Solving the Stefan problem for a solid phase growth on plane plate and spherical surfaces and testing of theoretical equations

Abstract

Solutions of the Stefan problem in the 2D space considering a moving boundary of a solid deposit growing under mass transfer control on either plane plate or spherical solid substrates are reported. In the former case, the displacement of the growth front at the plane plate occurs perpendicularly to the substrate, whereas for the latter it shifts radially. For both substrates, in the absence of convection and surface roughness effects, the phase growth kinetics is determined by diffusion and advection, the latter being due to the linear displacement of the growth front with time. For both geometric arrangements the theory predicts two limiting kinetic situations, namely a diffusion control when the time and/or the radius of the substrate approach zero, and an advection control for the reverse conditions. For the spherical substrate, when its radius tends to infinity, the kinetics of the process approaches that found at the plane plate substrate. Theoretical potentiostatic current density transients are tested utilising growth pattern data for the formation of 2D silver dense branching electrodeposits on a plane plate cathode in a quasi-2D cell, and silver electrodeposits on spherical cathodes employing a high viscosity plating solutions.