Nonmonotonous pressure as a function of the density in a fluid without attractive forces

A simple result for the pressure of a hard sphere fluid that was developed many years ago by Rennert is extended in a straightforward manner by adding the terms that are of the same form as the Rennert's formula. The resulting expression is moderately accurate but its accuracy does not necessarily improve as additional terms are included. This expression has the interesting consequence that the pressure can have a maximum, as the density increases, which is consistent with the freezing of hard spheres. This occurs solely as a consequence of repulsive interactions. Only the Born-Green-Yvon and Kirkwood theories show such a behavior for hard spheres and they require a numerical solution of an integral equation. The procedure outlined here is ad hoc but is, perhaps, useful just as the popular Carnahan-Starling equation for the hard sphere pressure is also ad hoc but useful.


Introduction
For a number of years, the author has been intrigued by a result obtained by Rennert [1] for a hard sphere fluid. For this fluid, the interaction potential, u(R 12 ), for a pair of spheres whose centers are located at r 1 and r 2 , vanishes if the separation of the centers, R 12 = |r 1 −r 2 |, exceeds their diameter, d. Since the spheres are hard, they cannot overlap and u(R) is infinite if R < d. To keep the discussion simple, here the hard spheres are all assumed to be of the same diameter. Hard spheres are important because the 'structure' of a simple liquid or dense gas, such as argon, is, apart from the small perturbing effect of the attractive dispersion forces, determined by hard sphere interactions [2].
The connection between the pressure of a fluid of N molecules and their interactions is provided by the configurational partition function, Q N , where β = 1/kT , and k and T are the Boltzmann constant and temperature, respectively. The interaction energy U is assumed to be the pairwise additive sum of the pair potentials, The challenge is to determine the configurational partition function, since once it is known, the density dependent part of the Helmholtz function, A, and the pressure, p, for a given value of the density, ρ = N /V , where V is the volume, can be obtained from Since N is enormous (of the order of Avogradro's number), this can be done only approximately.

Rennert's method
Rennert considers a collection of systems that have n hard spheres, where n is a variable. He defines Reasoning by analogy to one dimension, Rennert suggests the ansatz from which, after some algebra, it follows that where v d and ǫ are the parameters to be chosen. In one dimension, Rennert chooses because for ρd = 1, the hard particles fill space and a singularity is to be expected. The virial coefficients for one dimensional hard particles that result from equation (8) are They are correct with ǫ = 0. The resulting equation of state is which, for hard particles in one dimension, is exact.

Extension
A reasonable extension of equation (7) is which, neglecting the terms in the sum for i > 4 leads to the following extension of equation (8)

43001-2
Nonmonotonous pressure as a function of the density in a fluid without attractive forces and Keeping only ǫ 1 yields Rennert's result. Each of the ǫ i can be chosen to give B i+1 , which are known to high order [2]. The result for B i that follows from the above result has a pleasing form. It is The sum is terminated when a negative term is encountered. If one wishes to add an additional term in equation (12)  Results for p/ρkT as a function of ρ for hard spheres (three dimensions) are given in figure 1 when the series is terminated after ǫ 1 , ǫ 3 , and ǫ 4 . The results for the case where only ǫ 1 is included because this is Rennert expression. In this case, the pressure is monotonous. Results for the cases when the series terminates after ǫ 2 are not included because the pressure is monotonous. The displayed curves are 43001-3 compared with the results of computer simulations taken from [2]. The simulation curves consist of two branches, one for the fluid branch and one for the solid branch, which terminates at close packing. The agreement is quite reasonable. However, the most intriguing feature is not the numerical accuracy of this procedure that, for the fluid branch, is not quite as good as that of the ad hoc Carnahan-Starling [3] expression but the fact that with terms through ǫ 3 or ǫ 4 included in equation (12), there is a maximum in the pressure at approximately the location of the transition from the fluid to the solid phases. Better results are given with ǫ 4 included but neglecting the contribution of ǫ 4 does a better job of locating the transition. It is reasonable to regard this maximum as indicative of the freezing of the hard sphere fluid, since a system with a negative compressiblity would expand with an increase of pressure, which is unphysical. The Carnahan-Starling expression does not give any indication of the presence of this transistion and, in fact, is unphysical at very high densities since it continues to give results for densities past close packing.
To be sure, the procedure reported here is ad hoc. There is no guarantee that including more ǫ i will continue to give better results or even a maximum pressure. In fact, this is the situation in two dimensions where neglecting ǫ 4 yields a maximum while including ǫ 4 does not. The point of this article is to point out that this procedure does give an indication of a phase transition in a hard sphere fluid. To the author's knowledge, beyond the more complex Born-Green-Yvon [4][5][6] and Kirkwood [7] approximations, this is the only theory to do so. This paper might probably point the way to a simple non-empirical description of the hard sphere transition and be useful in this regard.