Cavities and free volume in hard-disc and hard-sphere systems

Abstract

The pressure of a hard-sphere system (in 1, 2 and 3 dimensions and at all densities) can be expressed in terms of the surface area 〈s_i〉 and volume 〈v_i〉 of an average cavity. In contrast, the analogous expression for the pressure in terms of free volume requires the average 〈s_f/v_f〉 over the free volume of each sphere, which is not expressible in terms of an average free volume. We show that 〈s_i〉/〈v_i〉=〈s_f/v_f〉.

Numerical values for the number, size and variance of cavities in the hard-disc fluid and solid are calculated.

An incidental finding is that deviations from the ideal gas equation are given to a good approximation by p/ρkT–1 =⅕ exp 5z for discs and by p/ρkT–1 =⅖ exp 5z for spheres, between the low density percolation transition and the freezing density.