as electrostatic interactions, should also be of no

imp ort ance.





















am


a m
cb


m



dO

OR



o




YU
cacr
em
++cr


5 E
y a
ooo
++ S
co
co


~66
k
HH
~ C~ C~
HH
m
~P~P~
o


O
cb
k


so










The rheometer used in all experiments was an ARES LS-1 strain controlled rheometer

(TA Instruments). Experiments were performed using both parallel-plate and Couette

geometries. The Couette geometry consisted of a rotating :34 mm diameter cup and a

stationary :32 mm diameter hoh, resulting in a gap of 1 mm. The hob height was 3:3

mm and to ensure consistent measurement, the reservoir gap was set to 10 mm in all

experiments. A 50 mm diameter parallel-plate was also used and the plate separation was

set at 1.0 mm in all cases reported. Note however, some experiments were performed for

different plate separations of 0.5 mm, 1.5 mm, and 2.0 mm. Results for plate separations

of 1.0 mm and larger matched within experimental error, indicating that boundary 1 m~lic

effects near the walls negligibly impact the reported results despite the relatively small

ratio of gap to particle diameter.

Since the theology of noncolloidal suspensions is sensitive to the microstructural

arrangement of particles and thus to the initial configuration, the suspensions were

presheared in each geometry before starting the oscillatory experiments. A steady preshear

of 24s- was emploi- II for a period of 120 seconds to reduce any effects caused by loading

the sample and to ensure that the oscillatory experiments began from a fairly consistent

configuration. This total strain of 2880 was sufficient to reach a short-time steady state in

the Couette geometry (Figure 2-2 inset), but not long enough to result in any significant

shear-induced migration in the vertical direction [91]. In the case of the Couette geometry,

a nonuniform concentration profile develops within the gap during sample loading. To

achieve a homogeneous state, a preshear is required to induce a redistribution of particles

in the radial direction [91]. Following the preshear, the oscillatory shear experiments were

performed. In each oscillatory shear experiment, the shear rate was set by specifying

the strain amplitude ,4 and frequency f. Based on the applied strain amplitude, a total

strain can he calculated by y = 4,4n, where n is the number of cycles. Unless noted

otherwise, all experiments were performed at the same frequency of f = 1.59 cycles/sec.










The temperature was maintained at 250 C using an external temperature bath. During a

typical experiment the temperature fluctuated less than 0.050 C.

Results

In this section, steady shear results are first given as a reference to the oscillatory

shear results. The oscillatory shear results are then provided in detail for a monodisperse

suspension system. Comparison of these results are made with other suspension systems to

evaluate the effects of particle size distribution and choice of particle type and suspending

fluid.

Steady Shear

The time evolution of the relative viscosity rl (defined in this work as the suspension

viscosity normalized by the short-time viscosity of the suspending liquid) of both

monodisperse and polydisperse suspension systems sheared in the Couette geometry

at a steady rate of y =24 s-l is shown in Figure 2-2. At short times the systems show

a small decrease in the viscosity followed by a short-time steady state (shown as an

inset in Figure 2-2). This short-time steady state corresponds to a microstructural

rearrangement of particles across the Couette gap [91]. From this short-time steady state,

the viscosity of the suspensions decreases continuously in time. For the suspension of

monodisperse polystyrene spheres in polyalkylene glycol, a long-time steady state occurs

after 15 hours and was found to be sustainable for at least 10 hours more. The transient

long-time decrease in the viscosity of suspensions of spheres in the Couette geometry has

been previously observed in experiments [37, 65, 109] and is the result of shear-induced

migration of particles out of the Couette gap into the stagnant reservoir [91]. Furthermore,

the long-time steady state value of the relative viscosity for the polystyrene suspension

system agrees well with the experiments of Gadala-Maria and Acrivos [65].

For the same time scale, no apparent steady state is reached for the other three

systems studied. In all cases, the observed behavior can be directly attributed to

characteristics of the suspending liquids. For example, the suspension containing













O /O
O 9-0
x D noO O O O
SO
x O 0 15 30 45 60
OA Oo time [sec]


OOO xxxxxxx




0 5 10 15 20 25
time [hours]

Figure 2-2. Relative viscosity plotted as a function of time for monodisperse PMMA
spheres in UCON/water/Nal (triangles), monodisperse poli--li-i. in.- spheres in
pl"li- I11:ylene glycol (squares), monodisperse PMMA spheres in EG/glycerol
(crosses), and polydisperse PMMA spheres in EG/glycerol (circles) undergoing
steady shear in the Couette geometry. The inset graph shows the short-time
evolution of the viscosity for the monodisperse poli--r i-i. in.- suspension. The
results are presented for 4 = 0.40 and ji=24 s-l


monodisperse PMMA spheres in the aqueous liquid shows a decrease in the relative

viscosity followed by a slow increase at long times. The slow increase is due to the

evaporation of water from the suspending liquid which eventually prevails over the

shear-induced particle migration that causes the initial decrease. In the case of the PMMA

spheres in ethylene glycol/glycerol, a small decrease in viscosity continued to occur

regardless of the size distribution of particles, although the effect was not as dramatic for

the monodisperse suspension. In both cases, the absence of a steady state for the ethylene

glycol/glycerol system can be attributed to the 5' decrease in viscosity with time for the

suspending liquid in the Couette geometry over the same time period of the experiments.

Figure 2-3 shows the time evolution of the relative viscosity for the same suspension

systems in the parallel-plate geometry for a steady shear rate of ji=24 s-l. Similar to

the observations in the Couette geometry, there are qualitative differences in the results

depending upon the system used. For example, for the polydisperse PMMA spheres in










the ethylene glycol/glycerol fluid, the relative viscosity decreases !II'.~ after 24 hours of

steady shear and no steady state was observed during this period. Over the same time

period of 24 hours, a larger decrease of 55' is observed in the relative viscosity of the

monodisperse PMMA spheres in the ethylene glycol/glycerol fluid. Since the qualitative

behavior of the suspension viscosity is the same regardless of particle size distribution,

polydispersity can be eliminated as the primary cause of the observed decrease. Another

possible origin of the behavior is related to problems with the suspending fluid; a 311l' .

decrease is observed for the suspending liquid over a period of 24 hours at a constant shear

rate of 24s- as seen in the inset of Figure 2-3. Normalizing the apparent viscosity of

the polydisperse suspension by the corresponding instantaneous viscosity of the ethylene

glycol/glycerol mixture gives a viscosity which is approximately independent of time (or

strain), -II__- -lin;~! that the decrease in viscosity is primarily due to the suspending liquid

rather than any particle migration. For the monodisperse suspension, the decrease in

viscosity of the ethylene glycol/glycerol fluid also accounts for most of the observed drop

in viscosity of the suspension system.

For the UCON oil/water system, the relative viscosity begins at a value 311' higher

than the other systems and doubles over a time scale of 24 hours; during this period,

no steady state was observed. The higher initial value of the relative viscosity is due to

the preparation of both the suspension and suspending liquid. The presence of particles

in the suspending liquid requires that the suspension be subjected to longer periods of

time under vacuum to remove bubbles as compared to the suspending liquid with no

particles. This prolonged exposure to vacuum removes water from the suspension. As a

result, an artificially high initial value of the relative viscosity is observed in the aqueous

system. This effect would be minimized at lower particle volume fractions than used here.

The increase in viscosity during shear is also due to the evaporation of water from the

suspending liquid, despite the use of a solvent trap. The extent of evaporation is shown

in Figure 2-4, in which a depletion of the suspending liquid was observed after 24 hours









35


32 0.os


20 060 12 24 ,aa A
time [hours] gaaa



0 ggy00000000000000000000000000000



0 5 10 15 20 25
time [hours]

Figure 2-3. Relative viscosity plotted as a function of time for monodisperse PMMA
spheres in UCON/water/Nal (triangles), monodisperse pohi-0 i-n in.- spheres in
pl"li- I11:ylene glycol (squares), monodisperse PMMA spheres in EG/glycerol
(crosses), and polydisperse PMMA spheres in EG/glycerol (circles) undergoing
steady shear in the parallel-plate geometry. The results are presented for
~=0.40 and ji=24 s-l. The inset graph shows the time evolution of the
viscosity for the EG/glycerol suspending liquid at a constant shear rate of
j=24 s-l


of shear. The increase of the relative viscosity with time occurs through two processes.

The suspending liquid viscosity increases as the concentration of water decreases through

evaporation, and the effective particle volume fraction increases due to a depletion in the

volume of the suspending liquid.

Whereas the other three systems showed significant changes in viscosity over time

due to the characteristics of the suspending liquid, the suspension of monodisperse spheres

suspended in pl~ili- I11:ylene glycol di; pl li-- 4I no change. The relative viscosity begins at a

value consistent with both the experiments performed in the Couette geometry as well as

the simulations of Sierou and Brady [122] and is constant over a time period of several

hours after a small change at short times. Polyalkylene glycol was subjected to the same

tests performed on the previous two suspending liquids and exhibited no change in the

viscosity over a 24 hour period.























Figure 2-4. Images of the UCON oil/water suspending liquid in the parallel-plate
geometry. The image on the left shows the sample initially loaded between
the plates. The image on the right shows the state of the sample after 24
hours of shearing at j= 24 s-l. The considerable depletion of the suspending
liquid is due to evaporation of water during shear.


Whereas the difficulties associated with measuring viscosity in the Couette geometry

are well established, whether a similar shear-migration phenomena effects measurements

in the parallel-plate geometry remains controversial. This issue is discussed in detail in the

next section, where these results are compared to other experiments and to theories.

Oscillatory Shear: Monodisperse Suspension

The strain evolution of the complex viscosity was evaluated as a function of the

applied strain amplitude. In small amplitude oscillatory shear, the calculation of

the complex viscosity requires that the waveform of the stress response is sinusoidal.

However, previous experiments [26, 65, 101] -11---- -r that at large amplitudes, the stress

output signal becomes nonlinear, leading to errors in the calculation of the complex

viscosity. As a result, the stress response was measured for several strain amplitudes at

small total strains and is presented in Figure 2-5. Slight nonlinearity is observed at an

amplitude-to-gap ratio A/H > 1 at the point where the shear direction reverses. This

behavior is related to microstructural changes upon shear reversal and is in agreement

with previous experiments investigating large amplitude oscillatory shear [101]. At A/H

< 1, the output signals are sinusoidal which is in agreement with previous observations at









low strain amplitudes [26, 65]. In this work, the complex viscosity is calculated assuming

a sinusoidal stress waveform regardless of the value of A/H. Consequently, the data

for A/H> 1 should be interpreted as an I1pp I.rent" complex viscosity as argued by

Breedveld et al. [26], and not as the complex viscosity as classically defined from the

linear theory of small amplitude oscillatory shear [98]. Additionally, the stress response is

closely in phase with the strain rate as opposed to the strain (900 phase lag) for all strain

amplitudes. As a result, the complex viscosity can be interchanged with the dynamic

viscosity rl', the portion of the response in phase with the strain rate. Although the out of

phase component rl" was finite in all experiments, it was vanishingly small relative to the

dynamic viscosity.

Figure 2-6 shows the relative complex viscosity rl; (defined as the complex viscosity

normalized by the viscosity of the suspending liquid) as a function of time measured

for a suspension of monodisperse polystyrene spheres in the Couette geometry. The

long-time results are shown for a frequency of 1.59 cycles/sec and an applied strain

amplitude (normalized by the gap width) of 0.05. Although the data presented in Figure

2-6 represents a single experiment, the theology was repeatable over multiple experiments.

Error bars included in Figure 2-6 reflect the variation between repeated experiments.

The oscillatory shear results for A/H = 0.05 show a long-time increase in the complex

viscosity. At t = 0 the observed complex viscosity is lower than the corresponding steady

shear viscosity. The complex viscosity then increases rapidly in time until a steady value,

also below the steady shear viscosity, is reached after approximately 5 hours. The time at

which this steady value is reached corresponds to a total strain of y=5000. This value of

the complex viscosity remained unchanged for at least 15 hours.

To determine the effect of the strain amplitude on the theological response, the

long-time results for A/H=0.05 are plotted with results obtained from four other strain

amplitudes while holding the frequency constant at f = 1.59 cycles/sec. Figure 2-7 shows

results over a strain amplitude ranging from A/H=0.01 to 1 and the results are plotted












































Figure 2-5.


Input strain waves (dashed line) and resulting output stress waves (solid line)
plotted versus time for the #=0.40 suspension of monodisperse polystyrene
spheres suspended in pulo~i ll:ylene glycol for A/H = 10 (a), 5 (b), 1 (c), 0.5
(d), 0.1 (e), and 0.05 (f). In all cases, the frequency of oscillation was 0.159
cycles per second. The input and output signals have been normalized by their
peak values.





















6-


5.5
0 5 10 15 20
time [hours]

Figure 2-6. Relative complex viscosity as a function of time for a monodisperse
suspension in the Couette geometry. The amplitude-to-gap ratio is 0.05
and the frequency is 1.59 cycles/sec. The experiments were performed
for suspensions of polyalkylene glycol containing polystyrene spheres at a
concentration of ~=0.40. Error hars reflect the variation observed between
repeated experiments.

as a function of total strain; the data for A/H=0.05 represents only a fraction of that

reported in Figure 2-6, which is plotted versus time. The error between measurements in

all cases is similar in magnitude to that shown in Figure 2-6.

The qualitative behavior of the response is dramatically affected by the value of ,4/H.
For example, at the highest strain amplitude (A/H=1), the complex viscosity decreases

slightly from its initial value at y=0 to a steady value that was sustainable over a large

total strain. As the strain amplitude decreases, the qualitative behavior changes such that

at A/H=0.1 the complex viscosity increases with total strain. A critical point exists at

0.1
the magnitude of the steady value also increases with decreasing amplitude such that the

steady complex viscosity nearly doubles by simply changing the applied strain amplitude

by two orders of magnitude. For example, at A/H=1 the steady value of the relative

complex viscosity is 4.6 whereas the value reaches 7.3 for A/H=0.01.


























0 5000 10000 15000


Figure 2-7. Relative complex viscosity as a function of strain and amplitude-to-gap ratio
for a monodisperse suspension in the Couette geometry. The experiments are
conducted at a constant frequency of 1.59 cycles/sec and strain amplitudes of
0.01 (crosses), 0.05 (circles), 0.1 (diamonds), 0.5 (squares), and 1 (triangles).
The experiments shown are for a suspension of pokli Ill:ylene glycol containing
polystyrene spheres at a concentration of #=0.40.


To evaluate the frequency dependence, the strain amplitude was held constant at

A/H = 0.5 and the short-time behavior was evaluated over a range of frequencies. The

complex viscosity at frequencies of 0.06, 0.159 and 1.59 cycles per second (corresponding

to 0.3768, 1 and 10 radians per second) is plotted in Figure 2-8 as a function of total

strain. The quantitative behavior of the suspension was found to be statistically

equivalent, thus independent of frequency over the range studied. The highest frequency

that is comparable with the values used by Gadala-Maria and Acrivos [65] ( f = 1.59

cycles/sec) was used in subsequent tests.

Oscillatory experiments were also conducted in a parallel-plate geometry with a gap

of 1 mm. As a result of the long times required to reach a steady complex viscosity for

the smaller strain amplitudes, only three amplitudes were studied in the parallel-plate

geometry. Figure 2-9 shows the normalized complex viscosity as a function of the total

strain for the monodisperse suspension in both geometries. For the A/H values shown,














5.2K [3--E] m= 0.06 cycles/sec
M w = 0.159 cycles/sec
I~ w = 1.59 cycles/sec

"5.1







4.9
0 100 200 300 400 500


Figure 2-8. Relative complex viscosity as a function of total strain and frequency for a
monodisperse suspension in the Couette geometry. The experiments shown
are for a suspension of polyalkylene glycol containing pokli-- i-t o spheres at a
concentration of ~=0.40. The experiment is performed at an amplitude-to-gap
ratio of 0.5.


the geometry has no qualitative effect on the time evolution of the complex viscosity. The

largest difference occurs at the lowest strain amplitude of 0.05 where the complex viscosity

observed in the parallel-plate geometry is slightly higher than that observed in the Couette

geometry. Although neither have reached a steady value by the end of the experiment at

y=7500, the greatest observed difference is only >' .

Oscillatory Shear: Comparisons Between Systems

The effect of the suspension characteristics on the oscillatory theology was studied.

The data for the monodisperse polystyrene suspension system presented in the previous

section (Figure 2-7) is first compared to a PMMA suspension system with a similar size

distribution. Figure 2-10 shows the oscillatory theology in the Couette geometry as a

function of total strain and A/H for the two monodisperse systems. The qualitative

behavior is independent of the type of particle used, as is the time scale required to reach

a steady value for each strain amplitude. Deviations from this trend do however occur at

















O
II
n
F
F


Comparison of the responses observed in parallel-plate (closed symbols) and
Couette (open symbols) geometries for a monodisperse suspension system. The
instantaneous complex viscosity is normalized by the complex viscosity at y=0
and is plotted versus total strain for amplitude-to-gap ratios of 0.05 (circles),
0.5 (squares), and 1.0 (triangles). The results are shown for a suspension
of polyalkylene glycol containing polystyrene spheres at a concentration of
~=0.40.


Figure 2-9.


the smallest strain amplitude, where the complex viscosity in the parallel-plate geometry

was observed to be lI' higher.

To determine the effect of size distribution on the oscillatory theology, a suspension

with a monomodal particle size distribution was compared to a suspension with a much

broader distribution. Figure 2-11 shows the results from suspensions of PMMA particles

dispersed in the aqueous suspending liquid as measured in the Couette geometry. The

qualitative behavior is independent of the size distribution of the suspending particles with

the largest difference occurring at A/H=0.05, where there is a noticeable difference in the

complex viscosity of the systems.

























0.9
0 2500 5000 7500


Figure 2-10. Comparison of the responses observed in the Couette geometry for
monodisperse suspension systems containing polystyrene (open symbols)
and PMMA (closed symbols). The instantaneous complex viscosity is
normalized by the complex viscosity at ,=0 and is plotted versus total strain
for amplitude-to-gap ratios of 0.05 (circles), 0.5 (squares), and 1.0 (triangles).
The results are shown for a suspension of publi I11:ylene glycol containing
polystyrene spheres and for a suspension of UCON oil/water/Nal containing
PMMA spheres, both at a concentration of #=0.40.


Discussion

Steady Shear

Although the shear-induced migration of particles in the Couette geometry is

well-documented, the existence of migration in the parallel-plate geometry is in dispute.

The diffusive flux model [91] predicts radial migration of particles to the center of the

plates where the shear rate is zero, though an additional flux term for curvature [87]

can lead to a net migration towards the edge of the plates [96]. However, the suspension

balance model [10:3] predicts no migration within the parallel-plate device. Alany have

found evidence of little or no migration [:36, :37, 87], but most recently Merhi et
did find evidence of shear-induced migration radially outward. Merhi et
the discrepancy with the previous experiments of C'!l 11pill 1' [:36] to an inadequate total

strain over which observations were made. Over a total strain of 2.4 x 106 the viscosity














1.2 -



S1.1-







0.9
0 2500 5000 7500


Figure 2-11. Comparison of the responses observed in the Couette geometry for
suspensions of PMMA spheres with monodisperse (closed symbols) and
polydisperse (open symbols) size distributions. The instantaneous complex
viscosity is normalized by the complex viscosity at ,*=0 and is plotted versus
total strain for amplitude-to-gap ratios of 0.05 (circles), 0.5 (squares), and 1.0
(triangles). The results are shown for suspensions of UCON oil/water/Nal
containing PMMA spheres at a concentration of #=0.40.


was observed to increase by 4(' by Merhi et
visual evidence of migration of particles towards the edge of the plate, but reported the

visualization for a much smaller total strain.

The results shown here indicate that there is no long term variation in the theology

within the parallel-plate geometry over a similar total strain of 106, indicating the lack of

any significant particle migration in the non-aqueous suspending liquid (see Figure 2-3).

The changes in viscosity with total strain for the other three systems shown in Figure 2-3

are not the result of a particle migration phenomena, but are instead related to identifiable

problems with the suspending liquids. The mixture of ethylene glycol and glycerol has a

viscosity that decreases with time whereas evaporation causes the increase in the viscosity

with time in the case of the aqueous suspending liquid (UCON oil mixture with water and

Nal).










The present study shows that aqueous suspending liquids are not suitable for

long-time studies in the parallel-plate geometry unless equipped with a better environmental

control than used here. Merhi et al. [96] made measurements in the parallel-plate

geometry using a suspending fluid containing an aqueous component, whether evaporation

affects their viscosity measurements and visualization studies is unclear. However, a

suspension system can be sheared between parallel-plates for very large strains without

observing a viscosity change arising from shear-induced migration of the particles.

Oscillatory Shear

The experiments show that there is a qualitative difference in the behavior depending

on the applied strain amplitude. This dependence of the complex viscosity on A/H is

more clearly demonstrated in Figure 2-12, which shows the change of viscosity between

the initial value and the final value at y=15000 for the poki-h-li-. to particle system and

y=7500 for the PMMA particle systems. The transition point for all of the suspension

systems studied occurred at a value of A/H between 0.1 and 0.5. Above these critical

values, the complex viscosity slightly decreases with total strain. At values of A/H less

than the transition point, the complex viscosity increases with total strain, with the

amount of change increasing with decreasing amplitude. This is the first time significant

changes in the complex viscosity at low amplitudes have been observed in oscillatory shear

flow. The underlying mechanism for this long-time change in viscosity is speculated upon

in the next section.

Origin of oscillatory behavior

The significant changes in complex viscosity occurring during oscillatory shear of the

suspensions are probably due to a microstructural change or rearrangement of particles

across the gap rather than a net migration of particles perpendicular to the flow direction.

Results from the Couette and parallel-plate geometries are qualitatively similar at small

amplitudes of oscillation. A direct comparison of the complex viscosity as evaluated at

y = 15000 for the polystyrene particles suspended in the polyalkalene glycol fluid appears










O Monodisperse (PS-Polyalkylene glycol)
a Monodisperse (PMMA-UCON/water/Nal)
0.4 IO Polydisperse (PMMA-UCON/water/Nal)




S0.2-








0.01 0.1 1
A/H

Figure 2-12. Difference between the final and initial values of the complex viscosity
normalized with respect to its initial value in the Couette geometry plotted as
a function of amplitude-to-gap ratio. The results are shown for all suspension
systems studied at a concentration of 95=0.40.


in Figure 2-13. For amplitudes of A/H < 5, the results are within 10'; of each other.

The close agreement at small amplitudes -II__- -0 that migration is either minimal or

has an equivalent effect in both geometries. Furthermore, experiments similar to those of

Leighton and Acrivos [91], in which the Couette reservoir was sealed with mercury, were

performed for the oscillatory shear case with A/H = 0.01. The complex viscosity was

slightly higher for the system with the sealed gap, but the qualitative behavior remained

unchanged, indicating that a net migration of particles between the Couette reservoir and

gap does not exist. Others have also concluded that shear-induced migration does not

p11l i- a significant role in determining the theology of noncolloidal suspensions exposed to

oscillatory shear at small strain amplitudes [26, 65].

Some previous studies have addressed the importance of the evolving microstructure

in determining the theology in oscillatory flows. An analogy between the theology of

suspensions under oscillatory conditions and the theology of suspensions undergoing

reversal of shear direction, where microstructural rearrangements are responsible for the
















S0.7-


S0.6-

(H Parallel-Plate
0.5 [3-E Couette



0$.o.0 0. 1 10
A/H

Figure 2-1:3. Complex viscosity (evaluated at y=15000) plotted as a function of applied
strain amplitude for the Couette and parallel-plate geometries. Values
are normalized by the viscosity for the corresponding geometry in steady
shear as evaluated at y=:3000. At all strain aniplitudes the frequency was
held constant at f=1.59 cycles/sec. The experiments were performed for
suspensions of polyalkylene glycol containing polystyrene spheres at a
concentration of ~=0.40.


transient behavior following reversal of shear [65, 101], has been made. For example,

Gadala-1\aria and Acrivos [65] successfully predicted the shape of oscillatory response

waves at high aniplitudes front results observed in the transient shear reversal experiments.

Direct observations of the microstructure in oscillatory flow have also been made. Gondret

and Petit [69] studied the oscillatory shear theology of a suspension at a single amplitude

and noticed a drift in the viscosity with time. These macroscopic observations were

compared to optical nicasurenients and the change in viscosity was attributed to a

microstructural ordering of particles in response to the oscillatory shear.

The changes in viscosity arising front the evolving microstructure are clearly due to

an irreversible process which occurs for even the smallest strain amplitude of ,4/H = 0.01;

if the suspension were reversible, no long-tinle change in the viscosity would occur. The

irreversibility, whether arising from small surface inhomogeneities on the particles [4:3, 114]










or directly from the complex multibody interactions [90], can be characterized as a

shear-induced self-diffusion of particles which exists even in the case of small amplitude

oscillations [106, 112]. Pine et al. [112] used particle tracking techniques to measure

particle displacements as a function of amplitude in oscillatory shear and found that

the resulting shear-induced self-diffusion becomes increasingly smaller with decreasing

amplitude. However, even at a strain amplitude of 0.05, the authors found a non-zero

value for the shear-induced diffusivity. Though not for simple shear flows, irreversibility

was also seen in the simulations of Morris [97] and experiments of Butler et al. [30] for

oscillatory flow in a tube. At sufficiently small amplitudes, the particles in suspension

migrated towards the bounding walls, instead of towards the centerline as occurs in steady

pressure-driven flows [29, 93, 125].

Strain dependence of the results

The attainment of a steady complex viscosity for these suspensions should depend

only upon the total strain, rather than time, since only hydrodynamic forces are assumed

to act upon the particles. For nearly all strain amplitudes and systems, the relative

viscosity quickly increases to a nearly stationary value within a total strain of 2500.

However, the relative complex viscosity changes a noticeable amount between the strain of

2500 and the last strain of either 7500 or 15000 for the smallest amplitudes. Consequently,

the relative viscosity values plotted in Figure 2-12 may not have reached a steady value,

though the suspensions have been sheared for very large times.

The dependence of the total strain required to reach a steady value for the complex

viscosity on the applied strain amplitude implies that at low strain amplitudes, previous

measurements in oscillatory flows may not have been reported with respect to steady

values. For example, Gadala-Maria and Acrivos [65] reported reaching a steady state

within 100 cycles, but the total duration of their experiments is unclear. One hundred

cycles, at the strain amplitudes studied by Gadala-Maria and Acrivos [65], corresponds

to only a fraction of the strain required to reach an unchanging complex viscosity found










in the experiments here. For example at a strain amplitude of 0.05, the complex viscosity

does not reach a steady value until 2.5 x 104 cycleS as shown in Figure 2-6. Breedveld

et
viscosity was nearly independent of strain amplitude for low amplitudes, whereas Figure

2-12 shows very different behavior. Once again though, it is uncertain whether or not the

measurements reported by Breedveld et
values since the total strain is not reported.

Other sources of these differences may be linked to the concentration differences

and frequencies emploi-, .1 in the studies. Breedveld et
on a suspension at a volume fraction of 0.50, which could exhibit significantly different

theology from the volume fraction of 0.40 studied here. We investigate the dependence of

the theology on concentration in the next section. Also, Breedveld et
measurements at higher frequencies for the smaller amplitude studies than were used

here. However, the study of dependency of the results on frequency shown in Figure 2-8

demonstrates independence with respect to frequency. Furthermore, the theology for

suspensions exposed to the same strain rate amplitude, but different amplitudes and

frequencies (for example, comparing the data for A/H= 0.05 and f = 1.59 Hz in Figure

2-7 with the data for A/H= 0.5 and f = 0.159 Hz in Figure 2-8) is different, -II__- -r;!

that the strain amplitude is the relevant parameter, not the strain rate amplitude. Finally,

another source of the differences might he the different systems used by the different

researchers. This possibility was also explored, as discussed next.

Effect of suspension characteristics

For concentrated suspensions, the theology is qualitatively independent of the

particle concentration. Figure 2-14 shows the complex viscosity at y,=15000 plotted as

a function of applied strain amplitude for concentrations ranging from #=0.20-0.45.

At all concentrations except ~=0.20, the viscosity is a nonmonotonic function of the

applied strain amplitude, with the behavior becoming more apparent as the concentration










~ =().45
n ~=().4()














0.1 1
A/H

Figure 2-14. Complex viscosity (evaluated at y=15000) plotted as a function of applied
strain amplitude and concentration for the parallel-plate geometry. Values are
normalized by the viscosity of the suspending liquid. At all strain amplitudes
the frequency was held constant at f=1.59 cycles/sec, and the experiments
were performed for suspensions of polyalkylene glycol containing polystyrene
spheres.


increases. Likewise, the minimum value of the complex viscosity occurs at A/H=1,

indicating that for 4 >0.20, the theology is qualitatively independent of concentration.

The results are also plotted versus volume fraction in Figure 2-15. As a reference, the data

is plotted along with results from the K~rieger-Dougherty relation [86],


Dr = 1 -(2-1)


with the volume fraction for maximum pI .11; :', =0.64. For all strain amplitudes,

the relative complex viscosity increases nonlinearly with concentration. For the largest

strain amplitude (,4/H=5.0) as well as for the smallest (A/H=0.05), the volume fraction

dependence is similar to predictions from steady shear.

The results also indicate that there is little dependence of the oscillatory theology

on the particle size distribution. Polydispersity has been found to phI i- a significant role

in steady shear theology highlighted by its effect on shear-induced migration. Recent





























Figure 2-15. Complex viscosity (evaluated at y=15000) plotted as a function of
concentration for the parallel-plate geometry. Values are normalized by the
viscosity of the suspending liquid. At all strain aniplitudes the frequency
was held constant at f=1.59 cycles/sec and the experiments were performed
for suspensions of pub~l- I11:ylene glycol containing polystyrene spheres. For
reference, results front the K~rieger-Dougherty relation [86] are also plotted
(solid line).


examples include particle size separation [81] and radial migration in a parallel-plate

geometry [87]. The results here show that for oscillatory shear, the size distributions

studied have no effect. This is in contrast to the oscillatory shear experiments performed

by Gondret and Petit [70] who found that bidisperse suspensions show different dynamic

viscosities depending on the size distribution of the particles. The disparity between this

work and the present work may be due to differences in the frequencies and concentrations

as discussed previously.

The oscillatory shear theology was affected by suspension characteristics only at low

strain aniplitudes. These differences never surpass 1CI' for any of the comparisons made,

-II--- _f l-r;:: that the theology is still relatively independent of the systeni chosen, although

the theology may be more sensitive to these factors at strain aniplitudes smaller than










those studied here. However, as the strain amplitude is lowered, the experimental error

increases which makes comparisons between systems difficult at small strain aniplitudes.

Conclusions

The theological behavior of noncolloidal suspensions in oscillatory shear flow was

evaluated by measuring the shear stress as a function of total strain. The theology was

evaluated for five different suspensions as well as for a variety of flow geometries including

Couette and parallel-plate. As a reference, the theology under steady shear was also

evaluated in both geometries. For steady shear in the Couette geometry, suspension

viscosities decreased with time in qualitative agreement with previous experimental

studies and models. The steady shear results in the parallel-plate geometry show that

shear-induced migration of particles is either non-existent or has a very small effect.

The oscillatory shear theology depends strongly on the magnitude of the applied

strain amplitude such that at high aniplitudes, the complex viscosity decreases with total

strain whereas at low aniplitudes, the complex viscosity increases with total strain. The

critical aniplitude-to-gap ratio at which the qualitative behavior changes occurred at

0.1 < A/H < 0.5. Although researchers have observed drift in the viscosity of suspensions

in oscillatory shear before, this work is the first to present evidence of non-nionotonic

behavior. Additionally, this work shows that the amount of strain required to reach

a steady complex viscosity is much larger than previously reported, ell__- -1 it.-; that

previous measurements may not have been reported with respect to their steady values.

Furthermore, the oscillatory shear results were independent of the geometry, -II---- -1 it,-

that shear-induced migration is of no consequence. Instead, the changes in complex

viscosity with strain are due to rearrangenients in the microstructure as it evolves with the

oscillating shear flow. To complement the work presented here, simulations are currently

being conducted to investigate details of the role of shear induced diffusion on the theology

of suspensions exposed to unsteady flow.










CHAPTER :3
CORRELATION BETWEEN STRESSES AND 1\ICROSTRITCTITRE IN
CONCENTRATED SUSPENSIONS OF NON-BROWNIAN SPHERES SITBJECT TO
UNSTEADY SHEAR FLOWS

Introduction

The steady shear theology of concentrated suspensions of non-Brownian spheres is

generally well-understood as a result of extensive experimental and computational research

as suninarized hv Stickell and Powell [126]. For unsteady shear the observed theology,

primarily provided by experiments, is less understood.

K~olli et
of shear flow and found that the shear stress decreases to a nxininiun value before

returning to the steady state value. A similar transition was found in the normal stress

difference, which changed sign before returning to the steady state value. Explanations

concerning the origin of the observed theology [65, 100] are largely confirmed by the

simulations performed in this chapter. In steady shear flow, an excess of particles develops

in the compressional quadrant upstream front a test particle [2:3, 12:3], as opposed to the

extensional quadrant downstream front the particle as illustrated in Figure :3-1. Upon

reversing the direction of flow, the excess of particles is found in the extensional quadrant

until the microstructure rearranges in accordance with the new flow direction. This sudden

microstructural change results in the transient theology observed in the shear reversal

experiments.

Interpreting the theology of suspensions undergoing oscillatory shear flow is more

problematic. One issue concerns the irreversibility of suspensions at small strain

aniplitudes. Pine et
hydrodynantic diffusivities and concluded that particle trajectories are irreversible only

above a critical strain amplitude. However, theological experiments by Bricker and Butler

[28] demonstrated irreversible behavior below the critical amplitude reported by Pine et




















Compressional Extensional
Quadrant Quadrant

Figure 3-1. Schematic showing particle pairs exposed to simple shear flow. The particle
with the dashed line border is approaching the stationary test sphere and
is in compression, whereas the particle with the solid line border is moving
away from the test sphere and is in extension. The particle anisotropy arises
as particles preferentially spend more time in the compressional quadrants
as opposed to the extensional quadrants. The schematic shown is symmetric
about the angle of 450.


concerns the non-monotonic dependence of the viscosity on the applied strain amplitude.

Bricker and Butler [28] found that as the strain amplitude decreases, the complex viscosity

decreases until a minimum value is observed at a strain amplitude of one. For smaller

strain amplitudes, the complex viscosity was found to be higher. Though clearly related to

microstructural changes, this theological behavior is not currently understood.

In this chapter, we elucidate issues concerning the theology of suspensions in unsteady

shear flow using Stokesian dynamics simulations of a munubt~ll.~ U. We present the theology

along with the corresponding microstructural details for flow with reversal of shear after

attainment of steady state and oscillatory flow. For oscillatory shear flow, we present

results for the viscosity as a function of strain amplitude and find qualitative agreement

with the non-monotonic behavior observed in experiments [28]. Additionally, we discuss

the issue of irreversibility for the suspension system. These findings are discussed in detail

along with an interpretation of more complex flows in terms of the theology observed here.

Rheology Simulations

The Stokesian dynamics method accounts for the motion of rigid spheres in a

Newtonian solvent by calculating the particle interactions necessary to describe the










hydrodynamic forces transmitted through the fluid [25]. This simulation method has

been used to calculate the theology for non-Brownian systems [24, 52, 5:3, 122, 12:3],

Brownian particulate systems [62, 111], and even for the simulation of the theology of

electrorheological systems [22].

In this work, the Stokesian dynamics method is used to simulate shear flows of

suspensions of non-Brownian spheres constrained to move in the velocity-gradient plane;

we refer to this as a (]r'risesl li. r) simulation. Extensive computational time is required to

complete simulations at lower strain amplitudes as a result of the large strains required

to achieve a steady state microstructure [28]. The simplification to a ]rn..rsul li.~ r provides

substantial savings in computational time by reducing the number of degrees of freedom

associated with each particle while still maintaining an accurate description of the

relevant physics [24, 97, 10:3, 111, 12:3]. In addition to predicting the theological behavior

of a variety of systems, monolayer simulations accurately predict the corresponding

microstructure. For example, ]ritursul li.~ r simulations correctly predict the formation of

'hydroclusters' in shear thickening colloidal suspensions [16]. For comparing ]risnes..I li.~ r

simulations to fully three-dimensional experiments, the corresponding volume fraction is

2/3 times the areal fraction [10:3, 111].

We briefly discuss the Stokesian dynamics method for bounded flows and provide

details concerning the implementation of the method, along with alterations made to

accommodate unsteady shear flows.

The Stokesian Dynamics IVethod

Shear flow is simulated by explicitly including walls in the calculations as done

by Nott and Brady [10:3] and 1\orris [97] for simulating pressure driven flows and by

Singh and Nott [12:3] for simulating shear flow. As sketched in Figure :3-2, the walls are

permeable and represented by chains of spheres which have an imposed velocity in the

.r-direction which can generally be a function of time t. The velocities of the wall particles

in the y and x directions are set to zero. Since periodic boundary conditions are used, a












CU,(t)

OO U Up 0o
O gap with bulkpatce



wall particles Uy=Up
*ll layer of clear fuid



Figure :3-2. The periodic cell used in the simulations. Wall particles are white and freely
suspended particles in the bulk of the suspension are filled. A clear 11s- fluid of the same height as the gap between the walls is introduced to enable
simulations using periodic boundary conditions.


clear 11s-c v of fluid is placed at the bottom of the cell as shown in Figure :3-2; the height

of this clear 111-c v is set equal to the height of the gap in which the bulk particles are
suspended.

The forces on the wall particles and response of the bulk particles are simulated using

the Stokesian dynamics method. In this method, a far-field mobility matrix, ~MO, is

formed, inverted to give a resistance matrix, and then modified to account for near-field
interactions ,

R = (~M")- + Rab R 3-1)

where R~b is a tensor containing the near-field 2-hody interactions for all particle pairs

and R" is the far-field component which is subtracted to avoid double counting. The

details of the formation of the grand resistance matrix, R, is discussed in detail hv
Durlofsky et al. [54].

Singh and Nott [12:3] replaced the 2-hody lubrication interactions between the

suspended and wall particles with the lubrication interactions for a sphere interacting with
a flat wall, at least for the force-velocity couplings; it is not clear that the stresslet-velocity










couplings were likewise altered. In the present work, no such alteration was made. The

interactions between the wall and suspended particles were treated using sphere-sphere

lubrication interactions. Direct comparisons between the results of Singh and Nott [123]

and the results produced using bumpy walls indicates that the impact on the solution for

the rheologfical properties is negfligfible.

The resistance matrix in Equation 3-1 relates the instantantaneous forces, torques,

velocities, rotational velocities, and stresslets using the equation


F R"," R"fy R"" E" sF E su

(3-2)




In this equation, the forces and torques, F, and stresslets, S, are separated into vectors

for the bulk and wall particles (denoted by a superscript s and w, respectively). The force,

F", on the suspended particles is zero, other than the inclusion of a short-range repulsive

force as discussed in the next section.

The velocities, U", of the wall particles are set as shown in Figure 3-2. Since the flow

is driven by the motion of the walls, the mean velocity (u) is set to zero to ensure that no

other driving forces act on the particles, preserving simple shear flow. Simulations were

also performed using the alternative constraint,


F" = 0, (3-3)
Nw

where NV, is the number of wall particles. No significant differences occurred as a result of

the constraint chosen. Similar to Singh and Nott [123], simulation results show that the

velocity profile is linear except small deviations near the wall. The rate of strain has been

set to zero since the motion of the wall particles generates the shear.










