Cavities in the hard sphere fluid and crystal and the equation of state

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Abstract
In a hard sphere system regions where there is sufficient space to accommodate another sphere are called cavities. Exact relations between the number, size and surface area of cavities, and thermodynamic properties are reviewed, and the prospect of developing a theory of the D-dimensional hard sphere fluid and crystal, and of melting, by analysing the statistical geometry of cavities, is investigated theoretically. The equation of state is expressed as pV/RT = 1 + a(z)/<ν>1/D , where z is the density relative to close packing, a(z) is a well-behaved cavity shape factor and <ν> is the average cavity size. We show that in the high density limit <ν> varies as (1 - z) D and pV/RT varies as D/(1 - z). A function F(z) is defined such that the number of cavities per sphere, n c, is given 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 show that it is a well-behaved function that tends to a finite constant value in the high-density limit, where ln n c, pV/RT, ln <ν> and ΔS all diverge. These results are used to recover the asymptotic form of the equation of state derived in Salsburg and Wood's polytope theory, but without having to assume the existence of ‘stable close-packed configurations’. They provide expressions for the asymptotic density dependence of <ν> and n c and more explicit expressions for the thermodynamic properties at high density.

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