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 βμex=c0+c1+c2+c3. Excellent agreement between the Carnahan-Starling and Carnahan-Starling-Kolafa theories with our MD data is evident.