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.@en

Atomic layer deposition (ALD) is a technique of choice for a uniform, conformal coating of substrates of complex geometries, owing to its characteristic self-limiting surface reactions upon sequential exposure to precursor vapors. In order to achieve a uniform coating, sufficient gas exposure needs to be provided. This requirement becomes particularly relevant for highly porous and high aspect-ratio substrates, where the gas transport into the substrate structure is limited by diffusion (diffusion-limited regime), or for ALD precursor systems exhibiting a low surface reaction rate (reaction-limited regime). This work reports how the distinction between diffusion- and reaction-limited ALD regimes is directly quantitatively related to the width of the reaction front and the profile of chemisorption coverage in a single-cycle ALD, all of them being determined by the natural length unit of the system. We introduce a new parametrization of the system based on its natural system of units, dictated by the scales of the physical phenomena governing the process. We present a range of scaling laws valid for a general porous substrate, which scale intuitively with the natural units of the system. The scaling laws describe (i) the coating depth in a diffusion-limited regime with respect to the gas exposure, (ii) the chemisorption coverage in a reaction-limited regime with respect to the gas exposure, and (iii) the width of the reaction zone in the diffusion-limited regime. For the first time, the distinction between diffusion- and reaction-limited ALD regimes is directly quantitatively related to the width of the reaction zone and the profile of chemisorption coverage in a single-cycle ALD. The model system for the multicycle diffusion-limited coating of random fibrous mats was validated with an experiment of ALD on a forest of tortuous carbon nanotubes. @en