Electro-osmotic instability of concentration enrichment in curved geometries for an aqueous electrolyte

We report that an electro-osmotic instability of concentration enrichment in curved geometries for an aqueous electrolyte, as opposed to the well-known one, is initiated exclusively at the enriched interface (anode), rather than at the depleted one (cathode). For this instability, the limitation of an unrealistically high material Peclet number in planar geometry is eliminated by the strong electric field arising from the line charge singularity. In a model setup of concentric circular electrodes, we show by stability analysis, numerical simulation, and experimental visualization that instability occurs at the inner anode, below a critical radius of curvature. The stability criterion is also formulated in terms of a critical electric field and extended to arbitrary (two-dimensional) geometries by conformal mapping. This discovery suggests that transport may be enhanced in processes limited by salt enrichment, such as reverse osmosis, by triggering this instability with needlelike electrodes.

Figure 1

(a) Sketch of the anode vortex near the inner anode in a circular channel. At limiting current, both the concentration enrichment (b) and an additional peak (${\overline{E}}^p$) of electric field (c) appear near the inner anode. (d) ${\overline{E}}^p$ as a function of $\chi$ in a log-log scale, demonstrating the tendency of developing a singularity as $\chi \to 0$.

Figure 2

Linear stability analysis. (a) The neutral stability curves versus dimensionless wave number $k$ for various radius ratios $\chi$, showing the critical material Peclet number ${\mathrm{Pe}}_{\min }$ for each curve. (b) Nearly inverse dependence of ${\mathrm{Pe}}_{\min }$ on the maximum electric field on the anode, ${\overline{E}}^{\text{max}}=-{(\chi ln\chi )}^{-1}$. For aqueous electrolyte with ${\mathrm{Pe}}_{\mathrm{aqu}}=0.5$, instability occurs for ${\overline{E}}^{\text{max}}>{\overline{E}}_w^{\text{max}}=7.17$ or $\chi <{\chi }_w=0.045$.

Figure 3

Numerical simulation. (a) Top panel for the concentration evolution. (b) Bottom panel for the magnified view around the inner anode, demonstrating the development of EOI and the formation of vortices indicated by the black streamlines in the concentration enrichment region. ($\chi =1/120, V=8, \varepsilon =3\times {10}^{-4}, p_1=2$, and time scaled by ${\chi }^2{R}_2^2/D$.)

Figure 4

Experimental observation of anode vortex in a circular channel. (a) The sketch of the electrochemical cell for a circular channel. (b) The observation of the anode vortex by the time-lapse imaging indicating the streamline ($\chi =1/60,I=5\mu A$). particle image velocimetry imaging of the flow field (c) near the inner anode ($\chi =1/120,\mathrm{\Phi }=6\text{V}$) and (d) near the inner cathode ($\chi =1/120,\mathrm{\Phi }=-6\text{V}$).