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Langmuir Article Figure 7. Water deionization by shock electrodialysis operating under OLC. (a) Schematic of the extraction flow and salt concentration profile. (b) Photograph of the button-cell device. (c) Conductivity (y axis) and salt concentration (green lines) of the extracted water versus flow rate, with theoretical scalings from eq 12. (d) Energy cost per volume with (dashed lines) and without (solid lines) the series resistance attributed to the reservoir and electrodes for the same experiments. data point (from the Supporting Information) is on the order of 10%, which is smaller than the point size in the log−log plots of Figure 5d−f showing power-law scalings. According to dilute solution theory, eq 2, the limiting current, Ilim, is linearly proportional to concentration. The fitted Ilim (Figure 5d) verifies this scaling at low concentration and deviates to lower values at high concentration, consistent with reduced Cu2+ activity and diffusivity.45 A simple estimate (red curve), using free-solution values46 D0 for D(c0) in eq 2, captures the scaling of the data for Ilim(c0) well. In the Bruggeman approximation, however, the macroscopic diffusiv- ity Dm = ε3/2D0 at low concentration is ≈13 times smaller than the apparent dispersion D from eq 2. The discrepancy partially reflects transient diffusion (an effectively smaller L) because the diffusion distance 2(Dmt)1/2 ≈ 0.5 mm when the limiting current is reached is somewhat smaller than the frit thickness, L =1mm. Consistent with our analysis of OLC below, the leading cause of the enhanced dispersion inferred from Ilim may be electroosmotic convection in the glass frit. Electroosmotic Figure 8. Overlimiting current and shock electrodialysis with 10 mM silver nitrate using the device in Figure 3. The current−voltage relation in plot a exhibits a similar constant overlimiting conductance, similar to that of 10 mM copper sulfate, but with a smaller limiting current and voltage scale, which lead to a larger dimensionless current in plot b. Bars indicate data taken at Vapp = 1.5 V. The deionization factor in plot c is larger in AgNO3 than in CuSO4 at the same voltage, flow rate, and salt concentration, as predicted by the scaling theory, eq 12 (solid curves and inset data collapse). flow toward the impermeable membrane/cathode structure is balanced by a pressure-driven back flow that produces dispersion.19 Taylor dispersion is negligible on the basis of the formula for a single cylindrical pore, DTaylor/Dm − 1 = εp3/2Pe12/48 ∼ 0.01, even using a large EOF velocity of U = 400 μm/sinthesingle-channelPećletnumberPe1=Uhp/D0.Fora network of pores, however, there is additional dispersion due to the randomness in streamline topology47 (also referred to as eddy diffusion or dispersion48). Indeed, the simple estimate 16172 dx.doi.org/10.1021/la4040547 | Langmuir 2013, 29, 16167−16177PDF Image | Overlimiting Current and Shock Electrodialysis in Porous Media
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