Optimal Packings of Superballs

TL;DR

This paper analytically determines the densest packings of superballs, a versatile family of convex and concave particles, revealing how packing density varies with shape parameter p and offering insights into their phase behavior.

Contribution

The paper provides the first analytical constructions of the densest known packings for all superball shapes, including convex and concave cases, expanding understanding beyond spherical packings.

Findings

Maximal packing density increases dramatically as p moves away from 1.

Candidate packings are certain families of Bravais lattice packings.

Density function is nonanalytic at p=1 (sphere point).

Abstract

Dense hard-particle packings are intimately related to the structure of low-temperature phases of matter and are useful models of heterogeneous materials and granular media. Most studies of the densest packings in three dimensions have considered spherical shapes, and it is only more recently that nonspherical shapes (e.g., ellipsoids) have been investigated. Superballs (whose shapes are defined by |x1|^2p + |x2|^2p + |x3|^2p <= 1) provide a versatile family of convex particles (p >= 0.5) with both cubic- and octahedral-like shapes as well as concave particles (0 < p < 0.5) with octahedral-like shapes. In this paper, we provide analytical constructions for the densest known superball packings for all convex and concave cases. The candidate maximally dense packings are certain families of Bravais lattice packings. The maximal packing density as a function of p is nonanalytic at the sphere-point (p = 1) and increases dramatically as p moves away from unity. The packing characteristics determined by the broken rotational symmetry of superballs are similar to but richer than their two-dimensional "superdisk" counterparts, and are distinctly different from that of ellipsoid packings. Our candidate optimal superball packings provide a starting point to quantify the equilibrium phase behavior of superball systems, which should deepen our understanding of the statistical thermodynamics of nonspherical-particle systems.