Pair correlation function characteristics of nearly jammed disordered and ordered hard-sphere packings

Abstract

We study the approach to jamming in hard-sphere packings and, in particular, the pair correlation function g(2) (r) around contact, both theoretically and computationally. Our computational data unambiguously separate the narrowing delta -function contribution to g(2) due to emerging interparticle contacts from the background contribution due to near contacts. The data also show with unprecedented accuracy that disordered hard-sphere packings are strictly isostatic: i.e., the number of exact contacts in the jamming limit is exactly equal to the number of degrees of freedom, once rattlers are removed. For such isostatic packings, we derive a theoretical connection between the probability distribution of interparticle forces P(f) (f) , which we measure computationally, and the contact contribution to g(2) . We verify this relation for computationally generated isostatic packings that are representative of the maximally random jammed state. We clearly observe a maximum in P(f) and a nonzero probability of zero force, shedding light on long-standing questions in the granular-media literature. We computationally observe an unusual power-law divergence in the near-contact contribution to g(2) , persistent even in the jamming limit, with exponent -0.4 clearly distinguishable from previously proposed inverse-square-root divergence. Additionally, we present high-quality numerical data on the two discontinuities in the split-second peak of g(2) and use a shared-neighbor analysis of the graph representing the contact network to study the local particle clusters responsible for the peculiar features. Finally, we present the computational data on the contact contribution to g(2) for vacancy-diluted fcc crystal packings and also investigate partially crystallized packings along the transition from maximally disordered to fully ordered packings. We find that the contact network remains isostatic even when ordering is present. Unlike previous studies, we find that ordering has a significant impact on the shape of P(f) for small forces.

Microstructure and macroscopic properties of polydisperse systems of hard spheres

“…3.8 are determined using Eq. (3.14) by mapping the diverging compressibility factor to finite values, as suggested by Torquato et al [181,201]. It must be noted that all φ J differ from ν max obtained in simulations by at most 10 −13 , which confirms that we are close to the infinite pressure limit.…”

“…In this section we show that sufficiently close to the jamming density, the compressibility factor is independent of the size distribution and depends on only one parameter, the jamming density itself, complementing an earlier study of monodisperse systems [181]. Figure 3.8 shows the compressibility factor scaled by equation (3.14) for systems with uniform size distributions.…”

“…Over time the additional energy created during collisions would accelerate the particles, but this is avoided by periodic rescaling of the average particle velocity to hold the mean temperature constant [180], such that the total change in kinetic energy of the system is kept small (below 1%). If the growing is sufficiently slow, the system will be in approximate equilibrium during the densification and one can rather efficiently gather quasi-equilibrium data as a function of density [116,181]. The number of particles N used in most simulations is 16 3 = 4096, if not explicitly stated otherwise.…”

“…The straightforward approach is to analyze the static structure factor, which is a direct measure of the local microstructure of particles [73]. The pair correlation function g(r) is also a popular quantity for the analysis of non-crystalline materials [181,237]. The problem with such quantities is that the detailed three-dimensional information is lost as a result of statistical averages, as also by considering only pair of particles.…”

Monte Carlo simulations of densely-packed athermal polymers in the bulk and under confinement

“…The later value, initially conjectured by Kepler and proven recently by Hales (Hales, 2005;Hales et al, 2010), corresponds to the packing density of the face-centered cubic (fee) lattice. While no such proof exists for the corresponding densest limit of random sphere packings the concept of the maximally random jammed (MRJ) state provides a precise mathematical and geometrical definition of the aforementioned state (Donev et al, 2005a(Donev et al, , 2005bTorquato et al, 2000).…”