Local stability of spheres via the convex hull and the radical Voronoi diagram

TL;DR

This paper introduces geometric methods to identify rattlers in sphere packings, simplifying the process and providing insights into local stability using convex hulls and radical Voronoi diagrams.

Contribution

It proposes two geometric classifications of rattlers based on convex hulls and radical Voronoi cells, generalizable to various packing and network types.

Findings

Convex hull and radical Voronoi methods effectively identify rattlers.

The geometric approach offers a simpler alternative to existing methods.

Applicability extends to hyperstatic packings and non-spherical particles.

Abstract

Jamming is an emergent phenomenon wherein the local stability of individual particles percolates to form a globally rigid structure. However, the onset of rigidity does not imply that every particle becomes rigid, and indeed some remain locally unstable. These particles, if they become unmoored from their neighbors, are called \textit{rattlers}, and their identification is critical to understanding the rigid backbone of a packing, as these particles cannot bear stress. The accurate identification of rattlers, however, can be a time-consuming process, and the currently accepted method lacks a simple geometric interpretation. In this manuscript, we propose two simpler classifications of rattlers based on the convex hull of contacting neighbors and the maximum inscribed sphere of the radical Voronoi cell, each of which provides geometric insight into the source of their instability. Furthermore, the convex hull formulation can be generalized to explore stability in hyperstatic soft sphere packings, spring networks, non-spherical packings, and mean-field non-central-force potentials.