Tethered-particle model: The calculation of free energies for hard-sphere systems

Abstract

Two methods for computing the entropy of hard-sphere systems using a spherical tether model are explored which allow the efficient use of event-driven molecular-dynamics simulations. An intuitive derivation is given that relates the rate of particle collisions, either between two particles or between a particle and its respective tether, to an associated hypersurface area which bounds the system's accessible configurational phase-space. Integrating the particle-particle collision rates with respect to sphere diameter (or, equivalently, density) or the particle-tether collision rates with respect to tether length then directly determines the volume of accessible phase space and, therefore, the system entropy. The approach is general and can be used for any system composed of particles interacting with discrete potentials in fluid, solid, or glassy states. The entropies calculated for the liquid and crystalline hard-sphere states using these methods are found to agree closely with the current best estimates in the literature, demonstrating the accuracy of the approach.

From Atoms to Colloids: Does the Frenkel Line Exist in Discontinuous Potentials?

“…In addition, simulations were performed along the isochore ρσ 3 = 1.3 for temperatures from k B T /ε = 1 to infinity (the hard-sphere limit). The free energy of the solid fcc phase could then be integrated with respect to the hard-sphere system, which was calculated using the equation of state of the hard-sphere solid taken from ref , and the residual Helmholtz free energy at ρσ 3 = 1.21 is 7.984 ± 0.001 …”

“…The free energy of the solid fcc phase could then be integrated with respect to the hard-sphere system, which was calculated using the equation of state of the hard-sphere solid taken from ref 40, and the residual Helmholtz free energy at ρσ 3 = 1.21 is 7.984 ± 0.001. 41 Based on the intersection of the vapor−liquid and solid− liquid coexistence curves, the triple point is estimated to be ρ t σ 3 ≈ 0.61, k B T/ε ≈ 2.16, and p c σ 3 /ε ≈ 0.056. Our melting curve is in good agreement with previous estimates using the cell model.…”

“…The free energy of the solid fcc phase could then be integrated with respect to the hard-sphere system, which was calculated using the equation of state of the hard-sphere solid taken from ref ( 40 ), and the residual Helmholtz free energy at ρσ 3 = 1.21 is 7.984 ± 0.001. 41 …”

Comparing four hard-sphere approximations for the low-temperature WCA melting line

“…Turning next to the freezing density, we first note that the HS fluid freezing density has been computed recently by Moir, Lue and Bannerman to the value [91] ρ…”