**Figure 3.** Representative images of the SA process of the studied CL microstructures.

It is important to highlight that all samples' SA reconstruction temperature was the same ( \(T=1 x {10}^{-6})\) and the reconstructions were stirring until the convergence of the annealing error \({E}_{SA}=1 x {10}^{-6}\). The C1 column indicates the bar-ordered CL system, \({I}_{xy}=0.00\pm 0.00\), while the C4 column represents the homogeneously agitated CL system \({I}_{xy}=1.00\pm 0.00\). C2 and C3 columns were selected to analyze their behavior in the results shown in Fig. 6. Each image in Fig. 3 represents a Ω ensemble, but the values are the averages and standard deviation obtained for W = 15.

Figure 4 presents plots based on \({I}_{SA}\) iterations: A) for the error variation of the reconstruction process \({E}_{SA}\) (Eq. 5), B) for the isotropy variation \({I}_{xy}\) (Eq. 9) and C) for \({D}_{eff}\) (Eq. 13), all graphs based on the \({I}_{SA}\) statistical iteration. The values presented in this figure include all the universe of ω realizations of the total samples studied.

Figure 4 (a) shows that the \({E}_{SA}\) value tends to zero, which indicates that the actual microstructure of the CL is converging to the values of the objective microstructure of the CL, as observed in the representative images of Fig. 3. As the \({I}_{SA}\) iterations increase, the overall error between the reconstructed microstructure and the target microstructure decreases, the reconstruction can be seen in Fig. 3, where in column 1 we have the initial microstructure arranged in aligned bars and in the following columns 2 to 4 the dispersion of the porous phase and the agglomerated phase is observed.

The \({I}_{xy}\) values in Fig. 4 (b) tend to unity, indicating that the error between the correlation functions \({S}_{j,x}\left(r\right)\) and \({S}_{j,y}\left(r\right)\) are decreasing, as indicated in Eq. 9. It is observed that \({I}_{xy}\) is more dependent of Φ than \({E}_{SA}\), this trend is similar to that reported by 36. Figure 4 (c) shows the response of \({D}_{eff}\)coefficient depending on the reconstruction process iteration \({I}_{SA}\). It is observed that the \({D}_{eff}\) variation is higher for the first \({I}_{SA}\) iterations.

Figure 5 shows the response of \({D}_{eff}\)compared to the evolution of isotropy \({I}_{xy}\). It must be pointed out that the numerical solution considers the black color as the pore phase and the white color as the agglomerate phase, both for a synthesized microstructure of a catalytic layer. The values presented in this figure includes all possibilities (points) and validation magnitudes (dotted lines).

Figure 5 (A) shows the evolution of \({D}_{eff}\) as a function of isotropy \({I}_{xy}\). The results from all examined samples (\({\Omega }\in \text{W}=15 \omega\)) are presented with markers and solid lines denoting their averages. It can be observed that when the \({I}_{xy}=0\) the microstructure corresponds to the porous phase and the agglomerate are vertically aligned. In this condition, it is observed that \({D}_{eff}\) values are equal to the analytical solution of a parallel diffusion resistances; these values their equations are highlighted in the gray shading of Fig. 5 (A). The values of \({D}_{eff}\)obtained by analytical solution are in the range of \(1.6 x {10}^{-5} {\text{m}}^{2}{\text{s}}^{-1}\)@ 50% \({D}_{O2,Pore}\)- 50% \({D}_{O2,Agg}\) and \(3.2 x {10}^{-6} {\text{m}}^{2}{\text{s}}^{-1}\)@ 10% \({D}_{O2,Pore}\)- 90% \({D}_{O2,Agg}\). Ceballos et al.29 reports effective oxygen diffusion values for a catalytic layer of a PEMFC with 30% porous phase surface fraction; the results reported in their work are obtained by solving different effective diffusion models, they report oxygen \({D}_{eff}\) in a range of \(1.87 x {10}^{-6} {\text{m}}^{2}{\text{s}}^{-1}\) for Nam and Kabiany model and \(8.91 x {10}^{-6} {\text{m}}^{2}{\text{s}}^{-1}\) for Tomadakis and Storirchos model; In this work, for the same porous phase surface fraction condition an analytical result of \(9.7x {10}^{-6} {\text{m}}^{2}{\text{s}}^{-1}\) is obtained. The variations in the reported values are attributed to the different microstructures of the evaluated electrodes.

As isotropy increases, \({D}_{eff}\) values exhibit an exponential decrease, the exponential decrease being particularly pronounced in the sample \(\varphi =10 \%\) (green color) and less pronounced in the sample \(\varphi =50 \%\) (red color). Figure 5 (B) shows a zoom section for details of the range \({D}_{eff}\le 1.5 x {10}^{-7} {\text{m}}^{2}{\text{s}}^{-1}\) and \({I}_{xy}\ge 0.3\). In this figure, the diffusion coefficient of oxygen in the agglomerate (data input: \({D}_{O2, Agg}=8.45 \text{x} {10}^{-9} {\text{m}}^{2}{\text{s}}^{-1}\)) is noted by the dashed blue line.

