## 4 Modelling the Structure-Property Relationships
### \"Solution-Diffusion\" Models
#### 4.1.2 Multiphase Models
When assuming \(\overline{c}_{-}<<\overline{Q}\) and \(\overline{c}_{+}\approx\overline{Q}\), Equation (24) can be simplified. In the case of a symmetrical electrolyte (\(z_{+}=-z_{-}=z\)) one gets [46; 148]:  
\[\overline{c}_{A}=\frac{K_{D}^{z}}{\overline{Q}}c_{A}^{2},\overline{c}_{1}= \overline{Q}+\overline{c}_{A} \tag{31}\]  
where subscript \"\(1\)\" is used for the counterion; \"A,\" for the co-ion; \(c_{A}\) refers to the co-ion concentration in the inter-gel solution (assumed to be the same as the external solution). The ions are present in both phases; the partition coefficient, \(K_{s}\), involving the co-ion concentration in the membrane, \(c_{A}^{s}\), can be found using Equation (31) [63]:  
\[K_{s}=\frac{c_{A}^{s}}{c_{A}}=f_{1}\frac{K_{D}}{Q^{g}}c_{A}+f_{2} \tag{32}\]  
in which the first term on the right-hand side stands for the contribution of the gel phase and the second term, for that of the inter-gel solution. \(Q^{g}\) and \(K_{D}\) are the IEC of the gel phase and the Donnan coefficient, respectively. Despite of small value of \(f_{2}\) (typically less than 0.1 in homogeneous membranes and close to 0.2 in heterogeneous ones), the sorption of electrolyte by the inter-gel spaces is dominant, especially in diluted solutions, because of the co-ion exclusion from the gel involving EDL in micro- and mesopores [46; 148; 149] is small. For conventional IEMs, Equation (31) is verified for external concentrations up to \(1-2\) M. Note that an equation similar to Equation (32) was proposed by [PERSON] et al. [150], where the second term on the right-hand side of this equation was attributed to an \"uncharged material,\" which can sorb electrolyte stronger than the charged gel. Equation (32) predicts a linear increase of \(K_{s}\) with increasing the external concentration. At low external electrolyte concentration \(K_{s}\) is close to \(f_{2}\), that is, it value is in the vicinity of 0.1 (or lower) for the homogeneous membranes and 0.2 for the heterogeneous ones [46; 148; 149]. To summarize, Equations (29)-(32) form the microheterogeneous model. There are six parameters (in the case of a single electrolyte): two static, \(K_{D}\) and \(Q^{g}\) (thermodynamics coefficients); two structural, \(f_{1}\) and \(\alpha\); and two kinetic ones, the diffusion coefficients of counterion, \(D_{1}^{g}\) and co-ion, \(D_{A}^{g}\), in the gel phase. The diffusion coefficients in the inter-gel solution, \(D_{i}^{s}\), are assumed the same as in free solution. When these parameters are known, \(L_{i}\) coefficients can be calculated. Then the transport membrane characteristics \(\kappa\), \(t_{i}\) and \(P\), can be found by using Equations (8)-(10) giving the links between the Onsager and Kedem-Katchalsky conductance coefficients. Equations (8)-(10) can be applied not only in the case, where the membrane is equilibrated with a bathing solution but also when there is a concentration gradient across the membrane. Then Equations (8)-(10) are applied at any coordinate in the membrane for a local concentration \(c\) of the inter-gel solution. In this way, it is possible to incorporate the microheterogeneous model into a boundary-value problem for modelling ion and solvent transport in membrane systems [46; 151]. To obtain the microheterogeneous model's parameters, some experiments should be carried out and the data obtained processed. To get all the six parameters listed above, the IEC, electrolyte uptake, conductivity and diffusion permeability should be determined as functions of the electrolyte concentration. The data processing algorithm is described in References [54; 152]. However, often only some among the six parameters are of interest. For instance, the volume fraction of the gel phase (or inter-gel solution) is a quite important membrane characteristic. It may be relatively easily obtained from the concentration dependence of the membrane conductivity (\(\kappa\)).