## 4 Modelling the Structure-Property Relationships
### \"Solution-Diffusion\" Models
#### 4.1.2 Multiphase Models
Equations (29) and (8) allows a simple approximation at \(\alpha\to 0\)[63]:  
\[\kappa=(\kappa^{g})^{f_{1}}(\kappa^{s})^{f_{2}} \tag{33}\]  
Since the conductivity of gel, \(\kappa^{g}\), only slightly depends on electrolyte concentration (since the co-ion uptake by the gel is quite low--when the external concentration is not high, usually \(<\)1 M), the \(lg\kappa-lg\kappa^{s}\) correlation, according to Equation (33), should be linear with \(f_{2}\) as the coefficient. Numerical calculations indicate that near the \"isoconductance point\" (where \(\kappa=\kappa^{g}=\kappa^{g}\)), \(\kappa\) is almost independent on \(\alpha\)[63]. In the range \(0.1c_{iso}<c<10c_{iso}\), the \(lg\kappa-lg\kappa^{s}\) dependence can be approximated by a straight line with slope \(f_{2}\), if \(|\alpha|\leq 0.2\). For most IEMs, \(\alpha\) lays in the range \(0.1-0.3\)[37; 54], hence, Equation (33) can be confidently applied, when the concentration is sufficiently close to the isoconductance concentration \(c_{iso}\). Figure 4 shows \(Lg\kappa-Lg\) dependencies for different IEMs taken from different papers. Since \(\kappa^{s}\) is approximately proportional to the concentration of bathing solution, \(c\), the slope of these curves should be close to \(f_{2}\). Taking into account that some approximations have been made, we will call the value of \(f_{2}\) found as the slope of \(Lg\kappa-Lg\) linear regression the \"apparent volume fraction of the inter-gel solution, \(f_{2\text{app}}\)\". It can be seen from Figure 4 that the slope of linear \(Lg\kappa-Lg\) approximations for the homogeneous membranes is significantly lower than this slope for the heterogeneous ones. The membranes are identified in Table 1, which contains also the values of \(f_{2\text{app}}\) and the conductivity of the gel phase, \(\kappa^{g}\). To a first approximation (justified when the contribution of co-ions in the conductivity of the gel can be neglected), the value of \(\kappa^{g}\) may be considered as a constant equal to the conductivity of the membrane at the isoconductance point. Figure 4: Conductivity of IEMs under AC (\(\kappa\)) as a function of bathing solution concentration, \(c\), in Lg-Lg coordinates for different membranes; 1 – CMX in KCl [141], 2 – MK-40 in KCl [141], 3 – Nafion 125 in NaCl [153], 4 – MF-45K-101 in NaCl [153], 5 – MA-41 in KCl [141], 6 – AMX in KCl [141], 7 – AMV in KCl [144], 8 – AMX in NaCl [154], 9 – AMX in KCl [144], 10 – CMS in NaCl [146], 11 – LNA in NaCl [154]. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Membrane**} & \multirow{2}{*}{**Type, Manufacturer**} & \multirow{2}{*}{**IEC,** & **Water Content,** & **Conductivity of Gel** & **Apparent Volume** \\  & & **meq/g** & **  
[PERSON] et al. [41; 148] have compared the values of \(f_{2}\) found from conductivity measurements when applying Equation (33) and when using other methods, namely electrolyte sorption, differential scanning calorimetry (DSC) and standard contact porosimetry [37; 38; 156]. Relatively close values of \(f_{2}\) were reported. For example, for a heterogeneous MK-40 membrane, the authors found \(f_{2}\) equal to 0.17 \(\pm\) 0.02 from conductivity measurements [41], 0.10 \(\pm\) 0.02 [148] and 0.26 \(\pm\) 0.02 [41] from sorption, 0.21 from DSC [41] and 0.23 \(\pm\) 0.05 from porosimetry [41]. These results show that the microheterogeneous model adequately describes the membrane properties when operating with reasonable structure and kinetic parameters. The ease of applying Equation (33) for processing experimental conductivity data, the possibility to determine the parameters having a clear sense, the ability to describe a set of transport properties, involving the diffusion permeability and ion transport numbers, makes this model suitable for IEMs characterization [8; 142; 143; 144; 157; 138; 145; 139; 160; 161].