State Variables in Granular Materials an Investigation of Volume and Stress Fluctuations

State Variables in Granular Materials An Investigation of Volume and Stress Fluctuations

James G. Puckett Abstract

PUCKETT, JAMES GRAHAM. State Variables in Granular Materials: an Investigation of Volume and Stress Fluctuations. (Under the direction of Karen E. Daniels.)

This thesis is devoted to the investigation of granular materials near the transition between solid-like and fluid-like behavior. We aim to understand the collective dynamics in dense driven systems, the role of geometry in the volume fluctuations, and the equilibration of granular temperatures. The experiments are conducted using two-dimensional materials composed of a single layer of disks which are supported by a thin layer of air. In driven granular systems, particle dynamics have commonly been quantified by the diffusion, even though this measure discards information about collective particle motion known to be important in dense systems. We draw inspiration from fluid mixing, and utilize the braid entropy, which provides a direct topological measure of the entanglement of particle trajectories and has been used to quantify mixing. We find that as the density or pressure increases, the dynamics slow and the braiding factor exhibits intermittency signifying a loss of chaos in the trajectories on the experimental timescale. In the same system, we experimentally measure the local volume fraction distribution, which we find to be independent of the boundary condition and the inter-particle friction coefficient. We extend the granocentric model to account for randomness in particle separations, which are important in dynamic systems. This model is in quantitative agreement with experimentally-measured local volume fraction distributions, indicating that geometry plays a central role in determining the magnitude of local volume fluctuations. Finally, we test whether the zeroth law of several ensemble-based granular temperatures is satisfied by two granular systems in contact. We calculate the compactivity and angoricity which are the temperature-like quantities associated with the volume and stress ensembles; we observe the compactivity does not satisfy the zeroth law test, while the angoricity does equilibrate between the two systems. c Copyright 2012 by James Graham Puckett

by James Graham Puckett

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

2012

APPROVED BY:

Karen E. Daniels Michael Shearer Chair of Advisory Committee

T. Matthew Evans Christopher M. Roland Dedication

To my wife and son.

ii Biography

The author was born December 23, 1982, in Salisbury, NC. In 2005, he received his B.S. in Physics at North Carolina State University and returned to continue his studies in 2007.

iii Acknowledgements

There many people I would like to thank for helping make this dissertation possible:

Karen Daniels. I acknowledge my deep gratitude for your continuous and enthusiastic support during my research, for indispensable guidance and advice, and for creating an enjoyable and enriching environment to conduct my studies.

My committee: Michael Shearer, Christopher Roland, and Matt Evans: for providing useful suggestions, comments and your kind consideration.

My collaborators: Frédéric Lechenault and Jean-Luc Thiffeault, for scientiﬁc expertise, pa- tience and co-authorship for much of the work in this dissertation.

My fellow group members, Eli Owens, Carlos Ortiz, Stephen Strickland, and Lake Bookman: for sharing in the trials of graduate student life and making the best of it.

North Carolina State University, the Physics department. Thank you for educating and supporting me.

My family. To my parents: for being there from the beginning and every day since. To my mother-in-law: for your tireless efforts and support. Thank you.

My wife Xingyi. For your love, encouragement, and sacriﬁce. For the gift of our son, Noah, who inspires and amazes me.

The work has been supported by NSF DMR-0644743.

iv Table of Contents

List of Figures ...... vii

Chapter 1 Granular materials ...... 1 1.1 Introduction...... 1 1.2 Jamming...... 2 1.3 Dynamics near jamming ...... 4 1.4 Volume fraction...... 7 1.5 Statistical mechanics for granular systems...... 10 1.5.1 Granular temperatures ...... 10 1.5.2 Edwards ensemble ...... 10 1.5.3 Stress ensemble...... 12 1.5.4 Test of granular ensembles...... 13 1.6 Overview of experiments...... 14

Chapter 2 Trajectory entanglement in dense granular materials ...... 17 2.1 Abstract...... 17 2.2 Introduction...... 18 2.3 Experiment ...... 20 2.4 Results...... 22 2.4.1 Self-diffusion...... 22 2.4.2 Braid entropy...... 23 2.4.3 Comparison...... 31 2.5 Discussion & Conclusion...... 32 2.6 Acknowledgements...... 33

Chapter 3 Local origins of volume fraction ﬂuctuations in dense granular materials 34 3.1 Abstract...... 34 3.2 Introduction...... 35 3.3 Experiment ...... 37 3.4 Results...... 40 3.5 Model...... 42 3.6 Comparison...... 47 3.7 Discussion...... 51 3.8 Conclusion ...... 52 3.9 Acknowledgments...... 53

Chapter 4 Experimental methods ...... 54 4.1 Apparatus...... 56 4.2 Particle positions ...... 59

v 4.3 Contact forces...... 61 4.3.1 The solution to the stress ﬁeld in a disc due to z forces ...... 65 4.3.2 General Solution to z forces on a disc ...... 67 4.3.3 Finding contact forces on a disc by optimization...... 69

Chapter 5 Do temperature-like variables equilibrate in jammed granular subsystems? ...... 75 5.1 Abstract...... 75 5.2 Introduction...... 75 5.3 Experiment ...... 77 5.4 Results Volume ...... 80 5.5 Results stress ...... 81 5.6 Discussion...... 83

Chapter 6 Conclusion ...... 86 6.1 Summary of results...... 86 6.2 Future work and open questions ...... 88

Bibliography ...... 89

vi List of Figures

Figure 1.1 Sketch of a proposed phase diagram for granular materials with the inverse volume fraction 1/Φ, stress Σ and temperature T axes, based on one published in [LN98]...... 3

Figure 1.2 Liquid, glassy, solid and crystalline phases are shown as a function of the volume fraction Φ for two-dimension granular materials...... 4

Figure 1.3 Particle trajectories for a granular packing vibrated at a constant amplitude and frequency at (a) Φ = 0.567, (b) 0.701 and (c) 0.749, (from [RIS07])...... 5

Figure 1.4 Mean squared displacement of a vibrated granular material at various Φ, ( from [RIS07] )...... 6

Figure 1.5 (a) Local volume deﬁned with a (a) Voronoi tessellation and a (b) radical Voronoi tessellation for a bidisperse packing. The distance ξ from the particle center to the cell boundary is shown for the (c) Voronoi and (d) radical Voronoi tessellations...... 8

Figure 1.6 Fluctuations in φ as a function of the density (a) from [Ast06], (b) from [Bri+08] and (c) from [LD10]...... 10

Figure 1.7 The dependence of Φ on tapping amplitude for a vibrated granular material...... 11

Figure 1.8 An image of force-chains in a two-dimensional granular packing of photoelastic discs under isotropic compression...... 13

Figure 1.9 (a) Photograph of the apparatus used for investigating dynamic granular materials. The piston provides constant pressure via pulleys and weights, or constant volume by ﬁxing its position to the surface of the table. (b) Image of isochromatic fringes in an array of discs biaxially compressed...... 16

Figure 2.1 (a) Schematic and (b) photograph of apparatus. The piston provides constant pressure via pulleys and weights, or constant volume by ﬁxing its position to the surface of the table...... 19

vii Figure 2.2 (a) Entanglement of space-time trajectories for N = 20 particles, which are (b) projected on the x-axis with four crossings marked with symbols. The crossings and 4 are topologically “over” (left over right), and the crossings ♦ and 5 are topologically “under” (left under right). Note that the crossings 5 and 4 undo each other, while ♦ and do not. 21 Figure 2.3 Representative diffusion relations σ2(τ) for constant pressure (CP) experiments with µ2 and φ = 0.74 to 0.80. The dashed line represents a slope of unity. The diffusion coefﬁcient, D, is calculated between the two + symbols on each curve...... 23

Figure 2.4 The braiding factor logL(t) for the same experimental runs shown in Figure 2.3, with φ = 0.74 to 0.79. The inset shows logL(t) divided by its value at the maximum time, T...... 24

Figure 2.5 The Sbraid per particle, shown for µ2 and CP systems, with the axis of projection θ = 0. The arrow shows increasing φ of the system...... 26

Figure 2.6 A plot of logL(t) for two systems with µ2 where the axis of projection is rotated between θ = [0,π/2], where (a) is φ = 0.77,P = 12 mN/m and (b) is φ = 0.78,P = 18 mN/m...... 28

Figure 2.7 Measured values of D as a function of φ and P in (a) and (b), respectfully. Error bars denote the magnitude of the standard error. The black × on the axis signiﬁes systems for which D is poorly deﬁned for that particular φ or P...... 29

Figure 2.8 Measured values of Sbraid as a function of φ and P in (a) and (b), respectfully. Error bars denote the magnitude of the standard error. The black × on the axis signiﬁes systems for which Sbraid is poorly deﬁned for that particular φ or P...... 30

Figure 2.9 A logarithmic plot of Sbraid and D for each system, where the arrow denotes increasing φ...... 31

Figure 3.1 Photograph (left) and schematic (right) of the apparatus, showing confining wall in constant pressure (CP) configuration, with weights m suspended from a pulley via a mono-filament line. Constant volume (CV) is obtained by fixing the wall to the surface of the table...... 38

Figure 3.2 Equation of state φ(P˜) for CP experiments at µ1 (+), µ2 (◦) and µ3 (×) on linear axes and inset with semilog axes...... 39

viii Figure 3.3 Representative P(φ) for three experimental runs at φ ∼ 0.780: + for CP and µ2, • for CP and µ3, and × for CV and µ2. For the run at CP and µ3, dashed and dotted lines show PL,S(φ) for large and small particles, respectively. All P(φ), whether or not they distinguish large and small particles, are normalized by the total number of measurements, so that P(φ) = PL(φ) + PS(φ)...... 41 Figure 3.4 Mean φ and variance σ2 of P(φ) measured from individual Voronoï 4 cells. Each point is for a single experimental run with & 10 configurations. Inter-particle friction is denoted by shape (µ1, ◦; µ2, O; µ3, ) and boundary condition by open or filled markers (CP, open; CV, filled). The line is a least squares fit with φ-intercept at φ = 0.842 ± 0.002. . . 42

Figure 3.5 Schematic of the central particle and one neighbor with radii rc and r j, respectively. The shortest distance between the edges of the two particles is s j. The shaded region is the contribution V from this neighbor to the total radical Voronoï volume...... 43 expt Figure 3.6 P (s) (dotted) measured for two experimental runs at CV and µ3, with φ = 0.747 (thick) and φ = 0.792 (thin). For comparison, exponen- λ s 1 e−s/λ s tial distribution P ( ) = λ (dashed) with λ = ¯ and δ-function distribution Pδ (s) = δ(s − s¯) (solid)...... 44 Figure 3.7 Scaled mean (∆) and standard deviation (×) of s, for all boundary condition (CP and CV) and µ, as a function of the local volume fraction φ...... 45

Figure 3.8 (a,b,c) Measured P(φ) (dashed line) and calculated granocentric prediction Pg(φ) (◦). The granocentric prediction is calculated using (a,b) Pexpt(s) from Fig. 3.6 and (c) Psim(s) for a simulated packing with N = 104 particles shown in (d)...... 48

Figure 3.9 Color online. (a) Mean and variance of P(φ) for all experiments (, same data as Fig. 3.4), compared to granocentric predictions drawn from several different s distributions: the experimentally-measured dis- expt tribution P (s) (•), the s < rS portion of the experimentally-measured expt,cut s λ s 1 e−s/λ distribution P ( ) (+), exponential distribution P ( ) = λ δ (N) where λ = sexpt, and P (s) = δ(s−sexpt) (). The ﬁlled markers represent distribution in the experimentally observed range of 0.154 < δ λ sexpt/rS < 0.339. The open markers, ♦ and 4 are P (φ) and P (φ) λ for s < sexpt. (b) The mean number of neighbors versus φ for P (s) and Pδ (s), with the same markers as in (a)...... 50

ix Figure 4.1 Schematic of apparatus of the air table used to study static granular packings. The fan provides the necessary pressure and volume of air to ﬂow through the polypropylene sheet creating a nearly frictionless surface...... 55

Figure 4.2 Schematic of air-table apparatus and reflective photoelasticity. Light shines through a linear polarizer (P) then a (wavelength matched) quarter- wave plate (Q) before passing through the material. The light then reflects off a mirror and passes through a quarter-wave plate and polarizer before being observed. Particles float on a thin layer of air provided by the pressure difference across the polypropylene (PP) sheet...... 56

Figure 4.3 Schematic of apparatus showing the arrangement of the pistons. . . . . 57

Figure 4.4 Particle centers are found using the (a) white light image of the particles, where the (b) particle edges are detected with a Sobel ﬁlter. The circular Hough transform of the edge image is shown for (c) small particles and (d) large particles. Each red peak is the center of the particle. . 59

Figure 4.5 The edges of the particles detected by the Hough transform are outlined in black...... 60

Figure 4.6 An image of the force data for an array of photoelastic discs using re- ﬂection photoelasticity...... 61

Figure 4.7 (a) A image of the isochromatic stress on a disc with z = 4 contacts, and (b) a schematic of possible contact forces acting on the disc. . . . . 62

Figure 4.8 A disc (R = 0.0055 m, Fσ = 100), with two equal forces of F = 0.4N with αi = 0 at βi = 0, and π. On the left is the image of fringe number every where on the disc. On the right is the image of the isochromatic fringes. The fringe number at the center of the disc is exactly 1...... 63

Figure 4.9 Calibration of Fσ using the Nf ringe and F...... 64 Figure 4.10 A semi-inﬁnite plate of thickness t with a force, F, at point O...... 66

Figure 4.11 A disc with a force, F, at point O1 at a direction α...... 68 Figure 4.12 Comparison of observed image (left) and ﬁtted image (right) for a set of eight diametrically compressed disc with r = 0.0055, Fσ = 100 under various forces [N] shown in the lower right in red...... 72

x Figure 4.13 Comparison of observed image (left) and ﬁtted image (right) for an array of isotropically compressed discs. Outline of discs are shown in red...... 73

Figure 5.1 Schematic of apparatus showing (a) two walls bi-axially compressing an array of disk-shaped particles composed of an outer subsystem (black, high µ) and an inner subsystem (red, low µ) and (b) reflective photoelasticity on air-floated particles. Light shines from green LEDs through a linear polarizer (P) a wavelength-matched quarter wave plate (Q) before entering the photoelastic material. A mirrored surface on the bottom of each particle reflects light back through the particle. A second quarter-wave plate and linear polarizer are mounted on the camera to resolve the photoelasticity. Three images of each configuration are recorded: (c) unpolarized white light for locating particle positions, (d) polarized green light showing isochromatic fringes for calculating contact forces and (e) an ultraviolet light for identifying the low-µ particles. 79

Figure 5.2 (a) Volume histograms, P(V), for Φ = 0.776 (N), 0.784 (), and 0.802 (•) with m = 48. (b) A semi-logarithmic plot of the ratio each histogram with respect to the Φ = 0.784 distribution, i.e. Pi(V)/Pi=2(V). (c) The inverse compactivity given by Eqn. 5.3 plotted as a function of the inverse volume fraction where µB are shown as black • and µS are red . Large/small symbols denote jammed/un-jammed conﬁgurations, respectively. Errorbars shown are uncertainties in P(V) and propagated through the calculation. The inverse compactivity given by the FDT method (Eqn. 5.4), is shown with the solid line for comparison. (d) The ratio of number of jammed/un-jammed conﬁgurations recorded at each Φ...... 81

Figure 5.3 (a) Distribution of σp where m = 8 and Γ = 0.0007 (H), 0.0010 (), 0.0015 (•), and 0.0024 Nm (N). A semi-logarithmic plot of the (b) ratio Pi(σp)/P j(σp) where the reference system j is Γ = 0.0015 Nm. The pressure angoricity AP and shear angoricity Aτ are shown as a function of Γ where the results using overlapping histograms for µB and µS are shown as black ◦ and are red ♦, respectively. The solid line is the angoricity calculated using FDT. The gray dashed lines provide a visual reference of the slopes 0.15 and 0.45, respectively. Inset: The scaled 2 variance hδσpi of the P j(σp) distribution, as a function of the cluster size m...... 82

xi Figure 6.1 The volume fraction axis showing liquid, glassy, solid and crystalline phases. The range of Φ examined in experiments on dense driven system (T > 0) are shown with black hatching on top, while static T = 0 isotropically compressed systems are shown in red hatching on bottom. The static system becomes rigid in between Φ = 0.78 and 0.79, where driven systems remain liquid-like above this value...... 87

xii Chapter 1

Granular materials

Granular materials are collections of discrete particles massive enough so that thermal fluctuations are negligible. While, the behavior of individual particles is governed by classical mechanics, collections of particles exhibit complex and fascinating behavior. Granular materials are present in every day life (e.g. sand, rice, cereal) and in industry (e.g. coal and powders). These materials are not solids, though are composed of solids and can behave collectively as a solid like a sand-pile. Nor are these materials liquids, though they can flow like grain in a hopper. What determines whether a granular material behaves like a solid or a liquid? This thesis is an investigation of these two phases and a search for relevant state variables for granular materials. Both granular and molecular systems are collections of interacting particles. Like molecular systems, granular materials have reproducible statistical properties, e.g. sugar poured into a container is expected to fill up a certain volume. While several more similarities can be identified, it is more helpful to explain how granular and molecular systems differ. Granu- lar materials can be distinguished by three key properties: massive, discrete, and dissipative. Molecular systems are also collections of discrete, massive particles but molecular volumes and masses are on a much smaller scale which proves to have great importance in the physical properties of the system. The next section briefly describes a selection of properties for which important differences are identified between granular and molecular systems.

