Probing surface structure via time-of-escape analysis of gas in Knudsen regime

Abstract

Continuing the study begun in [Feres, R., Yablonsky, G., 2004. Knudsen's cosine law and random billiards. Chemical Engineering Science 59, 1541–1556], we consider transport in the Knudsen regime of inert gases through straight channels and investigate how small-scale surface geometry of a macroscopically flat channel affects the diffusion characteristics of the gas. We show that the diffusivity constant contains information about the surface micro-geometry. Our investigation is carried out partly by analytical means and also through numerical simulation of what we call time-of-escape experiment. This is a type of multi-scattering experiment that measures the mean residence time in channels of varying lengths, allowing for an analysis of the transport process at various stages of development. We focus attention on the smoothness constant ξ = D / D 0 , defined as the ratio of diffusivity for a given micro-geometry, D, and diffusivity D 0 under “ideal roughness,” i.e., under the assumption that the Knudsen cosine law holds at each collision. The dependence of ξ on small-scale surface morphology is investigated using a variety of parametrized families of micro-geometries. We show that ξ can tell the presence of certain geometric features, such as curvature or sharp angles at the microscopic level. For example, for a simple two-dimensional model of atomically smooth surface made of a linear packing of spheres of radius R probed by gas molecules of radius A (Fig. 6), we obtain the approximate relation ξ ∝ 1 + A / R , with a proportionality constant of roughly 1.3. (See Section 2 for the range of parameters considered.) This relates to rapid diffusion phenomena recently observed by [Holt, J.K., Park, H.G., Wang, Y., Stadermann, M., Artyukhin, A.B., Grigoropoulos, C.P., Noy, A., Bakajin, O., 2006. Fast mass transport through sub-2-nanometer carbon nanotubes. Science 312, (1034)]. In the classical collision model of Maxwell–Smoluchowski, with no detailed regard of surface geometry, one always has ξ ⩾ 1 , but we observe that ξ can sometimes be less than 1 (Section 2.5). We obtain exact values for the mean number and duration of collisions as function of the micro-geometry (Section 3). This is needed to determine the relative importance of these quantities on diffusivity. We also extend the analysis of the scattering operator of random billiards of [Feres, R., Yablonsky, G., 2004. Knudsen's cosine law and random billiards. Chemical Engineering Science 59, 1541–1556] by giving a general criterion for the validity of Knudsen's cosine law, and describe some of the spectral properties of this operator.