Particle Motion in Colloidal Dispersions: Applications to ...· Particle Motion in Colloidal Dispersions:

Particle Motion in Colloidal Dispersions: Applications toMicrorheology and Nonequilibrium Depletion Interactions.

Thesis by

Aditya S. Khair

In Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

California Institute of Technology

Pasadena, California

2007

(Defended February 6, 2007)

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c 2007

Aditya S. Khair

All Rights Reserved

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Acknowledgements

The past four and one-third years I have spent at Caltech have been among the most enjoy-

able of my life. Many people have contributed to this; unfortunately, I cannot acknowledge

each and every person here. I offer my most sincere apologies to those whom I (through

ignorance) do not mention below.

Firstly, I wish to thank my thesis adviser, John Brady. John gave me a fantastic

research project to work on; allowed me generous vacation time to go back to England; and

was always available to talk. As a researcher, Johns integrity, curiosity, and rigor serve as

the standard that I shall aspire to throughout my professional life.

I would also like to thank the members of my thesis committee John Seinfeld, Todd

Squires, and Zhen-Gang Wang for their support and interest in my work. In particular,

Todd was instrumental in sparking my interest in microrheology and has given me much

advice on pursuing a career in academia.

I have been fortunate enough to share the highs and lows of graduate school with some

great friends. I thank the members of the Brady research group that I have overlapped

with: Josh Black, Ileana Carpen, Alex Leshansky, Ubaldo Cordova, James Swan, Manuj

Swaroop, and Andy Downard, for making the sub-basement of Spalding an enjoyable place

to work. I also thank Alex Brown, Justin Bois, Rafael Verduzco, Mike Feldman, and Akira

Villar for sharing many laughs (and many beers) over the years.

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All of the Chemical Engineering staff receive my most sincere thanks. In particular, the

graduate-student secretary, Kathy Bubash, and our computer expert, Suresh Guptha, have

helped me greatly.

A special thanks goes to Hilda and Sal Martinez, the parents of my fiancee, Vanessa

(see below). Hilda and Sal have treated me so kindly and always welcomed me into their

home. I will be forever grateful.

I would not have reached this point without the love of my parents. From an early age,

they emphasized to me the importance of a good education; furthermore, they provided me

every opportunity to get one. Their continued support and encouragement means more to

me than they will ever know.

The last, but certainly not the least, acknowledgment goes to the love of my life, Vanessa.

She is the one person who can always put on a smile on my face when I am feeling low. I

hope that many years from now we can both look back on my time at Caltech with fond

memories.

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Abstract

Over the past decade, microrheology has burst onto the scene as a technique to interro-

gate and manipulate complex fluids and biological materials at the micro- and nano-meter

scale. At the heart of microrheology is the use of colloidal probe particles embedded in

the material of interest; by tracking the motion of a probe one can ascertain rheological

properties of the material. In this study, we propose and investigate a paradigmatic model

for microrheology: an externally driven probe traveling through an otherwise quiescent col-

loidal dispersion. From the probes motion one can infer a microviscosity of the dispersion

via application of Stokes drag law. Depending on the amplitude and time-dependence of

the probes movement, the linear or nonlinear (micro-)rheological response of the dispersion

may be inferred: from steady, arbitrary-amplitude motion we compute a nonlinear micro-

viscosity, while small-amplitude oscillatory motion yields a frequency-dependent (complex)

microviscosity. These two microviscosities are shown, after appropriate scaling, to be in

good agreement with their (macro)-rheological counterparts. Furthermore, we investigate

the role played by the probes shape sphere, rod, or disc in microrheological experi-

ments.

Lastly, on a related theme, we consider two spherical probes translating in-line with

equal velocities through a colloidal dispersion, as a model for depletion interactions out of

equilibrium. The probes disturb the tranquility of the dispersion; in retaliation, the disper-

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sion exerts a entropic (depletion) force on each probe, which depends on the velocity of the

probes and their separation. When moving slowly we recover the well-known equilibrium

depletion attraction between probes. For rapid motion, there is a large accumulation of

particles in a thin boundary layer on the upstream side of the leading probe, whereas the

trailing probe moves in a tunnel, or wake, of particle-free solvent created by the leading

probe. Consequently, the entropic force on the trailing probe vanishes, while the force on

the leading probe approaches a limiting value, equal to that for a single translating probe.

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Contents

Acknowledgements iii

Abstract v

1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Single particle motion in colloidal dispersions: a simple model for active

and nonlinear microrheology 13

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Nonequilibrium microstructure . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Average velocity of the probe particle and its interpretation as a microviscosity 26

2.4 Nonequilibrium microstructure and microrheology at small Peb . . . . . . . 32

2.4.1 Perturbation expansion of the structural deformation . . . . . . . . . 32

2.4.2 Linear response: The intrinsic microviscosity and its relation to self-

diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4.3 Weakly nonlinear theory . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.5 Numerical solution of the Smoluchowski equation for arbitrary Peb . . . . . 46

2.5.1 Legendre polynomial expansion . . . . . . . . . . . . . . . . . . . . . 48

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2.5.2 Finite difference methods . . . . . . . . . . . . . . . . . . . . . . . . 51

2.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.6.1 No hydrodynamic interactions . . . . . . . . . . . . . . . . . . . . . 53

2.6.2 The effect of hydrodynamic interactions . . . . . . . . . . . . . . . . 57

2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Appendices to chapter 2 78

2.A The Brownian velocity contribution . . . . . . . . . . . . . . . . . . . . . . . 79

2.B Finite difference method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.C Boundary-layer equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

2.D Boundary-layer analysis of the pair-distribution function at high Peb in the

absence of hydrodynamic interactions . . . . . . . . . . . . . . . . . . . . . 85

2.E Boundary-layer analysis of the pair-distribution function at high Peb for b 1 88

3 Microviscoelasticity of colloidal dispersions 93

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.2.1 Smoluchowski equation . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.2.2 Average probe velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.3 Small-amplitude oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.4 Connection between microviscosity and self-diffusivity of the probe particle 110

3.5 Microstructure & microrheology: No hydrodynamic interactions . . . . . . . 112

3.6 Microstructure & microrheology: Hydrodynamic interactions . . . . . . . . 120

3.7 Scale-up of results to more concentrated dispersions . . . . . . . . . . . . . 129

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3.8 Comparison with experimental data . . . . . . . . . . . . . . . . . . . . . . 132

3.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

3.10 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Appendix to chapter 3 148

3.A High-frequency asymptotics with hydrodynamic interactions . . . . . . . . . 149

4 Microrheology of colloidal dispersions: shape matters 153

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

4.2 Nonequilibrium microstructure . . . . . . . . . . . . . . . . . . . . . . . . . 158

4.2.1 Prolate probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

4.2.2 Oblate probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

4.3 Microviscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

4.4 Analytical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

4.4.1 Near equilibrium Pe 1 . . . . . . . . . . . . . . . . . . . . . . . . 166

4.4.2 Far fr