On the relation between virial coefficients and the close-packing of hard disks and hard spheres

TL;DR

This paper investigates whether virial coefficients can predict the maximum packing fraction of hard disks and spheres, proposing an alternative method that supports the idea that this limit equals the crystalline packing density.

Contribution

It introduces a new approach using virial coefficients and the equation of state to estimate the divergence point, supporting the conjecture linking it to crystalline packing.

Findings

Direct Padé approximants are inconclusive for predicting divergence.

The alternative approach aligns with the conjecture that divergence occurs at crystalline packing.

Supports the idea that the fluid's divergence point equals maximum crystalline packing fraction.

Abstract

The question of whether the known virial coefficients are enough to determine the packing fraction $\eta_\infty$ at which the fluid equation of state of a hard-sphere fluid diverges is addressed. It is found that the information derived from the direct Pad\'e approximants to the compressibility factor constructed with the virial coefficients is inconclusive. An alternative approach is proposed which makes use of the same virial coefficients and of the equation of state in a form where the packing fraction is explicitly given as a function of the pressure. The results of this approach both for hard-disk and hard-sphere fluids, which can straightforwardly accommodate higher virial coefficients when available, lends support to the conjecture that $\eta_\infty$ is equal to the maximum packing fraction corresponding to an ordered crystalline structure.