![]() This method was used to quantitatively demonstrate that the FCC lattice is more stable than the HCP lattice for hard-sphere systems. 2,3 In crystalline states, the SO cell model and unmodified hard-sphere system are assumed to be equal, as the particles themselves confine each other to the centers of the Voronoi polyhedra, thus preventing interactions with the SO cell boundaries. Regardless, thermodynamic integration can be carried out over this small transition to connect the crystalline and ideal gas states, and this was successfully used to settle a long debate over the most stable structure for mono-sized spheres. This model lowers the entropy of the fluid state by preventing free movement, and this softens, but does not eliminate, the first order freezing transition. The center of each particle is then constrained to remain in its respective cell. In the SO model, space is partitioned into cells based on Voronoi polyhedra created from the sites of the crystal lattice under investigation, for example, the face-centered cubic (FCC) lattice. Hoover and Ree originally introduced the single occupancy (SO) cell model 1 to demonstrate the existence of a first-order melting transition for hard-sphere systems and quantitatively determined the fluid and solid coexistence densities. There have been a number of attempts to modify model fluids to construct a continuous thermodynamic path between an ideal state and the crystalline state. 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. ![]() ![]() The approach is general and can be used for any system composed of particles interacting with discrete potentials in fluid, solid, or glassy states. Integrating the particle–particle collision rates with respect to the sphere diameter (or, equivalently, density) or the particle–tether collision rates with respect to the tether length then directly determines the volume of accessible phase space and, therefore, the system entropy. An intuitive derivation is given, which 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. 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. ![]()
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