A Linear Programming Algorithm to Test for Jamming in Hard-Sphere Packings

TL;DR

This paper introduces a rigorous linear programming algorithm to determine if hard-sphere packings are jammed, applicable to both ordered and random packings with various boundary conditions, and provides new insights into dimensional differences in jamming.

Contribution

The paper presents a novel, efficient linear programming algorithm for testing jamming in hard-sphere packings, advancing beyond previous heuristic methods.

Findings

Random packings are strictly jammed in 3D but not in 2D.

The algorithm can identify unjamming motions when packings are not jammed.

Applicable to finite, regular, and random packings with different boundary conditions.

Abstract

Jamming in hard-particle packings has been the subject of considerable interest in recent years. In a paper by Torquato and Stillinger [J. Phys. Chem. B, 105 (2001)], a classification scheme of jammed packings into hierarchical categories of locally, collectively and strictly jammed configurations has been proposed. They suggest that these jamming categories can be tested using numerical algorithms that analyze an equivalent contact network of the packing under applied displacements, but leave the design of such algorithms as a future task. In this work we present a rigorous and efficient algorithm to assess whether a hard-sphere packing is jammed according to the afformentioned categories. The algorithm is based on linear programming and is applicable to regular as well as random packings of finite size with hard-wall and periodic boundary conditions. If the packing is not jammed, the algorithm yields representative multi-particle unjamming motions. We have implemented this algorithm and applied it to ordered lattices as well as random packings of disks and spheres under periodic boundary conditions. Some representative results for ordered and disordered packings are given, but more applications are anticipated for the future. One important and interesting result is that the random packings that we tested were strictly jammed in three dimensions, but not in two dimensions. Numerous interactive visualization models are provided on the authors' webpage.