Cavity formation energy in hard sphere fluids: An asymptotically correct expression

Exact geometrical relations valid for hard sphere (HS) fluids are used to derive analytical expressions for the cavity formation energy equal to the free energy cost of insertion of a HS solute into a HS solvent and the contact value of the solute-solvent pair distribution function (PDF) in the limit of the infinite solute dilution. In contrast to existing relations from the Boublik–Mansoori–Carnahan–Starling–Leland (BMCSL) equation of state, the derived expressions are self-consistent and result in correct asymptotics when the solute size goes to infinity. The proposed equations are tested against Monte Carlo simulations at diameter ratios $d$ in the range $\text{1⩽d⩽3.5}$ and three reduced densities 0.7, 0.8, and 0.9. The BMCSL theory is shown to systematically underestimate contact PDF values as compared to simulations both for finite solute concentrations and in the infinite dilution limit calculated by extrapolation of the results obtained at several concentrations. These infinite-dilution values of the solute-solvent PDF at contact calculated from simulations are in excellent agreement with the analytical expression derived in the paper. An analogy to the BMCSL equation for HS mixtures is used to extend this equation into the range of finite concentrations of the solute. The proposed equation is found to agree well with our simulation results.