Rigidity of Microsphere Heaps.

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ABSTRACT

ORTIZ, CARLOS POMPEYO. Rigidity Of Microsphere Heaps. (Under the direction of Karen Daniels and Robert Riehn.)

Rigidity emerges in a broad class of soft matter systems, relevant to many industrial and biolog-ical processes. In our experiments, we study a model soft matter system, hard-sphere Brownian

suspensions of submicron particles. Brownian suspensions lack rigidity in the absence of external

driving, but form flow-stabilized solid-like microsphere heaps under the influence of hydrodynamic forces. The overarching question driving my dissertation is "What is the nature of the rigidity of

these microsphere heaps?" Does the rigidity of the heaps follow from mechanical stability driven by

a sufficiently interconnected network of particle contacts? Or, does the rigidity of the heaps follow from a kinetic glass transition characterized by a diverging resistance to flow such that the time

necessary to observe rearrangements grows prohibitively large? We expect that insights into the

mechanism of rigidity of Brownian microsphere heaps are applicable to a wide class of soft matter systems.

In this thesis, we have overcome the limitations of previous experimental approaches. Namely,

we show that the rigidity of our heaps does not emerge from the effects of gravity, inertia, static friction, or van der Waals sticking. In Chapter 1 of thesis, we review the background literature.

In Chapter 2, we present the experimental, analytical, and computational methods used in the

remainder of the thesis. In Chapter 3, we investigate the onset of rigidity by characterizing the steady-state size of the heap versus the imposed flow conditions. We show that thermal fluctuations

and repulsive interparticle interactions, the dominant forces at the single-particle scale, suppress

the development of a rigid phase. These conditions imply that the onset of rigidity in involves many-body collective interactions. In Chapter 4, we measure the response of the heap to external

perturbations, which allows us to measure their elastic modulus and compare our results to hard

sphere theoretical expectations. We find bulk nonlinear elastic behavior. In Chapter 5, we study the particle displacements in response to external perturbations and quantify the local nonlinear

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Rigidity Of Microsphere Heaps

by

Carlos Pompeyo Ortiz

A dissertation submitted to the Graduate Faculty of North Carolina State University

in partial fulfillment of the requirements for the Degree of

Doctor of Philosophy

Physics

Raleigh, North Carolina

2013

APPROVED BY:

Albert Young Michael Shearer

Karen Daniels

Co-chair of Advisory Committee

Robert Riehn

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DEDICATION

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BIOGRAPHY

I was born in Buenaventura, a port-city in Colombia, on May 29, 1986. I moved to Columbia, SC, on the Spring of 2002, where I attended Irmo High School. The origin of my interest in experimental soft

matter began during my high school AP Chemistry lab experiments, where I developed an interest

in polymers and pursued curiosity-driven research on polymer flame tests. These studies enabled me to compete at the state and then the national level on the “Polymer Detectives” event of the High

School Science Olympiad on the Spring of 2003. That May, I received an Extended Studies Diploma from Irmo High School.

Four years later, I received a B.S. in Physics degree from Davidson College, NC. During my

undergraduate degree, my enthusiasm for experimental research grew while taking Electronics and Instrumentation, taught by Dan Boye. I went on to do research in his lab during the summer of 2006,

studying rare-earth doped sol-gel glasses. During my senior year, I decided to continue the research,

culminating in the writing of an honors thesis about the coupling between the structural properties of a disordered porous material, a sol-gel glass, and the spectral properties of rare-earth atoms and

rare-earth atom clusters dispersed throughout the glass.

During my senior year, I also won the Hearst Teaching Fellowship which allowed me to explore my sentiments for teaching as a profession by teaching Spanish for a year at Charlotte Country Day

School, an independent school. After teaching for a year, I decided to pursue a PhD in physics at

North Carolina State University. My advisors, Karen Daniels and Robert Riehn, allowed me to join the lab and do exploratory research on the summer of 2008, prior to my first semester graduate

classes. The results and insights from that summer allowed me to write a coherent research proposal

for the NSF Graduate Fellowship, which funded the bulk of my PhD after the spring of 2008. During my PhD, I presented my work in the form of posters and talks at local, regional and

national conferences and workshops: Triangle Soft Matter, SESAPS, DFD, NSHP, GRC, APS. I was

a teaching assistant for laboratories using the Matter and Interactions curriculum developed by physics faculty at NCSU. I was also the teaching assistant for Electronics and Senior Lab. I was a

grader for both introductory physics courses and two first year graduate classes (classical mechanics

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ACKNOWLEDGEMENTS

First, I would like to thank my professional colleagues, friends, and family, all of whom helped make this thesis possible, whether they realize it or not. I would like to thank my mother for allowing

me to tear apart even my most expensive toys, under the claim that I could rebuild them and

make them “work better.” I would especially like to thank my wife for believing in me and for her loving support during all the holiday-weekend experiments, the midnight paper revisions, and the

restaurant-napkin scribbling.

I would like to thank my advisors, Karen Daniels and Robert Riehn, not just for all their guidance

on thehowof doing experiments, but more importantly all their guidance on thewhysthat allow

you to reason and make decisions from your results. I am also deeply grateful for the many forms of mentoring they gave me, on teaching, writing, speaking, and others.

There are several faculty members in the department who graciously allowed me to use their

equipment, several technical and support staff members who went the extra mile to help me achieve a specific goal, several graduate students both in our groups and outside or groups who have made

contributions through our discussions. There are too many of these instances to list them all, which

is just a testament of the large degree of comradery that we enjoy in our Department.

I would like to end by thanking every single person who has or will read this thesis. I hope you

enjoy reading about the research as much as I enjoyed my time doing it.