Equation 3-2 can be solved for the unknown velocities of the suspended particles,


U" =(%",,)- [F %",", (U" (u))] + (u) ,(34


and the unknown forces acting upon the walls to maintain the velocity U",


F" = R", (U" (u)) + %"" -("-(). (3-5)


The stresslets of the bulk particles, S", also determined from Equation 3-2, are given by


SS = R", (US (u)) + Rsu (U" (u)) (3-6)


Evaluation of Rheology

The shear stress can be evaluated using either the forces on the wall particles [123]

or the stresslets acting on the suspended particles [24]. For the wall force evaluation, the

stress is



N,





where the stresslet has been separated into the hydrodyamic portion and a portion which

arises from application of the short range repulsions defined in Equation 3-15. Although

the calculation according to the stresslets represents the particle contribution to the stress,

the calculation based on the wall forces represents the total stress. For the remainder

of the paper, only the particle contribution to the stress will be reported, thus the total

stress in Equation 3-7 is reduced to the particle contribution by subtracting the portion

of the wall force F" corresponding to the prescribed velocity U" (second term given in

Equation 3-5).










The hydrodynantic stresslet is defined by Equation 5-12 with the velocities eliminated

front the equation and the terms due to the repulsive forces removed,





The non-hydrodynantic contribution is


s' = RM (RE,)- F" + (zFs) (3-10)


Furthermore, the mean hydrodynantic stresslet is given by [24, 12:3]


(sH __ H1


and the mean contribution to the stress front the interparticle force is


(S") = S. 312)


In addition to the shear stresses, direct evaluation of the normal stress acting on the

walls is given by
N,
o-, = F" .(:313)
i= 1 4,
The first normal stress difference NI~ can then he evaluated front the stresslets,


1V = Oz Opy

-,[ (sH P )( ~$-Y) H ) SP)

The added value of using walls is that the normal stress acting on the walls can he

evaluated directly front the wall forces. When not using wall particles, but fully periodic

systems with driven shear, the normal stress difference NI~ can he calculated with ease

front the difference in the stresslets acting on the particles. However, to directly calculate

o-, requires the value of the particle pressure, the isotropic portion of the normal stress

components.










Implementation

All lengths are made non-dimensional by the particle radius a, except for the strain

amplitude A, which is normalized by the gap spacing. Time is made dimensionless by the

inverse of the maximum shear rate y, and the components of the resistance matrix are

made non-dimensional following the scalings of Bossis and Brady [23]. Simulations were

performed using a dimensionless time step of at = 1.0 x 10-2. To prevent particle overlap

during the course of the simulations and to replicate the behavior of a more realistic

system [122], an interparticle force is incorporated between the bulk particles,


Frp) FO T a (3-15)


The separation distance, e, is given as e = |rop| 2, where rep is the magnitude of the

separation distance between the centers of sphere a~ and sphere P. For am 1! I G~~y of the

work, the values of the parameters specifying the range and magnitude of the force are set

to -r = 100 and Fo = 1.0 x 10-4, TOSpectively. The values chosen for -r and Fo are equivalent

to the work of Singh and Nott [123] and thus allow for easy comparison to their work for

steady shear flow. The effect of varying the range and magnitude of the repulsive force are

discussed in a later section. The repulsive force exists among the bulk particles, but not

between the bulk particles and the particles forming the wall.

The far-field interactions are updated every 10 time steps as in the work of Nott and

Brady [103], while the near-field lubrication interactions are updated at each time step.

In all studies, the number of wall particles is 14. Each simulation begins from a random

distribution of particles. For the steady shear and shear reversal cases, the gap is set to

a height of 30 and the theology is studied for a range of concentrations. For oscillatory

shear flow, the areal fraction is set to 4 = 0.60 and the gap is set to a height of 15 to

reduce computation time. Results for different gap heights in steady shear show that there

is little or no dependence of the theology on the two choices of gap spacing. For all shear

flows, the theology is reported as an average over several runs. Since the level of averaging










for each of the shear flows is different, each will be listed individually in the following

sections.

The boundary conditions depend on the shear flow. For steady shear flow, the top

and bottom wall velocities are assigned the same magnitude (UT; = |1|) but act in opposite

directions, thus initiating a simple shear flow. To evaluate the transient theology during

shear reversal, a suspension is sheared until the attainment of a steady state after which

the top and bottom wall velocities change sign but maintain their original magnitude. For

simulations of oscillatory shear flow, the walls are assigned a time dependent velocity in

the form of a sinusoid as done in the pressure-driven flow simulations of Morris [97],


Up' = UTcos (wt) (3-16)


where ET is the nmaxiniun wall velocity and w is the frequency of oscillation. The

nmaxiniun wall velocity at each strain amplitude is held constant and equal in magnitude

to that used in the steady shear case.

For oscillatory shear flow, stresses are calculated twice during each cycle. The forward

calculation occurs when the dintensionless top wall velocity is Up' = 1 while the backward

calculation occurs when Up' = -1. The theology is evaluated as a function of total strain,


Yo~to = E |eo~at) dt,(3-17)


and strain amplitude, ,4.

A fourth-order Runge-K~utta integration method is intpleniented in place of the

Adanis-Bashforth predictor method coninonly used in Stokesian dynamics [22, 122].

Simulations using this method confirm that particles do not overlap for the time step used

and tests show that the theological behavior is convergent.

Results

Using the Stokesian dynamics method as described earlier, we calculate the theology

of suspensions within steady shear flow and compare with results front Singh and Nott










[12:3] to verify the code and provide a reference to the results from unsteady shear flows.

We then investigate the transient theology following a reversal of shear after attainment of

steady state and oscillatory shear flow.

Steady Shear

Though studied extensively [24, 122, 12:3], simulation of the theology of noncolloidal

suspensions in steady shear flow is repeated using the present code. Each data point

shown in this section represents an average over a set of 10 runs calculated over a

dintensionless time of 5000. Each run consisted of a different initial configuration

of particles and the theology was calculated using data averaged over the last 2000

dintensionless time units.

Comparisons shown in Figure :3-:3 for the particle contribution to the shear stress, or,,.,

as a function of areal fraction, 4, demonstrate that the results closely match those of Singh

and Nott [12:3] for a gap of :30. The error bars on the calculated stresses represent the

variation in the mean of 10 individual runs. For the concentrated systems, the calculations

indicate a slightly higher value than Singh and Nott [12:3] for 4,,, perhaps due to the

use of hunipy walls. Furthermore, the results show that the stress can he calculated front

either the stresslets on the bulk particles, S ,., or front the tangential forces acting on the

wall particles, F". For the concentrated systems, the error between the two evaluation

techniques is approximately 5' .

The normal stress a,, as calculated front the normal force F, acting on the wall

particles is also in close agreement with the results of Singh and Nott [12:3], though there

is once again a noticeable difference for higher concentrations as seen in Figure :3-4. In this

case, the difference is most notable for 4 = 0.60 rather than at 4 = 0.65 where the results

agree fairly well. For concentrations lower than 4 = 0.30, the normal stress is nearly zero.

The first normal stress difference, NI~ = e.,:, a,,, is shown in :3-5. The value of the

normal stress difference is negative for steady shear flow at all concentrations, though the

value of NI~ is vanishingly small for 4 < 0.3, both of which are characteristics in agreement



























0 0.1 0.2 0.3 0.4 0.5 0.6


Figure 3-3. Particle contribution to the shear stress, a,,, as a function of areal fraction,
~, for steady shear flow. Results are shown from the calculation based
upon an analysis of the wall forces and the bulk particle stresslets. The
error bars represent the variation observed between 10 simulations. For all
concentrations, the gap is 30.


with the experimental observations of Zarraga et al. [142]. Aside from the total normal

stress difference, contributions from the hydrodynamic and repulsive force as defined

in Equations 3-11 and 3-12 are plotted. The repulsive contribution has little effect on

the total value for concentrations up to = 0.60. At this concentration, the repulsive

contribution is clearly positive and hence, the absolute value of the hydrodynamic

contribution exceeds the absolute value of the total. The normal stress differences are

compared to the results of Singh and Nott [123]. As with the previous shear stress results,

the NI~ values closely match.

In addition to the theology in steady shear flow, the microstructure was investigated

and compared to previous work. The radial dependence of the pair distribution function,

g(r), is plotted as a function of areal fraction in Figure 3-6. An excess of particles

exists at distances near contact. As expected, g(r) approaches 1 as the radial distance

increases. An areal fraction dependence is also apparent. As 4 decreases, the excess































0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
areal fraction, 4

Figure :3-4. Normal stress o-y, as a function of areal fraction for steady shear flow.
Results are shown from the calculation based upon an analysis of the wall
forces. Comparisons are made with the results of Singh and Nott [12:3].


S-1


0.1 0.2 0.3 0.4
areal fraction, 4


0.5 0.6 0.7


Figure :3-5.


Normal stress difference NI~ as a function of areal fraction 4 for steady shear
flow. Also shown are the contributions from the hydrodynamic and repulsive
forces. Results are shown from the calculation based upon an analysis of the
wall forces. Comparisons are made with the results of Singh and Nott [12:3].


E-E Total
u-g Total, Singh & Nott
GH Hydrodynamic
t Hydrodynamic, Singh & Nott
Mn Repulsive
0 Repulsive, Singh & Nott











500

$=40
400~ \ --- =50

30 -- ~6

200 \




20-



-3 -2 -1 0
log(r-2.0)

Figure :3-6. Pair distribution, g(r), as a function of radial distance and areal fraction
integrated over all angles. Data is presented for particles in the center of the
gap (1:3

near contact becomes less evident, a feature consistent with the previous work of

Sierou and Brady [122] who did three-dimensional Stokesian dynamics simulations of

noncolloidal suspensions. The agreement of the simulation of monolayers with those of

fully three-dimensional simulations indicates that the two-dimensional calculation produces

the correct trends.

The angular pair distribution near contact, g(0), is plotted as a function of areal

fraction in Figure :3-7. As evident in the simulations of Singh and Nott [12:3], results show

a clear .I-i-inin.! It-y in the angular pair distribution, with particle pairs found preferentially

in the compressional quadrant (Oo < < o0). This .l-i-int...~ I1y becomes progressively

less pronounced as the areal fraction decreases from = 0.65 to = 0.40, a result

consistent with the three-dimensional simulations of Serion and Brady [122]. Although

the pair distributions in Figure :3-7 are reported for particles located in the center of the

gap, calculations of g(0) as a function of position show that as particles approach the

walls, a deviation from bulk properties is apparent. This effect is likewise observed in



















S400-


200-



0 50 100 150
0 (degrees)

Figure 3-7. Angular pair distribution, g(0), at contact for particles located in the center
of the gap (13 simulations conducted with a gap of 30. The compressional quadrant is given
by Oo < 0 <900.


the simulations of Singh and Nott [123]. These results show that the inclusion of bumpy

walls, as opposed to smooth walls, has some effect on particles near the walls, as would be

expected.

Shear Reversal

Results are now presented for shear reversal flows in which the theology is evaluated

after reversing the direction of the shear (-y) following steady state at a shear rate y.

The responses are measured immediately following reversal of shear and are presented as a

function of total strain. All data represents an average over 10 runs, each beginning from a

different initial configuration.

The particle contribution to the shear stress is plotted in Figure 3-8 following shear

reversal for four areal fractions. The shear stress, normalized by the steady state shear

stress in steady shear flow, is calculated from the particle stresslets. The instantaneous

(7total = 0) shear recovery value of the shear stress is equal to that obtained in the

previous shea~r direction? at steady state (i.e. a,,/4temav = 1). From~ a dimelnsionless strain?


















0.9



I:~ 1 =0.60
0.8 -I ..
| --- =0.65



0.7
0 2 4 6 8 10
Ytotal

Figure 3-8. Particle contribution to the shear stress as a function of total strain upon
reversal of shear flow. The results are presented for a range of areal fractions
and for a gap of 30. The data represents an average over 10 simulations.


of yattl = 0, the shear stress decreases to a nxininiun value before rapidly increasing

to a steady state. For all areal fractions studied, the steady state value following shear

reversal equals the steady value prior to reversing the shear direction. Although not

presented here, simulations through a strain of astor~ = 20 were performed and show an

oscillation about o-,,./o-gody i consistent with the experimental work of K~olli et
[85], the nxininiun in the shear stress following shear reversal shifts to larger strains for

smaller concentrations. In addition, the magnitude of the nxininiun value decreases with

increasing concentration, a characteristic also in agreement with K~olli et
concentrations studied, a steady state value is reached by a strain of astor~ a 6 which is

higher than the value of 4 found by K~olli et
transient shear stress is evident in the experiments whereas the simulation results show a

slight overshoot for the smaller areal fractions studied.

The first normal stress difference, NI~, following shear reversal is normalized by

the steady state normal stress difference in steady shear flow and plotted in Figure



















/ -- #=0.40
-0.5- 7,--- #=0.50

-1 ---- #=0.65

-1.5-
0 24 6 810
Total

Figure 3-9. First normal stress difference as a function of total strain upon reversal of the
flow. The results are presented for a range of areal fractions and for a gap of
30. The data represents an average over 10 simulations.


3-9. The results are based on an analysis of the wall forces and the level of averaging

is the same as that reported in Figure 3-8. Immediately following reversal of the shear

direction, Ni~/Nistena" is negative. This corresponds to an NI~ that is initially positive

following a reversal of shear direction since in the steady shear case, Nistead" is negative

[123, 142]. Furthermore the magnitude of the relative value of NI~ at 7total = 0 varies with

concentration such that it increases with increasing areal fraction. For all areal fractions,

Ni/Nlste""" decreases in magnitude as 7total increases until Ni undergoes a transition in

which it reverses sign and becomes negative, eventually reaching the steady state value

achieved in the previous shear direction (NI~/N~te"d" = 1). The total strain at which

Ni~/Nistead" changes sign depends on the areal fraction. For example, at = 0.65 the first

normal stress difference changes sign prior to 7total = 1 whereas for 4 = 0.40, the sign

change occurs later, at a total strain of a 2.

Figure 3-10 shows the transition of the pair distribution function, g(0), near contact

as a function of total strain and angle upon reversing the shear direction for a suspension

at a concentration of = 0.60. At 7total = 0, the distribution is equivalent to the

















oi
a 400 Itotl=0.0


Mo ytotal=3.3
200-
Ctota,,,l=6.0O



0 20 40 60 80 100 120 140 160 180
0 (degrees)

Figure 3-10. Pair distribution, g(0), at contact for particles in the center of the gap
(13 direction. The compressional quadrant is given by 0"<0<900. The results
are for a simulation of #=0.60 and a gap of 30.


distribution at steady state under steady shear conditions (Figure 3-7), except the excess

of particles is now found in the extensional quadrant (900 < 0 < 1800) due to the sudden

change in shear direction. At 7total = 1, the pair distribution function is nearly symmetric,

showing little preference between the compressional and extensional quadrants. This

approximately corresponds to the strain at which the shear stress following shear reversal

is minimum (Figure 3-8) and the first normal stress difference is zero (Figure 3-9).

The microstructure rearranges rapidly for the new flow direction and by a strain of

total = 1.7, the .I-i-inin.! Ir y reappears, with particle pairs once again concentrated in the

compressional quadrant. The microstructure nearly reaches steady state at 7total = 3.3 and

the small changes taking place from 7total = 3.3 6.0 correspond to the strain over which

the shear stress reaches steady state. The same is true for NI~, however the sign of the

first normal stress difference is sensitive only to the nature of the .I-i-all...~ r -y in g(0). At

7total = 6, the microstructure is completely re-established and resembles the steady shear

microstructure. The maximum value of g(0) in the compressional quadrant is equivalent





















A =1.0





A =0.5

Figure 3-11. Input strain wave (dotted line) and the resulting output stress wave (solid
line) plotted versus strain for simulations at #=0.60 for a gap of 15. The
stress waves are normalized by their peak values.


to the maximum value observed in the extensional quadrant immediately following shear

reversal (y,,,, = 0).

Oscillatory Shear

Rheology

The input strain waves and resulting output stress waves are plotted versus strain for

different strain amplitudes 24 in Figure 3-11. The shear stress waves are reported as an

average over 8 experiments, each beginning from a different random initial configuration.

To avoid artifacts from the start-up of flow, the waves are reported for a time period

immediately following the first 10 cycles of shear. Stress waves presented for ,4 = 0.5 are

representative of the waveforms for A < 0.5 and are not reported. For the remainder of the

oscillatory shear flow analysis, the theology is reported with respect to the value when the

wall velocity is maximum in both the forward and backward directions.

The output stress waves are in phase with the strain rate (900 phase lag from the

strain input) at all strain amplitudes, which is consistent with experimental work [28, 64].

At ,4 = 5, the shear stress is noisy at the point where the instantaneous strain value is
















0.9-


0 */
It

S0.6: = .


--A 02


0.0 10000 20000 30000 40000
Ytotal

Figure :3-12. Particle contribution to the shear stress plotted versus total strain for
oscillatory shear flow at applied strain amplitudes of A= 0.1, 0.25, 0.5, 1,
and 2. The shear stresses are normalized by the steady state values of the
shear stress under steady flow conditions. Results are shown for simulations
at #=0.60 and a gap of 15.


zero (corresponding to maximum wall velocity). Below this value, the stress response

becomes smooth. For the range of strain amplitudes, there are no apparent nonlinearities

in the stress response such as those previously observed in the large amplitude oscillatory

shear experiments of Narumi et al. [101]. However, the stress waveform at A = 1 shows a

slightly flattened response when the instantaneous value of the strain is zero. Furthermore,

the noisy stress output at the largest strain amplitude makes it difficult to detect the

presence of a nonlinear response.

Figure :3-12 shows the strain evolution of the particle contribution to the shear stress

for different strain amplitudes. The shear stresses are normalized by the steady state

values of the shear stress for similar conditions (~ = 0.60, gap=15) under steady shear

flow. The shear stresses are reported for the forward direction (UY = 1 for the top wall)

only. Calculation of the shear stress was also made in the backward direction, and matches

those reported in Figure :3-12 for all strain amplitudes studied. The strain evolution of the
















"1-



0.8 .


0.6-



0.4
0.1 1 10
A

Figure 3-13. Large strain viscosity in oscillatory shear flow normalized by the steady state
viscosity in steady shear flow as a function of strain amplitude. Simulations
are for ~=0.60 and a gap of 15, and the error hars represent statistical error
between eight runs at each strain amplitude. For comparison, the simulation
results are plotted along with the normalized viscosities front the experiments
of Bricker and Butler [28].


shear stress for ,4 = 5 is not shown due to the large fluctuations observed at nmaxiniun

wall velocity (Figure 2-5).

The initial value of the shear stress in oscillatory shear flow is 45' of the steady

state value of the shear stress in steady shear flow. Fr-on this initial value, the shear stress

increases with ytotal until a steady state is observed after a total strain that depends on

the value of the applied strain amplitude. As the strain amplitude decreases, the total

strain required to reach a steady state value for the shear stress increases dramatically.

For example, the steady state shear stress for ,4 = 2 is achieved quickly as it increases

front ,os /eg ad" = 0.47 at ytortl = 0 to a steady state value of cT /ag at"d" = 0.76 after

a total strain of only 1600. At the smallest strain amplitude studied (,4 = 0.1), the shear

stress increases continuously front q;; /g ad" = 0.49 at 7jtortl = 0 anld no0 obSer^vable Steady

state is reached over a total strain of 40000.










The values of the shear stress observed at large 7total in Figure 3-12 are used to

evaluate the dependence of the viscosity on the applied strain amplitude. To obtain

these values for the shear stress, a moving average was calculated for each of the curves

in Figure 3-12. The values in Figure 3-13 corresponds to the point of minimum slope

over a portion of the curve at the end of each run. The shear stress values are averaged

over 8 runs beginning from different random initial configurations. The error bars on the

simulation results represent the error observed over the 8 different runs. The results are

presented along with the experimental work of Bricker and Butler [28] for comparison.

Although the simulations are for a ]rlrn..n .1w. r of spheres, an areal fraction of 4 = 0.60

roughly corresponds to the volume fraction of 4 = 0.40 used in the experiments of Bricker

and Butler [28].

The simulation results show non-monotonic behavior that is qualitatively similar to

the experiments. 1Vost noticeably, a minimum in the large strain viscosity is observed

at A = 1. At this strain amplitude, the value of the large strain viscosity observed in

the oscillatory shear flow is approximately equal to one-half of the steady shear value.

From this point, the large strain viscosity increases with increasing strain amplitude as

it approaches the steady shear value (pose ~~"des = 1). The simulation results over this

range of strain amplitudes show remarkable agreement with the experimental work. In

some cases, the error bars on the simulation values overlap the values obtained from

experiments using a parallel-plate geometry. For A < 0.5, the simulation results begin to

deviate from the experiments, showing a much steeper increase in viscosity with decreasing

strain amplitude.

Th~e stealdy state ~normal stress n~ormalized by th~e steady shear value (eosq eady)"

is plotted versus strain amplitude in Figure 3-14. The level of averaging is equivalent

to that done for the shear stress data in Figure 3-13. The normal stress in steady shear

flow has a negative value, indicating that the stress is compressive. The normal stress at

the highest strain amplitudel is alson comnpo;resiv (aOC seady 0.81). From A = 5, the
















0.5-



: o-



-0.5-

0.1 1 10
A

Figure 3-14. Normal stress normalized by the steady shear value plotted as a function of
strain amplitude for oscillatory shear flow for #=0.60 and a gap of 15. The
error hars represent statistical error front eight runs at each strain amplitude.


normal stress decreases rapidly as the strain amplitude decreases to intermediate values.

By a strain amplitude of 1, the value of the normal stress difference is essentially zero.

Front this point, o-g continues to decrease with decreasing strain amplitude. At ,4 = 0.5

the qualitative behavior of of~ changes such that a sign change occurs. For ,4 < 0.5 the

average value of the normal stress is tensile, however whether the stress changes sign or

becomes vanishingly small for 4 < 0.5 remains unknown even for the level of averaging

done here. Some consequences of a transition in the normal stress front compressive to

tensile under oscillatory flow conditions is discussed in a later section.

Suspension microstructure

To assist in understanding the theological behavior in a suspension undergoing

oscillatory shear flow, the corresponding microstructure is studied. Figure 3-15 shows the

second moment of the instantaneous particle distances front the centerline as a function

of strain amplitude. The data represents an average over 8 runs and the error bars reflect

the deviation among the different runs. The moments are calculated for a steady state

distribution of particles and as a reference, the value of the second nionent for a random
















S4.8-

V 4.6-

~4.4-

4.2-


0.1 1 10
A

Figure :3-15. Second moment of the particle distances (at steady state) from the centerline
as a function of strain amplitude. The results are for simulations at #=0.60
and a gap of 15. The error hars represent statistical error from eight runs at
each strain amplitude.


distribution is 4.2. As the strain amplitude increases, the particles move closer to the wall

on average. The particle distribution shifts away from the wall towards a more random

distribution at smaller strain amplitudes such that a 10I' difference is observed between

the second moment at A = 5 and ,4 = 0.1. Although not shown, the particle distributions

are symmetric about the centerline.

The pair distribution functions are evaluated for each strain amplitude. Figure :3-16

shows the radial pair distribution function, g(r), at steady state integrated over all angles

(0 < 0 < 180) as a function of radial distance, r. To avoid artifacts from the presence of

the wall, g(r) is calculated for particles located near the center of the gap (5.5 < y < 9.5)

and the results reflect averages over 8 runs at each strain amplitude. For A > 2 the radial

pair distribution shows an excess of particles at close contact. This is an expected result

and resembles g(r) for the case of steady shear (Figure :3-6). As the strain amplitude

decreases however, particle pairs become more separated. For example, the peak value of
































A = 5.0






4 -3 -2 -


A = 2.0






4 -3 -2 -1


'


400


200


0
10


I


300


150


0
10


A =1.0


A = 0.5






-2 -1


0L
0 -3


-2 -1


CW A = 0.25 A = 0.1
20 20-





10 0
-3 -2 -1 0 -3 -2 -1 0

log (r-2.0)


Figure 3-16. The radial pair distribution as a function of the radial distance integrated
over all angles (0<0<180) for a simulation at ~=0.60 and a gap of 15.

The particle pairs are calculated for particles near the center of the gap
(5.5









g(r) at A = 0.1 occurs at a separation distance two orders of magnitude larger than A = 5.

Notice also that the scale on the y-axis changes dramatically for A < 1.

Figure 3-17 shows the angular pair distribution, g(0), for all strain amplitudes at

radial distances corresponding to the maximum in Figure 3-16. For example, at A > 2 the

angular pair distribution is calculated for r < 2.001, whereas for A < 1 the distribution is

calculated for r < 2.1. The sole exception is for the case of A = 0.25, where the calculation

is performed for a larger radial distance of r < 2.5 as a result of the difficulty in defining

a peak value in g(r). This an~ lli--;- results in the difference in scales in Figure 3-17. The

data is averaged over eight runs and for each strain amplitude, g(0) is calculated for

different sections across the gap.

Three distinct microstructural regimes occur. At high strain amplitudes (2 < A < 5)

the angular pair distribution resembles the steady shear case (Figure 3-7), there is an

.I-i-all...~ r -y in g(0) as a result of an excess of particles in the compressional quadrant. As

the strain amplitude drops below A = 2, the microstructure changes significantly. The

angular pair distribution becomes symmetric for A < 1, and for 0.5 < A < 1, an excess of

particles is found at 8 = 0, 90, and 1800. The angular pair distribution is approximately

equal at all three angles indicating that there is no preference between particles arranging

themselves perpendicular to the flow direction as opposed to parallel to the flow direction.

A third regime is observed for low strain amplitudes (0.1 < A < 0.25), where particle pairs

are in excess at 8 = 0, 60, 120 and 1800. Again, there is no preference for finding pairs in

any of these orientations except for particles close to the bounding walls, where there is a

slight bias of pairs in the compressional and extensional quadrants corresponding to 8 = 60

and 1200. For all strain amplitudes, there is a noticeable effect of the presence of the wall

on g(0).
Discussion

We discuss key issues regarding oscillatory shear theology of noncolloidal suspensions

of spheres. Specifically, we concentrate on the irreversibility of the suspensions, the



























OUU


\.''''
A=5
.1 '. r
~
\


, .--'/-- A= 2


400 250



0 30 60 90 120 150 180 0 30 60 90 120 150 180
30001 2000

I~ I 1 /
bl) 150 -- 10


0 0
0 30 60 90 120 150 180 0 30 60 90 120 150 180
80001 5000
A = 0.25 A = 0.1

4000 -4 1-1 2500 .


0 0
0 30 60 90 120 150 180 0 30 60 90 120 150 180

6(degrees)


Figure 3-17. The angular pair distribution, g(0), at contact as a function of theta for
a simulation at ~=0.60 and a gap of 15. The angular pair distribution is
calculated for particles in three different regions of the, 7.5 line 9.52,

g(0) is calculated for separation distances of r<2.001 while for A<2, g(0) is
calculated for r<2.1. In the case of A=0.25, the angular pair distribution is
reported for particle separations of r<2.5. The compressional quadrant is
defined by Oo go0"












0.08-


S0.06-
V

c~0.04-


V 0.02 ~-- '



0 10 20 30 40
Ytotal

Figure :3-18. 1\ean squared displacements nondimensionalized by the particle radius, a,
plotted versus total strain for a strain amplitude of ,4=1. The results are for
simulations at #=0.60 and a gap of 15. Data is shown for both the flow (.r)
and gradient (y) directions. The solid lines represent linear regression of the
data.


non-monotonic dependence of viscosity on strain amplitude, and implications of the

change in sign of the normal stress on current shear-induced particle migration models.

In the last section, we briefly discuss relationships between shear reversal and oscillating

shear flows.

Strain Dependent Diffusivity

Precise reversibility is a delicate issue and the work presented here does not prove

or disprove reversibility of ideal suspensions of perfect, hard spheres interacting purely

through hydrodynamics. However, the simulations presented here do reproduce the

irreversibility observed in the experiments of Bricker and Butler [28]; irreversibility is

evident from the rheologfical changes that occur with strain amplitudes as low as 0.1.

Sources of irreversibility in the experiments could include the breakdown of lubrication

and subsequent contact between rough spheres [4:3, 1:38] as one example. Irreversibility in

experiments may also result from small effects of particle inertia or thermal fluctuations.










Numerous possible sources of irreversibility exist in the simulations as well. The inclusion

of a short-ranged repulsive force is the most prominent among these. Additional

possibilities include approximations inherent in the Stokesian dynamics method and

the update rate of the far-field calculations. Furthermore, an explicit numerical integration

scheme has been used.

To clearly quantify the level of irreversibility in these simulations, the hydrodynamic

diffusivities are calculated. Figure 3-18 shows mean square displacements plotted versus

strain for a strain amplitude of A = 1 for both the flow and gradient directions. The mean

square displacements are calculated for the first 10 cycles of shear and are averaged over

50 runs, each beginning from a different initial particle configuration. Differences in the

values of the mean square displacements for the x and y directions occur as a result of the

anisotropy of the flow. In both cases, the mean square displacements increase linearly with

strain. These characteristics are similar for all strain amplitudes except A < 1, where the

relationship between the mean square displacements and strain deviates slightly from a

linear dependence.

The mean square displacements are used to calculate dimensionless hydrodynamic

diffusivities in the flow direction according to [112]

(xy)-x~ +a)=a 2Dz. (3-18)


Similarly, Equation 3-18 is used to calculate diffusivities in the gradient direction, the

results are plotted for each strain amplitude in Figure 3-19. The diffusivities are based on

an average over 10 sets of experiments and the error bars never exceed the symbol size.

For A < 1, the diffusivity changes slightly with strain as a result of the nonlinearity of

the mean square displacements. For the purpose of this work, the diffusivity reported for

A < 1 is an average of the mean diffusivities calculated after each cycle. This additional

averaging is represented in Figure 3-19 as an open symbol. Although the calculations

in Figure 3-19 were performed using 64-bit double precision, additional calculations of











100m







S10-4


O O O



106 0.1 1
A

Figure 3-19. Dimensionless hydrodynamic diffusivities plotted versus strain amplitude for
simulations at #=0.60 and a gap of 15. The data is plotted as an average
over 10 sets of experiments and is shown for both the flow (x) and gradient
(y) directions. Open symbols represent diffusivities that have been calculated
from slightly nonlinear mean square displacements.


the hydrodynamic diffusivities at A = 0.1 were performed using both higher and lower

precision levels. No dependence of the diffusivities on the precision level of the calculation

was observed. Furthermore, the hydrodynamic diffusivities reported are convergent with

respect to the time step.

A strong dependence of the diffusivity on strain amplitude is apparent for 0.25 <

A < 2. Over this range, D, and D, decrease by 4 and 3 orders of magnitude, respectively.

This dependence weakens in the high (A > 2) and low (A < 0.25) strain amplitude limits

such that plateaus exist over the remaining range of strain amplitudes. The simulation

results are qualitatively similar to the work of Pine et al. [112] for A > 0.5, the lack

of quantitative agreement may be due to differences in the details of the simulations.

Most notably, Pine et al. [112] report diffusivities from three-dimensional simulations,

whereas our simulations are restricted to the flow and gradient directions. Additionally

in our work, suspensions are bounded by walls in the gradient direction, whereas in the










simulations of Pine et
also occur due to the form of the input strain wave. Pine et
square-wave protocol as opposed to the sinusoidal form used here. Other factors might

include the implementation of an interparticle force, the level of averaging performed, time

step, and frequency of far-field updates, all of which are not reported by Pine et
Pine et
with numerical accuracy.

Suspension Microstructure and Rheology

The steady state oscillatory shear viscosity has a nonmonotonic dependence on the

applied strain amplitude (Figure :3-13). The behavior of the steady state viscosity in

oscillatory shear can he understood in terms of the suspension microstructure. Depending

upon the applied strain amplitude, we observe one of three distinct microstructures:

hydroclusters, ordered 1~-;isu, or a phase consisting of a locally ordered ( of i-- We explain

each structure in detail in the context of previous studies and correlate the microstructures

to the observed theology.

Snapshots of the instantaneous particle configurations at steady state for various

strain amplitudes are shown in Figure :3-20. The microstructures are captured when the

wall velocity is maximum. An example of an initial configuration (^/total = 0) is provided

as a reference; the particle positions are random and no microstructure is observed. The

suspension microstructure evolves from this liquid-like configuration to one of three

different microstructural regimes.

The first regime occurs for high strain amplitudes (2 < A < 5), where chains of

particles are apparent in the compressional quadrant. The microstructure clearly resembles

hydle n 1I1,r 11),_ which dominates the steady shear of suspensions of noncolloidal spheres

[24, 104] and the high shear limit (Pe c o) of Brownian spheres [16, 110, 111]. Similarity

with steady shear is expected since the strain experienced by the suspension during one

























000 000

~O OOOD

O oO O


O


Initial configuration


A = 5.0


A =1.0


A = 0.1


Figure 3-20. Initial configuration of particles within the shear cell presented along with
the instantaneous steady state particle configurations for strain amplitudes
of A=5, 1, and 0.1. The results are reported for simulations at #=0.60 and
a gap of 15. The shaded particles illustrate the structures formed by the
oscillatory shear at different strain amplitudes. Arrows at the boundaries
indicate the instantaneous direction of the shear flow.














(a)"' (b)"'- '"
Figure ~ ~ ~ ~ ~ ~ ~~~'"' 3-21.'' Sceai soigth ldngmcaim a bere o itreiaesri
ampitde vesste:oke -: ehnim()osredfrlwsri
ampitde. The trjcoyo h cne fmso a tetprtce(dse
line). in.. oseillator shear.::'~ isgvnb tehl ine. The top rowof particles i
in"' a'"' shear pln ih ihrsha aethnterw eo n tearw







FInthe second rhegaime hwhing he ocusliigmcaim()os for intermediate strain apiue 05<, )

the microstrutures rerrnes ito rdjetred 11- ve alinted in th flow s diretion. Tdahis
strutureise simia to the ordee structr iien within Brown iane sythem trwh the pthermal
and~i hdonamicforce lane (Peh a 1)he [110, 111. Ihn t our simulation the onlyw

effet o alerngt the parameterso thae rpulsiv e focei disecussed uing thnexmlte sction.

Shear-indcled bordering has als o be alobstervdi ostr illato ryhearrg bexprienthesha on

noan-colloecidashee.GndtanPti[6]igdthstdytteptilitruio

of patice s undergoing, osiatoryrshr flow wanted notied tai qasiperioudies ordering

ofshe ires in 111-< re arrallel ando perpend li;icuart algnthe fow direction. Th reing

ws explareine sini trsof the indertia srcondry fhi Bows tat occur as a e reut of finite

Red ynolds numbe frce [105e(e~ )[10]. On ur simulations hwvrso thatti ye ofodrng lyr

ores noven ate Re= her inertiral fet are absient.e ndhdoynmcfrcs h





In tphe th ird reg-;ime pat alle st prain apitudes(0.




a crystal-like structure at steady state (Figure 3-20) as a result of the small oscillatory










displacements of the particles. Similarly, experiments on Brownian suspensions of

hard-spheres indicate that ordering can he induced by oscillatory shear even for cases

where the equilibrium structure is liquid-like. For example, Ackerson [:3] observed

shear-induced ordering in oscillatory shear flow for volume fractions as low as 0.47,

even though the transition between a liquid-like and crystal structure occurs at a critical

volume fraction of at least 0.49 [2]. In the non-Brownian system, shear-induced ordering

occurs at an areal fraction of 4 = 0.60, which is well helow the corresponding liquid phase

transition for two dimensions at 4 m 0.69 [6, 80] for Brownian particles at equilibrium.

Consequently, the results seem to indicate that oscillatory shear induced ordering is a high

Pe phenomena. For example, the ordering in Brownian systems is limited to conditions

where thermal fluctuations are weak and do not disrupt the shear-induced structure, such

as at higher volume fractions or shear rates (high strains) where the Pe is relatively large.

Each of these microstructures correlates with the observed theology. The enhanced

viscosity at high strain amplitudes results from the formation of hydroclusters which

are primarily controlled by lubrication forces. These types of structures are explained in

detail by Foss and Brady [62]. The minimum viscosity at A = 1 is due to the formation

of ordered l o,-;-is, which allow spheres to easily slide past one another. Figure :3-21 shows

the sliding mechanism that is responsible for the minimum viscosity. This mechanism

of particle 1.,-;-re sliding past one another results in a minimum viscosity in Brownian

systems [111] as well as for colloidal systems in which the particles are electrostatically

repulsive [71]. At low strain amplitudes, the viscosity is again enhanced as it is for high

strain amplitudes. However, the structure differs significantly. In this regime, the enhanced

viscosity is a result of local crystal-like ordering of spheres, which lock particles into a

hexagonal pattern (Figure :3-21). This crystalline structure inhibits particle motion, even

at small strain amplitudes, where maximum displacement is on the order of a particle

radius. Although for much stronger shear flows, the resulting theology resembles the












Mn Fo= 1.0 x10 t= 100
EO Fo= 1.0 x10~ ,t= 10
1- ( ) Fo=1.0x10-3, t= 10


0.8-


S0.6-


0.4-


0.2
0.1 1 10


Figure 3-22. Steady state viscosity in oscillatory shear flow normalized by the
corresponding steady state viscosity in steady shear flow as a function of
strain amplitude. The results are reported for a range of inter-particle
repulsive forces, which are determined by altering the range, -r, and
magnitude, Fo. All simulations are for #=0.60 and a gap of 15.


increase in viscosity in simulations of suspensions of sheared arrays of electrostatically

repulsive particles [71].

The poor agreement between the theology in simulations and experiments (Figure

3-13) at low strain amplitudes is possibly a consequence of limiting the simulations to a

]riselric. li.~ r. For example at low strain amplitudes, the steady state microstructure in the

experiments of Bricker and Butler [28] may be a face-centered cubic or body-centered

cubic structure, and thus not accessible in ]rn..sic.11i.~ r simulations. As a result, the planar

crystal-like ordering at low strain amplitudes corresponds to a steady state viscosity which

overshoots the value found experimentally.

Effect of the Repulsive Force on the Rheology

The theology is qualitatively insensitive to the repulsive force used. Regardless of the

repulsive force, the relative values of the steady state viscosity at each strain amplitude

are qualitatively similar, as are the angular pair distributions. The repulsive force does










however affect the relative particle separation distances at steady state as well as the

magnitude of the steady state viscosity. The results for different repulsive forces are

plotted in Figure 3-22. The repulsive force is determined by altering the parameters -r and

Fo in Equation 3-15, which correspond to the range and magnitude, respectively. Each

parameter was changed independently and the theology was evaluated for steady shear,

and for oscillatory shear for strain amplitudes of A = 0.1, 1, and 5. For comparison, the

results are plotted along with those obtained from setting Fo = 1.0 x 10-4 and -r = 100

(Figure 3-13).

The steady state viscosity is a nonmonotonic function of the applied strain amplitude

with a minimum observed at A = 1 regardless of the values chosen for -r and Fo.

The corresponding microstructure at each strain amplitude is also independent of the

repulsive force and resembles those reported in Figure 3-20. Specifically, the angular pair

distribution at each strain amplitude is independent of the repulsive force such that all

three microstructures of hydroclusters, ordered 1., r-is, and local crystal-like ordering are

observed at the same values of A.

The repulsive force does however have an effect on the magnitude of the steady

state viscosity. The effect is primarily controlled by the range of the repulsive force. In

general, the greatest difference in the steady state viscosities occurs for A = 0.1, where

rlosslsse end ranges from a 0.5 to a 1.2 depending upon the repulsive force used. At higher

strain amplitudes, the dependence becomes less evident. The radial pair distribution

is also affected by the repulsive force such that the particle separation distance varies

with varying repulsive force. As the repulsive force is increased by either increasing Fo

or decreasing -r, particle pairs become more separated and lubrication forces phI i- a less

significant role, leading to an overall decrease in the steady state viscosities.

