Modeling and simulation of 3D electrochemical phase formation under mixed kinetic-diffusion growth control

Abstract

The model has been developed for the case of instantaneous nucleation and mixed kinetic-diffusion growth control of new-phase nuclei on the surface of an indifferent electrode under potentiostatic conditions. A new approach has been proposed that does not use the planar diffusion zone concept. The model provides a correct relationship between the concentration profiles to the nucleus and to the electrode due to spherical and planar diffusion, respectively. The influence of charge transfer and mass transfer parameters and the overpotential on the calculated current density transients and the average size of the nuclei has been analyzed. The simulated current density transients have been compared with the results of the calculation according to the Altimari–Pagnanelli and Scharifker–Hills models. The simulation results show that the mutual position of the current density transients under mixed control differs from that under diffusion control. It has been confirmed that the use of the dimensionless dependences of the Scharifker–Hills model at non-pure diffusion growth control can lead to serious errors in determining the mechanisms and electrocrystallization parameters.

Appendix

After substituting the result of differentiation of Eq. (13) at x = 0, as well as Eq. (18) at $\xi$ = r with coefficients (19) into the boundary condition (8), we obtain

$${B}_{2}{a}_{2}+{B}_{1}{a}_{1}+{B}_{0}{c}_{\text{s}}=\mathrm{exp}(-\beta f\eta )\;,$$

(29)

where

$$\begin{array}{l}{B}_{2}=\frac{{S}_{\text{e}}\mathrm\;{exp}(\alpha f\eta )}{\pi {\text{c}}_{0}}\cdot \frac{3{r}^{2}{R}^{2}-{R}^{4}-3{r}^{4}}{3{R}^{2}\left({R}^{2}-2{r}^{2}\right)}\\ {B}_{1}=\frac{{S}_{\text{e}}\mathrm\;{exp}(\alpha f\eta )}{2\pi {\text{c}}_{0}}\cdot \left(\frac{3{r}^{2}{R}^{2}-{R}^{4}-3{r}^{4}}{3{R}^{2}\left({R}^{2}-2{r}^{2}\right)}+\frac{1}{R}-\frac{1}{r}\right)-\frac{{S}_{\text{e}}zeD}{s{i}_{0}}\\ {B}_{0}=\frac{{S}_{\text{e}}\mathrm\;{exp}(\alpha f\eta )}{\pi {\text{c}}_{0}{R}^{2}}\end{array}\;.$$

(30)

The Gibbs-Thomson effect was not considered in the calculations, since its contribution is negligibly small at t ≥ 10⁻⁵ s for the chosen parameter values.

To describe the concentration profile near the electrode using Eq. (13), we supplement Eq. (29) with two more equations for the following two points of the profile:

$${a}_{2}{x}_{1}^{2}+{a}_{1}{x}_{1}+{c}_{\text{s}}={c}_{1}\;,$$

(31)

$${a}_{2}{x}_{2}^{2}+{a}_{1}{x}_{2}+{c}_{\text{s}}={c}_{2}\;.$$

(32)

We now construct a matrix of coefficients for the system of Eqs. (29), (31), and (32), as well as its inverse matrix to determine the linear relationship between c₁, c₂, and c_s,

$$\mathbf{G}=\left(\begin{array}{ccc}{x}_{1}^{2}& {x}_{1}& 1\\ {x}_{2}^{2}& {x}_{2}& 1\\ {B}_{2}& {B}_{1}& {B}_{0}\end{array}\right),\ \mathbf{G}^{-1}=\left(\begin{array}{ccc}{g}_{11}& {g}_{12}& {g}_{13}\\ {g}_{21}& {g}_{22}& {g}_{23}\\ {g}_{31}& {g}_{32}& {g}_{33}\end{array}\right)\;.$$

(33)

which is reflected in the formula

$${c}_{1}{g}_{31}+{c}_{2}{g}_{32}+\mathrm{exp}(\alpha f\eta ){g}_{33}={c}_{\text{s}}\;.$$

(34)

Complementing Eq. (34) with a system of equations for calculating the mass transfer by the backward Euler method,

$$-W_\text{Lj}c_\text{L}^\text{new}+W_\text{j}c^{\mathrm{new}}-W_\text{Rj}c_\text{R}^\text{new}=c^\text{old}\;,$$

(35)

where

$$\begin{array}{l}{W}_{\text{Lj}}=\frac{2D\Delta t}{\Delta {x}_{\text{j}}\left(\Delta {x}_{\text{j}}+\Delta {x}_{{\text{j}}+1}\right)}\\ {W}_{\text{j}}=1+\frac{2D\Delta t}{\Delta {x}_{\text{j}}\Delta {x}_{{\text{j}}+1}}\\ {W}_{\text{Rj}}=\frac{2D\Delta t}{\Delta {x}_{{\text{j}}+1}\left(\Delta {x}_{\text{j}}+\Delta {x}_{{\text{j}}+1}\right)}\end{array}\;,$$

(36)

we obtain the matrix M and the vector columns L and C:

$$\mathbf{M}=\left(\begin{array}{ccccccc}1& -{g}_{31}& -{g}_{32}& 0& 0& \cdots & 0\\ -{W}_{{\text{L}}1}& {W}_{1}& -{W}_{{\text{R}}1}& 0& 0& \cdots & 0\\ 0& -{W}_{{\text{L}}2}& {W}_{2}& -{W}_{{\text{R}}2}& 0& \cdots & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 0& 0& 0& 0& 0& \cdots & 1\end{array}\right)\;,$$

(37)

$$\mathbf L=\begin{pmatrix}g_{33}\;\exp(-\beta f\eta)\\c_1^\text{old}\\c_2^\text{old}\\\vdots\\c_0\end{pmatrix}C=\begin{pmatrix}c_\text{s}^\text{new}\\c_1^\text{new}\\c_2^\text{new}\\\vdots\\c_0\end{pmatrix}\;,$$

(38)

where superscripts old and new refer to the previous (k − 1) and current (k) time step. Finally, the matrix equation M × C = L is solved by the Jordan-Gauss method to calculate the concentration profile at each next time step.