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 separates the narrowing delta-function contribution to $g_{2}$ due to emerging interparticle contacts from the background contribution due to near contacts. The data also shows 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 jammed random state. We clearly observe a maximum in $P_{f}$ and a nonzero probability of zero force. We computationally observe an unusual power-law divergence in the near-contact contribution to $g_{2}$, persistent even in the jamming limit, with an exponent of -0.4. We also present the first 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. Unlike previous studies, we find that ordering has a significant impact on the shape of $P_{f}$ for small forces.