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

Abstract

The Labík 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 Boublík-Mansoori-Carnahan-Starling-Leland and Boublík-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.

9 pages, 4 figures; v2: minor changes

References in corpus (2)

• Class of consistent fundamental-measure free energies for hard-sphere mixtures
• Chemical-Potential Route: A Hidden Percus-Yevick Equation of State for Hard Spheres

Cited by in corpus (3)

• Structural and Thermodynamic Properties of Hard-Sphere Fluids
• Carnahan Starling type equations of state for stable hard disk and hard sphere fluids
• Determining the chemical potential via universal density functional learning