Solving the inverse Knudsen problem: Gas diffusion in random fibrous media

About a century ago, Knudsen derived the groundbreaking theory of gas diffusion through straight pipes and holes, which since then found widespread application in innumerable fields of science and inspired the development of vacuum and related coating technologies, from academic research to numerous industrial sectors. Knudsen's theory can be straightforwardly applied to filter membranes with arrays of extended holes for example, however, for the inverse geometry arrangement, which arises when solid nanowires or fibers are arranged into porous interwoven material (like in carpets or brushes) the derivation of an analytical theory framework was still missing. In this paper, we have identified the specific geometric and thermodynamic parameters that determine the gas diffusion kinetics in arrays of randomly oriented cylinders and provide a set of analytical expressions allowing to comprehensively describe the gas transport in such structures. We confirmed analytical solutions by Monte Carlo simulations. We specify our findings for an atomic layer deposition process, the diffusion of trimethyaluminium molecules into a carbon nanotube array, but highlight the applicability of our derivation for other fields comprising gas diffusion membranes, composite materials, fuel cells and more.