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TABLE OF CONTENTS

LIST OF TABLES . . . .viii

LIST OF FIGURES. . . ix

Chapter 1 INTRODUCTION AND BACKGROUND . . . 1

1.1 What is Soft Matter? . . . 2

1.2 Motivating Questions . . . 5

1.3 Preview of Research . . . 6

1.4 Theories of Rigidity . . . 8

1.4.1 Glass Transition . . . 9

1.4.2 Maxwell’s Rigidity Criterion . . . 10

1.4.3 Critique of Mode Coupling Theory . . . 13

1.5 Hard Sphere Colloids . . . 14

1.5.1 Introduction to Colloids . . . 15

1.5.2 Single particle in a fluid . . . 16

1.5.3 Interparticle Interactions . . . 18

1.5.4 Bulk modulus from DLVO potential . . . 22

1.5.5 Steric stabilization: non-DLVO forces . . . 22

1.5.6 Our Interparticle Potential . . . 24

1.6 Phase Diagram of Hard Spheres . . . 28

1.6.1 Order in Hard Spheres and Disks . . . 30

1.6.2 Two-dimensional melting . . . 31

1.6.3 Quasi Two Dimensional . . . 35

1.6.4 Polydispersity suppresses crystal formation . . . 39

Chapter 2 METHODS AND INSTRUMENTATION . . . 41

2.1 General Considerations . . . 41

2.2 Device Assembly . . . 42

2.2.1 Device Layout . . . 43

2.2.2 Photolithography, Etching, and Sandblasting . . . 45

2.2.3 Silanization . . . 49

2.2.4 Bonding with PDMS coverglass . . . 52

2.2.5 Holder design . . . 55

2.2.6 Assembly Verification . . . 55

2.3 Characterization . . . 58

2.3.1 Sizing by DLS . . . 58

2.3.2 Zeta Potential by Zetasizer . . . 59

2.3.3 Density Matching . . . 60

2.4 List of Setups Used . . . 64

2.5 Imaging . . . 66

2.5.1 Acquiring Images with Charged Coupled Devices (CCDs) . . . 66

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2.5.3 Resolution . . . 68

2.5.4 Pixel size . . . 68

2.6 Image Analysis Methods . . . 72

2.6.1 Image Stitching . . . 72

2.6.2 Image Denoising . . . 72

2.6.3 Wiener Deconvolution . . . 73

2.6.4 Particle Tracking . . . 73

2.7 Hydrodynamics . . . 73

2.7.1 Hydrodynamics in microfluidic devices . . . 74

2.7.2 Flow in a Hele-Shaw Cell . . . 75

2.7.3 Equations of motion for hard sphere suspension in Hele-Shaw . . . 75

2.8 Fluid Stress on Microheap . . . 77

2.8.1 Calculation of Stress on the Heap . . . 80

2.8.2 Heap Permeability . . . 81

Chapter 3 ANGLE OF REPOSE. . . 84

3.1 Abstract . . . 84

3.2 Introduction . . . 85

3.3 Experimental Setup . . . 86

3.4 Results . . . 89

3.5 Discussion . . . 96

3.6 Acknowledgements . . . 99

Chapter 4 NONLINEAR ELASTICITY . . . .100

4.1 Abstract . . . 100

4.4 Results . . . 103

4.6 Acknowledgements . . . 111

Chapter 5 NON-AFFINE DEFORMATIONS. . . .113

5.1 Abstract . . . 113

5.4 Methods . . . 119

5.4.1 Large Timestep Affine Field . . . 119

5.4.2 Large Timestep Particle Tracking . . . 120

5.4.3 Small Timestep Particle Tracking . . . 120

5.5 Results . . . 124

5.5.1 Total Non-affine Response . . . 124

5.5.2 Dynamic Rearrangements . . . 130

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Chapter 6 CONCLUSIONS AND FUTURE WORK . . . .148

6.1 Summary . . . 148

6.2 Future Work . . . 150

BIBLIOGRAPHY . . . .153

APPENDIX . . . .182

Appendix A Static Microstructure . . . 183

A.1 Abstract . . . 183

A.2 Results . . . 184

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LIST OF TABLES

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LIST OF FIGURES

Figure 1.1 Overview of Apparatus . . . 7

Figure 1.2 DLVO Parameters . . . 21

Figure 1.3 Triton X-100 . . . 23

Figure 1.4 Interparticle Potential . . . 27

Figure 1.5 Hard Sphere Colloid Phase Diagram . . . 29

Figure 1.6 Hard-DiskPφDiagram . . . 33

Figure 1.7 RCP Ill Defined . . . 34

Figure 1.8 HS Structures Between Plates . . . 37

Figure 1.9 HS Phase Diagram Between Plates . . . 38

Figure 1.10 Polydispersity Phase Diagram . . . 40

Figure 2.1 Mask Layout . . . 43

Figure 2.2 Wafer Layout . . . 44

Figure 2.3 Wafer Processing . . . 45

Figure 2.4 Sandblasting . . . 48

Figure 2.5 FOTS . . . 50

Figure 2.6 Dewetting at Inlet due to Silanization . . . 50

Figure 2.7 Inlet Silanization Photolithography . . . 51

Figure 2.8 PDMS Coverglass . . . 53

Figure 2.9 Bonded Device . . . 54

Figure 2.10 Device Holder . . . 56

Figure 2.11 Wetting Characterization . . . 56

Figure 2.12 Profilometry of Silicon Microchannel . . . 57

Figure 2.13 Microsphere Size from DLS . . . 58

Figure 2.14 Microsphere Zeta Potential . . . 60

Figure 2.15 Density Matching Test . . . 61

Figure 2.16 Segregated Deposition . . . 63

Figure 2.17 Density Matching Effect on Segregated Deposition . . . 64

Figure 2.18 Pixel Size Calibration . . . 69

Figure 2.19 Barrier Size Characterization . . . 71

Figure 2.20 Velocity Profile Near Wall . . . 76

Figure 2.21 Streamlines Schematic . . . 78

Figure 2.22 Definition of Excluded Zone . . . 79

Figure 2.23 Simulated Fluid Flow Around Heap . . . 82

Figure 3.1 Microchip Geometry . . . 87

Figure 3.2 Density Fluctuations . . . 90

Figure 3.3 Time-Series of Heap Properties . . . 92

Figure 3.4 Steady-State Angle of Repose . . . 94

Figure 3.5 Staircase Experiment Time-Series . . . 95

Figure 4.1 Overview of Deformation Experiment . . . 101

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Figure 4.3 Stress-Strain Measurement Protocol . . . 106

Figure 4.4 Raw Stress-Strain Measurements . . . 107

Figure 4.5 Scaled Stress-strain Measurements . . . 108

Figure 5.1 Bidisperse Disordered Heap . . . 118

Figure 5.2 Particle Tracking Image Analysis Steps . . . 121

Figure 5.3 Particle Radius Identification . . . 122

Figure 5.4 Image Correlation Estimate of Affine Deformation Field . . . 126

Figure 5.5 Affine Field from Particle Tracks . . . 128

Figure 5.6 Full Deformations and Non-Affine Deformations . . . 129

Figure 5.7 Heap Center Before and After Compression . . . 131

Figure 5.8 Velocity of Compressed Front . . . 132

Figure 5.9 Particle Tracks and Packing Fraction . . . 135

Figure 5.10 Particle Trackingvs.PIV . . . 136

Figure 5.11 Affine Deformation Time-Series Analysis During Decompression . . . 138

Figure 5.12 Affine Deformation Time-Series Analysis During Compression . . . 139

Figure 5.13 Compression versus Decompression Dependence ofD_min2 (y,t). . . 140

Figure 5.14 Scaling ofD_min2 versus y and t . . . 142

Figure 5.15 Schematic of Compression . . . 146

Figure A.1 Radial Distribution Function forH=1.32d. . . 185

Figure A.2 Radial Distribution Function for Crystal inH=1.26d . . . 187

Figure A.3 Monodisperse Crystals . . . 190

Figure A.4 Schematic of Configurations forH =1.32d . . . 192

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CHAPTER

1 INTRODUCTION AND BACKGROUND

The study of the collective properties of many-body interacting matter continues[Col03]to be a “new physics” frontier, complementary to cosmology and particle physics, yet beautiful and

fundamental[And72; Gu09]unto itself. Remarkably, condensed matter systems with many internal

degrees of freedom have various phenomenological characteristics in common. Many systems have been formally classified into a small set of universality classes. However, in systems with

strong dissipation and non-ergodicity, we encounter great difficulties identifying the relevant order

parameters or even identifying all the relevant degrees of freedom. As long as these difficulties remain, the identification of universality classes in nonequilibrium systems will remain a major

unsolved problem. One approach to tackle these issues is to study a classically interacting system,

where tracking all the positions, velocities, orientations, and interparticle forces would in principle provide a full description of the internal degrees of freedom or hidden memory elements of the

material. While this complete description is possible in principle, it has yet to be realized in practice.

In this thesis, we study a soft condensed matter system out of equilibrium due to external driving, viscous dissipation, and the geometric inaccessibility of configuration transitions, all of which lead

to a breakdown of ergodicity. Single particles in our experimental system can be treated as a rigid body, so that the internal degrees of freedom of the single particle can be ignored. Except for van

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1.1. WHAT IS SOFT MATTER? CHAPTER 1. INTRODUCTION AND BACKGROUND

nature of the emergent rigidity of many-body systems by studying real-space positional degrees of freedom at discrete times.

