Thermodynamically self consistent radial distribution functions for inverse power potentials

Computation of virial coefficients from integral equations

A polynomial-time method of computing the virial coefficients from an integral equation framework is presented. The method computes the truncated density expansions of the correlation functions by series transformations, and then extracts the virial coefficients from the density components. As an application, the method was used in a hybrid-closure integral equation with a set of self-consistent conditions, which produced reasonably accurate virial coefficients for the hard-sphere fluid and Gaussian model in high dimensions.

Thermodynamic properties and entropy scaling law for diffusivity in soft spheres

The purely repulsive soft-sphere system, where the interaction potential is inversely proportional to the pair separation raised to the power n, is considered. The Laplace transform technique is used to derive its thermodynamic properties in terms of the potential energy and its density derivative obtained from molecular dynamics simulations. The derived expressions provide an analytic framework with which to explore soft-sphere thermodynamics across the whole softness-density fluid domain. The trends in the isochoric and isobaric heat capacity, thermal expansion coefficient, isothermal and adiabatic bulk moduli, Grüneisen parameter, isothermal pressure, and the Joule-Thomson coefficient as a function of fluid density and potential softness are described using these formulas supplemented by the simulation-derived equation of state. At low densities a minimum in the isobaric heat capacity with density is found, which is a new feature for a purely repulsive pair interaction. The hard-sphere and n = 3 limits are obtained, and the low density limit specified analytically for any n is discussed. The softness dependence of calculated quantities indicates freezing criteria based on features of the radial distribution function or derived functions of it are not expected to be universal. A new and accurate formula linking the self-diffusion coefficient to the excess entropy for the entire fluid softness-density domain is proposed, which incorporates the kinetic theory solution for the low density limit and an entropy-dependent function in an exponential form. The thermodynamic properties (or their derivatives), structural quantities, and diffusion coefficient indicate that three regions specified by a convex, concave, and intermediate density dependence can be expected as a function of n, with a narrow transition region within the range 5 < n < 8.

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.