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

Abstract

The Lab\'ik 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, $P_0(\eta,\sigma_0)$, of tracer hard-sphere (HS) particles of different diameters, $\sigma_0$, in a host HS fluid of diameter $\sigma$ and packing fraction, $\eta$, up to $0.5$. It is shown analytically that the only polynomial representation of $-\ln P_0(\eta,\sigma_0)$ consistent with the limits $\sigma_0\to 0$ and $\sigma_0\to\infty$ has necessarily a cubic form, $c_0(\eta)+c_1(\eta)\sigma_0/\sigma+c_2(\eta)(\sigma_0/\sigma)^2+c_3(\eta)(\sigma_0/\sigma)^3$. Our MD data for $-\ln P_0(\eta,\sigma_0)$ are fitted to such a cubic polynomial and the functions $c_0(\eta)$ and $c_1(\eta)$ are found to be statistically indistinguishable from their exact solution forms. Similarly, $c_2(\eta)$ and $c_3(\eta)$ agree very well with the Boubl\'ik-Mansoori-Carnahan-Starling-Leland and Boubl\'ik-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 $\sigma_{0}\to\sigma$, as $\beta\mu^{\text{ex}}=c_0+c_1+c_2+c_3$. Excellent agreement between the Carnahan-Starling and Carnahan-Starling-Kolafa theories with our MD data is evident.