## 4 Modelling the Structure-Property Relationships
### \"Solution-Diffusion\" Models  
#### 4.1.1 Teorell-Meyer-Sievers (TMS) Model  
Historically, the first model successfully applied to explain the main properties of charged membranes, was the model developed in thirties years of the last century independently by [PERSON] [103] and [PERSON] [104], called the TMS model. It provides the basis of mathematical description of the ion transport in membranes. The main component of the TMS model is the Nernst-Planck equation, Equation (17). The membrane is considered as a single homogeneous phase (charged gel), representing an aqueous solution of matrix polymer chains together with mobile and fixed ions. Along with Equation (17), the model involves the LEN assumption in the membrane, Equation (21), the equation expressing the electric current as a function of the individual ionic flux densities:  
\[i=(z_{+}J_{+}+z_{-}J_{-})F \tag{23}\]and the Donnan equilibrium at the left-hand (I) and right-hand (II) interfaces used as the boundary conditions:  
\[\frac{\left(\overline{c}_{+}^{k}\right)^{1/z_{+}}}{\left(\overline{c}_{-}^{k} \right)^{1/z_{-}}}=K_{D}\frac{\left(c_{+}^{k}\right)^{1/z_{+}}}{\left(c_{-}^{k} \right)^{1/z_{-}}},k=I,\Pi \tag{24}\]  
where \(K_{D}\) is the Donnan equilibrium coefficient; \(\overline{c}_{i}^{k}\) and \(c_{i}^{k}\) are the concentrations of ion \(i\) at interface \(k\), from the side of the membrane and solution, respectively [105]. Equations (17), (21)-(24) form a boundary-value problem for the electrodiffusion of ions across an IEM placed between two solutions of known component concentrations. The Donnan relation, Equation (24), is obtained from the condition of continuity of electrochemical potential written for each ion species \(i\) at the membrane/solution interface \(k\):  
\[RT\ln a_{i}^{k}+z_{i}F\varphi^{k}=RT\ln\overline{a}_{i}^{k}+z_{i}F\overline{ \varphi}^{k} \tag{25}\]  
The ion activity coefficients implicitly entering into Equation (25) are generally functions of the pressure in each phase. It is easy to derive the following two relations from Equation (25):  
\[\left(\overline{a}_{+}^{k}\right)^{1/z_{+}}/\left(\overline{a}_{-}^{k}\right) ^{1/z_{-}}=\left(a_{+}^{k}\right)^{1/z_{+}}/\left(a_{-}^{k}\right)^{1/z_{-}} \tag{26}\]  
and the other for the electric PD between two phases:  
\[\Delta\varphi_{D}=\overline{\varphi}^{k}-\varphi^{k}=-\frac{RT}{z_{+}F}\ln \frac{\overline{a}_{+}^{k}}{a_{+}^{k}}=-\frac{RT}{z_{-}F}\ln\frac{\overline{ a}_{-}^{k}}{\overline{a}_{-}^{k}} \tag{27}\]  
When introducing ionic concentrations (mmol cm\({}^{-3}\)) in Equation (26) instead of activities (\(a_{i}=c_{i}y_{i}\) and \(\overline{a}_{i}=\overline{c}_{i}\overline{y}_{i}\)), we arrive at Equation (24) with the Donnan equilibrium coefficient, \(K_{D}\), expressed through the ratio of mean ionic activity coefficients:  
\[K_{D}=\left(y_{\pm}/\overline{y}_{\pm}\right)^{1/z_{+}-1/z_{-}} \tag{28}\]  
where \(y_{\pm}=\left(y_{+}^{r_{+}}y_{-}^{r_{-}}\right)^{1/v}=\left(y_{+}^{1/z_{+}}y_ {-}^{-1/z_{-}}\right)^{(1/z_{+}-1/z_{-})^{-1}}\). For the mean ionic activity coefficient in the membrane \(\overline{y}_{\pm}\), a similar expression can be written. The activity coefficients in water solutions can be determined from the data in the literature [78; 106]. Recently, [PERSON], [PERSON] and [PERSON] [47] have calculated the ionic activity coefficients using [PERSON]'s counterion condensation theory. The theory was originally developed for polyelectrolyte solutions [51] and contains no adjustable parameters. It is one of the most recognized theories, which describe colligative properties of polyelectrolyte solutions [51]. In this theory, a polyelectrolyte chain is modelled as a linear, infinitesimally thin and infinitely long rod with fixed charges evenly spaced along the backbone. The mathematical description involves a single parameter, \(\zeta\), related to the linear charge density of the polyelectrolyte chain [51]. \(\zeta\) is a ratio of two length scales, the distance between neighbouring fixed charges and the Bjerrum length, \(\lambda_{\text{B}}\).