## 2 Materials and Methods  
Taking Ni-BZCY/BZCY-LSCF PCFC as an example, the electrochemical and multi-physical coupling model for the PCFC with H\({}^{+}\)/e\({}^{-}\)/O\({}^{2-}\) mixed conducting cathodes is developed. It should be noted that the model is not limited to Ni-BZCY/BZCY/BZCY-LSCF, but can also be used for other PCFCs with the similar working processes. The PCFC cell consists of three parts: (a) the anode is a porous structure consisting of Ni- and BZCY- particles, (b) the electrolyte is made of dense BZCY, and (c) the cathode is a porous structure consisting of LSCF- and BZCY- particles. Figure 1a displays the schematic diagram of the microstructure and the working processes of a typical PCFC. The corresponding equivalent circuit of e\({}^{-}\), O\({}^{2-}\), and H\({}^{+}\) electric processes and two activation overpotentials within the LSCF-BZCY composite cathode are also illustrated in Figure 1b. Because LSCF in the cathode can conduct both electrons and oxygen ions, the oxygen reduction half reaction (O\({}_{2}\)\(\rightarrow\)O\({}^{2-}\)) occurs on the surface of the percolated LSCF-pore DPBs. Percolated means that there is a connection path between the DPBs and the oxygen inlet and electronic current collector. Steam formation half reaction (O\({}^{2-}\) + 2H\({}^{+}\) =H\({}_{2}\)O) occurs at the percolated LSCF-BZCY-pore TPBs. Percolated TPBs means that there are three paths connected with TPBs, including vapor transport (pore) from the TPBs to the cathode inlet, proton transport (BZCY) from TPBs to the electrolyte layer, and electron transport (LSCF) from TPBs to the electronic current collector. O\({}^{2-}\) is the intermediate transform substance between electrons and protons. With hydrogen as the fuel and air as the oxidant, the basic electrochemical reactions based on the mutual electric charge conversions among the electrons, oxygen ions, and protons can be described as follows:  
\[\text{Anode}:\text{H}_{2}(\text{pore})=2\text{H}^{+}(\text{BZCY})+2\text{e}^{-} (\text{Ni}) \tag{1}\]  
\[\text{Cathode}:0.5\text{O}_{2}(\text{pore})+2\text{e}^{-}(\text{LSCF})=\text{O }^{2-}(\text{LSCF}) \tag{2}\]  
\[\text{O}^{2-}(\text{LSCF})+2\text{H}^{+}(\text{BZCY})=\text{H}_{2}\text{O}( \text{pore}) \tag{3}\]  
The contents in parentheses indicate the carriers of the reactants and products. As PCFC has quite different working processes from the conventional O\({}^{2-}\)-SOFC, the multi-physical model would be more complicated. There are two different electromotive forces (e\({}^{-}\)/O\({}^{2-}\) and O\({}^{2-}\)/H\({}^{+}\) interfaces) within the cathode side; thus, there are two different  
Figure 1: (**a**) Illustration of the electrochemical and physical coupling processes within a typical Ni-BZCY/ BZCY/ BZCY-LSCF PCFC button cell; (**b**) the corresponding equivalent circuit of e\({}^{-}\), O\({}^{2-}\), and H\({}^{+}\) electric processes within the LSCF-BZCY composite cathode. activation overpotentials for the charge transfer rates between \(\mathrm{e}^{-}\) and \(\mathrm{O}^{2-}\) and \(\mathrm{O}^{2-}\) and \(\mathrm{H}^{+}\), respectively. More details can be found from the Supplementary Materials, in which the relevant local electrochemical equilibriums are carefully illustrated. In order to verify the effectiveness of the model, the Ni-BZCY/BZCY/BZCY-LSCF PCFC reported by Ref. [34] is selected in this paper to develop and verify the numerical model. The percolation micro model in Ref. [16] is used to calculate the effective properties of the anode, electrolyte, and cathode in PCFC from the experiment-specified microstructure parameters [34]. Accurate and proper electrochemical and physical properties are essential to solve the coupled governing equations of physical-chemical processes. Table 1 lists the basic microstructure parameters of each PCFC component layer with the experimental sources specified. Most of the other effective electrode properties in Table 1 are evaluated by our previously developed percolation micro model [16] as a function of the reported microstructure parameters.