Many-body quantum systems are considered by some to be more interesting than classical

systems due to their unusual properties (e.g. superconductivity, superfluidity). However, far more information is experimentally accessible in many-body classical systems, allowing for much richer

tests of much of the same underlying physics. In fact, detailed isomorphisms[Cha81]between

quan-tum many-body systems and classical many-body systems can be constructed. For example, Gennes [Gen72]constructed a correspondence between a Landau-Ginzburg superconductor and a smectic A liquid crystal. Protein conformations[Fra91; Ste85]have been studied using spin-glass[Ste04]

Hamiltonians. More relevant to this thesis, the classical statistical mechanics of point particles interacting via two-body Yukawa forces is isomorphic[Fro76]to a massive Thirring model[Thi58]

quantum field theory, itself isomorphic[Del98; Col75]to a Sine- Gordon model, which has soliton

solutions. Rather than simple speculative analogies, these constructions are meant to be a powerful framework to arrive at quantitative solutions of condensed matter systems, depending on which

one is easier to study, the quantum mechanical or the classical case. In our case, experiments on

the classical system are easier.

1.1 What is Soft Matter?

In this thesis, we study phase transitions in a type of soft matter: Brownian hard sphere suspensions. Soft condensed matter physics, the study of soft matter, is a subfield that has emerged on the

premise that gaining a more general understanding of disordered many-body systems could enable

large-scale engineering of complex nanostructures, sensors, and synthetic biological systems with broad societal impact. The classic studies that sparked interest in the field are studies of the glass

transition, polymer physics, and liquid crystals[Lar98; JW03; CL00]. There is no well-established

definition of soft matter, but there has been little disagreement for the past fifty years about the following two aspects of soft matter. First, the relevant length-scales are neither atomic (0.1-1nm)

nor macroscopic (≥1mm), and are sometimes referred to as a mesoscopic length-scale [Lar98; PA06].

Second, the response to external forces (rheological response) of soft matter differs from liquids and solids. This rheological indeterminacy characteristic of soft matter is clearly expressed in one of

the classic texts on rheology. “Deformation, Strain and Flow,” published on 1949 by Markus Reiner, says, “strictly defined rheological divisions belong to ideal abstract bodies and not to real materials.

If we say that stone is a liquid, every builder will laugh at us and the structural engineer will dismiss

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elastic, a property absent in a Newtonian liquid” (p. 158)[Rei49].

While there has long been general agreement that soft matter should be studied by condensed

matter physicists, there has been broad disagreement as to what to call the field, partially because it

continues to expand its reach year after year[JW03]. Even today, the name “soft condensed matter” is not yet universally accepted. In his 1991 lecture after being awarded a Nobel Prize in Physics,

“for discovering that methods developed for studying order phenomena in simple systems can be

generalized to more complex forms of matter, in particular to liquid crystals and polymers,” de Gennes[De 92]points out that “Americans prefer to call it[soft matter]complex fluids.” In Web of

Science, the topic “complex fluids” returns more than 40,000 items, versus fewer than 6,000 items

for “soft matter.” We can also compare two classic texts in the field that use different terms: “The structure and rheology of complex fluids,” which prefers “complex fluids,” was published in 1999 by

Ronald Larson, and has 2531 citations (195 citations/year), which is more citations per year than the 2642 citations (155 citations/year) of “Principles of Condensed Matter Physics”, which prefers “soft matter”, and was published in 1995 by Paul Chaikin and Tom Lubensky. Wilson Poon bypasses the

name question and simply declares, “Soft condensed matter physics is concerned with the study of

complex fluids”[PA06]. It would not be surprising if the field takes on a different name years from now, but it is clear that the scientific goal to extend our understanding of condensed matter beyond

crystalline solids will remain central.

Part of the urge to pursue this field comes from observing most real materials fall under this

category. The slow relaxation times and the bulk solid-like responses of soft matter can be observed in

materials of such mundane character as toothpaste, mayonnaise, and shaving cream. In fact, modern microscopy and fabrication techniques have allowed us to realize that the majority of apparently

homogeneous, rigid materials have disordered rather than crystallized structures. A non-exhaustive

list of examples of such soft materials includes colloidal suspensions, glasses, emulsions, foams, granular materials, pastes, liquid crystals, polymer gels, protein suspensions, and cells[Che10a].

These soft materials have captured academic and industrial interest of a highly interdisciplinary

character. For a historical evolution of soft condensed matter physics and a survey of its central themes, see [Wit99; Cat04; Noz12].

According to [JW03], Chaikin and Lubesky’s 1995 textbook was the first to attempt to

com-prehensively cover the subject. Perhaps for that reason, it is one of the few that tackles head-on the apparent paradox of how soft condensed matter can test concepts from traditional or “hard”

condensed matter, despite the fact that often soft matter interparticle interactions are classical

rather than quantum mechanical. Chaikin and Lubensky argue that condensed matter physics deals in general with understanding the macroscopic properties of interacting many-body

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fundamental interactions between the constituents. Hard condensed matter physics studies the quantum properties of solids made up of atoms and molecules. Conversely, “soft” condensed matter

physics concerns the study of solids where the constituents can be described to interact classically

such that “ħh =0”[CL00]. While not all soft condensed matter systems interact classically at the microscopic level (e.g., liquid crystals), Chaikin and Lubensky argue that nothing fundamental is

lost in the study of many soft condensed matter systems such as granular materials, emulsions,

foams, non-Brownian suspensions, where the interactions can in fact be written classically. The following paragraph is a summarized presentation of their argument.

In general, the study of soft matter allows unique tests and extensions of the unifying concepts

of condensed matter physics, which is desirable because these unifying concepts have had broad impact in other areas of physics such as particle physics and cosmology. One such concept that is

not lost, arguably the most powerful unifying idea from the study of condensed matter, is that

macro-scopic properties are governed by conservation laws and broken symmetries. The key reason is that conservation laws and broken symmetries are equally important in classical and quantum systems.

Thus, the physical consequences of the underlying conservation laws and broken symmetries to

the macroscopic properties are largely independent of whether the underlying particle dynamics are classical or quantum mechanical, as long as one uses the appropriate language to express the

interactions and the appropriate averaging process to connect to the macroscopic properties. The point is that studying classical rather than quantum systems does not prevent one from learning

general truths about phases of matter and their transitions. Crucially, by focusing on materials other

than crystalline solids, soft condensed matter physics accesses a broader set of critical phenomena and macroscopic properties. This rich, novel set of phase transitions can be readily accessed

ex-perimentally using setups that are comparatively simpler than the vacuum chambers needed to

make plasmas or low temperatures needed to make Bose Einstein condensates. One of the main justifications for studying these new types of phase transitions is that by understanding the nature

of the phase transitions, we may expand the set of condensed matter physics concepts that we

consider fundamental[CL00].

Researchers have long realized the importance, relevance, and advantages of studying soft

matter, but the field has not been subject to intense study until this century, likely due to its high

complexity. This high complexity has become amenable to study due to technological advances and due to the theoretical successes of condensed matter systems in crystalline systems.Novel issues

tackled in the study of soft condensed matter typically include disordered structures, fluctuation

spectra with long tails, and complex structural dynamics. Much of these non-trivial properties of disordered matter emerge from collective phenomena coupling multiple length-scales and

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1.2. MOTIVATING QUESTIONS CHAPTER 1. INTRODUCTION AND BACKGROUND

has been a central question that has unified the eclectic and interdisciplinary approaches to study soft matter[DD01]. The systematic application of conservation laws and identification of broken

symmetries has had a broad impact across physics. Similarly, a fundamental understanding of how

global properties emerge in systems composed of subunits collectively interacting via simple rules and collective effects could also have a broad impact across physics. For example, it could advance

our understanding of the intricate connection[Bah10]between structure and function in biological

nano- machines.

