Molecular Dynamics. VI. Free-Path Distributions and Collision Rates for Hard-Sphere and Square-Well Molecules

Free-path distributions and collision rates for systems of hard-sphere and square-well molecules are computed and compared to simple models. Mean free paths for solids and dense fluids are found to be within 2% of those predicted by dividing the kinetic-theory mean free path by the relative probability of two molecules in contact. These mean free paths are much smaller than a mean intermolecular spacing or even the diameter of a free volume. For hard-sphere and hard-disk fluids, free-path distributions are found to be monotone decreasing, nearly exponential, and when scaled by the mean free path, nearly density independent for all fluid densities. This density independence precludes any essential difference in transport mechanisms between dilute and dense fluids based on independent free paths. Scaled free-path distributions for hard-sphere and hard-disk solids are found to agree with those of their fluid counterparts except for long free paths where the smaller free-path distribution for solids reflects their increased molecular localization. The free-volume theory and its simple extensions are shown to be qualitatively inconsistent with the shape of observed free-path distributions in solids. For fluids of square-well molecules, the rate of hard-core collisions is found to be insensitive to the presence of the square well, and so is the free-path distribution as well as selected values of the pair-distribution function. These features are in good agreement with the van der Waals hypothesis, which neglects soft collisions. For dense liquids, hard-core collisions are in the majority, invalidating the hypothesis that soft-core collisions cause a Brownian motion between successive hard-core collisions.