1.1 Introduction

In our experiments, a typical particle has a mass of several grams and diameter of several cen- −21 timeters. Each particle has a thermal energy Ethermal = kBT ≈ 10 J (at T = 300K), which is about ﬁfteen orders of magnitude smaller than the typical gravitational potential energy required to lift a particle by its own diameter and instead requires external forces to exhibit dynamics. Systems this strongly athermal are not driven by thermal ﬂuctuations. Individual grains interact dissipatively. Energy is lost through inelastic collisions and frictional interactions. Therefore, an external force must act on the system in order for a granular

1 system to flow and behave like a liquid. Otherwise, the system quickly comes to rest. Granular materials are collections of discrete macroscopic particles. In the dry granular systems discussed in this thesis, there are no attractive potentials between particles: no forces hold two particles together. In other words, the tensile strength for a dry granular material is zero. On the other hand, if two grains are pressed together, the grains experience a repulsive force. Grains do not overlap; the stress causes the particle to deform. The volume occupied by a particle is excluded from the rest of the particles of the system. The volume fraction, Φ, is the ratio the total volume of the particles and the system volume and by definition is bounded between zero and one. If the volume fraction is large enough, the system becomes rigid, behaving like a solid. Congested traffic, a confined crowd, or a suitcase are a few examples of this. Packing a suitcase with a few items leaves plenty of room for things to shift around; try to pack too many items, and you can be sure nothing is going to move. Granular materials offer challenges not only in industry but also for understanding natural and destructive phenomena. Avalanches, mudslides, and earthquakes costs millions of dollars and human lives. Understanding how granular materials fail is important in predicting these phenomena and mitigating damages.

1.2 Jamming

Under what conditions does a granular material flow or jam? In the suitcase example above, the volume fraction (how tightly packed) characterized the state of the items inside (fluid or solid). What other variables are important? Liu and Nagel proposed a jamming phase diagram for disordered materials like granular materials [LN98], much like in Figure 1.1. The axes are the inverse volume fraction (1/Φ), stress (Σb), and temperature, T, where the volume fraction is defined as

N ∑ Vi i Φ = =1 . (1.1) Vsystem where Vi is the volume of the ith particle. The shaded region near the origin, where Φ is large and Σ and T are small, represents the solid or jammed phase. The liquid or ﬂowing phase is outside the shaded region, where Φ is small and the applied stress or T is large. The axes in the jamming diagram mirror the familiar phase diagram for molecular systems where the axis 1/Φ

2 T

Σ̂

J 1/Φ

FIGURE 1.1. Sketch of a proposed phase diagram for granular materials with the inverse volume fraction 1/Φ, stress Σ and temperature T axes, based on one published in [LN98].

is like V and the axis Σb is like P. The T in the diagram the kinetic temperature which is relevant in colloidal systems and driven systems. Liu and Nagel [LN98] proposed that the transition to rigidity, or jamming, is universal for disordered systems. Disordered systems are not limited to granular materials but includes foams, emulsions, glasses and super-cooled liquids. In Figure 1.1, the point J suggests a critical value of Φ at which the system jams at zero temperature and pressure. Simulations and experiments provide strong evidence of a mixed ﬁrst and second order transition near point J [Hec10; LN10]. Below jamming, there is a disconti- nuity in the coordination number z, where below jamming z = 0 and above jamming z exhibits square root scaling with the distance to jamming ∆Φ = Φ − ΦJ [BW90; Dur95a; Dur95b; Mak+99; O’H+03]. Further, material properties like bulk and shear moduli show power law scaling with the distance to jamming [MW95; Lac+96; MGW97; Dur95a; Dur95b; O’H+02; O’H+03].

3 FIGURE 1.2. Liquid, glassy, solid and crystalline phases are shown as a function of the volume fraction Φ for two-dimension granular materials.

While experiments have shown deviations from the diagram exist, recently shear jamming [Zha+10; Bi+11], Figure 1.1 provides a visual for organizing thoughts on jamming. The work in this thesis explores both driven, dynamic systems (T > 0) and static jammed systems (T = 0). According to Figure 1.1, we have an expectation that driven systems will jam at higher Φ than un-driven systems. With external driving, the system remains dynamic and liquid-like at larger Φ than systems with no driving. In Chapter2, the behavior of individual particles is investigated using the diffusion and a topological measure called the braiding entropy. The external driving is held constant and the pressure or volume fraction is varied in the range of 0.71 < Φ < 0.81, as shown with black hatching in Figure 1.2. At T = 0, systems jam at lower Φ. In Chapter4, a narrow range of 0 .774 < Φ < 0.805, which spans ΦJ, is explored in an isotropically compressed system. Starting with an initially unjammed state, pistons quasi-statically compress the system, and the system jams at some value of Φ. Repeating this process many times, the system jams over a large range of 0.784 < Φ < 0.793. Simulations have shown that the width of the jamming transition (at Σb = 0 and T = 0) exhibits finite system size effects and is approximately 0.16N−1/2, independent of polydispersity and dimensionality [O’H+03]. The following sections introduce each topic of this thesis beginning with dynamics of particles near jamming, moving on to a geometric model for volume fluctuations and finally turning to granular ensembles and temperatures.

1.3 Dynamics near jamming

Due to the dissipative nature of granular materials, a bulk driving force is required in order to maintain any long-term dynamics. Grains pouring from a hopper are one example where

4 FIGURE 1.3. Particle trajectories for a granular packing vibrated at a constant amplitude and frequency at (a) Φ = 0.567, (b) 0.701 and (c) 0.749, (from [RIS07]).

gravity provides the driving force. Experimentally, individual grains can be identified in a granular material, and in turn the trajectory can be tracked for individual particles. Figure 1.3 shows trajectories taken at different Φ from a two-dimensional granular material vertically vibrated at a constant magnitude [RIS07]. In Figure 1.3, three systems are shown for a vibrated granular material at volume fractions Φ = 0.567, 0.701 and 0.749. Each figure shows the relative size of the particle and a single particle’s trajectory. At low Φ, the trajectory is clearly diffusive resembling that of a Brownian fluid. At high Φ, the trajectory is highly localized like a solid. However, there exists a range of Φ for which particles exhibit behavior as Figure 1.3b. An individual particle may spend a period of time localized, though after a long enough time, the particle makes a relatively large spatial jump. The type of behavior is commonly referred to as caged [Wee00; AD06], and is associated with glasses [DMB05; MD05]. To quantify the dynamics, a simple and conventional approach is to calculate the mean squared displacement, M(τ), as

2 2 M(τ) = h∆xi(t j + τ) + ∆yi(t j + τ) ii, j (1.2)

averaged is over all particles in the system i and over starting times t j for an ergodic system. M(τ) scales with the lag time as, M(τ) ∼ τα (1.3) where the exponent α is useful in characterizing the behavior of the system. At very short times, particles have not yet collided and behave ballistically giving α = 2. After a long time,

5 FIGURE 1.4. Mean squared displacement of a vibrated granular material at various Φ, ( from [RIS07] ). the trajectories of particles are diffusive giving α = 1. In dense granular materials, as shown in Figure 1.4 taken from [RIS07], α can be less than unity signifying particles are sub-diffusive. This behavior has also been observed in colloids which are thermal [Wee00]. As the volume fraction of the granular material increases, dynamics become heterogeneous [DMB05], the diffusion vanishes and viscosity diverges as a power law [Gre+08]. Par- ticles are localized spatially or caged for extended periods of time, see Figure 1.3. The cage weakens with time [WW02; MD05], and particles escape to different cage with a different set of neighbors in a string-like fashion [DMB05]. Waiting long enough, particles diffuse from cage to cage. The dynamics suggest that dense driven granular materials, colloids, and supercooled liquids or glasses have much in common.

6 Particle dynamics are commonly quantified using the diffusion coefficient. However, as Φ increases and the system approaches jamming, particles jump from cage to cage via cooperatively rearranging regions [CR05; Can+10] which induces spatial and temporal dynamical inhomogeneities [CD09; Dur+09]. The size of the cooperatively rearranging regions grows as the system approaches the jamming. Therefore, a multi-particle measure is needed when quantifying the dynamics. The four-point susceptibility χ4 is one such scalar which has been used to this end [GNSd00; DMB05; Key+07; Lec+08], yet χ4 does not quantify mixing. The topological braid entropy quantifies the degree of entanglement of particle trajectories and can be used to analyze the efficiency of mixing [BAS00; Thi05; GTF06; Thi10; AT12]. Chapter2 is based on [Puc+12], which quantifies the dynamics of particles in a dense driven granular material with both the diffusion coefficient and the braid entropy. The braid entropy is calculated from the braiding factor, a measure of the instantaneous entanglement of the particle trajectories. As the pressure on the system increases, the dynamics slow and the growth of the braiding factor becomes more intermittent, signifying a loss of chaos in the dynamics as the system becomes more glassy. This analysis introduces powerful tools to the field for identifying intermittent dynamics and quantifying the mixing.

1.4 Volume fraction

In the previous section, the volume fraction, Φ, was used to characterize the behavior of the material from ﬂuid (small Φ) to solid (large Φ). For a monodisperse packing of grains, the volume fraction has a well-known upper limit which depends strongly on the dimension. In two dimensions, a simple hexagonal lattice yields √ Φ = π/ 12 ≈ 0.91. In three dimensions, the densest packing, as conjectured by Kepler, is a √ HCP (or FCC) lattice where Φ = π/ 18 ≈ 0.74 [Hal05]. With a consistent protocol, one can generate a range of reproducible global volume fraction. Shaken, vibrated, or compressed, a monodisperse granular packing will achieve a minimum value of random loose packings have a volume fraction ΦRLP ≈ 0.6) [BM60; OL90] for random loose packings and a maximum of ΦRCP ≈ 0.64 [BM60; SK69; Fin70; Ber83; KL07; SWM08] for random close packings in three dimensions. In two dimensions, the values are higher with ΦRLP ≈ 0.775 [Mey+10] and ΦRCP ≈ 0.82 [Kau71; Ber83]. Though the value of Φ is repeatable with a given preparation protocol, the exact value of Φ is sensitive to the dimensionality, conﬁnement due to boundaries [DW09], and particle properties like: friction [SL03;

7 (a) (b)

r ξ r ξ A rB A rB

FIGURE 1.5. (a) Local volume deﬁned with a (a) Voronoi tessellation and a (b) radical Voronoi tessellation for a bidisperse packing. The distance ξ from the particle center to the cell boundary is shown for the (c) Voronoi and (d) radical Voronoi tessellations.

Sil10], polydispersity [Kau71], and particle shape [JST10]. It has be hypothesized that packings can have Φ > ΦRCP, though at the expense of the randomness in particle positions [TTD00]. In tapped or vibrated systems, Φ fluctuates about a mean value [Now+97; Now+98; SGS05; PNC07; Pug+10]. The magnitude of the agitation determines the mean and the size of the fluctuations. In general, as the volume fraction of the material increases, the size of the fluctuations decreases. The volume fraction can be defined globally or locally. To define the local volume fraction, a Voronoi tessellation, as shown in Figure 1.5a & c, is a natural choice. The Voronoi tessellation partitions the space so that the edge of the Voronoi cell is located at

ξ = d/2 (1.4)

That is, each cell contains the set of points closer to the associated site than any other site. The

8 boundary of the Voronoi cells are outlined in gray in Figure 1.5a. The local volume fraction for each particle i with radius ri and Voronoi volume Vi is simply

2 πri φi = (1.5) Vi

For a monodisperse packing (a single size of radii), the volume fraction is between 0 and 1. However, for a polydisperse granular packing, the edges of the Voronoi cells can be inside a particle giving φi > 1, as shown in Figure 1.5a & c. In order to keep φi bounded between 0 and 1, the distance to the cell boundary is weighted by the radii of the particles and is given by

d2 + r2 − r2 ξ = A B . (1.6) 2d

Compare the tessellation radical Voronoi tessellation in Figure 1.5b with the standard Voronoi in Figure 1.5a. The radical Voronoi tessellation always draws the cell boundary outside of the grains, therefore the volume fraction is bounded between 0 and 1. For a monodisperse packing, the radical Voronoi tessellation is identical to the standard Voronoi tessellation. Other ways of defining local volume fractions exist, for example: with tetrahedrons [Ast06], or navigation maps [Ric+01; Med+06]. Methods for defining local φ have been compared for polydisperse materials [Ric+01]. Fluctuations in the local volume distribution have been observed to decrease as the density increases both globally [SGS05] and locally [Ast06; AD08a; Bri+08; LD10; ST+10], shown in Figure 1.6. The relationship between the fluctuations in the local free volume and the density is very general. Whether the fluctuations in the free volume is calculated using tetrahedrons [Ast06], Voronoi tessellations [AD08a], or small windows [LD10], the relationship remains. Whether the system is jammed or dense, simulated packings in three dimensions or experimental packings in two dimensions, this relationship is quite robust. What determines the size of fluctuations in the local volume distribution? Does it depend on friction? Or is it determined by merely geometric constraints? In Chapter3, we experimentally measure the local volume fraction distribution, which we find is independent of the boundary conditions and the inter-particle friction coefficient. We extend the recently published granocentric model [Clu+09; Cor+10] to include the separation distribution and find agreement between the predicted and observed local volume fraction distributions. The success of the model signifies geometry plays a defining role in determining

9 (a) (b) (c)

FIGURE 1.6. Fluctuations in φ as a function of the density (a) from [Ast06], (b) from [Bri+08] and (c) from [LD10]. the magnitude of local volume ﬂuctuations in un-jammed packings, explaining why the phe- nomenon is so universally observed.

1.5 Statistical mechanics for granular systems

1.5.1 Granular temperatures

There are several properties of granular materials that are analogous to molecular systems. Both are collections of interacting particles with reproducible statistical properties. Both have phases: gas, liquid and solid-like behaviors. However, as mentioned in section 1.1, granular materials are athermal, highly dissipative and inherently non-equilibrium systems. Jammed granular systems have zero kinetic temperature (T = 0) from a strictly kinetic standpoint. From a statistical mechanics viewpoint, the system is locked into a single microstate. Without ergodicity and out of equilibrium, granular materials violate assumptions central to statistical mechanics; notwithstanding, there is strong motivation to use a granular statistical mechanics.

1.5.2 Edwards ensemble

By pouring or vibrating a granular material with a strict protocol, a certain volume or density is expected. While the kinetic temperature of the static packing is zero, the packing can be made to have a range of volumes by varying the strength of the vibration as in Figure 1.7. In the set of well-known Chicago experiments [Now+97; Now+98], the pack was initially in a loose state and the tapping amplitude was varied. By increasing the amplitude, loose voids in the system

10 ΦRCP reversible

ΦRLP irreversible

Tapping Amplitude

FIGURE 1.7. The dependence of Φ on tapping amplitude for a vibrated granular material.

collapse and Φ increases to ΦRLP (random loose packed) on an irreversible branch. However, decreasing the driving amplitude, increases Φ on a reversible branch to ΦRCP. The reversibility in this system reflects the lack of history dependence and suggests that these systems can be put into an ergodic state [EG02; MBE05]. Edwards proposed a statistical mechanics for jammed granular materials [EO89]. While relevant energy in jammed granular materials is zero (T = 0), Edwards proposed that the system volume V take the place of energy in conventional statistical mechanics. For a given number of particles, there are many possible ways particles can be arranged with the same macroscopic observable V. As such, there are many microstates or configurations of particles for a given volume. By analogy, the granular entropy is defined as

S = klogΩ(V) (1.7)

where the density of states is Ω(V) and k plays the role of the Boltzmann constant. By thermo-

11 dynamic analogy, the inverse of the granular temperature is deﬁned as

1 ∂S = . (1.8) X ∂V

This temperature measures the distance from the most compact random state. As such, X is called the compactivity. The densest random packing ΦRCP corresponds to X = 0, and the loosest random packing ΦRLP has X = ∞. Pressing the analogy further, the probability of ﬁnd a state with volume V and compactivity X is then given by a Boltzmann-like distribution,

Ω(V) P(V) = e−V/X (1.9) Z(X) where Z(X) is the partition function.