Normal Stresses

The simulation results show that normal stresses may change sign in oscillatory shear

flow (Figure 3-14). Such behavior potentially has implications for models of shear-induced










particle migration in more complex flows. For example, results from experiments [:30] and

simulations [97] on suspensions exposed to oscillating pipe flow show an anomalous

migration phenomenon in which particles migrate toward the wall for small strain

amplitudes. This contrasts with the case of unidirectional pipe flow, in which particles

migrate toward the center [29, 9:3, 125]. The migration phenomenon in oscillatory pipe

flow at low strain amplitudes is not captured by diffusion models of shear-induced particle

migration [91, 109]. However, 1\orris [97] so__~-r-- I that if the normal stresses change sign

during oscillatory pipe flow, the anomalous migration could be explained in terms of the

spatial variation of the particle-induced normal stresses as proposed in the suspension

balance model of Nott and Brady [10:3]. In unidirectional pipe flow the particle normal

stress is compressive; for the suspension balance model to predict a 'reverse' migration of

particles as seen in small amplitude oscillatory pipe flow, a rearrangement of the particle

microstructure must occur and result in a particle normal stress that is tensile.

In experiments, it is difficult to measure normal stresses in noncolloidal suspensions

as a result of their finite but small values, which are often close to the sensitivity limits of

the instrument [85, 124, 142]. Furthermore, only normal stress differences can he reported

as a result of the inability to resolve the isotropic component of the normal stress due

to the hydrostatic pressure. An advantage of simulations is that values of the normal

stresses can he reported. Statistically, the sign change in the value of the normal stress

in oscillating simple shear flow is marginal mainly because the magnitude of the normal

stress is so small. Nevertheless, the results indicate that the average value of the normal

stress transitions from compressive to tensile. Although a sign change in oscillatory

shear flow has not been measured experimentally, there are some indirect indications

that the transition does occur. Shear reversal experiments show a short-time transition

in the normal stress immediately following reversal of the shear direction; the normal

stress becomes tensile before rapidly returning to compressive. The relationship between

oscillatory and shear reversal theology is discussed in more detail in the next section.










Relationship Between Oscillatory and Shear Reversal Rheology

Several authors have attempted to link the transient theological response following

shear reversal to the oscillatory theology. Gadala-Maria and Acrivos [65] predicted the

shape of oscillatory response waves at high strain amplitudes from the transient results

observed in shear reversal experiments. Similarly, Narumi et al. [101] reported on the

theology of suspensions undergoing large amplitude oscillatory shear and -II_0-r-- -1.. that

the theology could be described by a rearrangement in the particle microstructure, as

it occurs in shear reversal experiments. As the suspension is sheared in one direction,

particles form weak hydroclusters in response to the flow. When the direction of the shear

flow changes, the microstructure is destroyed, after which a new microstructure develops

during the half-cycle.

The results for shear reversal show that for small strains, the transient behavior of

the stresses rapidly change corresponding to a simultaneous change in the microstructure.

In the previous section, we speculated on the possibility of the presence of tensile stresses

for oscillatory shear flow at small strain amplitudes. The simulation results for the

transient theology following shear reversal show that immediately upon reversing the

direction of shear, the normal stress is initially positive before quickly changing to a

negative steady state value. This change from a tensile to compressive stress in shear

reversal flows is explained by the pair distribution function. As the direction of shear is

reversed, the quadrants are switched such that the 1 in H. B~y of particles are temporarily

in extension, leading to a tensile stress. As the shear flow continues in this direction, the

pair distribution transitions back to a structure in which particles are preferentially in

compression, resulting in a compressive stress. Thus, the shear reversal results provide

further evidence that the sign change for the normal stress in oscillatory shear flow may

exist.

The behavior of the viscosity in oscillatory shear flow is more complex. The steady

state microstructure at high strain amplitudes resembles the steady shear case as does the










viscosity. In relation to the microstructural changes that take place in shear reversal flow,

this corresponds to the structure at steady state, where the microstructure has rearranged

for the new flow direction. At small strain amplitudes, the theology in oscillatory flow

becomes much more interesting. In this case, the correlation between the transient

response of the microstructure during shear reversal and the oscillatory theology is

less understood. Even at the lowest strain amplitude studied, the oscillatory shear

microstructure is much more complex than the instantaneous microstructural arrangement

following shear reversal.

Conclusions

We simulated a ]rlrn~ l..s.I r of non-Brownian spheres undergoing simple shear flows

using Stokesian dynamics. The theology and corresponding microstructure was evaluated

for steady shear, shear reversal, and oscillatory shear flow. Results from steady shear

flow verified that the method was implemented correctly and provide a reference to the

unsteady flow results. When exposed to shear reversal, the stresses undergo a transient

period immediately following reversal of shear direction before quickly returning to

the steady state value. Simulations are in agreement with experiments [85, 100] and

confirm that this transient theology corresponds to a rapid change in the suspension

microstructure that occurs upon reversing the shear direction.

For oscillatory shear flow, theological changes occur over the entire range of strain

amplitudes studied. Specifically, at each strain amplitude, the shear stress increases with

strain before attaining steady state. The irreversible behavior for the simulation conditions

studied agrees with the experimental findings of Bricker and Butler [28]. The steady state

shear stresses show a non-monotonic dependence on the applied strain amplitude that

qualitatively agrees with experimental results [28]. The steady state theology is correlated

with the steady state suspension microstructure and three distinct regimes are observed

over the range of strain amplitudes studied. At high strain amplitudes, the theology

resembles the steady shear theology. At intermediate strain amplitudes, a minimum in










the viscosity occurs corresponding to an ordering of the suspension microstructure in the

direction parallel to the flow. Surprisingly, at low strain amplitudes an enhanced viscosity

occurs corresponding to a microstructure with local ordering, where particles are partially

trapped by their nearest neighbors. The steady state normal stresses also depend on the

strain amplitude and become tensile at small strain amplitudes.









CHAPTER 4
RHEOLOGY OF SEMI-DILITTE SUSPENSIONS OF RIGID POLYSTYRENE
ELLIPSOIDS AT HIGH PECLET NITMBERS

Introduction

Experiments on the theology of senli-dilute suspensions of rigid fibers at large

rotational Peclet numbers consistently show a decrease in viscosity with increasing shear

rate [:35, 66, 119]. Distinctly different behavior is predicted by theories and simulations.

Theories evaluating the stress generated in a senli-dilute suspension of rigid non-Brownian

fibers [12, 47, 120] do not predict a dependence of the suspension viscosity on shear rate.

Likewise, simulations on senli-dilute suspensions of fibers [11:3, 141] find no dependence on

the shear rate, even when allowing mechanical contact between fibers [129]. Though the

theories and simulations are performed in the limit of infinite rotational Peclet number,

the results are expected to qualitatively predict the results of experiments at sufficiently

large rotational Peclet numbers.

To account for the shear rate dependence observed in some specihec experiments,

explanations such as fiber flexibility [18] and fiber adhesion [:35] have been offered.

However, the seemingly universal nature of the shear thinning phenomenon remains

unclear. Interpretation of results front experiments are further complicated by deviations

front model systems, which can effect the observed theology. For example, suspensions

are typically composed of heavy fibers suspended in a light liquid [66], resulting in fiber

sedimentation. Furthermore, fibers typically have large length scales which can result in

complications front fiber breakage if using fragile materials [119] and boundary effects [48].

Clearly, a study of the theology of well-defined fiber suspensions is needed.

In the following sections, we report on the theology of well characterized polystyrene

ellipsoid suspensions in the senli-dilute concentration regime. The theology of suspensions

of spheres with the same material properties as the ellipsoids are also evaluated and

compared. A description of the fabrication of the fibers is provided along with a

characterization of the particle size and distribution. Results are presented for two

































Figure 4-1. Scanning electron micrographs of polystyrene particles. The top image shows
the polystyrene spheres before processing. The bottom images show ellipsoids
with average aspect ratios of e4 (left) and =7 (right). The scale bar in each
image is 2 pm, with the exception of the bottom right image, which has a scale
bar of 1 pm.


different particle aspect ratios. We observe shear thinning behavior in suspensions of

rigid ellipsoids at rotational Peclet numbers greater than 103. A discussion of possible

mechanisms for the rate dependent theology along with scalings of the steady state

viscosity is given, and conclusions are presented in the last section.

Exp er ime nt

Polystyrene ellipsoids were manufactured using the method of Nagy and K~eller [99].

Monodisperse polystyrene spheres (Polysciences, Inc.) with a mean diameter of 1.06 +

0.02 p-m were added to a mixture of 5' by weight polyvinyl alcohol ( \! P Biomedicals,

Inc.) in water. The mixture was dried to make thin films which were deformed uniaxially

at a temperature of 1900C to a desired draw ratio. Ellipsoidal particles were obtained

by dissolving the stretched films in a t:I' by volume mixture of isopropanol in water.









To limit the amount of residual polyvinyl alcohol from the particle surface, the particles

were repeatedly washed and then centrifuged. The issue of the role of residual polyvinyl

alcohol on the observed theology is discussed in detail in a later section. A more detailed

description of the procedure is presented elsewhere [77, 99]. This particular method allows

for the production of ellipsoids with aspect ratios and size distributions which are highly

controllable [77].

To study the effect of varying the ellipsoid aspect ratio on the theology, different draw

ratios were used, resulting in ellipsoids with average aspect ratios, L/d (where L is the

length of the long axis and d the short axis), of 4.17 + 0.81 and 7.14 + 1.49. Scanning

electron micrographs of the spheres and resulting ellipsoids are shown in Figure 4-1.

Normalized size distributions of the particles are given in Figure 4-2, and average values

of the particle volume and surface area for each aspect ratio are tabulated in Table 4-1.

All length scales are obtained from scanning electron microscopy and the values represent

averages over 50 random particles.

The suspending liquid consisted of UCON 50-HB-5100 oil (Dow C'I. a!Im dls) blended

with 10I' by volume distilled water and 1 mM potassium chloride. The components were

chosen in part to match the density of the particles and suspending liquid. The measured

density of the suspending liquid was 1.06 g/cm3 and the reported particle density was

1.056 g/cm3, thus buoi- ill i- effects are negligible over the time scale of the experiments.

The suspending liquid slightly shear thins, exhibiting a decrease in viscosity from 1.93

Pa s to 1.75 Pa s over the range of shear rates 0.1 s-l < y < 400 s-l. Although the

difference over the shear rate range is less than 10I' the relative viscosity, rlr, reported in

the remainder of the paper is defined as the effective viscosity of the suspension at a given

shear rate normalized by the viscosity of the suspending liquid at the same shear rate. To

avoid effects from evaporation of the suspending liquid, the duration of the experiments

were minimized and a solvent trap was used in each experiment.





















Sco aC





L 00 a3


co3
++








E co

oco









E to
OO






co
++









All suspensions were prepared by gently hand mixing the ellipsoids in small

increments until a homogeneous state was reached. The suspensions were placed under

vacuum prior to testing to eliminate any air bubbles entrained within the suspension. In

the semi-dilute regime, defined by 1 < nL3 < L/d (where n is the number density of

ellipsoids) [51], the interparticle spacing is such that the ellipsoids are unable to rotate

freely without being impeded by neighboring ellipsoids. The theology was studied for

concentrations within the semi-dilute concentration regime corresponding to nL3 = 1.00 to

4.10 for ellipsoids with an aspect ratio of 4, and nL3 = 1.13 to 6.95 for ellipsoids with an

aspect ratio of 7.

The maximum particle-based Reynolds number for the suspension systems is Re 10-6,

so the effects of inertia are minimal. The minimum rotational Peclet number, Per = y/Dr,

calculated based on the rotational diffusion, Dr, of a prolate spheroid [27], is 103. The

Peclet number based on the translational diffusion of a prolate spheroid, Dt, is defined

as Pet = yL2/Dr. Since the rotational Peclet number represents the smaller of the two

Peclet numbers, Per is used in the remainder of the paper to define the flow strength. A

discussion of other possible nonhydrodynamic effects is given in a later section.

Experiments were performed using a 50 mm diameter parallel-plate geometry. The

gap between plates was set to 500 pm, resulting in a minimum gap-to-particle length

ratio of 125. Additional experiments performed at a larger gap of 1000 pm (H/L = 250,

where H is the gap height) were statistically equivalent. In the parallel-plate geometry,

the rate of strain varies in the radial direction, resulting in inhomogeneous flow. Since

the microstructure of fiber suspensions is strain dependent, additional experiments were

performed using a 50 mm diameter cone-and-plate geometry, which maintains a constant

shear rate in the radial direction. For all cone-and-plate measurements, the cone angle

was 0.04 radians, and the gap was set to 45.7 pm, resulting in a minimum gap-to-particle

length ratio of 12 at the apex of the cone and 270 at the edge.













0 0.2 0.4 0.6 0.8 1
1~d (pm)

0


~d0.5-

rb0


0 2 4 6 8 10
L/d

Figure 4-2. Normalized size distributions for ellipsoids with an average aspect ratio of m4
(solid) and =7 (dashed). In each case, particle numbers are normalized by
their respective peak values. Each distribution is calculated from 50 random
particles.


The rheometer used in all experiments was an ARES LS-1 strain controlled rheometer

(TA Instruments). The steady shear theology was investigated for a range of fiber

concentrations and aspect ratios. To ensure that each test began from a similar initial

state, the suspensions were presheared at a shear rate of y = 100 s-l for 300 seconds,

which was sufficient to enable the suspension to reach a steady state. The temperature

was maintained at 250C and the temperature fluctuation was less than 0.050C during a

typical experiment.

Results

Start-up experiments were performed in the parallel-plate geometry to evaluate

the transient behavior of ellipsoid suspensions. Figure 4-3 shows the relative viscosity

as a function of time and shear rate for suspensions of ellipsoids with L/d = 4. The

particle volume fraction, 4, of 0.103 falls within the semi-dilute concentration regime.

Each start-up experiment was performed immediately following a preshear at y = 100 s-]

for 300 s. For smaller shear rates, the transient response of the relative viscosity is more












5 _-- j= 0.1 sl

= 100 s ,










0 50 100 150 200 250 300
time (s)

Figure 4-:3. Relative viscosity as a function of time and shear rate for polystyrene
ellipsoids suspended in a polyalkylene glycol/water/K(Cl mixture. The
ellipsoids have an average aspect ratio of 4 and the volume fraction is 0.10:3.
For each shear rate, the suspensions are presheared at ji=100 s-l for :300 s.
Experiments were performed using a 50 nin parallel-plate geometry with a
gap of 500 pm.


dramatic. For example, at y= 100 s-l, the relative viscosity is already at steady state

and no additional changes are observed over the time scale of the experiment. At lower

shear rates, ty, increases continuously and does not reach steady state over the duration of

the experiment. Furthermore, although the viscosities at t=0 are equivalent, the viscosity

at ji=0.1 s-l increases to il, a 5.5 after :300 s whereas at ji=1 s-l, the increase is much

more modest (rlr a :3.5) over the same time period. The results are qualitatively similar to

experiments on suspensions of large aspect ratio (L/d a 40) fibers [:35, 49], which show an

increase in viscosity with time at low shear rates.

Figure 4-4 shows the relative viscosity at ji=0.251 s-l plotted as a function of time

and concentration for suspensions of fibers with L/d=4. The results are plotted for three

different volume fractions and represent the time evolution following a preshear at ji=100

s-l for :300 s. At all three volume fractions, the relative viscosity increases with time. For

the volume fractions studied, no steady state is observed through t=:300 s. Additional












= 0.103
= 0.079












0 50 100 150 200 250 300
time (s)

Figure 4-4. Relative viscosity as a function of time and volume fraction for polystyrene
ellipsoids suspended in a polyalkylene glycol/water/K(Cl mixture. The
ellipsoids have an average aspect ratio of 4 and all volume fractions lie
within the semi-dilute concentration regime. For each volume fraction, the
suspensions are sheared at a rate of ji=0.251 s-l immediately following
a preshear at ji=100 s-l. Experiments were performed using a 50 mm
parallel-plate geometry with a gap of 500 pm.


experiments were conducted over larger time scales for suspensions with #=0.079; no

steady state was observed and the viscosity continued to increase through t=2000 s.

To determine the effect of ellipsoid aspect ratio on the theology, suspensions of

ellipsoids with an average aspect ratio of L/d=7 were studied. The relative viscosity at

j=0.1 s-l, plotted as a function of time and concentration for suspensions of ellipsoids

with L/d=7, is shown in Figure 4-5. The results are plotted for three different volume

fractions and represent the time evolution following a preshear at ji=100 s-l. Similar

to the suspensions containing ellipsoids with a smaller aspect ratio (Figure 4-4), the

relative viscosity increases with time, with the exception of the lowest volume fraction,

~=0.012, which shows a steady value over the duration of the experiment. Comparison

of the transient response of suspensions of ellipsoids with different aspect ratios reveals

a dependence on L/d. Specifically, the viscosity increase over t= 300 s is greater for














2.5-




-' $ = 0.058
/~ -- 4 = 0.045
-- 4 = 0.012
1.5-




10 50 100 150 200 250 300
time (s)

Figure 4-5. Relative viscosity as a function of time and volume fraction for polystyrene
ellipsoids suspended in a polyalkylene glycol/water/K(Cl mixture. The
ellipsoids have an average aspect ratio of 7 and all volume fractions lie
within the senli-dilute concentration regime. For each volume fraction, the
suspensions are sheared at a shear rate of ji=0.1 s-l ininediately following
a preshear at ji=100 s-l. Experiments were performed using a 50 nin
parallel-plate geometry with a gap of 500 pm.


suspensions with ellipsoids of higher aspect ratio. For example, at


viscosity for suspensions of ellipsoids with L/d=7 increases hv t:;2 over a total time of :300

s, whereas for suspensions of ellipsoids with L/d=4 the relative viscosity increases by only

21 over the same time.

The dependence of the relative viscosity on shear rate (and Pe,) is plotted in

Figure 4-6 for suspensions of fibers with L/d=4. Rate sweeps were performed using

the parallel-plate geometry for suspensions at six different volume fractions spanning the

senli-dilute concentration regime. At each shear rate, the value of the relative viscosity

is recorded after :360 seconds of shear, which was chosen to reduce the total experiment

time and consequently, to nminintize possible evaporation of the suspending liquid. Since

the relative viscosity at each shear rate is sensitive to the time over which the suspension

is sheared, not all values represent steady state viscosities after :360 s. Thus, open symbols











103 104 105 106

a ) = 0.127
M 0 = 0.103
0 0 = 0.079
6~ 0 = 0.055
IB-E $ = 0.046
ct $= 0.031









0.1 1 10 100
J (S-1)

Figure 4-6. Relative viscosity as a function of shear rate and volume fraction for
polystyrene ellipsoids suspended in a pa~~i- ll:ylene glycol/water/K(Cl mixture.
Each data point represents the apparent viscosity after 360 seconds of shear
at each corresponding shear rate. The ellipsoids have an average aspect ratio
of 4 and all volume fractions lie within the semi-dilute concentration regime.
Experiments were performed using a 50 mm parallel-plate geometry with a
gap of 500 pm.


represent steady state values of the relative viscosity and closed symbols represent values

that have not attained a steady state value after t=360 s.

For all volume fractions studied, the suspensions show shear thinning behavior. The

dependence of the relative viscosity on shear rate becomes weaker as the volume fraction

decreases. For example, at the highest volume fraction (#=0.127), the value of the relative

viscosity decreases 3501' Over three decades of shear, while at the lowest volume fraction

(#=0.031), the relative viscosity decreases by only !I :' Over the same range of shear

rates. Below ji=1 s-l, the values of the relative viscosities reported in Figure 4-6 cease

to represent steady state values. However, even in shear rate ranges where the values of

the viscosity are at steady state (for example, 10 s-l < < 251 s-l), shear thinning

is apparent. Moreover, rate sweeps from high to low y were performed and show that











104 105 106


(H $ = 0.074
Mn 0 = 0.058
0 0 = 0.045
$t = 0.032
--0 0 = 0.012









0.1 1 10 100
j S-1

Figure 4-7. Relative viscosity as a function of shear rate and volume fraction for
polystyrene ellipsoids suspended in a pa~~i- ll:ylene glycol/water/K(Cl mixture.
Each data point represents the apparent viscosity after 360 seconds of shear
at each corresponding shear rate. The ellipsoids have an average aspect ratio
of 7 and all volume fractions lie within the semi-dilute concentration regime.
Experiments were performed using a 50 mm parallel-plate geometry with a
gap of 500 pm.


the shear thinning is reversible. Although rate dependent theology is not expected for

suspensions of fibers at high Peclet numbers [107], shear thinning similar to that observed

here is consistently observed in experiments [66].

Figure 4-7 compares the relative viscosities at L/d=7 for five different volume

fractions spanning the semi-dilute concentration regime. The values of the relative

viscosity at each shear rate are measured as they were in Figure 4-6. Open symbols

represent steady state values of the relative viscosity and closed symbols represent

viscosities which have not reached steady state. The range of shear rates studied is shifted

to enable comparison at similar Per for the two systems with different aspect ratios.

Similar to suspensions containing ellipsoids with L/d=4, higher aspect ratio ellipsoids

show shear thinning behavior for all volume fractions. Again, the shear rate dependence of









Pe
103 104 105 106

a 0 =0.103
0 = 0.079
P $ =0.055












0.1 1 10 100
j (S-1)

Figure 4-8. Comparison of the relative viscosity measured in the cone-and-plate (dotted
line) and parallel-plate (solid line) geometry for suspensions of polystyrene
ellipsoids with L/d=4 in a polyalkylene glycol/water/K(Cl mixture. The
relative viscosity is plotted as a function of shear rate and volume fraction.
Each data point represents the apparent viscosity after 360 seconds of shear at
each corresponding shear rate. The cone-and-plate geometry had a diameter
of 50 mm and a cone angle of 0.04 radians. The geometry gap was set at 45.7
p-m.


the relative viscosity becomes less noticeable as the concentration decreases. For example,

at the lowest volume fraction studied for either aspect ratio ( =0.012), the viscosity is

nearly independent of the shear rate over three decades of shear.

Since the theology of fiber suspensions depends on the spatial configuration and

orientation of fibers, which is strain dependent, the parallel-plate geometry may be

unsuitable for fiber suspension theology because the strain varies with radial position

[48]. To determine the effect of geometry on the theology, additional experiments were

performed in a cone-and-plate fixture which maintains a constant strain in the radial

direction. Figure 4-8 shows the relative viscosity in the cone-and-plate geometry plotted as

a function of shear rate for suspensions of fibers with L/d=4. Although experiments were










performed over the entire range of volume fractions presented in Figure 4-6, only three

are reported and compared to experiments in the parallel-plate geometry. For all volume

fractions studied, the values of the relative viscosity for suspensions in the cone-and-plate

geometry are consistently higher than those in the parallel-plate geometry. The differences

are most significant at the lowest shear rates, where the reported viscosities have not yet

attained steady state. Despite these differences, the qualitative behavior of our system is

insensitive to the geometry used. Due to the smaller gap-to-particle length ratio at the

apex in the cone-and-plate geometry, the theology was subject to stress jumps, possibly

due to particle jamming at the apex of the cone. An example of particle jamming occurs

for #=0.079, where the relative viscosity anomalously increases at y=1.58 s-]

Discussion

In this section, we discuss possible mechanisms for the rate dependence of the relative

viscosities. We also investigate scalings of the steady state viscosity in the semi-dilute

concentration regime and compare to results from simulations.

Rate Dependent Rheology

Experiments on fiber suspensions at large rotational Peclet numbers predominantly

show shear thinning behavior [35, 66, 119]. For large values of Per, hydrodynamic

interactions dominate, and theories and simulations within the hydrodynamic limit of

Per= co are expected to agree with experiments. However, rate dependent theology is not

predicted by theories and simulations [107]. In an attempt to elucidate the shear thinning

phenomenon in some specific experiments, explanations such as fiber flexibility [18] and

fiber adhesion [35] have been offered. To evaluate the role of colloidal interactions in our

system, suspensions of spheres with material properties identical to the ellipsoids are

studied.

A possible cause of shear thinning, unique to our system, is the competition between

hydrodynamic forces and the mechanical binding of particles due to residual PVA on the

particle surface. Polyvinyl alcohol binds strongly to polystyrene and can affect particle
























Figure 4-9. SE1\ images of processed polystyrene spheres before (left) and after (right)
washing. In each image, the scale bar represents 1 pm.


surface properties. Figure 4-9 shows scanning electron micrographs of polystyrene spheres

which were subjected to the same procedure described earlier, but not stretched. Prior to

washing the processed particles, much of the PVA matrix is still attached to the particle

surface, mechanically finding particles to one another. After the washing cycles however,

the processed particles resemble plain poli--r i-i. in.- spheres with little evidence of residual

PVA on the particle surface. Thus, it is clear that the extent of washing affects the

amount of residual PVA remaining on the particle surface for particles prepared in the

manner described earlier.

To determine the effect on the theology from residual polyvinyl alcohol on the particle

surface, the theology of two different suspensions of spheres were studied. Specifically,

comparisons were made between suspensions of plain polysytrene spheres (as received

from the manufacturer) and processed poli--li-1 in-- spheres following the washing cycles.

Both suspensions were prepared at a volume fraction of < =0.07, and the relative viscosity

as a function of shear rate is plotted in Figure 4-10. For suspensions containing plain

polystyrene spheres, the relative viscosity is constant over three decades of shear rate.

Suspensions containing processed spheres show a slight decrease in relative viscosity over

the same range of shear rates, however a nl I i} O~l y of the shear thinning occurs prior to

ji=1 s l, after which the relative viscosity remains constant and statistically equivalent





























Figure 4-10. Relative viscosity as a function of shear rate and aspect ratio for different
suspension systems. The results are from poli--i-1. in-- particles suspended in
a p..Ili- I11:ylene glycol/water/K(Cl mixture at a volume fractions of 4=0.07.
Experiments were performed using a 50 mm parallel-plate geometry with a
gap of 500 pm. The values represent averages over two individual runs. Error
hars are specified when the error is larger than the size of the symbol.


to the relative viscosity for suspensions containing plain polystyrene spheres. The results

indicate that the theology may be slightly affected by the presence of PVA on the particle

surface, but the qualitative difference is small and only apparent at the lowest shear rates.

Other possible mechanisms for the rate dependent theology can he evaluated through

a direct comparison of the theology of suspensions of spheres having identical material

properties as the ellipsoids. Chaouche and K~och [35] present a mechanism for the shear

thinning behavior of non-Brownian fibers in which hydrodynamic forces compete with

adhesive forces. At small values of the applied shear rate, contacting particles flocculate

due to adhesion. As the applied shear rate increases, flocs break apart, resulting in a

decrease in the apparent viscosity. Figure 4-10 shows the relative viscosity as a function of

shear rate for suspensions of ellipsoids at 4 =0.07 along with the theology of suspensions

of processed spheres. The theology of the processed spheres qualitatively differs from the

theology of suspensions of ellipsoids at similar volume fractions. Since the adhesive force










should depend primarily upon the physical and chemical properties of the suspending

liquid and particle surface [35], it is unlikely that adhesive forces would cause strong

shear thinning behavior for suspensions of ellipsoids, but not for suspensions of spheres,

thus eliminating the possibility of adhesive forces as the cause of the shear thinning

theology. The role of electrostatic interactions as a potential source for the shear thinning

behavior can also be eliminated. Electrostatic interactions cause rate dependent theology

in suspensions of spheres [71, 115], however Figure 4-10 shows that the relative viscosity

of suspensions of polystyrene spheres is independent of the shear rate, indicating that

electrostatic interactions are unimportant. Although the shape and surface properties of

polystyrene particles can alter the electrostatic effects [78], it is unlikely that the theology

of the ellipsoid suspensions is affected by electrostatic interactions.

Other factors that may contribute to the observed shear thinning behavior in our

system include particle inertia and fiber flexibility. Although there is a finite amount of

inertia in our experiments, the Reynolds number is low (Re=10-6) and inertia is presumed

to have a negligible effect on the theology. Fiber flexibility, which can result in rate

dependent theology [18], is also negligible since the polystyrene ellipsoids used in our study

are rigid. Scanning electron microscopy of the ellipsoids shows no indication of bent or

deformed particles for any of the aspect ratios studied. Shear-induced bending is also

unlikely. The maximum stress generated during the experiments is significantly less than

the minimum stress required to bend a single pohi- i-1. Im~ ellipsoid [61] by a factor of 10".

An alternative mechanism for the rate dependent theology may be particle migration

arising from hydrodynamic interactions with the bounding walls. For example, in

pressure-driven flow, rigid fibers at high, but not infinite, Peclet numbers undergo a

net migration away from the walls, with the magnitude of the drift increasing with

flow strength [117]. Since the magnitude of the drift due to particle migration is rate

dependent, such a mechanism may account for the observed theology in our system. At

large shear rates for example, particle migration may dominate, forcing a nonuniform










spatial distribution of particles as the ellipsoids migrate away from the wall towards

the center of the gap. In this case, a decrease in the total stress will occur, resulting in

a decrease in apparent viscosity. At lower shear rates, particle migration weakens and

thermal diffusion causes the suspension to relax to a uniform spatial distribution, resulting

in a transient increase in the apparent viscosity similar to that observed in Figure 4-3.

The role of shear-induced migration on the theology, however, remains unclear and

additional work is required to determine whether such a mechanism can account for the

rate dependence. For example, one might expect thermal motion to be unimportant at

Peclet numbers greater than 10 In this case, the theology would agree with results

from theories and simulations at Per of infinity. Indeed, suspensions containing spherical

particles cease to exhibit shear thinning behavior at Peclet numbers (based on the

translational diffusion of a sphere) larger than 102 [62, 111]. However, unlike suspensions

of spheres, a drift perpendicular to the wall can occur due to the interactions between

rodlike particles and boundaries which depends upon the particle orientation distribution.

The characteristic time scale for particle drift due to migration is small, and may be

comparable to the time scale for thermal diffusion, which is finite in our system. Such

a mechanism therefore -II--- -; that thermal diffusion, though small, may pili a role in

the observed theology and that theories and simulations in the limit of infinite Pe, are

not applicable. However, to determine if migration can result in shear thinning theology,

a calculation of the effect on the total stress from particle drift (relative to thermal

diffusion) is required. One might also expect that a mechanism hased on interactions

between walls and bulk particles would result in a dependence on the gap size. However in

this work, we report the theology for two different gap spacings and find no dependence on

confinement. The apparent lack of dependence on gap spacing however, does not discount

particle migration as a potential mechanism for the observed theology. In the absence of

any other mechanism available to explain the shear thinning theology, particle migration

warrants further study.









Scalings of rl, at High Per

Figure 4-11 shows the steady state relative viscosity at Per=106 aS a funCtiOn Of

volume fraction and aspect ratio. The steady state relative viscosity at Per=106 is an

increasing function of volume fraction for both aspect ratios. The viscosity increases with

volume fraction much faster for larger aspect ratios. For example, at 4 m0.07 the steady

state relative viscosity for L/d=7 is approximately one-and-a-half times greater than that

for L/d=4, even though rl, for both systems converge to similar values at low volume

fractions. The dependence of the steady state viscosity on aspect ratio agrees qualitatively

with results from experiments [49] and simulations [38]. Computational results [38] for

statistically homogeneous dispersions of spheroids with aspect ratios similar to those

studied here are plotted along with our experimental results in Figure 4-11. The results

agree reasonably well, though the experiments show a much stronger dependence of the

relative viscosity on aspect ratio compared to the simulation results of Claeys and Brady

[38]. The differences are not surprising considering that Claeys and Brady [38] evaluate

the viscosity for a static isotropic orientation distribution of spheroids, whereas the steady

state orientation distribution is unknown in the experiments.

Figure 4-12 shows the relative viscosity at Pe,= 104 and 106 aS a funCtiOn Of aSpect

ratio and dimensionless number density, nL3. For the aspect ratios studied, the values of

the steady state relative viscosity at Per=106 COllapse within a narrow band when plotted

versus nL3, indicating that for suspensions of small aspect ratio ellipsoids at high Per,

the dimensionless number density is the only controlling parameter. Additionally, the

experiments show that rl, scales linearly with nL3 at P6r=106. The linear dependence of

the viscosity on nL3 is COnSIStent With semi-dilute suspensions of homogeneous dispersions

of ellipsoids with similar aspect ratios [38] as well as isotropic suspensions of cylindrical

fibers [141]. However, Claeys and Brady [38] and Yamane et al. [141] both predict a much

stronger dependence of the relative viscosity on aspect ratio when plotted versus nL3. Our

results show a stronger dependence of rl, on aspect ratio at smaller values of Per. Figure











*L/d= 4
A L/d= 7
O Claeys & Brady (1993), L/d = 3
0 Claeys & Brady (1993), L/d = 6
2.5-
V Claeys & Brady (1993), L/d = 10




r 2-

1.5 -v

AV O O

0 0.05 0.1 0.15


Figure 4-11. Steady state relative viscosity at Pe,=106 as a function of volume fraction
and aspect ratio. The results are from polystyrene ellipsoids suspended in
a p..Ili- I11:ylene glycol/water/K(Cl mixture. The experimental results are
compared with results from simulations of isotropic suspensions of ellipsoidal
particles [38].


4-12 shows the relative viscosity (following 360 s of shear) at Pe,=104 plotted versus nL .

At Pe,=104, the viscosity no longer increases linearly with nL3 and there is an obvious

dependence on ellipsoid aspect ratio.

Conclusions

The theology of well characterized suspensions of rigid poli--r i-i. in.. ellipsoids at

rotational Peclet numbers greater than 10" was evaluated. P li-- i-i. in.. ellipsoids having

two different aspect ratios were fabricated using the method of Nagy and K~eller [99].

Similar to previous experiments on fiber suspensions at large Peclet numbers [66], ellipsoid

suspensions exhibit shear thinning behavior for both aspect ratios over all concentrations

spanning the semi-dilute concentration regime. The shear thinning behavior is reversible

and independent of the geometry used. Distinctly different behavior is predicted by

theories and simulations, which show no dependence of the viscosity on shear rate in the

limit of infinite Pe,.











*L/d = 4 P-e=104
3.5~ L/d = 7Pe =104
O L/d = 4, Per=106
3~ -O L/d= 7 Pe=106

F 2.5-



OO
1.5-
OOo

0 1 2 3 4 5 6 /
nL3

Figure 4-12. Steady state relative viscosity at Pe,=104 and 106 as a function of
dintensionless number density and aspect ratio. The results are front
polystyrene ellipsoids suspended in a polyalkylene glycol/water/K(Cl mixture.



Possible mechanisms for the shear thinning behavior were evaluated. Direct

comparison of the theology of suspensions of the ellipsoidal particles with suspensions

of spherical particles, processed in an identical fashion, -II---- -r that the method of

manufacturing the ellipsoids as well as other colloidal interactions are improbable

sources of the rate dependent theology. An alternative niechanisni for the observed

shear thinning was proposed which involves the competition between fiber migration due

to hydrodynantic interactions with the bounding walls and thermal diffusion. For large

Peclet numbers, the dependence of the relative viscosity on volume fraction is qualitatively

similar to simulations of isotropic suspensions of spheroids with similar aspect ratios [38].

Additionally, at the largest Peclet numbers studied, the relative viscosity scales linearly

with nL regardless of the ellipsoid aspect ratio. At lower values of the applied shear rate,

corresponding to lower values of the Peclet number, the relative viscosity no longer scales

linearly with nL3 and a dependence on the ellipsoid aspect ratio becomes apparent.









CHAPTER 5
CROSS-STREAM MIGRATION OF RIGID BROWNIAN FIBERS UNDERGOING
SIMPLE SHEAR FLOW NEAR A SOLID BOUNDARY

Introduction

Flowing suspensions of rodlike particles undergo unique phenomena which often result

in an inhomogeneous spatial distribution. Flexible polymers in solution, for example,

undergo a net migration towards the center of the channel in pressure-driven flow [75, 83,

133] as a result of hydrodynamic interactions between the polymer and bounding wall [94].

Migration in suspensions of rigid polymers have also been observed [10, 102, 118], though

the extent of the migration is noticeably weaker than the migration of flexible polymers

[117]. In the case of the migration of rigid polymers, the precise effects of hydrodynamic

interactions with the wall are unclear.

When disregarding hydrodynamic interactions with boundaries, models on suspensions

of rigid fibers undergoing pressure-driven flow show a net migration ::1:- li from the

channel centerline, toward the walls [10, 45, 102, 118]. However, recent simulation results

considering hydrodynamic interactions with the wall show that the net migration is

affected by such interactions [117]. Currently, a model comparable to the kinetic theory

for an elastic dumbbell developed by Ma and Graham [94], which includes hydrodynamic

interactions with the wall, is needed to accurately describe the net migration of rigid

fibers.

In this chapter, we present a model for the migration of rigid fibers undergoing simple

shear flow near a solid boundary in which hydrodynamic interactions with the wall are

considered. The basis of the model is introduced along with details of the approximations

made. The model predictions qualitatively differ from previous models which ignore

hydrodynamic interactions with the wall, highlighting their importance. Results from

the model are provided along with comparisons to previous models and a discussion of

the application of the model to the theology of suspensions of fibers. Conclusions are

presented in the last section.























Figure 5-1. Geometry and notation used to describe a fiber in solution undergoing simple
shear flow near a solid boundary.


Model

The geometry and notation used to model the migration of a Brownian fiber

undergoing simple shear flow near a solid wall is shown in Figure 5-1. The flow is in

the 1-direction and the geometry considered is bounded by a single wall at x2 = 0.

Although the model is based on a far-field approximation of a single wall, shear flow of a

rigid rod confined between two walls can in principle be calculated using the appropriate

Green's function [92] or can be approximated by superposing the Green's function for a

single wall [32, 94]. The slender-body model is used to model the rigid fiber, thus the fiber

length, L, is much larger than the diameter, d, such that A = L/d > 10. For the purpose

of this work, we set A = 10. The fibers are exposed to external Brownian forces balanced

by hydrodynamic forces, such that the net force is zero (i.e. no inertia).

The time evolution of a rigid fiber in solution is governed by a continuity equation for

the distribution function, W (xi, pi, t), of the center of mass, xi, and orientation, pi,





The probability distribution function is separated into a center of mass, n, and orientation

distribution function, ~, as done by Ma and Graham [94] and Butler et al. [32],


W (xi, pi, t) = n(xi, t) (xi, pi t) (5-2)









where

n (xit x, t) t p.(5-3)

Integrating Equation 5-1 over pi and solving for the steady result gives,




where the angle brackets <> indicate an ensemble average over pi,


< xi >=1' Jp~ (5-5)


To solve Equation 5-4 for the center of mass distribution, the averaged center of mass

velocity is required. To obtain an expression for the center of mass velocity which includes

contributions from the presence of a solid boundary, a far-field approximation is used

[21, 32, 94]. Instead of solving for the position-dependent orientation distribution function,

we evaluate the ensemble average of the center of mass velocity via simulations of a single

Brownian rod undergoing shear flow in which we ignore hydrodynamic interactions with

the wall.