1.2 Motivating Questions

One of the central themes in the study of soft condensed matter revolves around understanding the nature of the rigidity transition at these intermediate scales. There are many methods and concepts

used to study soft matter, but the foundational studies upon which to build further studies can be

grouped into three conceptual lines of inquiry. First, (i) how does the criterion of rigidity vary as a function of temperature, density and applied stress? Second, (ii) how is the long-range order or

disorder of the microstructure reflected in both translational and orientational order parameters?

Third, (iii) are there features in the dynamics of the structure that reveal details about the nature of the transition? These three lines of inquiry are the ones that we follow in this thesis. Chapter 3 is

devoted to the first question by studying an indirect measure of rigidity as a function of applied stress.

Chapter 4 is also devoted to the first question by studying a direct measure of rigidity as a function of applied stress. In Chapter 4, we also tackle the second question by studying the microstructure

directly. In Chapter 5, we study the last question by studying nonlinear behavior in the local structural

dynamics. There are further possible conceptual lines of inquiry possible that would build upon these studies by testing the existing understanding: for example, detection and prediction of heap

failures such as cracks and avalanches, increasing the complexity of the single-particle by changing

one of its static properties such as shape, dielectric polarizability, or magnetic moment, or the simultaneous formation and interactions between multiple microsphere heaps.

We study the questions (i-iii) in a Brownian system with nearly-hard sphere interparticle

poten-tial, under carefully controlled conditions. The aim of these questions is to probe the way the system explores phase space to understand the origin of the non-ergodic nature (the particles cease to

explore phase space uniformly) of the behavior at the level of the microstructure. A detailed under-standing of the origin of non-ergodicity is lacking, but it is known to exist in systems with dissipation,

confinement, and external driving. In this thesis, our Brownian suspension is non-ergodic due to all

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1.3. PREVIEW OF RESEARCH CHAPTER 1. INTRODUCTION AND BACKGROUND

applied pressure from a compressed air regulator. Information about this phase space exploration can be obtained by directly tracking the microstructural rearrangements in real space[BM97], by

making global measurements of the bulk[CW77]and shear modulus[Wil02], or by understanding

the quasi-localization of the vibrational eigenmodes and the deviations from crystallinity apparent in the vibrational density of states[Wya05; O’H03; Che10b].

1.3 Preview of Research

We seek a basic understanding of the physical properties of a wide class of soft materials including

cells, proteins, emulsions (e.g. mayonnaise), and colloidal suspensions (e.g. inks). These

technolog-ically important materials share a capacity of exhibiting both solid-like and liquid-like behavior. Despite a long history of prior studies, we lack a fundamental understanding to predict even basic

aspects of their behavior under external driving.

We perform experiments on hard sphere Brownian suspensions of submicron particles, an idealized soft matter system that lacks rigidity in the absence of external driving, see Fig. 1.1. A basic

description of our apparatus is as follows. A dilute suspension of hard spheres is pumped into a

shallow microchannel that confines the particle phase space to quasi-2D motion (see Fig. 1.1a). All our experiments use submicron particle sizes, and a large portion are focused on 530 nm particles.

For this size, single particles have a diffusion constant of 0.55µm2/s, and the single particle elastic modulus is∼4 GPa. We use a compressed air regulator to pump the suspension (see Fig. 1.1b), which allows for fine pressure control because it uses a piezoelectric crystal as its actuating element.

Example heaps of Brownian microspheres under flow are shown in Fig. 1.1c.

The overarching question driving my dissertation is "What is the nature of the rigidity of these heaps?" Does the rigidity of the heaps follow from static mechanical stability driven by a sufficiently

interconnected network of particle contacts? Or, does the rigidity of the heaps follow from a kinetic

glass transition characterized by a diverging resistance to flow such that the time necessary to observe rearrangements grows prohibitively large? We cannot address these questions in the most

general sense, but we address limited forms of these questions. In the remainder of Chapter 1, we

introduce relevant theories of rigidity, we describe how to experimentally tune colloidal systems to a hard sphere regime, and we review the phase diagram of hard spheres and hard disks. In Chapter

2, we describe the most relevant details of our experimental, analytical and computational methods. These methods have allowed us to overcome the limitations of previous experimental approaches.

Namely, the rigidity of our heaps does not emerge from the effects of gravity, inertia, static friction,

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1.3. PREVIEW OF RESEARCH CHAPTER 1. INTRODUCTION AND BACKGROUND

geometry

w≈ 100 d

W≈ 2 w

h≈ 1.5 d H≈ 1.8 d

(a)

100 µm

y

x

ξ

(b)

(a)

flow

v∞ = 33 μm/s

1.0mm

compressed air

regulator

atmosphere

hard sphere polystyrene

d = 0.530 μm D = 0.55 μm2_/s

K ~ 4 GPa

(b)

(c)

Figure 1.1Overview of Apparatus

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1.4. THEORIES OF RIGIDITY CHAPTER 1. INTRODUCTION AND BACKGROUND

conditions imply that the onset of rigidity involves many-body collective interactions. In Chapter 3, we study the steady-state size, angle of repose, and internal fluctuations of the heap versus the

im-posed flow conditions. In Chapter 4, we measure the response of the heap to external perturbations,

allowing us to quantify their elastic modulus. In Chapter 5, we study the microstructure of the heap and track the local particle displacements during deformations, allowing us to separate the affine

and non-affine components of the deformation. We expect that insights resulting from this thesis

into the mechanism of rigidity of our system are applicable to a wide class of soft matter systems.

1.4 Theories of Rigidity

There appear to be two generic, distinct mechanisms by which disordered, rigid matter emerges. One is aglass transitionof atomic-scale objects as a function of temperature, density, and pressure, where

a rigid phase emerges as a metastable state characterized by dynamics with extremely long

time-scales relative to human experimental time-time-scales (e.g., plastics, melt-glass). Another is amechanical transitionto static stability of macroscale objects as a function of shear stress, pressure, and geometry

of interactions (e.g., bridges, sandpiles). The relevance of collective effects to our understanding of

these two generic transition mechanisms of soft-matter is demonstrated by studying the simplest case, the hard sphere interparticle potential. With a hard potential, the existence of a rigid, disordered

phase is not driven by a free-energy difference, unlike crystal nucleation. Both experiments and

simulations of hard spheres find rich dynamics and bulk responses characteristic of glassy states and mechanically stable states, allowing the hard sphere system to become the ideal generic system

for studying the physics of the glass transition and the mechanical stability transition.

