Radial distribution function of penetrable sphere fluids to second order in density

TL;DR

This paper derives exact second-order density expansions for penetrable sphere fluids, analyzing their thermodynamic properties and comparing theoretical predictions with Monte Carlo results across different temperature regimes.

Contribution

It provides the first exact second-order density expansion for penetrable sphere fluids, including the cavity function and virial coefficient, with detailed comparison to HNC and PY theories.

Findings

Exact expressions for cavity function and virial coefficient derived.

PY theory better than HNC for low temperatures, but less accurate overall.

Virial coefficient changes sign around T* ≈ 1.1, indicating complex temperature dependence.

Abstract

The simplest bounded potential is that of penetrable spheres, which takes a positive finite value $\epsilon$ if the two spheres are overlapped, being 0 otherwise. In this paper we derive the cavity function to second order in density and the fourth virial coefficient as functions of $T^*\equiv k_BT/\epsilon$ (where $k_B$ is the Boltzmann constant and $T$ is the temperature) for penetrable sphere fluids. The expressions are exact, except for the function represented by an elementary diagram inside the core, which is approximated by a polynomial form in excellent agreement with accurate results obtained by Monte Carlo integration. Comparison with the hypernetted-chain (HNC) and Percus-Yevick (PY) theories shows that the latter is better than the former for $T^*\lesssim 1$ only. However, even at zero temperature (hard sphere limit), the PY solution is not accurate inside the overlapping region, where no practical cancelation of the neglected diagrams takes place. The exact fourth virial coefficient is positive for $T^*\lesssim 0.73$, reaches a minimum negative value at $T^*\approx 1.1$, and then goes to zero from below as $1/{T^*}^4$ for high temperatures. These features are captured qualitatively, but not quantitatively, by the HNC and PY predictions. In addition, in both theories the compressibility route is the best one for $T^*\lesssim 0.7$, while the virial route is preferable if $T^*\gtrsim 0.7$.