## 2 Oxygen Reduction in SOFCs  
### Oxygen Reduction Reaction Steps and Pathways in SOFC Cathodes  
Among the entity of charge transport and transfer processes in an operating SOFC, the activation loss \(\eta\) associated with the ORR at the cathode is a main contribution to the overall voltage drop [36]. In MIEC cathodes, ABO\({}_{3}\) oxides are generally known as oxygen-deficient perovskites because oxygen vacancies are their dominant anion defect while RP oxides can be both oxygen-deficient and oxygen-excess, depending upon their majority oxygen defects. In the case of oxygen-deficient RP oxides, oxygen nonstoichiometry (\(\delta\)) arises from oxygen vacancies, whereas oxygen interstitials result in oxygen hyperstochiometric RP oxides. Therefore, two ORR reactions may occur in MIEC cathodes depending on majority oxygen defects [13; 37; 38]. The overall ORR kinetics for oxygen-excess RP oxides can be expressed as:  
\[\mathrm{O_{2}(g)+4e^{-}\to 2O_{i}}^{\mathrm{\ \ }} \tag{1}\]  
where \(\mathrm{e^{-}}\) is an electron and \(\mathrm{O_{i}}^{\mathrm{\ \ }}\)\" is an oxygen interstitial. Unlike oxygen-excess RP oxides, the following ORR reaction can occur in ABO\({}_{3}\) and oxygen deficient RP oxides.  
\[\mathrm{O_{2}(g)+4e^{-}+2V_{o}^{\mathrm{\ \ }}\cdots\to 2O_{o}}^{\mathrm{\ \ } } \tag{2}\]  
where \(V_{o}^{\mathrm{\ \ }}\) is an oxygen vacancy and \(O_{o}^{\mathrm{\ \ }}\) represents an oxygen ion in the cathode or electrolyte lattice.  
However, ORR reaction is composed of numerous elementary reaction steps. These steps can be surface or bulk related reactions such as adsorption, dissociation, electronation, and diffusion (Figure 1). As shown in Figure 1, pure electronic conductors such as Pt and LSM\({}_{113}\) have the surface pathway for the possible ORR reaction steps, which are adsorption and dissociation of molecular oxygen on the cathode surface, surface diffusion of adsorbed oxygen atoms to the TPB, formation of oxide ions by electron transfer and incorporation into the electrolyte. Previous studies [39; 40; 41; 42; 43; 44] identified using polycrystalline LSM\({}_{113}\) samples that the ORR occurs only at the one-dimensional TPB, where cathode, electrolyte, and gas phase are in contact in pure electronic conductors.  
Figure 1: Schematic of possible elementary reaction steps during oxygen reduction reaction (ORR) and possible pathways for two different classes of cathode materials; (**a**) pure electronic conductor and (**b**) mixed ionic and electronic conducting (MIEC) cathodes.  
In contrast, the surface pathway is complemented by a bulk pathway in MIEC cathodes, where adsorbed oxygen species are reduced at the cathode surface and transported to the electrolyte by solid-state diffusion through the cathode interior regardless of the dominant oxygen defects. Consequently, the active area for the ORR on MIEC cathodes spans over the entire cathode surface. The overall rate of the multitude of reaction steps in MIEC cathodes is determined by a single rate determining step. Then, all remaining steps can be considered to be close to equilibrium so that the voltage loss associated with the rate-determining step constitutes the cathodic activation loss \(\eta\).