## 2 Materials and Methods
### Electrochemical Characterization  
The electrochemical test-cell that was used for electrochemical characterization (concentration potential and impedance spectroscopy measurements) is schematized in the Supplementary Information (Figure S2a). The studied samples were placed between both half-cells, which were filled with NaCl aqueous solutions, stirred at 540 rpm to minimize interfacial effects, and a reversible Ag/AgCl electrode was introduced in each half-cell. Measurements were carried out at room temperature (25 \(\pm\) 2 \({}^{\circ}\)C) and standard pH (5.8 \(\pm\) 0.3). Figure S2b in the Supplementary Information shows a scheme of solid sample/electrolyte electrical distribution and interfacial effects.  
Concentration potentials (\(\Delta\)E) measurements were performed with different NaCl solutions (concentrations c\({}_{\text{c}}\) and c\({}_{\text{v}}\)) in each half-cell (c\({}_{\text{c}}\) = 0.01 M for all measurements, while c\({}_{\text{v}}\) ranged between 0.002 M and 0.1 M), and the Ag/AgCl electrodes were connected to a digital voltmeter (Yokohama 7552, 1 G\(\Omega\) input resistance, Tokyo, Japan) to measure electrical potential values [39]. When reversible electrodes are used in this kind of measurements, \(\Delta\)E values also include the electrode potential, \(\Delta\Phi_{\text{elec}}\) = (RT/zF)In(c\({}_{\text{f}}\)/c\({}_{\text{v}}\)), where z represents the valence of reversible ion, while R and F are the gas and Faraday constant and T is the temperature of the system [40]. Consequently, the nanostructure contribution can be determined by subtraction of electrode potential to measured values: \(\Delta\Phi=\Delta\text{E}-\Delta\Phi_{\text{elec}}\). The analysis of \(\Delta\Phi\) values was carried out by taking into account potentials for both the solution/sample interface and diffusion in the pores, plus electroneutrality condition, [26] and it is described in detail in Section 3.3. Electrochemical Characterization.  
Electrochemical impedance spectroscopy (EIS) measurements were carried out by connecting the electrodes to a Frequency Response Analyzer (FRA, Solartron 1260, Farnborough, England) controlledby a computer. Up to 100 different frequencies ranging between 1 Hz and 10\({}^{7}\) Hz at maximum voltage of 0.01 V were recorded for the system: electrode/0.002 M NaCl solution/NPAS sample/0.002 M NaCl solution/electrode [41]. EIS data were corrected by software and the influence of connecting cables and other parasite capacitances. ZView 2 data analysis program (Scribner, Southern Pines, NC, USA) was used for electrical parameters determination.  
The impedance (Z) is a complex number, \(Z=Z_{\text{real}}+\text{j}\,Z_{\text{img}}\), and real (\(Z_{\text{real}}\)) and imaginary (\(Z_{\text{img}}\)) parts can be separately obtained by algebra rules. These components are related to the transport of charge across the system (electrical resistance, R) and charge storage (capacitance, C) by the following expressions:  
\[Z_{\text{real}}=\text{R}/[1+(\omega\text{RC})^{2}] \tag{2}\]  
\[Z_{\text{img}}=-\omega\text{R}^{2}\text{C}/[1+(\omega\text{RC})^{2}] \tag{3}\]  
where \(\omega=2\pi\text{f}\) is the angular frequency. The analysis of the impedance data was performed by using the Nyquist plot in the complex plane (\(-Z_{\text{img}}\) versus \(Z_{\text{real}}\)), where a parallel resistance-capacitance circuit (RC) corresponds to a semi-circle with intercepts on the \(Z_{\text{real}}\) axis at R\({}_{\infty}\) (\(\omega\)\(\rightarrow\)\(\infty\)) and R\({}_{0}\) (\(\omega\)\(\rightarrow\)0), with R = 0.5(R\({}_{0}\)-R\({}_{\infty}\)) being the resistance of the system, while the maximum of the semi-circle occurs at such a frequency that \(\omega\)RC = 1 [42]. This description is valid for systems with a unique relaxation time (homogeneous systems), while a depressed semi-circle due to a distribution of relaxation times is obtained in the case of non-homogeneous systems [42; 43].