Divergence of the virial series for hard discs and hard spheres at closest packing

Abstract

Using a new series acceleration method, the authors find that the virial expansion for the pressure of hard discs and hard spheres is a monotonically increasing function of the number density rho and diverges at the density of closest packing with the critical exponent gamma =1. The virial series describes a metastable fluid phase for rho 0< rho < rho c, where rho 0 is the density of the fluid-solid transition. In the case of hard discs the known values of the virial coefficients Bn are well represented by Bn=(C-Dn-1) rho c-(n-1) for n>or=4, with C and D being positive constants. Adopting this approximation for n>or=7 leads to a simple, closed-form equation of state for the fluid phase which yields values of the pressure throughout the regime 0< rho < rho 0 which are only very slightly larger than the series acceleration estimates.