Sharp-interface model of electrodeposition and ramified growth

We present a sharp-interface model of two-dimensional ramified growth during quasisteady electrodeposition. Our model differs from previous modeling methods in that it includes the important effects of extended space-charge regions and nonlinear electrode reactions. The electrokinetics is described by a continuum model, but the discrete nature of the ions is taken into account by adding a random noise term to the electrode current. The model is validated by comparing its behavior in the initial stage with the predictions of a linear stability analysis. The main limitations of the model are the restriction to two dimensions and the assumption of quasisteady transport.

Figure 1

Sketch of the initial geometry of the system. Two coplanar metal electrodes of width $W$ are placed a distance of $2L$ apart. The gap between them is filled by an electrolyte with cation concentration $c_{+}$ and anion concentration $c_{-}$. A voltage difference of $V_0$ is applied between the electrodes.

Figure 2

Sketch of the electrode growth. The electrode surface at time $t_i$ is indicated with a solid line. In the time step $t_{i+1}-t_i$ an amount of material $\mathrm{\Delta }L$, which may vary with position, is deposited on the electrode. On the basis of the deposited material, the geometry at time $t_{i+1}$ is created (indicated with a dashed line).

Figure 3

Three-dimensional extension of our two-dimensional model. The electrode interface can vary in the $xy$ plane according to the calculated ion currents, but it has a fixed depth $\mathrm{\Delta }h$ in the $z$ direction. The interface is also divided into a number of bins of width $\mathrm{\Delta }s$ in the $xy$ plane. Each bin thus has the area $\mathrm{\Delta }h\mathrm{\Delta }s$.

Figure 4

Example of the simplifying cutting procedure. The reduced interface (thick black line) divides the domain into an active region (white) and a passive region (light gray). The dark gray area shows the real cathode. The example is taken from a simulation with $c_0=1\text{mM}$ and $V_0=10$ after deposition for 31 h and 28 min.

Figure 5

Example of domain meshing at varying magnification. The example is taken from a simulation with $c_0=1\text{mM}$ and $V_0=10$ after deposition for 7 h and 50 min. The wiggly black line is the cathode surface. The light gray lines are the mesh boundaries and the dark (red) lines show the sections that are magnified. The mesh elements above the cathode surface are only used for storing the solution between time steps.

Figure 6

Power spectra averaged over 50 runs for three different mesh sizes $\mathrm{\Delta }s=\{0.1{\lambda }_{\mathrm{c}}^{},0.2{\lambda }_{\mathrm{c}}^{},0.4{\lambda }_{\mathrm{c}}^{}\}$. In each run we used $M=100$ time steps of $\mathrm{\Delta }t=0.64\text{s}$ and the parameter values $c_0=10\text{mM}, L=100\mu \text{m}$, and $V_0=30$. The solid black line shows the analytical result and the dashed black lines show the analytical standard error on the mean. The result for $\mathrm{\Delta }s=0.1{\lambda }_{\mathrm{c}}^{}$ is shown in dark gray (red), the result for $\mathrm{\Delta }s=0.2{\lambda }_{\mathrm{c}}^{}$ is shown in medium gray (red), and the result for $\mathrm{\Delta }s=0.4{\lambda }_{\mathrm{c}}^{}$ is shown in light gray (red).

Figure 7

Electrodeposits in the $V_0-c_0$ plane obtained for $L=100\mu \text{m}, c_0=\{1\text{mM},10\text{mM},100\text{mM}\}$, and $V_0=\{10,20,30\}$. The aspect ratio varies between the panels, since the width $W$ of the simulated region is always set to $200{\lambda }_{\mathrm{c}}$. The gray area has different shades corresponding to times $t_0$ (light), $0.75t_0$ (darker), $0.5t_0$ (darker yet), $0.25t_0$ (darkest). The white line indicates the reduced surface at time $t_0$. The contours in the liquid represent the relative magnitude of the cation current.

Figure 8

Three simulations of electrodeposits using the same parameter values $L=100\mu \text{m}, c_0=1\text{mM}$, and $V_0=10$. The electrodeposits are clearly different from one another, but they do share some general features.

Figure 9

Electrodeposits in the $V_0-c_0$ plane extended to an equal width of $100\mu \text{m}$ by mirror repetition of the original part from Fig. 7 (marked by the horizontal black line) between $0\mu \text{m}$ and the first mirror axis (vertical dashed line). Here, in contrast to Fig. 7, the aspect ratio is constant across the panels.

Figure 10

(a) Instability length scale ${\delta }_{\mathrm{inst}}$ obtained from the simulations plotted versus the most unstable wavelength ${\lambda }_{\max }$. Also, a linear fit highlighting the roughly linear dependence is shown. (b) Characteristic length scale ${\delta }_{\mathrm{char}}$ obtained from the simulations plotted versus the most unstable wavelength ${\lambda }_{\max }$. Also, a linear fit highlighting the roughly linear dependence is shown.

Figure 11

Electrode interface at two different time steps for $c_0=1\text{mM}$ and $V_0=10$. In this simulation no part of the domain was rendered passive at any time. The gray area indicates the electrode after 30 h and 22 min and the black line indicates the electrode interface after 24 h and 2 min. In the upper part of the domain the electrode interface is virtually unchanged between the two times.

Figure 12

Instability growth rate $\mathrm{\Gamma }$ plotted versus perturbation wavelength $\lambda$. The solid (black) line is the numerical result and the dashed (red) line is the analytical result. The critical wavelength ${\lambda }_{\mathrm{c}}^{}$ and the most unstable wavelength ${\lambda }_{\max }^{}$ are indicated in the figure. The parameter values used in the model are those in Table 1 as well as $c_0=10\text{mM}, L=100\mu \text{m}$, and $V_0=30$.

Figure 13

Critical wavelength ${\lambda }_{\mathrm{c}}^{}$ (solid line) and the most unstable wavelength ${\lambda }_{\max }^{}$ (dashed line) plotted versus $V_0$. In addition to the values listed in Table 1, the parameter values used are $L=100\mu \text{m}$ and $c_0=\{1\text{mM},10\text{mM},100\text{mM}\}$.

Figure 14

Illustration of the box-counting method leading to the Minkowski dimension for the electrodeposit obtained for $c_0=10\text{mM}$ and $V_0=20$. The boxes that cover part of the deposit perimeter are shown in gray and the remaining boxes are shown in white. In this example the grid size is $\epsilon =0.85\mu \text{m}$ and the number of boxes it takes to cover the perimeter is $N\left(\epsilon \right)=234$.

Figure 15

Number $N\left(\epsilon \right)$ of boxes it takes to cover the electrodeposit plotted vs the box side length $\epsilon$. A linear fit is shown in each of the two approximately linear regions and the Minkowski dimension in each region is indicated. The crossing point between the linear fits is marked by an arrow and the characteristic dimension ${\delta }_{\mathrm{char}}=0.50\mu \text{m}$ is calculated based on this crossing point.