## 2 Oxygen Reduction in SOFCs
### Oxygen Diffusion and Surface Exchange Kinetics in MIEC Cathodes  
The total electrical conductivity \(\sigma_{tot}\) of perovskite oxides can be expressed as:  
\[\sigma_{tot}=\sigma_{d}+\sigma_{ion} \tag{3}\]  
where \(\sigma_{d}\) and \(\sigma_{ion}\) are electronic conductivity and ionic conductivity, respectively. Depending on the relative amount of \(\sigma_{el}\) and \(\sigma_{ion}\), perovskite oxides are classified into pure electronic conductors or MIECs.  
Oxide cathode materials are typically extrinsic semiconductors, where \(\sigma_{el}\) results from the movement of holes or excited electrons through the surrounding lattice. In addition, MIEC cathodes conduct O\({}^{2-}\) ions via oxygen vacancies or interstitials at a significant rate. The mobility of O\({}^{2-}\) ions in MIEC cathodes is correlated with the diffusion coefficient (\(D^{\bullet}\)) for O\({}^{2-}\). In addition, the oxygen surface exchange coefficient (\(k^{\bullet}\)) is also a key parameter for determining the rate of oxygen transport through a MIEC cathode material. Therefore, \(D^{\bullet}\) and \(k^{\bullet}\) play a determining role in the ORR kinetics on MIEC cathodes.  
In the case of thin film MIEC cathodes, the ORR kinetics can be limited by either oxygen diffusion or oxygen surface exchange kinetics depending on the critical film thickness, \(t_{\text{crit}}\), which can be defined as \(D^{\bullet}/k^{\bullet}\)[45]. The ORR kinetics of thick film cathodes are mainly governed by the bulk transport oxygen whereas in thin film cathodes the ORR kinetics can be governed by the surface exchange of oxygen. In general, \(t_{\text{crit}}\) is in the range of 100 \(\upmu\)m for many perovskite oxides [46]. Therefore, the ORR kinetics of thin film MIEC cathodes can be enhanced by an increase in the oxygen surface exchange kinetics.  
Electrochemical impedance spectroscopy (EIS) measurements are commonly used to analyze the kinetics of oxygen surface exchange on thin film cathodes [6; 26; 28; 47]. Typical EIS tests are conducted in the frequency range from \(\sim\)10 MHz to \(\sim\)100 \(\upmu\)Hz using an AC voltage amplitude with zero DC bias in a wide range of temperatures and oxygen partial pressures. Without dc bias, the Nyquist plot represents three well-separated features analyzed by an equivalent circuit [48], which are denoted as high frequency (HF), medium frequency (MF) and low frequency (LF) feature. The HF feature (10\({}^{4}\)-10\({}^{5}\) Hz) is associated with the transport of O\({}^{2-}\) ions through an electrolyte, while the MF features (10\({}^{3}\)-10\({}^{4}\) Hz) is associated with the ionic transfer resistance at the electrode/electrolyte interface. In addition, the magnitude of the capacitance at the MF feature is relatively small (\(\sim\)10\({}^{-6}\) F) compared to the LF feature (\(\sim\)10\({}^{-3}\) F), which is the main resistive process representing the surface exchange reaction [6; 49; 50; 51; 52]. In the case of thin film cathodes, the magnitude of capacitance is due to the oxygen content change in the films, and therefore the oxygen surface reaction on thin film electrodes generally corresponds with the LF feature. The electrical oxygen surface exchange coefficient (\(k^{\prime}\)) can be determined using the expression [53; 54],  
\[k^{\prime}=R\cdot T/4F^{2}\cdot R_{ORR}\cdot A_{electrode}\cdot c_{o} \tag{4}\]  
where \(R\) is the universal gas constant (8.314 J\(\cdot\)mol\({}^{-1}\cdot\)K\({}^{-1}\)), \(T\) is the absolute temperature, \(F\) is the Faraday's constant (96,500 C\(\cdot\)mol\({}^{-1}\)), \(R_{ORR}\) is the resistance obtained from the LF feature, \(A_{electrode}\) is the area of the electrode, and \(c_{o}\) is the lattice oxygen concentration in thin film cathodes.