Experimental demonstration of nonuniform frequency distributions of granular packingsGuo-Jie Gao,1Jerzy Blawzdziewicz,1,2Corey S. O’Hern,1,2and Mark Shattuck31Department of Mechanical Engineering, Yale University, New Haven, Connecticut 06520-8286, USA2Department of Physics, Yale University, New Haven, Connecticut 06520-8120, USA3Benjamin Levich Institute and Physics Department, The City College of the City University of New York,New York, New York 10031, USA共Received 28 March 2009; revised manuscript received 3 September 2009; published 17 December 2009兲We developed an experimental method to generate mechanically stable 共MS兲 packings of frictionless disksand performed coordinated experiments and simulations to characterize MS packings in small systems. For agiven system geometry, MS packings occur as discrete, well-separated points in configuration space withprobabilities that vary by many orders of magnitude and are robust with respect to the packing preparation.Over a continuous range of system geometries, MS packings occur as distinct geometrical families and only asmall fraction of families are sampled via quasistatic dynamics. These results suggest that the most frequentMS packings may dominate the structural and mechanical properties of dense granular media.DOI: 10.1103/PhysRevE.80.061304 PACS number共s兲: 83.80.Fg, 61.43.⫺j, 05.20.⫺y, 64.70.kjEquilibrium statistical mechanics provides a powerfulmethod to determine the macroscopic state of thermal sys-tems by counting the number of microstates. Several recentstudies have applied similar methods to describe densegranular materials 关1–4兴. Many of these have used Edwards-ensemble descriptions, which are based on an assumptionthat all static configurations of a granular system under agiven set of macroscopic constraints are equally likely 关5兴.Despite the fact that granular media are dissipative andrequire external driving forces 共not thermal fluctuations兲 toexplore configuration space, there has been surprising suc-cess in describing static and slowly evolving granular sys-tems using statistical methods based on the Edwards’ as-sumption 关6–9兴. However, the assumption of equalmicrostate probability has not been tested explicitly and therelevant microstates have not been clearly defined. We advo-cate a “bottom-up” approach to constructing statistical-mechanics descriptions of dense granular materials—onewhere we enumerate the microstates and accurately measuretheir probabilities 关10,11兴.To study microstate statistics, we performed experimentsand simulations of static packings in two dimensions 共2D兲.We focused on small frictionless systems to enable enumera-tion of nearly all mechanically stable 共MS兲 packings 共i.e.,microstates兲. Frictionless MS disk packings possess two im-portant characteristics: 共1兲 force balance is achieved on allgrains and 共2兲 all possible single and collective particle dis-placements 共except those arising from rattler particles兲 leadto particle overlaps and increases in energy 关10兴. In experi-ments, an ensemble of packings is obtained by randomizingthe system using large-amplitude vibrations and in simula-tions packings are generated using deposition under gravityfrom random initial positions. To generate frictionless pack-ings in experiments, we developed a technique where fric-tional forces are relaxed using small-amplitude, high-frequency vibrations. This method allows one to differentiatethe effects of geometrical constraints from friction and thus ithas broad applicability.We find the following four key results concerning mi-crostate distributions for frictionless MS packings: 共1兲 Thesets of MS packings found in experiments and simulationsare nearly identical. 共2兲 For a specific system geometry, thereis a finite number of discrete MS packings. These packingspossess highly nonuniform frequencies 共contrary to the Ed-wards’ hypothesis兲 that are relatively insensitive to packingpreparation. 共3兲 Over a continuous range of system geom-etries, MS packings can be classified using a finite number ofgeometrical families characterized by the particle contactnetwork. 共4兲 During uniaxial quasistatic compression, thesystem samples a small fraction of families. The fractiondecreases with increasing strain, which indicates highly non-ergodic evolution.The apparatus for generating frictionless disk packings isdepicted in Fig. 1. A mixture of thin disks with two differentdiameters 共diameter ratio d =␴l/␴s=1.252兲 was confined be-tween two glass plates. The cell rests on a thin plunger and isconnected to an electromagnetic shaker through a slot in theLElectromagnetic ShakerrjFIG. 1. Schematic of the experiment used to generate mechani-cally stable frictionless granular packings.PHYSICAL REVIEW E 80, 061304 共2009兲1539-3755/2009/80共6兲/061304共5兲 ©2009 The American Physical Society061304-1bottom. The shaker enabled us to apply vertical vibrations atvariable amplitude and frequency to repeatedly generate MSpackings. The mixtures consisted of an odd number of grains共N=5 and 7兲 with one more small than large disk. The sepa-ration L between the side walls can vary from zero to 6␴sbyfixing one wall and moving the other via a stepper motor. Aforce transducer measures the applied force.We employed two experimental procedures to study MSpackings: 共a兲 enumeration of nearly all packings at fixed cellwidth L and 共b兲 quasistatic changes in L to study dynamicsfrom one MS packing to another. To generate an ensemble ofpackings at a given width, we first oscillated at high ampli-tude and low frequency 共50 Hz兲 for at least1storandomizeparticle positions and then relaxed the system under gravitywith the shaker turned off for 400 ms. We applied a low-amplitude, high-frequency 共400 Hz兲 oscillation for 500 ms,which excites particle rotation and relaxes frictional forces.Finally, oscillations were turned off and particles came torest. Particle positions were captured to an accuracy of⌬/␴s=6⫻10−6. The MS packings were classified using theset of particle positions Rជi=兵rជ1,rជ2,...,rជN其 for configurationi, where rជjgives the coordinates of the N particles.For quasistatic dynamics, we initialized the system in oneof the MS packings at large wall separation Lmax/␴s=1+4冑d. We compressed the system by successively decreasingL in 100 small increments. During each step, we