Velocity Expressions

The motion of each fiber is described by a line distribution of Stokeslets according

to the slender-body theory [11, 41, 42]. In the limit of Stokes flow, the center of mass

velocity, xi, and rotational velocity, pi, are related to the disturbance velocity, ui(s), and a

line distribution of point forces, fj, by

In(2A)
xi + slii U,(s) =(s+pp)fy),(5-6)
4x p

where p is the fluid viscosity and s is any point along the rod evaluated from the center

of the rod. Integration of Equation 5-6 over the length of the fiber, L, gives the center of

mass velocity as,
1 rL/2 In(2A)
xi = ui(s)ds + (6ij + pipj) Fy. (5-7)
L -L/2 4xpl-L









The rotational velocity is given by taking the cross product of Equation 5-6 with spi and

integrating over L,

12 *L/2 3 In(2A)
Pi=L3 JL/2 ;p-L3 (s ay y 58

where the integrated force is
/L/2
J-L/2
and Fj is the weighted force corresponding to the torque, ~,


Eaikjk Pk i F,) (5-10)

We linearize the force distribution using Legendre polynomials following Butler and

Shaqfeh [31] and Saintillan and Shaqfeh [117] and retain only the first two terms,

L1 12s (u'i, i "I
fi(s) = LFe 3(s y),+pS,(-1

where S is the stresslet of a single fiber,

2xy~ rL/2
S = spiuids, (5-12)
In(2A) J-L/2

which is a scalar quantity arising from the inability of a fiber to stretch or compress along

its usl r ~ axis.

The disturbance velocity in Equations 5-7, 5-8, and 5-12 is the sum of the imposed

velocity field, u@o), and a contribution to the velocity field due to the presence of the wall,



as = Uo) (s) + U,(s), (5-13)

which, for shear flow, becomes

/L/2
J-L/2









where Gij(s, s') is the Green's function for a single wall [21, 32, 94]. We note here that s'

is the location of the point source on the fiber, whereas s is the point of evaluation of the

disturbance.

Combining Equations 5-7, 5-8, 5-12, and 5-14, substituting Equation 5-11 for fy (S'),

and eliminating S gives,


_CsL4 ln2A) G yk L228 d
1 (0o) 24xy~ (1) (1)1
99)+L2 Go cL6 In(2A) GkP~GsF


CsL2818n( A) G pjpzl(2)Pmlmk k FI


(5-15)


Pi ii L/ 22 () c L26818n( A)rGp f L/ 22 (0)

PG3 o) csL2818n( A) Py tmGkF


31ln(2A) 144 ()3456x~ ,(2)
xpL3PilL6 Psy Jjk Pkl CsL9 In(2A) Ps2j Pfu 2m Pmli


(5-16)


where


24xy- ,(2)
cs=1 +i psG pj
L3 1n(2A)


and


G 0
G
G j

G 1


Gay (xl + sp xl + s'pl) ds'ds


sGij (xl + p It+sp)d's


s'Gsy (xl + sp xl + s'pl)ds'ds


as'Gsy (xl + sp xl + s'pl)ds'ds.


(5-19)


(5-20)


(5-21)


(5-22)


xi =~ _La L/22 0l)d

I rn(2A) n,


G Pjk -


r L/2

J -L/2
r L/2

L/ I
J -L/2
r L/2

~ -L/2









Gij(s, s') is linearized about the center of the rod under the assumption that x2 is
much greater than L [32] to give,


Gij(s, s') a Gij(Zk, k~) +9 [GyrZ y2 81 G(k, o

(5-23)
resulting in the simplification of Equations 5-19, 5-20, 5-21, and 5-22,

G32 ~ (bs 4_4._ (5-24)


Gel= C [(s + 4 A)p2 i9 bipj +P .] (5-25)


G(2) 0. (5-27)

The resulting expressions for Gij are then substituted into Equation 5-15 to obtain

the center of mass velocity,


xi = ( ,s~ _. + iiA(Z2 i iZjFj i ljPjkFk~, (5-28)

where

1281n(2A) x59
The vectors and tensors associated with the hydrodynamic interactions appearing in

Equation 5-28 are,

Ui = plp2 [6i2~ p292) 2pip2] (5-30)
In(2A) 31
Melj ~~ (6 y + pipj) 32r-1 (bg; + 4_.1


32xy~ 256 In(2A)3 x~(
2p2( p9292 (r + j6i2)+( p292 26i26j2] (5-31)

64s r = (ia b2j2 92 Gi29) + r) ~.4.] (5-32)









Similarly, the rotational velocity becomes,

31ln(2A), 53
fig = ~Yp2 il pip1) CO w2 ikFk, 3 i+(-3

where

co(Z2) =~f (5-34)

and Wik, is a tenSOT aSSociated with the hydrodynamic interactions,

Wik (ik 6i26k2 2pipk)p2 I', (p292)pibk2. (5-35)

Probability Distribution

For the purpose of this work, we consider only the center of mass distribution

perpendicular to the bounding wall. Equation 5-28 is substituted into Equation 5-5 and
multiplied by a to get the center of mass flux, 'T,


14 = n < xi > = njZ2 ~il 2 jX~ Ui

(~ ( 8k~ In W~ 8 In~a -k a~k T,(-6

where the forces and torques in the third and fourth terms, respectively, were replaced

with the Brownian forces and torques written in terms of distribution functions,

(Br) 8 In W~
F. -kT (5-37)

FT kT (5-38)

We rewrite the third term in Equation 5-36 by factoring W into a product of n and ~,

(~~~ ( n n n 8 1n
~M y (-kT = n~~) -nkT Ms 'l -kT M '.
8in 8 1k n ~iX
-kT < Ma > kTM (5-39)









The last term of Equation 5-39 is ignored as done by Ma and Graham [94], since we
disregard hydrodynamic interactions with the wall when evaluating the orientation
distribution. Using this approximation, the orientation distribution is a function of pi only.
Similarly, the fourth term in Equation 5-36 becomes,
SIn W 4
Myi31jk kT= kTPjk ij > nkT jki

,,,,,,,( 6 In(2A);~L ( a ipp 6a

(5-40)

Unlike the dumbbell case [32, 94], the fourth term in Equation 5-36 is nonzero, revealing
an additional contribution to the migration of rigid fibers.
The flux normal to the wall (in the 2-direction) is zero,

31 D li~i) U>i:.4 iL2>~ +il8n~ l) 6 In(2A);~L
(5-41)
As an approximation, we include only the first term of the self mobility, < M221~
kTl(1+~~f(- < pJ2P2 >), an~d ignore efctls of interactlions on this termn. Substitution and

rearrangement gives,
8 In n A(Z2)
82L2[Pe (a~) + 24 (P)] (5-42)

where the Peclet number is defined as,

4xy~ L3y
Pe =.(5-43)
kT In(2A)

Equation 5-42 contains two contributions: a contribution, Pe(a~), from the imposed shear
flow and a contribution, 24(P), arising from the Brownian torque,

1+l~>- pppp 1 < p292>(54

and

(1 3 < p292>+< ~ (5-45)









Ensemble averages of the orientation moments in Equations 5-44 and 5-45 are obtained

numerically.

Calculation of Orientation Moments

To calculate the orientation moments in Equations 5-44 and 5-45, a numerical

simulation of a single Brownian rod under simple shear flow was performed and the

orientations were sampled and averaged. Since we are only concerned with the fiber

orientation, which we assume is independent of position, we consider only rotational

Brownian motion without wall hydrodynamic interactions. The equation of motion for a

single Brownian rod undergoing simple shear flow in the 2-direction can be derived from

Equation 5-8,
31ln(2A) B
Pi = yp2 (il pip1> 3ii F3 'l, (5-46)

where Fl"' is th~e weigh~ted force correspon~ding to a Browniann torqlue which satisfies th~e

fluctuation-dissipation theorem and therefore can be expressed for the discrete time step



i 31In(2A)At jk Fj~k)n (5-47
where I,, is a random vector of length three with unit variance.

Combining Equations 5-46 and 5-47 and adding a correction term for numerical

integration by a modified Euler method [39, 58], gives


pi = p2 il ( PeiP1 1/2 O3 pipj) my ps (5-48)

which is made dimensionless using the characteristic time scale jiland the rotational

Peclet number,
xpL3 PC
Per (5-49)
3kT In(2A) 12











0.3 -




~0.2-

0.1


1010101210 0
Pe
rn~s
Figre5-. umeicll clcuatd nsmbl aergeof heorenatin omnt a
a ~ ~ fucino e iuainswr efredo rwia ii ie




The orienais at2 tueime ='ar calculated usinleavrg Equatheoion 5-48 nd sampedt for




times. For example, the average moment for plp2 becomes,



< pip2> (5-50)


Specifically, the averaging is performed over 1000 simulation time increments of t = 5 until

a final time step of t = 5000. The time step was set to at = 5 x 10 ', and the number

of samples taken within each increment of time was NV = 104. This level of averaging was

repeated five times using different random seed numbers to arrive at the final averaged

value of the moments. The standard deviation of the calculated moments was less than



Figure 5-2 shows the ensemble average of the moments in Equations 5-44 and 5-45

presented as a function of Per. The < p292 > moment has the largest value at low Per,

and decreases as the flow strength increases. The < plp2 > and < plp29292 > moments

approach zero at low Per and increase with increasing flow strength until reaching a












0.4-


0.2-


S0-


-0.2-


-0.4-


0 20 40 60 80 100 120 140 160 180
6 (de g)

Figure 5-3. Effect of fiber orientation on the center of mass drift velocity, x2. The drift
velocity is normalized by yXand 8 represents the fiber angle with respect to
the wall. A positive value for the drift velocity corresponds to the drift of a
rigid fiber away from the wall, while a negative value corresponds to a drift
towards the wall.


maximum at Per a 10. All moments decrease at large values of Per as a result of the

p2 COmponent, which vanishes as fibers align in the direction of the flow. The results for

< plp2 > are quantitatively similar to results for suspensions of spheroids [9].

Results and Discussion

In this section, we present results from the model described in the previous section.

The migration of rigid fibers undergoing simple shear flow near a single boundary is

presented in terms of the center of mass velocity and distribution. We also discuss the

applicability of these results to the theology of suspensions of rigid fibers.

Center of Mass Velocity

The first two terms in Equation 5-28 represent the the response of the center of mass

velocity to an imposed shear flow. The expression for the velocity in the 2-direction due to

shear flow in the 1-direction becomes,


x2 = YXX 12 (192 3p2932). (5-51)










6 = 900



S= 144.740 6 = 35.260



-

6 = 180" 'st s= Oo





Fiur 54.Scemti smarzig heefec o ibe retto nth rfeoiy
The~~~ sin niaetedrc iono irto;apstv increpnst







wl.Figure 5-3. schowacs 22/a otedasafuctono the angle, 0 of the ro orientation o h rf eoiy
with espet the 1-n idctetedirection For Oo< 3.20atind 90 < 0 < 144.740 thre velocity
is ostiean alcits away from the wall. Forl al oegthervale sg threspns velocityisngtvad
acts toward the wall.Thmxiuveoiyoraprilmoigwyfomteal

occudrs when o 1730 the corrtydeens ponding maximumto value for e movemen pet towr the wl

ocus hn 6.3. Figure 5-4 hw ay lte asumarfncizes the depndlence of the rdirec nation o h






drift velocity on the instantaneous fiber orientation. The results quantitatively agree with

the work of Saintillan et al. [117], as does the expression for x2-

Center of Mass Distribution

Since the orientation moments are calculated numerically assuming no dependence on

position, a~ and p are constants depending only on Per (or Pe), and Equation 5-42 can be











1000


.~ 100 HDPe(o + 24(P)


o1 /



0. 1 -.

S0.01 I


0.1o~ 1 10 100 1000 10000
Pe

Figure 5-5. Contribution of the shear flow (Pe(a~)) and Brownian torque (24(P)) to the
migration of a rigid fiber plotted as a function of Per-


integrated directly,
n~x) e~,I-L Pe(a~) + 24(P)18n2 ) 5 )

where the integration constant is set equal to 1 such that the bulk value n(Z2 i OO) = i.

The individual contributions, Pe(a~) and 24(P), to the center of mass distribution

function are plotted in Figure 5-5 as a function of Per. The value of 24(P) is positive

for Per > 0 and approaches a limiting value of 24, whereas the value of a~ is weakly

negative for values of Pe, less than 1, as confirmed by a low Peclet expansion of the

distribution function. Consequently, Eq. 5-52 predicts a weak migration towards the wall

for sufficiently small shear rates. However, Pe(a~) increases indefinitely over the range of

Pe studied and dominates the migration behavior for strong shearing flows.

Figure 5-6 shows the center of mass distribution as a function of the distance from the

wall. The distribution is calculated for Pe ranging from 7.2 x 102 to 4.8 x 10s. In general,

a net migration away from the wall is observed. For Pe < 7.2 x 102, the probability

distribution is approximately uniform at all distances from the wall. The concentration

near the wall is higher than expected since excluded volume effects are not included in our



















,' I P=.x0
I / / .
*1 / Pe=1.2x10x10
0.2
-I / ', /' P= .- Pe4.8



0 1 2 3i 4 5 6



Figure 5-6. Center of mass distribution, n(Z2), plotted as a function of x2/L and Pe.


calculations. As the center of mass Peclet number increases, a noticeable depletion lIns-cr

occurs in the vicinity of the wall as fibers drift away from the boundary. The imbalance

of the center of mass distribution increases as the flow strength increases, such that at the

highest Pe, there are no particles occupying a space one fiber length away from the wall.

The depletion thickness is plotted as a function of Pe in Figure 5-7. The depletion

LIn-;-r thickness is defined as the distance from the wall where n(Z2) = 0.1 and n(Z2) = 0.5.

As the flow strength increases, the depletion 1 e. -r increases. At the lowest Pe, the

depletion thickness defined according to n(Z2) = 0.1 is equivalent to 0.01L. Increasing

Pe three orders of magnitude results in a depletion lIn-;-r thickness equivalent to 4L.

The results are qualitatively similar to simulation results for rigid fibers undergoing

pressure-driven flow [117].

The migration of rigid fibers away from the wall during shear flow qualitatively

differs from models of rigid fibers which neglect hydrodynamic interactions with the

walls. Nitsche and Hinch [102] predict a migration of fibers toward the wall for arbitrary

values of Pe when neglecting both steric and hydrodynamic interactions with the wall.

Though limited to weak flows, the model of Schiek and Shaqfeh [118], which includes steric


















O

a, 0.1


Pe

Figure 5-7. Depletion thickness normalized by fiber length plotted as a function of Pe.
The depletion lI T-c r is defined separately as the distance from the wall where
n (-2) = 0.1 (circles), and n (12) = 0.5 (squares).


wall effects, also predicts migration towards the wall. The qualitative differences in the

prediction of the direction of migration imply that hydrodynamic interactions with the

wall pIIl v, a significant role.

The model prediction of migration of rigid fibers away from the wall during simple

shear flow has implications in terms of the theology of rigid fiber suspensions. As

discussed in the previous chapter, suspensions of spheroids at large Pe, exhibit rate

dependent theology even though theories and simulations predict otherwise. The results

from the model may partially explain the shear thinning theology observed in experiments.

The model predicts a migration of rigid fibers away from the wall, with the magnitude

of the migration depending upon the flow strength. A depletion 1.,-:-r near the boundary

may effect theological measurements, which evaluate the stress at the wall. Specifically,

the magnitude of the stress should decrease as the depletion 1 ,-c -r grows. A depletion 1 .,-c :

which grows with flow strength as predicted in Figure 5-7 will thus result in a stress which

decreases with flow strength.










Conclusions

In this chapter, a model for the migration of a rigid Brownian fiber near a single

boundary was presented. The relationship between the direction of migration and the

orientation of a fiber was investigated and the center of mass distribution function

was derived as a function of Pe. The migration is controlled by two contributions; the

Brownian force acting on the fiber and the imposed shear flow. At high Pe,, the imposed

shear flow dominates and results in a lift of the fiber
l a-;r thickness increases as the flow strength increases. The model predictions may provide

an explanation for the apparent anomalous rate dependent theology of suspensions of rigid

fibers [35, 66, 119].

Though the model presents general trends for a rigid fiber undergoing simple shear

flow in the presence of a single boundary, work must he done to extend the model to

results from experiments and simulations. For example, we neglect excluded volume

effects which results in an enhanced concentration of fibers near the wall, especially at low

Pe. To gain a more accurate distribution, these effects can he taken into account using

results which include the effect of excluded volume [118]. This study will also be extended

to evaluate the stress. Once the stress calculation is completed, the model results can

he compared to the experimental work presented in ChI Ilpter 4 as well as the dynamic

simulation of a Brownian rod near a solid boundary.









CHAPTER 6
CONCLUSIONS

The work presented in this dissertation significantly contributes to the knowledge

of the dynamics of suspension systems. Suspensions exhibit av .vi'. iv of properties that

are utilized in many technological and industrial applications, which include paints and

inks, food thickeners, body armor, and brakes and clutches, among others (C'!s Ilter 1).

To enhance the quality of products derived from suspension systems, studies must be

conducted to develop a complete understanding of their dynamics. This work specifically

addresses 1) the theology of noncolloidal suspensions of spheres undergoing oscillatory

shear flow and 2) the dynamics of suspensions of rigid fibers.

Despite studies on the theology of noncolloidal suspensions in oscillatory shear flow

[26, 65, 69, 101], the theology is not well-understood. For example, a drift in viscosity

is observed prior to steady state at all strain amplitudes [65, 69]. Similarly, the results

presented in Chapter 2 show that the viscosity increases with total strain for small strain

amplitudes and decreases for large strain amplitudes, with the transition point at which

the qualitative behavior changes occurring at an amplitude-to-gap ratio between 0.1 and

0.5. Explanations for the drift have been offered [105, 106], yet the origin remains unclear.

Results from the first dynamic simulation of noncolloidal spheres in unsteady shear flows

((C!s Ilter 3) resolve this issue. The drift in viscosity at each strain amplitude occurs as a

result of microstructural changes that disrupt the equilibrium structure over large total

strains. The large strains required to reach steady state -II__- -r that previous results from

experiments may not have been reported with respect to their steady state values.

Another illI i ~r contribution of this work is the discovery of a nonmonotonic

dependence of the steady state viscosity on strain amplitude, which has not been reported

previously. Specifically, simulations and experiments show that at large strain amplitudes,

the viscosity approaches the value observed in steady shear. As the strain amplitude

decreases, the steady value of the viscosity decreases until a minimum is observed at a










strain amplitude of unity. The steady value of the viscosity then increases as the strain

amplitude decreases. Dynamic simulations reveal three distinct microstructural regimes

which depend upon the applied strain amplitude. Hydroclusters [16] are observed at

large strain amplitudes, where the theology closely resembles steady shear. At a strain

amplitude of 1, where a minimum in the viscosity is observed, ordered 1 swr-is of particles

form both parallel and perpendicular to the flow direction, allowing particles to easily slide

past one another during shear. The formation of < sof .1111,- structures are observed at the

lowest strain amplitudes studied. The most significant of these steady state structures is

the formation of a crystalline phase at low strain amplitudes, even though the particle

concentration is low enough such that the equilibrium structure is liquid-like. This is a

surprising result considering that at low strain amplitudes, particle displacements are

small and interactions are short-ranged, occurring only between nearest neighbors. In

this case, one might expect particle trajectories to be reversible [112], and the equilibrium

structure to remain undisturbed. Our experiments and simulations, however, show that

particle trajectories are irreversible over large total strains, leading to the formation of

steady structures which are highly ordered.

This work would benefit greatly from direct experimental evidence of the steady state

microstructure as a function of strain amplitude. Though results from optical experiments

are available for Brownian suspensions undergoing oscillatory shear flow [2-4], no such

experiments exist for suspensions of noncolloidal spheres. Similarly, model predictions

are not available for oscillatory shear flow. Currently, it is unclear whether or not current

models [68, 108, 127] on the theology of noncolloidal suspensions can account for the

behavior presented in this dissertation. The 1!! i 1- difficulty in applying these models

to oscillatory shear flow has been the lack of experiments; the comprehensive results

presented in Chapter 2 should allow for an adequate comparison. As a result, these

models need to be extended to oscillatory shear flows. Lastly, fully three-dimensional










simulations must he conducted to account for more realistic structures, especially at low

strain amplitudes.

An additional contribution of this work concerns the theology of fiber suspensions

at large rotational Peclet numbers. In ('!! Ilpter 4, we presented results from a model

suspension of polystyrene ellipsoids with moderate aspect ratios. Similar to previous

experiments on fiber suspensions at large rotational Peclet numbers [35, 66, 119],

polystyrene ellipsoid suspensions exhibit shear thinning behavior. Theories and simulations,

however, do not predict a dependence of the theology on shear rate [107]. As a result,

explanations have been offered to account for the rate dependence observed in some

specific experiments [18, 35], however the origin of shear thinning in experiments

remains unclear. The 1!! I r~~ advantage of this work is that we are able to evaluate

and compare the theology of suspensions of spheres having material properties identical

to the ellipsoids. In this manner, effects associated with the method of manufacturing the

ellipsoids, as well as other colloidal interactions, were found to be unlikely causes of the

rate dependent theology. In lieu of these, a mechanism, involving competition between

particle drift arising from hydrodynamic interactions with the bounding walls and thermal

diffusion, is proposed to explain the observed shear thinning phenomenon in our system.

To compliment the work on the theology of fiber suspensions, ('!s Ilter 5 presents

an outline of calculations to determine the center of mass distribution of slender bodies

in bounded simple shear flow. Preliminary results indicate that a drift occurs away from

the bounding walls in simple shear flow. The magnitude of the drift velocity depends

on the flow strength relative to the applied Brownian force, and fiber orientation. To

determine whether hydrodynamic interactions between the particles and bounding walls

can account for the rate dependent theology observed in experiments, a calculation of the

total stress for slender bodies in simple shear flow is required. Furthermore, to correlate

more precisely with the experiments, calculations must he performed for ellipsoids as

opposed to slender bodies. Further evidence of the migration should be obtained from










optical microscopy experiments on the poli-- i- 1. Im ellipsoid suspensions flowing through a

microchannel, allowing direct measurement of the center of mass distribution as a function

of flow strength.









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BIOGRAPHICAL SKETCH

.Jonathan Mark Bricker was born in Orlando, Florida on February 24, 1980 to Paul

and Mariann. His family briefly relocated to San Antonio, Texas before moving on to

Greenville, South Carolina, where .Jonathan spent most of his adolescence. He attended

Eastside High School in Greenville and graduated 9 ^ of 215 students.

.Jonathan chose to attend the Honors College at the University of South Carolina,

where he graduated with honors and received a Bachelor of Science degree in chemical

engineering. While attending ITSC, .Jonathan conducted research under the direction of

Dr. Francis Gadala-Maria in the area of complex fluids.

.Jonathan continued his education as a research assistant at the University of Florida.

He undertook research with Dr. .Jason E. Butler in the area of complex fluids. Specifically,

.Jonathan focused on describing the dynamics of suspension systems using experimental

and computational means. .Jonathan received a Doctor of Philosophy degree in chemical

engineering, and now looks forward to beginning the next chapter in his life as he applies

the skills developed in graduate school to the workplace.





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Iamfortunatetohavemetmanyspecialpeopleandfriendswhohavecontributedtomyresearchovertheyears.Withoutthem,mysuccesswouldnothavebeenpossibleandIextendaheartfeltappreciationtotheseindividuals.Foremost,Imustthankmyresearchadvisor,JasonE.Butler.Overtheyears,Jasonhastaughtmethescienceofuidmechanicsthroughbothexperimentalandcomputationalmeans.Hehasinstilledinmethequalitiesofagoodresearcher;namelyhehastaughtmetobepersistentandinquisitiveconcerningallthingsknownandunknown.Hispatienceallowedmetodiscoverconceptsonmyownandhispassionanddriveinspiredandchallengedme.Jasonallowedmetomatureasaresearcher,andasaperson.Forthesethings,Iamgrateful.Ialsobenettedgreatlyfromtheadviceanddirectionofmycommitteemembers.Specically,IextendthankstoDr.AnthonyLaddforhishelpfuldiscussionsandcriticismofmyworkovertheyearsandDr.RangaNarayananwhohastaughtmetoconstantlyquestionthestatusquo.Althoughnotonmycommittee,IthankDr.KirkZieglerforhisinsightonlifeingeneral.Iowethankstomycolleagues,PhilipD.Cobb,JoontaekPark,andHyun-OkParkfortheirwillingnesstolistenandtheirpatienceinteachingme.IextendspecialthankstoO.BerkUsta,whowasinstrumentalintheprogressionofmywork.Intheprocess,hehasbecomeavaluablefriendwhosesincerityandunselshnessIwillneverforget.IwouldalsoliketothankstamembersDennisVinceandJimHinnantfortheirhelpinkeepingthingsoperationalinthelaboratory.Theirworkhasbeeninvaluableovertheyears.IamfortunatetohavedevelopedmanyfriendshipsattheUniversityofFlorida.IwouldliketothankMattMonroe,PatrickMcKinney,MichaelJune,andDarrenMcDufortheirfriendshiponandothecourt.IalsoextendthankstoColinSturmforhiswisdomandhumor.Ilookforwardtoconductingbusinesswithhiminthefuture. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 13 CHAPTER 1THERHEOLOGYOFSUSPENSIONS:PASTEXPERIMENTS,SIMULATIONS,ANDTHEORIES ......................... 15 Introduction ...................................... 15 SuspensionsofSpheres ................................ 17 SuspensionsofFibers ................................. 22 2OSCILLATORYSHEAROFSUSPENSIONSOFNONCOLLOIDALPARTICLES ..................................... 25 Introduction ...................................... 25 Experiment ...................................... 27 Results ......................................... 32 SteadyShear .................................. 32 OscillatoryShear:MonodisperseSuspension ................. 36 OscillatoryShear:ComparisonsBetweenSystems .............. 41 Discussion ....................................... 43 SteadyShear .................................. 43 OscillatoryShear ................................ 45 Originofoscillatorybehavior ..................... 45 Straindependenceoftheresults .................... 48 Eectofsuspensioncharacteristics .................. 49 Conclusions ...................................... 52 3CORRELATIONBETWEENSTRESSESANDMICROSTRUCTUREINCONCENTRATEDSUSPENSIONSOFNON-BROWNIANSPHERESSUBJECTTOUNSTEADYSHEARFLOWS ................... 53 Introduction ...................................... 53 RheologySimulations ................................. 54 TheStokesianDynamicsMethod ....................... 55 EvaluationofRheology ............................. 58 Implementation ................................. 60 Results ......................................... 61 SteadyShear .................................. 62 6

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................................. 66 OscillatoryShear ................................ 70 Rheology ................................. 70 Suspensionmicrostructure ....................... 74 Discussion ....................................... 77 StrainDependentDiusivity .......................... 79 SuspensionMicrostructureandRheology ................... 82 EectoftheRepulsiveForceontheRheology ................ 86 NormalStresses ................................. 87 RelationshipBetweenOscillatoryandShearReversalRheology ....... 89 Conclusions ...................................... 90 4RHEOLOGYOFSEMI-DILUTESUSPENSIONSOFRIGIDPOLYSTYRENEELLIPSOIDSATHIGHPECLETNUMBERS ................... 92 Introduction ...................................... 92 Experiment ...................................... 93 Results ......................................... 97 Discussion ....................................... 104 RateDependentRheology ........................... 104 ScalingsofratHighPer 109 Conclusions ...................................... 110 5CROSS-STREAMMIGRATIONOFRIGIDBROWNIANFIBERSUNDERGOINGSIMPLESHEARFLOWNEARASOLIDBOUNDARY ... 112 Introduction ...................................... 112 Model ......................................... 113 VelocityExpressions .............................. 114 ProbabilityDistribution ............................ 118 CalculationofOrientationMoments ...................... 120 ResultsandDiscussion ................................ 122 CenterofMassVelocity ............................ 122 CenterofMassDistribution .......................... 123 Conclusions ...................................... 127 6CONCLUSIONS ................................... 128 REFERENCES ....................................... 132 BIOGRAPHICALSKETCH ................................ 142 7

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Table page 2-1Summaryofthemonodisperseandpolydispersesuspensionsystemsstudied ... 30 4-1Summaryofthecharacteristiclengthscales,volumeperparticle,andsurfaceareaperparticleofpolystyreneparticles ...................... 95 8

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Figure page 1-1ImagesofwovenKevlarfabrics ........................... 16 1-2Electrorheologicalsuspensioncomposedofsilicaspheresincornoil ....... 17 1-3Summaryofobservationsofthenetbulkmigrationofparticlesundergoingoscillatorypipeowversusdimensionlessstrainamplitude ............ 18 1-4PhasediagramtakenfromtheworkofAckerson[ 3 ]forBrowniansuspensionsundergoingoscillatoryshearowasafunctionofvolumefractionandstrainamplitude ....................................... 19 1-5Dimensionlessdiusivityintheow(closed)andgradient(open)directionasafunctionofstrainamplitudeforconcentratednoncolloidalsuspensions ...... 21 1-6Schematicshowingthesteadystatespatialandorientationaldistributionofberspriorto(left)andfollowing(right)simpleshearow ............ 23 1-7Relativeviscosityofsemi-concentratedsuspensionsofbersasafunctionofshearrate ....................................... 24 2-1SEMimagesandrespectivesizedistributionsoftheparticlesusedintheoscillatoryexperiments ................................ 28 2-2RelativeviscosityplottedasafunctionoftimeformonodispersePMMAspheresinUCON/water/NaI(triangles),monodispersepolystyrenespheresinpolyalkyleneglycol(squares),monodispersePMMAspheresinEG/glycerol(crosses),andpolydispersePMMAspheresinEG/glycerol(circles)undergoingsteadyshearintheCouettegeometry ........................ 33 2-3RelativeviscosityplottedasafunctionoftimeformonodispersePMMAspheresinUCON/water/NaI(triangles),monodispersepolystyrenespheresinpolyalkyleneglycol(squares),monodispersePMMAspheresinEG/glycerol(crosses),andpolydispersePMMAspheresinEG/glycerol(circles)undergoingsteadyshearintheparallel-plategeometry ..................... 35 2-4ImagesoftheUCONoil/watersuspendingliquidintheparallel-plategeometry 36 2-5Inputstrainwaves(dashedline)andresultingoutputstresswaves(solidline)plottedversustimeforthe=0.40suspensionofmonodispersepolystyrenespheressuspendedinpolyalkyleneglycol ...................... 38 2-6RelativecomplexviscosityasafunctionoftimeforamonodispersesuspensionintheCouettegeometry ............................... 39 2-7Relativecomplexviscosityasafunctionofstrainandamplitude-to-gapratioforamonodispersesuspensionintheCouettegeometry .............. 40 9

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................. 41 2-9Comparisonoftheresponsesobservedinparallel-plate(closedsymbols)andCouette(opensymbols)geometriesforamonodispersesuspensionsystem .... 42 2-10ComparisonoftheresponsesobservedintheCouettegeometryformonodispersesuspensionsystemscontainingpolystyrene(opensymbols)andPMMA(closedsymbols) ............................... 43 2-11ComparisonoftheresponsesobservedintheCouettegeometryforsuspensionsofPMMAsphereswithmonodisperse(closedsymbols)andpolydisperse(opensymbols)sizedistributions .............................. 44 2-12DierencebetweenthenalandinitialvaluesofthecomplexviscositynormalizedwithrespecttoitsinitialvalueintheCouettegeometryplottedasafunctionofamplitude-to-gapratio .......................... 46 2-13Complexviscosity(evaluatedat=15000)plottedasafunctionofappliedstrainamplitudefortheCouetteandparallel-plategeometries .............. 47 2-14Complexviscosity(evaluatedat=15000)plottedasafunctionofappliedstrainamplitudeandconcentrationfortheparallel-plategeometry ........... 50 2-15Complexviscosity(evaluatedat=15000)plottedasafunctionofconcentrationfortheparallel-plategeometry ............................ 51 3-1Schematicshowingparticlepairsexposedtosimpleshearow .......... 54 3-2Theperiodiccellusedinthesimulations ...................... 56 3-3Particlecontributiontotheshearstress,yx,asafunctionofarealfraction,,forsteadyshearow ................................. 63 3-4Normalstressyyasafunctionofarealfraction ..... 64 3-5NormalstressdierenceN1asafunctionofarealfraction 64 3-6Pairdistribution,g(r),asafunctionofradialdistanceandarealfractionintegratedoverallangles ............................... 65 3-7Angularpairdistribution,g(),atcontactforparticleslocatedinthecenterofthegap(13
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69 3-11Inputstrainwave(dottedline)andtheresultingoutputstresswave(solidline)plottedversusstrainforsimulationsat=0.60foragapof15 .......... 70 3-12ParticlecontributiontotheshearstressplottedversustotalstrainforoscillatoryshearowatappliedstrainamplitudesofA=0.1,0.25,0.5,1,and2 ...... 71 3-13Largestrainviscosityinoscillatoryshearownormalizedbythesteadystateviscosityinsteadyshearowasafunctionofstrainamplitude .......... 72 3-14Normalstressnormalizedbythesteadyshearvalueplottedasafunctionofstrainamplitudeforoscillatoryshearowfor=0.60andagapof15 ...... 74 3-15Secondmomentoftheparticledistances(atsteadystate)fromthecenterlineasafunctionofstrainamplitude .......................... 75 3-16Theradialpairdistributionasafunctionoftheradialdistanceintegratedoverallangles(0<<180)forasimulationat=0.60andagapof15 ......... 76 3-17Theangularpairdistribution,g(),atcontactasafunctionofthetaforasimulationat=0.60andagapof15 ........................ 78 3-18Meansquareddisplacementsnondimensionalizedbytheparticleradius,a,plottedversustotalstrainforastrainamplitudeofA=1 ............. 79 3-19Dimensionlesshydrodynamicdiusivitiesplottedversusstrainamplitudeforsimulationsat=0.60andagapof15 ....................... 81 3-20InitialcongurationofparticleswithintheshearcellpresentedalongwiththeinstantaneoussteadystateparticlecongurationsforstrainamplitudesofA=5,1,and0.1 ....................................... 83 3-21Schematicshowingtheslidingmechanism(a)observedforintermediatestrainamplitudesversusthe`lockedlayer'mechanism(b)observedforlowstrainamplitudes ....................................... 84 3-22Steadystateviscosityinoscillatoryshearownormalizedbythecorrespondingsteadystateviscosityinsteadyshearowasafunctionofstrainamplitude ... 86 4-1Scanningelectronmicrographsofpolystyreneparticles .............. 93 4-2Normalizedsizedistributionsforellipsoidswithanaverageaspectratioof4(solid)and7(dashed) ............................... 97 4-3Relativeviscosityasafunctionoftimeandshearrateforpolystyreneellipsoidssuspendedinapolyalkyleneglycol/water/KClmixture .............. 98 4-4Relativeviscosityasafunctionoftimeandvolumefractionforpolystyrene 11

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.......................... 99 4-5Relativeviscosityasafunctionoftimeandvolumefractionforpolystyreneellipsoidssuspendedinapolyalkyleneglycol/water/KClmixture.Theellipsoidshaveanaverageaspectratioof7 .......................... 100 4-6Relativeviscosityasafunctionofshearrateandvolumefractionforpolystyreneellipsoidssuspendedinapolyalkyleneglycol/water/KClmixture.Eachdatapointrepresentstheapparentviscosityafter360secondsofshearateachcorrespondingshearrate.Theellipsoidshaveanaverageaspectratioof4 .... 101 4-7Relativeviscosityasafunctionofshearrateandvolumefractionforpolystyreneellipsoidssuspendedinapolyalkyleneglycol/water/KClmixture.Eachdatapointrepresentstheapparentviscosityafter360secondsofshearateachcorrespondingshearrate.Theellipsoidshaveanaverageaspectratioof7 .... 102 4-8Comparisonoftherelativeviscositymeasuredinthecone-and-plate(dottedline)andparallel-plate(solidline)geometryforsuspensionsofpolystyreneellipsoidswithL=d=4inapolyalkyleneglycol/water/KClmixture ........ 103 4-9SEMimagesofprocessedpolystyrenespheresbefore(left)andafter(right)washing ........................................ 105 4-10Relativeviscosityasafunctionofshearrateandaspectratiofordierentsuspensionsystems .................................. 106 4-11SteadystaterelativeviscosityatPer=106asafunctionofvolumefractionandaspectratio ...................................... 110 4-12SteadystaterelativeviscosityatPer=104and106asafunctionofdimensionlessnumberdensityandaspectratio ........................... 111 5-1Geometryandnotationusedtodescribeaberinsolutionundergoingsimpleshearownearasolidboundary .......................... 113 5-2NumericallycalculatedensembleaverageoftheorientationmomentsasafunctionofPer 121 5-3Eectofberorientationonthecenterofmassdriftvelocity,_x2 122 5-4Schematicsummarizingtheeectofberorientationonthedriftvelocity .... 123 5-5Contributionoftheshearow(Pe())andBrowniantorque(24())tothemigrationofarigidberplottedasafunctionofPer. ............... 124 5-6Centerofmassdistribution,n(x2),plottedasafunctionofx2=LandPe 125 5-7DepletionthicknessnormalizedbyberlengthplottedasafunctionofPe 126 12

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Theworkpresentedinthisdissertationprovidesasignicantcontributiontounderstandingtherheologicalbehaviorofsuspensionssystems.Specically,thisworkprovidesresultsfromacomprehensivesetofexperimentsontherheologyofsuspensionsofnoncolloidalspheresundergoingoscillatoryshearow.Theexperimentsarecomplimentedbyresultsfromtherstdynamicsimulationofsuspensionsofnoncolloidalspheresinunsteadyshearows.Additionally,resultsfromexperimentsonthesteadyshearrheologyofmodelsuspensionsofrigidellipsoidsarepresented. Experimentsonsuspensionsofspheresundergoingoscillatoryshearowshowthattherheologyisstronglyinuencedbytheappliedstrainamplitude.Ateachamplitude,thesteadyvalueofthecomplexviscositydecreaseswithtotalstrainforhighstrainamplitudesandincreasesforlowamplitudes.Thetransitionpointatwhichthequalitativebehaviorchangesoccursatanamplitude-to-gapratiobetween0.1and0.5andisindependentoftheparticlesizedistributionandsuspensionsystem.Thesteadystatevalueofthecomplexviscosityisanonmonotonicfunctionoftheappliedstrainamplitude,withaminimumviscosityobservedatanamplitude-to-gapratioof1.Theexperimentssuggestthatshear-inducedmigrationisofnoconsequence,andthattheobservedbehaviorisinsteadduetochangesinthesuspensionmicrostructure.TherheologyobservedintheexperimentsislargelyconrmedbyStokesiandynamicssimulations.Simulationsofsuspensionsofnoncolloidalspheresundergoingunsteadysimpleshearowsare 13

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Therheologyofsemi-dilutesuspensionsofrigidpolystyreneellipsoidsatrotationalPecletnumbersgreaterthan103wasstudiedfortwodierentaspectratios.Theellipsoidsuspensionsexhibitshearthinningbehaviorforbothaspectratios.Althoughinagreementwithpreviousexperiments,ratedependentrheologyisnotpredictedbytheoriesandsimulations.Directcomparisonoftheresultswithrheologicalmeasurementsonsuspensionsofsphereswithmaterialpropertiesidenticaltotheellipsoidssuggestthatcolloidalinteractionsareimprobablesourcesoftheshearthinningbehavior.Amechanismfortheobservedshearthinningbehaviorisproposedwhichinvolvesacompetitionbetweenparticledriftduetoshear-inducedmigration,andthermaldiusion.Forthelargestvalueoftheshearrate,therelativeviscosityscaleslinearlywithdimensionlessnumberdensityregardlessoftheaspectratio.Atthelowestshearrate,therelativeviscositydoesnotscalelinearlywithdimensionlessnumberdensityandadependenceontheellipsoidaspectratioisapparent. 14

