Analysis of probability of inserting a hard spherical particle with small diameter in hard-sphere fluid

The probability of inserting, without overlap, a hard spherical particle of diameter σ in a hard-sphere fluid of diameter σ₀ and packing fraction η determines its excess chemical potential at infinite dilution, μ^ex(σ, η). In our previous work [R. L. Davidchack and B. B. Laird, J. Chem. Phys. 157, 074701 (2022)], we used Widom’s particle insertion method within molecular dynamics simulations to obtain high precision results for μ^ex(σ, η) with σ/σ₀ ≤ 4 and η ≤ 0.5. In the current work, we investigate the behavior of this quantity at small σ. In particular, using the inclusion-exclusion principle, we relate the insertion probability to the hard-sphere fluid distribution functions and thus derive the higher-order terms in the Taylor expansion of μ^ex(σ, η) at σ = 0. We also use direct evaluation of the excluded volume for pairs and triplets of hard spheres to obtain simulation results for μ^ex(σ, η) at σ/σ₀ ≤ 0.2247 that are of much higher precision than those obtained earlier with Widom’s method. These results allow us to improve the quality of the small-σ correction in the empirical expression for μ^ex(σ, η) presented in our previous work.