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Figure 1-2: Regimes of Concentration Polarization. Reprinted with permission from Mani et al [68]. Copyright 2009 American Chemical Society. by ∗∗I∗ ∗∗I∗ (1−V )cu + hc∗u +1 = (1−V )cd + hc∗d +1 (1.11) In the system at hand, the initial conditions in the microchannels and nanochannels are very important in determining whether the concentration polarization propagates (a shock is formed) or whether it does not propagate. Figure 1-2 shows a plot of the various regimes together with the steady-state values of the concentration in the anodic microchannel (subscript 1), the nanochannel (subscript n), and the cathodic microchannel (subscript 5). It is clear from figure 1-2 that a shock always propagates when the characteristic velocities in the nanochannel are negative (subcritical). The remaining conditions come into play when the initial characteristic velocities in the nanochannel are pos- itive (supercritical) and are heavily dependent on the dimensionless coion mobility in the nanochannel (ν∗ ). One important conclusion from this theory is that an ex- 2n perimentalist cannot arbitrarily vary and control the ion density in the nanochannel in steady state. It is completely dependent on the surface chemistry and hence the regime in figure 1-2 of the problem. It can only be varied by altering the surface chemistry. The theory by Mani et al presented here was experimentally verified by Zangle et al [121]. Furthermore, Kim et al [55] investigated this phenomenon exper- 36PDF Image | Shock Electrodialysis for Water Purification and Electrostatic Correlations
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