1.5.3 Stress ensemble

In a granular material, the stress is not homogeneously distributed in the packing [Dan57; Liu+95; JNB96; HBV99]. In fact, a fraction of the particles carry most of the stress. Figure 1.8 shows the heterogeneity in the spatial distribution of forces in an isotropically jammed granular system of photoelastic discs, where regions of high stress are bright and regions of low stress are dark. While several theorists show the distribution should be exponential [EGB03; HC09], other models predict steeper than exponential distributions [Tig+10]. In experiments, the difference can be difficult to distinguish. The stress distribution is influenced by the boundary stresses; when isotropically compressed the tail of the stress distribution is exponential, and the tail is steeper than exponential when sheared [MB05; Maj06]. The stress ensemble [Edw05; HOC07; HC09; BE09; Cha10] uses force and torque balance conservation laws to enumerate the number of valid configurations for a given boundary stress. In the stress ensemble, the angoricity Ab plays the role of the temperature, and is given by,

1 ∂S = . (1.10) Ab ∂Σb

12 FIGURE 1.8. An image of force-chains in a two-dimensional granular packing of photoelastic discs under isotropic compression.

where Σb = ∑~ri j~fi j is the stress-moment tensor. The probability of ﬁnd a state Σb is given by

Ω(Σb) −Tr Σ A P(Σb) = e (b/b). (1.11) Z(Ab)

1.5.4 Test of granular ensembles

Both volume and stress ensembles provide an intensive temperature-like measure which quantifies the size of the fluctuations. Granular temperatures (compactivity X and angoricity Ab) have been measured in simulations and experiments using two methods: the fluctuation-dissipation theorem and the ratio of histograms. By measuring the size of the global volume fluctuations for packings generated with a strict protocol, the compactivity can be measured [Now+98; SGS05]. The size of the fluctuations in the global volume decrease as Φ increases, and X can be measured using an analogue of the fluctuation-dissipation theorem, as given by

1 1 Z V2 dV − = . (1.12) X X 2 1 2 V1 σV

2 where σV is the variance in the volume and represents a speciﬁc heat-like quantity. This method

13 gives 1/X up to an additive constant. A second method for calculating X involves taking the ratio of histograms [DL03] as in Eqn. 1.9. Experimental and numerical experiments show the ratio of histograms to be exponential in V [McN+09]. This method was used with success in the stress ensemble as well [HOC07; HC09]. These results provide strong evidence supporting the Boltzmann-like distribution predicted by the volume and stress ensembles. Though granular temperatures can be measured and quantify the size of fluctuations in the system, an important question remains to be answered. Do granular temperatures behave like the temperature found in equilibrium statistical mechanics? In the most fundamental sense, do granular temperatures obey the zeroth law and equilibrate between systems? While the compactivity has been shown to equilibrate between different parts of a monodisperse packing [McN+09] and the angoricity shown to equilibrate in different portions of an isotropically compressed packing of frictionless particles [HOC07; HC09], there is no test examining whether X equilibrates between systems which are dissimilar in some way (e.g. polydispersity, friction). In Chapter5, two granular subsystems are placed in contact which differ only in inter-particle friction. We find that the temperatures calculated using the fluctuation dissipation theorem agrees with the ratio of the histograms. Finally, we observe that the compactivity does not equilibrate, while the angoricity does equilibrate between systems with different friction. Angoricity is therefore a more appropriate state variable for use in frictional systems.

1.6 Overview of experiments

We experimentally study granular materials and the relevance of non-equilibrium statistical mechanics from various viewpoints. In this work, we investigate dense dynamic granular materials, volume ﬂuctuations and granular temperatures. To do this, we use two separate but related apparatus. While each apparatus takes advantage of a nearly frictionless surface provided by pressurized air through a porous medium, one is designed for dynamic systems [LD10; PLD11; ND12; Puc+12] and one for static systems (Chapter5). In Chapters2 and3, the granular materials studied are dynamic, driven at the boundary. Figure 1.9 is a photograph of the apparatus used for these experiments. Individual particle trajectories were recorded over long periods of time. A two-dimensional granular material rests on the nearly frictionless surface of the air table. The particles are driven by an array

14 of bumpers on the perimeter of the table which are randomly triggered at 10 Hz. The piston provides a constant pressure (via pulleys and weights) or constant volume (by fixing its position to the table) boundary conditions. Close examination of Figure 1.9 reveals a multi-colored set of nine dots on each of the particles in the system. The dots encode each particles identity and provide confident tracking of particle trajectories over the entire experimental time-scale (up to 50 hours). With this apparatus, we observe dynamics as a function of the volume fraction, pressure, and the inter-particle friction. Testing ensemble based granular temperatures requires a very different apparatus. Since, the volume and stress ensembles are only defined for static granular materials. Particle positions and contact stresses are needed to measure the granular temperatures for the volume and stress ensembles, respectively. While several reliable methods for finding particle positions are available, contact forces are more difficult to measure. Photoelasticity can be used to accurately measure both normal and tangential forces using the image of isochromatic fringes [Maj06; Maj+07]. The method is described in detail in Chapter4. Two pistons are positioned by stepper-motors and can be moved independently. The work presented in Chapter5 uses the pistons to apply isotropic compression to the system. The photoelastic discs are supported by a thin layer of air driven through a microporous sheet. Starting with an initially dilute system, the pistons alternately compress and dilate the system generating many independent configurations of jammed discs. With this apparatus, we can examine static configurations of discs on a nearly frictionless surface near the jamming transition. We measure both the compactivity and the angoricity. By placing two subsystems in contact (a large outer bath and a small inner subsystem) which differ only in the inter-particle friction, we test whether granular temperatures equilibrate. The experiments presented in this dissertation provide novel insight to the dynamics near jamming (Chapter2), the role of geometry in the volume fluctuations (Chapter3), and granular temperatures and ensembles (Chapter4&5).

15 FIGURE 1.9. (a) Photograph of the apparatus used for investigating dynamic granular materials. The piston provides constant pressure via pulleys and weights, or constant volume by ﬁxing its position to the surface of the table. (b) Image of isochromatic fringes in an array of discs biaxially compressed.

16 Chapter 2

Trajectory entanglement in dense granular materials

This chapter is based on the following publication: James G. Puckett1, Frédéric Lechenault2, Karen E. Daniels1, Jean-Luc Thiffeault3. Journal of Statistical Mechanics (2012) P06008 1Department of Physics, NC State University, Raleigh, NC, 27695 USA 2Laboratoires de Physique Statistique, École Normale Supérieure, 24 rue Lhomond, 75005 Paris, France 3Department of Mathematics, University of Wisconsin, Madison, WI, 53706 USA

2.1 Abstract

The particle-scale dynamics of granular materials have commonly been characterized by the self-diffusion coefﬁcient D. However, this measure discards the collective and topological information known to be an important characteristic of particle trajectories in dense systems. Direct measurement of the entanglement of particle space-time trajectories can be obtained via

the topological braid entropy Sbraid, which has previously been used to quantify mixing efficiency in fluid systems. Here, we investigate the utility of Sbraid in characterizing the dynamics of a dense, driven granular material at packing densities near the static jamming point φJ. From particle trajectories measured within a two-dimensional granular material, we typically observe that Sbraid is well-defined and extensive. However, for systems where φ & 0.79, we find that Sbraid (like D) is not well-defined, signifying that these systems are not ergodic on the experimental timescale. Both Sbraid and D decrease with either increasing packing density or confining pressure, independent of the applied boundary condition. The related braiding factor provides a means to identify multi-particle phenomena such as collective rearrangements. We discuss possible uses for this measure in characterizing granular systems.

17 2.2 Introduction

Recently, there have been extensive efforts to draw parallels between dense granular systems, foams, emulsions, and glassy molecular systems [LN10; Hec10]. For such systems approach- ing the glass/jamming transition, the dynamics slow down due to an increasing intrication of the available phase space, with rearrangements taking the form of cage jumps [Wee00; WW02; PBN03; VL04; MD05; RIS07]. In this regime, particles are trapped by their neighbors at short timescales and resume diffusive behavior at long timescales due to eventual loss of correlation in the succession of jumps. Such jumps involve cooperatively-rearranging regions which in- duce dynamical heterogeneities [CR05; Can+10], and the particles pass each other only rarely. The growth in size of these rearranging regions is believed to be associated with the rigidity and loss of ergodicity exhibited by fragile glasses. Unlike glassy molecular systems, granular materials permit the tracking of individual particles. The average dynamics have traditionally been characterized by the mean squared displacement, σ2(τ). When the experimental timescale is long enough compared to the viscous timescale, permitting a complete decorrelation in the particle motion, it is possible to calculate the self-diffusion constant D. This coefﬁcient is a single-particle measure in which all information regarding relative motions of particles is discarded. In contrast, recent experiments and simulations [Wee00; RIS07; Ber+05; DMB05; Key+07; Lec+08; CDB09; CD10; Dur+09; Can+10] have shown that the dynamics near the glass transition become heterogeneous in both space and time. As such, it is necessary to take relative motions into account when formulating the correct average description of the state. To this end, multi-particle scalars have been devel-

oped, including the four-point susceptibility χ4 [DMB05; Key+07; Lec+08]. However, to our knowledge, no scalars have been introduced which characterize mixing effects, which may be relevant to quantifying the cooperative motion of particles. The central observation of this paper is that the trajectories of a two-dimensional assembly of grains take the form of a braid when plotted in a space-time diagram, as shown in Fig- ure 2.2. Each strand of the braid represents the space-time trajectory of a single particle in the system, with the length of the braid corresponding to time and the entanglement of the strands arising from dynamics. The topological braid entropy, Sbraid, is a measure of the degree of this entanglement. This approach has been successfully used to analyze the mixing efficiency in two-dimensional fluid flows [BAS00; Thi05; GTF06; TF06; Thi10; AT12]. Specifically, the degree of lagrangian chaos created by the motion of rods in a fluid is quantitatively related to

18 FIGURE 2.1. (a) Schematic and (b) photograph of apparatus. The piston provides constant pressure via pulleys and weights, or constant volume by ﬁxing its position to the surface of the table. the topology of the corresponding braid. We anticipate that this notion might also be applica- ble to characterizing caging and dynamical heterogeneity in granular systems, by providing a quantitative measure of how close a system is to departing from ergodicity. Here, we experimentally investigate the behavior of the average topological braid entropy extracted from the long-time dynamics of particles in a dense, driven granular material, examining the dependence on boundary conditions (constant pressure, CP, and constant volume,

19 CV) and the inter-particle friction coefﬁcient µ. We analyze the dynamics with two methods: a single-particle method (the self-diffusion constant, D) and a topological multi-particle method

(the braid entropy, Sbraid). We ﬁnd that both D and Sbraid are sensitive to the packing density φ and pressure P of the system in the same qualitative way. However, at high φ and P, both D and

Sbraid are inaccessible due to insufficiently long experimental timescales or ill-definedness. In contrast, the braiding factor can be readily computed, with computational expense comparable to the diffusion coefficient, and offers an instantaneous view of the magnitude and intermittency of the rearrangements in the system.

2.3 Experiment

We conduct experiments on a two-dimensional granular assembly supported upon a nearly- frictionless horizontal air table and agitated by bumpers around the boundary. The granular material is bi-disperse, consisting of large and small particles with diameter of rL = 83.6 mm and rS = 55.8 mm and mass of mL = 8.1 g and mS = 3.5 g, respectively. The concentration of small/large grains is ﬁxed at NS/NL = 2, so as to occupy similar areas. The apparatus is shown in Figure 2.1. The boundary condition and the inter-particle friction µ can be changed for each experimental run. The boundary condition is established by the conﬁguration of the piston

(mpiston = 95 mS). By attaching weights to the piston via a low-friction pulley, the system is confined under constant pressure (CP) with a range of pressures 5.7 < P < 80 mN/m. The constant volume (CV) boundary condition is established by fixing the piston to the table, where a range of packing densities φ = 0.72 to 0.81 is explored by removing units of particles (two small and one large), maintaining the relative concentration. The inter-particle friction is selected by changing the material around the particles’ outer edge, with µ1 = 0.1 for PTFE (Teflon) wrapping, µ2 = 0.5 for bare polystyrene, and µ3 = 0.85 for rubber. Around the boundary, an array of bumpers agitates the system on three sides. Bumpers are triggered pairwise via a computer at a high frequency ( f = 10 Hz) to keep particles in motion despite energy dissipated in collisions. A triggered pair consists of a bumper and its corresponding bumper on the opposite wall. We have previously measured [ND12] that the effect of this driving system is to provide a thermal-like bath which maintains the granular material at constant average kinetic energy. More information on the driving and kinetics of the system can be found in references [ND12; LD10; PLD11].

20 FIGURE 2.2. (a) Entanglement of space-time trajectories for N = 20 particles, which are (b) projected on the x-axis with four crossings marked with symbols. The crossings and 4 are topologically “over” (left over right), and the crossings ♦ and 5 are topologically “under” (left under right). Note that the crossings 5 and 4 undo each other, while ♦ and do not.

Images are taken τcamera = 2 or 5 seconds apart depending on the φ-dependent dynamical timescales [LD10]. We record the positions of particles and map trajectories in time using unique particle identifiers [LD10]. The unique identifiers allow a relatively large τcamera, so we can confidently track the trajectories of individual particles and thereby be certain that topological dynamics do not erroneously result from the exchange of particle identities due to 2 tracking mistakes. The packing density φ is calculated as φ = hπri /Viii with ri the particle radius and Vi its local Voronoi cell area, calculated with Voro++ [Ryc09]. The average is taken over all particles in the central 20% area of the table to reduce ordering effects induced by the boundaries [ND12; DW09].

21 2.4 Results

2.4.1 Self-diffusion

The diffusion behavior is quantiﬁed using the mean-squared displacement

2 2 2 σ (τ) = h∆xi(t j + τ) + ∆yi(t j + τ) ii, j (2.1) where the average h·i is performed over all particles i and all times t j. As can be seen in Figure 2.3, the slope of σ2(τ) is not constant for all timescales τ. It is useful to define a local curve σ2(τ) ∝ τα , where the value of α characterizes the average individual dynamics associated with that τ. For short timescales (τ < τD), the trajectories remain sub-diffusive, with α < 1. This arises because the particles are locally confined, or caged, by neighboring particles [Wee00; WW02]. At longer timescales (τ > τD), each particle has experienced multiple cage- breaking events and thus resumes diffusive behavior. For timescales at which α = 1, we can 1 d 2 measure the two-dimensional self-diffusion coefficient D = 4 dτ σ (τ). Finally, if the diffusion process is fast enough, a typical particle covers a distance of many particle diameters within the duration of the experiment. In such cases, the exponent α can again fall below unity at long times, due to the finite size of the experiment. This effect is visible in Figure 2.3 for the lowest φ at which we perform experiments. In general, it is possible to calculate D using σ2(τ) for experiments which are of long enough duration to reach the α = 1 regime. For this system, we find that this regime is reached even at values of φ near static random loose packing [LD10]. A convenient means to measure D is to specify a tolerance on α which defines a diffusive regime, marked by + symbols in Figure 2.3. We calculate D by fitting to the largest continuous region where the smoothed 2 derivative of σ (τ) has a slope |α − 1| < 0.1. The diffusive timescale τD is the timescale at which the particle motion first achieves diffusive behavior. With increasing packing density φ, τD becomes longer, as seen in Figure 2.3. For systems with large φ and P, the range of τ over which σ2(τ) was used to find D (marked with crosses in Figure 2.3) becomes less than a decade. Given the constraint that |α − 1| < 0.1 and the finite duration of our experiment ≈ 104 s, it is not always possible to calculate a diffusion constant for systems with large P or φ. One such example (φ = 0.80, P = 80 mN/m) is shown in Figure 2.3. A more conservative tolerance specification on α would further decrease the experiments for which we could calculate

22 FIGURE 2.3. Representative diffusion relations σ2(τ) for constant pressure (CP) experiments with µ2 and φ = 0.74 to 0.80. The dashed line represents a slope of unity. The diffusion coefﬁcient, D, is calculated between the two + symbols on each curve. a meaningful value of D. In many experiments where we would wish to quantify the self- diffusion of particles, D is in fact poorly-deﬁned.

2.4.2 Braid entropy

The braid entropy is a quantitative measure of mixing, by means of the degree of entanglement of trajectories [Thi05; TF06; Thi10]. To illustrate this notion of entanglement, one can think the means by which three strands of hair can be plaited. The starting conﬁguration of this braiding process is three, un-entangled parallel strands, with upper ends attached and bottom ends at positions labeled 1, 2 and 3. Braiding in this context proceeds with two alternating moves: ﬁrst, the strand in position 1 is passed above the strand in position two, and second, the strand now in position 2 is passed below the one in position 3. The result is a regular braid. From a mathematical point of view, these moves correspond to group elements, the over and under moves being the mathematical inverse of one another for a given pair of neighboring strands.

23 FIGURE 2.4. The braiding factor logL(t) for the same experimental runs shown in Figure 2.3, with φ = 0.74 to 0.79. The inset shows logL(t) divided by its value at the maximum time, T.

The corresponding braid group can be similarly deﬁned for arbitrary number of strands; for- mal details of the group structure can be found in [Bir75] or most textbooks on knot theory. Importantly, the length of the resulting braid is not speciﬁed only by the number of moves, since successive over/under moves can in fact cancel each other. An example of this situation is shown by the triangular symbols in Figure 2.2. We will quantify the topological notion of entanglement by computing the braiding factor L for each sequence. To gain a better physical insight into the meaning of this quantity, one can think about it in the following way. Consider an imaginary rubber band enclosing the strings at the bottom of the braid shown in Figure 2.2a. As we slide the rubber band upwards along the strands, its length L(t) will tend to grow, as it is caught on the strings and is not allowed to traverse them. The ratio of L(t) to the initial length of the rubber band (which we can assume is one) is the braiding factor. If the trajectories are chaotic, they will build up a non-trivial

24 braid, and L(t) will grow exponentially. The growth rate is then related to the largest Lyapunov exponent of the underlying dynamical system.

