## 3 Results and Discussion  
### Electrochemical Characterizations  
Ion diffusive transport across membranes, which depends on membrane material electrical characteristics and pore size, as well as the electrolyte and solution concentrations, can be studied by analyzing membranes' potential values. This analysis allows the estimation of membrane effective fixed charge concentration (Xef) and ion transport numbers (ti), or fraction of the total current transported through the membrane by one ion (t = I\({}_{1}\)/I\({}_{7}\)) [38]; for an ideal positively charged membrane (anion exchanger), t \(-\) = 1 and t + = 0, that is, only negative charges can pass through the membrane, while for cation exchangers or ideal negatively charged membranes, t + = 1 and t \(-\) = 0. Effective fixed charge controls Donnan (or interfacial) potential (\(\Delta\)\(\omega_{\text{Don}}\)), while values of ion transport numbers in the membrane also depend on pore size and are directly related to diffusion potential (\(\Delta\)\(\omega_{\text{off}}\) ). The expressions for Donnan and diffusion potentials are [38]:  
\[\text{Donnan potential, }\Delta\omega_{\text{Don}}=(\text{RT}/\text{F})\text{ln}[( \text{wX}_{\text{eff}}/2\text{C})+[(\text{wX}_{\text{eff}}/2\text{C})^{2}+1)^{ 1/2}] \tag{1}\]  
\[\text{Diffusion potential, }\Delta\varphi_{\text{off}}=-\ (\text{RT}/\text{F})\text{[(t}- -\text{t}_{+})]\text{ln}(C_{\text{v}}/\text{C}_{\text{f}})=(\text{RT}/\text{F })[(1-2t-)]\text{ln}(C_{\text{v}}/\text{C}_{\text{f}}) \tag{2}\]with w = \(-1\)/+1 for negatively/positively charged membranes, while C\({}_{\text{i}}\) indicates solution concentration (i = v for variable solution concentration and i = f for fixed solution concentration), and R, F, and T have been previously identified. According to the Teorell-Meyer-Sievers model [41; 42], membrane potential consists of the sum of two Donnan potentials (one for each membrane/solution interface) and the diffusion potential in the membrane. Figure 2 shows a scheme of ion transport through two porous membranes of the same material (similar positive charge) but different pore size and porosity, as is the case for the Sf-NPAM/TiO\({}_{2}\) and Ox-NPAM/TiO\({}_{2}\) samples. Figure 2a shows a membrane with small pore size, which behaves as an almost ideal anion-exchanger, avoiding the pass of practically all cations (membrane co-ions), but the increase of pore size favors the inclusion of electrolyte solution into the pores as shown in Figure 2b, which raises the transport of cations and increases t\({}_{*}\) value. Because ion transport numbers depend not only on membrane charge but also electrolyte electrochemical parameters, membrane ionic permselectivity (PS\({}_{\text{i}}\)), which represents the relative variation of counter ions in the membrane with respect to their values in solution, is usually considered; then, for positively charged membranes and NaCl solutions, PSC\({}_{\text{Cl}}\)\({}^{-}\), is expressed as [43]: PSC\({}_{\text{Cl}}\)\({}^{-}\) = (t\({}_{\text{Cl}}\)\({}^{-}\) - t\({}_{\text{Cl}}\))/t\({}_{\text{Na}}\)\({}^{+}\), where t\({}_{\text{Cl}}\)\({}^{-}\) is the transport number of the counter-anion in the membrane pores, while t\({}_{\text{Cl}}\)\({}^{-}\) and t\({}_{\text{Na}}\)\({}^{+}\) represent the transport numbers of the anion and cation, respectively, in the NaCl solution. Figure 3a shows the dependence of membrane potential values on solution concentration ratio for clean Sf-NPAM/TiO\({}_{2}\) and Ox-NPAM/TiO\({}_{2}\) membranes, but concentration dependence for solution diffusion potential (dashed line) obtained using solution transport numbers in Equation (2) (t\({}_{\text{Na}}\)\({}^{+}\)\({}^{\text{o}}\) and t\({}_{\text{Cl}}\)\({}^{-}\)\({}^{\text{o}}\)[44], dashed line); theoretical \(\Delta\Phi_{\text{mbr}}\) values for an ideal anion-exchanger membrane (t\({}^{-}\) = 1, solid line) are also represented for comparison.