## 3 Results and Discussion
### Transport Numbers in SZY Membranes
#### 3.2.1 Transport Numbers of Oxygen Ions in the Bulk and Film SZY Membranes
The high-frequency semicircle with a characteristic capacitance of \(10^{-9}\) F cm\({}^{-2}\) which is typical for the grain boundary response in ceramic electrolytes was ascribed to the response of the grain boundaries of SZY. The characteristic capacitance of the low-frequency arc (\(10^{-6}\) F cm\({}^{-2}\)) is typical for an electrolyte-electrode interface, so it can be considered as the electrode response. The oxygen-ion transport numbers were calculated using Equation (5) and the obtained values of \(R_{\text{T}}\) and \(R_{\text{E}}\) (\(R_{\text{E}}\) = \(R_{\text{b}}\) + \(R_{\text{gb}}\)). Figure 4: Impedance spectra of **(a)** Cell 1 with the bulk SZY membrane, **(b)** Cell 2 with the film membrane exposed to \(p\)O\({}_{2}\)(air)/\(p\)O\({}_{2}\)(H\({}_{2}\)) gradient (in the insert: CV curves for Cell 2); **(c,d)**—equivalent circuits for Cell 1 and Cell 2. For Cell 2, the Nyquist plots are characterized by the appearance of a small fragment of a high-frequency arc related to the electrolyte response and two overlapping arcs at lower frequencies with characteristic capacitances of 10\({}^{-6}\) and 10\({}^{-3}\) F cm\({}^{-2}\), which were ascribed to the effects of the electrode (Figure 4b). So far as the impedance spectra of Cell 2 were far from completed in the low-frequency region (see Figure 4b), the values of \(R_{\rm T}\) were determined from the measurement of the tangents to the CV curves of the concentration cell at low current values (the CV curves are shown in the insert to Figure 4b). The significantly different shapes of the electrode response of Cell 1 and Cell 2 were caused by the different compositions and geometries of the anode; the polarization resistance of the Ni-YSZ-anode supported cell was much larger than that of the electrolyte-supported cell (see Figure 4a,b). The spectra have been fitted to the equivalent circuit \(R_{\rm b}(R_{\rm gb}Q_{\rm gb})(R_{\rm el,1}Q_{\rm el,1})(R_{\rm el,2}Q_{\rm el,2})\). The equivalent circuits for Cells 1 and 2 are given in Figure 4c,d. In oxidizing atmospheres, the measurements were performed on Cell 1 supplied with air and argon with the same amount of humidity (\(p\)H\({}_{2}\)O = 3365 Pa) to prevent proton transport. So, the cell configuration may be designated as \(p\)O\({}_{2}\)(air), Pt/SZY/Pt, \(p\)O\({}_{2}\)(Ar). Typical impedance spectra of the cell in the Nyquist representation are depicted in Figure 5. The spectra consist of a fragment of a high-frequency semicircle with a characteristic capacitance of 10\({}^{-9}\) F cm\({}^{-2}\) related with the electrolyte response and a low-frequency semicircle with a characteristic capacitance of 10\({}^{-6}\)-10\({}^{-5}\) F cm\({}^{-2}\) ascribed to the polarization of the electrodes. The area-specific values of \(R_{\rm E}\) and \(R_{\rm T}\) obtained from the impedance data and used for calculation of transport numbers as functions of inverse temperature are presented in Figure S1. As can be seen, the film membrane resistance is about one order of magnitude smaller than the resistance of the bulk membrane (Figure S1a), however, the difference between the total resistances of the cells is much smaller (Figure S1b), which is caused by a higher polarization resistance of the electrode-supported cell. The values of the measured OCV (\(E_{\rm m}\)) and the theoretical EMF (\(E_{\rm N}\)) for the gas concentration cells are shown in Figure S2. The temperature dependences of \(E_{\rm m}\) and \(E_{\rm N}\) are somewhat different, since \(E_{\rm m}\), unlike \(E_{\rm N}\), is affected by the temperature-dependent electrode polarization. Figure 6 illustrates temperature dependences of the effective conductivity \(\overline{\sigma}\) of the bulk and film SZY membranes exposed to \(p\)O\({}_{2}\)(air)/\(p\)O\({}_{2}\)(H\({}_{2}\)) and \(p\)O\({}_{2}\)(air)/\(p\)O\({}_{2}\)(Ar) gradients, which were calculated using the following relation:  
\[\overline{\sigma}=\frac{l}{R_{\rm E}S} \tag{6}\]  
where \(l\) is a membrane thickness, \(S\) is a surface area of electrodes.