The braid entropy Sbraid is deﬁned as the asymptotic growth rate of logL(t),

d S = lim logL(t), (2.2) braid t→∞ dt where L(t) is maximized over the choice of our imaginary rubber band. (We describe below how the rubber band is in practice encoded symbolically.) This rubber band viewpoint is well-illustrated by taffy pullers and dough-kneading devices, where material is made to stretch exponentially by repeated folding [FT11]. The braid entropy provides a measure of how effi- ciently the dynamics mixes the system: the characteristic timescale of mixing is proportional to 1/Sbraid. When trajectories arise from a continuous dynamical system, then the braid entropy is a lower bound on the topological entropy of the system. The topological entropy is a measure of the loss of information about the identity of trajectories — it is an entropy in the sense of information theory. As more trajectories are included, the braid entropy converges to the topological entropy [CF07]. However, in a granular medium such as the one considered here there is no underlying continuous dynamical system: the braid entropy is not necessarily related to some intrinsic topological entropy. We follow the method described in [Thi10] to calculate a good approximation to the braid entropy, using a symbolic approach based on earlier work [Dyn02; Mou06]. By projecting 2D trajectories along a single axis, as shown in Figure 2.2, we can find all times at which any two particle trajectories cross along the chosen axis; these times are specified as the crossing times. At each crossing, the particle initially on the left passes either over or under the trajectory of the particle initially on the right and the crossing is given a sign of ±1, respectively. The corresponding algebraic representation is a sequence of braid group generators, with one generator per crossing. The sequence contains all of the information necessary to re-generate the braid projected along the chosen axis. A hypothetical loop (corresponding to the rubber band described above) drawn around the braid and acted upon by the generators will grow in length as a function of time. This can be used to compute the braiding factor L(t), using a purely symbolic description of the loop. This symbolic description relies on the fact that two-dimensional loops can be expressed succinctly in terms of the number of crossings with fixed reference lines, and that an action of the braid group on these coordinates can be defined. Sample code to compute the braiding factor is provided in [Thi10].

25 FIGURE 2.5. The Sbraid per particle, shown for µ2 and CP systems, with the axis of projection θ = 0. The arrow shows increasing φ of the system.

We calculate the braiding factor L(t) using the trajectories of N = 20 particles. Because particles can enter and leave the region captured by the camera, this choice of number provides a compromise between having too many particles with missing points in the trajectory, and too few particles with the maximum trajectory duration. We find that this choice of N is sufficient for L(t) to capture the dynamics of the whole system, as will be discussed in more detail below. The braid entropy Sbraid (see Equation 2.2) is measured from the growth rate of the logarithm of the braiding factor. To obtain a single value for a particular run, we calculate a linear fit to logL(t). This entropy will be well-defined if logL(t) meets three conditions: it grows linearly in time, is an extensive quantity, and is independent of the choice of projection axis. We can directly check each of these conditions. In Figure 2.4, we show logL(t) for systems identical to Figure 2.3, omitting the (φ = 0.80, P = 80 mN/m) system as too few persistent crossings occur. We find logL(t) grows, on average, linearly with time for φ . 0.79. In the inset, logL(t) is divided by its value at the maximum time, to highlights how the degree of linearity present at different values of φ.

To test the extensivity of Sbraid, we repeat the calculation described above for different

26 numbers of particle trajectories, and test whether S/N is a constant. Figure 2.5 shows the results of this test for a series of CP experiments at different φ. For N & 15, we observe that S/N is quite flat, independent of boundary condition, µ, φ, and P. This observation is consistent with Sbraid being an extensive quantity. For very small ensembles of trajectories, N . 15 (less than 10% of the system size), we find that the braid is too sparse to be representative of the dynamics of the system. In addition, it is difficult to test for extensivity in systems with low φ. In this regime, particles have a larger diffusion constant and are therefore more likely to enter and leave the region being monitored by the camera. Therefore, it is not possible to find N & 25 particles which all satisfy the constraint on the minimum duration of continuous trajectories required to make a calculation of Sbraid. We compute L(t) for 9 projections along axes oriented from θ = 0 to π/2, and compare the results in Figure 2.6 for two characteristic systems, both confined at CP and with µ2 but at different φ and P. For the example at low φ, we observe that logL(t) is quite linear, even on short timescales, and that the choice of projection axis has little effect on the slope of logL(t) or Sbraid. However, in the higher-φ example, logL(t) grows more quickly for the (θ = π/2)- projection than for the (θ = 0)-projection, where the difference in entropy is ≈ 2%. Increasing N decreases the difference in entropy between projections. In the limit of long experiment duration, such differences in calculated entropies would be expected to vanish. In examining the runs at large φ and large P, we note the occurrence of two important features in logL(t) which are not present in the diffusion measurements: flat regions and steep increases. In Figure 2.6b (representative of such systems), we observe that logL(t) may remain 4 constant for an extended period of time, as in this case for the interval 0.7 . t . 1.2 × 10 s. Such a flat L(t) curve signifies that particles are not exchanging neighbors. The second feature to note in Figure 2.6b is the sharp increases in logL(t) beginning around t ≈ 1.5×104 s, for all choices of projection axis. Such steps correspond to a large number of crossings, no matter the chosen orientation, and therefore unambiguously signify a collective rearrangement of particles (cage-breaking event). Another such event can be seen in Fig. 2.4 just after t = 1 × 104 sec. Due to these events, the slope of logL(t) can be much larger on short timescales. As such, the assumption that logL(t) grows linearly in time is not valid for short timescales. However, we find Sbraid is well-defined for systems where (φ . 0.79, P . 50) as logL(t) is linear on the experimental timescale, extensive, and independent of axis of projection.

27 FIGURE 2.6. A plot of logL(t) for two systems with µ2 where the axis of projection is rotated between θ = [0,π/2], where (a) is φ = 0.77,P = 12 mN/m and (b) is φ = 0.78,P = 18 mN/m.

28 FIGURE 2.7. Measured values of D as a function of φ and P in (a) and (b), respectfully. Error bars denote the magnitude of the standard error. The black × on the axis signiﬁes systems for which D is poorly deﬁned for that particular φ or P.

29 FIGURE 2.8. Measured values of Sbraid as a function of φ and P in (a) and (b), respectfully. Error bars denote the magnitude of the standard error. The black × on the axis signiﬁes systems for which Sbraid is poorly deﬁned for that particular φ or P.

30 FIGURE 2.9. A logarithmic plot of Sbraid and D for each system, where the arrow denotes increasing φ.

2.4.3 Comparison

Based on prior work [WW02; AD06; LD10], we expect D to decrease as φ increases. Indeed, in Figure 2.7ab, D decreases with increasing φ and P, independent of boundary condition

(CP and CV). Similarly, we find that Sbraid decreases with increasing φ and P, as shown in Figure 2.8ab, respectively. Error bars represent the average standard error of Sbraid from the set of 9 projections on axes oriented from θ = 0 to π/2, where the magnitude of the standard error of Sbraid is ≈ 10% and does not depend on φ and P. We observe little dependence on the inter-particle friction coefficient µ on D and Sbraid, though qualitatively D decreases with increasing µ. As discussed earlier, an experimental measurement of D using σ2 requires a constraint on the value of α to be used to find D, which was selected to be |α − 1| < 0.1. For systems with high P, this constraint was not met and therefore D is poorly defined, as shown in Figure 2.7ab.

Similarly, Sbraid is also poorly-deﬁned for high P systems as logL(t) no longer grows linearly

31 within the experimental timescale. For systems with φ & 0.79 and P & 50 mN/m, neither D nor Sbraid are useful for measuring the dynamics even with long experimental timescales of 5 × 104s (≈ 14 hr).

2.5 Discussion & Conclusion

We have investigated the behavior of the global braid entropy Sbraid associated with the long- time dynamics of a dense granular assembly, and ﬁnd it to be a well-deﬁned quantity. Notably,

Sbraid was observed to be extensive, making it a promising candidate to play a role in the elaboration of a proper equation of state for dense granular systems. The breakdown in the measurement of Sbraid coincides with the loss of diffusive dynamics, which is to be expected due to the association of the onset of caging behavior with a loss of ergodicity. The measurement of both Sbraid and the single-particle diffusion coefficient D provide a quantitative measure of the degree to which slower individual dynamics also results in poorer mixing. In spite of the fact that D is a single-particle measurement, we find that it is a good predictor of the degree of mixing. One benefit of characterizing the average dynamics with Sbraid instead of D is that no arbitrary fitting parameters, such as α which is used to identify the diffusive regime, are necessary: the method is instead fully specified by the trajectories themselves. In addition to providing a measure of the average behavior, the braiding factor L(t) provides a simple way to identify temporal events governing the dynamics, via the crossings. This information is unavailable to single-particle measurements such as D. While there are a few methods available in the literature to identify collective motions, the most common of which are four-point correlation functions, these are computationally-costly, slowly-converging, and averaged in space and time. In comparison, L(t) is less computationally-demanding and provides an instantaneous measurement. We observe that it displays increasing intermittency as the packing density or pressure is increased, with successions of plateaus and jumps indicative of collective neighborhood changes, or structural rearrangements. In fact, this departure from linearity can provide a measure of the chaoticity/ergodicity of the dynamics. The duration required to reach an acceptable degree of linearity, independent of projection axis, is a measurement of a characteristic dynamical timescale of the system. From this point of view, the experimental timescales over which we have investigated our dense granular systems were comparable or larger than this equilibration time, with the exception of the highest P and φ systems we tested. For our experimental timescale, a loss of ergodicity was observed for sys-

32 tems with close proximity to the static jamming transition. The size of each jump is likely a function of the number of particles participating in the collective event. This investigation represents a ﬁrst step towards a more systematic and controlled char- acterization of the braid entropy in two-dimensional granular systems. The braiding factor L(t) may be a means of obtaining system trajectories which could be analyzed with the recently developed thermodynamics of histories [Gar+09]. Quantitative benchmarking of Sbraid in thermal and glassy systems, together with a complete comparison with the usual dynamical susceptibilities are promising directions for future investigations. Extensions to three dimensional-ﬂow may also be possible, by projecting along a plane appropriate to the direction of stirring/shearing.

2.6 Acknowledgements

The authors are grateful to the National Science Foundation for providing support under grant numbers DMR-0644743 (JGP and KED) and DMS-0806821 (J-LT).

33 Chapter 3

Local origins of volume fraction ﬂuctuations in dense granular materials

This chapter is based on the following publication: James G. Puckett1, Frédéric Lechenault2, Karen E. Daniels1. Phys. Rev. E 83, 041301 (2011) 1Department of Physics, NC State University, Raleigh, NC, 27695 USA 2LCVN, UMR 5587 CNRS-UM2, Université Montpellier II, place Eugène Bataillon, 34095 Montpellier, France

3.1 Abstract

Fluctuations of the local volume fraction within granular materials have previously been observed to decrease as the system approaches jamming. We experimentally examine the role of boundary conditions and inter-particle friction µ on this relationship for a dense granular material of bidisperse particles driven under either constant volume or constant pressure. Using a radical Voronoï tessellation, we find the variance of the local volume fraction φ monotonically decreases as the system becomes more dense, independent of boundary condition and µ. We examine the universality and origins of this trend using experiments and the recent granocentric model [Clusel et al. Nature 2009, Corwin et al. Soft Matter 2010.] , modified to draw particle locations from an arbitrary distribution P(s) of neighbor distances s. The mean and variance of the observed P(s) are described by a single length scale controlled by φ¯. Through the granocentric model, we observe that diverse functional forms of P(s) all produce the trend of decreasing fluctuations, but only the experimentally-observed P(s) provides quantitative agreement with the measured φ fluctuations. Thus, we find that both P(s) and P(φ) encode similar information about the ensemble of observed packings, and are connected to each other by the local granocentric model.

34 3.2 Introduction

Recent measurements of granular systems show that densely packed aggregates exhibit smaller fluctuations in their local volume fraction φ than more loosely-packed systems. This has been observed for several static systems [AD08b; Bri+08; ST+10], as well as for a dense driven granular system [LD10] where particles with different frictional properties each exhibited the same quantitative relationship. Measurements of fluctuations in the global volume fraction Φ (or total volume) also demonstrate a similar trend in the fluctuations around a steady state value [Now+98; SGS05; PCN06; Pug+10]. However, in several cases, the shape of the relationship between Φ and its variance did not monotonically decrease [SGS05; Pug+10]. This bulk behavior may be related to the onset of cooperative effects and a possible phase transition [Sch+07]. In contrast, for local measurements of φ of static [AD08b; Bri+08; ST+10] and dynamic [LD10] packings, the decrease in the variance was monotonic in φ. The observations of this trend span experiment and simulation, various preparation protocols and particle properties, and both two and three dimensions, suggesting that a universal explanation might underly the observation. For dense systems, the decrease in the fluctuations of global Φ (or local φ) is suggestive of a decreasing number of valid configurations as the system approaches random close packing. Heuristically, this can be understood by considering six nearest neighbor particles arranged in a ring surrounding a central particle. As the size of this ring shrinks (corresponding to increasing φ locally), the number of possible configurations for the neighbors decreases until there is a only single configuration possible in a hexagonally close packed state. One framework in which to describe the global fluctuations in the volume fraction is the Edwards approach to the statistical mechanics of static granular systems [EO89]. The entropy S(V) = klogΩ(V) increases with the number of mechanically valid, static configurations Ω, which is a function of the system’s volume V for a constant number of particles. The change in this entropy as a function of V provides a temperature-like quantity, compactivity X ≡ ∂V/∂S, for which X → 0 as Φ → ΦRCP (random close packing) and X → ∞ as Φ → ΦRLP (random loose packing). The variance in either V or Φ is associated with the compactivity-analog of specific heat. It remains an open question how to connect local, statistical measurements of φ to a ‘thermodynamics’ of the bulk system for jammed systems [EGB03; BE03]. It is even less clear how one might apply such descriptions for dynamic or even slightly unjammed configurations, as in the experiments presented here.

35 On the particle-scale, the local volume fraction φ is defined as the ratio of the volume occupied by the particle to the total locally-available space. One method for partitioning space is the radical Voronoï tessellation (also known as Laguerre cells or power diagrams), which con- structs cells according to the locations and radii of the neighboring particles [Ric+01; Lec+06]. Each Voronoï cell contains a single particle i, with the boundaries of the cell enclosing the set 2 2 of points for which the distance di to the particle i satisfies di < dk + ri − rk (k are the indices of all other particles in the packing). This tessellation tiles all space, with one cell for each particle; neighboring particles are defined as those having cells which share an edge. The use of the radical tessellation is desirable for dense polydisperse systems in order to ensure φ < 1. This choice is not unique, and alternate methods for partitioning space at the particle-scale have also been utilized to similar effect [Ric+01; BB02; BE03; Clu+09]. Given a complete set of 2 cells which tile the volume, we can define a local volume fraction φi = vi/Vi, where vi = πri is th the volume of the i particle and Vi is the volume of its Voronoï cell. The mean and variance of this distribution are denoted φ and σ2 = h(φ − φ)2i, respectively. A bar over a local variable indicates a mean over all particles. For spheres in two or three dimensions, hexagonal close packed order provides the dens- √ est packing with a volume fraction of φ 2D = π/ 12 ≈ 0.91 in two dimensions and φ 3D = √ i i π/ 18 ≈ 0.74 in three dimensions [Hal05] . For both of these ordered packings, only a single local configuration is possible (φi = φ = const.) and the variance of the distribution P(φ) is 2 thus σ = 0. In disordered systems, a jamming transition occurs at global volume fraction ΦJ α and mean coordination number zJ which obey the relationship z−zJ ∝ (Φ−ΦJ) for packings above the jamming transition [O’H+02; O’H+03]. In the presence of disorder, static packings exhibit local variations in both z and φ. As a result, increasing the volume fraction and coordination number of the aggregate decreases the translational and rotational randomness as well as the anisotropy [TTD00; ST+10]. Correspondingly, σ2 is one measure of how much disorder is present in a system, and has the important advantage of not requiring knowledge of whether or not two neighboring particles are in mechanical contact. This makes it an experimentally- tractable state variable. Due to the prevalence of the trend of decreasing σ2 with increasing φ in multiple jammed and unjammed experiments and simulations, we investigate how it arises. We perform experiments on an unjammed, driven system to probe the robustness of this trend with regard to boundary condition (constant pressure and constant volume) and the frictional properties of the particles (inter-particle friction coefficients µ = 0.04, 0.50, and 0.85). We measure φ for

36 individual particles, and observe that σ2 decreases linearly with increasing φ, independent of boundary condition or µ. As this trend is quite similar to observations in jammed systems, we suggest that geometry plays an important role, rather than driving. Thus, we examine the origins of this trend via a generalization of the granocentric model [Clu+09; Cor+10], whereby we introduce randomness through the nearest neighbor distance distribution P(s). We find that while various models of P(s) all produce the trend of decreasing fluctuations, only the experimentally-observed P(s) provides quantitative agreement with the measured φ fluctuations.

