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channel. Such phenomenon is allowed only until the concentration at the wall in the depleted layer becomes zero. In this condition, the so-called “limiting current density” is achieved, which, from eq. (11), can be expressed as: 𝑖 = ± 𝐶𝑧𝐹𝐷 = ± 𝑆h 𝐶𝑧𝐹𝐷 𝛿𝑇 − 𝑡 𝐷𝑇 − 𝑡 (13) According to the sign conventions cited above, ilim will be positive if directed towards the positive y axis. Thus the limiting current density depends on hydrodynamic conditions, channel thickness and salt transport numbers. Interestingly, once the values of ilim and 𝑇 are known, the DBL thickness and Sh could be easily calculated from eq. (13), but it should be noted that phenomena occurring in limiting current conditions, such as electroconvection, may significantly affect the value of Sh determined with this equation [139]. Therefore, if the hydrodynamics features and mass transport coefficients have to be determined in the underlimiting range, where the above mentioned phenomena do not play a role, chronopotentiometric and EIS-based methods are more suitable. From eqs. (11) and (13), it follows that, for a current density i < ilim, the concentrations at the IEM- solution interface can be expressed as [169, 177, 179, 190, 205] 𝐶 = 𝐶 1 ∓ 𝑖 𝑖 (14) Values of limiting current density can be obtained experimentally by current-voltage curves. As the voltage increases, the current increases more and more slowly due to the higher boundary layer effects that increase the resistance (see Section 4.3), until current increases only slightly with large voltage increments, indicating the achievement of limiting current condition. However, when the limiting current is approached, experimental observations deny the presence of the plateau theoretically postulated as a curve saturation (see Figure 7), so that the classical theory of concentration polarization is not anymore considered valid since the early 1970s [40, 169, 170]. Instead, only a narrow flat region or even an inflection point are found, along with a further increase of current (overlimiting current), due to the onset of further phenomena that were not observed at electrode- solution interfaces. Water splitting was considered responsible of the further charge transport, thus causing such deviation. However, many other mechanisms have been more recently theorized, discussed and experimentally proven (see Section 4.3). Figure 7 shows a typical S-shaped current-voltage curve, exhibiting three regions [175, 188, 190, 191, 206–210]. Region I has been often defined as Ohmic, though this may be misleading due to the coexistence of both Ohmic and non-Ohmic phenomena. In the first small tract, the curve can be well approximated by a straight line; however, as the current increases, polarization (non-Ohmic) effects 27PDF Image | Electrodialysis for water desalination
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