Squarehyphen;well gas collision integrals and diffusion in a squarehyphen;well Lorentz gas

摘要

A representation for collision cross sections of a squarehyphen;well dilute gas mixture is developed that reduces their computation to the evaluation of onehyphen;dimensional integrals with integrands of elementary closed form. Consequently, the computation of collision integrals and Chapmanndash;Enskog and Kihara transporthyphen;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 coefficientDLof a Lorentz gas with squarehyphen;well potential interactions as a function of both squarehyphen;well width and temperature, i.e., squarehyphen;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 exactDLis compared with its first and second Chapmanndash;Enskog approximations,D1andDCE2, as well as its second Kihara approximation,DK2. (The first Kihara approximation coincides with the first Chapmanndash;Enskog approximation so that it is unnecessary to attach a superscript toD1to distinguish between the two approximation schemes.) TheDK2is 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 squarehyphen;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 lsquo;lsquo;softrsquo;rsquo; scattering are used to evaluateD1. For wide wells the latter yields an excellent approximation (which, in turn, is a good approximation toDLfor such wells). This approximation is no longer useful for narrow wells. As the temperature goes to zero,DLapproaches the diffusion coefficient of a hardhyphen;sphere Lorentz gas with interaction diameter equal to the hardhyphen;core diameter plus the squarehyphen;well width. This represents aspecial case of the general limiting coincidence of squarehyphen;well and squarehyphen;mound scattering in the infinitehyphen;strength limit.