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

Abstract

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.

Commentary and Notes on the Original Derivations of the Scharifker-Hills Model

“…For further reading regarding the local nucleation kinetics and concentration field around the nuclei, the reader is encouraged to follow up on papers. 21,[24][25][26] A.5. Calculation of the apparent occupation fraction of the diffusion zones using the KJMA model (box-8,9 in Fig.…”

“…Several references have been referred to develop the derivation process in the entire appendix. [1][2][3][4]8,[14][15][16][17][18][19][20][21][22][23][24][25][26] The List of Symbols and the definition of those symbols are presented at the beginning of the appendix of this commentary. Different from the perspectives in the literature, in Appendix A, we focus on building the physics picture of the S-H model, rather than a review of different past efforts in improving it.…”

Study of the Silicon Electrochemical Nucleation in LiCl-KCl-CsCl-K2SiF6 Melt

“…The presence of linear sections on both dependences and the probability of electric generation proceeding by any of the possible mechanisms are observed. 37,38 To clarify the nature of silicon electrocrystallization on a GC electrode, the dimensionless experimental dependences (i/i m ) 2 -t/t m were compared with similar model dependences derived by Scharifker and Hills for the cases of instantaneous and progressive nucleation. 36 The corresponding expressions for the model dependencies are given below:…”

Mechanism and Kinetics of the Phase Formation and Dissolution of NaxWO3 on a Pt Electrode in a Na2WO4–WO3 Melt

“…Obviously the mechanism of deposit formation in our case is more complex, so none of these models can be directly used for treatment of chronoamperograms. Nevertheless, a comparison of the shape of I(t) dependencies in Figure 3b with [53,61] apparently allows us to exclude the hypothesis of purely diffusion controlled growth. In addition, the low filling of the substrate with the deposit at t close to the time corresponding to the current maximum contradicts the theory of growth controlled by the discharge of depositing ions under potentiostatic conditions [53,57].…”