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33 ],andhouseholdgoodssuchasinks,paints,andcement[ 40 88 131 ].Asaresultoftheubiquitousnatureofthisparticularclassofcomplexuids,manystudieshavebeenconductedtounderstandthedynamicsofsuspensionsystems. Therheologyofsuspensionsystemsvarieswidelydependingonthenatureoftheindividualphasesandtheinteractionsamongthem.Ingeneral,theadditionofsolidparticlestoasuspendingliquidcontributestothetotalstressofthesuspensionresultinginanenhancedviscosity,evenatlowvolumefractions[ 13 56 57 86 ].Incertainapplications,particlesarespecicallyusedtoaltertheviscosityofasubstance.Forexample,xanthamgumisarigidrod-likepolysaccharidecommonlyusedasafoodthickener.Incertainsuspensionsystems,theviscositydependsontherateofdeformation.ExamplesofsuspensionswhichtypicallyexhibitratedependentrheologyincludesystemswhichcontainBrownianparticles[ 3 16 17 62 111 ]orparticleswhichinteractthroughelectrostaticinteractions[ 71 115 134 135 ].Shearthinningphenomenaoccurswhentheappliedshearratebecomelargeenoughsuchthatthestressesdisturbtheequilibriumparticledistribution.Householdpaints,whichthinuponapplicationwitharoller,areshearthinninguidscomposedofcolloidalsilicaspheresinanorganicsolvent[ 40 ].Athighervolumefractionsandshearstresses,shearthickeningoccurs.Thispropertyisparticularlyadvantageousinbodyarmortechnology[ 44 79 130 ].Forexample,Leeetal.[ 89 ]showthatbyimpregnatingKevlarwovenfabricswithashearthickeningsuspensionofcolloidalsilicainethyleneglycol,theeectivenessofbodyarmorissignicantlyenhancedasshowninFigure 1-1 .Electrorheologicaluidsareanotherexampleinwhichinteractions 15

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ImagesofwovenKevlarfabricswithout(left)andwith(right)impregnationofacolloidalshearthickeningsuspensionfollowingballisticstesting.TheimageistakenfromtheworkofLeeetal.[ 89 ],whofoundthattheadditionofashearthickeninguidtoKevlardramaticallyimprovestheeciencyofbulletproofvests. betweenparticlesaectthesuspensionrheology.Electrorheologicaluidscontainpolarizableparticlesinanonpolarsuspendingmedium.Whenanelectriceldisappliedtothesuspension,theparticlesformdipoles,allowingbrousparticlechainstoforminthedirectionparalleltotheelectriceldasshowninFigure 1-2 .Thesechainsdramaticallyincreasetheviscosityandcausethesuspensiontohaveayieldstressproportionaltothesquareoftheelectriceld[ 22 84 139 ].Theuniquepropertiesoftheseuidsareutilizedinautomotiveapplicationssuchasshockabsorbersandvariable-dierentialtransmissions,andinindustrialapplicationssuchasvariableowpumps[ 72 ].Otheruidproperties,suchaselectricalandthermalconductivity,canbealteredbytheadditionofparticlessuchascarbonnanotubes,whichformlightweighthigh-strengthcompositeswithsuperiormaterialproperties[ 15 63 67 140 ]. Clearly,suspensionshaveawidevarietyofusesowingtotheuniquepropertiesthatdevelopasaresultoftheinteractionsbetweenphases.Understandingtherheologicalpropertiesofsuspensionsystemsiscrucialtoimprovingconsumerproductsandindustrialprocesses.Forsuspensionsofsphericalparticles,theliteratureisrichandmuchisknownabouttherheology,includingmacro-andmicroscopicdetails.Ontheother 16

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ElectrorheologicalsuspensioncomposedofsilicaspheresincornoiltakenfromtheworkofKlingenbergetal.[ 84 ].Theleftimageshowstherandomdistributionofparticlespriortoapplicationofanelectriceldandtherightimageshowstheformationofchainsunderthepresenceofanelectriceld.ERuidsareusedinavarietyofautomotiveandindustrialapplications[ 72 ]. hand,suspensionsofbersarenotfullyunderstoodanddisparitiescurrentlyexistintheliterature.SuspensionsofSpheres 126 ].Therheologyintime-dependentowssuchasshearreversalandoscillatoryshearowislessunderstood.Forexample,inoscillatoryshearow,therheologyisoftentimesunexpectedandqualitativelydiersfromsteadyshear. Thephenomenonofshear-inducedparticlemigrationcanbequalitativelydierentdependingonthenatureoftheow.Shear-inducedparticlemigrationisaphenomenoninwhichparticlesexposedtoaninhomogeneousshearowundergoabulkmigrationfromregionsofhighshearratetoregionsoflowshearrate[ 5 ].Theroleofshear-inducedparticlemigrationonsuspensionrheologycanbeprofoundandoftentimesleadstoanomalousresults[ 116 ].Shear-inducedmigrationhasbeenobservedinsteadyshearowssuchaspressure-drivenow[ 7 29 73 93 125 ],narrow-gapCouetteow[ 28 37 65 91 ],andwide-gapCouetteow[ 1 109 ].Therearecurrentlytwocompetingmodelswhichdescribeshear-inducedparticlemigrationinsteadyows.Inthe`diusiveux'model[ 91 17

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Summaryofobservationsofthenetbulkmigrationofparticlesundergoingoscillatorypipeowversusdimensionlessstrainamplitude.TheschematicistakenfromtheworkofButleretal.[ 30 ].Forreference,particlesinsteadypipeowundergoanetmigrationtothecenterofthepipeandreversemigrationdoesnotoccur. 109 ],theparticleuxarisesduetotheself-diusionofparticles,whileinthe`suspensionbalance'model[ 103 ],theparticleuxarisesdirectlyfromtheparticlestress.Bothmodelssuccessfullydescribethesteadystateconcentrationprolesinanumberofsteadyows.However,inoscillatorypressure-drivenow,experiments[ 30 ]andsimulations[ 97 ]showthatshear-inducedparticlemigrationisafunctionoftheappliedstrainamplitudeassummarizedinFigure 1-3 .Forlargeamplitudesofoscillation,particlesmigratetothecenterofthepipe,astheydoinsteadypipeow.Atlowstrainamplitudeswhereonemightexpectauniformconcentrationprole,particlesmigratetothewallsofthepipe.Currentlyitisunclearifexistingmodelscanaccountforthereversemigrationobservedinoscillatoryshearow,orifasimilarreversemigrationphenomenaoccursforotheroscillatoryshearows,suchasnarrow-gapCouetteow. Inadditiontomacroscopicrheologicaldierences,oscillatoryandsteadyowofsuspensionscanyielddistinctbehavioratthemicroscopiclevel.InconcentratedsuspensionsofBrownianspheres,thesuspensionmicrostructuredependsonthebalancebetweenBrownianandhydrodynamicforces,aswellastheowtype[ 135 ].Insteadyshear,simulations[ 62 110 111 ]andexperiments[ 3 4 ]showthreedistinctmicrostructural 18

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PhasediagramtakenfromtheworkofAckerson[ 3 ]forBrowniansuspensionsundergoingoscillatoryshearowasafunctionofvolumefractionandstrainamplitude.Thesymbolscorrespondtoamorphous(A),FCCcrystal(C),andslidinglayer(S)structures.ThecrystallinestructuresareuniquetooscillatoryshearandarenotobservedinthesteadyshearofBrowniansuspensionsatvolumefractionsinwhichtheequilibriumphaseisliquid-like(<0:47). phasesdependingupontheratioofhydrodynamictoBrownianforces.Amorphousstructures,string-likestructures,andhydroclusters[ 17 46 95 ]canallbeinducedbyalteringtheratioofhydrodynamictoBrownianforces.Whenexposedtooscillatoryshearow,Browniansuspensionsshowanadditionalmicrostructuralphase.Figure 1-4 showsthestructuresobtainedfromlightscatteringexperimentsonBrowniansuspensionsundergoingoscillatoryshearowasafunctionofconcentrationandstrainamplitude[ 2 { 4 ].Atcertainstrainamplitudes,crystallinestructuresareformedevenwhentheequilibriumstructureisliquid-like.Suchstructurescannotbeinducedbysteadyshear,unlesstheparticlevolumefractionislargeenoughsuchthattheequilibriumstructureiscrystalline.AlthoughmicrostructuraldierencesinsuspensionsofBrownianspheresarewell-dened,itisunclearwhetherornotsimilardierencesoccurinnoncolloidalsuspensions. 19

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5 55 126 ].Ecksteinetal.[ 55 ]foundthathydrodynamicinteractionsamongparticlesleadtotheirregularmotionofparticlessimilartorandomwalks.Infact,thehydrodynamicdiusivityofatracerparticlehasbeenmeasuredbytrackingthemotionofasingleparticleinsuspension[ 55 90 ].Severalsourcesexistfortheirreversibility.Forexample,althoughthetrajectoriesoftwoisolatedsmoothparticlesinteractinginStokesowaresymmetricandreversible,three-bodyinteractionscanleadtoasymmetricirreversibleparticledisplacements[ 26 53 91 ].Irreversibilityofparticletrajectoriescanalsooccurasaresultofinteractionsbetweenroughspheres[ 8 43 114 ].Inoscillatoryshearow,thehydrodynamicdiusivityofnoncolloidalparticlesisafunctionofappliedstrainamplitude.Figure 1-5 showsresultsfromexperimentsandsimulationsonnoncolloidalsuspensionsperformedbyPineetal.[ 112 ].Asthestrainamplitudedecreases,thehydrodynamicdiusivitydecreasesdramatically.Asaresult,Pineetal.[ 112 ]identifyawell-denedthresholdstrainamplitudebelowwhichparticletrajectoriesarereversible.Abovethecriticalthreshold,particlesdiuseirreversiblywithtime,asmostphysicalprocessesdo[ 121 ].TheresultsofPineetal.howeverseematoddswithexperimentsontheoscillatoryshearofnoncolloidalsuspensions[ 65 69 ]whichshowchangesinthesuspensionviscosityoverlargestrainsevenatsmallstrainamplitudes,aswellaswithexperimentsandsimulationsontheshear-inducedmigrationofnoncolloidalsuspensionsinoscillatorypipeow[ 30 97 ],whichshowanetmigrationofparticlesatlowstrainamplitudes.Asaresult,itisunclearwhetherornotathresholdforirreversibilitytrulyexists. Thisworkaddressessomeoftheunansweredquestionsassociatedwiththerheologyandmicrostructureofnoncolloidalsuspensionsofspheresundergoingoscillatoryshearow.Chapter 2 containsacomprehensivesetofresultsfromexperimentswhichshowthattherheologyisastrongfunctionoftheappliedstrainamplitude.Rheological 20

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Dimensionlessdiusivityintheow(closed)andgradient(open)directionasafunctionofstrainamplitudeforconcentratednoncolloidalsuspensions.Resultsfromexperiments(black)andsimulations(gray)aretakenfromPineetal.[ 112 ]. changesoccuroverlargetotalstrainsforallstrainamplitudesstudied.Thetransientrheologyisattributedtochangesinthesuspensionmicrostructureasopposedtoashear-inducedmigrationphenomenawhichoccursonlyforlargestrainamplitudes.Theexperimentsalsoshowevidenceofanonmonotonicdependenceofthesteadystateviscosityonstrainamplitude,whichhasnotbeenreportedpreviously.TherheologicalobservationsareconrmedbyStokesiandynamicssimulationsinChapter 3 .Inadditiontothemacroscopicbehavior,thesuspensionmicrostructurecanbeevaluated.Thisistherststudytoidentifyandcorrelatetherheologicalbehaviorofnoncolloidalsuspensionsundergoingoscillatoryshearowtothesuspensionmicrostructure.Thesimulationsshowthatthenonmonotonicbehaviorofthesteadystateviscositycorrelatestothreedistinctmicrostructuresinducedbyoscillatoryshearow.Dependingontheappliedstrainamplitude,crystallinestructures,orderedlayers,orhydroclustersareobserved.Additionally,simulationresultsshowthattheparticlediusivityisniteforallstrain 21

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137 ].Thesedimentationofsuspendedparticlesalsodependsonparticleshape.Studiesonthesedimentationofrigidnon-Brownianbers[ 31 76 ]showthatasaninitiallyhomogeneoussuspensionofberssettles,clustersformasaresultofinstabilitiesfromhydrodynamicinteractionsbetweenindividualbers;similarinstabilitiesarenotobservedinsuspensionsofspheres.Asaresultoftheaddedcomplexityofsuspensionscontainingnon-sphericalorientableparticles,thereisconsiderableinterestinunderstandingthedynamicsofsuchsystems. Muchworkhasbeendonetostudythedynamicsofrigidbersuspensions.Inthediluteconcentrationregime,wherethemotionofoneberisnotaectedbythepresenceofneighboringbers,therheologyiswellunderstood[ 19 82 120 ],andquantitativelyagreeswiththeories[ 136 ].Inthesemi-diluteconcentrationregime,wheretherotationofeachberisseverelyrestrictedbyneighboringbers,therheologyisnotaswell-described.ForBrownianbers,predictionsoftheshearrheology,whicharebasedontheoriesofdiusion[ 50 59 60 ],qualitativelypredictcorrecttrends,suchasshearthinningbehavior[ 136 ],howeverthequantitativeagreementwithexperimentsispoor.Intheabsenceofthermalmotion,whereonlyhydrodynamicinteractionsbetweenbersareconsidered,muchlargerdisparitiesexistamongtheories,simulations,andexperiments. Theoriesonsemi-dilutesuspensionsofnon-Brownianberspredictaviscositywhichdependsupontheconcentrationandorientationdistributionofbers[ 12 47 120 ].Theavailabletheoriesdonotpredictadependenceoftherheologicalpropertiesonshearrate.Insimpleshearow,suspensionsofberswilldisplayadecreaseinviscositywithtime, 22

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Schematicshowingthesteadystatespatialandorientationaldistributionofberspriorto(left)andfollowing(right)simpleshearow.Fibersrotateandalignintheowdirectionasaresultofanappliedshearow.Alignmentoccursatsteadystateregardlessofthemagnitudeoftheshearrate. regardlessofthemagnitudeoftheshearrate,astheorientationdistributiontransitionsfromisotropictooneinwhichtheparticlesarealignedintheowdirection,asshownschematicallyinFigure 1-6 .Thesteadystatealignmentofbersisindependentoftheappliedshearrate.Indeed,studiesontheorientationdistributionofnon-Brownianbersinsimpleshearowsshowthatbersorientwiththeowdirectionindenitely[ 128 ]withtheexceptionofperiodicippingmotionofindividualbers[ 82 ].Asaresult,therheologyofnon-Browniansuspensionsofbersisnotexpectedtoberatedependent.Simulationsofnon-Brownianbersuspensionsalsoshownodependenceofthesteadystaterheologyonshearrate[ 113 129 141 ].Experimentalresultshowever,qualitativelydisagreewiththeoriesandsimulations.Figure 1-7 showsacompilationofexperimentsonnon-Brownianbersuspensions,whichconsistentlyshowshearthinningbehavior.Interpretationofexperimentalresultsareoftencomplicatedbythepresenceofnon-idealcharacteristicssuchaslargesizedistributions,boundaryeects,andothernonhydrodynamiceects.Whetherornotaratedependenceisobservedinamodelsuspensionofbers,isunknown.Clearly,astudyoftherheologyofwell-denedbersuspensionsisneeded. Thisworkaddressestheissueoftheratedependentrheologyofnon-Brownianbersuspensions.Resultsfromexperimentsonsuspensionsofpolystyreneellipsoidsinthesemi-diluteconcentrationregimeareprovidedinChapter 4 .Thisistherststudyin 23

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Relativeviscosityofsemi-concentratedsuspensionsofbersasafunctionofshearrate.Resultsarefromareviewofexperimentsperformedpriorto1985[ 66 ].Allexperimentsshowshearthinningrheologydespitetheoriesandsimulationswhichpredictnoratedependencefortherheologyofrigidnon-Brownianbers[ 107 ]. whichtherheologyofsuspensionsofrod-likeparticlesisevaluatedwithrespecttotherheologyofsuspensionsofsphereshavingidenticalmaterialproperties.Similartopreviousexperimentsonbersuspensions,shearthinningbehaviorisobservedatallconcentrationsandaspectratios.InChapter 5 ,abriefexplanationisoeredfortheratedependentrheologywhichinvolvesacompetitionbetweenparticledriftarisingfromshear-inducedmigration,andthermaldiusion. 24

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34 65 132 ],thustherheologyofthesesuspensionsisfairlywell-denedasrecentlyreviewedbyStickellandPowell[ 126 ].Insteadyow,noncolloidalsuspensionsatvanishingReynoldsnumbersundergoadiusive-likemotionwhichisduesolelytohydrodynamicinteractions.Whereasthetrajectoriesofparticlesundergoingsimplepairwiseinteractionsaresymmetricandreversible[ 14 ],inhomogeneitiesintheparticlesurfacecancausepermanentdisplacementsofparticlesfromtheiroriginalstreamlines,resultinginirreversibleinteractions[ 43 114 ]. Shear-induceddiusionofparticlesinsteadyshearowhasbeenobservedinbothexperiments[ 26 55 90 ]andsimulations[ 52 53 ].Theshear-induceddiusionexplainstheapparentlyrandomtrajectoriesthatparticlesundergoinconcentratedsuspensionsexposedtosteadyshearandhasbeenusedtoexplainsuchphenomenaasshear-inducedparticlemigration[ 91 109 ].Thischaoticmotionisintimatelyrelatedtothetimeandspatialcongurationoftheparticles,closelylinkingtherheologytothemicrostructure. Whereastherheologyofsuspensionsinsteadysheariswell-understood,thebehaviorofnoncolloidalsuspensionsinoscillatoryowislessunderstood.Thisisdueinparttofewexperimentsinvestigatingunsteadyows.Gadala-Maria[ 64 ]studiedsuspensionsexposedtooscillatoryshear,ndingthataboveacertainstrainamplitude,thestressresponsebecamenonlinear.Therheologywasfoundtobeindependentofthefrequencyandtheamplitudeofthestressresponsedriftedlowerintimeuntilasteadystatewasachievedafterapproximately100cycles[ 65 ].GondretandPetit[ 69 ]foundasimilardriftforastrainamplitudeof0.1,howeverasteadystatewasachievedafteramuchlargernumberofcycles.GondretandPetit[ 69 ]explainedtheobserveddecreaseinviscositywithtimeasbeingcausedbymicrostructuralchangesduetoaregularorderingofparticlesintolayers 25

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26 ]performedacomprehensivesetofoscillatoryshearexperimentsonnoncolloidalsuspensionsinaCouettegeometryandfoundthattherheologywasbothfrequencyandamplitudedependentandthatplateausexistedatbothsmallandlargeamplitudeswhichwereseparatedbyatransientperiodinwhichtheresponsesignalswerenonlinear.IncontrasttoGondretandPetit[ 69 ],Breedveldetal.[ 26 ]foundthatatsmallstrainamplitudes,theparticlepositionsremainedunchangedwithnoevidenceofashear-inducedmicrostructure.SimilartoBreedveldetal.[ 26 ],Pineetal.[ 112 ]reportedresultsonhydrodynamicdiusivitiesfromsimulationsandexperimentsindicatingthatatsmallstrainamplitudestheparticlediusivitybecameincreasinglysmallsuchthatparticlesreturnedtotheirinitialpositionsaftereachoscillation.Pineetal.[ 112 ]alsofoundatransitionathighstrainamplitudesabovewhichtheowbecomesirreversible.Narumietal.[ 101 ]reportontherheologyofsuspensionsundergoinghighamplitudeoscillatoryshearandexplaintheunusualoutputwaveformsintermsofthetransientresponsesobservedinshearreversalexperiments[ 85 100 ]. Signicantgapsthereforeexistinunderstandingthecharacteristicbehaviorofnoncolloidalsuspensionsinoscillatoryshearow.Inthischapter,suspensionsofnoncolloidalparticlesareexposedtooscillatorysheartoinvestigatetheresponseasafunctionoftheappliedstrainamplitude.Thegoalistogainsomeunderstandingofthedynamicswithinunsteadyows.Inthefollowingsections,wepresentresultsfromexperimentsofnoncolloidalsuspensionsexposedtobothsteadyandoscillatoryshearow.TheresultsarepresentedforvedierentsystemsinbothCouetteandparallel-plategeometriesusingthesameinstrument.Adetaileddescriptionoftheexperimentisprovidedalongwithdetailsconcerningthecharacterizationofeachsuspensionsystem.Resultsarepresentedforthecasesofsteadyshearowandoscillatoryow,andconclusionsoftheworkarepresentedinthelastsection. 26

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2-1 showsSEMimagesoftwosetsoftheparticleswithsizedistributionsforallthreesets.ThedistributionsweremeasuredusingaCoultercounterandwereveriedusingscanningelectronmicroscopy.ThepolystyreneandPMMAparticlesobtainedfromSapidyneIndustrieswerefoundtobemonodispersewithsimilarlyshapedsizedistributionsandmeandiametersof996mand1008m,respectively.ThePMMAparticlesobtainedfromBangsLaboratorieshadamuchbroadersizedistributionandameandiameterof6116masmeasuredinthelaboratory,thoughthemanufacturerreportedameandiameterof83m.Thesurfaceroughnesswasevaluatedusingscanningelectronmicroscopy,andforallofthespheresused,thesizeofthesurfaceroughnessfeaturesneverexceeded1m.Thesphericitywasalsoevaluatedusingscanningelectronmicroscopy.AformfactorwasevaluatedaccordingtotheequationF=4Ap=P2,whereApistheparticleareaandPistheperimeter.Foreachsetofparticles,asimilarformfactorofF0:91wasmeasured,whereasafactorof1correspondstoaperfectsphere. ThreedierentsuspendingliquidswereinvestigatedandarelistedinTable 2-1 .Forthepolystyrenespheres,apolyalkyleneglycoloil(Aldrich)wasusedandfoundtobeNewtonianwithaviscosityof1.87Pasandameasureddensityof1.052g/cm3.TwodierentsuspendingliquidswerepreparedforthePMMAspheres.Therstconsistedofamixtureofequalamountsbyvolumewaterand75-H-90000UCONoil(DowChemical).ToeectivelymatchthedensitiesofthePMMAandsuspendingliquid,sodiumiodide(25%byweight)wasaddedtothewaterpriortomixing.Thecompositionofthesuspendingliquidiscomparabletopreviousexperiments[ 29 30 125 ].AlthoughtheUCONoil/watersuspendingliquidwasfoundtohaveaconstantshort-timeviscosityof0.97Pas,asignicantdriftintheviscositywithtimewasobservedintheparallel-plategeometryduetoaslowevaporationofthesuspendingliquid.Thisdriftwasfoundto 27

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SEMimagesandrespectivesizedistributionsoftheparticlesusedintheoscillatoryexperiments.TheleftimageshowsthemonodispersePSparticles(alsorepresentativeofthemonodispersePMMAparticles).ThedistributionsshownareformonodispersePS(solidline)andPMMA(dashedline).TheparticlesshownontherightarethepolydispersePMMAparticlesandtheirrespectivedistribution.ThedistributionsweredeterminedusingaCoultercounterandveriedusingSEMimaging.Thenumberdistributionsarenormalizedbythepeakvaluesandthescalebarontheaccompanyingimagesis200m. 28

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74 ]wasused.Inthissystem,amixtureofapproximatelyequalamountsbymassofethyleneglycolandglycerolweremixedbasedonthemeasureddensityoftheparticles.Theshort-timeviscosityofthesuspendingliquidwasfoundtobe0.0924PasandalthoughthesuspendingliquidhasbeenreportedtobeNewtonianinpreviousstudies[ 74 ],a30%decreaseintheviscositywasobservedover24hourswhenshearedintheparallel-plategeometryataconstantshearrateof24s1.Asmallerdecreaseof5%wasobservedintheCouettegeometryoverthesametimeperiodandshearrate.Thedecreaseprobablyresultsfromabsorptionofmoisturefromthesurroundingair,whichisminimizedintheCouettegeometrysincetheexposedsurfaceareatovolumeratioissmaller. AsummaryofthesuspensionsystemsstudiedisprovidedinTable 2-1 .Toconrmclosematchingofdensitiesbetweenthesuspendingliquidandparticulatephases,suspensionsatavolumefractionof=0:10weresetoutforaperiodof24hours.Inallsystems,noapparentsedimentationorbuoyancywasobservedduringthisperiod.Unlessnotedotherwise,thevolumefractionofparticleshenceforthwassetat=0:40forallexperiments.Allsuspensionswerepreparedbygentlyhandmixingtheparticlesinsmallincrementsuntilahomogeneousstatewasreached.Thesuspensionswereplacedundervacuumpriortotestingtoeliminateanyairbubblesentrainedwithinthesuspension. Themaximumparticle-basedReynoldsnumberforthesystemsusedis107,sotheeectsofinertiaareexpectedtobeminimal.Likewise,thermaldiusionisnegligiblefortherelativelylargeparticlesizessincetheminimumparticle-basedPecletnumberis107.Othercolloidalinteractions,suchaselectrostaticinteractions,shouldalsobeofnoimportance. 29

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Summaryofthemonodisperseandpolydispersesuspensionsystemsstudied.Theshort-timeviscosityofthesuspendingliquidsisgivenbys. SuspendingLiquidPhaseParticlePhaseParticleDiameter,ds

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Sincetherheologyofnoncolloidalsuspensionsissensitivetothemicrostructuralarrangementofparticlesandthustotheinitialconguration,thesuspensionswerepreshearedineachgeometrybeforestartingtheoscillatoryexperiments.Asteadypreshearof24s1wasemployedforaperiodof120secondstoreduceanyeectscausedbyloadingthesampleandtoensurethattheoscillatoryexperimentsbeganfromafairlyconsistentconguration.Thistotalstrainof2880wassucienttoreachashort-timesteadystateintheCouettegeometry(Figure 2-2 inset),butnotlongenoughtoresultinanysignicantshear-inducedmigrationintheverticaldirection[ 91 ].InthecaseoftheCouettegeometry,anonuniformconcentrationproledevelopswithinthegapduringsampleloading.Toachieveahomogeneousstate,apreshearisrequiredtoinducearedistributionofparticlesintheradialdirection[ 91 ].Followingthepreshear,theoscillatoryshearexperimentswereperformed.Ineachoscillatoryshearexperiment,theshearratewassetbyspecifyingthestrainamplitudeAandfrequencyf.Basedontheappliedstrainamplitude,atotalstraincanbecalculatedby=4An,wherenisthenumberofcycles.Unlessnotedotherwise,allexperimentswereperformedatthesamefrequencyoff=1:59cycles/sec. 31

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2-2 .Atshorttimesthesystemsshowasmalldecreaseintheviscosityfollowedbyashort-timesteadystate(shownasaninsetinFigure 2-2 ).Thisshort-timesteadystatecorrespondstoamicrostructuralrearrangementofparticlesacrosstheCouettegap[ 91 ].Fromthisshort-timesteadystate,theviscosityofthesuspensionsdecreasescontinuouslyintime.Forthesuspensionofmonodispersepolystyrenespheresinpolyalkyleneglycol,along-timesteadystateoccursafter15hoursandwasfoundtobesustainableforatleast10hoursmore.Thetransientlong-timedecreaseintheviscosityofsuspensionsofspheresintheCouettegeometryhasbeenpreviouslyobservedinexperiments[ 37 65 109 ]andistheresultofshear-inducedmigrationofparticlesoutoftheCouettegapintothestagnantreservoir[ 91 ].Furthermore,thelong-timesteadystatevalueoftherelativeviscosityforthepolystyrenesuspensionsystemagreeswellwiththeexperimentsofGadala-MariaandAcrivos[ 65 ]. Forthesametimescale,noapparentsteadystateisreachedfortheotherthreesystemsstudied.Inallcases,theobservedbehaviorcanbedirectlyattributedtocharacteristicsofthesuspendingliquids.Forexample,thesuspensioncontaining 32

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RelativeviscosityplottedasafunctionoftimeformonodispersePMMAspheresinUCON/water/NaI(triangles),monodispersepolystyrenespheresinpolyalkyleneglycol(squares),monodispersePMMAspheresinEG/glycerol(crosses),andpolydispersePMMAspheresinEG/glycerol(circles)undergoingsteadyshearintheCouettegeometry.Theinsetgraphshowstheshort-timeevolutionoftheviscosityforthemonodispersepolystyrenesuspension.Theresultsarepresentedfor=0:40and_=24s1. monodispersePMMAspheresintheaqueousliquidshowsadecreaseintherelativeviscosityfollowedbyaslowincreaseatlongtimes.Theslowincreaseisduetotheevaporationofwaterfromthesuspendingliquidwhicheventuallyprevailsovertheshear-inducedparticlemigrationthatcausestheinitialdecrease.InthecaseofthePMMAspheresinethyleneglycol/glycerol,asmalldecreaseinviscositycontinuedtooccurregardlessofthesizedistributionofparticles,althoughtheeectwasnotasdramaticforthemonodispersesuspension.Inbothcases,theabsenceofasteadystatefortheethyleneglycol/glycerolsystemcanbeattributedtothe5%decreaseinviscositywithtimeforthesuspendingliquidintheCouettegeometryoverthesametimeperiodoftheexperiments. Figure 2-3 showsthetimeevolutionoftherelativeviscosityforthesamesuspensionsystemsintheparallel-plategeometryforasteadyshearrateof_=24s1.SimilartotheobservationsintheCouettegeometry,therearequalitativedierencesintheresultsdependinguponthesystemused.Forexample,forthepolydispersePMMAspheresin 33

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2-3 .Normalizingtheapparentviscosityofthepolydispersesuspensionbythecorrespondinginstantaneousviscosityoftheethyleneglycol/glycerolmixturegivesaviscositywhichisapproximatelyindependentoftime(orstrain),suggestingthatthedecreaseinviscosityisprimarilyduetothesuspendingliquidratherthananyparticlemigration.Forthemonodispersesuspension,thedecreaseinviscosityoftheethyleneglycol/glyceroluidalsoaccountsformostoftheobserveddropinviscosityofthesuspensionsystem. FortheUCONoil/watersystem,therelativeviscositybeginsatavalue30%higherthantheothersystemsanddoublesoveratimescaleof24hours;duringthisperiod,nosteadystatewasobserved.Thehigherinitialvalueoftherelativeviscosityisduetothepreparationofboththesuspensionandsuspendingliquid.Thepresenceofparticlesinthesuspendingliquidrequiresthatthesuspensionbesubjectedtolongerperiodsoftimeundervacuumtoremovebubblesascomparedtothesuspendingliquidwithnoparticles.Thisprolongedexposuretovacuumremoveswaterfromthesuspension.Asaresult,anarticiallyhighinitialvalueoftherelativeviscosityisobservedintheaqueoussystem.Thiseectwouldbeminimizedatlowerparticlevolumefractionsthanusedhere.Theincreaseinviscosityduringshearisalsoduetotheevaporationofwaterfromthesuspendingliquid,despitetheuseofasolventtrap.TheextentofevaporationisshowninFigure 2-4 ,inwhichadepletionofthesuspendingliquidwasobservedafter24hours 34

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RelativeviscosityplottedasafunctionoftimeformonodispersePMMAspheresinUCON/water/NaI(triangles),monodispersepolystyrenespheresinpolyalkyleneglycol(squares),monodispersePMMAspheresinEG/glycerol(crosses),andpolydispersePMMAspheresinEG/glycerol(circles)undergoingsteadyshearintheparallel-plategeometry.Theresultsarepresentedfor=0.40and_=24s1.TheinsetgraphshowsthetimeevolutionoftheviscosityfortheEG/glycerolsuspendingliquidataconstantshearrateof_=24s1. ofshear.Theincreaseoftherelativeviscositywithtimeoccursthroughtwoprocesses.Thesuspendingliquidviscosityincreasesastheconcentrationofwaterdecreasesthroughevaporation,andtheeectiveparticlevolumefractionincreasesduetoadepletioninthevolumeofthesuspendingliquid. Whereastheotherthreesystemsshowedsignicantchangesinviscosityovertimeduetothecharacteristicsofthesuspendingliquid,thesuspensionofmonodispersespheressuspendedinpolyalkyleneglycoldisplayednochange.TherelativeviscositybeginsatavalueconsistentwithboththeexperimentsperformedintheCouettegeometryaswellasthesimulationsofSierouandBrady[ 122 ]andisconstantoveratimeperiodofseveralhoursafterasmallchangeatshorttimes.Polyalkyleneglycolwassubjectedtothesametestsperformedontheprevioustwosuspendingliquidsandexhibitednochangeintheviscosityovera24hourperiod. 35

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ImagesoftheUCONoil/watersuspendingliquidintheparallel-plategeometry.Theimageontheleftshowsthesampleinitiallyloadedbetweentheplates.Theimageontherightshowsthestateofthesampleafter24hoursofshearingat_=24s1.Theconsiderabledepletionofthesuspendingliquidisduetoevaporationofwaterduringshear. WhereasthedicultiesassociatedwithmeasuringviscosityintheCouettegeometryarewellestablished,whetherasimilarshear-migrationphenomenaeectsmeasurementsintheparallel-plategeometryremainscontroversial.Thisissueisdiscussedindetailinthenextsection,wheretheseresultsarecomparedtootherexperimentsandtotheories.OscillatoryShear:MonodisperseSuspension 26 65 101 ]suggestthatatlargeamplitudes,thestressoutputsignalbecomesnonlinear,leadingtoerrorsinthecalculationofthecomplexviscosity.Asaresult,thestressresponsewasmeasuredforseveralstrainamplitudesatsmalltotalstrainsandispresentedinFigure 2-5 .Slightnonlinearityisobservedatanamplitude-to-gapratioA/H>1atthepointwherethesheardirectionreverses.Thisbehaviorisrelatedtomicrostructuralchangesuponshearreversalandisinagreementwithpreviousexperimentsinvestigatinglargeamplitudeoscillatoryshear[ 101 ].AtA/H1,theoutputsignalsaresinusoidalwhichisinagreementwithpreviousobservationsat 36

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26 65 ].Inthiswork,thecomplexviscosityiscalculatedassumingasinusoidalstresswaveformregardlessofthevalueofA/H.Consequently,thedataforA/H>1shouldbeinterpretedasan\apparent"complexviscosityasarguedbyBreedveldetal.[ 26 ],andnotasthecomplexviscosityasclassicallydenedfromthelineartheoryofsmallamplitudeoscillatoryshear[ 98 ].Additionally,thestressresponseiscloselyinphasewiththestrainrateasopposedtothestrain(90phaselag)forallstrainamplitudes.Asaresult,thecomplexviscositycanbeinterchangedwiththedynamicviscosity0,theportionoftheresponseinphasewiththestrainrate.Althoughtheoutofphasecomponent00wasniteinallexperiments,itwasvanishinglysmallrelativetothedynamicviscosity. Figure 2-6 showstherelativecomplexviscosityr(denedasthecomplexviscositynormalizedbytheviscosityofthesuspendingliquid)asafunctionoftimemeasuredforasuspensionofmonodispersepolystyrenespheresintheCouettegeometry.Thelong-timeresultsareshownforafrequencyof1.59cycles/secandanappliedstrainamplitude(normalizedbythegapwidth)of0.05.AlthoughthedatapresentedinFigure 2-6 representsasingleexperiment,therheologywasrepeatableovermultipleexperiments.ErrorbarsincludedinFigure 2-6 reectthevariationbetweenrepeatedexperiments. TheoscillatoryshearresultsforA=H=0:05showalong-timeincreaseinthecomplexviscosity.Att=0theobservedcomplexviscosityislowerthanthecorrespondingsteadyshearviscosity.Thecomplexviscositythenincreasesrapidlyintimeuntilasteadyvalue,alsobelowthesteadyshearviscosity,isreachedafterapproximately5hours.Thetimeatwhichthissteadyvalueisreachedcorrespondstoatotalstrainof=5000.Thisvalueofthecomplexviscosityremainedunchangedforatleast15hours. Todeterminetheeectofthestrainamplitudeontherheologicalresponse,thelong-timeresultsforA/H=0.05areplottedwithresultsobtainedfromfourotherstrainamplitudeswhileholdingthefrequencyconstantatf=1:59cycles/sec.Figure 2-7 showsresultsoverastrainamplituderangingfromA/H=0.01to1andtheresultsareplotted 37

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Inputstrainwaves(dashedline)andresultingoutputstresswaves(solidline)plottedversustimeforthe=0.40suspensionofmonodispersepolystyrenespheressuspendedinpolyalkyleneglycolforA/H=10(a),5(b),1(c),0.5(d),0.1(e),and0.05(f).Inallcases,thefrequencyofoscillationwas0.159cyclespersecond.Theinputandoutputsignalshavebeennormalizedbytheirpeakvalues. 38

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RelativecomplexviscosityasafunctionoftimeforamonodispersesuspensionintheCouettegeometry.Theamplitude-to-gapratiois0.05andthefrequencyis1.59cycles/sec.Theexperimentswereperformedforsuspensionsofpolyalkyleneglycolcontainingpolystyrenespheresataconcentrationof=0.40.Errorbarsreectthevariationobservedbetweenrepeatedexperiments. asafunctionoftotalstrain;thedataforA/H=0.05representsonlyafractionofthatreportedinFigure 2-6 ,whichisplottedversustime.TheerrorbetweenmeasurementsinallcasesissimilarinmagnitudetothatshowninFigure 2-6 ThequalitativebehavioroftheresponseisdramaticallyaectedbythevalueofA=H.Forexample,atthehigheststrainamplitude(A/H=1),thecomplexviscositydecreasesslightlyfromitsinitialvalueat=0toasteadyvaluethatwassustainableoveralargetotalstrain.Asthestrainamplitudedecreases,thequalitativebehaviorchangessuchthatatA/H=0.1thecomplexviscosityincreaseswithtotalstrain.Acriticalpointexistsat0:1

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Relativecomplexviscosityasafunctionofstrainandamplitude-to-gapratioforamonodispersesuspensionintheCouettegeometry.Theexperimentsareconductedataconstantfrequencyof1.59cycles/secandstrainamplitudesof0.01(crosses),0.05(circles),0.1(diamonds),0.5(squares),and1(triangles).Theexperimentsshownareforasuspensionofpolyalkyleneglycolcontainingpolystyrenespheresataconcentrationof=0.40. Toevaluatethefrequencydependence,thestrainamplitudewasheldconstantatA=H=0:5andtheshort-timebehaviorwasevaluatedoverarangeoffrequencies.Thecomplexviscosityatfrequenciesof0.06,0.159and1.59cyclespersecond(correspondingto0.3768,1and10radianspersecond)isplottedinFigure 2-8 asafunctionoftotalstrain.Thequantitativebehaviorofthesuspensionwasfoundtobestatisticallyequivalent,thusindependentoffrequencyovertherangestudied.ThehighestfrequencythatiscomparablewiththevaluesusedbyGadala-MariaandAcrivos[ 65 ](f=1:59cycles/sec)wasusedinsubsequenttests. Oscillatoryexperimentswerealsoconductedinaparallel-plategeometrywithagapof1mm.Asaresultofthelongtimesrequiredtoreachasteadycomplexviscosityforthesmallerstrainamplitudes,onlythreeamplitudeswerestudiedintheparallel-plategeometry.Figure 2-9 showsthenormalizedcomplexviscosityasafunctionofthetotalstrainforthemonodispersesuspensioninbothgeometries.FortheA=Hvaluesshown, 40