3.3 Experiment

The experiments are performed on a single layer of particles which are supported by an air hockey table from below and driven to rearrange by an array of sixty bumpers which form the perimeter (see Fig. 3.1, the same apparatus as [LD10]). The particles are a bidisperse mixture of particles with large rL = 43 mm and small rS = 29 mm radii with masses mL = 8.4 g and mS = 3.4 g respectively. The ratio of the number of particles is fixed at 1 large particle to every 2 small particles. The particles are prepared to have one of three inter-particle frictional coefficients: µ1 = 0.04 (PTFE wrapped), µ2 = 0.50 (bare polystyrene), or µ3 = 0.85 (rubber wrapped). The air jets provide nearly frictionless contact between the particles and the base, and the table is leveled so that particles do not drift to one side in the absence of bumper driving. While particles not experiencing collisions with either the bumpers or neighboring particles can still drift slightly due to the air jets and/or local heterogeneities in the table, these velocities are small compared with the dynamics induced by the bumpers. Each experimental run consists of at least 104 configurations captured by a camera mounted above the surface of the table. Images are collected every 2 to 5 seconds, according to the Φ- dependent dynamical timescale of the system. The configurations are generated by the agitation from the perimeter of the packing by bumpers of width ∼ 3.2rS. A pair of bumpers at the same position on opposite walls are simultaneously triggered so that no net torque is exerted on the system. Every 0.1 s, two pairs of bumpers are randomly selected and fired, maintaining ongoing dynamics in the aggregate. The bumpers generate evolving dense particle configurations at global volume fractions within the range 0.71 < Φ < 0.81. For reference, we previously measured the static random loose packing for these particles to be ΦRLP ≈ 0.81 for µ2,3 [LD10]. Experiments and sim-

37 (a)

(b) x

Pressure e

gravity

bumpers piston FIGURE 3.1. Photograph (left) and schematic (right) of the apparatus, showing confining wall in constant pressure (CP) configuration, with weights m suspended from a pulley via a mono- filament line. Constant volume (CV) is obtained by fixing the wall to the surface of the table. ulations [O’H+02; Maj+07; Lec+08] with similar bidispersity have observed random close packing ΦRCP ≈ 0.84. The particle dynamics, driven by the bumpers at the perimeter, are caged at short time-scales and diffusive at long time-scales [LD10]. As global Φ approaches jamming, the dynamics slow down sharply, as when the glass transition is approached in thermal systems. The system is also well-mixing, as measured by the braiding factor [Thi10] of

38 FIGURE 3.2. Equation of state φ(P˜) for CP experiments at µ1 (+), µ2 (◦) and µ3 (×) on linear axes and inset with semilog axes.

the trajectories growing exponentially in time [PLD09]. Thus, this apparatus is well-suited for generating a large number of sterically-valid (non-overlapping) configurations. The boundary condition is determined by a movable wall on one side of an approximately 1 m × 1 m region along with bumpers on the other three sides of the aggregate. The wall, of mass 95mS, extends the width of the table and can be configured to provide either constant pressure (CP) or constant volume (CV) boundary conditions. For CP conditions, shown in Fig. 2.1, the wall functions like a piston and is pulled towards the aggregate by a weight of mass m suspended from a low-friction pulley and a mono-filament line. For a fixed number of particles N = 186, we perform experiments for a range of scaled pressures P˜ = m/mS from P˜ = 0.17 (0.58 g) to 2.41 (8.14 g). This driven granular aggregate behaves as a compressible fluid, with φ(P˜) shown in Fig. 3.2. For CV conditions, the wall is fixed to the table so that the particles are confined within an approximately square region. We vary the number of particles

39 N from 180 to 204 (altering the global Φ) while keeping the 2:1 (small:large) ratio for all N. From each image, we extract the center and radius of each particle, and perform our analysis on only the inner 20% (about 40 particles), which reduces ordering effects due to the boundary [DW09]. From the particle positions, we calculate φ using the Voro++ [Ryc+06] implemen- tation of the radical Voronoï tessellation. These local measurements allow us to consider the probability density function (PDF) P(φ), as well as its mean φ and variance σ2, as a function of the three values of µ and two boundary conditions. In addition, we record the neighbors for each particle together with the inter-particle distance s which separates the edges of the two particles. The distribution P(s) will provide a key input to the granocentric model.

3.4 Results

As has been previously observed for static granular media, the P(φ) functions arise from the inverse of a gamma-like function of the free volume [Lec+06; AD08a; AD08b]. Remarkably, for experiments with similar φ, but different boundary conditions and µ, we find the P(φ) to have similar mean, variance, and shape, as shown in Fig. 3.3. This observation typically holds at other φ as well. The bimodal volume distribution seen in [Lec+06] is also present in this φ- distribution but is less dramatic due to scaling by particle size. Across all experimental runs, we observe a one-to-one relationship between the mean and variance for φ over the entire range of global Φ, three values of µ, and two different boundary conditions. Taking the moments φ and σ2 from each distribution, we observe a single, linearly-decreasing trend, shown in Fig. 3.4. The linear fit gives the intercept φ = 0.842 ± 0.002 for σ2 = 0, which is close to 2 ΦRCP. Similar universality in the relationship σ (φ) was observed in the same apparatus using a different technique for measuring φ and σ2 [LD10]. In addition, a similar trend, perhaps with discontinuities or changes in slope, was previously observed in experimental and numerical three dimensional, monodisperse, jammed packings [AD08b; Bri+08; ST+10]. The φ-distribution is sensitive to the location of neighbors, where each neighbor determines the boundary of one side of the Voronoï cell. The probability distribution of s is shown for two example runs at different φ in Fig. 3.6, where s is the distance measured from particle edge to particle edge (see Fig. 3.5). In our experiment, the shape of Pexpt(s) is similar in all our expt data regardless of boundary condition or µ. After a peak in P (s) near smax = 0.05rS (1.8 pixels), the probability falls towards a flat value until a knee at s ≈ rS, after which it starts to fall off exponentially with a decay set by s. A closer examination of Pexpt(s) reveals that

40 FIGURE 3.3. Representative P(φ) for three experimental runs at φ ∼ 0.780: + for CP and µ2, • for CP and µ3, and × for CV and µ2. For the run at CP and µ3, dashed and dotted lines show PL,S(φ) for large and small particles, respectively. All P(φ), whether or not they distinguish large and small particles, are normalized by the total number of measurements, so that P(φ) = PL(φ) + PS(φ).

the location of smax is affected by whether the neighboring particles are large or small. For large-large pairs, smax ≈ 0.075rS (0.051rL) and for small-small pairs smax ≈ 0.026rS, while the knee remains at s = rS and the exponential tail of the distribution remains unchanged. These values are independent of φ. For simplicity, we ignore the distinction between large and small particles for Pexpt(s) with little loss in accuracy of the model results. (This choice may be inappropriate in highly polydisperse systems.) In spite of the peculiar shape of the observed distribution, both the mean and standard deviation of s appear to be smoothly set by φ, as shown in Fig. 3.7. This suggests that a single length scale controls the distribution.

41 FIGURE 3.4. Mean φ and variance σ2 of P(φ) measured from individual Voronoï cells. 4 Each point is for a single experimental run with & 10 configurations. Inter-particle friction is denoted by shape (µ1, ◦; µ2, O; µ3, ) and boundary condition by open or filled markers (CP, open; CV, filled). The line is a least squares fit with φ-intercept at φ = 0.842 ± 0.002.

3.5 Model

Due to the universality of the σ2(φ) trend with regard to boundary condition and inter-particle friction, as well as the observation of a similar relationship present for other dimensionality, polydispersity, and protocol [AD08b; Bri+08; Pug+10; LD10], we seek a geometric explanation for the relationship. We choose as our starting point the recent granocentric model [Clu+09; Cor+10] which considers the inherently local origins of the volume fraction. The model takes the ﬁnite amount of angular space available in the vicinity of a single particle, and considers a random walk which ﬁlls this available space with particles drawn from a known distribution. We explore suitable extensions to the model and examine the implications for the

42 (a) sj r c r Θ j j sj/2

FIGURE 3.5. Schematic of the central particle and one neighbor with radii rc and r j, respectively. The shortest distance between the edges of the two particles is s j. The shaded region is the contribution V from this neighbor to the total radical Voronoï volume. universal trend in shown in Fig. 3.4. Three parameters govern the model: the size distribution of the particles P(r), maximum available angular space Θc (2π in two dimensions, 4π in three dimensions), and fraction of n neighbors in mechanical contact, pz ≡ z/n. The z contacts are those which provide mechanical stability for the central particle, and contribute to the z = 2d condition for isostaticity in d dimensions. For measurements of the jammed emulsions which formed the inspiration for the model, pz ≈ 0.4 was observed and the constant separation s between non-contact neighbors the free parameter used to fit the predicted P(φ) to experiment [Clu+09]. The model invokes randomness through both the radius distribution and pz, and finds P(φ) in quantitative agreement with experimental measurements. To apply the granocentric model to un-jammed systems (Φ < ΦJ) such as this one, we consider several modifications to the inclusion of P(s). The original model draws s from a binomial distribution containing s = 0 and a tunable s = const. with probability pz and 1 − pz, respectively. As shown in Fig 3.6, a much wider distribution of s is observed in our un-jammed and driven system. Examining P(s) for s ≈ 0, we observe that mechanical contacts are rare; an upper bound of pz = 0.04 is set by threshold of the image resolution. Therefore, we do not

43 expt FIGURE 3.6. P (s) (dotted) measured for two experimental runs at CV and µ3, with φ = λ s 1 e−s/λ 0.747 (thick) and φ = 0.792 (thin). For comparison, exponential distribution P ( ) = λ (dashed) with λ = s¯ and δ-function distribution Pδ (s) = δ(s − s¯) (solid). treat contacts and neighbors separately. Following the original formulation of the granocentric model [Clu+09; Cor+10], there is a maximum angular space Θc available around a central particle with radius rc which can be occupied by neighbors. Each neighbor of radius r j with its edge s j away from the edge of the central particle occupies an amount of space

r j Θ j = 2arcsin (3.1) r j + rc + s j which provides a theoretical range 0 ≤ Θ j ≤ π; in the experiments, only the range 0.11π < Θ j < 0.76π is observed. For a collection of n randomly-selected neighbors, the total angular n space occupied is ∑1 Θ j(rc,r j,s j), and in two-dimensions this sum must be less than Θc = 2π.

44 FIGURE 3.7. Scaled mean (∆) and standard deviation (×) of s, for all boundary condition (CP and CV) and µ, as a function of the local volume fraction φ.

∗ Each neighbor contributes Vj to the Voronoï cell Vi of the central particle, shown as the shaded region in Fig. 3.5 and given by

s j 2 ∗ r j(rc + 2 ) Vj = q 2 2 (rc + r j + s j) − r j s j 2 Θ j = rc + tan (3.2) 2 2

To compare with the observed P(φ), we perform a Monte Carlo simulation which draws particles from the 2:1 bidisperse size distribution and s j from a speciﬁed P(s). For each rc, neighbors are sequentially selected at random from these two distributions. For each neighbor, we calculate the Θ-contribution according to Eq. 3.1. The random process continues

45 n+1 until ∑1 Θ j > 2π, at which point insufﬁcient angular spaces available for the last randomly- n ∗ selected particle. Only the n neighbors are retained and used to calculated Vi = ∑1Vj for the central particle. This process is repeated for 104 different seeds in order to obtain a distribution 2 ∗ of local volume fractions φi = πrc /Vi . ∗ In calculating Vj , we make one additional adjustment to account for the fact that the rejec- n tion of the n + 1 neighbor leaves behind a neighbor-less gap of size Θex ≡ 2π − ∑1 Θ j. Failure to account for this gap leads to an overestimation of φ, which we correct by apportioning Θex among the neighbors in proportion to the angular space they already occupy. The adjusted angle occupied by each neighbor becomes Θex Θ˜ j = Θ j 1 + (3.3) Θc

n ˜ so that Θc = ∑1 Θ j = 2π. Using this adjusted value, Eq. 3.2 becomes

2 ˜ ∗ s j Θ j V˜ = rc + tan (3.4) j 2 2

2 n ˜ and φ ≡ πrc /∑1Vj provides a better model of the local Voronoï volume, as the angular space surrounding the particle is now completely occupied by neighbors. Note that the construction of the boundary at the half-distance (rc + s j/2) between the edges of the particles does not result in a cell with realistic Voronoï shape. This is also true for a boundary drawn at a more 1 2 2 2 Voronoï-like distance 2((rc + r j + s j) + rc − r j )/(rc + r j + s j) from the central particle. In either case, we ﬁnd that the model reproduces the observed φ¯, but the half-distance construction quantitatively predicts σ2(φ) better than the Voronoï-like construction. Therefore, we use the half-distance construction for the granocentric model in the results that follow. We start from four different P(s) distributions: (1) the experimentally-measured Pexpt(s); expt,cut expt (2) P (s) = P (s < rS), the experimentally-measured distribution without the low- λ s 1 e−s/λ ≡ s probability knee; (3) an exponential distribution P ( ) = λ with λ and (4) a delta function Pδ (s) = δ(s − s). The effect of the low probability but large-s tail of Pexpt(s) is illustrated by Pexpt,cut(s). While Pλ (s) and Pδ (s) may not be physically realistic, these artiﬁcial distributions are chosen to show the strong dependence of P(φ) on the functional form of P(s). Each of the four distributions is shown in Fig. 3.6 for comparison. In order to examine the Pg(φ) which arise from these four P(s), a Monte Carlo process draws a th random rc to form the basis for the i cell of the distribution. Around that central particle,

46 sequential random neighbors of size r j are placed at random separations s j until the available angular space is used up; this collection of neighbors provides a value φi for that cell. 4 This same Monte Carlo method is repeated to generate 10 values of φi and thereby compute Pg(φ) for of the four P(s) for each experimental run. Two advantages of a Monte Carlo simulation over the semi-analytical techniques of [Clu+09; Cor+10] are that it permits the use of the experimentally-measured Pexpt(s) as an input to the model, in addition to the ability to redistribute Θex among the randomly selected neighbors.

3.6 Comparison

Using Pexpt(s) as the input to the granocentric model, as described above, provides a prediction for Pg(φ) which is in quantitative agreement with the experimental results. This comparison is shown in Fig. 3.8ab, for the same two runs as in Fig. 3.6. As expected, the width of the distribution narrows with increasing φ towards the random close packed limit. In Fig 3.8, all three panels compare the P(φ) observed in the experiment and simulation with the granocentric prediction Pg(φ). In Fig. 3.8abc, we observe quantitative agreement in the peak and shape of P(φ). However, for denser packings (φ & 0.75), the model is systematically high on the low-φ side of P(φ). This comparison is performed on a small number of particles over many conﬁgurations. To test the model for a larger number of particles in a single conﬁguration with a different preparation protocol, we use the YADE discrete element model [KD09; KD08] with 104 particles, the

same bidispersity as the experiment, and inter-particle friction coefficient µ2. The simulation is similar to the Lubachevsky-Stillinger algorithm [LS90], where each particle starts as a point with r = 0 and is grown linearly in time proportional to the desired radius with a damping coefficient of η = 0.3. For the purposes of comparison with experimental data (which is not in a jammed configuration) we end the inflation algorithm when the volume fraction reaches φ = 0.742. Using the particle positions and sizes, we repeat the same φ and s measurements as for the experiments.

47 (a)

(b)

FIGURE 3.8. (a,b,c) Measured P(φ) (dashed line) and calculated granocentric prediction Pg(φ) (◦). The granocentric prediction is calculated using (a,b) Pexpt(s) from Fig. 3.6 and (c) Psim(s) for a simulated packing with N = 104 particles shown in (d).

48 The measured Psim(s) is shown in Fig 3.8d and exhibits a similar shape to the Pexpt(s) sim shown in Fig. 3.6, with an exponential-like decay for r > rS. Note that the peak in P (s) is located at s = 0, whereas the peak in Pexpt(s) is finite (but small). The difference may be due to the differing preparation protocols. In the simulations, as the radii of particles grow, the overall energy in the system would increase unless kept finite by viscous damping. Thus, the dynamics of the system are slowed as the desired global Φ is reached. Since Fig. 3.8d measures the final result of the simulation once all particle-overlaps are eliminated, we ensure s & 0; due to the slow dynamics, most contacts remain at s = 0. In contrast, the configurations in the experiment arise through collisions and the configuration s = 0 is less likely. Finally, we are able to compare how well the model can explain the experimental results shown in Fig. 3.4. The results are shown in Fig. 3.9a, where the black squares are all of the experimentally-measured σ2(φ) from in Fig. 3.4. We observe that of the four proposed distributions of s, the full Pexpt(s) produces the best agreement and is able to capture not only the decreasing linear trend, but nearly the correct quantitative values. We find that the exponentially-rare neighbors with s > rS provide an important contribution to both the mean and variance: when they are removed from Pexpt(s) to form Pexpt,cut(s), the moments of P(φ) are less-accurately reproduced. For the two heuristic models of P(s), we are also able to produce the trend of decreasing variance, but without quantitative agreement with experiments. The exponential distribution Pλ (s) is a function of a single free parameter λ = s¯, where the mean and variance are equal to s¯ ands ¯2, respectively. By smoothly varying λ over a range of values consistent with Pexpt(s), we obtain systematically lower variance in φ, indicating that the particular shape of P(s) is important for quantitative agreement. Even for the constant s provided by Pδ (s) (likely more consistent with jammed systems than the driven ones described here), a linear relationship for σ2(φ) remains, although with significantly less variance than observed in the experiments. Using Pδ (s) causes σ2(φ) to become discontinuous, unlike experimental observations. This can be understood as arising from the low degree of polydispersity in the system. In Fig 3.9b, the discontinuities in σ2(φ) and the average number of neighbors, n, occur at the same values of φ, which is controlled bys ¯. Ass ¯ increases (φ decreases), the number of neighbors is approximately constant until Θex is greater than the mean Θ˜ , at which point there is on average room for one more neighbor which appears as a step in both plots. For distributions of P(s) with sufficient variance, σ2(φ) andn ¯ are observed to be continuous. Although the experiment does not probe φ & 0.8 due to lengthening timescales [LD10],

49 (a) Expt.

(b)

FIGURE 3.9. Color online. (a) Mean and variance of P(φ) for all experiments (, same data as Fig. 3.4), compared to granocentric predictions drawn from several differ- expt ent s distributions: the experimentally-measured distribution P (s) (•), the s < rS portion of the experimentally-measured distribution Pexpt,cut(s) (+), exponential distribution λ s 1 e−s/λ s δ s s − s P ( ) = λ (N) where λ = expt, and P ( ) = δ( expt) (). The ﬁlled markers represent distribution in the experimentally observed range of 0.154 < sexpt/rS < 0.339. The δ λ open markers, ♦ and 4 are P (φ) and P (φ) for s < sexpt. (b) The mean number of neighbors versus φ for Pλ (s) and Pδ (s), with the same markers as in (a).

we can explore Pδ (s) and Pλ (s) for larger φ (smaller s). These points are shown as the open markers in Fig. 3.9. In the limit s → 0, σ2 approaches the same non-zero value for both models. For comparison, a very loose arrangement of particles prepared to have φ < 0.6 by the same Lubachevsky-Stillinger algorithm used in Fig. 3.8c exhibits an increase in σ2(φ) through

50 a maximum near φ ≈ 0.5. Comparing the model Pg(φ) with Psim(φ) for low φ, the model underestimates the variance even though φ¯ is calculated with good agreement. Thus, we ﬁnd that the shape and moments of P(s) strongly affect the prediction of the local distribution of φ. Very narrow distributions such as Pδ (s), or distributions which re- semble only a portion of the experimentally measured distribution of s, failed to predict our experimentally measured φ. Nonetheless for dense granular systems, all P(s) produced a monotonic decreasing relationship between σ2 and φ, suggesting that this trend is quite robust.

