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 tem- peratures. 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 distribu- tion, 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 separa- tions, 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 sys- tems 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 All Rights Reserved State Variables in Granular Materials: an Investigation of Volume and Stress Fluctuations 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 sup- porting 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 subsys- tems? ..................................... 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=F, stress S 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 F for two-dimension granular materials....... 4 Figure 1.3 Particle trajectories for a granular packing vibrated at a constant am- plitude and frequency at (a) F = 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 F, ( from [RIS07] ). ........................... 6 Figure 1.5 (a) Local volume defined with a (a) Voronoi tessellation and a (b) rad- ical Voronoi tessellation for a bidisperse packing. The distance x 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 f as a function of the density (a) from [Ast06], (b) from [Bri+08] and (c) from [LD10]....................... 10 Figure 1.7 The dependence of F on tapping amplitude for a vibrated granular ma- terial. ................................... 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 gran- ular 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 sym- bols. 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 s2(t) for constant pressure (CP) ex- periments