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RelativecomplexviscosityasafunctionoftotalstrainandfrequencyforamonodispersesuspensionintheCouettegeometry.Theexperimentsshownareforasuspensionofpolyalkyleneglycolcontainingpolystyrenespheresataconcentrationof=0.40.Theexperimentisperformedatanamplitude-to-gapratioof0.5. thegeometryhasnoqualitativeeectonthetimeevolutionofthecomplexviscosity.Thelargestdierenceoccursattheloweststrainamplitudeof0.05wherethecomplexviscosityobservedintheparallel-plategeometryisslightlyhigherthanthatobservedintheCouettegeometry.Althoughneitherhavereachedasteadyvaluebytheendoftheexperimentat=7500,thegreatestobserveddierenceisonly8%.OscillatoryShear:ComparisonsBetweenSystems 2-7 )isrstcomparedtoaPMMAsuspensionsystemwithasimilarsizedistribution.Figure 2-10 showstheoscillatoryrheologyintheCouettegeometryasafunctionoftotalstrainandA/Hforthetwomonodispersesystems.Thequalitativebehaviorisindependentofthetypeofparticleused,asisthetimescalerequiredtoreachasteadyvalueforeachstrainamplitude.Deviationsfromthistrenddohoweveroccurat 41

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Comparisonoftheresponsesobservedinparallel-plate(closedsymbols)andCouette(opensymbols)geometriesforamonodispersesuspensionsystem.Theinstantaneouscomplexviscosityisnormalizedbythecomplexviscosityat=0andisplottedversustotalstrainforamplitude-to-gapratiosof0.05(circles),0.5(squares),and1.0(triangles).Theresultsareshownforasuspensionofpolyalkyleneglycolcontainingpolystyrenespheresataconcentrationof=0.40. thesmalleststrainamplitude,wherethecomplexviscosityintheparallel-plategeometrywasobservedtobe4%higher. Todeterminetheeectofsizedistributionontheoscillatoryrheology,asuspensionwithamonomodalparticlesizedistributionwascomparedtoasuspensionwithamuchbroaderdistribution.Figure 2-11 showstheresultsfromsuspensionsofPMMAparticlesdispersedintheaqueoussuspendingliquidasmeasuredintheCouettegeometry.ThequalitativebehaviorisindependentofthesizedistributionofthesuspendingparticleswiththelargestdierenceoccurringatA/H=0.05,wherethereisanoticeabledierenceinthecomplexviscosityofthesystems. 42

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ComparisonoftheresponsesobservedintheCouettegeometryformonodispersesuspensionsystemscontainingpolystyrene(opensymbols)andPMMA(closedsymbols).Theinstantaneouscomplexviscosityisnormalizedbythecomplexviscosityat=0andisplottedversustotalstrainforamplitude-to-gapratiosof0.05(circles),0.5(squares),and1.0(triangles).TheresultsareshownforasuspensionofpolyalkyleneglycolcontainingpolystyrenespheresandforasuspensionofUCONoil/water/NaIcontainingPMMAspheres,bothataconcentrationof=0.40. SteadyShear 91 ]predictsradialmigrationofparticlestothecenteroftheplateswheretheshearrateiszero,thoughanadditionaluxtermforcurvature[ 87 ]canleadtoanetmigrationtowardstheedgeoftheplates[ 96 ].However,thesuspensionbalancemodel[ 103 ]predictsnomigrationwithintheparallel-platedevice.Manyhavefoundevidenceoflittleornomigration[ 36 37 87 ],butmostrecentlyMerhietal.[ 96 ]didndevidenceofshear-inducedmigrationradiallyoutward.Merhietal.[ 96 ]attributedthediscrepancywiththepreviousexperimentsofChapman[ 36 ]toaninadequatetotalstrainoverwhichobservationsweremade.Overatotalstrainof2:4106theviscosity 43

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ComparisonoftheresponsesobservedintheCouettegeometryforsuspensionsofPMMAsphereswithmonodisperse(closedsymbols)andpolydisperse(opensymbols)sizedistributions.Theinstantaneouscomplexviscosityisnormalizedbythecomplexviscosityat=0andisplottedversustotalstrainforamplitude-to-gapratiosof0.05(circles),0.5(squares),and1.0(triangles).TheresultsareshownforsuspensionsofUCONoil/water/NaIcontainingPMMAspheresataconcentrationof=0.40. wasobservedtoincreaseby40%byMerhietal.[ 96 ].Merhietal.[ 96 ]alsoprovidedvisualevidenceofmigrationofparticlestowardstheedgeoftheplate,butreportedthevisualizationforamuchsmallertotalstrain. Theresultsshownhereindicatethatthereisnolongtermvariationintherheologywithintheparallel-plategeometryoverasimilartotalstrainof106,indicatingthelackofanysignicantparticlemigrationinthenon-aqueoussuspendingliquid(seeFigure 2-3 ).ThechangesinviscositywithtotalstrainfortheotherthreesystemsshowninFigure 2-3 arenottheresultofaparticlemigrationphenomena,butareinsteadrelatedtoidentiableproblemswiththesuspendingliquids.Themixtureofethyleneglycolandglycerolhasaviscositythatdecreaseswithtimewhereasevaporationcausestheincreaseintheviscositywithtimeinthecaseoftheaqueoussuspendingliquid(UCONoilmixturewithwaterandNaI). 44

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96 ]mademeasurementsintheparallel-plategeometryusingasuspendinguidcontaininganaqueouscomponent;whetherevaporationaectstheirviscositymeasurementsandvisualizationstudiesisunclear.However,asuspensionsystemcanbeshearedbetweenparallel-platesforverylargestrainswithoutobservingaviscositychangearisingfromshear-inducedmigrationoftheparticles.OscillatoryShear 2-12 ,whichshowsthechangeofviscositybetweentheinitialvalueandthenalvalueat=15000forthepolystyreneparticlesystemand=7500forthePMMAparticlesystems.ThetransitionpointforallofthesuspensionsystemsstudiedoccurredatavalueofA/Hbetween0.1and0.5.Abovethesecriticalvalues,thecomplexviscosityslightlydecreaseswithtotalstrain.AtvaluesofA=Hlessthanthetransitionpoint,thecomplexviscosityincreaseswithtotalstrain,withtheamountofchangeincreasingwithdecreasingamplitude.Thisisthersttimesignicantchangesinthecomplexviscosityatlowamplitudeshavebeenobservedinoscillatoryshearow.Theunderlyingmechanismforthislong-timechangeinviscosityisspeculateduponinthenextsection.Originofoscillatorybehavior 45

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DierencebetweenthenalandinitialvaluesofthecomplexviscositynormalizedwithrespecttoitsinitialvalueintheCouettegeometryplottedasafunctionofamplitude-to-gapratio.Theresultsareshownforallsuspensionsystemsstudiedataconcentrationof=0.40. inFigure 2-13 .ForamplitudesofA=H5,theresultsarewithin10%ofeachother.Thecloseagreementatsmallamplitudessuggeststhatmigrationiseitherminimalorhasanequivalenteectinbothgeometries.Furthermore,experimentssimilartothoseofLeightonandAcrivos[ 91 ],inwhichtheCouettereservoirwassealedwithmercury,wereperformedfortheoscillatoryshearcasewithA=H=0:01.Thecomplexviscositywasslightlyhigherforthesystemwiththesealedgap,butthequalitativebehaviorremainedunchanged,indicatingthatanetmigrationofparticlesbetweentheCouettereservoirandgapdoesnotexist.Othershavealsoconcludedthatshear-inducedmigrationdoesnotplayasignicantroleindeterminingtherheologyofnoncolloidalsuspensionsexposedtooscillatoryshearatsmallstrainamplitudes[ 26 65 ]. Somepreviousstudieshaveaddressedtheimportanceoftheevolvingmicrostructureindeterminingtherheologyinoscillatoryows.Ananalogybetweentherheologyofsuspensionsunderoscillatoryconditionsandtherheologyofsuspensionsundergoingreversalofsheardirection,wheremicrostructuralrearrangementsareresponsibleforthe 46

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Complexviscosity(evaluatedat=15000)plottedasafunctionofappliedstrainamplitudefortheCouetteandparallel-plategeometries.Valuesarenormalizedbytheviscosityforthecorrespondinggeometryinsteadyshearasevaluatedat=3000.Atallstrainamplitudesthefrequencywasheldconstantatf=1.59cycles/sec.Theexperimentswereperformedforsuspensionsofpolyalkyleneglycolcontainingpolystyrenespheresataconcentrationof=0.40. transientbehaviorfollowingreversalofshear[ 65 101 ],hasbeenmade.Forexample,Gadala-MariaandAcrivos[ 65 ]successfullypredictedtheshapeofoscillatoryresponsewavesathighamplitudesfromresultsobservedinthetransientshearreversalexperiments.Directobservationsofthemicrostructureinoscillatoryowhavealsobeenmade.GondretandPetit[ 69 ]studiedtheoscillatoryshearrheologyofasuspensionatasingleamplitudeandnoticedadriftintheviscositywithtime.Thesemacroscopicobservationswerecomparedtoopticalmeasurementsandthechangeinviscositywasattributedtoamicrostructuralorderingofparticlesinresponsetotheoscillatoryshear. ThechangesinviscosityarisingfromtheevolvingmicrostructureareclearlyduetoanirreversibleprocesswhichoccursforeventhesmalleststrainamplitudeofA=H=0:01;ifthesuspensionwerereversible,nolong-timechangeintheviscositywouldoccur.Theirreversibility,whetherarisingfromsmallsurfaceinhomogeneitiesontheparticles[ 43 114 ] 47

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90 ],canbecharacterizedasashear-inducedself-diusionofparticleswhichexistseveninthecaseofsmallamplitudeoscillations[ 106 112 ].Pineetal.[ 112 ]usedparticletrackingtechniquestomeasureparticledisplacementsasafunctionofamplitudeinoscillatoryshearandfoundthattheresultingshear-inducedself-diusionbecomesincreasinglysmallerwithdecreasingamplitude.However,evenatastrainamplitudeof0.05,theauthorsfoundanon-zerovaluefortheshear-induceddiusivity.Thoughnotforsimpleshearows,irreversibilitywasalsoseeninthesimulationsofMorris[ 97 ]andexperimentsofButleretal.[ 30 ]foroscillatoryowinatube.Atsucientlysmallamplitudes,theparticlesinsuspensionmigratedtowardstheboundingwalls,insteadoftowardsthecenterlineasoccursinsteadypressure-drivenows[ 29 93 125 ].Straindependenceoftheresults 2-12 maynothavereachedasteadyvalue,thoughthesuspensionshavebeenshearedforverylargetimes. Thedependenceofthetotalstrainrequiredtoreachasteadyvalueforthecomplexviscosityontheappliedstrainamplitudeimpliesthatatlowstrainamplitudes,previousmeasurementsinoscillatoryowsmaynothavebeenreportedwithrespecttosteadyvalues.Forexample,Gadala-MariaandAcrivos[ 65 ]reportedreachingasteadystatewithin100cycles,butthetotaldurationoftheirexperimentsisunclear.Onehundredcycles,atthestrainamplitudesstudiedbyGadala-MariaandAcrivos[ 65 ],correspondstoonlyafractionofthestrainrequiredtoreachanunchangingcomplexviscosityfound 48

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2-6 .Breedveldetal.[ 26 ]measuredthedynamicviscosityasafunctionofamplitudeandfoundthattheviscositywasnearlyindependentofstrainamplitudeforlowamplitudes,whereasFigure 2-12 showsverydierentbehavior.Onceagainthough,itisuncertainwhetherornotthemeasurementsreportedbyBreedveldetal.[ 26 ]werereportedwithrespecttotheirsteadyvaluessincethetotalstrainisnotreported. Othersourcesofthesedierencesmaybelinkedtotheconcentrationdierencesandfrequenciesemployedinthestudies.Breedveldetal.[ 26 ]performedmeasurementsonasuspensionatavolumefractionof0.50,whichcouldexhibitsignicantlydierentrheologyfromthevolumefractionof0.40studiedhere.Weinvestigatethedependenceoftherheologyonconcentrationinthenextsection.Also,Breedveldetal.[ 26 ]performedmeasurementsathigherfrequenciesforthesmalleramplitudestudiesthanwereusedhere.However,thestudyofdependencyoftheresultsonfrequencyshowninFigure 2-8 demonstratesindependencewithrespecttofrequency.Furthermore,therheologyforsuspensionsexposedtothesamestrainrateamplitude,butdierentamplitudesandfrequencies(forexample,comparingthedataforA/H=0:05andf=1:59HzinFigure 2-7 withthedataforA/H=0:5andf=0:159HzinFigure 2-8 )isdierent,suggestingthatthestrainamplitudeistherelevantparameter,notthestrainrateamplitude.Finally,anothersourceofthedierencesmightbethedierentsystemsusedbythedierentresearchers.Thispossibilitywasalsoexplored,asdiscussednext.Eectofsuspensioncharacteristics 2-14 showsthecomplexviscosityat=15000plottedasafunctionofappliedstrainamplitudeforconcentrationsrangingfrom=0.20-0.45.Atallconcentrationsexcept=0.20,theviscosityisanonmonotonicfunctionoftheappliedstrainamplitude,withthebehaviorbecomingmoreapparentastheconcentration 49

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Complexviscosity(evaluatedat=15000)plottedasafunctionofappliedstrainamplitudeandconcentrationfortheparallel-plategeometry.Valuesarenormalizedbytheviscosityofthesuspendingliquid.Atallstrainamplitudesthefrequencywasheldconstantatf=1.59cycles/sec,andtheexperimentswereperformedforsuspensionsofpolyalkyleneglycolcontainingpolystyrenespheres. increases.Likewise,theminimumvalueofthecomplexviscosityoccursatA=H=1,indicatingthatfor>0.20,therheologyisqualitativelyindependentofconcentration.TheresultsarealsoplottedversusvolumefractioninFigure 2-15 .Asareference,thedataisplottedalongwithresultsfromtheKrieger-Doughertyrelation[ 86 ], m[2:5]m(2{1) withthevolumefractionformaximumpacking,m=0.64.Forallstrainamplitudes,therelativecomplexviscosityincreasesnonlinearlywithconcentration.Forthelargeststrainamplitude(A=H=5.0)aswellasforthesmallest(A=H=0.05),thevolumefractiondependenceissimilartopredictionsfromsteadyshear. Theresultsalsoindicatethatthereislittledependenceoftheoscillatoryrheologyontheparticlesizedistribution.Polydispersityhasbeenfoundtoplayasignicantroleinsteadyshearrheologyhighlightedbyitseectonshear-inducedmigration.Recent 50

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Complexviscosity(evaluatedat=15000)plottedasafunctionofconcentrationfortheparallel-plategeometry.Valuesarenormalizedbytheviscosityofthesuspendingliquid.Atallstrainamplitudesthefrequencywasheldconstantatf=1.59cycles/secandtheexperimentswereperformedforsuspensionsofpolyalkyleneglycolcontainingpolystyrenespheres.Forreference,resultsfromtheKrieger-Doughertyrelation[ 86 ]arealsoplotted(solidline). examplesincludeparticlesizeseparation[ 81 ]andradialmigrationinaparallel-plategeometry[ 87 ].Theresultshereshowthatforoscillatoryshear,thesizedistributionsstudiedhavenoeect.ThisisincontrasttotheoscillatoryshearexperimentsperformedbyGondretandPetit[ 70 ]whofoundthatbidispersesuspensionsshowdierentdynamicviscositiesdependingonthesizedistributionoftheparticles.Thedisparitybetweenthisworkandthepresentworkmaybeduetodierencesinthefrequenciesandconcentrationsasdiscussedpreviously. Theoscillatoryshearrheologywasaectedbysuspensioncharacteristicsonlyatlowstrainamplitudes.Thesedierencesneversurpass10%foranyofthecomparisonsmade,suggestingthattherheologyisstillrelativelyindependentofthesystemchosen,althoughtherheologymaybemoresensitivetothesefactorsatstrainamplitudessmallerthan 51

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Theoscillatoryshearrheologydependsstronglyonthemagnitudeoftheappliedstrainamplitudesuchthatathighamplitudes,thecomplexviscositydecreaseswithtotalstrainwhereasatlowamplitudes,thecomplexviscosityincreaseswithtotalstrain.Thecriticalamplitude-to-gapratioatwhichthequalitativebehaviorchangesoccurredat0:1
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126 ].Forunsteadysheartheobservedrheology,primarilyprovidedbyexperiments,islessunderstood. Kollietal.[ 85 ]andNarumietal.[ 100 ]measuredthestressresponseafterareversalofshearowandfoundthattheshearstressdecreasestoaminimumvaluebeforereturningtothesteadystatevalue.Asimilartransitionwasfoundinthenormalstressdierence,whichchangedsignbeforereturningtothesteadystatevalue.Explanationsconcerningtheoriginoftheobservedrheology[ 65 100 ]arelargelyconrmedbythesimulationsperformedinthischapter.Insteadyshearow,anexcessofparticlesdevelopsinthecompressionalquadrantupstreamfromatestparticle[ 23 123 ],asopposedtotheextensionalquadrantdownstreamfromtheparticleasillustratedinFigure 3-1 .Uponreversingthedirectionofow,theexcessofparticlesisfoundintheextensionalquadrantuntilthemicrostructurerearrangesinaccordancewiththenewowdirection.Thissuddenmicrostructuralchangeresultsinthetransientrheologyobservedintheshearreversalexperiments. Interpretingtherheologyofsuspensionsundergoingoscillatoryshearowismoreproblematic.Oneissueconcernstheirreversibilityofsuspensionsatsmallstrainamplitudes.Pineetal.[ 112 ]performedexperimentsandsimulationsinvestigatinghydrodynamicdiusivitiesandconcludedthatparticletrajectoriesareirreversibleonlyaboveacriticalstrainamplitude.However,rheologicalexperimentsbyBrickerandButler[ 28 ]demonstratedirreversiblebehaviorbelowthecriticalamplitudereportedbyPineetal.[ 112 ],asevidentfromthelong-timechangesinthecomplexviscosity.Anotherissue 53

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Schematicshowingparticlepairsexposedtosimpleshearow.Theparticlewiththedashedlineborderisapproachingthestationarytestsphereandisincompression,whereastheparticlewiththesolidlineborderismovingawayfromthetestsphereandisinextension.Theparticleanisotropyarisesasparticlespreferentiallyspendmoretimeinthecompressionalquadrantsasopposedtotheextensionalquadrants.Theschematicshownissymmetricabouttheangleof45. concernsthenon-monotonicdependenceoftheviscosityontheappliedstrainamplitude.BrickerandButler[ 28 ]foundthatasthestrainamplitudedecreases,thecomplexviscositydecreasesuntilaminimumvalueisobservedatastrainamplitudeofone.Forsmallerstrainamplitudes,thecomplexviscositywasfoundtobehigher.Thoughclearlyrelatedtomicrostructuralchanges,thisrheologicalbehaviorisnotcurrentlyunderstood. Inthischapter,weelucidateissuesconcerningtherheologyofsuspensionsinunsteadyshearowusingStokesiandynamicssimulationsofamonolayer.Wepresenttherheologyalongwiththecorrespondingmicrostructuraldetailsforowwithreversalofshearafterattainmentofsteadystateandoscillatoryow.Foroscillatoryshearow,wepresentresultsfortheviscosityasafunctionofstrainamplitudeandndqualitativeagreementwiththenon-monotonicbehaviorobservedinexperiments[ 28 ].Additionally,wediscusstheissueofirreversibilityforthesuspensionsystem.Thesendingsarediscussedindetailalongwithaninterpretationofmorecomplexowsintermsoftherheologyobservedhere.RheologySimulations 54

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25 ].Thissimulationmethodhasbeenusedtocalculatetherheologyfornon-Browniansystems[ 24 52 53 122 123 ],Brownianparticulatesystems[ 62 111 ],andevenforthesimulationoftherheologyofelectrorheologicalsystems[ 22 ]. Inthiswork,theStokesiandynamicsmethodisusedtosimulateshearowsofsuspensionsofnon-Brownianspheresconstrainedtomoveinthevelocity-gradientplane;werefertothisasa`monolayer'simulation.Extensivecomputationaltimeisrequiredtocompletesimulationsatlowerstrainamplitudesasaresultofthelargestrainsrequiredtoachieveasteadystatemicrostructure[ 28 ].Thesimplicationtoamonolayerprovidessubstantialsavingsincomputationaltimebyreducingthenumberofdegreesoffreedomassociatedwitheachparticlewhilestillmaintaininganaccuratedescriptionoftherelevantphysics[ 24 97 103 111 123 ].Inadditiontopredictingtherheologicalbehaviorofavarietyofsystems,monolayersimulationsaccuratelypredictthecorrespondingmicrostructure.Forexample,monolayersimulationscorrectlypredicttheformationof`hydroclusters'inshearthickeningcolloidalsuspensions[ 16 ].Forcomparingmonolayersimulationstofullythree-dimensionalexperiments,thecorrespondingvolumefractionis2=3timesthearealfraction[ 103 111 ]. WebrieydiscusstheStokesiandynamicsmethodforboundedowsandprovidedetailsconcerningtheimplementationofthemethod,alongwithalterationsmadetoaccommodateunsteadyshearows.TheStokesianDynamicsMethod 103 ]andMorris[ 97 ]forsimulatingpressuredrivenowsandbySinghandNott[ 123 ]forsimulatingshearow.AssketchedinFigure 3-2 ,thewallsarepermeableandrepresentedbychainsofsphereswhichhaveanimposedvelocityinthex-directionwhichcangenerallybeafunctionoftimet.Thevelocitiesofthewallparticlesintheyandzdirectionsaresettozero.Sinceperiodicboundaryconditionsareused,a 55

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Theperiodiccellusedinthesimulations.Wallparticlesarewhiteandfreelysuspendedparticlesinthebulkofthesuspensionarelled.Aclearlayerofuidofthesameheightasthegapbetweenthewallsisintroducedtoenablesimulationsusingperiodicboundaryconditions. clearlayerofuidisplacedatthebottomofthecellasshowninFigure 3-2 ;theheightofthisclearlayerissetequaltotheheightofthegapinwhichthebulkparticlesaresuspended. TheforcesonthewallparticlesandresponseofthebulkparticlesaresimulatedusingtheStokesiandynamicsmethod.Inthismethod,afar-eldmobilitymatrix,M1,isformed,invertedtogivearesistancematrix,andthenmodiedtoaccountfornear-eldinteractions, whereR2bisatensorcontainingthenear-eld2-bodyinteractionsforallparticlepairsandR12bisthefar-eldcomponentwhichissubtractedtoavoiddoublecounting.Thedetailsoftheformationofthegrandresistancematrix,R,isdiscussedindetailbyDurlofskyetal.[ 54 ]. SinghandNott[ 123 ]replacedthe2-bodylubricationinteractionsbetweenthesuspendedandwallparticleswiththelubricationinteractionsforasphereinteractingwithaatwall,atleastfortheforce-velocitycouplings;itisnotclearthatthestresslet-velocity 56

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123 ]andtheresultsproducedusingbumpywallsindicatesthattheimpactonthesolutionfortherheologicalpropertiesisnegligible. TheresistancematrixinEquation 3{1 relatestheinstantantaneousforces,torques,velocities,rotationalvelocities,andstressletsusingtheequation Inthisequation,theforcesandtorques,F,andstresslets,S,areseparatedintovectorsforthebulkandwallparticles(denotedbyasuperscriptsandw,respectively).Theforce,Fs,onthesuspendedparticlesiszero,otherthantheinclusionofashort-rangerepulsiveforceasdiscussedinthenextsection. Thevelocities,Uw,ofthewallparticlesaresetasshowninFigure 3-2 .Sincetheowisdrivenbythemotionofthewalls,themeanvelocityhuiissettozerotoensurethatnootherdrivingforcesactontheparticles,preservingsimpleshearow.Simulationswerealsoperformedusingthealternativeconstraint, whereNwisthenumberofwallparticles.Nosignicantdierencesoccurredasaresultoftheconstraintchosen.SimilartoSinghandNott[ 123 ],simulationresultsshowthatthevelocityproleislinearexceptsmalldeviationsnearthewall.Therateofstrainhasbeensettozerosincethemotionofthewallparticlesgeneratestheshear. 57

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3{2 canbesolvedfortheunknownvelocitiesofthesuspendedparticles, andtheunknownforcesactinguponthewallstomaintainthevelocityUw, Thestressletsofthebulkparticles,Ss,alsodeterminedfromEquation 3{2 ,aregivenby 123 ]orthestressletsactingonthesuspendedparticles[ 24 ].Forthewallforceevaluation,thestressis whereasthestresscanalsobeevaluatedfromthestressletsaccordingto wherethestresslethasbeenseparatedintothehydrodyamicportionandaportionwhicharisesfromapplicationoftheshortrangerepulsionsdenedinEquation 3{15 .Althoughthecalculationaccordingtothestressletsrepresentstheparticlecontributiontothestress,thecalculationbasedonthewallforcesrepresentsthetotalstress.Fortheremainderofthepaper,onlytheparticlecontributiontothestresswillbereported,thusthetotalstressinEquation 3{7 isreducedtotheparticlecontributionbysubtractingtheportionofthewallforceFwcorrespondingtotheprescribedvelocityUw(secondtermgiveninEquation 3{5 ). 58

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5{12 withthevelocitieseliminatedfromtheequationandthetermsduetotherepulsiveforcesremoved, Thenon-hydrodynamiccontributionis Furthermore,themeanhydrodynamicstressletisgivenby[ 24 123 ] 11 andthemeancontributiontothestressfromtheinterparticleforceis Inadditiontotheshearstresses,directevaluationofthenormalstressactingonthewallsisgivenby TherstnormalstressdierenceN1canthenbeevaluatedfromthestresslets, Theaddedvalueofusingwallsisthatthenormalstressactingonthewallscanbeevaluateddirectlyfromthewallforces.Whennotusingwallparticles,butfullyperiodicsystemswithdrivenshear,thenormalstressdierenceN1canbecalculatedwitheasefromthedierenceinthestressletsactingontheparticles.However,todirectlycalculateyyrequiresthevalueoftheparticlepressure,theisotropicportionofthenormalstresscomponents. 59

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23 ].Simulationswereperformedusingadimensionlesstimestepoft=1:0102.Topreventparticleoverlapduringthecourseofthesimulationsandtoreplicatethebehaviorofamorerealisticsystem[ 122 ],aninterparticleforceisincorporatedbetweenthebulkparticles, Theseparationdistance,,isgivenas=jrj2,whereristhemagnitudeoftheseparationdistancebetweenthecentersofsphereandsphere.Foramajorityofthework,thevaluesoftheparametersspecifyingtherangeandmagnitudeoftheforcearesetto=100andF0=1:0104,respectively.ThevalueschosenforandF0areequivalenttotheworkofSinghandNott[ 123 ]andthusallowforeasycomparisontotheirworkforsteadyshearow.Theeectofvaryingtherangeandmagnitudeoftherepulsiveforcearediscussedinalatersection.Therepulsiveforceexistsamongthebulkparticles,butnotbetweenthebulkparticlesandtheparticlesformingthewall. Thefar-eldinteractionsareupdatedevery10timestepsasintheworkofNottandBrady[ 103 ],whilethenear-eldlubricationinteractionsareupdatedateachtimestep.Inallstudies,thenumberofwallparticlesis14.Eachsimulationbeginsfromarandomdistributionofparticles.Forthesteadyshearandshearreversalcases,thegapissettoaheightof30andtherheologyisstudiedforarangeofconcentrations.Foroscillatoryshearow,thearealfractionissetto=0:60andthegapissettoaheightof15toreducecomputationtime.Resultsfordierentgapheightsinsteadyshearshowthatthereislittleornodependenceoftherheologyonthetwochoicesofgapspacing.Forallshearows,therheologyisreportedasanaverageoverseveralruns.Sincethelevelofaveraging 60

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Theboundaryconditionsdependontheshearow.Forsteadyshearow,thetopandbottomwallvelocitiesareassignedthesamemagnitude(Uwx=j1j)butactinoppositedirections,thusinitiatingasimpleshearow.Toevaluatethetransientrheologyduringshearreversal,asuspensionissheareduntiltheattainmentofasteadystateafterwhichthetopandbottomwallvelocitieschangesignbutmaintaintheiroriginalmagnitude.Forsimulationsofoscillatoryshearow,thewallsareassignedatimedependentvelocityintheformofasinusoidasdoneinthepressure-drivenowsimulationsofMorris[ 97 ], whereUisthemaximumwallvelocityand!isthefrequencyofoscillation.Themaximumwallvelocityateachstrainamplitudeisheldconstantandequalinmagnitudetothatusedinthesteadyshearcase. Foroscillatoryshearow,stressesarecalculatedtwiceduringeachcycle.TheforwardcalculationoccurswhenthedimensionlesstopwallvelocityisUwx=1whilethebackwardcalculationoccurswhenUwx=1.Therheologyisevaluatedasafunctionoftotalstrain, andstrainamplitude,A. Afourth-orderRunge-KuttaintegrationmethodisimplementedinplaceoftheAdams-BashforthpredictormethodcommonlyusedinStokesiandynamics[ 22 122 ].Simulationsusingthismethodconrmthatparticlesdonotoverlapforthetimestepusedandtestsshowthattherheologicalbehaviorisconvergent.Results 61

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123 ]toverifythecodeandprovideareferencetotheresultsfromunsteadyshearows.Wetheninvestigatethetransientrheologyfollowingareversalofshearafterattainmentofsteadystateandoscillatoryshearow.SteadyShear 24 122 123 ],simulationoftherheologyofnoncolloidalsuspensionsinsteadyshearowisrepeatedusingthepresentcode.Eachdatapointshowninthissectionrepresentsanaverageoverasetof10runscalculatedoveradimensionlesstimeof5000.Eachrunconsistedofadierentinitialcongurationofparticlesandtherheologywascalculatedusingdataaveragedoverthelast2000dimensionlesstimeunits. ComparisonsshowninFigure 3-3 fortheparticlecontributiontotheshearstress,yx,asafunctionofarealfraction,,demonstratethattheresultscloselymatchthoseofSinghandNott[ 123 ]foragapof30.Theerrorbarsonthecalculatedstressesrepresentthevariationinthemeanof10individualruns.Fortheconcentratedsystems,thecalculationsindicateaslightlyhighervaluethanSinghandNott[ 123 ]foryx,perhapsduetotheuseofbumpywalls.Furthermore,theresultsshowthatthestresscanbecalculatedfromeitherthestressletsonthebulkparticles,Ssyx,orfromthetangentialforcesactingonthewallparticles,Fwx.Fortheconcentratedsystems,theerrorbetweenthetwoevaluationtechniquesisapproximately5%. ThenormalstressyyascalculatedfromthenormalforceFyactingonthewallparticlesisalsoincloseagreementwiththeresultsofSinghandNott[ 123 ],thoughthereisonceagainanoticeabledierenceforhigherconcentrationsasseeninFigure 3-4 .Inthiscase,thedierenceismostnotablefor Therstnormalstressdierence,N1=xxyy,isshownin 3-5 .Thevalueofthenormalstressdierenceisnegativeforsteadyshearowatallconcentrations,thoughthevalueofN1isvanishinglysmallfor 62

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Particlecontributiontotheshearstress,yx,asafunctionofarealfraction,,forsteadyshearow.Resultsareshownfromthecalculationbaseduponananalysisofthewallforcesandthebulkparticlestresslets.Theerrorbarsrepresentthevariationobservedbetween10simulations.Forallconcentrations,thegapis30. withtheexperimentalobservationsofZarragaetal.[ 142 ].Asidefromthetotalnormalstressdierence,contributionsfromthehydrodynamicandrepulsiveforceasdenedinEquations 3{11 and 3{12 areplotted.Therepulsivecontributionhaslittleeectonthetotalvalueforconcentrationsupto 123 ].Aswiththepreviousshearstressresults,theN1valuescloselymatch. Inadditiontotherheologyinsteadyshearow,themicrostructurewasinvestigatedandcomparedtopreviouswork.Theradialdependenceofthepairdistributionfunction,g(r),isplottedasafunctionofarealfractioninFigure 3-6 .Anexcessofparticlesexistsatdistancesnearcontact.Asexpected,g(r)approaches1astheradialdistanceincreases.Anarealfractiondependenceisalsoapparent.As 63

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Normalstressyyasafunctionofarealfraction 123 ]. NormalstressdierenceN1asafunctionofarealfraction 123 ]. 64

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Pairdistribution,g(r),asafunctionofradialdistanceandarealfractionintegratedoverallangles.Dataispresentedforparticlesinthecenterofthegap(13
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Angularpairdistribution,g(),atcontactforparticleslocatedinthecenterofthegap(13
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Particlecontributiontotheshearstressasafunctionoftotalstrainuponreversalofshearow.Theresultsarepresentedforarangeofarealfractionsandforagapof30.Thedatarepresentsanaverageover10simulations. oftotal=0,theshearstressdecreasestoaminimumvaluebeforerapidlyincreasingtoasteadystate.Forallarealfractionsstudied,thesteadystatevaluefollowingshearreversalequalsthesteadyvaluepriortoreversingthesheardirection.Althoughnotpresentedhere,simulationsthroughastrainoftotal=20wereperformedandshowanoscillationaboutyx=steadyyx=1.ConsistentwiththeexperimentalworkofKollietal.[ 85 ],theminimumintheshearstressfollowingshearreversalshiftstolargerstrainsforsmallerconcentrations.Inaddition,themagnitudeoftheminimumvaluedecreaseswithincreasingconcentration,acharacteristicalsoinagreementwithKollietal.[ 85 ].Forallconcentrationsstudied,asteadystatevalueisreachedbyastrainoftotal6whichishigherthanthevalueof4foundbyKollietal.[ 85 ].Additionally,noovershootinthetransientshearstressisevidentintheexperimentswhereasthesimulationresultsshowaslightovershootforthesmallerarealfractionsstudied. Therstnormalstressdierence,N1,followingshearreversalisnormalizedbythesteadystatenormalstressdierenceinsteadyshearowandplottedinFigure 67

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Firstnormalstressdierenceasafunctionoftotalstrainuponreversaloftheow.Theresultsarepresentedforarangeofarealfractionsandforagapof30.Thedatarepresentsanaverageover10simulations. 3-9 .TheresultsarebasedonananalysisofthewallforcesandthelevelofaveragingisthesameasthatreportedinFigure 3-8 .Immediatelyfollowingreversalofthesheardirection,N1=Nsteady1isnegative.ThiscorrespondstoanN1thatisinitiallypositivefollowingareversalofsheardirectionsinceinthesteadyshearcase,Nsteady1isnegative[ 123 142 ].FurthermorethemagnitudeoftherelativevalueofN1attotal=0varieswithconcentrationsuchthatitincreaseswithincreasingarealfraction.Forallarealfractions,N1=Nsteady1decreasesinmagnitudeastotalincreasesuntilN1undergoesatransitioninwhichitreversessignandbecomesnegative,eventuallyreachingthesteadystatevalueachievedintheprevioussheardirection(N1=Nsteady1=1).ThetotalstrainatwhichN1=Nsteady1changessigndependsonthearealfraction.Forexample,at=0:65therstnormalstressdierencechangessignpriortototal=1whereasfor=0:40,thesignchangeoccurslater,atatotalstrainof2. Figure 3-10 showsthetransitionofthepairdistributionfunction,g(),nearcontactasafunctionoftotalstrainandangleuponreversingthesheardirectionforasuspensionataconcentrationof=0:60.Attotal=0,thedistributionisequivalenttothe 68

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Pairdistribution,g(),atcontactforparticlesinthecenterofthegap(13
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Inputstrainwave(dottedline)andtheresultingoutputstresswave(solidline)plottedversusstrainforsimulationsat=0.60foragapof15.Thestresswavesarenormalizedbytheirpeakvalues. tothemaximumvalueobservedintheextensionalquadrantimmediatelyfollowingshearreversal(total=0).OscillatoryShear Rheology 3-11 .Theshearstresswavesarereportedasanaverageover8experiments,eachbeginningfromadierentrandominitialconguration.Toavoidartifactsfromthestart-upofow,thewavesarereportedforatimeperiodimmediatelyfollowingtherst10cyclesofshear.StresswavespresentedforA=0:5arerepresentativeofthewaveformsforA<0:5andarenotreported.Fortheremainderoftheoscillatoryshearowanalysis,therheologyisreportedwithrespecttothevaluewhenthewallvelocityismaximuminboththeforwardandbackwarddirections. Theoutputstresswavesareinphasewiththestrainrate(90phaselagfromthestraininput)atallstrainamplitudes,whichisconsistentwithexperimentalwork[ 28 64 ].AtA=5,theshearstressisnoisyatthepointwheretheinstantaneousstrainvalueis 70

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ParticlecontributiontotheshearstressplottedversustotalstrainforoscillatoryshearowatappliedstrainamplitudesofA=0.1,0.25,0.5,1,and2.Theshearstressesarenormalizedbythesteadystatevaluesoftheshearstressundersteadyowconditions.Resultsareshownforsimulationsat=0.60andagapof15. zero(correspondingtomaximumwallvelocity).Belowthisvalue,thestressresponsebecomessmooth.Fortherangeofstrainamplitudes,therearenoapparentnonlinearitiesinthestressresponsesuchasthosepreviouslyobservedinthelargeamplitudeoscillatoryshearexperimentsofNarumietal.[ 101 ].However,thestresswaveformatA=1showsaslightlyattenedresponsewhentheinstantaneousvalueofthestrainiszero.Furthermore,thenoisystressoutputatthelargeststrainamplitudemakesitdiculttodetectthepresenceofanonlinearresponse. Figure 3-12 showsthestrainevolutionoftheparticlecontributiontotheshearstressfordierentstrainamplitudes.Theshearstressesarenormalizedbythesteadystatevaluesoftheshearstressforsimilarconditions(=0:60,gap=15)understeadyshearow.Theshearstressesarereportedfortheforwarddirection(Uwx=1forthetopwall)only.Calculationoftheshearstresswasalsomadeinthebackwarddirection,andmatchesthosereportedinFigure 3-12 forallstrainamplitudesstudied.Thestrainevolutionofthe 71

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Largestrainviscosityinoscillatoryshearownormalizedbythesteadystateviscosityinsteadyshearowasafunctionofstrainamplitude.Simulationsarefor=0.60andagapof15,andtheerrorbarsrepresentstatisticalerrorbetweeneightrunsateachstrainamplitude.Forcomparison,thesimulationresultsareplottedalongwiththenormalizedviscositiesfromtheexperimentsofBrickerandButler[ 28 ]. shearstressforA=5isnotshownduetothelargeuctuationsobservedatmaximumwallvelocity(Figure 2-5 ). Theinitialvalueoftheshearstressinoscillatoryshearowis45%ofthesteadystatevalueoftheshearstressinsteadyshearow.Fromthisinitialvalue,theshearstressincreaseswithtotaluntilasteadystateisobservedafteratotalstrainthatdependsonthevalueoftheappliedstrainamplitude.Asthestrainamplitudedecreases,thetotalstrainrequiredtoreachasteadystatevaluefortheshearstressincreasesdramatically.Forexample,thesteadystateshearstressforA=2isachievedquicklyasitincreasesfromoscyx=steadyyx=0:47attotal=0toasteadystatevalueofoscyx=steadyyx=0:76afteratotalstrainofonly1600.Atthesmalleststrainamplitudestudied(A=0:1),theshearstressincreasescontinuouslyfromoscyx=steadyyx=0:49attotal=0andnoobservablesteadystateisreachedoveratotalstrainof40000. 72

