How dense can one pack spheres of arbitrary size distribution?

TL;DR

This paper introduces a systematic algorithm to estimate the maximum packing density of spheres with arbitrary size distributions, applicable in materials like ceramics and concrete, using an Apollonian filling rule.

Contribution

The paper presents the first algorithm for estimating maximum sphere packing density across arbitrary size distributions, including real-world and empirical data, with applications in material engineering.

Findings

Power-law distributions yield densest packings

Algorithm effectively estimates packing densities for various distributions

Proposes schemes for low-porosity grain size distributions

Abstract

We present the first systematic algorithm to estimate the maximum packing density of spheres when the grain sizes are drawn from an arbitrary size distribution. With an Apollonian filling rule, we implement our technique for disks in 2d and spheres in 3d. As expected, the densest packing is achieved with power-law size distributions. We also test the method on homogeneous and on empirical real distributions, and we propose a scheme to obtain experimentally accessible distributions of grain sizes with low porosity. Our method should be helpful in the development of ultra-strong ceramics and high performance concrete.