These two generic paths to rigidity provide complimentary pictures for how to relate the mi-crostructure to the macroscopic dynamics and it has even been proposed that they might be

expressions of the same underlying transition–the jamming transition. Here, as used by Liu and

Nagel[LN10], van Hecke[Hec10], and O’Hern[O’H03], “jamming” refers to the generalized rigidity transition, be it glassy or mechanical. It is of historical value that the first instance of the term

“jamming” in the literature in relation to the glass transition comes from Gibbs[Gor76], who refers

to the glassy state as “the nonequilibrium, jammed state.” The mechanical transition as we have dis-cussed it refers to the athermal rigidity transition versus density and applied stress. The zero-stress

limit of the critical point of the mechanical transition is known as Point J, a point that has been well-studied due to its theoretical and numerical accessibility and the generality of arguments made

about it. While the unified “jamming" proposal has gained traction, we take the neutral position of

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reasonable, just like as one studies electricity and magnetism it becomes clear that their relationship is so intertwined that pursuing a unified treatment is warranted. Nevertheless, theoretical[Mar09]

and experimental[Lec08]objections to unifying the transition do exist. Indeed, while increasing the

packing fraction, the fraction of volume occupied by spheres relative to the total volume available to the structure, in vibrated granular system[Cou13], two distinct transitions are observed. One is a

structural transition occurring atφSrelated to the number of interparticle contacts, and the other is a dynamical transition occurring atφDrelated to the number of rearranging particles. However, as the vibrational excitations driving the granular material are reduced,φSandφDconverge to a single value that may be the generalized Point J[Cou13].

1.4.1 Glass Transition

Temperature, density, and pressure are control variables of the glass transition of thermal hard

sphere systems. It has been known for millennia[MM04]that glass can be formed by fast cooling of

a liquid. Alexandrov showed experimentally that the glass transition temperature of a polymer, TG, is controlled by the number density of the molecular constituents, independent of the molecular

weight and degree of polymerization of the polymer[AL44]Gibbs & DiMarzio[GD58]. Alexandrov’s

results provided early experimental evidence for using free volume as a central concept in the glass transition. More modern treatments indicate that the polymer’s molecular weight has an effect on

the glass transition temperature, but retain the importance of the free volume, as summarized by the empirical Flory-Fox equation[FF50]:

TG=TG,∞− V_F

M (1.1)

whereVFis an empirical measure of free volume andMis the molecular weight of the polymer. Gibbs & DiMarzio[GD58]established theoretically that the density dependence should be understood in

terms of the free volume in the system. Understanding that free volume is an important physical

variable controlling the transition, Gibbs realized that deeper insights could be obtained by studying hard sphere systems where the dynamics are dictated primarily by free-volume. In hard sphere

simulations, Gibbs et al.[Gor76]demonstrated glassy dynamics under a slow compression in the limit

of finite but arbitrarily smallT/P. A non-trivial insight in terms of how these variables are related in the hard sphere case has been achieved by showing that temperature and pressure equally affect the

rate of slowing down the dynamics. Nagel et al.[Xu09]showed that the dimensionless relaxation time τ/p

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type of hard potential used: harmonic, Hertzian, Weeks-Chandler-Andersen, or true hard-sphere potential, which gives the result broad applicability to our real colloidal hard sphere system.

Despite many advances in our understanding of the glass transition reviewed by Hunter &

Weeks[HW12], our understanding of the glass transition is far from complete. Current[Mar11] theoretical approaches remain at odds about the molecular processes by which liquids acquire

amorphous rigidity upon cooling. Also, aside from temperature, density and pressure, there may

be other relevant control variables. For example, classical descriptions of glassy systems using T,ρ, and P achieve quantitative accuracy in certain systems, particularly colloidal glasses, but

molecular, electronic, and magnetic system can form glassy states where quantum mechanical

effects significantly inhibit or enhance glassy dynamics[Mar11].

External stress (shear and pressure) and density are the control variables of the mechanical

transition of athermal hard sphere systems. Information about this transition can be obtained by the

glass transition of thermal hard spheres if one drives the thermal system to the limitT/P →0[Xu09]. The terminology is that at this limit one arrives at the zero temperature jamming transition of

frictionless spheres with finite ranged repulsions[Xu09]. As an example of this limit, da Cruz et

al.[Cru05]simulate hard spheres at zero temperature and show that mechanical stability depends on ratio of shear stress to normal stress. Their simulations are relevant because they reproduce

experimental observations dating back to Dantu[Dan57]of heterogeneous force distributions along chain-like paths, also known as force chains. In addition, they observe quasi-static phases with a

macroscopic angle of friction of≈6.6◦despite having put in frictionless interparticle interactions.

1.4.2 Maxwell’s Rigidity Criterion

Maxwell[Max64]provided a microscopic explanation where a sufficiently large number of contacts leads to mechanical stability, leading to the modern understanding that friction is not necessary to

form rigid packs. Here, pack refers to a generic assembly of particles without regard for the specific

geometry. Microsphere heaps are a type of thermal pack of spherical particles with a specific bulk geometry. The seminal studies of this rigidity criterion were made in granular materials, which are

athermal packs of frictionally interacting particles[Meh07]. One measure of mechanical stability

in granular systems is their angle of repose, which is the angle that a sandpile makes with the horizontal[Meh07]. Euler[Eul65]was the first to write a condition relating the angle of repose,θ, to µ, the coefficient of static friction. But heaps of frictionless particles are not solid blocks and their macroscopic angle of repose would imply static friction coefficients that are nearly zero, much less

than the bulk static friction coefficient of the polymer that makes up the particles (polystyrene).

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repose just due to the interlocking of grains into a stable structure. Coulomb’s insight was formalized by Maxwell in the argument that follows.

By counting the number of constraints relative to the number of degrees of freedom, Maxwell

showed[Max64]that rigidity emerges at a sharply defined (isostatic) value of the average number of contacts per particle, ziso. GivenN particles ind dimensions, there areN d Newton’s equations of motion or degrees of freedom. Givenz average contacts per particle, there areN z/2 constraints (because a contact shared by two particles isoneconstraint). Rigidity develops if the number of constraints is greater than or equal to the number of equations, meaning ifN z/2≥N d⇒z≥2d. This inequality is Maxwell’s original criterion. Isostaticity, a sharply defined onset of rigidity, develops

because the particles considered are hard spheres, so there is a second inequality to satisfy the non-deformation of particle surfaces. For each particle, hard surfaces mean there cannot be more

than two contacts per degree of freedom; symbolically,z/d≤2. Together, these two inequalities imply that ziso=2d. Note that torques are not considered because we are considering frictionless spherical particles, such that contact forces do not apply a torque and spheres are free to rotate

without disrupting the rigidity of the structure.

One way to inspect whether the previous discussion of isostatic frictionless spheres is relevant to our experimental system of colloidal hard spheres is to consider a dimensionless number that weighs

the importance of particle deformations. Moukazel showed[Mou98; Mou02]that contact networks in packs of slightly polydisperse particlesmustbe isostatic when the particle stiffness is large. One

way to quantify this is by using a dimensionless number that measures the ratio of the elastic stress

necessary to deform particles up to strains on the order of the size variation between particles to the total stress supplied by the applied force. Such a dimensionless number is the isostaticity parameter

I =kδR/fL, which measures if packs are isostatic if the number is much greater than one, wherek is the single particle spring constant,δR is the polydispersity, andfLis the load force. One way to compute the isostaticity parameter starting from dimensional units is to measurek in units of the material’s bulk modulus, which directs one to measureδRas a strain-like quantity in dimensionless units of the size variation between particles relative to the mean particle size, and to measure the applied load force in pressure-like units of force/area. For us, the material’s bulk modulus is on

the order of a few GPa, the polydispersity measured as the width of the particle size distribution

divided by its mean is no greater than six percent, and the applied stress on the entire structure is on the order of 100 Pa, such that no single particle feels a force that exceeds that stress. Putting these

together, we get a lower bound on our isostatic number, which is well in the isostatic limit (I >>1):

I ≈(1 GPa) _0.06

100 Pa

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Isostatic rigid states are not the only rigid states possible. Rigid hypostatic packs withz <ziso can be readily accessed with frictional particles, and hyperstatic packs withz>z_isocan be readily accessed with soft particles. The degree to which frictional particles and ellipsoidal particles affect

the transition is reviewed by van Hecke[Hec10], but notably the existence of a specific minimum contact number remains, although the ability to predict that contact number by counting arguments

is less exact. For hyperstatic packs, simulations find a square root scaling of excess contact number

versus excess packing fraction that is independent of dimension[O’H03].