3.7 Discussion

Independent of boundary condition and inter-particle friction µ, we have found that as the mean local volume fraction φ increases, the variance monotonically decreases in our driven granular system. This result is reminiscent of similar trends observed in static granular systems [Now+98; SGS05; PCN06; AD08b; Bri+08; ST+10], in spite of the special nature of jammed systems. In particular, many properties of static granular systems are due to the inter-particle friction. Increasing µ in jammed systems decreases the required number of contacts from z = 2d for µ = 0 to z = d + 1 for µ = ∞. In a model of local mechanical stability by [SL03], increasing µ increases both the tangential force and the maximum angle for which two particles can be mechanically stable. This would presumably lead to µ-dependence in the distribution of local φ. However, in driven systems the number of conﬁgurations may well be independent of µ as there is no constraint on z. In unjammed systems, the lack of µ-dependence on the local φ- distribution also suggests that dissipation does not play an important role in determining local φ. However, prior observations in the same apparatus [LD10] indicate that driven, equilibrating subsystems in fact have ensembles for which Ω = Ω(µ), which appears to be in contrast with this idea. The lack of dependence on boundary condition is also surprising, particularly given two possible sources of anisotropy in the system: only three of the four walls provide driving in both CP and CV conditions, and the piston introduces a compressive force in the case of CP. Nonetheless, little anisotropy was observed in the angular distribution of neighbors: the bond

angle order parameter Q6 [TTD00] was indistinguishable for CP and CV systems at a the same φ. Additionally, we tested P(s) for angular dependence with respect to a reference vector in the lab frame and found that P(s) was rotationally symmetric for both boundary conditions.

51 In global (rather than local) measurements of system volume V, Schröeter et al. [SGS05] observed a non-monotonic relationship for static aggregates prepared through sedimentation 2 of a fluidized bed at different flow rates. There, σ fell from φRLP towards a transition point, then again rose on approach to φRCP. This rise was attributed to cooperative effects within a finite number of statistically-independent regions, and may be related to the presence of spatial correlations during the approach to φRCP [TTD00]. Aste et al. [AD08b] compared the standard deviation as function of φ for a wide variety of experiments and simulations, and found non- monotonic behavior for the aggregated data across the transition from un-jammed to jammed configurations, although not for individual controlled experiments/simulations. Fig. 3.9 suggests the interpretation that different experiments/simulations produce different P(s) and thus fall on different σ2(φ) curves, each of which has its own monotonic relation.

3.8 Conclusion

We find a geometric explanation for the universally-observed trend of decreasing fluctuations with increasing volume fraction. By generalizing the granocentric model [Clu+09; Cor+10] to take the neighbor distribution P(s) as the single input to the model, we find that the variance of local φ measurements exhibits a smoothly decreasing trend as long as the first two moments of P(s) are also smoothly decreasing. Therefore, it is not surprising that σ2(φ) has been observed to decrease with φ for a variety of preparation protocols and particle properties in both two and three dimensions. When P(s) is chosen to be the experimentally-observed distribution, the granocentric model provides quantitative agreement with the observed shape of P(φ) and σ2(φ) without reference to any spatial correlations in local φ due to cooperative effects. Interestingly, this suggests that P(s) and P(φ) both encode similar information about the distribution of free volume, with the first two moments set by φ. While P(φ) (and the related free-volume distribution) has been well-studied in granular systems, P(s) has the advantage of being closely related to the radial distribution function, with the caveat of only considering neighbors instead of all particles. In conclusion, the decreasing relationship σ2(φ) with respect to increasing φ reveals the key role played by the constrained availability of angular space in the vicinity of a single particle. This constraint holds regardless of whether the system is static or dynamic, jammed or unjammed, mono- or polydisperse, or two or three dimensional, and requires no information about the force chains or dynamics of the system. Therefore, this relationship provides a way

52 of examining the distribution of free volume, and perhaps ultimately the full ensemble of valid conﬁgurations, in both jammed and driven granular materials.

3.9 Acknowledgments

We are grateful to Matthias Schröter, Mark Shattuck, Raphael Blumenfeld, and Eric Corwin for useful discussions, and to the NSF for support under DMR-0644743.

53 Chapter 4

Experimental methods

In equilibrium statistical mechanics, the temperature of a system can be calculated in several ways (e.g. the entropy, Stokes-Einstein, energy fluctuations) all of which give the same temperature. One of the most fundamental statements of thermodynamics is that two systems in thermal contact, given time to come to thermal equilibrium, will have the same temperature. The zeroth law of thermodynamics relates this transitively; if system A and B are both in equilibrium with system C, then A and B are in equilibrium and thus have the same temperature. In granular materials, the volume and stress ensembles provide quantities analogous to the temperatures which quantify the size of volume and stress fluctuations, respectively. While researchers have shown these temperatures equilibrate in different parts of the same packing [McN+09; HOC07; HC09], there has been no test of whether these temperatures obey the zeroth law if two systems differ in some way (such as polydispersity, shape or friction). Our zeroth law test is made by placing two static granular systems in contact and measuring the granular temperature for each. Chapter5 details this investigation of granular equilibration and provides description of the apparatus designed and built for this study, and the experimental methods used to calculate the inter-particle contact forces necessary to compute the granular stress temperature. In section 1.5, the volume and stress ensembles were introduced as were their respective temperatures the compactivity and angoricity. Models have been used to measure the compactivity [SL03; BA11], as have simulations [SWM08; Bri+08] and experiments [Now+98; SGS05; Rib+07; McN+09]. The angoricity [HOC07; HC09; Loi+09; Cha10; Wan+10], by comparison, has received less attention. While particle positions can be located with precision in two and three dimensions using a variety of techniques, measuring contact forces poses a more difficult problem. In general, particles are too rigid to measure forces through deformations, though this has been done with success in emulsions [Bru+03; Bru+07; Zho06; Des+12]. At present, using photoelasticity uniquely offers a quantitative measurement of normal and tangential forces only in two dimensional granular materials. In section 4.3, this technique is explained in detail, from the solution for the stress field on a disk [Mic00; Fro41; TG51] to calculating contact forces through a

54 Porous 24” Polypropylene Pore size = 120µm

Fan, ½ hp 20” D = 24” 36”

”

FIGURE 4.1. Schematic of apparatus of the air table used to study static granular packings. The fan provides the necessary pressure and volume of air to flow through the polypropylene sheet creating a nearly frictionless surface. non-linear least-squares optimization [Maj06; Maj+07]. This thesis focuses on granular materials near the jamming transition. Previous work [Maj06; Maj+07] examined bidisperse packings under isotropic strain or pure shear, where particles rested on a plexiglass sheet and force data was imaged using transmission photoelasticity. Majmudar noted frictional forces with the plexiglass were small but not negligible, e.g. when moving the boundaries, particles would stop before colliding with other particles. There was also evidence of frozen stresses at the meso-scopic scale which were especially prevalent in the low force regime. For systems very close to the jamming transition, contact forces are weak so friction between the base and particles cannot be neglected. This chapter describes the apparatus I designed and constructed to overcome these diffi- culties and the experimental methods involved in experimentally locating particle positions and calculating contact forces in weakly jammed granular materials. The apparatus allows the generation of many independent un-jammed and jamming configurations.

55 LED camera LED P Q

reﬂective surface photoelastic particle porous PP

pressurized air FIGURE 4.2. Schematic of air-table apparatus and reflective photoelasticity. Light shines through a linear polarizer (P) then a (wavelength matched) quarter-wave plate (Q) before passing through the material. The light then reflects off a mirror and passes through a quarter-wave plate and polarizer before being observed. Particles float on a thin layer of air provided by the pressure difference across the polypropylene (PP) sheet.

4.1 Apparatus

Frictional forces between particles and the substrate can be signiﬁcantly reduced using an air table. A sketch of the apparatus used is in Figure 4.1. An exhaust fan (FanTech 1/2 hp, 24” diameter, 835 RPM) provides the pressure difference used to supply a steady stream of air through a porous sheet. The thin porous polypropylene sheet (PP-sheet) has a width, length and thickness of 61.0 × 61.0 × 0.3 cm3, respectively, and a nominal pore size ≈ 120 µm. The thickness and pore size of the PP-sheet represents a balance between the volume and pressure

56 oaie.Tersligiaei fteiohoai rne hc srcre yacamera. a by recorded is another which and fringes plate isochromatic quarter-wave the a of surface is through mirrored image and A polarizer resulting material The the linear material. polarizer. through a photoelastic back the through light through shown the then reflects is plate then Light wave quarter 4.2. a by Figure followed in shown is photoelasticity reflective Therefore, opaque. is sheet polypropylene through solely transmitted be to considered contacts. particles are inter-particle the stresses between boundary and friction effective negligible, is the the base apparatus, support the this and to With is air surface. air of the of flow on flow enough about the driven large However, are a air. of provides layer fan thin The a fan. on the particles by produced and flow PP-sheet air The the side. in one to cluster preferentially by not separated drift are do not fan particles do the particles that that enough so flat leveled is (1 and sheet valleys thin The in A below. from particle. sheet each the supporting supports aluminum air of mini-streams several by provided pistons. the of arrangement the showing apparatus of Schematic 4.3. FIGURE sn h i al,hwvr etit h s ftasiso hteatct,a h porous the as photoelasticity, transmission of use the restricts however, table, air the Using gravity ≈ 1 . fdcig itnelreeog ordc inhomogeneities reduce to enough large distance a ducting, of m 8 piston reflective 57 1 piston hteatct sue.Ashmtcof schematic A used. is photoelasticity not olreta atce vrunor overturn particles that large so / ”tik rdof grid thick) 8” The particles are cut from PSM-4 (Vishay) sheets that are 0.125 inches thick. This material was chosen for its quality and high sensitivity (large stress-optic coefficient). However, the material is soft, with a bulk modulus ≈ 4 MPa, and sheets are not manufactured with a reflective backing (in anticipation that large strains cause the reflective backing to crack or peel). We coat the bottom of particles with a thin layer of metallic paint to provide a reflective surface without damaging the material’s optical qualities. In this work, only small strains are applied to particles so the reflective coating does not crack. In Figure 4.3, the apparatus is shown in a top-down view (gravity is into the page). The surface of the table (the PP-sheet described in the previous section) is level and flat so that particles do not preferentially drift. The table is rectangular in shape with maximal dimensions 50 × 50 cm. Two pistons bi-axially compress the system and are independently controlled so that isotropic boundary stresses can be applied. The work in Chapter5 focuses on isotropic compression. The pistons are positioned by stepper motors and bi-axially compress the system by a series of small steps. The step-size is set to ∆x ≈ 0.3 mm, corresponding to ∆Φ ≈ 0.0009.

58 (a) (b)

FIGURE 4.4. Particle centers are found using the (a) white light image of the particles, where the (b) particle edges are detected with a Sobel ﬁlter. The circular Hough transform of the edge image is shown for (c) small particles and (d) large particles. Each red peak is the center of the particle.

4.2 Particle positions

The particle centers are found using a bright white light without the polarizers, as shown in Figure 4.4a. Though uneven illumination is present, a Sobel ﬁlter is sufﬁcient to detect edges of particles as shown in Figure 4.4b. A Hough transform generalized for circles is ideal for detecting circles with known radii [KBS75; Bal81]. The software used [You10] is implemented in MATLAB. For each true pixel in the binary edge-detected image, the algorithm draws a circle with radius r around the point and the result is accumulated in an array. Each of the

59 FIGURE 4.5. The edges of the particles detected by the Hough transform are outlined in black. pixels detected on the edge of a circle then contributes to the vote at the center of the circle on the accumulator array to a maximum vote of 2πr. The accumulator array for r = rS and r = rL is shown in Figure 4.4c and d, respectively, where the peaks associated with small particles and large particles can be located and distinguished. Using the center of mass of each peak can give sub-pixel resolution for particle centers, where the result the center detection algorithm is shown in Figure 4.5.

60 FIGURE 4.6. An image of the force data for an array of photoelastic discs using reﬂection photoelasticity.

4.3 Contact forces

In Figure 4.6, an image using reflection photoelasticity is shown for a packing of jammed, photoelastic discs. The bright-dark fringe patterns on the discs are due to the stresses applied to the discs. In general, the bright and dark region corresponds to large and small forces, though the fringe pattern gives much more than this qualitative observation. In section 4.3, the process is described in detail where normal and tangential forces are extracted from the fringe pattern using a non-linear least squares optimization [Maj06; Maj+07]. Contact forces are found by examining the isochromatic fringes in images of photoelastic discs using reflective photoelasticity. The fringes correspond to optical interference, due to stress dependent refractive index in the material [Fro41]. Given the image of the fringes as in Figure 4.7, the location, angle and magnitude of contact forces can be calculated. The calculation of contact forces starts with identifying possible contacts. A naïve definition of a contact is given by

|~xi −~x j| < ri + r j + dtol (4.1)

61 f ,α

f ,α

f ,α (a) (b)

FIGURE 4.7. (a) A image of the isochromatic stress on a disc with z = 4 contacts, and (b) a schematic of possible contact forces acting on the disc. where the distance between particle centers is used with a threshold. Even with sub-pixel resolution, this method is inadequate for determining force-bearing contacts. For each disc, n ≥ z and the dtol must be set generously to overcome uncertainty in particle positions, where n is the number of neighbors satisfying Eqn. 4.1 and z is the number of force-bearing contacts.

Others have used the gradient of Iobs to reduce n [Maj06; Maj+07], however, this method was found unreliable due to insufficient resolution and contrast. The uncertainty in z can be overcome ex post facto. By observing that a contact with approximately zero force, fi ≈ 0, is identical to a null contact, work can be done on a larger parameter space 2n−3 ≥ 2z−3. From this point, we assume z = n and use the solution ( f ,α) to determine the contacts that satisfy f > ftol. The location of the contact must be along the line between the two particle centers, and is specified by the polar angle β. The isochromatic fringe pattern using the reflective photoelastic method outlined in section 4.1 is a function of the difference in principal stresses,

π(σ − σ ) I = sin2 1 2 . (4.2) Fσ where Fσ is the stress optic coefﬁcient. For an unstressed disc σ1 = σ2 = 0 everywhere in the

62 FIGURE 4.8. A disc (R = 0.0055 m, Fσ = 100), with two equal forces of F = 0.4N with αi = 0 at βi = 0, and π. On the left is the image of fringe number every where on the disc. On the right is the image of the isochromatic fringes. The fringe number at the center of the disc is exactly 1. disc, the difference in the phase is zero and thus the isochromatic image is dark. The first bright fringe is when the difference in principal stress is σ1 −σ2 = Fσ /2. The bright and dark fringes on the disc arise from the periodic function (sin2 x) in the phase retardation. The stress optic coefficient depends on optical properties of the material C, the wavelength of light used λ, and the thickness of the material, t, and is given by Fσ = λ/Ct. The stress optic coefficient for a disc can be empirically determined by diametrically compressing the disc, recording the force, and the fringe number, which we now define. The fringe number is found by counting the number of fringes between the edge of the disk (at the equator, if the force is applied at the poles) and the center of the disk. The fringe number at the boundary is zero. Moving toward the center, each bright/dark band increments the fringe number by 0.5. In Figure 4.8, the isochromatic image is shown on the right. There is one bright band and one dark band moving from the edge toward the center. So the fringe number for the disc is 1.

63 FIGURE 4.9. Calibration of Fσ using the Nf ringe and F.

The stress at the center of a disc under diametric strain is [Fro41; TG51]

2F σx = (4.3) 2πR −6F σy = 2πR σxy = 0

Thus, σy = −3σx. The principal difference in stress is the difference in the eigenvalues, which in the absence of shear is, σ1 − σ2 = |σx − σy|. Therefore, the stress-optic coefﬁcient is given by σ1 − σ2 4σx 4F Fσ = = = (4.4) Nf ringe Nf ringe πRNf ringe

So, ﬁnding Fσ involves counting the fringe number at the center of a diametrically compressed disc and measuring the force. Using linear least-squares to ﬁnd the a in the linear relation F = aNf ringe gives a = πFσ R/4, which gives Fσ ≈ 100 for the discs used in this thesis.

