Revisiting Maxwell-Smoluchowski theory: low surface roughness in straight channels

TL;DR

This paper refines the Maxwell-Smoluchowski theory for gas diffusion in straight channels by linking the diffuse collision fraction to detailed micro- and macro-geometric parameters, improving understanding of gas-surface interactions in the Knudsen regime.

Contribution

It provides a new expression for the diffuse collision fraction  in terms of micro- and macro-geometric parameters, enhancing the classical MS theory with microgeometry considerations.

Findings

 can be explicitly expressed using geometric parameters.

The surface microgeometry influences the velocity space diffusion operator.

The refined model improves predictions of gas transport in microchannels.

Abstract

The Maxwell-Smoluchowski (MS) theory of gas diffusion is revisited here in the context of gas transport in straight channels in the Knudsen regime of large mean free path. This classical theory is based on a phenomenological model of gas-surface interaction that posits that a fraction $\vartheta$ of molecular collisions with the channel surface consists of diffuse collisions, i.e., the direction of post-collision velocities is distributed according to the Knudsen Cosine Law, and a fraction $1-\vartheta$ undergoes specular reflection. From this assumption one obtains the value $\mathcal{D}=\frac{2-\vartheta}{\vartheta}\mathcal{D}_K$ for the self-diffusivity constant, where $\mathcal{D}_K$ is a reference value corresponding to $\vartheta=1$. In this paper we show that $\vartheta$ can be expressed in terms of micro- and macro-geometric parameters for a model consisting of hard spheres colliding elastically against a rigid surface with prescribed microgeometry. Our refinement of the MS theory is based on the observation that the classical surface scattering operator associated to the microgeometry has a canonical velocity space diffusion approximation by a generalized Legendre differential operator whose spectral theory is known explicitly. More specifically, starting from an explicit description of the effective channel surface microgeometry -- a concept which incorporates both the actual surface microgeometry and the molecular radius -- and using this operator approximation, we show that $\vartheta$ can be resolved into easily obtained geometric parameters.