## 4 Modelling the Structure-Property Relationships
### \"Solution-Diffusion\" Models
#### 4.1.2 Multiphase Models
This formulation addresses the effective-medium approach [91; 126] intensively developed in relation to a great variety of systems and transfer phenomena: electrical conduction [127], diffusion [128], heat transfer [129], optics and other [130; 131]. This approach was firstly applied by [PERSON] [132], who, as early as in 1873, has theoretically found the conductivity of a system containing a conductive continuum medium and dispersed spheres, the conductivity of which differed from that of the medium. Later, [PERSON], [PERSON], [PERSON], [PERSON], [PERSON] [133] and other [134; 135] contributed to the development of effective-medium approach. The hypothesis called \"principle of generalized conductivity\" [135] is often applied for generalization of experimental results. According to this hypothesis, the function relating \(L_{i}\) to \(L_{i}^{j}\) does not depend on both the nature of applied force and the nature of the transported species. Regarding the application of this approach to IEM, several forms of functions were proposed, linking the effective conductivity of the medium with that of individual phases. Some of them relate to the diffusion [133; 134], the others, the electrical conductance [134; 136; 137; 124]. [PERSON] et al. [63; 74; 135] developed a comprehensive model called the \"microheterogeneous model\" which deals with conductance coefficients \(L_{i}\) and \(L_{i}^{j}\) instead of particular transport coefficients. Along with the membrane electrical conductivity [138; 139; 140; 141; 54; 142; 143], this approach also allows one to find electrolyte diffusion permeability [54; 114], permselectivity (ion transport numbers) [54; 144] and some other properties [145; 146; 74; 147]. In the case of electrolyte sorption, conductivity, diffusion permeability and transport numbers, it is possible to quantitatively describe all these four characteristics as functions of bathing solution concentration starting from a single set of structural and kinetic parameters [147]. In the framework of the microheterogeneous model, the effective membrane conductance coefficient, \(L_{i}\), is expressed through \(L_{i}^{j}\) in the following form [63; 135]:  
\[L_{i}=\left[f_{1}\left(L_{i}^{g}\right)^{\alpha}+f_{2}\left(L_{i}^{g}\right)^ {\alpha}\right]^{1/\alpha} \tag{29}\]  
where \(L_{i}^{g}\) refers to the gel phase and \(L_{i}^{g}\) to the inter-gel electroneutral solution;\(f_{1}\) and \(f_{2}\) are the volume fractions of the corresponding phases: \(f_{1}=V_{g}/V_{m}f_{2}=V_{s}/V_{m},f_{1}+f_{2}=1\) (where \(V_{g}\), \(V_{s}\) and \(V_{m}\) are the volumes of gel, inter-gel solution and the membrane, respectively, see Figure 3); \(\alpha\) is the structural parameter depending on the position of the phases with respect to the axis of transport: when the phases are parallel to this axis, \(\alpha=1\); when they are in serial disposition, \(\alpha=-1\); in other cases \(-1<\alpha<1\). When describing the transport in the gel phase, all the assumptions made in the TMS model are accepted. The properties of the inter-gel solution are assumed the same as those of the bathing equilibrium solution; the Nernst-Planck and the LEN assumption are applied there. \(L_{i}^{s}\) and \(L_{i}^{g}\) are expressed as functions of the ionic diffusion coefficients, \(D_{i}^{s}\) and \(D_{i}^{g}\) and the concentrations, \(c_{i}^{s}\) and \(c_{i}^{g}\) in the corresponding phase [54; 63]:  
\[L_{i}^{s}=D_{i}^{s}c_{i}^{s}/RT,L_{i}^{S}=D_{i}^{g}c_{i}^{g}/RT \tag{30}\]  
Concentrations \(c_{i}^{s}\) and \(c_{i}^{g}\) are linked by the Donnan relations, Equation (24), the local equilibrium being assumed between both phases. Usually the concentration of fixed ions (IEC, \(\overline{Q}\)) in ion-exchange materials is rather high, close to 1 mol (L wet membrane)\({}^{-1}\) or higher. Co-ions are therefore strongly excluded, so that their concentration in the gel phase is small compared to \(\overline{Q}\), at least in relatively dilute solutions.