A computer simulation study of cavities in the hard disc fluid and crystal

Abstract

The number and size of the cavities in a hard disc fluid and crystal are calculated in a computer simulation experiment. A cavity is a region where there is sufficient space to insert another disc. In the higher-density fluid and in the crystal the number of cavities per disc, n _c, closely follows the exact one-dimensional result n _c = exp(-zpV/RT), where z is the density relative to close packing, over 40 orders of magnitude. The average size of the cavities, <ν>, varies by only 3·5 orders of magnitude in the same density range, and, to within about 20%, <ν> varies as <ν> = [σ/(pV/RT - 1)]², where σ is the disc diameter. Across the freezing transition n _c <ν> is exactly constant. When the crystal melts to a fluid the number of cavities increases by about 50% and their size decreases in proportion, but their surface-to-volume ratio only decreases by 5%, showing that they have a more compact shape in the fluid. Above one-half of the close-packed density the computed values of n _c and <ν> are represented precisely by ln n _c = 1 - pV/RT - F(z) and ln <ν> = ΔS/R - ln(N/V) + F(z), where ΔS is the entropy relative to the ideal gas. F(z) is exactly zero in one dimension, and we find empirically that in two dimensions F(z) = -0·25 + 2 ln z in the crystal and F(z) = -2·2 - 2 ln z in the dense fluid. The number of vacancies per disc, n _v, in the crystal is measured and can be represented by n _v = n _c/[2(z - 0·75)]. At low density there is a large cavity that percolates. At z = 0·237 ± 0·003 there is equal probability of a cavity or a cluster percolating. The number of cavities reaches a maximum of one for every three discs at z = 0·38. Relations between cavity, cell and free volume theories are discussed empirically and theoretically.