Morphological instability during steady electrodeposition at overlimiting currents

We present a linear stability analysis of a planar metal electrode during steady electrodeposition. We extend the previous work of Sundstrom and Bark by accounting for the extended space-charge density, which develops at the cathode once the applied voltage exceeds a few thermal voltages. In accordance with Chazalviel's conjecture, the extended space-charge region is found to greatly affect the morphological stability of the electrode. To supplement the numerical solution of the stability problem, we have derived analytical expressions valid in the limit of low and high voltage, respectively.

Figure 1

Sketch of the studied system with lower and upper electrode surfaces at $x=f_{\ell }(y,t)$ and $x=2+f_u(y,t)$, respectively. The coordinates are given relative to the moving frame of reference, following the mean rate of deposition on the electrode surfaces. The coordinates are also normalized by half the electrode spacing $L$.

Figure 2

Zeroth-order cation concentrations $c_{+}^0$ shown in full (black) lines and zeroth-order charge densities ${\rho }^0/Z$ shown in dashed (red) lines. The inset shows the fields close to the electrode. In the simulation the parameter values $c_0=10\text{mM},L=10\mu \mathrm{m}$, and $V_0=\{1,5,12,30\}$ were used.

Figure 3

The growth rate $\mathrm{\Gamma }$ plotted versus the perturbation wavelength $\lambda$ for $V_0=30,c_0=10\text{mM}$, and $L=10\mu \mathrm{m}$. The full (black) line shows the growth rate obtained from numerical simulations, and the dashed (red) line shows the growth rate according to the analytical model [Eq. (29)]. For perturbation wavelengths smaller than the critical wavelength ${\lambda }_{\mathrm{c}}=51\text{nm}$ the system is stable and for larger wavelengths it is unstable. At the most unstable wavelength ${\lambda }_{\max }=110\text{nm}$ the growth rate is ${\mathrm{\Gamma }}_{\max }=0.0193$.

Figure 4

The growth rate $\mathrm{\Gamma }$ plotted versus the perturbation wavelength $\lambda$ and voltage $V_0$ for $c_0=10\text{mM}$, and $L=10\mu \mathrm{m}$. The (cyan) space curves are plots of $\mathrm{\Gamma }$ versus $\lambda$ for $V_0=\{5,10,15,20,25,30\}$. The shade of the in plane contour plot is based on the logarithm of $\mathrm{\Gamma }$, which is why there are no contours in the low $\lambda$ limit where $\mathrm{\Gamma }$ is negative. The thick (blue) in plane line marks the crest of the hill; i.e., it marks the most unstable wavelength for each value of $V_0$.

Figure 5

Contour plots of $\mathrm{\Gamma }$ plotted versus wavelength $\lambda$ and voltage $V_0$ for $c_0=\{1\text{mM},10\text{mM},100\text{mM}\}$ and $L=\{1\mu \mathrm{m},10\mu \mathrm{m},100\mu \mathrm{m},1\text{mm}\}$. In each plot, $\mathrm{\Gamma }$ is normalized by its maximum value, and the contours are logarithmically spaced. The maximum value ${\mathrm{\Gamma }}_{\max }$ of $\mathrm{\Gamma }$ is given on top of each plot, and the point where the maximum value is attained is indicated with a dark (red) circle. The three thin black lines in each plot indicate contours where $\mathrm{\Gamma }$ equals $0.01,0.2$, and $0.7$ times ${\mathrm{\Gamma }}_{\max }$. The thick bright (yellow) line marks ${\lambda }_{\max }$ for each value of $V_0$, and the dashed (blue) lines mark the two corresponding analytical limits. The thick black line marks ${\lambda }_{\mathrm{c}}$ for each value of $V_0$, and the dotted (green) lines mark the two corresponding analytical limits.

Figure 6

The instability length scale $L_{\mathrm{\Gamma }}$ (full line) and most unstable wavelength ${\lambda }_{\max }$ (dashed line) plotted versus bias voltage $V_0$. The concentration varies between the values $c_0=\{1\text{mM},10\text{mM},100\text{mM}\}$ and the length $L=100\mu \mathrm{m}$ was used.