## 4 Modelling the Structure-Property Relationships
### Modelling of Properties of Ion-Exchange (IEMs) Containing Nanoparticles  
Recently, IEMs containing immobilized nanoparticles have attracted a keen interest in the literature. This type of membrane modification was found promising for improving IEMs used in fuel cells. It was established that Nafion based composite membranes with incorporated SiO\({}_{2}\), ZrO\({}_{2}\), TiO\({}_{2}\) and other nanoparticles showed better water retention and proton conductivity, especially at low water vapor pressure, in a rather large temperature interval up to 140 \({}^{\circ}\)C [162; 163; 164; 165; 166; 167], which sometimes was accompanied with increasing IEC [168]. [PERSON] et al. [169; 170; 171; 161; 162; 163; 164; 165; 166; 167] reported that along with increasing water content and conductivity, the hybrid nanoparticle-containing membranes showed a decrease in diffusion permeability and an increase in permselectivity. It has been shown that IEMs modification with different nanoparticles is very important not only for a better performance of fuel cells but also for improving separation processes in different industries [172]. Namely, the modification of RO, NF and ultrafiltration membranes by immobilization of inorganic nanoparticles increases their antifouling resistance [172]. [PERSON] [169] suggested an interpretation of the change of membrane transport characteristics, in particular, the increase in \(\kappa\) by proposing a physicochemical model named the model of semielasticity of membrane pore walls. The main impact of nanoparticles on the membrane properties, according to this model, is due to the expansion of the pores and channels caused by the nanoparticles. The increase in the size of membrane channels leads to growing ion conductivity. Nevertheless, when the volume fraction of nanoparticles is too large, they block the ion conducting channels, which results in a decrease of the conductivity. Quantitative description of the effect of nanoparticles on the transport properties of IEMs is a rather complicated problem. [PERSON] et al. [115; 116; 117] developed a fine-pore model of ion transfer in an IEM with immobilized nanoparticles. The membrane is considered as quasi-homogeneous medium as in the TMS theory and applies the approach known as the \"fine porous membrane\" model suggested by [PERSON], [PERSON] and [PERSON] [173]. The simulation was compared with experimental data on diffusion permeability of a hybrid MF-4 SK membrane containing silica nanoparticles. This comparison allowed to the authors [115; 116; 117] to find in what extent the ion effective diffusion coefficients change in the membrane due to incorporation of silica. By applying the \"fine porous membrane\" model, the main membrane physicochemical parameters can be determined. In addition to the diffusion coefficients, it is possible to find the equilibrium distribution coefficients of electrolyte [117]. [PERSON] et al. [174] proposed a mathematical model describing the effect of nanoparticles on the transport properties of membranes on the basis of the microheterogeneous model [63; 54]. In addition to two phases (the gel phase and the electroneutral inter-gel solution) present in commercial membranes, there are nanoparticles. The core of a nanoparticle is non-permeable to ions and water. However, the EDL, which forms around the particle in the case where its surface is charged (Figure 5), contributes to the selective permeability of the membrane. It is assumed that the nanoparticles are localized in the meso- and macropores; they replace there the bulk-like electroneutral solution. If the surface of nanoparticle is charged (e.g., in the case of SiO\({}_{2}\) or ZrO\({}_{2}\), where it has a negative charge), the EDL around the particle is considered to be complementary to the EDL occurring at the charged pore walls (Figure 5).