The rigidity of a solid-like pack is strongly influenced by the existence of boundaries. Here,

pack refers to a generic assembly of particles without regard for the specific geometry. In

partic-ular, because our microsphere heaps are an experimental system, they must be finite-sized and consequently must have an edge. The existence of this edge has repercussions because particles

along the edge are more prone to rearrangement due to having fewer neighbors than particles in

the interior. The interior must therefore have more average constraints per particle than an infinite system to achieve isostatic mechanical stability[Wya05]. This “cutting argument” by [Wya05]allows

one to define a lengthscale`∗below which the pack behaves as an isostatic solid and above which it

behaves as an elastic solid. The lengthscale`∗is related to the number of extra contacts∆z in the heap’s interior by[Wya05]:

`∗_∝ 1

∆z (1.2)

Applying the argument to our heap geometry leads to an estimate that`∗∝1/sinθ. It also leads

directly to two questions: what determines the elastic properties of the solid above`∗? And, what determines the local value of the number of contacts and of`∗? Due to the combination of thermal

energy, short-range lubrication forces, and steric effects due to a surfactant coating on the particles,

the surfaces of particles do not normally touch and the average number of contactsz per particle in the interior of the heap does not carry the same meaning as in macroscopic systems. Instead, here

z refers to the average number of interacting neighbors per particle, such that these interactions are valid time-averaged mechanical constraints. The lengthscale`∗and the number of extra contacts can be spatially heterogeneous, given that they satisfy the following local criterion for rigidity[BW06;

Wya05]:

∆z≥C₁p

p (1.3)

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as you increase the distance from the barrier. This hypothesis of a depth-dependent importance of cooperative motions motivates our studies of the local deformations in the heap at different

y-positions, which we pursue in Chapter 5.

1.4.3 Critique of Mode Coupling Theory

An old adage[Cha]states that the number of theories of the glass transition exceeds the number of glass transition theorists. Interested readers may refer to the following reviews of seminal

thermody-namic theories[Gor76], seminal hydrodynamic, spin, and tiling theories[Fre88]and more recent mean-field, kinetically constrained, and out-of-equilibrium theories[BB11].

Mode-coupling theory is arguably the most successful theory of the glass transition[LN01; ST05].

We will not use mode-coupling theory to analyze our experiments, but as the most successful attempt at a microscopic theory of kinetic arrest of hard spheres, it merits attention here. Kob

& Schilling[KS91]review the basic results of MCT including its approximations and errors that

indicated MCT might not be a mean field theory after all. For a detailed derivation of MCT, see Reichman & Charbonneau[RC05]. MCT is derived by finding the Liouville equation of motion of the

dynamical structure factor by using the projection operator technique, then finding an ergodic to

non-ergodic transition. There are few ways to make adjustments to the theory and there are no fitting parameters; however, extensions to the theory are often introduced to improve agreement with

experiments by introducingad hoccouplings to currents (kinetic energy terms)[RC05]. Brownian

hard sphere systems, such as the one of interest in this thesis, are particularly challenging for MCTs because the currents that are typically introduced simply do not exist[Sza03; RC05]. Thus, there is

no path to make progress beyond simple MCT, which underestimates[Sza03]the critical packing

fraction for rigidity by 10% in 3D. This underestimating is a serious failure because it amounts to predicting kinetic arrest to a non-ergodic phase at conditions where the system is still[RC05]ergodic

and liquid.

The most striking success of MCT is the experimental verification[EB02; Pha02; Bar02]of its non-trivial prediction of a re-entrant glass/gel/glass transition in hard sphere systems with a short-range

attractive potential.

The original MCT has been formally recast as a mean-field theory: random first order transition theory[LW07]. While this development has greatly increased its range of applicability, it has also

allowed us to understand more clearly its limitations. The mean-field approach is to reduce a many-body problem to an effective one-many-body problem by replacing a fluctuating local order parameter

with a spatially uniform coarse-grained order parameter[CL00]. To obtain sensible results from

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1.5. HARD SPHERE COLLOIDS CHAPTER 1. INTRODUCTION AND BACKGROUND

compared with the value of the order parameter itself. The physical intuition is that MFTs become more accurate in a many-body system in which each body is significantly coupled with many others.

Increasing dimensions allows more neighbors to interact, such that in the limit of infinitely large

dimensions, MFTs should become exact.

For any given MFT, there is an upper critical dimension above which it gives sensible results,

and below which it is grossly inaccurate. For instance, the mean field Ising models with nearest

neighbor interactions and a fixed external field predicts a finite temperature phase transition in 1D, but Ising showed analytically[Isi25]that there is none. The Ginzburg criterion[Gin60; Ami74]is

the seminal method of finding these upper critical dimensions. Landau mean field theory for the

Ising model has an upper critical dimension of 4. Also, the Random Field Ising Model with nearest-neighbor interaction spontaneously magnetizes[LP13]for dimensions greater than or equal to 3D,

by following the Imry-Ma argument[IM75]. Following this classic approach, Biroli & Bouchaud [BB04]showed that MCT has an upper critical dimension of 6, using the Ginzburg criterion.

However, MCT has failed a number of recent numerical simulations[BT10; IM10; SS10; Cha11],

casting doubt on whether an upper critical dimension exists for MCT at all, since inconsistencies

appear for dimensions greater than 4 and the errors gets worse with increasing dimension. These results suggest that the successes of MCT when compared with experiments are either a coincidence

at low dimensions or an artifact of the custom coupling schemes created for specific systems. Nevertheless, MCT makes predictions of the bulk response for 2D disordered packs that agree

qualitatively well with simulations[WH11]. While there is reasonable qualitative agreement between

experimental rheometer stress-strain curves and MCT flowing glass predictions[Ama13], the de-viations emphasize the importance of experimentally-observed features that are not modeled by

MCT: the contribution from local rearrangements to slow structural relaxation and the ability of

slow structural relaxations to melt to solid state[Ama13]. Because MCT tests have been made mostly with simulations and because MCT agrees better with weakly coupled supercooled liquids than

Brownian hard spheres deep in the glassy phase, it isinterestingto pursue experimental studies well

into the rigid regime as we do in this thesis, not for the purpose of testing MCT, but for the purpose of providing the basis for the next generation of theories.

1.5 Hard Sphere Colloids

Colloidal systems are dispersions of a type of phase of matter into another. The careful study of

colloidal systems has yielded profound insights. Einstein was inspired to estimate of the size of the

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experimental determination of Avogadro’s number, and earning Einstein the 1926 Nobel Prize in Physics. Casimir’s original calculation[Cas48; CP48]of the Casimir effect was inspired by an

observation of a discrepancy[VO48]between the scaling of the attractive potential in colloidal

systems and the scaling of Hamaker’s non-retarded calculations of the van der Waals (vdW) force. It was also with “colloidal suspensions of hard particles under shear” in mind that Cates et al.[Cat98]

wrote the paper that, together with its commentary [LN98], arguably catalyzed the recent interest

in the jamming transition[LN10; Hec10].

