## 4 Modelling the Structure-Property Relationships
### \"Solution-Diffusion\" Models
#### 4.1.2 Multiphase Models  
As it was discussed previously (Section 2), charged membranes are porous materials having generally micro-, meso- and macropores. For the definition of these types of pores, several characteristics can be applied (such as the length of action of adsorption forces) [123]. However, here we will use only one parameter to distinguish the pore type: the ratio of the pore radius, \(r\), to the Debye length, \(\lambda\), characterizing the EDL thickness at the inner pore wall. In the case of \(r<\lambda\), we have a micropore (\(r<1\) nm); if \(r>\lambda\) but \(r\) and \(\lambda\) being of the same order of magnitude, we deal with a mesopore and if \(r>>\lambda\) (\(r>50\) nm), it is a macropore. Despite the complicated structure of charged membranes, it is important that mesopores and macropores contain electrically neutral solution in their central part, which can be considered as a separate phase (Figure 3). The properties of this solution (dielectric permittivity, ion and water diffusion coefficients) are very close to the corresponding properties of the external free solution [5; 29; 44]. Note that heterogeneous IEMs contain macropores, while homogeneous IEMs do not [37]. Therefore, one can expect that the volume fraction of electroneutral solution in the homogeneous membranes is essentially lower than in the heterogeneous membranes. As soon as the first electroneutral phase is defined, the remaining volume can be attributed to the second phase. This phase, called \"gel phase\" [63; 124], includes the polymer matrix with fixed charged groups together with mobile ions and water within the EDLs, which compensate the charge of the fixed ions. Since the central parts of the relatively large pores are excluded from the gel phase, it can be considered as a microporous relatively homogeneous medium. Sometimes, within the gel phase, one distinguishes the third phase, being the aggregate of interlaced hydrophobic polymer chains, which is not permeable for ions and solvent [63]. This hydrophobic phase may also involve the (non-conducting) inert reinforcing binder, such as polyethylene in heterogeneous membranes. There is no distinct boundary between the inter-gel electroneutral solution and the EDL, which belongs to the gel phase. Nevertheless, the separation of the membrane material into different phases is a useful method, which allows one to simplify the mathematical description. [PERSON] et al. [125] carried out such a separation when applying a continuous space charge model for calculation of the membrane conductivity. The main idea for modelling the transport within a membrane considered as a multiphase medium is in attributing some physico-chemical properties to each phase and then to describe the properties characterizing the whole membrane as functions of the single phase properties. This idea is the central one in the approach called effective medium theory (or model) [91]. Let us consider a macroscopic volume in the form of a layer of thickness \(dx\). This layer should be sufficiently large to be \"representative\" and contain all phases of the membrane (Figure 3). Simultaneously, it should be small enough to allow differentiation. We neglect the detailed variations of concentrations and potential within \(dx\) and operate with the averaged quantities in the phase elements; the phases are assumed in equilibrium with one another within the \(dx\) layer. When ion and water transfer is described in accordance with irreversible thermodynamics, by Equations (1) or (17), the problem is to obtain the effective transport coefficient \(L_{i}\), which characterizes a membrane layer of thickness \(dx\) as a function of coefficients \(L_{i}^{j}\) (characterizing an individual phase \(j\)), as well as the structural and geometric parameters, describing the shape and mutual position of the phases.