Kinetics of island growth in the framework of “planar diffusion zones” and “3D nucleation and growth” models for electrodeposition

Observation of Weibull, Lognormal, and Gamma Distributions in Electrodeposited Cu and Cu-Ag Particles

In this work, the nearest-neighbor distances and Voronoi cell features of Cu-Ag deposits were analyzed and fitted with Lognormal, Weibull, and Gamma distributions. The nearest-neighbor distance distributions of the samples were compared with those of complete spatially random points, showing spatial inhomogeneity due to the nucleation exclusion effect. The radial distribution function was calculated, showing both influences from the grain size and the nucleation exclusion effect. Voronoi cells were generated based on the shape of the grains. The size, occupancy, and coordination of the Voronoi cells were examined and fitted. The results show that although the Cu-Ag deposits seemed to be governed by the instantaneous nucleation mode, the spatial distribution of the nuclei was more impacted by the nucleation exclusion effect than the Cu-only samples. This behavior is also justified by the grain size distribution generated with Voronoi cell size and occupancy distributions.

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.