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Langmuir Article Deddy/Dm ≈ Pee/2 ≈ 10 could explain the fitting result for D above where we use the same velocity and a mean loop size of hloop = 25 μm, consistent with Figure 3c in estimating the eddy Pećletnumber,Pee=Uhloop/Dm.Wehavealsoconfirmedthat replacing the silica glass frit with a loopless porous medium (an anodic aluminum oxide membrane with straight pores of 200 nm diameter) leads to a reduced Ilim consistent with Dm ≈ D0, as will be reported elsewhere (Han, J.-H.; Bai, P.; Khoo, E.; Bazant, M. Z., to be submitted for publication). Series Resistance. The series resistance Rs can be attributed to two primary sources: the Ohmic resistance of the reservoir, Rres, at low salt and the Faradaic resistance of the anode, RF, at high salt. (The cathode Faradaic resistance is reduced by contact with the Nafion membrane.) Neglecting ICP and assuming equal ionic diffusivities, we estimate the reservoir resistance using 2πR (ze) D(c0)c0 Assuming linearized Butler−Volmer kinetics, we estimate the occuratathicknessof8μm(forc0 =1M,L=1mm),12 which is comparable to the mean effective cylindrical pore diameter of the silica frit, 4hp = 2.3 μm. As noted above, the single-channel analysisunderestimatesthetrueeffectofEOFina heterogeneous porous medium because there are eddies around loops in the pore network (Figure 2) whose size he ≫ hp could reach tens of micrometers in our silica frit, thus making EOF dominant. Indeed, the EOF scaling theory is consistent with the experimental data. If we fit he to the lowest concentration data point using the theoretical formula eq 5 with Dm = (0.4)3/2D0 and D0 = 8.5 × 10−6 cm2/s46 and all other parameters known, then we obtain an eddy size of he = 100 μm, comparable to the thickness of the depleted region,12 (1 − Ilim/I)L for I = 1.1Ilim, but this neglects the unknown numerical prefactor in eq 5. We expect this prefactor to be smaller than 1 in order to obtain a mean eddy size in the 5−50 μm range, consistent with the scale of loops in the glass frit (set by aggregates of sintered particles seen in the SEM images (Figure 3)). This range is also consistent with the characteristic eddy size inferred from the effects of hydrodynamic dispersion on Ilim above. Remarkably, without any adjustable parameters, the EOF scaling theory (eq 5) accurately predicts the observed dependence on salt concentration, varying over 4 orders of magnitude, c0 = 0.1 mM−1.0 M. With constant qs (red dashed line in Figure 5f), the experiments reveal the nontrivial scaling, σOLC ∼ c04/5, at low salt concentration, and the predicted effect of surface charge regulation σOLC ∼ qs(c0)2/5 also captures the trend at high concentration (red solid curve in Figure 5f). We conclude that EOF is the likely mechanism for OLC in our experiments. Surface Charge Modification. The dependence on surface charge density, qs, was tested indirectly by varying the pH and modeling proton adsorption,34,41,42 but two chemical modifications, silanization49,50 and charged polymer deposi- tion,51,52 provide more direct evidence. (See the Supporting Information for details.) The current−voltage responses (Figure 6) are again fit to eqs 3 and 8. As expected, because Ilim is associated with bulk transport and Rs is associated with reservoir and electrode resistances, each is independent of qs. Theory predicts that σOLC is largest for qs < 0, and indeed we find an order-of-magntiude reduction for both surfaces with qs > 0. An unexpected finding is that the positive frits exhibit OLC, although it is much smaller than that of the negative silica frit. This should not occur in positive nanopores where SC acts to oppose the current. For larger, positive pores, however, OLC is possible because EOF eddies can still transport salt faster than diffusion, only rotating in the opposite sense and fighting against SC. Water Deionization. To complement the electrochemical evidence, we experimentally verify the extreme salt depletion associated with OLC12 by driving a flow that produces a deionization shock.27 In microfluidic devices with tiny (nano- liter) volumes, salt concentration is typically inferred by the optical detection of fluorescent particles,18,20,22,23,25 but here we can directly extract macroscopic (0.01−1.0 mL) samples of deionized water from the glass frit (Figure 7a) and test their conductivity by impedance spectroscopy. A proof-of-concept device (Figure 7b) is equipped with a circular outlet (d = 0.5 mm in diameter) at one point on the side of the frit just above the membrane, leading to an annular collection channel connected to the device outlet (Figure 3e). The volumetric kBTL Rres = 2 2 (9) (10) anode Faradaic resistance as RF = kBT = kBT neI 2eK c 1/2 0 00 where I = K cαc is the exchange current density. The transfer 000 coefficient is αc = 1/2 for the rate-limiting transfer of 1 out of n = 2 electrons.38,39 The data in Figure 5e is quantitatively consistent with the theory, Rs = Rres + RF (red curve), with a fitted prefactor of K0 = 2.8 A/m2 (for molar c0). The measured exchange current I0 = 3.9 A/m2 at c0 = 0.5 M is close to I0 = 9.7 A/m2 from recent experiments with CuSO4 at neutral pH43 (which is below I0 in strong acids38,39). We are thus able to attribute the remaining voltage V in eq 8, which exhibits OLC (eq 3), to the glass frit. Overlimiting Current. The SC and EOF mechanisms for OLC are distinguished by different scalings in eq 6 with salt concentration c0 and surface charge density, qs. In the absence of flow, the overlimiting conductance from surface conduction is given by eq 4.12 Using D = εp3/2Dm, the theoretical line with constant qs (green dashed line) roughly matches the experimental value at the lowest salt concentration but lies far below the data at higher concentrations (blue points) in Figure 5f. This is consistent with the prediction that SC becomes important only at low salt concentrations,12 but we must also consider the effect of charge regulation. It is well known that that the surface charge of silica is regulated by the dissociation of silanol groups,34,41,42 −+ SiOH↔SiO +H (pK=7.5) (11) Using the Gouy−Chapman−Stern model to obtain the surface pH,41 we can calculate the surface charge from the measured bulk pH versus salt concentration, assuming pK = 7.5 for silica. (Details are in the Supporting Information, where it is also shown that the pK has little effect on the prediction of OLC.) As shown in Figure 5f (green dotted curve), the decrease in qs with increasing c0 leads to the opposite trend from the experimental data. We conclude that SC is not the primary mechanism for OLC in our experiments, although it may contribute at low salt concentration. Because the data cannot be explained by SC, the theory suggests that EOF is the likely mechanism. In a parallel-plate microchannel, the transition from SC to EOF is predicted to 16173 dx.doi.org/10.1021/la4040547 | Langmuir 2013, 29, 16167−16177PDF Image | Overlimiting Current and Shock Electrodialysis in Porous Media
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