Effect of the combined action of Faradaic currents and mobility differences in ac electro-osmosis

In this work, we extend previous analyses of ac electro-osmosis to account for the combined action of two experimentally relevant effects: (i) Faradaic currents from electrochemical reactions at the electrodes and (ii) differences in ion mobilities of the electrolyte. In previous works, the ac electro-osmotic motion has been analyzed theoretically under the assumption that only forces in the diffuse (Debye) layer are relevant. Here, we first show that if the ion mobilities of a 1-1 aqueous solution are different, the charged zone expands from the Debye layer to include the diffusion layer. We later include the Faradaic currents and, as an attempt to explore both factors simultaneously, we perform a thin-layer, low-frequency, linear analysis of the system. Finally, the model is applied to the case of an electrolyte actuated by a traveling-wave signal. A steady liquid motion in opposite direction to the applied signal is predicted for some ranges of the parameters. This could serve as a partial explanation for the observed flow reversal in some experiments.

Figure 1

(Color online) (a) Diagram showing a microelectrode array subjected to a four-phase ac signal to produce a traveling-wave potential. (b) Scheme of traveling-wave electro-osmosis (TWEO) mechanism: the induced charge in the DL lags behind the applied signal due to the finite DL charging time. The tangential field produces a force in the direction of the wave for both positive and negative ions, resulting in steady liquid motion.

Figure 2

(Color online) Structure of the extended double layer (XDL) on an array of electrodes subjected to an ac potential. Three distinct layers are indicated: compact layer (formed by fixed adsorbed ions and water molecules), diffuse layer (where thermal and electric forces are of the same order), and diffusion layer (associated to the finite diffusion penetration length for the different species). The outer Helmholtz plane (OHP) is localized between the compact and diffuse layers.

Figure 3

(Color online) Real part of the decay factors for (a) an aqueous solution with equal ion mobilities and (b) one with very different ones. The eigenvalues have been plotted against the rescaled frequency $\Omega =\omega \left(1-{\gamma }^2\right)/\delta$. For equal ion mobilities, the decay lengths (thicknesses of the diffuse and diffusion layers) are distinct, but for highly asymmetrical diffusion they are closer and can even intersect: the distinction between the layers vanishes.

Figure 4

(Color online) Contributions to the electro-osmotic velocity, (a) $f_2⪡f_1$ and $\text{sgn}\left(f_2\right)=\text{sgn}\left(f_1\right)$, (b) $f_2⪡f_1$ and $\text{sgn}\left(f_2\right)=-\text{sgn}\left(f_1\right)$, and (c) $f_2\sim f_1$ and $\text{sgn}\left(f_2\right)=-\text{sgn}\left(f_1\right)$, where $f_1$ and $f_2$ are proportional to the voltage drop across the diffuse and diffusion layers, respectively. In the first and second cases, the effect of the diffusion layer is small and cannot reverse the flow. In the third case, the diffusion layer stresses can overcome the diffuse layer ones and produce flow reversal.

Figure 5

(Color online) Slip velocity for equal diffusivities and ideally polarizable electrodes for different compact layer thicknesses ${\lambda }_{\text{s}}$. Increasing ${\lambda }_{\text{s}}$ results in smaller slip velocity and a displacement of the maximum to higher frequencies.

Figure 6

(Color online) Slip velocity for ideally polarizable electrodes, with different compact layer thicknesses and with and without asymmetry. The effect of asymmetry is almost negligible in this case of zero Faradaic currents.

Figure 7

(Color online) Slip velocity in the presence of Faradaic currents, with (a) a small compact layer and (b) a large one, for equal ion mobilities. When the resistance to reaction decreases there appears a secondary maximum at low frequencies, associated to the Warburg impedance and characterized, in dimensionless form, condition ${\omega }^{1/2}\sim \delta$. For a thick compact layer, the primary maximum is displaced and lowered compared with the case ${\lambda }_{\text{s}}=0$, as in Fig. 5, due to the increase in compact layer thickness (notice the different vertical scales). The secondary maximum also reduces its amplitude, but the position of the maximum is not greatly affected by the change in ${\lambda }_{\text{s}}$.

Figure 8

(Color online) Effect of concentration of neutral species on slip velocity for equal ion mobilities with vanishing reaction resistance. For small $N$ (bath of neutral species), the behavior is dominated by diffusion of the charged species. For large $N$ (scarce neutral species) the reaction is blocked and the system behaves as an ideally polarizable one.

Figure 9

(Color online) Comparison of the complete linear analysis (solid line) and the thin-layer approximation (dashed line) for (a) negative and (b) positive asymmetries in the presence of Faradaic currents. The approximate solution has a catastrophic behavior for very low frequencies, but the complete solution shows that this is just a consequence of the approximations made and that the divergence does not happen in the regions of physical interest (for $\Omega \ge {10}^{-2}$).

Figure 10

(Color online) Dependence of the electro-osmotic velocity with the reaction conductance for a negative asymmetry [(a) case of NaOH] and a positive one [(b) case of HCl]. When the more mobile species is the nonreacting one, there is not predicted reversal except for extremely low frequencies (in dimensional form, around 1 Hz or less). If the more mobile species is the reacting one, the model predicts reverse flow when the reaction resistance is low enough.

Figure 11

(Color online) Dependence on the asymmetry for a solution in the facile kinetic regime $\left(G\to \infty \right)$. In this limit, the model predicts reverse flow for both limits of the asymmetry, but only in the case in which the reactive ion is the more mobile one $\left(\gamma \sim +1\right)$, the reverse flow lies in the observable range.

Figure 12

(Color online) Dependence on the compact layer thickness for a solution with positive asymmetry in the facile kinetic regime. If the compact layer is very thin there is no reverse flow, but if it is of same order or larger than the diffuse layer, there could be flow reversal since the voltage drop in the diffuse (Debye) layer is greatly reduced and can be overcome by the opposite voltage drop across the diffusion layer.

Figure 13

(Color online) Dimensional results for the case discussed in Sec. 6D. The parameters correspond to a 0.1 mM solution of HCl in water over an electrode array of wavelength $160\mu \text{m}$.

Figure 14

(Color online) Illustration of the matching condition for two values of a small parameter using as example the function $y=0.8\text{exp}\left(-x/\delta \right)+0.2\text{sin}\left(x\right)+0.5\text{cos}\left(x\right)$. Notice that ${\Phi }_0$ is the point where the straight line crosses the $x$ axis and can be different from the value of the potential on the intermediate region.