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3-12 areusedtoevaluatethedependenceoftheviscosityontheappliedstrainamplitude.Toobtainthesevaluesfortheshearstress,amovingaveragewascalculatedforeachofthecurvesinFigure 3-12 .ThevaluesinFigure 3-13 correspondstothepointofminimumslopeoveraportionofthecurveattheendofeachrun.Theshearstressvaluesareaveragedover8runsbeginningfromdierentrandominitialcongurations.Theerrorbarsonthesimulationresultsrepresenttheerrorobservedoverthe8dierentruns.TheresultsarepresentedalongwiththeexperimentalworkofBrickerandButler[ 28 ]forcomparison.Althoughthesimulationsareforamonolayerofspheres,anarealfractionof=0:60roughlycorrespondstothevolumefractionof=0:40usedintheexperimentsofBrickerandButler[ 28 ]. Thesimulationresultsshownon-monotonicbehaviorthatisqualitativelysimilartotheexperiments.Mostnoticeably,aminimuminthelargestrainviscosityisobservedatA=1.Atthisstrainamplitude,thevalueofthelargestrainviscosityobservedintheoscillatoryshearowisapproximatelyequaltoone-halfofthesteadyshearvalue.Fromthispoint,thelargestrainviscosityincreaseswithincreasingstrainamplitudeasitapproachesthesteadyshearvalue(oscss=steadyss=1).Thesimulationresultsoverthisrangeofstrainamplitudesshowremarkableagreementwiththeexperimentalwork.Insomecases,theerrorbarsonthesimulationvaluesoverlapthevaluesobtainedfromexperimentsusingaparallel-plategeometry.ForA<0:5,thesimulationresultsbegintodeviatefromtheexperiments,showingamuchsteeperincreaseinviscositywithdecreasingstrainamplitude. Thesteadystatenormalstressnormalizedbythesteadyshearvalue(oscyy=steadyyy)isplottedversusstrainamplitudeinFigure 3-14 .ThelevelofaveragingisequivalenttothatdonefortheshearstressdatainFigure 3-13 .Thenormalstressinsteadyshearowhasanegativevalue,indicatingthatthestressiscompressive.Thenormalstressatthehigheststrainamplitudeisalsocompressive(oscyy=steadyyy=0:81).FromA=5,the 73

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Normalstressnormalizedbythesteadyshearvalueplottedasafunctionofstrainamplitudeforoscillatoryshearowfor=0.60andagapof15.Theerrorbarsrepresentstatisticalerrorfromeightrunsateachstrainamplitude. normalstressdecreasesrapidlyasthestrainamplitudedecreasestointermediatevalues.Byastrainamplitudeof1,thevalueofthenormalstressdierenceisessentiallyzero.Fromthispoint,oscyycontinuestodecreasewithdecreasingstrainamplitude.AtA=0:5thequalitativebehaviorofoscyychangessuchthatasignchangeoccurs.ForA0:5theaveragevalueofthenormalstressistensile,howeverwhetherthestresschangessignorbecomesvanishinglysmallforA0:5remainsunknownevenforthelevelofaveragingdonehere.Someconsequencesofatransitioninthenormalstressfromcompressivetotensileunderoscillatoryowconditionsisdiscussedinalatersection.Suspensionmicrostructure 3-15 showsthesecondmomentoftheinstantaneousparticledistancesfromthecenterlineasafunctionofstrainamplitude.Thedatarepresentsanaverageover8runsandtheerrorbarsreectthedeviationamongthedierentruns.Themomentsarecalculatedforasteadystatedistributionofparticlesandasareference,thevalueofthesecondmomentforarandom 74

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Secondmomentoftheparticledistances(atsteadystate)fromthecenterlineasafunctionofstrainamplitude.Theresultsareforsimulationsat=0.60andagapof15.Theerrorbarsrepresentstatisticalerrorfromeightrunsateachstrainamplitude. distributionis4:2.Asthestrainamplitudeincreases,theparticlesmoveclosertothewallonaverage.Theparticledistributionshiftsawayfromthewalltowardsamorerandomdistributionatsmallerstrainamplitudessuchthata10%dierenceisobservedbetweenthesecondmomentatA=5andA=0:1.Althoughnotshown,theparticledistributionsaresymmetricaboutthecenterline. Thepairdistributionfunctionsareevaluatedforeachstrainamplitude.Figure 3-16 showstheradialpairdistributionfunction,g(r),atsteadystateintegratedoverallangles(0<<180)asafunctionofradialdistance,r.Toavoidartifactsfromthepresenceofthewall,g(r)iscalculatedforparticleslocatednearthecenterofthegap(5:5
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Theradialpairdistributionasafunctionoftheradialdistanceintegratedoverallangles(0<<180)forasimulationat=0.60andagapof15.Theparticlepairsarecalculatedforparticlesnearthecenterofthegap(5.5
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Figure 3-17 showstheangularpairdistribution,g(),forallstrainamplitudesatradialdistancescorrespondingtothemaximuminFigure 3-16 .Forexample,atA2theangularpairdistributioniscalculatedforr<2:001,whereasforA1thedistributioniscalculatedforr<2:1.ThesoleexceptionisforthecaseofA=0:25,wherethecalculationisperformedforalargerradialdistanceofr<2:5asaresultofthedicultyindeningapeakvalueing(r).ThisanalysisresultsinthedierenceinscalesinFigure 3-17 .Thedataisaveragedovereightrunsandforeachstrainamplitude,g()iscalculatedfordierentsectionsacrossthegap. Threedistinctmicrostructuralregimesoccur.Athighstrainamplitudes(2A5)theangularpairdistributionresemblesthesteadyshearcase(Figure 3-7 );thereisanasymmetrying()asaresultofanexcessofparticlesinthecompressionalquadrant.AsthestrainamplitudedropsbelowA=2,themicrostructurechangessignicantly.TheangularpairdistributionbecomessymmetricforA1,andfor0:5A1,anexcessofparticlesisfoundat=0;90;and180.Theangularpairdistributionisapproximatelyequalatallthreeanglesindicatingthatthereisnopreferencebetweenparticlesarrangingthemselvesperpendiculartotheowdirectionasopposedtoparalleltotheowdirection.Athirdregimeisobservedforlowstrainamplitudes(0:1A0:25),whereparticlepairsareinexcessat=0;60;120and180.Again,thereisnopreferenceforndingpairsinanyoftheseorientationsexceptforparticlesclosetotheboundingwalls,wherethereisaslightbiasofpairsinthecompressionalandextensionalquadrantscorrespondingto=60and120.Forallstrainamplitudes,thereisanoticeableeectofthepresenceofthewallong().Discussion 77

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Theangularpairdistribution,g(),atcontactasafunctionofthetaforasimulationat=0.60andagapof15.Theangularpairdistributioniscalculatedforparticlesinthreedierentregionsofthe,7.5
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Meansquareddisplacementsnondimensionalizedbytheparticleradius,a,plottedversustotalstrainforastrainamplitudeofA=1.Theresultsareforsimulationsat=0.60andagapof15.Dataisshownforboththeow(x)andgradient(y)directions.Thesolidlinesrepresentlinearregressionofthedata. non-monotonicdependenceofviscosityonstrainamplitude,andimplicationsofthechangeinsignofthenormalstressoncurrentshear-inducedparticlemigrationmodels.Inthelastsection,webrieydiscussrelationshipsbetweenshearreversalandoscillatingshearows.StrainDependentDiusivity 28 ];irreversibilityisevidentfromtherheologicalchangesthatoccurwithstrainamplitudesaslowas0:1.Sourcesofirreversibilityintheexperimentscouldincludethebreakdownoflubricationandsubsequentcontactbetweenroughspheres[ 43 138 ]asoneexample.Irreversibilityinexperimentsmayalsoresultfromsmalleectsofparticleinertiaorthermaluctuations. 79

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Toclearlyquantifythelevelofirreversibilityinthesesimulations,thehydrodynamicdiusivitiesarecalculated.Figure 3-18 showsmeansquaredisplacementsplottedversusstrainforastrainamplitudeofA=1forboththeowandgradientdirections.Themeansquaredisplacementsarecalculatedfortherst10cyclesofshearandareaveragedover50runs,eachbeginningfromadierentinitialparticleconguration.Dierencesinthevaluesofthemeansquaredisplacementsforthexandydirectionsoccurasaresultoftheanisotropyoftheow.Inbothcases,themeansquaredisplacementsincreaselinearlywithstrain.ThesecharacteristicsaresimilarforallstrainamplitudesexceptA<1,wheretherelationshipbetweenthemeansquaredisplacementsandstraindeviatesslightlyfromalineardependence. Themeansquaredisplacementsareusedtocalculatedimensionlesshydrodynamicdiusivitiesintheowdirectionaccordingto[ 112 ] Similarly,Equation 3{18 isusedtocalculatediusivitiesinthegradientdirection;theresultsareplottedforeachstrainamplitudeinFigure 3-19 .Thediusivitiesarebasedonanaverageover10setsofexperimentsandtheerrorbarsneverexceedthesymbolsize.ForA<1,thediusivitychangesslightlywithstrainasaresultofthenonlinearityofthemeansquaredisplacements.Forthepurposeofthiswork,thediusivityreportedforA<1isanaverageofthemeandiusivitiescalculatedaftereachcycle.ThisadditionalaveragingisrepresentedinFigure 3-19 asanopensymbol.AlthoughthecalculationsinFigure 3-19 wereperformedusing64-bitdoubleprecision,additionalcalculationsof 80

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Dimensionlesshydrodynamicdiusivitiesplottedversusstrainamplitudeforsimulationsat=0.60andagapof15.Thedataisplottedasanaverageover10setsofexperimentsandisshownforboththeow(x)andgradient(y)directions.Opensymbolsrepresentdiusivitiesthathavebeencalculatedfromslightlynonlinearmeansquaredisplacements. thehydrodynamicdiusivitiesatA=0:1wereperformedusingbothhigherandlowerprecisionlevels.Nodependenceofthediusivitiesontheprecisionlevelofthecalculationwasobserved.Furthermore,thehydrodynamicdiusivitiesreportedareconvergentwithrespecttothetimestep. Astrongdependenceofthediusivityonstrainamplitudeisapparentfor0:252)andlow(A<0:25)strainamplitudelimitssuchthatplateausexistovertheremainingrangeofstrainamplitudes.ThesimulationresultsarequalitativelysimilartotheworkofPineetal.[ 112 ]forA0:5;thelackofquantitativeagreementmaybeduetodierencesinthedetailsofthesimulations.Mostnotably,Pineetal.[ 112 ]reportdiusivitiesfromthree-dimensionalsimulations,whereasoursimulationsarerestrictedtotheowandgradientdirections.Additionallyinourwork,suspensionsareboundedbywallsinthegradientdirection,whereasinthe 81

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112 ],fullyperiodicsuspensionswereused.Dierencesmayalsooccurduetotheformoftheinputstrainwave.Pineetal.[ 112 ]implementedasquare-waveprotocolasopposedtothesinusoidalformusedhere.Otherfactorsmightincludetheimplementationofaninterparticleforce,thelevelofaveragingperformed,timestep,andfrequencyoffar-eldupdates,allofwhicharenotreportedbyPineetal.[ 112 ].Pineetal.[ 112 ]didnotcalculatediusivitiesforstrainamplitudesbelow0:5,citingissueswithnumericalaccuracy.SuspensionMicrostructureandRheology 3-13 ).Thebehaviorofthesteadystateviscosityinoscillatoryshearcanbeunderstoodintermsofthesuspensionmicrostructure.Dependingupontheappliedstrainamplitude,weobserveoneofthreedistinctmicrostructures:hydroclusters,orderedlayers,oraphaseconsistingofalocallyorderedcrystal.Weexplaineachstructureindetailinthecontextofpreviousstudiesandcorrelatethemicrostructurestotheobservedrheology. SnapshotsoftheinstantaneousparticlecongurationsatsteadystateforvariousstrainamplitudesareshowninFigure 3-20 .Themicrostructuresarecapturedwhenthewallvelocityismaximum.Anexampleofaninitialconguration(total=0)isprovidedasareference;theparticlepositionsarerandomandnomicrostructureisobserved.Thesuspensionmicrostructureevolvesfromthisliquid-likecongurationtooneofthreedierentmicrostructuralregimes. Therstregimeoccursforhighstrainamplitudes(2A5),wherechainsofparticlesareapparentinthecompressionalquadrant.Themicrostructureclearlyresembleshydroclustering,whichdominatesthesteadyshearofsuspensionsofnoncolloidalspheres[ 24 104 ]andthehighshearlimit(Pe!1)ofBrownianspheres[ 16 110 111 ].Similaritywithsteadyshearisexpectedsincethestrainexperiencedbythesuspensionduringone 82

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InitialcongurationofparticleswithintheshearcellpresentedalongwiththeinstantaneoussteadystateparticlecongurationsforstrainamplitudesofA=5,1,and0.1.Theresultsarereportedforsimulationsat=0.60andagapof15.Theshadedparticlesillustratethestructuresformedbytheoscillatoryshearatdierentstrainamplitudes.Arrowsattheboundariesindicatetheinstantaneousdirectionoftheshearow. 83

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Schematicshowingtheslidingmechanism(a)observedforintermediatestrainamplitudesversusthe`lockedlayer'mechanism(b)observedforlowstrainamplitudes.Thetrajectoryofthecenterofmassforatestparticle(dashedline)inoscillatoryshearisgivenbytheboldline.Thetoprowofparticlesisinashearplanewithahighershearratethantherowbelowandthearrowsdenotetheexpectedtrajectoryineachdirectionduringonecompletecycle. half-cycleofoscillationasA!1allowsthestructuretorearrangebeforetheshearowchangesdirection. Inthesecondregime,whichoccursforintermediatestrainamplitudes(0:5A1),themicrostructurerearrangesintoorderedlayersalignedintheowdirection.ThisstructureissimilartotheorderedstructurewithinBrowniansystemswhenthethermalandhydrodynamicforcesbalance(Pe1)[ 110 111 ].Inoursimulationstheonlyforcesinvolvedaretheshort-rangerepulsiveforcesandhydrodynamicforces.Theeectofalteringtheparametersoftherepulsiveforceisdiscussedinthenextsection.Shear-inducedorderinghasalsobeenobservedinoscillatoryshearexperimentsonnon-colloidalspheres.GondretandPetit[ 69 ]imagedthesteadystatespatialdistributionofparticlesundergoingoscillatoryshearowandnoticedaquasi-periodicorderingofspheresinlayersparallelandperpendiculartotheowdirection.TheorderingwasexplainedintermsoftheinertialsecondaryowsthatoccurasaresultofaniteReynoldsnumber[ 105 ].OursimulationshowevershowthatthistypeoforderingoccursevenatRe=0whereinertialeectsareabsent. Inthethirdregimeatsmallstrainamplitudes(0:1A0:25),particlesorderintoacrystal-likestructureatsteadystate(Figure 3-20 )asaresultofthesmalloscillatory 84

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3 ]observedshear-inducedorderinginoscillatoryshearowforvolumefractionsaslowas0.47,eventhoughthetransitionbetweenaliquid-likeandcrystalstructureoccursatacriticalvolumefractionofatleast0:49[ 2 ].Inthenon-Browniansystem,shear-inducedorderingoccursatanarealfractionof=0:60,whichiswellbelowthecorrespondingliquidphasetransitionfortwodimensionsat0:69[ 6 80 ]forBrownianparticlesatequilibrium.Consequently,theresultsseemtoindicatethatoscillatoryshearinducedorderingisahighPephenomena.Forexample,theorderinginBrowniansystemsislimitedtoconditionswherethermaluctuationsareweakanddonotdisrupttheshear-inducedstructure,suchasathighervolumefractionsorshearrates(highstrains)wherethePeisrelativelylarge. Eachofthesemicrostructurescorrelateswiththeobservedrheology.Theenhancedviscosityathighstrainamplitudesresultsfromtheformationofhydroclusterswhichareprimarilycontrolledbylubricationforces.ThesetypesofstructuresareexplainedindetailbyFossandBrady[ 62 ].TheminimumviscosityatA=1isduetotheformationoforderedlayers,whichallowspherestoeasilyslidepastoneanother.Figure 3-21 showstheslidingmechanismthatisresponsiblefortheminimumviscosity.ThismechanismofparticlelayersslidingpastoneanotherresultsinaminimumviscosityinBrowniansystems[ 111 ]aswellasforcolloidalsystemsinwhichtheparticlesareelectrostaticallyrepulsive[ 71 ].Atlowstrainamplitudes,theviscosityisagainenhancedasitisforhighstrainamplitudes.However,thestructuredierssignicantly.Inthisregime,theenhancedviscosityisaresultoflocalcrystal-likeorderingofspheres,whichlockparticlesintoahexagonalpattern(Figure 3-21 ).Thiscrystallinestructureinhibitsparticlemotion,evenatsmallstrainamplitudes,wheremaximumdisplacementisontheorderofaparticleradius.Althoughformuchstrongershearows,theresultingrheologyresemblesthe 85

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Steadystateviscosityinoscillatoryshearownormalizedbythecorrespondingsteadystateviscosityinsteadyshearowasafunctionofstrainamplitude.Theresultsarereportedforarangeofinter-particlerepulsiveforces,whicharedeterminedbyalteringtherange,,andmagnitude,F0.Allsimulationsarefor=0.60andagapof15. increaseinviscosityinsimulationsofsuspensionsofshearedarraysofelectrostaticallyrepulsiveparticles[ 71 ]. Thepooragreementbetweentherheologyinsimulationsandexperiments(Figure 3-13 )atlowstrainamplitudesispossiblyaconsequenceoflimitingthesimulationstoamonolayer.Forexampleatlowstrainamplitudes,thesteadystatemicrostructureintheexperimentsofBrickerandButler[ 28 ]maybeaface-centeredcubicorbody-centeredcubicstructure,andthusnotaccessibleinmonolayersimulations.Asaresult,theplanarcrystal-likeorderingatlowstrainamplitudescorrespondstoasteadystateviscositywhichovershootsthevaluefoundexperimentally.EectoftheRepulsiveForceontheRheology 86

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3-22 .TherepulsiveforceisdeterminedbyalteringtheparametersandF0inEquation 3{15 ,whichcorrespondtotherangeandmagnitude,respectively.Eachparameterwaschangedindependentlyandtherheologywasevaluatedforsteadyshear,andforoscillatoryshearforstrainamplitudesofA=0:1;1;and5.Forcomparison,theresultsareplottedalongwiththoseobtainedfromsettingF0=1:0104and=100(Figure 3-13 ). ThesteadystateviscosityisanonmonotonicfunctionoftheappliedstrainamplitudewithaminimumobservedatA=1regardlessofthevalueschosenforandF0.ThecorrespondingmicrostructureateachstrainamplitudeisalsoindependentoftherepulsiveforceandresemblesthosereportedinFigure 3-20 .Specically,theangularpairdistributionateachstrainamplitudeisindependentoftherepulsiveforcesuchthatallthreemicrostructuresofhydroclusters,orderedlayers,andlocalcrystal-likeorderingareobservedatthesamevaluesofA. Therepulsiveforcedoeshoweverhaveaneectonthemagnitudeofthesteadystateviscosity.Theeectisprimarilycontrolledbytherangeoftherepulsiveforce.Ingeneral,thegreatestdierenceinthesteadystateviscositiesoccursforA=0:1,whereoscss=steadyssrangesfrom0:5to1:2dependingupontherepulsiveforceused.Athigherstrainamplitudes,thedependencebecomeslessevident.Theradialpairdistributionisalsoaectedbytherepulsiveforcesuchthattheparticleseparationdistancevarieswithvaryingrepulsiveforce.AstherepulsiveforceisincreasedbyeitherincreasingF0ordecreasing,particlepairsbecomemoreseparatedandlubricationforcesplayalesssignicantrole,leadingtoanoveralldecreaseinthesteadystateviscosities.NormalStresses 3-14 ).Suchbehaviorpotentiallyhasimplicationsformodelsofshear-induced 87

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30 ]andsimulations[ 97 ]onsuspensionsexposedtooscillatingpipeowshowananomalousmigrationphenomenoninwhichparticlesmigratetowardthewallforsmallstrainamplitudes.Thiscontrastswiththecaseofunidirectionalpipeow,inwhichparticlesmigratetowardthecenter[ 29 93 125 ].Themigrationphenomenoninoscillatorypipeowatlowstrainamplitudesisnotcapturedbydiusionmodelsofshear-inducedparticlemigration[ 91 109 ].However,Morris[ 97 ]suggestedthatifthenormalstresseschangesignduringoscillatorypipeow,theanomalousmigrationcouldbeexplainedintermsofthespatialvariationoftheparticle-inducednormalstressesasproposedinthesuspensionbalancemodelofNottandBrady[ 103 ].Inunidirectionalpipeowtheparticlenormalstressiscompressive;forthesuspensionbalancemodeltopredicta`reverse'migrationofparticlesasseeninsmallamplitudeoscillatorypipeow,arearrangementoftheparticlemicrostructuremustoccurandresultinaparticlenormalstressthatistensile. Inexperiments,itisdiculttomeasurenormalstressesinnoncolloidalsuspensionsasaresultoftheirnitebutsmallvalues,whichareoftenclosetothesensitivitylimitsoftheinstrument[ 85 124 142 ].Furthermore,onlynormalstressdierencescanbereportedasaresultoftheinabilitytoresolvetheisotropiccomponentofthenormalstressduetothehydrostaticpressure.Anadvantageofsimulationsisthatvaluesofthenormalstressescanbereported.Statistically,thesignchangeinthevalueofthenormalstressinoscillatingsimpleshearowismarginalmainlybecausethemagnitudeofthenormalstressissosmall.Nevertheless,theresultsindicatethattheaveragevalueofthenormalstresstransitionsfromcompressivetotensile.Althoughasignchangeinoscillatoryshearowhasnotbeenmeasuredexperimentally,therearesomeindirectindicationsthatthetransitiondoesoccur.Shearreversalexperimentsshowashort-timetransitioninthenormalstressimmediatelyfollowingreversalofthesheardirection;thenormalstressbecomestensilebeforerapidlyreturningtocompressive.Therelationshipbetweenoscillatoryandshearreversalrheologyisdiscussedinmoredetailinthenextsection. 88

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65 ]predictedtheshapeofoscillatoryresponsewavesathighstrainamplitudesfromthetransientresultsobservedinshearreversalexperiments.Similarly,Narumietal.[ 101 ]reportedontherheologyofsuspensionsundergoinglargeamplitudeoscillatoryshearandsuggestedthattherheologycouldbedescribedbyarearrangementintheparticlemicrostructure,asitoccursinshearreversalexperiments.Asthesuspensionisshearedinonedirection,particlesformweakhydroclustersinresponsetotheow.Whenthedirectionoftheshearowchanges,themicrostructureisdestroyed,afterwhichanewmicrostructuredevelopsduringthehalf-cycle. Theresultsforshearreversalshowthatforsmallstrains,thetransientbehaviorofthestressesrapidlychangecorrespondingtoasimultaneouschangeinthemicrostructure.Intheprevioussection,wespeculatedonthepossibilityofthepresenceoftensilestressesforoscillatoryshearowatsmallstrainamplitudes.Thesimulationresultsforthetransientrheologyfollowingshearreversalshowthatimmediatelyuponreversingthedirectionofshear,thenormalstressisinitiallypositivebeforequicklychangingtoanegativesteadystatevalue.Thischangefromatensiletocompressivestressinshearreversalowsisexplainedbythepairdistributionfunction.Asthedirectionofshearisreversed,thequadrantsareswitchedsuchthatthemajorityofparticlesaretemporarilyinextension,leadingtoatensilestress.Astheshearowcontinuesinthisdirection,thepairdistributiontransitionsbacktoastructureinwhichparticlesarepreferentiallyincompression,resultinginacompressivestress.Thus,theshearreversalresultsprovidefurtherevidencethatthesignchangeforthenormalstressinoscillatoryshearowmayexist. Thebehavioroftheviscosityinoscillatoryshearowismorecomplex.Thesteadystatemicrostructureathighstrainamplitudesresemblesthesteadyshearcaseasdoesthe 89

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85 100 ]andconrmthatthistransientrheologycorrespondstoarapidchangeinthesuspensionmicrostructurethatoccursuponreversingthesheardirection. Foroscillatoryshearow,rheologicalchangesoccurovertheentirerangeofstrainamplitudesstudied.Specically,ateachstrainamplitude,theshearstressincreaseswithstrainbeforeattainingsteadystate.TheirreversiblebehaviorforthesimulationconditionsstudiedagreeswiththeexperimentalndingsofBrickerandButler[ 28 ].Thesteadystateshearstressesshowanon-monotonicdependenceontheappliedstrainamplitudethatqualitativelyagreeswithexperimentalresults[ 28 ].Thesteadystaterheologyiscorrelatedwiththesteadystatesuspensionmicrostructureandthreedistinctregimesareobservedovertherangeofstrainamplitudesstudied.Athighstrainamplitudes,therheologyresemblesthesteadyshearrheology.Atintermediatestrainamplitudes,aminimumin 90

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91

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35 66 119 ].Distinctlydierentbehaviorispredictedbytheoriesandsimulations.Theoriesevaluatingthestressgeneratedinasemi-dilutesuspensionofrigidnon-Brownianbers[ 12 47 120 ]donotpredictadependenceofthesuspensionviscosityonshearrate.Likewise,simulationsonsemi-dilutesuspensionsofbers[ 113 141 ]ndnodependenceontheshearrate,evenwhenallowingmechanicalcontactbetweenbers[ 129 ].ThoughthetheoriesandsimulationsareperformedinthelimitofinniterotationalPecletnumber,theresultsareexpectedtoqualitativelypredicttheresultsofexperimentsatsucientlylargerotationalPecletnumbers. Toaccountfortheshearratedependenceobservedinsomespecicexperiments,explanationssuchasberexibility[ 18 ]andberadhesion[ 35 ]havebeenoered.However,theseeminglyuniversalnatureoftheshearthinningphenomenonremainsunclear.Interpretationofresultsfromexperimentsarefurthercomplicatedbydeviationsfrommodelsystems,whichcaneecttheobservedrheology.Forexample,suspensionsaretypicallycomposedofheavyberssuspendedinalightliquid[ 66 ],resultinginbersedimentation.Furthermore,berstypicallyhavelargelengthscaleswhichcanresultincomplicationsfromberbreakageifusingfragilematerials[ 119 ]andboundaryeects[ 48 ].Clearly,astudyoftherheologyofwell-denedbersuspensionsisneeded. Inthefollowingsections,wereportontherheologyofwellcharacterizedpolystyreneellipsoidsuspensionsinthesemi-diluteconcentrationregime.Therheologyofsuspensionsofsphereswiththesamematerialpropertiesastheellipsoidsarealsoevaluatedandcompared.Adescriptionofthefabricationofthebersisprovidedalongwithacharacterizationoftheparticlesizeanddistribution.Resultsarepresentedfortwo 92

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Scanningelectronmicrographsofpolystyreneparticles.Thetopimageshowsthepolystyrenespheresbeforeprocessing.Thebottomimagesshowellipsoidswithaverageaspectratiosof4(left)and7(right).Thescalebarineachimageis2m,withtheexceptionofthebottomrightimage,whichhasascalebarof1m. dierentparticleaspectratios.WeobserveshearthinningbehaviorinsuspensionsofrigidellipsoidsatrotationalPecletnumbersgreaterthan103.Adiscussionofpossiblemechanismsfortheratedependentrheologyalongwithscalingsofthesteadystateviscosityisgiven,andconclusionsarepresentedinthelastsection.Experiment 99 ].Monodispersepolystyrenespheres(Polysciences,Inc.)withameandiameterof1.060.02mwereaddedtoamixtureof5%byweightpolyvinylalcohol(MPBiomedicals,Inc.)inwater.Themixturewasdriedtomakethinlmswhichweredeformeduniaxiallyatatemperatureof190Ctoadesireddrawratio.Ellipsoidalparticleswereobtainedbydissolvingthestretchedlmsina30%byvolumemixtureofisopropanolinwater. 93

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77 99 ].Thisparticularmethodallowsfortheproductionofellipsoidswithaspectratiosandsizedistributionswhicharehighlycontrollable[ 77 ]. Tostudytheeectofvaryingtheellipsoidaspectratioontherheology,dierentdrawratioswereused,resultinginellipsoidswithaverageaspectratios,L=d(whereListhelengthofthelongaxisanddtheshortaxis),of4.170.81and7.141.49.ScanningelectronmicrographsofthespheresandresultingellipsoidsareshowninFigure 4-1 .NormalizedsizedistributionsoftheparticlesaregiveninFigure 4-2 ,andaveragevaluesoftheparticlevolumeandsurfaceareaforeachaspectratioaretabulatedinTable 4-1 .Alllengthscalesareobtainedfromscanningelectronmicroscopyandthevaluesrepresentaveragesover50randomparticles. ThesuspendingliquidconsistedofUCON50-HB-5100oil(DowChemicals)blendedwith10%byvolumedistilledwaterand1mMpotassiumchloride.Thecomponentswerechoseninparttomatchthedensityoftheparticlesandsuspendingliquid.Themeasureddensityofthesuspendingliquidwas1.06g/cm3andthereportedparticledensitywas1.056g/cm3,thusbuoyancyeectsarenegligibleoverthetimescaleoftheexperiments.Thesuspendingliquidslightlyshearthins,exhibitingadecreaseinviscosityfrom1.93Pasto1.75Pasovertherangeofshearrates0:1s1_400s1.Althoughthedierenceovertheshearraterangeislessthan10%,therelativeviscosity,r,reportedintheremainderofthepaperisdenedastheeectiveviscosityofthesuspensionatagivenshearratenormalizedbytheviscosityofthesuspendingliquidatthesameshearrate.Toavoideectsfromevaporationofthesuspendingliquid,thedurationoftheexperimentswereminimizedandasolventtrapwasusedineachexperiment. 94

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Summaryofthecharacteristiclengthscales,volumeperparticle,andsurfaceareaperparticleofpolystyreneparticleswithaspectratiosofL=d=1,4,and7. AspectRatiodLVolume/particleSurfaceArea/particle 11:060:02m1:060:02m0:630:03m33:550:13m24:170:810:650:06m2:710:38m0:610:11m34:480:59m27:141:490:550:07m3:870:62m0:630:17m35:331:03m2

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51 ],theinterparticlespacingissuchthattheellipsoidsareunabletorotatefreelywithoutbeingimpededbyneighboringellipsoids.Therheologywasstudiedforconcentrationswithinthesemi-diluteconcentrationregimecorrespondingtonL3=1.00to4.10forellipsoidswithanaspectratioof4,andnL3=1.13to6.95forellipsoidswithanaspectratioof7. Themaximumparticle-basedReynoldsnumberforthesuspensionsystemsisRe106,sotheeectsofinertiaareminimal.TheminimumrotationalPecletnumber,Per=_=Dr,calculatedbasedontherotationaldiusion,Dr,ofaprolatespheroid[ 27 ],is103.ThePecletnumberbasedonthetranslationaldiusionofaprolatespheroid,Dt,isdenedasPet=_L2=Dt.SincetherotationalPecletnumberrepresentsthesmallerofthetwoPecletnumbers,Perisusedintheremainderofthepapertodenetheowstrength.Adiscussionofotherpossiblenonhydrodynamiceectsisgiveninalatersection. Experimentswereperformedusinga50mmdiameterparallel-plategeometry.Thegapbetweenplateswassetto500m,resultinginaminimumgap-to-particlelengthratioof125.Additionalexperimentsperformedatalargergapof1000m(H=L=250,whereHisthegapheight)werestatisticallyequivalent.Intheparallel-plategeometry,therateofstrainvariesintheradialdirection,resultingininhomogeneousow.Sincethemicrostructureofbersuspensionsisstraindependent,additionalexperimentswereperformedusinga50mmdiametercone-and-plategeometry,whichmaintainsaconstantshearrateintheradialdirection.Forallcone-and-platemeasurements,theconeanglewas0.04radians,andthegapwassetto45.7m,resultinginaminimumgap-to-particlelengthratioof12attheapexoftheconeand270attheedge. 96

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Normalizedsizedistributionsforellipsoidswithanaverageaspectratioof4(solid)and7(dashed).Ineachcase,particlenumbersarenormalizedbytheirrespectivepeakvalues.Eachdistributioniscalculatedfrom50randomparticles. TherheometerusedinallexperimentswasanARESLS-1straincontrolledrheometer(TAInstruments).Thesteadyshearrheologywasinvestigatedforarangeofberconcentrationsandaspectratios.Toensurethateachtestbeganfromasimilarinitialstate,thesuspensionswerepreshearedatashearrateof_=100s1for300seconds,whichwassucienttoenablethesuspensiontoreachasteadystate.Thetemperaturewasmaintainedat25Candthetemperatureuctuationwaslessthan0:05Cduringatypicalexperiment.Results 4-3 showstherelativeviscosityasafunctionoftimeandshearrateforsuspensionsofellipsoidswithL=d=4.Theparticlevolumefraction,,of0.103fallswithinthesemi-diluteconcentrationregime.Eachstart-upexperimentwasperformedimmediatelyfollowingapreshearat_=100s1for300s.Forsmallershearrates,thetransientresponseoftherelativeviscosityismore 97

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Relativeviscosityasafunctionoftimeandshearrateforpolystyreneellipsoidssuspendedinapolyalkyleneglycol/water/KClmixture.Theellipsoidshaveanaverageaspectratioof4andthevolumefractionis0.103.Foreachshearrate,thesuspensionsarepreshearedat_=100s1for300s.Experimentswereperformedusinga50mmparallel-plategeometrywithagapof500m. dramatic.Forexample,at_=100s1,therelativeviscosityisalreadyatsteadystateandnoadditionalchangesareobservedoverthetimescaleoftheexperiment.Atlowershearrates,rincreasescontinuouslyanddoesnotreachsteadystateoverthedurationoftheexperiment.Furthermore,althoughtheviscositiesatt=0areequivalent,theviscosityat_=0.1s1increasestor5:5after300swhereasat_=1s1,theincreaseismuchmoremodest(r3:5)overthesametimeperiod.Theresultsarequalitativelysimilartoexperimentsonsuspensionsoflargeaspectratio(L=d40)bers[ 35 49 ],whichshowanincreaseinviscositywithtimeatlowshearrates. Figure 4-4 showstherelativeviscosityat_=0.251s1plottedasafunctionoftimeandconcentrationforsuspensionsofberswithL=d=4.Theresultsareplottedforthreedierentvolumefractionsandrepresentthetimeevolutionfollowingapreshearat_=100s1for300s.Atallthreevolumefractions,therelativeviscosityincreaseswithtime.Forthevolumefractionsstudied,nosteadystateisobservedthrought=300s.Additional 98

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Relativeviscosityasafunctionoftimeandvolumefractionforpolystyreneellipsoidssuspendedinapolyalkyleneglycol/water/KClmixture.Theellipsoidshaveanaverageaspectratioof4andallvolumefractionsliewithinthesemi-diluteconcentrationregime.Foreachvolumefraction,thesuspensionsareshearedatarateof_=0.251s1immediatelyfollowingapreshearat_=100s1.Experimentswereperformedusinga50mmparallel-plategeometrywithagapof500m. experimentswereconductedoverlargertimescalesforsuspensionswith=0.079;nosteadystatewasobservedandtheviscositycontinuedtoincreasethrought=2000s. Todeterminetheeectofellipsoidaspectratioontherheology,suspensionsofellipsoidswithanaverageaspectratioofL=d=7werestudied.Therelativeviscosityat_=0.1s1,plottedasafunctionoftimeandconcentrationforsuspensionsofellipsoidswithL=d=7,isshowninFigure 4-5 .Theresultsareplottedforthreedierentvolumefractionsandrepresentthetimeevolutionfollowingapreshearat_=100s1.Similartothesuspensionscontainingellipsoidswithasmalleraspectratio(Figure 4-4 ),therelativeviscosityincreaseswithtime,withtheexceptionofthelowestvolumefraction,=0.012,whichshowsasteadyvalueoverthedurationoftheexperiment.ComparisonofthetransientresponseofsuspensionsofellipsoidswithdierentaspectratiosrevealsadependenceonL=d.Specically,theviscosityincreaseovert=300sisgreaterfor 99

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Relativeviscosityasafunctionoftimeandvolumefractionforpolystyreneellipsoidssuspendedinapolyalkyleneglycol/water/KClmixture.Theellipsoidshaveanaverageaspectratioof7andallvolumefractionsliewithinthesemi-diluteconcentrationregime.Foreachvolumefraction,thesuspensionsareshearedatashearrateof_=0.1s1immediatelyfollowingapreshearat_=100s1.Experimentswereperformedusinga50mmparallel-plategeometrywithagapof500m. suspensionswithellipsoidsofhigheraspectratio.Forexample,at0.055,therelativeviscosityforsuspensionsofellipsoidswithL=d=7increasesby32%overatotaltimeof300s,whereasforsuspensionsofellipsoidswithL=d=4therelativeviscosityincreasesbyonly21%overthesametime. Thedependenceoftherelativeviscosityonshearrate(andPer)isplottedinFigure 4-6 forsuspensionsofberswithL=d=4.Ratesweepswereperformedusingtheparallel-plategeometryforsuspensionsatsixdierentvolumefractionsspanningthesemi-diluteconcentrationregime.Ateachshearrate,thevalueoftherelativeviscosityisrecordedafter360secondsofshear,whichwaschosentoreducethetotalexperimenttimeandconsequently,tominimizepossibleevaporationofthesuspendingliquid.Sincetherelativeviscosityateachshearrateissensitivetothetimeoverwhichthesuspensionissheared,notallvaluesrepresentsteadystateviscositiesafter360s.Thus,opensymbols 100

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Relativeviscosityasafunctionofshearrateandvolumefractionforpolystyreneellipsoidssuspendedinapolyalkyleneglycol/water/KClmixture.Eachdatapointrepresentstheapparentviscosityafter360secondsofshearateachcorrespondingshearrate.Theellipsoidshaveanaverageaspectratioof4andallvolumefractionsliewithinthesemi-diluteconcentrationregime.Experimentswereperformedusinga50mmparallel-plategeometrywithagapof500m. representsteadystatevaluesoftherelativeviscosityandclosedsymbolsrepresentvaluesthathavenotattainedasteadystatevalueaftert=360s. Forallvolumefractionsstudied,thesuspensionsshowshearthinningbehavior.Thedependenceoftherelativeviscosityonshearratebecomesweakerasthevolumefractiondecreases.Forexample,atthehighestvolumefraction(=0.127),thevalueoftherelativeviscositydecreases350%overthreedecadesofshear,whileatthelowestvolumefraction(=0.031),therelativeviscositydecreasesbyonly43%overthesamerangeofshearrates.Below_=1s1,thevaluesoftherelativeviscositiesreportedinFigure 4-6 ceasetorepresentsteadystatevalues.However,eveninshearraterangeswherethevaluesoftheviscosityareatsteadystate(forexample,10s1_251s1),shearthinningisapparent.Moreover,ratesweepsfromhightolow_wereperformedandshowthat 101

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Relativeviscosityasafunctionofshearrateandvolumefractionforpolystyreneellipsoidssuspendedinapolyalkyleneglycol/water/KClmixture.Eachdatapointrepresentstheapparentviscosityafter360secondsofshearateachcorrespondingshearrate.Theellipsoidshaveanaverageaspectratioof7andallvolumefractionsliewithinthesemi-diluteconcentrationregime.Experimentswereperformedusinga50mmparallel-plategeometrywithagapof500m. theshearthinningisreversible.AlthoughratedependentrheologyisnotexpectedforsuspensionsofbersathighPecletnumbers[ 107 ],shearthinningsimilartothatobservedhereisconsistentlyobservedinexperiments[ 66 ]. Figure 4-7 comparestherelativeviscositiesatL=d=7forvedierentvolumefractionsspanningthesemi-diluteconcentrationregime.ThevaluesoftherelativeviscosityateachshearratearemeasuredastheywereinFigure 4-6 .Opensymbolsrepresentsteadystatevaluesoftherelativeviscosityandclosedsymbolsrepresentviscositieswhichhavenotreachedsteadystate.TherangeofshearratesstudiedisshiftedtoenablecomparisonatsimilarPerforthetwosystemswithdierentaspectratios.SimilartosuspensionscontainingellipsoidswithL=d=4,higheraspectratioellipsoidsshowshearthinningbehaviorforallvolumefractions.Again,theshearratedependenceof 102

