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Langmuir Article where Dm is the macroscopic diffusivity, −qs is the (negative) pore surface charge per area, and hp = ap−1 is the mean pore size equal to the inverse of ap, the internal area density (area/ volume)).12,29 In the absence of flow, we have Dm = εpD0/τp, where D0 is the molecular diffusivity in free solution, εp is the porosity, and τp is the tortuosity. Electroosmotic convection in the double layers contributes a small ∼10% correction to the surface conductivity in a dead-end channel,12 which can be neglected in thin pores. The primary effect of EOF is hydrodynamic dispersion, which increases the effective D and provides a second mechanism for OLC. Mechanism 2: Electroosmotic Flow. As the pore width is increased, convection by EOF eventually dominates SC. The possibility of OLC sustained by EOF was first proposed by Yaroshchuk et al. on the basis of a Taylor-Aris dispersion model, but the assumption of slow convection compared to the electroosmotic flow scalings above, we find lc ∼ uhe2/Dm ∼ qsλDjhe2/σbηDm. Assuming the same current−voltage relation as for a single microchannel (eq 8 in Dydek et al.12), the overlimiting conductance, σOLC ∼ 2(ze)2cdDmA/kBTL, is set by the mean salt concentration cd in the depleted region (Figure 2a). Although this region contains fingers of nonuniform salt concentration, we use the mean value cd to define the local bulk conductivity, σb = εD/λD2 and λD2 = εkBT/2(ze)2c, respectively, as a first approximation. All that remains then is to determine cd. Numerical simulations in a straight microchannel exhibit the scaling cd/c0 ≈ lc/L close to the limiting current.12 If the same relation also holds in a porous medium for j ∼ jlim ∼ 2zec0Dm/L, then we can eliminate lc and arrive at a scaling law for the overlimiting conductance, (5) where there is an unknown numerical prefactor that is independent of all the parameters. The best way to determine the prefactor, once the scalings are validated, is by experiment (below). transverse diffusion is typically violated. 19 Dydek et al. then 4/5 2/5 6/5 1/5 3/5 (2ch) q (ze) ε D A showed that EOF can support OLC by fast electroosmotic convection in the depleted region (Figure 2a), leading to nonuniform salt profiles with a thin boundary layer12 (Figure 2b). Rubinstein and Zaltzman analyzed this new mode of dispersion in the simpler case of a neutral solute with constant slip velocity on the side walls and described “wall fingers” transitioningtospiralstructureswithincreasingPećletnumber .33 As a first approximation for EOF OLC in a porous medium, we adapt the microchannel scaling analysis of Dydek et al.12 Electroosmotic flow scales as u ∼ εζE/η, where the zeta potential, ζ ∼ qsλD/ε, is related to the surface charge density qs using the thin diffuse-layer capacitance (C = ε/λD). The mean tangential electric field, E, is related to the local mean current density, j, via E ∼ j/σb, where λD(c) is the Debye length and σb(c) is the bulk conductivity, each depending on the local bulk salt concentration c. Combining these equations, we obtain the EOF velocity scaling, u ∼ qsλDj/σbη. The porous medium is pressed against an impermeable, ideally selective membrane for counterions (opposite to the pore charge). To ensure zero mean flow, a pressure-driven backflow balances EOF and leads to hydrodynamic disper- sion.19 For a regular microstructure (Figure 2b), the sum of these flows is a vortex pair of width he ∼ hp that produces parallel wall fingers.12,33 For an irregular microstructure (Figure 2c), variations in hydraulic resistance lead to nonuniform backflow that can exceed the electroosmotic flow in the larger pores. In that case, the mean eddy size is set by connected loops between pores of high and low hydraulic resistance, he ∼ hl. As the current increases, the eddy fingers extend across larger distances, and the flow may even become chaotic, as with electroosmotic instability in free solution.36 Nevertheless, the following simple theory manages to predict the scalings in our experiments. Consider fast electroosmotic convection in the depleted region leading to eddy fingers of transverse thickness, he (set by the mean size of either pores or loops), and axial length lc, set by the mean distance from the membrane to the eddy centers, as shown in Figure 2. As in boundary-layer analysis of forced convection in a pipe,37 the convection−diffusion equation, u·⃗ ∇c = Dm∇2c, then yields the scaling, u/lc ∼ Dm/he2. As the eddy size increases at larger overlimiting currents, the effective diffusivity D incorporates porosity and tortuosity factors as well as corrections due to microscopic hydrodynamic dispersion (such as the Taylor dispersion) on length scales smaller than the eddy size. Combining the convection−diffusion scaling with σEOF∼ 0e s m OLC (ηkBT)2/5L9/5 ■ Apparatus. The apparatus is designed to test the theoretical current−voltage relation (eq 3) for a charged porous medium and extract the overlimiting conductance. By choosing a copper electrolytic cell38−40 with known Faradaic reaction resistance RF(c0) and a porous silica glass frit with known surface charge qs(pH) in water,34,41,42 the Figure 3. Prototype “button cell” for shock electrodialysis. (a) Sketch of the frit/membrane/electrode sandwich structure (not to scale), (b) photograph of prototype, (c) scanning electron microscopy (SEM) image of the glass frit showing the distribution of pores, and (d) enlarged micrographs consistent with the mean pore size of around 500 nm. (e) Cross-section drawing to scale. (Right) Enlargement showing the radial outlet for fresh water extraction. 16169 dx.doi.org/10.1021/la4040547 | Langmuir 2013, 29, 16167−16177 EXPERIMENTPDF Image | Overlimiting Current and Shock Electrodialysis in Porous Media
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