Multi scale and physics models for intermediate and low temperatures H⁺-solid oxide fuel cells with H⁺/e^-/O^2- mixed conducting properties: Part A, generalized percolation theory for LSCF-SDC-BZCY 3-component cathodes

H⁺ based solid oxide fuel cell (SOFC) composite cathodes are generally agreed to be of quite different relationships among the microstructure parameters, electrode properties and detailed working processes from the conventional O^2--SOFC composite cathodes. In this paper, the percolation theory is significantly generalized and developed to suit most of the typical H⁺-SOFC composite cathodes with e^-/H⁺, e^-/O^2- or e^-/H⁺/O^2- mixed conducting characteristics; not just limited to the BCZY, SDC and LSCF materials. It provides an easy way to investigate the effect of microstructure parameters on the H⁺-SOFC electrode characteristics in quantity. The studied electrode properties include: i) the potential coexisting sites of O₂, e^-, and O^2- transport paths for the oxygen reduction; ii) the potential coexisting sites of O^2-, H⁺ and H₂O transport paths for the vapor formation; iii) the effective e^-, O^2-, and H⁺ conducting and gas diffusing capabilities of the composite cathodes, and so on. It will be helpful for the H⁺-SOFC composite cathode manufacture to achieve the expected properties. Furthermore, it is also an important step for the developing of the multiphysics-model in manuscript part B to study the effect of the microstructure parameters on the H⁺-SOFC working details.