Particles dispersed in a fluid are affected by multiple interparticle interactions: long-range

elec-trostatic and hydrodynamic forces, short-range van der Waals and surface forces, Brownian motion,

and excluded volume entropic forces. The interactions can be tuned by the appropriate choice of particle and fluid properties. Even the excluded volume can be tuned by using thermoresponsive

materials, such as PNIPAM microgel particles[Men12; Nor10; Zha09; Han08; Han08; Men12; Sol97; SR99; Mat09; Sno96; Kra00; Rom10], that swell and shrink in response to temperature changes.

Below we review the literature on how to tune a colloidal suspension to yield a nearly hard sphere

interaction. We begin with an introduction to colloids (Sec. 1.5.1). We review the single-particle

behavior in Sec. 1.5.2 prior to reviewing the particle-particle interactions. Out of the many single particle effects that we could address, we choose Brownian motion (Sec. 1.5.2.1) and sedimentation

(Sec. 1.5.2.2) due to their high importance. We then review passive methods for tuning the strength of interparticle interactions (Sec. 1.5.3) by modifying the dispersing fluid. These passive methods

must be informed by an understanding of the theory of van der Waals interaction (Sec. 1.5.3.1)

and the DLVO theory (Sec. 1.5.3.2). In Sec. 1.5.4, we point out that although the DLVO theory has been successful, it does not describe well the bulk modulus of suspensions. Last, we write down

an explicit argument for the specific form of the interaction potential for our experimental system,

which has some softness (Sec. 1.5.6).

1.5.1 Introduction to Colloids

Most general modern usage refers to colloids as a dispersion of one phase into another, without

regard for whether the phases are gas, liquid, or solid, so foams, aerosols, emulsions, gels,

suspen-sions are all types of colloidal dispersuspen-sions. The first usage of the termcolloids(from Gk. kolla “glue” +-oeides “form”) and the most widely-cited “start” of the field is the 1861 paper “Liquid Diffusion Applied to Analysis” by Thomas Graham[Gra61], who classified substances into crystalloids (sugar, salt, able to crystallize, able to diffuse through membrane) and colloids (not able to crystallize or

diffuse through membrane)[Rus92].

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are rules of thumb, not precise values. Particulate dispersions of polymeric microspheres are most frequently classified by size. Large particles, greater than 100µm, referred to as granular or

non-Brownian suspensions, which typically have a nearly-hard sphere interparticle potential. These

granular suspensions have primarily been used to study the rheological effects of the interstitial fluid on the properties of dry granular materials composed of the solid phase. Small particles, up to 1µm,

referred to as colloidal suspensions, typically have a soft potential and their behavior is strongly

influenced by Brownian motion. Because colloidal suspensions typically have a Leonnard-Jones interparticle potential, they allow the colloidal particle to be treated as a model atom[Poo04]and

the ability to model processes that are difficult to observe otherwise at the atomic scale, such as

crystallization[Sur06]and melting.

We study suspensions with particles of intermediate size, with diameters at the boundary

be-tween granular and colloidal suspensions, bebe-tween 0.5 and 1.0µm, with both the nearly hard sphere

potential and Brownian motion being important. Brownian suspensions are used to denote particle of size in the intermediate regime, to distinguish from the small and large particle suspensions. Here,

the nearly-hard sphere potential allows us to model the particle as a “grain,” yet the importance

of Brownian motion and various other features clearly puts the system in a class separate from granular matter. This is a good thing! It is precisely this distinction that makes both glass transition

arguments and mechanical stability arguments relevant. The following energy scales reflect the nearly-hard sphere character and the low importance of inertia, signaling a high importance of

geometric concepts and entropic concepts to determine the stability of a rigid phase:

repulsive force/Brownian force≈108 (1.4) repulsive force/viscous drag force≈105 (1.5) inertial force/viscous drag force≈10−12 (1.6)

1.5.2 Single particle in a fluid 1.5.2.1 Brownian Motion

In the absence of external driving, small particles in a fluid move randomly, a phenomenon known

as Brownian motion. The literature on Brownian motion is old and extensive, starting with the

discovery of Brownian motion by Robert Brown[Bro28]. The history of the subject and various well-established results are reviewed in Russel[Rus81]and Hänggi & Marchesoni[HM05]. Brownian

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given by the Stokes-Einstein-Sutherland[Ein05b; Sut05]relation:

D=kBT/6πηR (1.7)

whereT is the fluid temperature, typically room temperature,ηis the fluid viscosity,Ris the particle radius, andk_B is Boltzmann’s constant. Note, the relation is commonly referred to as the Stokes-Einstein relation, but Sutherland should also be credited for arriving at the relation independently

and submitting for publication earlier (Marchvs.May, 2005)[Wee07; SQ05]. The corresponding

fluctuation-dissipation theorem[Kub66]gives the relation between the mean squared displacement and this diffusion constant〈(x−x(0))2_〉₌_{2D t}_{. Thus, the viscosity of the fluid controls the diffusive} dynamics of the single particle.

1.5.2.2 Sedimentation

Under the influence of gravity, suspensions sediment over time. However, because of Brownian motion, the particles do not all lie at the bottom of the container. At equilibrium, the packing fraction φ=N V1/Vtotal(the fraction of available volume occupied by the particles), follows a Boltzmann-like distribution versus depth:

φ(z) =φ0(−z/z0) (1.8)

wherez₀is set by the ratio of gravitational to thermal energy:

z0=

kBT ∆ρ4

3πR3g

(1.9)

and where∆ρis the density difference between the particles and the fluid,Ris the particle radius, andg is the gravitational acceleration. At low Reynolds numbers, the rate of sedimentation is given by balancing gravity with the Stokes drag:

vsed= 2 9

∆ρR2_g

η (1.10)

For more details on the sedimentation distribution or sedimentation velocity, see Weeks[Wee07].

Note, both of the relations given here are valid only in the dilute limit, and observations deviate from these expressions for higher packing fractions. Even more drastic deviations occur in polydisperse

systems and non-spherically shaped particles, due to the prevalence of unusual velocity fluctuations

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1.5.3 Interparticle Interactions 1.5.3.1 Van der Waals force

The van der Waals (vdW) interaction contributes a large fraction of the energetic effects in colloidal

and biological systems[Lan74; Eli91]. The vdW force is the primary source of instability in colloidal

suspensions, thus minimizing the influence of the vdW force is the main challenge of creating suspensions with nearly hard sphere interactions. Misconceptions abound about the vdW force,

and the existing reviews do not review the issues at a level appropriate for this thesis, so we will review the vdW force in greater detail than any other interparticle interaction to make up for this

gap in the existing literature.

The existence of an attractive force between neutral atoms was first proposed by van der Waals in his PhD thesis[Waa73; Van88]. Originally, all types of dipole-dipole forces were referred as vdW

forces: orientational, inductional, and dispersive forces. But the most common usage today is to use

vdW force to refer only to the dispersion force. The traditional explanation of the origin of the vdW force is by the semiclassical London theory of dispersion forces, in which the force arises due to

the cross-correlation of fluctuating polarizations[EL30; Lon30; Lon37]. Using London theory as the

atomic vdW force (U ∝R16) and assuming pair-wise additivity, Hamaker obtained the vdW force

between spherical particles[Ham37]. Although significant deviations from Hamaker theory have

been reported for the value of the vdW force in aqueous suspensions[NP71; Smi73; TGR01], the

general tendency remains to use Hamaker theory (see section 1.5.3.2). Rather than use Hamaker theory as a predictive theory, the standard approach is to compute a Hamaker functional form for

specific geometries and to use an experimentally derived value for the Hamaker constant.

