## 2 Materials and Methods
\(j^{\rm a}_{\rm TPB}\) is the e\({}^{-}\)-H\({}^{+}\) charge transfer rate per unit TPB length, which can be evaluated by Butler-Volmer equation,  
\[j^{\rm a}_{\rm TPB}=j^{\rm a}_{\rm TPB,0,ref}\exp\biggl{(}-\frac{E_{\rm H_{2}}} {R}\biggl{(}\frac{1}{T}-\frac{1}{T_{\rm ref}}\biggr{)}\biggr{)}\biggl{(}\frac {p_{\rm H_{2}}}{p_{0}}\biggr{)}\biggl{[}\exp\biggl{(}\frac{2\alpha^{\rm a}_{t }F}{RT}\eta^{\rm a}_{\rm act}\biggr{)}-\exp\biggl{(}-\frac{2\beta^{\rm a}_{t }F}{RT}\eta^{\rm a}_{\rm act}\biggr{)}\biggr{]} \tag{8}\]  
where \(\eta^{\rm a}_{\rm act}\) is the anodic activation; more details can be found in the Supplementary Materials. \[\eta^{\rm a}_{\rm act}=\Phi_{\rm e}-\hat{\Phi}_{\rm H^{+}}+E^{st}+\frac{RT}{2F }\ln\frac{p_{\rm H_{2}}}{1{\rm atm}} \tag{9}\]  
\(i^{\rm V,c}_{\rm e^{-}-O^{2-}\_LSCF}\) is the conversion rate between e\({}^{-}\) and O\({}^{2-}\) electric charges over the percolated LSCF surface per cathode volume. \[i^{\rm V,c}_{\rm e^{-}-O^{2-}\_LSCF}=i_{\rm LSCF}S^{\rm V}_{\rm LSCF},\ [A\ {\rm m }^{-3}\Bigr{]} \tag{10}\]  
where \(S^{\rm V}_{\rm LSCF}\) is the LSCF-pore DPB area per unit cathode volume (m\({}^{-1}\)). It is determined by detailed electrode microstructure parameters [16]. The e\({}^{-}\)-O\({}^{2-}\) charge transfer rate per LSCF-particle area can be analogously evaluated as follows:  
\[i_{\rm LSCF}=i_{\rm LSCF0,ref}\exp\biggl{(}-\frac{E_{\rm O_{2}}}{R}\biggl{(} \frac{1}{T}-\frac{1}{T_{\rm ref}}\biggr{)}\biggr{)}\biggl{(}\frac{p_{\rm O_{2} }}{p_{0}}\biggr{)}^{0.25}\biggl{[}\exp\biggl{(}\frac{2\alpha_{\rm LSCF}F}{RT} \eta^{\rm c,1}_{\rm act}\biggr{)}-\exp\biggl{(}-\frac{2\beta_{\rm LSCF}F}{RT} \eta^{\rm c,1}_{\rm act}\biggr{)}\biggr{]} \tag{11}\]  
where the local cathode activation overpotential-c1 referring to the oxygen-reducing half reaction O\({}_{2}\) = 2O\({}^{2-}\) + 2e\({}^{-}\) can be estimated as follows (more details can be found in the Supplementary Materials):  
\[\eta^{\rm c,1}_{\rm act}=\hat{\Phi}_{\rm O^{2-}}-\Phi_{\rm e^{-}}+\frac{RT}{4 F}\ln\frac{p_{\rm O_{2}}}{1\,{\rm atm}} \tag{12}\]  
where \(i^{\rm V,c}_{\rm H^{+}-O^{2-}\_L-B}\) is the conversion rate of about H\({}^{+}\)-O\({}^{2-}\) electric charges over the BZCY-LSCF-pores TPB lengths per cathode volume. \[i^{\rm V,c}_{\rm H^{+}-O^{2-}\_L-B}=j_{\rm L-B}\lambda^{\rm V}_{\rm L-B},\ [A\ {\rm m }^{-3}\Bigr{]} \tag{13}\]  
where \(\lambda^{\rm V}_{\rm L-B}\) is the LSCF-BZCY-pore TPB lengths per unit volume. The H\({}^{+}\)-O\({}^{2-}\) charge transfer rate per LSCF-particle area can be evaluated by Butler-Volmer equation as follows:  
\[j_{\rm L-B}=j_{\rm L-B,0,ref}\exp\biggl{(}-\frac{E_{\rm O-H}}{R}\biggl{(} \frac{1}{T}-\frac{1}{T_{\rm ref}}\biggr{)}\biggr{)}\biggl{(}\frac{p_{\rm H_{2} }O}{p_{0}}\biggr{)}\biggl{[}\exp\biggl{(}\frac{2\alpha^{\rm c}_{t}F}{RT}\eta^{ \rm c,2}_{\rm act}\biggr{)}-\exp\biggl{(}-\frac{2\beta^{\rm c}_{t}F}{RT}\eta^{ \rm c,2}_{\rm act}\biggr{)}\biggr{]} \tag{14}\]  
where the local cathode activation overpotential-c2 referring to the vapor-producing half reaction O\({}^{2-}\) + 2H\({}^{+}\) = H\({}_{2}\)O can be estimated as follows (more details can be found in the Supplementary Materials),  
\[\eta^{\rm c,2}_{\rm act}=\hat{\Phi}_{\rm H^{+}}-\hat{\Phi}_{\rm O^{2-}}-\frac {RT}{2F}\ln\frac{p_{\rm H_{2}O}}{1\,{\rm atm}} \tag{15}\]  
The extended [PERSON]'s model is adopted to describe the species transfers within the porous mediums. \[\
abla\cdot N_{\alpha}=R_{\alpha} \tag{16}\]  
\[j^{\rm a}_{\rm TPB}=j^{\rm a}_{\rm TPB,0,ref}\exp\biggl{(}-\frac{E_{\rm H_{2}}} {R}\biggl{(}\frac{1}{T}-\frac{1}{T_{\rm ref}}\biggr{)}\biggr{)}\biggl{(}\frac {p_{\rm H_{2}}}{p_{0}}\biggr{)}\biggl{[}\exp\biggl{(}\frac{2\alpha^{\rm a}_{t }F}{RT}\eta^{\rm a}_{\rm act}\biggr{)}-\exp\biggl{(}-\frac{2\beta^{\rm a}_{t }F}{RT}\eta^{\rm a}_{\rm act}\biggr{)}\biggr{]} \tag{8}\]0where \(c_{\alpha}\) is the molarity of species \(\alpha\). \(D_{\alpha}^{\mathrm{eff}}\) is the effective diffusion coefficient.