Atomic Layer Deposition on Porous Substrates: From General Formulation to Fibrous Substrates and Scaling Laws

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.