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, βμ₀^ex, 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 βμ₀^ex 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, βa^ex, 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 βμ₀^ex, Z, and βa^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 Boublík-Carnahan-Starling-Kolafa formulas and an interpolation between two Percus-Yevick routes capture well in different ranges of the system parameter space.