Jammed Hard-Particle Packings: From Kepler to Bernal and Beyond

TL;DR

This review explores the diversity, classification, and properties of jammed configurations of hard particles in Euclidean spaces, highlighting recent advances and open questions in the field.

Contribution

It provides a comprehensive classification scheme for jammed packings, introduces the concept of maximally random jamming, and discusses recent progress in various particle types and dimensions.

Findings

Classification of jammed states into local, collective, and strict categories

Introduction of maximally random jamming (MRJ) as a precise concept

Analysis of jammed packings across different particle shapes and dimensions

Abstract

This review describes the diversity of jammed configurations attainable by frictionless convex nonoverlapping (hard) particles in Euclidean spaces and for that purpose it stresses individual-packing geometric analysis. A fundamental feature of that diversity is the necessity to classify individual jammed configurations according to whether they are locally, collectively, or strictly jammed. Each of these categories contains a multitude of jammed configurations spanning a wide and (in the large system limit) continuous range of intensive properties, including packing fraction $\phi$, mean contact number $Z$, and several scalar order metrics. Application of these analytical tools to spheres in three dimensions (an analog to the venerable Ising model) covers a myriad of jammed states, including maximally dense packings (as Kepler conjectured), low-density strictly-jammed tunneled crystals, and a substantial family of amorphous packings. With respect to the last of these, the current approach displaces the historically prominent but ambiguous idea of ``random close packing" (RCP) with the precise concept of ``maximally random jamming" (MRJ). This review also covers recent advances in understanding jammed packings of polydisperse sphere mixtures, as well as convex nonspherical particles, e.g., ellipsoids, ``superballs", and polyhedra. Because of their relevance to error-correcting codes and information theory, sphere packings in high-dimensional Euclidean spaces have been included as well. We also make some remarks about packings in (curved) non-Euclidean spaces. In closing this review, several basic open questions for future research to consider have been identified.