Because colloidal experiments showed the attractive interaction decays with distance faster than the London theory prediction used in DLVO theory, Overbeek[VO48]speculated that the discrepancy

was due to retardation effects at large separations due to the finite speed of light[CP48]. Inspired

by this[CP48], Casimir corrected for retardation effects in London theory to yield the expressions that are regarded as the Casimir force[Cas48; CP48]. Experiments by Tabor and Winterton

demon-strated the transition from non-retarded to retarded functional form at separations greater than

20 nm[TW68]. These results are the basis of the modern understanding of the vdW force as a type of Casimir force. One important prediction that has been confirmed experimentally[Mil96; Lee01;

Mun09; BS12]is the possibility of repulsive vdW forces between macroscopic objects depending on geometry and material properties. These repulsive vdW surfaces have been used to produce

ultra-low friction contact between colloidal particles and surfaces[Cap07; Fei08], which could have

further applications in novel sensors.

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as a depletion force of electromagnetic modes. This allowed Lifshitz to overcome the assumption of pair-wise additivity by packaging the material response in a frequency dependent dielectric

susceptibility. The dielectric susceptibility allows accounting for the many-body interactions specific

to the material. Lifshitz theory contains as limiting cases both London theory and the Casimir corrections. Hough[HW80]reviews in practical detail the methods needed to calculate a Hamaker

constant using Lifshitz theory and applies the methods to compute the Hamaker constant for a

polystyrene-water-polystyrene system, the material choices relevant to this thesis. He finds two values, 0.911 and 0.950×10−20J, depending on which reference he uses for the material constants.

For other material combinations, the Lifshitz theory of the Hamaker constant gives[Lar98]:

A=3 4kBT

" A−"B "A+"B

2 + 3

16 hν_e

p 2

n_A2₋_n B2 (nA2−nB2)3/2

(1.11)

where"i is the static dielectric constant of materiali (i =A or B),ni, is the index of refraction of materialiandνe is the UV absorption frequency of the materials (assumed to be identical). In the classical picture, this frequency represents the rate of spontaneous fluctuation of the dipoles, and it

typically has a value of 3×1015 Hz[Lar98]. The first term in Eq. 1.11 includes the zero-frequency orientation (Keesom) and induction (Debye) polarization terms, while the second term is the London dispersion contribution. See Chapter 6 of “Intermolecular and Surface Forces” by Israelachvili[Isr11]

for an accessible treatment of the Hamaker constant derivation in Lifshitz theory.

1.5.3.2 DLVO model

Derjaguin-Landau-Verwey-Overbeek[DL41; DL93; Ver47; VO48](DLVO) theory is the standard model of interparticle colloidal forces. DLVO theory computes the total interaction energyU_DLVO between two finite-size spheres in a fluid, by adding the vdW interactionUvdWand the screened electrostatic interactionUDLbetween the two spheres. Simplifying assumptions are used to calculate each term, which limit the generality of their result. The vdW interaction is computed by using the

Derjaguin approximation. The screened electrostatic interaction uses a mean-field approximation,

in the limit of low surface potentials. The main result of DLVO theory is that the form of the interaction potential between two spheres in a fluid is a Yukawa potential, with interaction length given by

the Debye-Huckel screening length 1/κ. Despite its simplicity, the theory has good experimental

support, with a broad range of validity. For a review of the historical development of the theory, see Wall[Wal10]. For a survey of experimental tests of the theory, see McLaughlin[McL89]. For a review

of applications of DLVO theory to unconventional geometries, see Hansen & Lowen[HL00].

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The screening occurs because the counterion distribution around each sphere is not uniform. In the dilute limit, the counterion distribution around a single-sphere is well-described as a double-layer.

As its name suggests, the structure of this equilibrium double layer has two layers. The inner layer,

known as the ““compact layer” or the “Helmholtz layer” or the “Stern layer”, includes low-mobility ions that are tightly bound or adsorbed to the surface and its width is of atomic dimension. The outer

layer, the “diffuse layer” or the “Gouy-Chapman layer” contains anions and cations (counterions)

distributed according to the Poisson-Boltzmann equation. The surface potential at the shear plane, the interface between the Stern layer and the Gouy-Chapman layer, is the experimentally accessible

surface potential known as theζ-potential.

The main result of the Guoy-Chapman-Stern[Hel53; Gou09; Cha13; DH23b; DH23a; Ste24; Lev90] theory of the diffuse layer is that, ignoring effects from overlapping diffuse layers, the screened

repulsive interaction between two spheres follows a Yukawa potential[Bus85]:

UR(h) =K e−κh

h+2σ, (1.12)

whereh=r−2σis the intersurface separation (surface-to-surface separation, see Fig. 1.2),r is the center-to-center distance,K =Z2_λ

B e

κσ

(1+κσ)2 =ζσ,Z is the bare surface charge,λB is the Bjerrum length of the medium,σis the sphere radius, andζis theζ-potential.

The vdW force expression that is standard for the DLVO theory is derived from Hamaker theory

and yields[Bus85]:

UvdW(x) =−A 12

₁ x2+

1

x2₋₁+2ln

1− 1 x2

, (1.13)

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h

𝜎𝜎

𝜅𝜅

−1

A

B

Figure 1.2DLVO Parameters

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1.5.4 Bulk modulus from DLVO potential

In this section we discuss how, unlike atomic crystals, theU(r)interaction potential from DLVO theory fails to universally account for the elastic properties of colloidal crystals, even in cases where

interparticle separation is on the order of the particle size and mechanical contact forces can be ignored[Wei98]. In colloidal crystals with long Debye lengths, the interparticle separation can be

tuned. For an interparticle separation of 6d, Weiss et al. find the interparticle coupling in the crystal

weak enough for the bulk modulus to be well-described byU(r)alone. However, reducing the interparticle separation from 6d to 4d, still long-ranged by typical colloidal standards, yields a

strong enough coupling to enable nonadditive interactions. These collective effects produce bulk moduli much weaker than expected from pairwise DLVO interactions alone. More recent elasticity

measurements[Rei07]have confirmed that non-central forces are necessary to explain the elastic

behavior of an fcc lattice of a charged suspension. Theoretical precedents[Dob03; Rus02; Hyn03] support the concept that the bulk modulus in these dense suspensions reflects more thanU(r)and includes many-body interactions. This failure motivates our exploration of the bulk modulus in

Chapter 4.

1.5.5 Steric stabilization: non-DLVO forces

On close approach between suspended particles, adsorbed polymer layers on particles repel each

other, preventing aggregation by vdW forces. Reports of steric[HP54]stabilization of colloidal

dispersions by an adsorbed polymer or surfactant layer date back to carbon black inks made by the ancient Egyptians circa 2500 B.C. [Rus92; Nap83]. The clearest experimental verification that vdW

forces could be overcome at short-range to produce a net-repulsive interaction in polymer-coated

surfaces goes back to Cain et al.[Cai78] [Van88]. This stabilization process is very important in a number of practical applications such as oil recovery, food industry, paints, etc[KL07b; Nap83].

Russel et al.[Rus92]and Napper[Nap83]provide a detailed historical review of the literature in polymeric stabilization and detailed experimental results on the structure and interactions between

adsorbed layers. The physical origin[Gra02]of this interaction is an entropic force, due to the

reduced configurational entropy of the polymer chains attached to the surface.

According to the DLVO model, particles in suspensions are electrostatically stabilized against

aggregation. However, even in the absence of external driving, a finite aggregation rate due to

thermal fluctuations is generally observed. This flaw is to be expected, since the DLVO model is a mean-field model without regard for large fluctuations in the surface charge of the dispersed

particles. The inclusion of non-DLVO steric stabilization forces makes the interparticle potential at