Square-well gas collision integrals and diffusion in a square-well Lorentz gas

Abstract

A representation for collision cross sections of a square-well dilute gas mixture is developed that reduces their computation to the evaluation of one-dimensional integrals with integrands of elementary closed form. Consequently, the computation of collision integrals and Chapman–Enskog and Kihara transport-coefficient approximations, which depend on the collision cross sections, is analytically and numerically simplified. These results are then applied to the limiting case of a Lorentz gas, i.e., a dilute gas of mobile particles diffusing through a bed of scatterers that can be regarded as fixed. In particular, our results are used to evaluate the exact diffusion coefficient DL of a Lorentz gas with square-well potential interactions as a function of both square-well width and temperature, i.e., square-well depth. (We treat here only the case in which the scatterers are dilute enough for the problem to be accurately described by a Boltzmann equation.) The exact DL is compared with its first and second Chapman–Enskog approximations, D1 and DCE2, as well as its second Kihara approximation, DK2. (The first Kihara approximation coincides with the first Chapman–Enskog approximation so that it is unnecessary to attach a superscript to D1 to distinguish between the two approximation schemes.) The DK2 is found to be of high accuracy over the whole range of parameters studied, suggesting that it represents the most accurate of these three approximations available for the binary diffusion coefficient of a dilute square-well gas. Finally, a simple approximation for the collision integrals linear in well depth (i.e., inverse temperature) as well as one that adds to the linear term all contributions from ‘‘soft’’ scattering are used to evaluate D1. For wide wells the latter yields an excellent approximation (which, in turn, is a good approximation to DL for such wells). This approximation is no longer useful for narrow wells. As the temperature goes to zero, DL approaches the diffusion coefficient of a hard-sphere Lorentz gas with interaction diameter equal to the hard-core diameter plus the square-well width. This represents a special case of the general limiting coincidence of square-well and square-mound scattering in the infinite-strength limit.