Chemical potential and surface free energy of a hard spherical particle in hard-sphere fluid over the full range of particle diameters

The excess chemical potential μ^ex(σ, η) of a test hard spherical particle of diameter σ in a fluid of hard spheres of diameter σ₀ and packing fraction η can be computed with high precision using Widom's particle insertion method [B. Widom, J. Chem. Phys. 39, 2808 (1963)] for σ between 0 and just larger than 1 and/or small η. Heyes and Santos [J. Chem. Phys. 145, 214504 (2016)] analytically showed that the only polynomial representation of μ^ex consistent with the limits of σ at zero and infinity has a cubic form. On the other hand, through the solvation free energy relationship between μ^ex and the surface free energy γ of hard-sphere fluids at a hard spherical wall, we can obtain precise measurements of μ^ex for large σ, extending up to infinity (flat wall) [R. L. Davidchack and B. B. Laird, J. Chem. Phys. 149, 174706 (2018)]. Within this approach, the cubic polynomial representation is consistent with the assumptions of morphometric thermodynamics. In this work, we present the measurements of μ^ex that combine the two methods to obtain high-precision results for the full range of σ values from zero to infinity, which show statistically significant deviations from the cubic polynomial form. We propose an empirical functional form for the μ^ex dependence on σ and η, which better fits the measurement data while remaining consistent with the analytical limiting behavior at zero and infinite σ.