## 2 The Ion Exchange Membrane Structure and Ion Transfer  
Description of the ion-exchange membrane structure should be started from one of the benchmark materials--perfluorosulfonic acid membranes of the Nafion type. Nafion is a random copolymer of tetrafluoroethylene and sulfonyl fluoride vinyl ether [59; 60]. The first of them forms a hydrophobic perfluorinated backbone, and the second forms a side chain with -SO\({}_{3}\)H (-SO\({}_{3}\)M) terminal group. The latter are hydrophilic and, as result of self-organization, form clusters, which absorb water from the environment. As a result, there is an extended network of pores (4-5 nm in size in a swollen state) and channels filled with water in the membrane. Dissociation of functional groups leads to the formation of an aqueous solution containing dissociated cations [61; 62; 63; 64]. On the contrary, fixed -SO\({}_{3}\)\({}^{-}\) ions are localized on the pore walls, giving them a negative charge. Due to the electrostatic interaction, most of the cations are localized in a thin Debye layer near the pore walls, and the so-called electrically neutral solution is located in the pore centers (Figure 1) [32; 65]. It is generally accepted that the composition of this electrically neutral solution is close to that of the solution contacting with the membrane. When membranes are used in electrodialysis or reverse electrodialysis, the concentration of contacting solution can be rather high [16; 31; 66]. Moreover, in the central part of the pore, there are a lot of cations and anions which can contribute significantly to the conductivity. On the other hand, it is this pore central part that determines the significant contribution of anionic conductivity, which reduces the selectivity of transport processes. On the contrary, the high field strength in the Debye layer leads to the anion displacement from it, and this layer performs predominantly selective cationic transport [65; 67]. When used in fuel cells, the membranes contact with water vapor only. In this case, the electrically neutral solution in their pore centers should be almost pure water with a low cation concentration. This water should freeze near 0 \({}^{\circ}\)C. However, due to the small size and high curvature radius, its freezing temperature decreases significantly. Freezing of this water was found in grafted membranes with a very high water uptake and a large pore size [68]. Moreover, its enthalpy value shows that not all water freezes at 0 \({}^{\circ}\)C, and about 20 remaining molecules per functional group gradually freeze during further cooling [69]. The water uptake of most ion-exchange membranes is less, therefore, phase transformations are usually not observed near 0 \({}^{\circ}\)C.  
The water freezing in perfluorinated Nafion membranes, the water uptake of which is usually much less, has been studied in detail. The authors of [60,70] constructed a phase diagram for these membranes at subzero temperatures, from which one can conclude that water freezing near 0 \({}^{\circ}\)C is possible for the water uptake of about 20-30 molecules per -SO\({}_{3}\)H group. [PERSON] et al. [71] estimated this water uptake as 50 molecules per functional group. Water in the Nafion membrane pores usually freezes only at subzero temperatures. Most researchers divided this water into bound water, as these water molecules do not freeze due to electrostatic interactions, and free water that have been observed to be freezable [60,72,73,74,75]. However, as the temperature decreases, their ratio changes significantly due to the freezing of water occupying the outer solvating shells of H\({}_{3}\)O\({}^{+}\) ions [70,72,73]. With a water uptake of four or less per functional group, water freezing is not observed at all [60]. This corresponds to the formation of a stable H\({}_{9}\)O\({}_{4}\)\({}^{+}\) ion. From the plot of total water vs. free water, [PERSON] et al. [60] estimated amount of non-freezable water as 4.75 molecules per proton of -SO\({}_{3}\)H group.