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A semi empirical expression of the virial coefficients for hard sphere fluids at all density

We propose a new semi empirical expression of the virial coefficients for a hard sphere fluid which is valid in the disordered phase over the whole density range. The results are in good agreement with the numerical data and better than those of the well-known Carnahan&Starling equation of state in the domain of validity of the later; moreover this new expression accounts for the singularity which is required for the equation of state at the close random packing limit.


Introduction
The random packing of a very high number of identical hard spheres in a large container is a useful model for simple fluids: rare gases at high density or liquid metals are examples of such fluids; it is also a model for colloids, granular and amorphous materials. Consequently, an enormous quantity of theoretical [1,2] as well as numerical [3,4,5,6] and experimental [7] works have been dedicated to the equation of state of hard sphere fluids.
An essential parameter is the packing factor ξ , which is the ratio of the actual volume of a system of spheres to the total volume they occupy. The pioneering experiments on model systems performed by Bernal et al. [8] and Scott et al. [9] have shown that the random packing density for hard spheres is very sensitive to experimental conditions. ξ can be increased up to a self-blocking situation ("loose random packing") whose precise value critically depends on the way the "fluid" has been poured into the container; by gently shaking or vibrating the container so as to simulate thermal agitation and to allow for local rearrangements, the packing density comes closer to an upper limit 0 ξ ("close random packing") which is found to be about 0.637 [10,11] when one extrapolates the experimental points to an infinite sample i.e. when the boundary effects become negligible.
Molecular dynamics and Monte Carlo-type calculations have confirmed those results but actually extensive numerical simulations have evidenced a more complex behaviour [12]: in the absence of any attractive term in the molecular interaction, there is no thermally induced liquid -gas phase transition but freezing of the fluid phase can occur when the packing density exceeds a value f 0.495 ξ ≅ . Above 0 ξ only the ordered phase exists, its density between ultimately capped by the fcc packing density fcc 2 / 6 0.7405 ξ = π ≅ . Between f ξ and 0 ξ the fluid phase is metastable [13,14]: a randomly packed fluid is very sensitive to shearing, in contrast with a regularly packed medium, and the close random packing limit 0 ξ can be exceeded if some shearing, locally creating ordered domains, is applied; the actual system then presents a mixture of ordered and disordered areas, the so-called "jammed states" [15,16,17]. Melting of the ordered phase occurs when the density goes down backwards below m 0.545 ξ ≅ .
Quite generally the fluid pressure can be expressed as an expansion in powers of the density, the "virial expansion". For a system of N identical hard spheres of diameter D in a container of volume V it can be written as So far a large number of theoretical or semi-empirical equations of state, usually expressed as have been proposed for hard sphere systems [2]. In particular the famous Carnahan & Starling equation of state [18] is obtained by rounding the known virial coefficients as That expression is known to work remarkably well in the stable phase, i.e. for ξ 0.5 <≅ , and even a little beyond in the metastable phase [19]. Now it is obvious that at high density it does not work any longer since it has a triple pole divergence for 1 = ξ , whereas the equation of state should exhibit a simple pole for 0 ξ ξ = ; actually it is known from the asymptotic expression of the virial coefficients [20] that the equation of state at high density takes the following form [21,22]: where λ is a numerical coefficient whose value is about 2.8; in the high density limit the virial coefficients are then given by ( ) In the present paper, starting from the existing numerical results for the virial coefficients, we intend to exhibit a compact expression for the equation of state which is simultaneously compatible both with the low and medium density behaviour (disordered stable phase) and with the asymptotic high density behaviour (disordered metastable phase). We will see that our resulting equation of state only depends on the following quantities: the virial coefficients 2 3 4 B , B , B , which can be exactly computed, the random packing density 0 ξ , which is imposed by statistical geometry, and the residue of the pole in 0 ξ .

Derivation of the virial coefficients
In order to account for the pole in 0 ξ ξ = it is necessary for the equation of state to have the following form: We develop 1 f (ξ) as a series of the density By construction, that equation of state is consistent with the low density limit (ξ 0) → and the high density limit 0 (ξ ξ ) → . In particular it is known that for hard spheres [1,2] i.e. for p 4 ≥ ( ) Obviously if p → ∞ then p b 0 → so that ( ) The p a coefficients and the derived ( ) and then smoothly go down to 0 for p 9 > .
(the low order virial coefficients are a priori more accurate than the high order ones).
Henceforth we will treat 1 ξ and β as adjustable independent parameters.
We then obtain the following equation of state:

Discussion
The following discussion will thus be restricted to the fluid phases.   and thus for the coefficients μ and ν although in all cases ν 3μ 1 + ≈ and ν 2 ≈ .) Equ. (24) can be re-written as: With ξ 0.7 < the second bracket in the numerator of equ. (25) is at least one of order of magnitude smaller than the first one; if we neglect it we finally get ( ) Noting that However it may well be a numerical coincidence.
Our The fact that 1 ξ is significantly above 0.77 means that the behaviour of the high order virial coefficients (beyond p 12 = ) changes with respect to the behaviour of the low order coefficients; actually we had shown in [20] that the transition between the low and medium We had also shown in [20] that the transition between the low and medium density regime and the high density regime is smooth and occurs around the freezing density. proposed for hard sphere fluids, many of them involving a more or less large set of parameters fitted with respect to the molecular dynamics calculations and/or to the numerically computed virial coefficients, e.g. via a Padé approximant approach (see in ref. [2] chapter 3 by Mulero et al., and more recently [26,27,28,29] [30]. The latter can be derived from the following free energy F : In ref. [30] the above expression is tentatively replaced by α , β and 0 ξ are supposed to be universal parameters; fitting those 3 quantities with respect to the molecular dynamics results of [4] and [5] gives 1 α ≅ , Although that equation of state shows good numerical results it predicts a double pole for the singularity in 0 ξ whereas a simple pole is expected and the fitted value for the random close packing limit is too high.

Conclusion
We have worked out a new compact semi empirical expression of the equation of state for a hard sphere fluids which is valid in the disordered phases over the whole density range below (stable phase) and above (metastable phase) the freezing point. The results are in a fairly good agreement with the available numerical data.