## 4 Modelling the Structure-Property Relationships
### Modelling of Properties of Ion-Exchange (IEMs) Containing Nanoparticles
However, the factor of increasing pore size and water content (determined experimentally and not taken into account in the model [174]) could contribute in additional increase in conductivity. Though, as far as we know, there are no publications reporting experimental data suitable for comparison of the membrane conductivity at different concentration, in the cases where the nanoparticles are present or absent. Known publications [166; 170; 175] provides the conductivity of the membrane equilibrated in pure water. As for the membrane permselectivity (quantified by the counterion transport number, \(t_{1}\)), this parameter should be higher in the membranes doped with nanoparticles, since even when the EDL thickness is low, the increase in \(t_{1}\) would be conditioned by the fact that a nonconductive nanoparticle replaces the electroneutral solution present in the inter-gel spaces. Therefore, contribution of the non-selective ion transport in this solution will be reduced in favour of highly selective transport in the charged gel phase. The content (volume fraction) of nanoparticles in the membrane, \(f_{p}\), is an important parameter. With increasing \(f_{p}\), the membrane conductivity initially increases (because the total EDL volume increases), then it passes through a maximum and declines. The latter is due to the saturation of the pore space available for nanoparticles. When \(f_{p}\) becomes large enough, the nanoparticles block ion conductivity in the pores, which they occupy. Then ion transport occurs only across the microporous medium (the \"gel phase\"), which is assumed not available for nanoparticles. An example of such a dependence is given in Figure 6a. It involves experimental data for a Nafion(r) membrane containing nanoparticles of caesium acid salt of phosphotungstic heteropolyacid [170] and the results of calculation according to the model described above [174]. Note that a similar dependence was found for a perfluorinated sulfocation-exchange MF-4 SK membrane (analogue of Nafion(r)): the proton conductivity curve as a function of additive content has a maximum at 2.6 vol%, when it is doped with SiO\({}_{2}\) and 1.5 vol%, when it is doped with PAni [175]. [PERSON] et al. [176] studied an experimental IEM modified with ZnO nanoparticles. Similarly as [PERSON] et al. [170] and [PERSON] et al. [175], they found that the dependence of water content, membrane potential, transport number and selectivity of this membrane initially increased with increasing the content of added ZnO nanoparticles, then these characteristics reach their maximum and declined when the nanoparticles' content was rather large. The presence of nanoparticles within the pores reduces their permeability towards the electrolyte diffusion (Figure 6b). Both the particle's core and the EDL around it contributes to a decrease in the diffusion permeability. However, there is another factor, the expansion of pores (not taken into account in the model [174]), which gives rise to increasing membrane diffusion permeability. Which factor is determinant, depends on the membrane material (which can be more or less rigid) and the incorporated nanoparticles. [PERSON], [PERSON] and [PERSON] [115] reports an increase in the diffusion permeability of a perfluorinated sulfocation-exchange MF-4 SK membrane after its doping with silica nanoparticles. Similar data are provided in Refs. [168; 175]. An increase in conductivity and a decrease in diffusion permeability essentially improve the permselectivity of modified membranes towards counterions [170; 174].