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Comparisonoftherelativeviscositymeasuredinthecone-and-plate(dottedline)andparallel-plate(solidline)geometryforsuspensionsofpolystyreneellipsoidswithL=d=4inapolyalkyleneglycol/water/KClmixture.Therelativeviscosityisplottedasafunctionofshearrateandvolumefraction.Eachdatapointrepresentstheapparentviscosityafter360secondsofshearateachcorrespondingshearrate.Thecone-and-plategeometryhadadiameterof50mmandaconeangleof0:04radians.Thegeometrygapwassetat45.7m. therelativeviscositybecomeslessnoticeableastheconcentrationdecreases.Forexample,atthelowestvolumefractionstudiedforeitheraspectratio(=0.012),theviscosityisnearlyindependentoftheshearrateoverthreedecadesofshear. Sincetherheologyofbersuspensionsdependsonthespatialcongurationandorientationofbers,whichisstraindependent,theparallel-plategeometrymaybeunsuitableforbersuspensionrheologybecausethestrainvarieswithradialposition[ 48 ].Todeterminetheeectofgeometryontherheology,additionalexperimentswereperformedinacone-and-platexturewhichmaintainsaconstantstrainintheradialdirection.Figure 4-8 showstherelativeviscosityinthecone-and-plategeometryplottedasafunctionofshearrateforsuspensionsofberswithL=d=4.Althoughexperimentswere 103

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4-6 ,onlythreearereportedandcomparedtoexperimentsintheparallel-plategeometry.Forallvolumefractionsstudied,thevaluesoftherelativeviscosityforsuspensionsinthecone-and-plategeometryareconsistentlyhigherthanthoseintheparallel-plategeometry.Thedierencesaremostsignicantatthelowestshearrates,wherethereportedviscositieshavenotyetattainedsteadystate.Despitethesedierences,thequalitativebehaviorofoursystemisinsensitivetothegeometryused.Duetothesmallergap-to-particlelengthratioattheapexinthecone-and-plategeometry,therheologywassubjecttostressjumps,possiblyduetoparticlejammingattheapexofthecone.Anexampleofparticlejammingoccursfor=0.079,wheretherelativeviscosityanomalouslyincreasesat_=1.58s1.Discussion 35 66 119 ].ForlargevaluesofPer,hydrodynamicinteractionsdominate,andtheoriesandsimulationswithinthehydrodynamiclimitofPer=1areexpectedtoagreewithexperiments.However,ratedependentrheologyisnotpredictedbytheoriesandsimulations[ 107 ].Inanattempttoelucidatetheshearthinningphenomenoninsomespecicexperiments,explanationssuchasberexibility[ 18 ]andberadhesion[ 35 ]havebeenoered.Toevaluatetheroleofcolloidalinteractionsinoursystem,suspensionsofsphereswithmaterialpropertiesidenticaltotheellipsoidsarestudied. Apossiblecauseofshearthinning,uniquetooursystem,isthecompetitionbetweenhydrodynamicforcesandthemechanicalbindingofparticlesduetoresidualPVAontheparticlesurface.Polyvinylalcoholbindsstronglytopolystyreneandcanaectparticle 104

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SEMimagesofprocessedpolystyrenespheresbefore(left)andafter(right)washing.Ineachimage,thescalebarrepresents1m. surfaceproperties.Figure 4-9 showsscanningelectronmicrographsofpolystyrenesphereswhichweresubjectedtothesameproceduredescribedearlier,butnotstretched.Priortowashingtheprocessedparticles,muchofthePVAmatrixisstillattachedtotheparticlesurface,mechanicallybindingparticlestooneanother.Afterthewashingcycleshowever,theprocessedparticlesresembleplainpolystyrenesphereswithlittleevidenceofresidualPVAontheparticlesurface.Thus,itisclearthattheextentofwashingaectstheamountofresidualPVAremainingontheparticlesurfaceforparticlespreparedinthemannerdescribedearlier. Todeterminetheeectontherheologyfromresidualpolyvinylalcoholontheparticlesurface,therheologyoftwodierentsuspensionsofsphereswerestudied.Specically,comparisonsweremadebetweensuspensionsofplainpolysytrenespheres(asreceivedfromthemanufacturer)andprocessedpolystyrenespheresfollowingthewashingcycles.Bothsuspensionswerepreparedatavolumefractionof=0.07,andtherelativeviscosityasafunctionofshearrateisplottedinFigure 4-10 .Forsuspensionscontainingplainpolystyrenespheres,therelativeviscosityisconstantoverthreedecadesofshearrate.Suspensionscontainingprocessedspheresshowaslightdecreaseinrelativeviscosityoverthesamerangeofshearrates,howeveramajorityoftheshearthinningoccurspriorto_=1s1,afterwhichtherelativeviscosityremainsconstantandstatisticallyequivalent 105

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Relativeviscosityasafunctionofshearrateandaspectratiofordierentsuspensionsystems.Theresultsarefrompolystyreneparticlessuspendedinapolyalkyleneglycol/water/KClmixtureatavolumefractionsof0.07.Experimentswereperformedusinga50mmparallel-plategeometrywithagapof500m.Thevaluesrepresentaveragesovertwoindividualruns.Errorbarsarespeciedwhentheerrorislargerthanthesizeofthesymbol. totherelativeviscosityforsuspensionscontainingplainpolystyrenespheres.TheresultsindicatethattherheologymaybeslightlyaectedbythepresenceofPVAontheparticlesurface,butthequalitativedierenceissmallandonlyapparentatthelowestshearrates. Otherpossiblemechanismsfortheratedependentrheologycanbeevaluatedthroughadirectcomparisonoftherheologyofsuspensionsofsphereshavingidenticalmaterialpropertiesastheellipsoids.ChaoucheandKoch[ 35 ]presentamechanismfortheshearthinningbehaviorofnon-Brownianbersinwhichhydrodynamicforcescompetewithadhesiveforces.Atsmallvaluesoftheappliedshearrate,contactingparticlesocculateduetoadhesion.Astheappliedshearrateincreases,ocsbreakapart,resultinginadecreaseintheapparentviscosity.Figure 4-10 showstherelativeviscosityasafunctionofshearrateforsuspensionsofellipsoidsat0.07alongwiththerheologyofsuspensionsofprocessedspheres.Therheologyoftheprocessedspheresqualitativelydiersfromtherheologyofsuspensionsofellipsoidsatsimilarvolumefractions.Sincetheadhesiveforce 106

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35 ],itisunlikelythatadhesiveforceswouldcausestrongshearthinningbehaviorforsuspensionsofellipsoids,butnotforsuspensionsofspheres,thuseliminatingthepossibilityofadhesiveforcesasthecauseoftheshearthinningrheology.Theroleofelectrostaticinteractionsasapotentialsourcefortheshearthinningbehaviorcanalsobeeliminated.Electrostaticinteractionscauseratedependentrheologyinsuspensionsofspheres[ 71 115 ],howeverFigure 4-10 showsthattherelativeviscosityofsuspensionsofpolystyrenespheresisindependentoftheshearrate,indicatingthatelectrostaticinteractionsareunimportant.Althoughtheshapeandsurfacepropertiesofpolystyreneparticlescanaltertheelectrostaticeects[ 78 ],itisunlikelythattherheologyoftheellipsoidsuspensionsisaectedbyelectrostaticinteractions. Otherfactorsthatmaycontributetotheobservedshearthinningbehaviorinoursystemincludeparticleinertiaandberexibility.Althoughthereisaniteamountofinertiainourexperiments,theReynoldsnumberislow(Re=106)andinertiaispresumedtohaveanegligibleeectontherheology.Fiberexibility,whichcanresultinratedependentrheology[ 18 ],isalsonegligiblesincethepolystyreneellipsoidsusedinourstudyarerigid.Scanningelectronmicroscopyoftheellipsoidsshowsnoindicationofbentordeformedparticlesforanyoftheaspectratiosstudied.Shear-inducedbendingisalsounlikely.Themaximumstressgeneratedduringtheexperimentsissignicantlylessthantheminimumstressrequiredtobendasinglepolystyreneellipsoid[ 61 ]byafactorof103. Analternativemechanismfortheratedependentrheologymaybeparticlemigrationarisingfromhydrodynamicinteractionswiththeboundingwalls.Forexample,inpressure-drivenow,rigidbersathigh,butnotinnite,Pecletnumbersundergoanetmigrationawayfromthewalls,withthemagnitudeofthedriftincreasingwithowstrength[ 117 ].Sincethemagnitudeofthedriftduetoparticlemigrationisratedependent,suchamechanismmayaccountfortheobservedrheologyinoursystem.Atlargeshearratesforexample,particlemigrationmaydominate,forcinganonuniform 107

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4-3 Theroleofshear-inducedmigrationontherheology,however,remainsunclearandadditionalworkisrequiredtodeterminewhethersuchamechanismcanaccountfortheratedependence.Forexample,onemightexpectthermalmotiontobeunimportantatPecletnumbersgreaterthan103.Inthiscase,therheologywouldagreewithresultsfromtheoriesandsimulationsatPerofinnity.Indeed,suspensionscontainingsphericalparticlesceasetoexhibitshearthinningbehavioratPecletnumbers(basedonthetranslationaldiusionofasphere)largerthan102[ 62 111 ].However,unlikesuspensionsofspheres,adriftperpendiculartothewallcanoccurduetotheinteractionsbetweenrodlikeparticlesandboundarieswhichdependsupontheparticleorientationdistribution.Thecharacteristictimescaleforparticledriftduetomigrationissmall,andmaybecomparabletothetimescaleforthermaldiusion,whichisniteinoursystem.Suchamechanismthereforesuggeststhatthermaldiusion,thoughsmall,mayplayaroleintheobservedrheologyandthattheoriesandsimulationsinthelimitofinnitePerarenotapplicable.However,todetermineifmigrationcanresultinshearthinningrheology,acalculationoftheeectonthetotalstressfromparticledrift(relativetothermaldiusion)isrequired.Onemightalsoexpectthatamechanismbasedoninteractionsbetweenwallsandbulkparticleswouldresultinadependenceonthegapsize.Howeverinthiswork,wereporttherheologyfortwodierentgapspacingsandndnodependenceonconnement.Theapparentlackofdependenceongapspacinghowever,doesnotdiscountparticlemigrationasapotentialmechanismfortheobservedrheology.Intheabsenceofanyothermechanismavailabletoexplaintheshearthinningrheology,particlemigrationwarrantsfurtherstudy. 108

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4-11 showsthesteadystaterelativeviscosityatPer=106asafunctionofvolumefractionandaspectratio.ThesteadystaterelativeviscosityatPer=106isanincreasingfunctionofvolumefractionforbothaspectratios.Theviscosityincreaseswithvolumefractionmuchfasterforlargeraspectratios.Forexample,at0.07thesteadystaterelativeviscosityforL=d=7isapproximatelyone-and-a-halftimesgreaterthanthatforL=d=4,eventhoughrforbothsystemsconvergetosimilarvaluesatlowvolumefractions.Thedependenceofthesteadystateviscosityonaspectratioagreesqualitativelywithresultsfromexperiments[ 49 ]andsimulations[ 38 ].Computationalresults[ 38 ]forstatisticallyhomogeneousdispersionsofspheroidswithaspectratiossimilartothosestudiedhereareplottedalongwithourexperimentalresultsinFigure 4-11 .Theresultsagreereasonablywell,thoughtheexperimentsshowamuchstrongerdependenceoftherelativeviscosityonaspectratiocomparedtothesimulationresultsofClaeysandBrady[ 38 ].ThedierencesarenotsurprisingconsideringthatClaeysandBrady[ 38 ]evaluatetheviscosityforastaticisotropicorientationdistributionofspheroids,whereasthesteadystateorientationdistributionisunknownintheexperiments. Figure 4-12 showstherelativeviscosityatPer=104and106asafunctionofaspectratioanddimensionlessnumberdensity,nL3.Fortheaspectratiosstudied,thevaluesofthesteadystaterelativeviscosityatPer=106collapsewithinanarrowbandwhenplottedversusnL3,indicatingthatforsuspensionsofsmallaspectratioellipsoidsathighPer,thedimensionlessnumberdensityistheonlycontrollingparameter.Additionally,theexperimentsshowthatrscaleslinearlywithnL3atPer=106.ThelineardependenceoftheviscosityonnL3isconsistentwithsemi-dilutesuspensionsofhomogeneousdispersionsofellipsoidswithsimilaraspectratios[ 38 ]aswellasisotropicsuspensionsofcylindricalbers[ 141 ].However,ClaeysandBrady[ 38 ]andYamaneetal.[ 141 ]bothpredictamuchstrongerdependenceoftherelativeviscosityonaspectratiowhenplottedversusnL3.OurresultsshowastrongerdependenceofronaspectratioatsmallervaluesofPer.Figure 109

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SteadystaterelativeviscosityatPer=106asafunctionofvolumefractionandaspectratio.Theresultsarefrompolystyreneellipsoidssuspendedinapolyalkyleneglycol/water/KClmixture.Theexperimentalresultsarecomparedwithresultsfromsimulationsofisotropicsuspensionsofellipsoidalparticles[ 38 ]. 4-12 showstherelativeviscosity(following360sofshear)atPer=104plottedversusnL3.AtPer=104,theviscositynolongerincreaseslinearlywithnL3andthereisanobviousdependenceonellipsoidaspectratio.Conclusions 99 ].SimilartopreviousexperimentsonbersuspensionsatlargePecletnumbers[ 66 ],ellipsoidsuspensionsexhibitshearthinningbehaviorforbothaspectratiosoverallconcentrationsspanningthesemi-diluteconcentrationregime.Theshearthinningbehaviorisreversibleandindependentofthegeometryused.Distinctlydierentbehaviorispredictedbytheoriesandsimulations,whichshownodependenceoftheviscosityonshearrateinthelimitofinnitePer. 110

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SteadystaterelativeviscosityatPer=104and106asafunctionofdimensionlessnumberdensityandaspectratio.Theresultsarefrompolystyreneellipsoidssuspendedinapolyalkyleneglycol/water/KClmixture. Possiblemechanismsfortheshearthinningbehaviorwereevaluated.Directcomparisonoftherheologyofsuspensionsoftheellipsoidalparticleswithsuspensionsofsphericalparticles,processedinanidenticalfashion,suggestthatthemethodofmanufacturingtheellipsoidsaswellasothercolloidalinteractionsareimprobablesourcesoftheratedependentrheology.Analternativemechanismfortheobservedshearthinningwasproposedwhichinvolvesthecompetitionbetweenbermigrationduetohydrodynamicinteractionswiththeboundingwallsandthermaldiusion.ForlargePecletnumbers,thedependenceoftherelativeviscosityonvolumefractionisqualitativelysimilartosimulationsofisotropicsuspensionsofspheroidswithsimilaraspectratios[ 38 ].Additionally,atthelargestPecletnumbersstudied,therelativeviscosityscaleslinearlywithnL3,regardlessoftheellipsoidaspectratio.Atlowervaluesoftheappliedshearrate,correspondingtolowervaluesofthePecletnumber,therelativeviscositynolongerscaleslinearlywithnL3andadependenceontheellipsoidaspectratiobecomesapparent. 111

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75 83 133 ]asaresultofhydrodynamicinteractionsbetweenthepolymerandboundingwall[ 94 ].Migrationinsuspensionsofrigidpolymershavealsobeenobserved[ 10 102 118 ],thoughtheextentofthemigrationisnoticeablyweakerthanthemigrationofexiblepolymers[ 117 ].Inthecaseofthemigrationofrigidpolymers,thepreciseeectsofhydrodynamicinteractionswiththewallareunclear. Whendisregardinghydrodynamicinteractionswithboundaries,modelsonsuspensionsofrigidbersundergoingpressure-drivenowshowanetmigrationawayfromthechannelcenterline,towardthewalls[ 10 45 102 118 ].However,recentsimulationresultsconsideringhydrodynamicinteractionswiththewallshowthatthenetmigrationisaectedbysuchinteractions[ 117 ].Currently,amodelcomparabletothekinetictheoryforanelasticdumbbelldevelopedbyMaandGraham[ 94 ],whichincludeshydrodynamicinteractionswiththewall,isneededtoaccuratelydescribethenetmigrationofrigidbers. Inthischapter,wepresentamodelforthemigrationofrigidbersundergoingsimpleshearownearasolidboundaryinwhichhydrodynamicinteractionswiththewallareconsidered.Thebasisofthemodelisintroducedalongwithdetailsoftheapproximationsmade.Themodelpredictionsqualitativelydierfrompreviousmodelswhichignorehydrodynamicinteractionswiththewall,highlightingtheirimportance.Resultsfromthemodelareprovidedalongwithcomparisonstopreviousmodelsandadiscussionoftheapplicationofthemodeltotherheologyofsuspensionsofbers.Conclusionsarepresentedinthelastsection. 112

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Geometryandnotationusedtodescribeaberinsolutionundergoingsimpleshearownearasolidboundary. 5-1 .Theowisinthe1-directionandthegeometryconsideredisboundedbyasinglewallatx2=0.Althoughthemodelisbasedonafar-eldapproximationofasinglewall,shearowofarigidrodconnedbetweentwowallscaninprinciplebecalculatedusingtheappropriateGreen'sfunction[ 92 ]orcanbeapproximatedbysuperposingtheGreen'sfunctionforasinglewall[ 32 94 ].Theslender-bodymodelisusedtomodeltherigidber,thustheberlength,L,ismuchlargerthanthediameter,d,suchthatA=L=d10.Forthepurposeofthiswork,wesetA=10.ThebersareexposedtoexternalBrownianforcesbalancedbyhydrodynamicforces,suchthatthenetforceiszero(i.e.noinertia). Thetimeevolutionofarigidberinsolutionisgovernedbyacontinuityequationforthedistributionfunction,(xi;pi;t),ofthecenterofmass,xi,andorientation,pi, @xi(_xi)@ @pi(_pi):(5{1) Theprobabilitydistributionfunctionisseparatedintoacenterofmass,n,andorientationdistributionfunction,,asdonebyMaandGraham[ 94 ]andButleretal.[ 32 ], (xi;pi;t)=n(xi;t)(xi;pi;t);(5{2) 113

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IntegratingEquation 5{1 overpiandsolvingforthesteadyresultgives, @xi(n<_xi>)=0;(5{4) wheretheanglebrackets<>indicateanensembleaverageoverpi, TosolveEquation 5{4 forthecenterofmassdistribution,theaveragedcenterofmassvelocityisrequired.Toobtainanexpressionforthecenterofmassvelocitywhichincludescontributionsfromthepresenceofasolidboundary,afar-eldapproximationisused[ 21 32 94 ].Insteadofsolvingfortheposition-dependentorientationdistributionfunction,weevaluatetheensembleaverageofthecenterofmassvelocityviasimulationsofasingleBrownianrodundergoingshearowinwhichweignorehydrodynamicinteractionswiththewall.VelocityExpressions 11 41 42 ].InthelimitofStokesow,thecenterofmassvelocity,_xi,androtationalvelocity,_pi,arerelatedtothedisturbancevelocity,ui(s),andalinedistributionofpointforces,fj,by _xi+s_piui(s)=ln(2A) 4(ij+pipj)fj(s);(5{6) whereistheuidviscosityandsisanypointalongtherodevaluatedfromthecenteroftherod.IntegrationofEquation 5{6 overthelengthoftheber,L,givesthecenterofmassvelocityas, _xi=1 4L(ij+pipj)Fj:(5{7) 114

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5{6 withspiandintegratingoverL, _pi=12 wheretheintegratedforceis and~Fjistheweightedforcecorrespondingtothetorque,Tj, WelinearizetheforcedistributionusingLegendrepolynomialsfollowingButlerandShaqfeh[ 31 ]andSaintillanandShaqfeh[ 117 ]andretainonlythersttwoterms, L3h(ijpipj)~Fj+piSi;(5{11) whereSisthestressletofasingleber, whichisascalarquantityarisingfromtheinabilityofabertostretchorcompressalongitsmajoraxis. ThedisturbancevelocityinEquations 5{7 5{8 ,and 5{12 isthesumoftheimposedvelocityeld,u(0)i,andacontributiontothevelocityeldduetothepresenceofthewall,u0i(s), which,forshearow,becomes 115

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21 32 94 ].Wenoteherethats0isthelocationofthepointsourceontheber,whereassisthepointofevaluationofthedisturbance. CombiningEquations 5{7 5{8 5{12 ,and 5{14 ,substitutingEquation 5{11 forfj(s0),andeliminatingSgives, _xi="1 csL4ln(2A)G(10)ijpjpkZL=2L=2su(0)kds#+ln(2A) 4L(ij+pipj)+1 csL5ln(2A)G(10)ikpkplG(1)ljFj+12 csL7ln(2A)G(10)ijpjplG(2)lmPmk~Fk _pi="12 csL6ln(2A)PijG(2)ijpkplZL=2L=2su(0)lds#+12 csL7ln(2A)PijG(2)jlplpmG(1)mkFk+3ln(2A) csL9ln(2A)PijG(2)jkpkpnG(2)nmPml~Fl where L3ln(2A)piG(2)ijpj;(5{17) and 116

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32 ]togive, @rl[Gij(rl;xk)]rk=xk+s0pl@ @r(0)lhGij(xk;r(0)k)ir(0)k=xk+;(5{23) resultinginthesimplicationofEquations 5{19 5{20 5{21 ,and 5{22 (5{24) (5{25) (5{26) TheresultingexpressionsforGijarethensubstitutedintoEquation 5{15 toobtainthecenterofmassvelocity, _xi=_x2i1+_(x2)Ui+MijFj+NijPjk~Fk;(5{28) where ThevectorsandtensorsassociatedwiththehydrodynamicinteractionsappearinginEquation 5{28 are, (5{30) 4L(ij+pipj)3 321 32L3 (5{31) 641 117

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_pi=_p2(i1pip1)+!(x2)WikFk+3ln(2A) where 641 andWikisatensorassociatedwiththehydrodynamicinteractions, 5{28 issubstitutedintoEquation 5{5 andmultipliedbyntogetthecenterofmassux,Ji, wheretheforcesandtorquesinthethirdandfourthterms,respectively,werereplacedwiththeBrownianforcesandtorqueswrittenintermsofdistributionfunctions, ~F(Br)j=kT@ln WerewritethethirdterminEquation 5{36 byfactoringintoaproductofnand, @xjdpknkTZMij@ln @xjdpk=kT@n @xjnkTMij@ln @xj: 118

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5{39 isignoredasdonebyMaandGraham[ 94 ],sincewedisregardhydrodynamicinteractionswiththewallwhenevaluatingtheorientationdistribution.Usingthisapproximation,theorientationdistributionisafunctionofpionly. Similarly,thefourthterminEquation 5{36 becomes, @pknkTPjk@Nij Unlikethedumbbellcase[ 32 94 ],thefourthterminEquation 5{36 isnonzero,revealinganadditionalcontributiontothemigrationofrigidbers. Theuxnormaltothewall(inthe2-direction)iszero, @x2+nkT(x2)6ln(2A) Asanapproximation,weincludeonlythersttermoftheselfmobility,=kTln(2A) 4L(1+),andignoreeectsofinteractionsonthisterm.Substitutionandrearrangementgives, @x2=(x2) wherethePecletnumberisdenedas, kTln(2A):(5{43) Equation 5{42 containstwocontributions:acontribution,Pe(),fromtheimposedshearowandacontribution,24(),arisingfromtheBrowniantorque, and 119

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5{44 and 5{45 areobtainednumerically.CalculationofOrientationMoments 5{44 and 5{45 ,anumericalsimulationofasingleBrownianrodundersimpleshearowwasperformedandtheorientationsweresampledandaveraged.Sinceweareonlyconcernedwiththeberorientation,whichweassumeisindependentofposition,weconsideronlyrotationalBrownianmotionwithoutwallhydrodynamicinteractions.TheequationofmotionforasingleBrownianrodundergoingsimpleshearowinthe2-directioncanbederivedfromEquation 5{8 _pi=_p2(i1pip1)+3ln(2A) where~FBrjistheweightedforcecorrespondingtoaBrowniantorquewhichsatisestheuctuation-dissipationtheoremandthereforecanbeexpressedforthediscretetimestepas ~FBrj=2kTL3 wherewkisarandomvectoroflengththreewithunitvariance. CombiningEquations 5{46 and 5{47 andaddingacorrectiontermfornumericalintegrationbyamodiedEulermethod[ 39 58 ],gives _pi=p2(i1pip1)+2 whichismadedimensionlessusingthecharacteristictimescale_1andtherotationalPecletnumber, 120

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NumericallycalculatedensembleaverageoftheorientationmomentsasafunctionofPer.SimulationswereperformedonaBrownianrigidberundergoingsimpleshearow. Theorientationsattimet=t0arecalculatedusingEquation 5{48 andsampledforNtimes.Forexample,theaveragemomentforp1p2becomes, Specically,theaveragingisperformedover1000simulationtimeincrementsoft=5untilanaltimestepoft=5000.Thetimestepwassettot=5107,andthenumberofsamplestakenwithineachincrementoftimewasN=104.Thislevelofaveragingwasrepeatedvetimesusingdierentrandomseednumberstoarriveatthenalaveragedvalueofthemoments.Thestandarddeviationofthecalculatedmomentswaslessthan1%. Figure 5-2 showstheensembleaverageofthemomentsinEquations 5{44 and 5{45 presentedasafunctionofPer.ThemomenthasthelargestvalueatlowPer,anddecreasesastheowstrengthincreases.TheandmomentsapproachzeroatlowPerandincreasewithincreasingowstrengthuntilreachinga 121

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Eectofberorientationonthecenterofmassdriftvelocity,_x2.Thedriftvelocityisnormalizedby_andrepresentstheberanglewithrespecttothewall.Apositivevalueforthedriftvelocitycorrespondstothedriftofarigidberawayfromthewall,whileanegativevaluecorrespondstoadrifttowardsthewall. maximumatPer10.AllmomentsdecreaseatlargevaluesofPerasaresultofthep2component,whichvanishesasbersaligninthedirectionoftheow.Theresultsforarequantitativelysimilartoresultsforsuspensionsofspheroids[ 9 ].ResultsandDiscussion 5{28 representthetheresponseofthecenterofmassvelocitytoanimposedshearow.Theexpressionforthevelocityinthe2-directionduetoshearowinthe1-directionbecomes, _x2=_(x2)(p1p2)(13p2p2):(5{51) 122

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Schematicsummarizingtheeectofberorientationonthedriftvelocity.Thesignsindicatethedirectionofmigration;apositivesigncorrespondstoavelocityawayfromthewallwhileanegativesigncorrespondstoavelocitytowardthewall. Thedirectionofthevelocitydependsupontheorientationoftheberwithrespecttothewall.Figure 5-3 shows_x2=_plottedasafunctionoftheangle,,oftherodorientationwithrespecttothe1-direction.For0<<35:26and90<<144:74,thevelocityispositive,andactsawayfromthewall.Forallothervalues,thevelocityisnegativeandactstowardthewall.Themaximumvelocityforaparticlemovingawayfromthewalloccurswhen=117:37;thecorrespondingmaximumvalueformovementtowardthewalloccurswhen=62:63.Figure 5-4 summarizesthedependenceofthedirectionofthedriftvelocityontheinstantaneousberorientation.TheresultsquantitativelyagreewiththeworkofSaintillanetal.[ 117 ],asdoestheexpressionfor_x2.CenterofMassDistribution 5{42 canbe 123

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Contributionoftheshearow(Pe())andBrowniantorque(24())tothemigrationofarigidberplottedasafunctionofPer. integrateddirectly, x2Pe()+24() 128ln(2A);(5{52) wheretheintegrationconstantissetequalto1suchthatthebulkvaluen(x2!1)=1. Theindividualcontributions,Pe()and24(),tothecenterofmassdistributionfunctionareplottedinFigure 5-5 asafunctionofPer.Thevalueof24()ispositiveforPer>0andapproachesalimitingvalueof24,whereasthevalueofisweaklynegativeforvaluesofPerlessthan1,asconrmedbyalowPecletexpansionofthedistributionfunction.Consequently,Eq. 5{52 predictsaweakmigrationtowardsthewallforsucientlysmallshearrates.However,Pe()increasesindenitelyovertherangeofPestudiedanddominatesthemigrationbehaviorforstrongshearingows. Figure 5-6 showsthecenterofmassdistributionasafunctionofthedistancefromthewall.ThedistributioniscalculatedforPerangingfrom7:2102to4:8105.Ingeneral,anetmigrationawayfromthewallisobserved.ForPe7:2102,theprobabilitydistributionisapproximatelyuniformatalldistancesfromthewall.Theconcentrationnearthewallishigherthanexpectedsinceexcludedvolumeeectsarenotincludedinour 124

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Centerofmassdistribution,n(x2),plottedasafunctionofx2=LandPe. calculations.AsthecenterofmassPecletnumberincreases,anoticeabledepletionlayeroccursinthevicinityofthewallasbersdriftawayfromtheboundary.Theimbalanceofthecenterofmassdistributionincreasesastheowstrengthincreases,suchthatatthehighestPe,therearenoparticlesoccupyingaspaceoneberlengthawayfromthewall. ThedepletionthicknessisplottedasafunctionofPeinFigure 5-7 .Thedepletionlayerthicknessisdenedasthedistancefromthewallwheren(x2)=0:1andn(x2)=0:5.Astheowstrengthincreases,thedepletionlayerincreases.AtthelowestPe,thedepletionthicknessdenedaccordington(x2)=0:1isequivalentto0:01L.IncreasingPethreeordersofmagnituderesultsinadepletionlayerthicknessequivalentto4L.Theresultsarequalitativelysimilartosimulationresultsforrigidbersundergoingpressure-drivenow[ 117 ]. Themigrationofrigidbersawayfromthewallduringshearowqualitativelydiersfrommodelsofrigidberswhichneglecthydrodynamicinteractionswiththewalls.NitscheandHinch[ 102 ]predictamigrationofberstowardthewallforarbitraryvaluesofPewhenneglectingbothstericandhydrodynamicinteractionswiththewall.Thoughlimitedtoweakows,themodelofSchiekandShaqfeh[ 118 ],whichincludessteric 125

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DepletionthicknessnormalizedbyberlengthplottedasafunctionofPe.Thedepletionlayerisdenedseparatelyasthedistancefromthewallwheren(x2)=0:1(circles),andn(x2)=0:5(squares). walleects,alsopredictsmigrationtowardsthewall.Thequalitativedierencesinthepredictionofthedirectionofmigrationimplythathydrodynamicinteractionswiththewallplayasignicantrole. Themodelpredictionofmigrationofrigidbersawayfromthewallduringsimpleshearowhasimplicationsintermsoftherheologyofrigidbersuspensions.Asdiscussedinthepreviouschapter,suspensionsofspheroidsatlargePerexhibitratedependentrheologyeventhoughtheoriesandsimulationspredictotherwise.Theresultsfromthemodelmaypartiallyexplaintheshearthinningrheologyobservedinexperiments.Themodelpredictsamigrationofrigidbersawayfromthewall,withthemagnitudeofthemigrationdependingupontheowstrength.Adepletionlayerneartheboundarymayeectrheologicalmeasurements,whichevaluatethestressatthewall.Specically,themagnitudeofthestressshoulddecreaseasthedepletionlayergrows.AdepletionlayerwhichgrowswithowstrengthaspredictedinFigure 5-7 willthusresultinastresswhichdecreaseswithowstrength. 126

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35 66 119 ]. Thoughthemodelpresentsgeneraltrendsforarigidberundergoingsimpleshearowinthepresenceofasingleboundary,workmustbedonetoextendthemodeltoresultsfromexperimentsandsimulations.Forexample,weneglectexcludedvolumeeectswhichresultsinanenhancedconcentrationofbersnearthewall,especiallyatlowPe.Togainamoreaccuratedistribution,theseeectscanbetakenintoaccountusingresultswhichincludetheeectofexcludedvolume[ 118 ].Thisstudywillalsobeextendedtoevaluatethestress.Oncethestresscalculationiscompleted,themodelresultscanbecomparedtotheexperimentalworkpresentedinChapter 4 aswellasthedynamicsimulationofaBrownianrodnearasolidboundary. 127

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Theworkpresentedinthisdissertationsignicantlycontributestotheknowledgeofthedynamicsofsuspensionsystems.Suspensionsexhibitavarietyofpropertiesthatareutilizedinmanytechnologicalandindustrialapplications,whichincludepaintsandinks,foodthickeners,bodyarmor,andbrakesandclutches,amongothers(Chapter 1 ).Toenhancethequalityofproductsderivedfromsuspensionsystems,studiesmustbeconductedtodevelopacompleteunderstandingoftheirdynamics.Thisworkspecicallyaddresses1)therheologyofnoncolloidalsuspensionsofspheresundergoingoscillatoryshearowand2)thedynamicsofsuspensionsofrigidbers. Despitestudiesontherheologyofnoncolloidalsuspensionsinoscillatoryshearow[ 26 65 69 101 ],therheologyisnotwell-understood.Forexample,adriftinviscosityisobservedpriortosteadystateatallstrainamplitudes[ 65 69 ].Similarly,theresultspresentedinChapter 2 showthattheviscosityincreaseswithtotalstrainforsmallstrainamplitudesanddecreasesforlargestrainamplitudes,withthetransitionpointatwhichthequalitativebehaviorchangesoccurringatanamplitude-to-gapratiobetween0.1and0.5.Explanationsforthedrifthavebeenoered[ 105 106 ],yettheoriginremainsunclear.Resultsfromtherstdynamicsimulationofnoncolloidalspheresinunsteadyshearows(Chapter 3 )resolvethisissue.Thedriftinviscosityateachstrainamplitudeoccursasaresultofmicrostructuralchangesthatdisrupttheequilibriumstructureoverlargetotalstrains.Thelargestrainsrequiredtoreachsteadystatesuggestthatpreviousresultsfromexperimentsmaynothavebeenreportedwithrespecttotheirsteadystatevalues. Anothermajorcontributionofthisworkisthediscoveryofanonmonotonicdependenceofthesteadystateviscosityonstrainamplitude,whichhasnotbeenreportedpreviously.Specically,simulationsandexperimentsshowthatatlargestrainamplitudes,theviscosityapproachesthevalueobservedinsteadyshear.Asthestrainamplitudedecreases,thesteadyvalueoftheviscositydecreasesuntilaminimumisobservedata 128

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16 ]areobservedatlargestrainamplitudes,wheretherheologycloselyresemblessteadyshear.Atastrainamplitudeof1,whereaminimumintheviscosityisobserved,orderedlayersofparticlesformbothparallelandperpendiculartotheowdirection,allowingparticlestoeasilyslidepastoneanotherduringshear.Theformationofcrystallinestructuresareobservedattheloweststrainamplitudesstudied.Themostsignicantofthesesteadystatestructuresistheformationofacrystallinephaseatlowstrainamplitudes,eventhoughtheparticleconcentrationislowenoughsuchthattheequilibriumstructureisliquid-like.Thisisasurprisingresultconsideringthatatlowstrainamplitudes,particledisplacementsaresmallandinteractionsareshort-ranged,occurringonlybetweennearestneighbors.Inthiscase,onemightexpectparticletrajectoriestobereversible[ 112 ],andtheequilibriumstructuretoremainundisturbed.Ourexperimentsandsimulations,however,showthatparticletrajectoriesareirreversibleoverlargetotalstrains,leadingtotheformationofsteadystructureswhicharehighlyordered. Thisworkwouldbenetgreatlyfromdirectexperimentalevidenceofthesteadystatemicrostructureasafunctionofstrainamplitude.ThoughresultsfromopticalexperimentsareavailableforBrowniansuspensionsundergoingoscillatoryshearow[ 2 { 4 ],nosuchexperimentsexistforsuspensionsofnoncolloidalspheres.Similarly,modelpredictionsarenotavailableforoscillatoryshearow.Currently,itisunclearwhetherornotcurrentmodels[ 68 108 127 ]ontherheologyofnoncolloidalsuspensionscanaccountforthebehaviorpresentedinthisdissertation.Themajordicultyinapplyingthesemodelstooscillatoryshearowhasbeenthelackofexperiments;thecomprehensiveresultspresentedinChapter 2 shouldallowforanadequatecomparison.Asaresult,thesemodelsneedtobeextendedtooscillatoryshearows.Lastly,fullythree-dimensional 129

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AnadditionalcontributionofthisworkconcernstherheologyofbersuspensionsatlargerotationalPecletnumbers.InChapter 4 ,wepresentedresultsfromamodelsuspensionofpolystyreneellipsoidswithmoderateaspectratios.SimilartopreviousexperimentsonbersuspensionsatlargerotationalPecletnumbers[ 35 66 119 ],polystyreneellipsoidsuspensionsexhibitshearthinningbehavior.Theoriesandsimulations,however,donotpredictadependenceoftherheologyonshearrate[ 107 ].Asaresult,explanationshavebeenoeredtoaccountfortheratedependenceobservedinsomespecicexperiments[ 18 35 ],howevertheoriginofshearthinninginexperimentsremainsunclear.Themajoradvantageofthisworkisthatweareabletoevaluateandcomparetherheologyofsuspensionsofsphereshavingmaterialpropertiesidenticaltotheellipsoids.Inthismanner,eectsassociatedwiththemethodofmanufacturingtheellipsoids,aswellasothercolloidalinteractions,werefoundtobeunlikelycausesoftheratedependentrheology.Inlieuofthese,amechanism,involvingcompetitionbetweenparticledriftarisingfromhydrodynamicinteractionswiththeboundingwallsandthermaldiusion,isproposedtoexplaintheobservedshearthinningphenomenoninoursystem. Tocomplimenttheworkontherheologyofbersuspensions,Chapter 5 presentsanoutlineofcalculationstodeterminethecenterofmassdistributionofslenderbodiesinboundedsimpleshearow.Preliminaryresultsindicatethatadriftoccursawayfromtheboundingwallsinsimpleshearow.ThemagnitudeofthedriftvelocitydependsontheowstrengthrelativetotheappliedBrownianforce,andberorientation.Todeterminewhetherhydrodynamicinteractionsbetweentheparticlesandboundingwallscanaccountfortheratedependentrheologyobservedinexperiments,acalculationofthetotalstressforslenderbodiesinsimpleshearowisrequired.Furthermore,tocorrelatemorepreciselywiththeexperiments,calculationsmustbeperformedforellipsoidsasopposedtoslenderbodies.Furtherevidenceofthemigrationshouldbeobtainedfrom 130

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JonathanMarkBrickerwasborninOrlando,FloridaonFebruary24,1980toPaulandMariann.HisfamilybrieyrelocatedtoSanAntonio,TexasbeforemovingontoGreenville,SouthCarolina,whereJonathanspentmostofhisadolescence.HeattendedEastsideHighSchoolinGreenvilleandgraduated9thof215students.JonathanchosetoattendtheHonorsCollegeattheUniversityofSouthCarolina,wherehegraduatedwithhonorsandreceivedaBachelorofSciencedegreeinchemicalengineering.WhileattendingUSC,JonathanconductedresearchunderthedirectionofDr.FrancisGadala-Mariaintheareaofcomplexuids.JonathancontinuedhiseducationasaresearchassistantattheUniversityofFlorida.HeundertookresearchwithDr.JasonE.Butlerintheareaofcomplexuids.Specically,Jonathanfocusedondescribingthedynamicsofsuspensionsystemsusingexperimentalandcomputationalmeans.JonathanreceivedaDoctorofPhilosophydegreeinchemicalengineering,andnowlooksforwardtobeginningthenextchapterinhislifeasheappliestheskillsdevelopedingraduateschooltotheworkplace. 142


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