Jamming II: Edwards’ statistical mechanics of random packings of hard spheres

Abstract

The problem of finding the most efficient way to pack spheres has an illustrious history, dating back to the crystalline arrays conjectured by Kepler and the random geometries explored by Bernal in the 60's. This problem finds applications spanning from the mathematician's pencil, the processing of granular materials, the jamming and glass transitions, all the way to fruit packing in every grocery. There are presently numerous experiments showing that the loosest way to pack spheres gives a density of ∼ 55% (named random loose packing, RLP) while filling all the loose voids results in a maximum density of ∼ 63 − 64% (named random close packing, RCP). While those values seem robustly true, to this date there is no well-accepted physical explanation or theoretical prediction for them. Here we develop a common framework for understanding the random packings of monodisperse hard spheres whose limits can be interpreted as the experimentally observed RLP and RCP. The reason for these limits arises from a statistical picture of jammed states in which the RCP can be interpreted as the ground state of the ensemble of jammed matter with zero compactivity, while the RLP arises in the infinite compactivity limit. We combine an extended statistical mechanics approach 'a la Edwards' (where the role traditionally played by the energy and temperature in thermal systems is substituted by the volume and compactivity) with a constraint on mechanical stability imposed by the isostatic condition. We show how such approaches can bring results that can be compared to experiments and allow for an exploitation of the statistical mechanics framework. The key result is the use of a relation between the local Voronoi volumes of the constituent grains (denoted the volume function) and the number of neighbors in contact that permits to simple combine the two approaches to develop a theory of volume fluctuations in jammed * Corresponding author

Edwards thermodynamics of the jamming transition for frictionless packings: Ergodicity test and role of angoricity and compactivity

“…Or, s = (z max − κ/w)/z * . Thus, we have verified that the inclusion of fluctuations in the coordination number does not change the shape of the jamming phase diagram obtained in [29,63]. These fluctuations may affect the probability of the microstates according to the density of states proposed in [29].…”

“…where z min = Z and z max = 6, β = 1/X, and κ = 2 √ 3. We follow the notation and concepts from [29,[62][63][64]. We define x = ( i z i )/N , thus:…”

Positronium Probing of Pores in Zirconia Nanopowders

“…The measured mass densities are far below a value of 74% which would be expected for a system of ideally packed equal-sized hardsphere particles, implied by solutions of the Kepler conjecture problem [12]. Models of random packing of uniform hard spheres have also been studied both theoretically as well as experimentally and maximum densities of ≈ 55 and ≈ 63−64% of the theoretical density value are typically encountered in cases of so called random loose packing (RLP) and random close packing (RCP), respectively [13]. The experimental mass densities shown in Table II seem to be in a reasonable consistency with a value of ≈ 2.8 g/cm 3 expected for a nanopowder composed of RCP packed aggregates of ideally packed initial nanoparticles.…”

Towards the Development of a Universal Expression for the Configurational Entropy of Mixing

“…The limit ρ RCP value is located in a narrow range around 0.64: (i) 0.634, deduced in [51]; (ii) 0.6366, related to Buffon's constant 2/π in [52]; and (iii) 0.640˘0.006 given in [53]. The value ρ RCP = 0.634 is selected in this work; however, the conclusions are still valid if another value is chosen.…”