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

A detailed comparison between the Boublík-Mansoori-Carnahan-Starling-Leland (BMCSL) equation of state of hard-sphere mixtures is made with Molecular Dynamics (MD) simulations of the same compositions. The Labík and Smith simulation technique [S. Labík and W. R. Smith, Mol. Simul. 12, 23-31 (1994)] was used to implement the Widom particle insertion method to calculate the excess chemical potential, βμ0ex, of a test particle of variable diameter, σ0, immersed in a hard-sphere fluid mixture with different compositions and values of the packing fraction, η. Use is made of the fact that the only polynomial representation of βμ0ex which is consistent with the limits σ0 → 0 and σ0 → ∞ has to be of the cubic form, i.e., c0(η)+c¯1(η)σ0/M1+c¯2(η)(σ0/M1)2+c¯3(η)(σ0/M1)3, where M1 is the first moment of the distribution. The first two coefficients, c0(η) and c¯1(η), are known analytically, while c¯2(η) and c¯3(η) were obtained by fitting the MD data to this expression. This in turn provides a method to determine the excess free energy per particle, βaex, in terms of c¯2, c¯3, and the compressibility factor, Z. Very good agreement between the BMCSL formulas and the MD data is found for βμ0ex, Z, and βaex 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 Boublík-Carnahan-Starling-Kolafa formulas and an interpolation between two Percus-Yevick routes capture well in different ranges of the system parameter space.