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∆𝜑 = ∆𝜑 −∆𝜑 +∆𝜑 (3) where ∆𝜑 is the Donnan potential on the left side, ∆𝜑 is the Donnan potential on the right side and ∆𝜑 is the diffusion potential arising from the concentration gradient within the membrane and from the different diffusivity of the two ions. The simplest expression of the membrane potential that can be derived from the TMS theory is [49, 62, 63] ∆𝜑 = 2 𝑡 − 1 𝑅𝑇 𝑧𝐹 𝑙𝑛 𝑎, (4) 𝑎, where 𝑡 is the transport number of the counter-ion in the membrane, while 𝑎, and 𝑎, are the activities in the right and left solution, respectively. The one-dimensional TMS model is based on several simplifying assumptions [63, 64] and predicts the membrane potential well within a certain range of solutions concentration, while deviating from experimental data in other conditions, e.g. with large concentration ratios [64, 65]. Models accounting for the effect of the membrane nano-pore radius have been developed, such as the “uniform potential” model and the “space charge” model [64, 65]. The discussion of these models goes beyond the scope of this work; however, the simple mathematical formulation of the TMS theory makes it the most commonly adopted for practical uses. Therefore, in the following the membrane potential will be intended as derived from the TMS theory. 3.2 IEMs preparation and classification Based on their morphology, IEMs can be classified into two main categories: homogenous and heterogeneous [57, 66]. A more detailed classification based on the degree of homogeneity of the structure can be made [66, 67], which is particularly useful for IEMs characterized by micro- inhomogeneities. Homogeneous membranes are the most commonly used in ED and other electro-driven processes [66, 68–72]. Such membranes appear homogeneous at the microscopic scale, but at the nanoscopic scale they can be observed as multiphase (i.e. microheterogeneous) systems composed by at least two phases: the polymer matrix with the fixed charges and the interstitial (electroneutral) salt solution. A more rigorous description, as given by Zabolotsky et al. [73], recalls the presence of a gel phase, generated by a thin layer of solution, and of fixed charges located at the internal walls of interstitial vescicles (Figure 5). This phase is also related to the formation of an electrical double layer (EDL), where electroneutrality is not maintained due to the split of anions and cations concentration profiles, which actually generates the Donnan potential at the interface [60, 66, 74, 75]. The 13PDF Image | Electrodialysis for water desalination
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