64 4.3.1 The solution to the stress ﬁeld in a disc due to z forces

Background

In this section, we derive the solution to the stress ﬁeld inside an elastic disc in mechanical equilibrium with z forces on the boundary. We review the theory of elasticity necessary to obtain the solution of z concentrated loads on a disk, as shown in Figure 4.7. The solution presented here largely follows the work by [Mic00; Fro41; TG51]. For a linear elastic solid with boundary forces, we have three basic assumptions:

• Condition of equilibrium (force and torque balance)

• Constitutive relation (linear stress-strain)

• Condition of compatibility (Saint-Venant’s condition)

In two dimensions, the condition of equilibrium for a static solid, in the absence of body forces, is given by

∂σxx ∂σxy + = 0 (4.5) ∂x ∂y ∂σyy ∂σxy + = 0 ∂y ∂x

Or more compactly, ∇ · σ = 0. The second assumption is a linear relation between the strain and stress tensors, σ = kε. (4.6)

The third assumption is Saint-Venant’s condition: the solid deforms continuously leaving no gaps and creating no overlaps [Mis45]. For plane strain in the absence of body forces, the compatibility of inﬁnitesimal strains is given by ! ∂ 2 ∂ 2 + (σxx + σyy) = 0. (4.7) ∂x2 ∂y2

The solution to Eqn. 4.7 in two dimensions is solved using Airy stress functions, ϕ, where the components of the stress tensor are derivatives of ϕ

∂ 2ϕ ∂ 2ϕ ∂ 2ϕ σxx = , σyy = , σxy = − (4.8) ∂x2 ∂y2 ∂x∂y

65 Then the compatibility equation of the stress function is biharmonic ! ∂ 2 ∂ 2 ∂ 2ϕ ∂ 2ϕ + ( + ) = 0 ∂x2 ∂y2 ∂x2 ∂y2 ∇4ϕ = 0 (4.9)

In general, the solution to Eqn. 4.9 is guessed by using symmetries or adjusting a known solution to the boundary conditions.

Semi-Inﬁnite Plane

α O θ Y r

FIGURE 4.10. A semi-inﬁnite plate of thickness t with a force, F, at point O.

In this section, the solution of the stress tensor for a concentrated force on an inﬁnite plane with thickness t is presented here. The solution to z force on a disc is derived from this solution. By symmetry, one could guess an Airy stress function of the form, ϕ = Crθ sinθ, which satisﬁes the compatibility equation. Eqn. 4.8 in polar coordinates gives σθθ = 0, σrθ = 0 and

1 ∂ϕ 1 ∂ 2ϕ 2C cosθ σrr = + = (4.10) r ∂r r2 ∂θ 2 r To determine the value of the constant C in Equation 4.10, we use Newton’s third law by setting

66 the applied force equal to the integral of the stress exerted by the solid , where F, r and θ as shown in Figure 4.10 Z π/2 F = −2 (σrr cosθ)rdθ (4.11) 0 Solving for C gives,

Z π/2 2cosθ C = −F/ 2 cosθrdθ 0 r Z π/2 = −F/ 4 cosθdθ 0 F = − (4.12) π

and the components of the stress tensor are

−2F cosθ σrr = (4.13) π r σθθ = 0

σrθ = 0

4.3.2 General Solution to z forces on a disc

Consider a stress on a disc, as shown in Figure 4.11, with radius R and thickness t. The solution follows that of the semi-inﬁnite plane but with an additional term due to the constraint that the disc must have a stress-free boundary. We deﬁne the origin at point O1 where the direction of pressure is O1O2, with angles θ1 and r1 as shown in Figure 4.11. Points on the circumference 2F cosθ 0 of the circle have a radial stress σrr = − π r in the direction r1. The angle between σrr and the tangent is θ2. The components of the stress tensor are

0 2 σrr = σrr sin θ2 (4.14) 0 σrθ = σrr sinθ2 cosθ2 0 σθθ = 0

67 Y F

O1 θ 1 α σ ' r1 n A β σ rr' θ C 2 σ X r2 rθ'

θ 2

FIGURE 4.11. A disc with a force, F, at point O1 at a direction α.

Substituting σrr and using θ = θ1 and r1 = 2Rsinθ2 everywhere on the circumference, then F σ0 = − [sin(θ + θ ) + sin(θ − θ )] rr πR 1 2 1 2 F σ0 = − [cos(θ + θ ) + cos(θ − θ )] (4.15) rθ πR 1 2 1 2

These two tensors are the superposition of three stresses at point A on the circumference.

F • normal stress −πR sin(θ1 + θ2) F • tangential stress −πR cos(θ1 + θ2) F • uniform stress in the direction of r1 with normal and tangential components −πR sin(θ2 − F θ1) and −πR cos(θ2 − θ1), respectively. z The disc is force and torque balanced. Torque balance implies ∑i Fi cos(θi,1 + θi,2) = 0. The sum of the uniform tension must also be zero. Therefore, the remaining term is the sum of z Fi the normal tensions at the boundary ∑i πR sin(θi,1 + θi,2) along the circumference, which is π π simpliﬁed using the inscribed angle theorem θ1 + θ2 = 2 ± α and the identity sin( 2 ± α) = cos(±α) = cosα. To ensure the boundary of the disc is stress free, a uniform tension of

68 F magnitude πR cosα is added to the radial stress distribution. The solution for z concentrated forces on a disc where θi is the angle between Fi and ri and α as shown in Figure 4.11 is

z −2F cosθ z F σ = i i + i cosα . (4.16) rr ∑ R r ∑ R i i=1 π i i=1 π

All other components of σ are zero. The solution is in z-polar coordinates. To write the solution in a single reference frame, we use a cartesian coordinate system with the origin at the center of the disc. Let β be the angle between the positive Y-axis and the line segment CO1, as shown in Figure 4.11. We compute σrr as in Eqn. 4.16 and transforming coordinate systems with a rotation by θi = −(βi +αi) as

z −1 σ = ∑ T(θi) σi,rr T (θi) (4.17) i=1

where T(θi) is the rotation matrix. The eigenvalues of σ are

q 1 2 2 σ = (σxx − σyy) ± (σxx − σyy) + 4σ (4.18) 1,2 2 xy

We have presented the solution to z-forces on a disc and given a calibration of Fσ for the photoelastic disc. The next section details the numerical methods involved in taking this solution and ﬁnding the contact forces in a array of discs.

4.3.3 Finding contact forces on a disc by optimization

The goal of this section is to take an image of the isochromatic fringes and the location of the contacts β and solve for the magnitude of the forces f and directions α as labeled in Figure 4.11. The solution to the stress on a disc is presented in detail in the previous section, which follows closely to the work in [Mic00; Fro41; TG51]. Using the known locations β and a guess at the contact forces ( f ,α) and solving for the stress field on a mesh, the image I( f ,α,β) of the isochromatic fringes is found using Eqn. 4.2. This image can be compared to the observed image of the disc and a fitness function defined as

e( f ,α,β) = kIobs − I( f ,α,β)k (4.19)

69 where Iobs is the observed image. The solution to contact forces for an array of discs involves locating particle centers, imaging the force data and solving for contact forces for N-discs. The force data is captured in the form of the isochromatic fringes using reflective photoelasticity and light from green LEDs. The method for calculating ( f ,α) have been done [Maj06; Maj+07] by finding a local minimum in Eqn. 4.19 using the non-linear squares optimiztion algorithm known as Levenberg- Marquardt [Lev44; Mar63]. The algorithm starts with an initial guess ( f 0,α0) and uses finite- difference to estimate the Jacobian and find a local minimum in Eqn. 4.19. The success and speed of convergence are greatly sensitive to the initial guess and the number of unknown parameters. In the next section, force-balance equations are detailed which reduce the number of unknowns in ( f ,α) to 2z − 3.

Force and torque balance on a disc

For z contacts, the force and torque balance constraints are given by

z z ∑ fi,x = ∑ fi sin(αi + βi) = 0 (4.20) i=1 i=1 z z ∑ fi,y = ∑ fi cos(αi + βi) = 0 i=1 i=1 z z ∑ τi = ∑ fi sinαi = 0 i=1 i=1

For z = 2, the above equations reduce to

f2 = f1 (4.21)

α2 = −α1

By examining Figure 4.11, α can be determined geometrically by the internal angles in the triangle O1O2C. The magnitude of the contact force f is the only free parameter. The initial test of the algorithm for minimizing e( f ,α) is used on a set of diametrically compressed discs, where Iobs is the image of the disc and fapplied is known. In Figure 4.12, the result of the best ﬁt I( f ,α) is compared with Iobs. For z = 2, initial guesses can be quite far from the ( f ,α), however higher fringe numbers requires better initial guesses. The error in the magnitude of contact force was < 5% and did not depend on the magnitude of the force.

70 The solution for z > 2 requires a bit more work. The force and torque constraints in Eqn. 4.21, reduce the number of free parameters by 3. To solve for these constrained parameters, one could do a coordinate transformation from ( f ,α) to the cartesian ( fx, fy), however this is not necessary. The solution requires rotating the coordinates the disc, βi = βi −βz. After, ( f ,α) are found, we simply rotate the coordinates back. Solving Eqn. 4.21 for fz−1, fz, αz, we have

z−2 z−2 − ∑ fi sinαi + ∑ fi sin(αi + βi − βZ) i=1 i=1 fz−1 = (4.22) sinαz−2 − sin(αz−2 + βz−2 − βz)

v u !2 !2 u z−1 z−1 fz = t ∑ fi sin(αi + βi − βz) + ∑ fi cos(αi + βi − βz) i=1 i=1

 z−1  − ∑ f sinα  i i  −1  i=1  αz = sin    fz 

71 FIGURE 4.12. Comparison of observed image (left) and ﬁtted image (right) for a set of eight diametrically compressed disc with r = 0.0055, Fσ = 100 under various forces [N] shown in the lower right in red.

72 FIGURE 4.13. Comparison of observed image (left) and ﬁtted image (right) for an array of isotropically compressed discs. Outline of discs are shown in red.

73 Final comments on the algorithm

The finding the contact forces for large z and f can be expensive. With a large number of free parameters, the Levenberg-Marquardt (LM) algorithm by no means guarantees a global minimum in e( f ,α). Many initial guesses are required. Given enough, distributed initial guesses, the algorithm gives small e( f ,α) signifying a good estimate to the contact forces. For an array of discs as used in this work, the process is computationally demanding. To improve the efficiency of this process, the work is done in waves. For each wave, several initial guesses give e( f ,α) for each disc. The best of these guesses is supplied to the LM-algorithm and a local minimum found. For some discs (likely those with z = 2 or small f ), this first wave is enough to give e( f ,α) < etol. These discs are then classified as good discs and set aside, as more guesses are unlikely to improve the estimate. This set of discs are used to improve the initial guesses for neighboring discs by applying Newton’s third law. However, force balance between neighboring discs is not enforced. The next wave begins and follows the same process. In this work, three waves were sufficient. Compare the preprocessed image of the forces with the solution from the algorithm in Figure 4.13. Features and details of the software is described and distributed http://nile.physics.ncsu.edu/pub/peDiscSolve.

The technique is limited by resolution and contrast in Iobs and is governed by the number (and quality) of initial guesses provided to the algorithm. In the low force limit, Iobs is dark in the center of the disk and contact show as small dim spots near the perimeter. Greater image resolution and contrast can improve the accuracy of the contact forces. The resolution of the image also limits how well large forces are calculated. Large forces (and large z) are more

computationally expensive as more initial guesses are required to give e( f ,α) < etol.

74 Chapter 5

Do temperature-like variables equilibrate in jammed granular subsystems?

5.1 Abstract

Although jammed granular systems are athermal, several thermodynamic-like descriptions have been proposed which make quantitative predictions about the distribution of volume and stress within a system and provide a corresponding temperature-like variable. We perform experiments with an apparatus designed to generate a large number of independent, jammed, two-dimensional configurations. Each configuration consists of a single layer of photoelastic disks supported by a gentle layer of air. New configurations are generated by alternately di- lating and re-compacting the system through a series of boundary displacements. Within each configuration, a bath of particles surrounds a smaller subsystem of particles with a different inter-particle friction coefficient than the bath. The use of photoelastic particles permits us to find all particle positions as well as the vector forces at each inter-particle contact. By comparing the temperature-like quantities in both systems, we find compactivity (conjugate to the volume) does not equilibrate between the systems, while the angoricity (conjugate to the stress) does. Both independent components of the angoricity are linearly dependent on the hydrostatic pressure, in agreement with predictions of the stress ensemble.

5.2 Introduction

Granular materials are a collection of discrete, athermal particles. In the absence of an external driving force, these materials relax into a mechanically stable jammed state and cannot move into another configuration since thermal fluctuations are negligible [JNB96]. While these materials are therefore inherently non-equilibrium, preparing a configuration with a strict protocol nonetheless yields different microscopic states with the same, reproducible volume [Kni+95]. Edwards proposed that the system volume (a conserved quantity) could be used to write a granular density of states, a corresponding entropy, and a temperature-like variable conjugate

75 to the volume [EO89]. However, a complete granular statistical mechanics should describe the distribution of contact forces as well as the volumes. Subsequent theoretical advances have proposed that a stress-based ensemble [BB02; God04; Edw05; HOC07; HC09; BE09; Loi+09; TV11] is likely required for a full treatment. In the Edwards ensemble, the volume V plays a role analogous to that of energy in equilibrium statistical mechanics. A granular temperature, dubbed the compactivity, is defined as X ≡ (∂S/∂V)−1, and has been successfully measured in models [SL03; BA11], simulations [SWM08; Bri+08], and experiments [Now+98; SGS05; Lec+06; Rib+07; McN+09; ZS12]. Similarly, the stress ensemble considers force and torque constraints on individual particles, and writes the density of states as a function of the stress-tensor Σb = ∑~ri j~fi j, where the ~ri j are the vectors pointing from the center of each particle to its contacts, and ~fi j is the corresponding contact force. The conjugate variable is then a tensorial temperature known as the angoricity, and is defined to be Ab = (∂S/∂Σb)−1. A minimal test of such temperature-like variables is to consider whether they obey the zeroth law of thermodynamics. In experiments and simulations, the compactivity [McN+09] has previously been shown to be equal in different parts of the same packing, and in different packings generated with the same particles under identical conditions. Simulations show this is also satisfied by the angoricity [HOC07; HC09]. However, no test has been made of whether two dissimilar systems can equilibrate either X or Ab. We provide such a test in a real granular system subject to isotropic compression, and find that while the compactivity fails this simple test, the angoricity equilibrates in a temperature-like way. Our experiments are conducted on a bi-disperse granular monolayer of photoelastic disks resting on a nearly frictionless surface provided by a thin layer of pressurized air. The assembly of particles is comprised of an inner subsystem and a larger bath which differ only in the inter- particle friction coefficient (see Figure 5.1). Starting from a dilute state, the monolayer is bi-axially compressed by outer walls in a series of short steps. At some global volume fraction Φ, the system jams and for all further steps the pressure on the system increases. Finally, the walls re-dilate to permit large scale rearrangements before the next series begins. By repeating this protocol many times, we generate an ensemble of configurations for which we record particle positions and calculate contact forces using methods similar to [Maj06; Maj+07]. With this information, we calculate the compactivity and angoricity for both the bath and the inner subsystem. In the canonical volume ensemble [EO89], the probability of finding a system with volume

76 V and compactivity X is proposed to be given by a Boltzmann-like distribution

Ω(V) P(V) = e−V/X (5.1) Z(X) where the density of states is Ω(V) is defined for an ensemble of jammed configurations, and the partition function is Z(X). The stress ensemble similarly proposes a Boltzmann-like distribution Ω(Σb) −Tr Σ A P(Σb) = e (b/b) (5.2) Z(Ab) for the stress-moment tensor Σb; the angoricity Ab is therefore also a tensor. To calculate either X or Ab, we use two methods: the method of overlapping histograms [DL03; HOC07; McN+09] and the fluctuation-dissipation theorem (FDT) [Now+98; SGS05; ZS12]. The ratio of P(V) between two systems is exponential in V and is given by

1 1 P (V) Z(X ) X −X V 1 = 2 e 2 1 . (5.3) P2(V) Z(X1)

By taking the logarithm of this ratio, one obtains a term linear in V, where the coefﬁcient is the difference in the inverse temperatures. This method determines 1/X up to an additive constant: 1/X → 1/X +CX . The FDT method also provides a relative measurement. Using the measured variance hδV 2i of P(V), we compute

1 1 Z V2 dV − = 2 (5.4) X1 X2 V1 hδV i

to obtain values of X, also up to a constant. The calculation of Ab utilizes equations analogous to Eqn. 5.3 and Eqn. 5.4; the tensorial aspects will be discussed in more detail below. Each of these methods is used separately on both the subsystem and the bath, in order to test for equilibration.

