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 $\mu^\mathrm{ex}(\sigma,\eta)$ of a test hard spherical particle of diameter $\sigma$ in a fluid of hard spheres of diameter $\sigma_0$ and packing fraction $\eta$ can be computed with high precision using Widom's particle insertion method [J.~Chem.~Phys.~{\bf 39}, 2808 (1963)] for $\sigma$ between 0 and just larger than 1 and/or small $\eta$. Heyes and Santos [J.~Chem.~Phys.~{\bf 145}, 214504 (2016)] showed analytically that the only polynomial representation of $\mu^\mathrm{ex}$ consistent with the limits of $\sigma$ at zero and infinity has a cubic form. On the other hand, through the solvation free energy relationship between $\mu^\mathrm{ex}$ and the surface free energy $\gamma$ of hard-sphere fluid at a hard spherical wall, we can obtain precise measurements of $\mu^\mathrm{ex}$ for large $\sigma$, extending up to infinity (flat wall) [J.~Chem. Phys.~{\bf 149}, 174706 (2018)]. Within this approach, the cubic polynomial representation is consistent with the assumptions of Morphometric Thermodynamics. In this work, we present measurements of $\mu^\mathrm{ex}$ that combine the two methods to obtain high-precision results for the full range of $\sigma$ values from zero to infinity, which show statistically significant deviations from the cubic polynomial form. We propose an empirical functional form for $\mu^\mathrm{ex}$ dependence on $\sigma$ and $\eta$ which better fits the measurement data while remaining consistent with the analytical limiting behaviour at zero and infinite $\sigma$.