## 3 Results and Discussion
### Transport Numbers in SZY Membranes
#### 3.2.3 Transport Numbers of Protons in the Bulk SZY Membrane in Oxidizing Atmospheres  
The effective transport numbers of protons, \(\vec{t}_{\rm OH^{\bullet}}\), in the bulk SZY membrane in air were measured using a water vapor concentration cell \(p\)H\({}_{2}\)O\({}^{\prime}\), Pt/SZY/Pt, \(p\)H\({}_{2}\)O\({}^{\prime\!/}\). Air flows with the different humidities (\(p\)H\({}_{2}\)O\({}^{\prime}\) = 40 Pa, \(p\)H\({}_{2}\)O\({}^{\prime\!/}\) = 610, 3365 and 4240 Pa) at a flow rate of approximately 2 L h\({}^{-1}\) were supplied to the gas chambers of the cell. EIS and OCV measurements were performed in the temperature range of 500-800 \({}^{\circ}\)C. For illustration, the impedance spectra of the water vapor concentration cell at 750 \({}^{\circ}\)C are shown in Figure 10.  
To account for the electrode polarization, Equation (5) and the data of OCV and IS measurements were used for calculation of the effective proton transport numbers as well as it was done for oxygen ions. The Nernst EMF of the water vapor concentration cell can be written as:  
\[E_{\rm N}=\frac{RT}{2F}\ln\frac{p\rm{H_{2}O}^{\prime\!/}}{p\rm{H_{2}O}^{\prime \!/}} \tag{13}\]  
The values of \(R_{\rm T}\) and \(R_{\rm E}\) (\(R_{\rm E}=R_{\rm b}+R_{\rm gb}\)) obtained by fitting the impedance data to the equivalent circuit \(R_{\rm b}(R_{\rm gb}Q_{\rm gb})(R_{\rm e,1}Q_{\rm e,1})(R_{\rm e,2}Q_{\rm e,2})\) are given in Figure S3. Figure S4 shows the temperature dependences of E\({}_{\rm N}\) and E\({}_{\rm m}\) for the water vapor concentration cells.  
The evolution of the effective proton transport numbers, \(\vec{t}_{\rm OH^{\bullet}}\), as a function of \(p\)H\({}_{2}\)O\({}^{\prime\!/}\) and temperature are presented in Figure 11. As can be seen the effective transport number of protons increases with increasing \(p\)H\({}_{2}\)O\({}^{\prime\!/}\) which can be explained as follows. In wet atmospheres, the dissolution of protons may be described by the hydration reaction:  
\[\rm{H_{2}O(g)+V^{\bullet\bullet}_{O}+O^{\times}_{O}\rightleftarrows 2 OH^{\bullet}_{O}} \tag{14}\]  
where O\({}^{\prime\!\times}_{O}\) is an oxygen ion in a normal lattice site, V\({}^{\bullet\bullet}_{O}\) is an oxygen vacancy, OH\({}^{\bullet}_{O}\) denotes a proton localized on an oxygen ion, or a proton defect. It follows from reaction in Equation (13) that the concentration of proton defects has to increase proportionally to \(p\)H\({}_{2}\)O\({}^{\prime\!/2}\) at the cost of oxygen vacancies. Therefore, the effective proton transport number in the membrane exposed to the water vapor gradient should increase with a rise of \(p\)H\({}_{2}\)O\({}^{\prime\!/}\).  
The effective proton transport number increases with decreasing temperature reaching saturation at T \(\approx\) 600 \({}^{\circ}\)C. This fact is in accordance with a common idea that at low temperatures the concentrations of proton defects equal the saturation limit and are therefore independent on temperature [44]. The largest value of the effective proton transport number (\(\tilde{t}_{\text{OH}^{\ast}}\)= 0.87) in the ceramic SZY membrane exposed to a gradient of \(p\)H\({}_{2}\)O was obtained at \(p\)H\({}_{2}\)O\({}^{\prime}\) = 40 Pa, \(p\)H\({}_{2}\)O\({}^{\prime\prime}\) = 4240 Pa in the temperature range from 500 to 600 \({}^{\circ}\)C. Thus, SZY shows predominant proton conductivity in wet air at intermediate temperatures. The data on the proton transport numbers obtained in our research are in good agreement with those reported for SrZr\({}_{0.9}\)Y\({}_{0.1}\)O\({}_{3-8}\) in [45]: the proton transport numbers in wet air under \(p\)H\({}_{2}\)O-gradients are close to 0.8 at 700 \({}^{\circ}\)C, and also increases with increasing \(p\)H\({}_{2}\)O-gradient and decreasing temperature.  
Figure 11: Effective proton transport number in the bulk SZY membrane exposed to \(p\)H\({}_{2}\)O gradient in air as a function of: (**a**) \(p\)H\({}_{2}\)O\({}^{\prime\prime}\) (\(p\)H\({}_{2}\)O\({}^{\prime}\) = 40 Pa) and (**b**) temperature.