## 3 Results and Discussion
### Distribution Function of Relaxation Times (DFRT) Analysis
The analytical procedure (Figure 6a,b) was executed using the following expression for the \(R\mid\mid\text{Q}\) contributions [44; 45]:  
\[R_{\text{i}}\cdot\text{G}(\tau)_{\text{RQ}}=\frac{R}{2\pi}\cdot\frac{\sin((1- \phi)\pi)}{\cosh\left(n\ln\left(\frac{\tau}{\tau_{0}}\right)\right)-\cos((1- \phi)\pi)} \tag{14}\]  
with \(\tau_{0}=\omega_{0}^{-1}=\sqrt[6]{R_{\text{i}}\cdot Y_{0}}\). Figure 6: Distribution function of relaxation times (DFRTs) obtained from an EQM of a linear combination of (\(R\mid\text{Q}\)) and (\(G\)) elements and using the _Bayesian_ regularisation of the EIS data obtained in (**a**) wet (\(p_{\text{H2O}}\sim 10^{-2}\) atm) and (**b**) low humidity (\(p_{\text{H2O}}\sim 10^{-5}\) atm) conditions at 450 \({}^{\circ}\)C in O\({}_{2}\) atmosphere (the 99% credibility interval is presented in the shadow area around the _Bayesian_ curves). An alternative approach to obtain a DFRT based on an EQM with a linear series of (\(R\mid\text{Q}\))-circuits, designated as multi-fit procedure (\(m\)(RQ)fit), is depicted in (**c**,**d**) for, respectively, wet and low humidity atmospheres. However, an alternative procedure was also adopted for simulating the _Gerischer_ dispersion (G) [44,45]:  
\[R_{\text{i}}\cdot\text{G}(\tau)_{\text{G}}=\frac{R}{\pi}\cdot\sqrt{\frac{\tau}{ \tau_{0}-\tau}},\tau\leq\tau_{0}\wedge R_{\text{i}}\cdot\text{G}(\tau)_{\text{G }}=0,\tau>\tau_{0} \tag{15}\]  
The DFRT is then the sum of all separate contributions [44,45]:  
\[R_{\text{p}}\cdot\text{G}(\tau)=\sum_{i}R_{\text{i}}\cdot\text{G}(\tau) \tag{16}\]  
with \(R_{\text{p}}=\sum_{i}R_{\text{i}}\), where the area under the curve of \(R\cdot\text{G}(\tau)\) equals \(R_{\text{p}}\). The analytical DFRT in wet conditions is composed of a first peak with a broad distribution for \(\tau_{0}=1.13\times 10^{-4}\) s, as a result of the low value of \(\phi=0.25\), followed by an asymmetric peak that is related to the _Gerischer_ dispersion for \(\tau_{0}=0.85\) s. Regarding the measurement in low humidity conditions, similar time constants were detected, but an additional contribution with a large time constant (\(\tau_{0}=7.06\) s) is observed. In this case, a sharper peak can be observed due to the higher value of the exponent (\(\phi=0.76\)). The full width at half maximum (FWHM) depends strongly on the value of the exponent \(\phi\)[44,45]. The \(\text{G}(\tau)\) of the _Gerischer_ element is unbounded to the left of \(\tau_{0}\), i.e., \(\lim\limits_{\tau\to\tau_{0}}\gamma(\tau)=\infty\) that relates to the diffusion part, followed by a straight vertical line with a discontinuity at \(\tau_{0}\) (right side of the peak), which relates to the capacitive part. It is worth saying that the analytical procedure is very sensitive to this behaviour, where the capacitive processes appear in the form of a narrower peak. Such complexity of the _Gerischer_ dispersion makes the DFRT of this contribution quite challenging. In this respect, the _Bayesian_ regression has been recently introduced [46,47], and it is here used as an attempt to facilitate the deconvolution of this response. In Figure 6a,b, the DFRT obtained by this method is also presented for comparison with the corresponding confidence interval (CI), by carrying out 2000 iterations. Indeed, the _Bayesian_ framework is a powerful tool to evaluate the credibility of the obtained DFRT solution, i.e., the narrower the CI, the more confident is the solution under the given hypothesis [46,47]. Such differences are highlighted in Figure 7a-c for a different regularisation parameter (\(\lambda\)), which was varied from \(10^{-3}\) to \(0.1\) for low humidity conditions. It follows that, for lower \(\lambda\) values, larger CI are observed, as well as more peaks, resulting in higher oscillation of the DFRT curve. Lower \(\lambda\) values usually lead to a higher degree of oscillation, creating more peaks, however with a better reconstructed EIS (Figure 7d) and lower residuals (Figure 7e).