## 3 Results and Discussion
### Electrochemical Characterizations
As can be observed, membrane potential values for the sample with lower pore radii, Sf-NPAM/TiO\({}_{2}\), hardly differ from those corresponding to the ideal positively charged membrane, but reduced \(\Delta\Phi_{\text{mbr}}\) values were obtained for the Ox-NPAM/TiO\({}_{2}\) membrane (according to Figure 2 explanations), as the difference is more significant at high concentration ratio values as the result of a fixed charge screening effect. On the other hand, dependence of \(\Delta\Phi_{\text{mbr}}\) values on the electrolyte solution can be observed in Figure 3b, where a comparison of data points for the Sf-NPAM/TiO\({}_{2}\) membrane using KCl and CaCl\({}_{2}\) is shown. The combination of both membrane geometry and surface material can be observed in Supplementary Information (Figure S3), where a comparison of membrane  
Figure 2: Scheme of ion transport through two positively charged nanoporous membranes with the same thickness and material but different pore size and porosity. **(a)** Pore charge practically rejects all co-ion favoring counter-ion transport in membranes with reduced pore radii; **(b)** higher pore size favors co-ion inclusion, increasing co-ion transport number. potential values measured with KCl solutions for Sf-NPAM, Ox-NPAM, Sf-NPAM/TiO\({}_{2}\), and Ox-NPAM/TiO\({}_{2}\) samples is presented [36], while the effect of pore size, porosity, and structure asymmetry for NPAMs has already been reported [35; 45]. The fit of the experimental values can be performed taking into account both Donnan and diffusion potentials as well as membrane electronentrality, and it allows the determination of the effective membrane fixed charge concentration (X\({}_{\rm{eff}}\)), anion/cation diffusion coefficient ratio (D\({}_{\rm{Cl}}\)\(-\)/D\({}_{\rm{Na}}\)+, because (t\({}_{\rm{Cl}}\)\(-\)/t\({}_{\rm{Na}}\)+) = (D\({}_{\rm{Cl}}\)\(-\)/D\({}_{\rm{Na}}\)+) [44]) or anion permselectivity, as has already been explained in previous papers [46; 47; 48]. The following values were determined: X\({}_{\rm{eff}}\) = + 0.015 M, D\({}_{\rm{Cl}}\)\(-\)/D\({}_{\rm{Na}}\)+ = 3.74, and P\({}_{\rm{Cl}}\)\(-\) = 45.2% for Sf-NPAM/TiO\({}_{2}\) membrane, but X\({}_{\rm{eff}}\) = + 0.012 M, D\({}_{\rm{Cl}}\)\(-\)/D\({}_{\rm{Na}}\)+ = 2.47, and P\({}_{\rm{Cl}}\)\(-\) = 25.3% for Ox-NPAM/TiO\({}_{2}\) membrane. These results show the reduction in pore size increases for ionic diffusion coefficient ratio, which tends to a value more similar to that for NaCl solution, (D\({}_{\rm{Cl}}\)\(-\)/D\({}_{\rm{Na}}\)+)\({}^{\rm{o}}\) = 1.60), and logically for anionic permselectivity, with a reduction of 44%, both associated with higher electrolyte content into the pores of the Ox-NPAM/TiO\({}_{2}\) membrane. Figure 4 shows a comparison of experimental and fitted values as well as  
Figure 3: Membrane potential as a function of solution ratio for: (**a**) Sf-NPAM/TiO\({}_{2}\) membrane (\(\square\)), Ox-NPAM/TiO\({}_{2}\) membrane (\(\lozenge\)) solution diffusion potential (dashed line) for NaCl electrolyte, and (solid line) ideal positively charged membrane (t\(-\) = 1); (**b**) Sf-NPAM/TiO\({}_{2}\) membrane and KCl solutions (\(\Delta\)), CaCl\({}_{2}\) solutions (\(\bigtriangledown\)). the separate contribution of Donnan and diffusion potentials as a function of variable concentration C\({}_{\rm v}\) for Sf-NPAM/TiO\({}_{2}\) and Ox-NPAM/TiO\({}_{2}\) membranes, where the agreement between experimental and theoretical values support the suitability of these results. On the other hand, a correlation between effective fixed charge concentration (X\({}_{\rm ef}\)) and surface charge density (\(\sigma_{\rm s}\)) for cylindrical pores was already proposed by the following expression [49]: \(\sigma_{\rm s}=\) (X\({}_{\rm ef,F.r_{\rm p}}\)/2), where F represents Faraday constant. Surface charge density values for the studied membranes were 7.2 mC/m\({}^{2}\) for the Sf-NPAM/TiO\({}_{2}\) sample and 7.5 mC/m\({}^{2}\) for the Ox-NPAM/TiO\({}_{2}\) sample.