5.3 Experiment

Our experimental apparatus is shown to scale in Figure 5.1. The granular monolayer consists of 1004 bi-disperse photoelastic (Vishay PhotoStress PSM-4) disks with a thickness ≈ 3.1 mm and diameters dS = 11.0 mm and dL = 15.4 mm, in equal concentrations. The particles are

77 supported on a thin layer of air provided by a steady flow of pressurized air through a porous polypropylene sheet with a nominal pore size of 120 µm. This minimizes the effect of friction between the particles and the surface, but does not otherwise cause significant dynamics once the system is jammed. The sheet is leveled (particles do not drift to one side) and flat (particles do not cluster). The system consists of an outer bath NB = 904 and an inner subsystem NS = 100. Particles in the bath have a friction coefficient µB ≈ 0.8, while particles in the inner subsystem are wrapped with a thin layer of PTFE tape with a µS < 0.1. Images of the particle positions, photoelastic images for measuring vector contact forces, and identification of the subsystem particles are recorded with three separate images captured by a single CCD camera located above the apparatus (see Figure 5.1b). Particle positions are identified using a white light image (see Figure 5.1c), from which the centers are detected with an accuracy of ≈ 0.01dS using a Hough transform. The photoelastic images (see Figure 5.1d) are captured using reflective photoelasticity, in which the silvered back side of each particle reflects polarized light back to the camera. Photoelasticity allows for the numerical determination of the normal and tangential forces at each contact point, as required to measure Σb. Similar to the methods pioneered by [Maj06; Maj+07], we minimize the error between the observed and fitted image of the particle using a non-linear least-squares optimization. Details and source code are available for download at [Puc12]. The third image is taken using black-light illumination to identify the subsystem particles, which are tagged with ultraviolet-sensitive ink (see Figure 5.1e). The subsystem comprises all low-µ particles which are Voronoï neighbors with at least one other particle in the subsystem. The particles are confined within a square region (maximally 50 × 50 cm) imposed by two stationary walls positioned by stepper motors, as shown in Figure 5.1a. The system is initially in a dilute, un-jammed state, with the global volume fraction Φ . 0.6. The two walls bi-axially compress the system by a series of small steps of constant size (∆Φ = 0.0009, equivalently ∆x = 0.3 mm or 0.02 rL). With each step of the wall, the three images are recorded, and data is collected over a series of volumes corresponding to 0.775 < Φ < 0.805, giving 30 different volumes for each compression cycle. Steps continue until the gradient squared of the force image [HBV99] indicates a pressure threshold has been reached; this reduces the risk of particles buckling out of plane. The walls then re-dilate to the dilute state, and the particles are then rearranged while maintaining subsystem continuity; this protocol is repeated 100 times. During the compression phase of each quasi-static cycle, we observe the percolation of force chains at a value Φperc. As the system is further compressed beyond this point, the

78 h e f10cce,ti hehl cusoe ag 0 range a over occurs For threshold increases. this particle cycles, per contacts 100 of of number set average the the and strength in grow forces contact identifying for light showing ultraviolet light an green (e) polarized and (d) forces low- contact positions, the calculating recorded: particle are for locating conﬁguration fringes for each isochromatic mounted of light are images white polarizer Three reﬂects linear unpolarized photoelasticity. particle and the (c) each plate resolve quarter-wave to of second camera bottom A the the on particle. on the surface through before mirrored back (Q) A plate light wave material. quarter wavelength-matched photoelastic a the (P) entering polarizer linear a through LEDs array green high an (black, compressing low subsystem bi-axially outer (red, walls an tem of two composed (a) particles showing disk-shaped apparatus of of Schematic 5.1. FIGURE (c) (a) µ gravity particles. µ

n b eetv hteatct narflae atce.Lgtsie from shines Light particles. air-floated on photoelasticity reflective (b) and ) piston 1 piston (d) 79 1 (b) Q P . 782 oosPP porous LED (e) reflective surface < Φ perc rsuie air pressurized µ n ninrsubsys- inner an and ) camera 1 < photoelastic 0 particle . 9,weethe where 792, LED width of the distribution is indicative of finite size effects [O’H+03; SOS12]. The ratio of un- jammed to jammed systems at a given Φ is shown in Figure 5.2d. We define random loose packing as ΦRLP = hΦperci ≈ 0.787 as the center of this distribution.

5.4 Results Volume

We calculate the distribution of local volumes P(Vm) over clusters of size m, using the sum of individual radical Voronoï volumes obtained from the Voro++ software [Ryc+06]. Each cluster is defined as the m − 1 nearest neighbors surrounding a central particle. For m = 1, P(Vm) has two distinct peaks which correspond to small and large particles [Lec+06]. With increasing cluster size, the bimodal aspect of P(Vm) disappears, but even for large cluster sizes (m > 100), the distribution remains asymmetric and non-Gaussian [Lec+06]. In Figure 5.2a, we show P(Vm) for three values of Φ with m = 48; the value of m is large enough so that P(Vm) does not show any features arising from bi-dispersity. In Figure 5.2b, we show the ratio Pi(V)/P j(V) where the reference system j is Φ = 0.784. In practice this can be done with any two systems so long as there is sufficient overlap between their histograms. As the ratio of Pi(V)/P j(V) is well-approximated by an exponential in V, the compactivity can be calculated using Eqn. 5.3. The inverse compactivity, 1/X, is also calculated using FDT using Eqn. 5.4, where the integrand is approximated using a third order polynomial. Each method determines 1/X only up to an additive constant, which is adjusted so that XRLP = ∞. In Figure 5.2c, the inverse compactivity is shown for both the bath (1/XB) and the subsystem (1/XS). We find good agreement between X(Φ) given by the overlapping histogram method and by the fluctuation dissipation theorem. In addition, for 4 < m < 50, we observe X to be approximately independent of m. However, we find that the compactivity of the bath is not equal to that of the subsystem (XB(Φ) =6 XS(Φ)), even considering adjustments of the additive constant. This represents a failure of the zeroth law for X. We can take further advantage of the accessibility of both jammed and un-jammed states within in the center of the range of explored Φ. While the Edwards ensemble is not defined for un-jammed systems, we can nonetheless carry out the histogram analysis as performed on the jammed systems. In this regime, we find that the P(Vm) histograms cannot distinguish between the jammed and un-jammed states. Furthermore, the measured values of X decrease continuously from above ΦRLP to below; this is an undesirable characteristic.

80 −4 10 101

(a) j (b) −5

10 / P i P P(V) −6 Φ = 0.776 10 100 Φ = 0.784 Φ = 0.802 10−7 1.4 1.5 1.6 1.7 1.8 1.9 2 1.4 1.5 1.6 1.7 1.8 1.9 2 V/ (m Vmin) V/ (m Vmin)

0.1 µB µ 0.05 S

0 /X S

V −0.05

−0.1 (c) 1 0.5 Un-Jammed Jammed (d)

0.775 0.78 0.785 0.79 0.795 0.8 Φ

FIGURE 5.2. (a) Volume histograms, P(V), for Φ = 0.776 (N), 0.784 (), and 0.802 (•) with m = 48. (b) A semi-logarithmic plot of the ratio each histogram with respect to the Φ = 0.784 distribution, i.e. Pi(V)/Pi=2(V). (c) The inverse compactivity given by Eqn. 5.3 plotted as a function of the inverse volume fraction where µB are shown as black • and µS are red . Large/small symbols denote jammed/un-jammed conﬁgurations, respectively. Errorbars shown are uncertainties in P(V) and propagated through the calculation. The inverse compactivity given by the FDT method (Eqn. 5.4), is shown with the solid line for comparison. (d) The ratio of number of jammed/un-jammed conﬁgurations recorded at each Φ.

5.5 Results stress

The stress ensemble also provides a Boltzmann-like distribution in the stress, as given in Eqn. 5.2. In the case of frictionless grains, the angoricity Ab is a scalar due the off-diagonal 81 Γ = 0.7 x 10-3 Γ = 1.0 x 10-3 3 (a) (b) 10 Γ = 1.5 x 10-3 102 Γ = 2.0 x 10-3 Γ = 2.4 x 10-3 101 j

) 2

p 10 0 / P σ i 10 ( P

P 1 −1 10 10 10−2 0 10 0 0.004 0.008 0.012 0.016 0 0.002 0.004 0.006 0.008 0.01

σp [Nm] σp [Nm]

−3 x 10 (c)

2 2 50 µB 1 µS 10 > / Γ

2 5 1 p A τ 1.5 1 <δ σ 0.5 1 2 5 10 20 50 A 1 m

0.5 A p

0 0 0.5 1 1.5 2 2.5 3 3.5 4 −3 Γ [Nm] x 10

FIGURE 5.3. (a) Distribution of σp where m = 8 and Γ = 0.0007 (H), 0.0010 (), 0.0015 (•), and 0.0024 Nm (N). A semi-logarithmic plot of the (b) ratio Pi(σp)/P j(σp) where the reference system j is Γ = 0.0015 Nm. The pressure angoricity AP and shear angoricity Aτ are shown as a function of Γ where the results using overlapping histograms for µB and µS are shown as black ◦ and are red ♦, respectively. The solid line is the angoricity calculated using FDT. The gray dashed lines provide a visual reference of the slopes 0.15 and 0.45, respectively. 2 Inset: The scaled variance hδσpi of the P j(σp) distribution, as a function of the cluster size m.

82 components in Σb being zero. In any real granular system, friction is present and a shear- free state is not readily obtained. Therefore, Σb is a symmetric tensor with non-zero off- diagonal components and can be reduced to two independent components related to the pressure and shear stress. The pressure angoricity Ap and the shear angoricity Aτ are conjugate to σp = (σ1 + σ2)/2 and the στ = (σ1 − σ2)/2, respectively [HC09], where σ1,2 are the principal stresses. The average hydrostatic pressure per particle in the system is given by Γ = Tr Σb/N. Both Ap and Aτ are obtained using the method of overlapping histograms (analogous to Eqn. 5.3) and the FDT (analogous to Eqn. 5.4). With each method, A is calculated up to an additive constant so that 1/A → 1/A +CA, where CA satisfies A → ∞ as Γ → ∞. In Figure 5.3a, the local distribution of pressure P(σp) is shown for m = 8 on configurations over a range 0.0006 < Γ < 0.0025 Nm. The ratio Pi(σp)/P j(σp) is exponential in σp (see Figure 5.3b, similar results for στ not shown), as required by the stress ensemble analogue of Eqn. 5.3. In addition, we observe that the variance of σp is proportional to m, which is consistent with S being an extensive entropy (see Figure 5.3c). We are therefore able to measure both the pressure angoricity Ap the shear angoricity Aτ using their corresponding distributions, shown in Figure 5.3c as a function of Γ. We find that Ap,τ are independent of m for m > 3, as also observed in simulations [HOC07; HC09], and that values obtained from the histogram method (points) and the FDT method (solid line) are in approximate agreement. Finally, we find that for either the shear or compressional angoricity, the values measured in the bath and in the subsystem are equivalent, signifying the angoricity is equilibrating between the subsystems. Nonetheless, the values of Aτ and Ap do not match each other, with the shear angoricity growing faster as a function of Γ. We find the angoricity is given by A = b Γ for both pressure angoricity and shear angoricity, where bp = 0.153 ± 0.004 and bτ = 0.450 ± 0.020, respectively. For a two-dimensional frictionless shear-free system, the stress ensemble predicts bp = 0.5 at the isostatic point [HOC07; HC09]. Above the isostatic point, the stress ensemble predicts bp to be a function of the average contact number. The disagreement between the frictional and frictionless values of bp implies friction significantly affects the density of states.

5.6 Discussion

We have measured both compactivity X (conjugate to volume in the Edwards ensemble), and angoricity Ab (conjugate to the stress tensor in the stress ensemble), in a laboratory granular

83 system using particle-scale characterizations. While we found that while the value of X calculated using the overlapping histogram method was consistent with the value found using the fluctuation-dissipation theorem, it failed to equilibrate between non-identical systems, making it a poor state variable. A similar failure is likely behind previous measurements by Schröter et al. [SGS05], in which two granular materials with different frictional properties, prepared using the same protocol, were found to have different globally-measured values of X. In contrast, we observed that the temperature-like variable Ab does successfully equilibrate between a subsystem and bath with dissimilar inter-particle friction coefficients, as would be required in order to have a valid zeroth law. Moreover, we find agreement with the prediction that angoricity should scale linearly the hydrostatic pressure [HC09]. These successes make angoricity a promising state variable for frictional granular systems. One downside to using angoricity as a state variable, particularly in experiments, is that its calculation requires the determination of both normal and tangential forces. While there has been a long history of measuring normal forces at the boundaries of granular systems [MJN98; LMF99; Bla+01; MJS00; CJN05], particle-scale measurements have seen more limited de- velopment. Outside of photoelastic particles such as those used here, measurements typically exist only for normal forces, whether the systems are frictional (tangential forces are neglected) [MP11; Saa+12] or frictionless [Bru+03; Zho+06; Des+12]. It is possible to understand the success of the stress ensemble over the Edwards (volume) ensemble by considering the underlying physics behind the conserved quantities in each. Under Newton’s third law, forces and torques must be strictly balanced at each force contact, while volume is merely constrained globally. As a result, our subsystem differed from the bath not only in the measured X, but more conventionally in the mean local volume fraction. In fact, the full canonical Edwards ensemble [Edw05] unifies the volume and stress ensembles, where the density of states depends on both V and Σb, and it has recently been ar- gued [Wan+12; BJE12] that the two should not be considered separately. The classic phe- nomenon of Reynolds dilatancy [Rey85] under which shear induces a bulk expansion similarly suggests that such a coupling is important. Nonetheless, we observed here that angoricity can be independently equilibrated, and future experiments should more fully investigate the relationship between ensembles, the relative importance of shear and compression, and the role of friction on the density of states.

Acknowledgements: The authors are grateful for ﬁnancial support under from the National Science Foundation (DMR-0644743), and for illuminating discussions with Dapeng Bi, Bulbul

84 Chakraborty, Silke Henkes, Brian Tighe, Matthias Schröter, and Song-Chuan Zhao.

85 Chapter 6

Conclusion

6.1 Summary of results

In this dissertation, particle dynamics in dense driven granular systems, the relationship between geometry and volume fluctuations, and granular temperatures were examined in granular materials near the transition to rigidity. This research searches for relevant and well-defined state variables in dynamic and jammed granular systems. We explored the dynamics using a novel method called the braid entropy [Thi10] which quantifies the collective rearrangement topologically, in Chapter2. The braiding factor gave an instantaneous measure of the dynamics, useful in identifying times of collective rearrangements. We compared the braid entropy with the diffusion coefficient D. Through D is a single particle measure, the diffusion D was found to be a good predictor of the degree of mixing. The braiding factor shows greater intermittency for systems closer to the transition to rigidity, signifying larger collective rearrangements, and loss of exponential growth (on the experimental length scale). This can provide a measure of the ergodicity in the dynamics. In Chapter3, we examined the origin of local volume fluctuations. We measured local volume fluctuations, which were found to decrease as the volume fraction was increased. Experimentally, we found neither inter-particle friction nor boundary condition play a role in determining the magnitude of the volume fluctuations. Building on a simple geometric model [Clu+09; Cor+10], we showed that the volume distribution could be reproduced using the distribution of separation distances between particles. The separation distribution is similar to the more familiar pair correlation function. In Chapter5, we tested whether granular temperatures equilibrate. To do this, we identified particle centers and measured contact forces using photoelastic particles and the method described in section 4.3. Two granular systems were placed in contact, a larger outer system with a high inter-particle friction, and a smaller inner subsystem with a small friction coefficient. A large number of independent configurations were generated. We used two methods for calculating granular temperatures: overlapping histograms and fluctuation-dissipation. While the two methods for finding temperatures agreed with one another and further the temperatures

86 FIGURE 6.1. The volume fraction axis showing liquid, glassy, solid and crystalline phases. The range of Φ examined in experiments on dense driven system (T > 0) are shown with black hatching on top, while static T = 0 isotropically compressed systems are shown in red hatching on bottom. The static system becomes rigid in between Φ = 0.78 and 0.79, where driven systems remain liquid-like above this value.

were equivalent in different parts of the same packing, we found the granular temperature for the volume ensemble (compactivity) did not obey the zeroth law. On the other hand, the temperature associated with the stress ensemble (angoricity) did obey the zeroth law. This lays the groundwork for identifying an equation of state in frictional granular systems. In Figure 6.1, compare the range of Φ which was examined for driven liquid-like systems 0.71 < Φ < 0.81 in Chapter2, with the static systems from Chapter50 .774 < Φ < 0.805. The static systems become rigid in the range 0.78 < Φ < 0.79, were driven systems remains fluid-like though very glassy up to Φ ≈ 0.81. In understanding the transition to rigidity, the role of a kinetic temperature and driving can not be neglected. For the liquid-like systems below jamming in Chapter3, we found the impact of inter- particle friction and boundary condition on the size of volume fluctuations to be negligible. Further, we extended the granocentric model [Clu+09; Cor+10] and found geometry alone was sufficient to quantify the magnitude of the volume fluctuations, but found in Chapter5 that this is not true for jammed systems. In Chapter5, we measured volume fluctuations in static granular systems in un-jammed and jammed states. Below jamming, we found the compactivity (the granular temperature associated with the size of the volume fluctuations) did not equilibrate between jammed systems with different inter-particle friction coefficients. This finding highlights an important feature in the transition to rigidity. Below jamming, friction plays a negligible role in determining volume fluctuations; for jammed systems, the dependence of the compactivity on Φ is sensitive to the inter-particle friction.

87 6.2 Future work and open questions

The work presented here provides a test ground for many open theoretical and experimental questions. Some of the future directions which may be pursued include:

• The exponential growth of the braiding factor begins to breaks down for glassy systems. However, waiting much longer may show that the braiding factor regains the exponential growth. What are the time-scales necessary for this? Do the time-scales diverge like those associated with the characteristic time of diffusion?

• How does the braid entropy compare with molecular systems? How does this compare with obstructed and un-obstructed random walks?

• The apparatus described in Chapter4 has given access to volume and stress statistics very near the jamming transition. More analysis can probe scaling relations between contact number and pressure with the distance from jamming. Does friction (or polydispersity) change the scaling exponents? What about particle shape?

• More analysis should be done on the compactivity of the system. We found the compactivity did not equilibrate between the bath and subsystem which had different friction coefﬁcients. Schröeter et al. proposed this was due to a larger number of accessible conﬁgurations were available to the system with larger friction [SGS05]. The sensitivity of the compactivity with friction were also shown using a statistical model [SL03]. With a larger dataset including volume and stress data, this conjecture could be proved experimentally.

• The role of polydispersity on the compactivity could be explored. This could help resolve whether friction alone increases the phase space of available conﬁgurations or whether other properties are important.

• Angoricity and compactivity should be considered jointly in the full canonical ensemble. By generating a larger number of jammed conﬁgurations, we could look deeper in to the density of states for frictional granular systems.

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