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

A detailed comparison between the Boubl\'ik-Mansoori-Carnahan-Starling-Leland (BMCSL)equation of state of hard-sphere mixtures is made with Molecular Dynamics (MD) simulations of the same compositions. The Lab\'ik and Smith simulation technique [S. Lab\'ik and W. R. Smith, Mol. Simul. \textbf{12}, 23-31 (1994)] was used to implement the Widom particle insertion method to calculate the excess chemical potential, $\beta \mu_0^\text{ex}$, of a test particle of variable diameter, $\sigma_0$, immersed in a hard-sphere fluid mixture with different compositions and values of the packing fraction, $\eta$. Use is made of the fact that the only polynomial representation of $\beta \mu_0^\text{ex}$ which is consistent with the limits $\sigma_0\to 0$ and $\sigma_0\to\infty$ has to be of the cubic form, i.e., $c_0(\eta)+\overline{c}_1(\eta)\sigma_0/M_{1}+\overline{c}_2(\eta)(\sigma_0/M_{1})^2+\overline{c}_3(\eta)(\sigma_0/M_{1})^3$, where $M_{1}$ is the first moment of the distribution. The first two coefficients, $c_0(\eta)$ and $\overline{c}_1(\eta)$, are known analytically, while $\overline{c}_2(\eta)$ and $\overline{c}_3(\eta)$ were obtained by fitting the MD data to this expression. This in turn provides a method to determine the excess free energy per particle, $\beta a^\text{ex}$, in terms of $\overline{c}_2$, $\overline{c}_3$, and the compressibility factor, $Z$. Very good agreement between the BMCSL formulas and the MD data is found for $\beta \mu^\text{ex}_0$, $Z$, and $\beta a^\text{ex}$ for binary mixtures and continuous particle size distributions with the top-hat analytic form. However, the BMCSL theory typically slightly underestimates the simulation values, especially for $Z$, differences which the Boubl\'ik-Carnahan-Starling-Kolafa formulas and an interpolation between two Percus-Yevick routes capture well in different ranges of the system parameter space.