Closed virial equations for hard parallel cubes and squares

TL;DR

This paper derives closed-form virial equations for hard parallel cubes and squares, revealing their limitations in representing fluid phases near freezing and highlighting similarities with sphere packings.

Contribution

It introduces closed-virial equations for cubes and squares based on observed behaviors of virial coefficients and their relation to close-packing.

Findings

Virial coefficients Bn show similar behavior to spheres and disks.

Virial pressures deviate from MD pressures below crystallization densities.

Limitations of Mayer cluster expansion in fluid phase modeling are confirmed.

Abstract

A correlation between maxima in virial coefficients (Bn), and "kissing" numbers for hard hyper-spheres up to dimension D=5, indicates a virial equation and close-packing relationship. Known virial coefficients up to B7, both for hard parallel cubes and squares, indicate that the limiting differences Bn-B(n-1) behave similar to spheres and disks, in the respective expansions relative to maximum close packing. In all cases, the increment Bn-B(n-1) will approach a negative constant with similar functional form in each dimension. This observation enables closed-virial equations-of-state for cubes and squares to be obtained. In both the 3D and 2D cases, the virial pressures begin to deviate from MD thermodynamic pressures at densities well-below crystallization. These results consolidate the general conclusion, from previous papers on spheres and disks, that the Mayer cluster expansion cannot represent the thermodynamic fluid phases up to freezing as commonly assumed in statistical theories.