Zhao et al.48 report experimentally obtained magnitudes of the effective diffusivity coefficient for catalytic layers. The reported averages of \({D}_{eff}\) for catalytic layers range from \(4.2 \pm 0.9 x {10}^{-7} {\text{m}}^{2}{\text{s}}^{-1}\) @ 25°C to \(4.6 \pm 0.5 x {10}^{-7} {\text{m}}^{2}{\text{s}}^{-1}\) @ 75°C. Experimental samples exhibited a surface fraction of the porous phase within a range of \(\varphi\) = 30% to \(\varphi\) = 40%. In this study, the reported values for different surface fractions of the porous phase and agglomerates are in the range of magnitudes of the order of \({10}^{-6}\) when \({I}_{xy}\) is equal to zero and \({10}^{-8}\) when \({I}_{xy}\) is equal to one. Our numerical model takes into account the diffusive coefficient of oxygen within the agglomerate, attributing that as the porous phase decreases the \({D}_{eff}\) values approach the reported value of \({D}_{O2, Agg}=8.45 \text{x} {10}^{-9} {\text{m}}^{2}{\text{s}}^{-1}\). Similarly, Shen et al.49 report experimentally obtained magnitudes of \({D}_{eff}\) for a catalytic layer with different thicknesses; the average value obtained is \(1.47 \pm 0.05 x {10}^{-7} {\text{m}}^{2}{\text{s}}^{-1}\), a figure that also falls within the range of results obtained in this study. Chen et al.50 presents results of the effective diffusivity coefficient of oxygen for a sample with a surface fraction of 17% for the porous phase using the lattice Boltzmann method, reporting a \({D}_{eff}\) of \(7.71 x {10}^{-8} {\text{m}}^{2}{\text{s}}^{-1}\). In our work, the result obtained for a similar surface fraction of the porous phase is \(1.4 x {10}^{-8} {\text{m}}^{2}{\text{s}}^{-1}\) when it is 20% − 8\({D}_{O2,Pore}\)0% \({D}_{O2,Agg}\). The variation in results is attributed to the difference in the isotropy value between synthetic and experimental materials. Microstructural isotropy directly influences the behavior of the effective diffusivity coefficient. The results show that the exponential tendency is preserved at higher isotropy ranges. The box highlighted in the gray shading of Fig. 5 (B) presents \({D}_{eff} @ {I}_{xy}=1\).

Figure 6 shows \({D}_{eff}\) as a function of pore surface fraction \(\varphi\), for three selected isotropies referred to in columns of Figs. 3 and 5: the solid black line for \({I}_{xy}=0.0\), the gay line for \({I}_{xy}=1.0\)and dashed black line for \({I}_{xy}=0.5\).

Figure 6 shows the results obtained for \({D}_{eff}\) as a function of the superficial fraction of the porous phase. The results from all examined samples (\({\Omega }\in \text{W}=15 \omega\)) are presented with markers and solid lines denoting their averages. For \({I}_{xy}\)= 0.0 and \({I}_{xy}\)=0.5 the \({D}_{eff}\)ranges are on the left side of the graph; for \({I}_{xy}\)= 1.0 the \({D}_{eff}\)ranges are on the right side of the graph.

The results are presented within \(\varphi\)=10% to \(\varphi\) =50%, with 10% intervals. The solid black line, labeled as \({I}_{xy}\)=0.0, corresponds to the average of all samples when the microstructure is vertically aligned. Numerical results indicate that the effective diffusion coefficient is directly related to the porosity of the catalyst layer microstructure. Higher porosity has a positive impact on \({D}_{eff}\) enhancement, this trend agrees with empirical correlation models of the effective diffusion coefficient in porous materials (Bruggeman 51, Neale and Nader 52, Tomadakis and Sotirchos 53, Mezedur et al.54, Zamel et al. 55, Das et al. 56). The results suggest that the isotropy of the microstructure can also have a significant effect on the \({D}_{eff}\) as shown in Fig. 6. An increasing trend is observed when the porosity of the microstructure is more significant. When the porosity is minor, oxygen molecules collide more frequently at the pore surface, resulting in higher diffusion resistance, this behavior is known as the Knudsen effect 48. The Knudsen coefficient is proportional to the pore diameter size48,57,58. \({D}_{eff}\)values range from 3.26 x 10− 6 m2 s-1@ \(\varphi\) = 10% to 1.60 x 10− 5 m2 s-1@ \(\varphi\) = 50%. This behavior is attributed to the reduced diffusive resistance when the phases are vertically aligned. As isotropy in the microstructure increases, \({D}_{eff}\)diminishes due to the agitation within the microstructure. When \({I}_{xy}\)= 0.5, \({D}_{eff}\)values are within the range of 3.74 x 10− 8 m2 s-1@ \(\varphi\) = 10% and 6.53 x 10− 6 m2 s-1@ \(\varphi\) = 50%. In the case of a randomly monodisperse microstructure (\({I}_{xy}\)= 1.0), the value of \({D}_{eff}\)varies between 1.08 x 10− 8 m2 s-1@ \(\varphi\) = 10% and 6.22x 10− 8 m2 s-1@ \(\varphi\) = 50%. Accordingly, microstructures with lower isotropy (aligned bars) and higher porosity have lower diffusion resistance, which translates as higher \({D}_{eff}\).