ac electrokinetic micropumps: The effect of geometrical confinement, Faradaic current injection, and nonlinear surface capacitance

Recent experiments have demonstrated that ac electrokinetic micropumps permit integrable, local, and fast pumping (velocities $\sim \mathrm{mm}∕\mathrm{s}$) with low driving voltage of a few volts only. However, they also displayed many quantitative and qualitative discrepancies with existing theories. We therefore extend the latter theories to account for three experimentally relevant effects: (i) vertical confinement of the pumping channel, (ii) Faradaic currents from electrochemical reactions at the electrodes, and (iii) nonlinear surface capacitance of the Debye layer. We report here that these effects indeed affect the pump performance in a way that we can rationalize by physical arguments.

Figure 1

Sketch of the device geometry. The interdigitated electrode array is biased with an ac voltage that induces a buildup of a Debye screening layer and electro-osmotic fluid motion.

Figure 2

Equivalent circuit diagram for the electrokinetic system: $R_0$ is the bulk electrolyte Ohmic resistance, $C_0$ is the double-layer capacitance, and $R_{\mathrm{ct}}$ is the charge-transfer resistance for the electrode reaction. The characteristic time for charging the Debye layer through the electrolyte is ${\tau }_0=R_0{C}_0$, and the time scale for decharging the layer through Faradaic electrode reaction is ${\tau }_{\mathrm{ct}}=R_{\mathrm{ct}}{C}_0$.

Figure 3

Time average slip velocity $⟨{\tilde{u}}_s⟩$ across the electrodes obtained for $\tilde{\omega }=1$ and $K=0$ (no Faradaic current) in a geometry with ${\tilde{W}}_1=1.5$, ${\tilde{W}}_2=7$, ${\tilde{G}}_2=5$, and $\tilde{H}⪢\tilde{L}$.

Figure 4

Pumping velocity $\tilde{U}$ as a function of frequency $\tilde{\omega }$. Dashed line: analytical result at low frequency where $\tilde{U}\propto {\tilde{\omega }}^2$.

Figure 5

Time average slip velocity $⟨{\tilde{u}}_s⟩$ for both electrodes against the relative position on the electrode $x∕W_i$. Symbols: results of Ramos et al. [15] for the narrow (▵) and wide (◻) electrodes, respectively. Solid lines: our numerical solution.

Figure 6

Contour plot of the maximal pumping velocity over frequency ${\max }_{\tilde{\omega }}\left\{\tilde{U}(\tilde{\omega },\dots )\right\}$ as a function of ${\tilde{W}}_1$ and ${\tilde{W}}_2$ for ${\tilde{G}}_2=5$. When ${\tilde{W}}_1={\tilde{W}}_2$ the net pumping vanishes due to symmetry. There is a unique optimum close to ${\tilde{W}}_1=1.5$ and ${\tilde{W}}_2=7$ with ${\tilde{U}}_{\max }=0.00358$ and ${\tilde{\omega }}_{\max }=0.8$.

Figure 7

Contour plot of pumping velocity $\tilde{U}$ as a function of frequency $\tilde{\omega }$ and degree of confinement $L∕H$.

Figure 8

Contour plot of the pumping velocity $\tilde{U}$ as a function of frequency $\tilde{\omega }$ and Faradaic conductance $K$.

Figure 9

Contour plot of $\tilde{U}$ in the zero-frequency limit $\tilde{\omega }\sim 0$ with asymmetric current injection: $K_1$ and $K_2$ denote the Faradaic conductance on the narrow and wide electrodes, respectively. The slip velocity $⟨{\tilde{u}}_s⟩$ at the points (a)–(d) is shown in Fig. 10.

Figure 10

Time average slip velocity $⟨{\tilde{u}}_s⟩$ on the electrodes for the parameter values marked in Fig. 9: (a) $K_1=1$, $K_2={10}^2$; (b) $K_1=K_2={10}^2$; (c) $K_1=K_2=1$; (d) $K_1={10}^2$, $K_2=1$.

Figure 11

Contour plot of the pumping velocity $\tilde{U}$ as a function of frequency $\tilde{\omega }$ and driving voltage ${\tilde{V}}_0$ with capacitance ratio $\delta =0.1$ and no Faradaic current, $K=0$. The white dashed line shows the frequency ${\tilde{\omega }}_{\max }$ at which $\tilde{U}$ is maximized as a function of ${\tilde{V}}_0$. At large driving voltage ${\tilde{\omega }}_{\max }$ shifts from $\sim 1\text{to}\sim {\tilde{\omega }}_{\infty }$, reflecting the larger relaxation time when the double-layer capacitance is dominated by the Stern layer.

Figure 12

Maximal pumping velocity ${\tilde{U}}_{\max }$ as a function of ${\tilde{V}}_0$, as evaluated along the white dashed line in Fig. 11. The data are scaled with ${\tilde{V}}_0^2$ to recover the physical dependence on driving voltage. For comparison, the dashed lines show linear and quadratic scaling with ${\tilde{V}}_0$, while the dash-dotted line is a fit to ${\tilde{V}}_0\log {\tilde{V}}_0$.

Figure 13

Contour plot of the pumping velocity $\tilde{U}$ as a function of driving frequency $\tilde{\omega }$ and voltage ${\tilde{V}}_0$ with capacitance ratio $\delta =0.1$ and Faradaic conductance $K=0.1$. The dashed line shows the optimum driving frequency ${\tilde{\omega }}_{\max }$ as a function of ${\tilde{V}}_0$. Notice the pumping in the low-frequency limit for ${\tilde{V}}_0\gtrsim 1$.

Figure 14

Pumping velocity $\tilde{U}$ as a function of ${\tilde{V}}_0$ in the low-frequency limit $\tilde{\omega }⪡1$. The data are scaled with ${\tilde{V}}_0^2$ to recover the physical dependence on driving voltage.