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

Abstract

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, $P_0(η,σ_0)$, of tracer hard-sphere (HS) particles of different diameters, $σ_0$, 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 $-\ln P_0(η,σ_0)$ consistent with the limits $σ_0\to 0$ and $σ_0\to\infty$ has necessarily a cubic form, $c_0(η)+c_1(η)σ_0/σ+c_2(η)(σ_0/σ)^2+c_3(η)(σ_0/σ)^3$. Our MD data for $-\ln P_0(η,σ_0)$ are fitted to such a cubic polynomial and the functions $c_0(η)$ and $c_1(η)$ are found to be statistically indistinguishable from their exact solution forms. Similarly, $c_2(η)$ and $c_3(η)$ 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 $σ_{0}\toσ$, as $βμ^{\text{ex}}=c_0+c_1+c_2+c_3$. Excellent agreement between the Carnahan-Starling and Carnahan-Starling-Kolafa theories with our MD data is evident.