“Linear diffusion domain” approach for modeling the kinetics of electrodeposition: a two-dimensional study

Electrode Surface Coverage with Deposit Generated Under Conditions of Electrochemical Nucleation and Growth. A Mathematical Analysis

The theory of the diffusion limited electrochemical nucleation and growth of a deposit consisting of isolated 3D hemispherical nuclei has been re-analysed. The analysis focuses on a widely discussed model which assumes formation of “diffusion zones” around the growing nuclei. It has been proposed in the literature that the deposit-free fraction of the surface area of the substrate can be directly calculated from the substrate coverage with the “diffusion zones”. The aim of this work is to analyse whether such an approach can be applied for the growth of isolated 3D hemispherical nuclei. This is accomplished by evaluation of equations which describe nuclei radii at various stages of the deposition process. The formulae allow determining the substrate surface coverage with the growing deposit. This, in turn, allows simulating and analysing faradaic currents due to other than the electrodeposition reactions which take place at the deposit-free fraction of the substrate surface. Both instantaneous and progressive modes of the nucleation are discussed and the influence of the nucleation type on the faradaic currents is outlined. A comparison with other approaches reported in the literature indicates that the deposit-free fraction of the substrate surface may not always be determined by means of recalculation of the substrate coverage with the “diffusion zones”.

Graphical abstract

Simulation of Diffusion-Controlled Growth of Interdependent Nuclei under Potentiostatic Conditions

The problem of diffusion-controlled growth following an instantaneous nucleation event was studied within the framework of a new numerical model, considering the spatial distribution of hemispherical nuclei on the electrode surface and the mutual influence of growing nuclei via the collision of 3D diffusion fields. The simulation of the diffusion-controlled growth of hexagonal and random ensembles was performed at the overpotential-dependent number density of nuclei. The diffusion flow to each nucleus within a random ensemble was simulated by the finite difference method using the derived analytical expressions for the surface areas and the volumes formed at the intersection of 3D diffusion fields with the side faces of a virtual right prism with a Voronoi polygon base. The implementation of this approach provides an accurate calculation of concentration profiles, time dependences of the size of nuclei, and current transients. The results, including total current density transients, growth exponents, and nucleus size distribution, were compared with models developed within the concept of planar diffusion zones, the mean-field approximation and the Brownian dynamics simulation method, as well as with experimental data from the literature. The prospects of the model for studying the initial stages of electrocrystallization were discussed.