Shear Flow of an Electrically Charged Fluid by Ion Concentration Polarization: Scaling Laws for Electroconvective Vortices

We consider electroconvective fluid flows initiated by ion concentration polarization (ICP) under pressure-driven shear flow, a scenario often found in many electrochemical devices and systems. Combining scaling analysis, experiment, and numerical modeling, we reveal unique behaviors of ICP under shear flow: a unidirectional vortex structure, its height selection, and vortex advection. Determined by both the external pressure gradient and the electric body force, the dimensionless height of the sheared electroconvective vortex is shown to scale as $({\varphi }^2/U_{\mathrm{HP}}{)}^{1/3}$, which is a clear departure from the previous diffusion-drift model prediction. To the best of our knowledge, this is the first microscopic characterization of ion concentration polarization under shear flow, and it firmly establishes electroconvection as the mechanism for an overlimiting current in realistic, large-area ion exchange membrane systems such as electrodialysis. The new scaling law has significant implications on the optimization of electrodialysis and other electrochemical systems.

Figure 1

Schematic diagram of sheared EC vortices (black ellipses) and ion concentration profiles $C$ initiated by ICP under shear flow in the microscale ED system. When the pressure-driven HP flow (with flow rate $Q$ and average velocity $U_{\mathrm{HP}}$) is applied, with a sufficient high voltage $V$, the EC zone $d_{\mathrm{ec}}$ and the corresponding circular depletion boundary layer $d_{\mathrm{bl}}$ are observed on both CEM and AEM. The vortices have only one direction (clockwise on CEM and counterclockwise on AEM) instead of symmetric pairs. By combining the unidirectional vortices and HP flow, meandering fluid flows are induced above the EC vortices. A typical concentration profile (at the maximum height of the vortex) near the AEM, along with the thicknesses of $d_{\mathrm{ec}}$, $d_{\mathrm{bl}}$, and $\delta$, are shown in the inset. $C_0$ indicates the bulk concentration at a given location, which gradually decreases downstream of the channel by electrodialysis [19]. The direction of electric field $E$ and current $I$ is from AEM to CEM (black arrows). The detailed fabrication and operating procedures are described in the Supplemental Material [21] and Kwak et al. [19].

Figure 2

Sheared ICP is visualized by showing local ion concentration profiles (a) and (c) and EC vortices (b) and (d) from the experiment (at $V=10\mathrm{V}$ and $U_{\mathrm{HP}}=0.83\mathrm{mm}/\mathrm{s}$ ($Q=10\mu \mathrm{L}/\min $). (a) and (b) and the simulation (at $V=25V_0$ and $U_{\mathrm{HP}}=80U_0$, where $V_0=25\mathrm{mV}$ and $U_0=29.66\mu \mathrm{m}/\mathrm{s}$; see Supplemental Material [21], Table 1) (c) and (d) CEM (AEM) is located at the bottom (top) of the channel, and the ratio of channel width to length is $1:5$ in both the experiment and the simulation. In the experiment, the local ion concentration was tracked with $10\mu \mathrm{M}$ Rhodamine 6G (R6G), and circular depletion boundary layers $d_{\mathrm{bl}}$ are observed as a dark region caused by the depletion of R6G (a); in simulation, ion depletion is represented as a blue region (c). Fluid flows were visualized by stacking the time-lapse images of $10\mu \mathrm{m}$ polystyrene (PS) beads (b). The vortex evolution is clearly visualized at the entrance region in both the experiment and the simulation (yellow and white dotted lines).

Figure 3

Dimensionless thickness of the EC vortex zone $d_{\mathrm{ec}}/w$ plotted against the scaling factor $({\varphi }^2/U_{\mathrm{HP}}{)}^{1/3}$ at various applied voltages and flow rates for both experiment $V=4–20\mathrm{V}$ and $Q=5–50\mu \mathrm{L}/\min $ ($U_{\mathrm{HP}}=0.42–4.17\mathrm{mm}/\mathrm{s}$) and simulation ($V=21V_0–41V_0$ and $U_{\mathrm{HP}}=60U_0–120U_0$, where $V_0=25\mathrm{mV}$ and $U_0=29.66\mu \mathrm{m}/\mathrm{s}$). Dotted lines are the best fitting straight lines for two different sets of experiment data on CEM and AEM and one set of simulation data.

Figure 4

(a) Space-time plot of vortex advection visualized by the corresponding circular depletion boundary layer. The time interval between adjacent images is 0.2 s. Even when neighboring vortices merge or separate (white dotted arrows), their advection with constant speed is clearly observed (white solid arrow). (b) Vortex advection speed $U_{\mathrm{ec}}$ according to the average flow velocity $U_{\mathrm{HP}}$ rectangle: experiments at $U_{\mathrm{HP}}=0.42–2.5\mathrm{mm}/\mathrm{s}$ ($Q=5–30\mu \mathrm{L}/\min$); triangle: simulations at $U_{\mathrm{HP}}=1.78–3.56\mathrm{mm}/\mathrm{s}$ ($U_{\mathrm{HP}}=60U_0–120U_0$, $U_0=29.66\mu \mathrm{m}/\mathrm{s}$). The best fitting curves for both experiment and simulation results are the same,$y=1.058x$ (the $x$ intercept is set at 0).