Chemical potential of a test hard sphere of variable size in a hard-sphere fluid

The Labík and Smith Monte Carlo simulation technique to implement the Widom particle insertion method is applied using Molecular Dynamics (MD) instead to calculate numerically the insertion probability, P₀(η,σ₀), of tracer hard-sphere (HS) particles of different diameters, σ₀, in a host HS fluid of diameter σ and packing fraction, η, up to 0.5. It is shown analytically that the only polynomial representation of -ln⁡P₀(η,σ₀) consistent with the limits σ₀→0 and σ₀→∞ has necessarily a cubic form, c₀(η)+c₁(η)σ₀/σ+c₂(η)(σ₀/σ)²+c₃(η)(σ₀/σ)³. Our MD data for -ln⁡P₀(η,σ₀) are fitted to such a cubic polynomial and the functions c₀(η) and c₁(η) are found to be statistically indistinguishable from their exact solution forms. Similarly, c₂(η) and c₃(η) agree very well with the Boublík-Mansoori-Carnahan-Starling-Leland and Boublík-Carnahan-Starling-Kolafa formulas. The cubic polynomial is extrapolated (high density) or interpolated (low density) to obtain the chemical potential of the host fluid, or σ₀→σ, as βμ^ex=c₀+c₁+c₂+c₃. Excellent agreement between the Carnahan-Starling and Carnahan-Starling-Kolafa theories with our MD data is evident.