## 4 Modelling the Structure-Property Relationships
### Modelling of Properties of Ion-Exchange (IEMs) Containing Nanoparticles
To take into account the presence of nanoparticles in a pore, the inter-gel domain is considered as a \"two-phase\" system containing a solution with the volume fraction \(f_{sin}=\frac{V^{sol}}{V^{unerged}}\) and a solid nonconductive inclusion (the core of the nanoparticle) with the volume fraction \(f_{pin}=\frac{V^{particle}}{V^{unerged}}\), where \(f_{sin}+f_{pin}=1\). The effective conductance of the inter-gel domain containing a solution and a nanoparticle, \(L_{i}^{s+p}\), may be presented, in accordance to the approach expressed by Equation (29), as  
\[L_{i}^{s+p}=\left[f_{pin}(L_{i}^{p})^{\beta}+f_{sin}(L_{i}^{s1})^{\beta} \right]^{1/\beta}=\left(1-f_{pin}\right)^{1/\beta}(L_{i}^{s1}) \tag{34}\]  
where the particle core conductance \(L_{i}^{p}=0\) (the non-conductive medium), \(\beta\) is a structural parameter, whose role is similar to that of \(\alpha\) in Equation (29). \(L_{i}^{s1}\) characterizes the solution within a pore, which can be considered again as a two-phase domain, containing the bulk electroneutral solution and the EDL, formed by the nanoparticle. \(L_{i}^{s1}\) is found by using the same two-phase approach:  
\[L_{i}^{s1}=\left[f_{EDLp}(L_{i}^{EDL})^{\beta}+\left(1-f_{EDLp}\right)(L_{i}^{ s})^{\beta}\right]^{\frac{1}{\beta}} \tag{35}\]  
where \(f_{EDLp}=\frac{V^{EDLp}}{V^{gal+EDLp}}\) is the volume fraction of the nanoparticle's EDL within this domain, \(\left(1-f_{EDLp}\right)\) is the volume fraction of the electroneutral solution whose conductivity is, as before, assumed equal to that of the free solution. To find \(V^{EDLp}\), it is assumed that each nanoparticle is a sphere with diameter \(d\); the size of all particles is the same. The growth of their volume fraction is achieved by increasing the number of pores where they are present. The thickness of the diffuse part of the EDL is presented as follows:  
\[L_{D}=\sqrt{\varepsilon\epsilon_{0}RT/2c_{i}^{s}z_{i}^{2}F^{2}} \tag{36}\]  
where \(\varepsilon\) is the relative dielectric permittivity, \(\varepsilon_{0}\) is the vacuum permittivity, \(z_{i}\) is the ion charge number, \(F\) is the Faraday constant. It is assumed that the diffusion coefficients in the nanoparticle's EDL are the same as in the free solution, while the concentrations of cation and anion are found using the Donnan relation, as in the gel phase. Figure 5: Schematic representation of a fragment of a mesoporous IEM containing a charged nanoparticle surrounded by an EDL. Adapted with permission from [174], Elsevier, 2016. When applying Equations (34)-(36), it is possible to calculate the conductance coefficients \(L_{i}\) and then the membrane conductivity, electrolyte diffusion permeability, transport numbers. The EDL formed around a nanoparticle plays an important role. It contributes to an increase in the membrane counterion conductivity, since the counterion concentration there is essentially higher than in the bulk pore solution. Increasing counterion conductivity and reduction of the concentration of co-ions (which are excluded from the EDL) results in higher membrane permselectivity. As the EDL thickness increases with decreasing the salt concentration in the external solution, its role increases; hence, the difference in membrane conductivity and permselectivity between commercial membranes and the membranes doped with nanoparticles increases with decreasing the concentration of the external solution. However, the solid core of the nanoparticle is an insulator and this contributes to loss of the membrane conductivity at relatively high concentrations (>1 M), where the EDL thickness and, consequently, its role decrease. Thus, according to the model [174], the conductivity of nanoparticles containing membranes should be higher than that of the pristine membranes in dilute solutions. On the contrary, their conductivity should be lower than that of the pristine membranes in concentrated solutions.