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

TL;DR

This study compares the BMCSL equation of state with Molecular Dynamics simulations to accurately determine the chemical potential of a test particle in hard-sphere mixtures, revealing good agreement and some underestimations.

Contribution

The paper introduces a method to fit MD data to a cubic polynomial form for the excess chemical potential, improving the understanding of hard-sphere mixture thermodynamics.

Findings

Good agreement between BMCSL and MD data for chemical potential and compressibility factor.

BMCSL slightly underestimates simulation values, especially for Z.

Alternative formulas better capture differences in certain parameter ranges.

Abstract

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.