On some properties of the hard-sphere radial distribution function

Thermodynamic properties of inverse power fluids

The local scaling behavior of the radial distribution function of the soft sphere or inverse power, r-n potential, fluid leads to a formula for the equation of state. From this formula different analytic forms for the compressibility factor, Z, have been derived. In the first, Z is expressed as a product of three functions, the hard sphere equation of state and two other functions incorporating the effects of the potential softness. In the second formula, the compressibility factor is cast in terms of the position and height of the first peak in the radial distribution function. In the final form, Z can be expressed as an exponential function which depends entirely on a combination of the virial coefficients. In each case Z is an explicit expression which has the correct low density limiting behavior and is accurate up to the freezing density for all packing fractions and circa n>or=12. Expressions are derived for the various component functions required for the different forms of Z, and relations between them are established. The compressibility factor manifests a maximum value or "ridge" when plotted as contours on the density-softness plane. It starts for the softer fluids at lower densities, increases with particle stiffness, and crosses the freezing line at n congruent with 33. From the compressibility factor other thermodynamic quantities can be obtained and the density-softness dependence of the infinite frequency limit elastic properties been determined. A self-consistent expression is derived for the effective hard sphere packing fraction (or equivalently, diameter), valid for all packing fractions and circa n>12. The effective hard-sphere diameter is compared with the formulas of Barker and Henderson, and Wheatley.

First derivative of the hard-sphere radial distribution function at contact

Molecular dynamics simulations have been carried out of the radial distribution function of the hard sphere fluid for a range of densities in the equilibrium fluid and just into the metastable region. The first derivative of the hard-sphere radial distribution function at contact was computed and its density dependence fitted to a simple analytic form. Comparisons were made with semi-empirical formulae from the literature, and of these the formula proposed by Tao et al (1992 Phys. Rev. A 46 8007) was found to be in best agreement with the simulation data, although it slightly underestimates the derivative at the higher packing fractions in excess of about 0.45. Close to contact, within a few per cent of the particle diameter, the radial distribution function can be represented well by a second order polynomial. An exponential function, which has some useful analytic features, can also be applied in this region.