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

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 $T^{\ast }\equiv k_{B}T/\epsilon$ (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 $T^{\ast }\lesssim 1$ 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 $T^{\ast }\lesssim 0.73$, reaches a minimum negative value at $T^{\ast }\approx 1.1$, and then goes to zero from below as $1/{T^{\ast }}^4$ 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 $T^{\ast }\lesssim 0.7$, while the virial route is preferable if $T^{\ast }\gtrsim 0.7$.

10 pages, 2 figures; v2: minor changes; to be published in PRE

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• Alternative Approaches to the Equilibrium Properties of Hard-Sphere Liquids
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• The effective colloid interaction in the Asakura-Oosawa model. Assessment of non-pairwise terms from the virial expansion
• Molecular dynamics simulation study of self-diffusion for penetrable-sphere model fluids
• Thermodynamic consistency between the energy and virial routes in the mean spherical approximation for soft potentials
• Simple relationship between the virial-route hypernetted-chain and the compressibility-route Percus--Yevick values of the fourth virial coefficient