## 3 Results and Discussion
### Equivalent Circuit Model (EQM)  
Figure 3a,b depicts the EIS dispersions obtained for the LNO+BCY10 composite cathode in both wet (\(p_{\text{H2O}}\sim 10^{-2}\) atm) and low humidity (\(p_{\text{H2O}}\sim 10^{-5}\) atm) conditions at 450 \({}^{\circ}\)C in O2 atmosphere. Significant differences can be observed between the two dispersions. In wet conditions, a well-defined spectrum with a high-frequency inductive tail can be observed, followed by a small contribution resembling a depressed semicircle. The presence of one or more processes at high frequencies has been previously observed in the SOFC literature, which are usually ascribed to interfacial charge-transfer processes [5; 19; 40; 41]. These processes can be appropriately described by \(R\mid Q\) elements, with impedance given by:  
\[Z_{\text{Q}}=Q_{\text{j}}^{-1}(\text{i}\omega)^{-\phi} \tag{2}\]  
Figure 2: SEM analysis of the (**a**) top-view and (**b**) fracture cross-section of the LNO+BCY10 composite cathode. where \(\omega\) corresponds to the angular frequency, \(Q_{\rm i}\) to the pseudo-capacitance, and \(\phi\) to the exponent of the angular frequency. The apparent capacitance is given by [4; 5; 42]:  
\[C=R^{(1-n)/n}Q^{1/n} \tag{3}\]  
With decreasing frequency, a straight line evolves, characteristic of diffusion limitations, followed by a smooth transition to an \(R\mid C\) shape, indicating a capacitive behaviour at lower frequencies. This shape resembles a _Gerischer_ contribution (\(G\)), which has been used to represent diffusion coupled to a reaction process in MIECs, such as the case of lanthanum nickelate cathodes [9] and also for a combination of a MIEC and an electrolyte phase [43]. The impedance of the _Gerischer_ contribution is given by:  
\[Z_{\rm G}(\omega)=\frac{Z_{0}}{\sqrt{K_{\rm G}+j\omega}} \tag{4}\]  
where \(Z_{0}\) is proportional to the chemical diffusion coefficient \(\left(\sqrt{\rm D}\right)^{-1}\), \(j\) is an imaginary number \(\left(j=\sqrt{-1}\right)\), and \(\omega\) is the radial frequency \((\omega=2\pi f)\). \(K_{\rm G}\) is related to the oxygen surface exchange rate, being independent of the geometry of the electrode. Hence, the _Gerischer_ resistance, \(R_{\rm G}\), is given by:  
\[R_{\rm G}=\frac{1}{Y_{0}\sqrt{K_{\rm G}}}\ [\Omega] \tag{5}\]  
where \(Y_{0}\) is a parameter related to the diffusion coefficient. The capacitance is calculated by:  
\[C_{\rm G}=\frac{1}{2\pi\cdot f_{max}\cdot R_{\rm G}} \tag{6}\]  
The resultant equivalent circuit model (EQM) is depicted in Figure 3c, which is composed by an inductor (\(L_{0}\)), an off-set resistance (\(R_{\infty}\)), a high-frequency electrode polarisation  
Figure 3: Impedance dispersions obtained under (**a**) wet (\(p_{\rm H2O}\sim 10^{-2}\) atm) and (**b**) low humidity (\(p_{\rm H2O}\sim 10^{-5}\) atm) conditions at 450 \({}^{\circ}\)C in O\({}_{2}\) atmosphere (numbers indicate the decades of the measuring frequencies). Equivalent circuit models (EQMs) used to fit the dispersions in (**c**) wet and (**d**) low humidity conditions. resistance (\(R_{\rm high}\)), and a _Gerischer_ element. Conversely, in low humidity conditions, the low-frequency region cannot be resolved within the measured frequency ranges (1 MHz to \(10^{-2}\) Hz), where an incomplete arc can be observed at low frequencies. For these conditions, a third element set composed of further \(R\mid\mid Q\) contributions was used (Figure 3d). The fitting curves are shown in Figure 3a,b, from where it is possible to observe a good match to the experimental points. However, this information is not sufficient, since if the fit model contains any parameter that has a minor influence on the form of the fitted curve, the precision of determination of this parameter is unclear.