Close packing density of polydisperse hard spheres

TL;DR

This paper introduces a simple, accurate approximation for the random close packing density of polydisperse hard spheres, validated through extensive simulations, and applicable to various size distributions.

Contribution

It presents a novel approximation method for packing density of polydisperse spheres based on a mapping to a one-dimensional problem, validated by simulations.

Findings

The approximation accurately predicts packing densities for various distributions.

Simulation results show weak dependence of packing density on fluid viscosity and particle size.

The theory matches known limits for large size ratios.

Abstract

The most efficient way to pack equally sized spheres isotropically in 3D is known as the random close packed state, which provides a starting point for many approximations in physics and engineering. However, the particle size distribution of a real granular material is never monodisperse. Here we present a simple but accurate approximation for the random close packing density of hard spheres of any size distribution, based upon a mapping onto a one-dimensional problem. To test this theory we performed extensive simulations for mixtures of elastic spheres with hydrodynamic friction. The simulations show a general (but weak) dependence of the final (essentially hard sphere) packing density on fluid viscosity and on particle size, but this can be eliminated by choosing a specific relation between mass and particle size, making the random close packed volume fraction well-defined. Our theory agrees well with the simulations for bidisperse, tridisperse and log-normal distributions, and correctly reproduces the exact limits for large size ratios.