Studies in Molecular Dynamics. V. High-Density Equation of State and Entropy for Hard Disks and Spheres

TLDR

The equations of state for hard disks and hard spheres in the solid phase have been accurately determined and used to evaluate the coefficients in the pressure expansion in powers of the relative free volume α. To expand other thermodynamic properties, the entropy constant was evaluated in the close‑packed limit by integrating the equation of state over the entire density range. The resulting pressure expansions are pV/NkT = 2/α + 1.90 + 0.67α for disks and 3/α + 2.56 + 0.56α for spheres, with cell models that include nearest‑neighbor correlations performing best; the Lennard‑Jones–Devonshire cell‑theory estimates are nearly correct, and no detectable pressure difference between hexagonal and face‑centered cubic packings leaves their relative stability unresolved.

Abstract

The equations of state for periodic systems of hard disks and hard spheres in the solid phase have been accurately determined and used to evaluate the coefficients in the expansion of the pressure in powers of the relative free volume, α = (V − V0) / V0, where V0 is the close-packed volume. For disks pV / NkT = 2 / α + 1.90 + 0.67α + O(α2) and for spheres pV / NkT = 3 / α + 2.56 + 0.56α + O(α2). These coefficients are compared to cell models, and those models which include correlations between neighboring particles work best. An equivalent expansion of other thermodynamic properties requires the entropy constant to be evaluated in the close-packed limit. This constant is obtained here by integrating the equation of state over the entire density region. The Lennard-Jones–Devonshire cell-theory estimates of the entropy constant are nearly correct; that is, the cell-theory estimate is too small by 0.06Nk for disks and too large by 0.24Nk for spheres. The pressure difference and hence the entropy difference between the hexagonal and face-centered cubic packings of spheres could not be detected, and thus the relative stability of these two phases remains an open question.