## 4 Cross-Linking of Polymer Membranes  
It can be assumed that there is a certain optimal size of pores and channels connecting them to ensure a good combination of ionic conductivity and selectivity of ion-exchange membranes. On the one hand, to provide high ionic conductivity of the membranes, large channels and pores are required, and hence a high water uptake. However, if we refer to the example with grafted membranes, the fact that membranes with intermediate water uptake have optimal transport properties can be seen [84]. At low water uptake, the whole pore/pore volume is covered by a double electric layer and the coion concentration in them is negligible. Therefore, the selectivity should be very high. But narrow channels and weak pore connection determine too low conductivity of such materials. Even at a high water uptake of the ion-exchange membranes equilibrated with water, the counterion concentration in the inter-pore solution is quite high (about 2-3 mol/L). Therefore, as a first approximation, the Debye layer thickness is assumed to be constant. With increasing water uptake, the pore size increases mainly due to an increase in the volume of the electrically neutral solution. Thus, with water uptake increasing, the counterion transport numbers, corresponding to the membrane selectivity, should decrease. Another factor determining the conductivity is the charge carrier concentration [97], which is determined by the counterion concentration. Since in most applications, membranes with functional groups based on strong acids and bases are used, it can be assumed that in the hydrated state the dissociation degree of functional groups is close to one and the carrier concentration is determined by the ion-exchange capacity. However, an increase in the concentration of functional groups in the hydrophobic polymer matrix unambiguously leads to an increase in water uptake and, hence, to an increase in the carrier mobility. Since water in the membrane matrix acts as a plasticizer, the elasticity of the matrix, which determines its swelling degree, increases with water uptake increasing. Moreover, a lot of ion-exchange materials with a high ion-exchange capacity lose their mechanical properties or even become water-soluble. This limits their practical application. In fact, this means that strategies aiming at increasing the membrane conductivity by increasing its ion-exchange capacity or water uptake should definitely decrease its selectivity and the mechanical properties [107; 153; 154; 155]. One of the solutions is the cross-linking of polymer chains. This limits the membrane swelling degree and, hence, leads to an improvement in their selectivity and mechanical properties [156; 157; 158; 159]. In a number of synthetic approaches, cross-linked agents, e.g., divinylbenzene (DVB), can be added to the ionomer during polymerization. Using such methods, highly selective membranes are often obtained. For example, the Neosepta(r) CMX membrane based on a sulfonated styrene-divinylbenzene copolymer (s-PS-DVB) has apparent cation transport number of 99% (0.5 M/0.1 M NaCl), low permeability for sodium chloride and contains 7-9 water molecules per functional group in the Na\({}^{+}\)-form [160]. [PERSON] et al. [128] showed that an increase in the cross-linking degree for grafted membranes based on sulfonated polystyrene can halve the membrane hydration degree as well as reduce its methanol permeability by 1.8 times. Generally, ion-exchange membrane manufacturing methods, including solvent casting stage, do not imply the possibility of medium or highly cross-linked materials preparation. Cross-linking method requires additional processing steps or the introduction of special groups (reactive sites) [3; 161; 162; 163]. Such approaches are rather relevant for membranes based on polyetheretherketones, polysulfones or block copolymers based on polystyrene and polyolefins. It is worth noting that this can significantly increase the membrane cost.