Phase separation between conductive and insulative materials induced by the electric field

To demonstrate that phase separation is a main mechanism of pattern formation for one of the spatiotemporal patterns emerging in the Ag and Sb electrodeposition system, I performed numerical simulations to model the mixed system of conductive and insulative materials under a steady electric field. For such a dissipative system, I derived the extended Cahn-Hilliard equation using Onsager's variational principle. My results demonstrate that conductive and insulative materials phase separate spatially under the constant-current mode.

Figure 1

Schematic drawing of the film formed by the insulative and conductive materials A and B under the constant-current mode.

Figure 2

Time evolution of the calculation derived from the numerical simulation using Eq. (9). Step time is $\mathrm{\Delta }\mathit{t}=3.0\times {10}^{-7}$. Maximum steps are $6.0\times {10}^5$. $a=0.5$. $b=1.0$. $\varepsilon =0.05$. The initial concentration of ${\mathit{C}}_{\mathit{A}}$, $C_{\mathrm{int}}=0.5$. $L=1.0$. (a)–(f) The conventional Cahn-Hilliard equation without the consumption energy [the last term in Eq. (9)] emerging by the resistance of the film. (a) 0th step. (b) $1.0\times {10}^5\mathrm{th}$ step. (c) $2.0\times {10}^5\mathrm{th}$ step. (d) $3.0\times {10}^5\mathrm{th}$ step. (e) $5.0\times {10}^5\mathrm{th}$ step. (f) $6.0\times {10}^5\mathrm{th}$ step. At $5.6\times {10}^5\mathrm{th}$ step, the phase separation is completed and, until the $6.0\times {10}^5\mathrm{th}$ step, the condition is stable. (g)–(l) The extended Cahn-Hilliard equation (9) including the consumption energy emerging by the resistance of the film under the constant-current mode. ${\mathit{I}}_{\mathit{0}}=2.1$. ${\mathit{A}}_{\mathit{0}}{\mathit{wR}}_{\mathit{A}}=5.0$. ${\mathit{A}}_{\mathit{0}}{\mathit{wR}}_{\mathit{B}}=1.0$. Therefore, the ratio of resistivity of ${\mathit{R}}_{\mathit{A}}$ to ${\mathit{R}}_{\mathit{B}}$ is 5.0. (g) 0th step. (h) $2.50\times {10}^4\mathrm{th}$ step. (i) $5.00\times {10}^4\mathrm{th}$ step. (j) $7.50\times {10}^4\mathrm{th}$ step. (k) $1.00\times {10}^5\mathrm{th}$ step. (l) $1.25\times {10}^5\mathrm{th}$ step.

Figure 3

Time evolution of the entire resistance $\mathit{R}$ of the film under the constant-current mode, using the extended Cahn-Hilliard equation [Eq. (9)]. The parameters are the same as those of Figs. 2(g)–2(l).