Abstract
Reducing the working temperature of solid oxide fuel cells is critical to their increased commercialization but is inhibited by the slow oxygen exchange kinetics at the cathode, which limits the overall rate of the oxygen reduction reaction. We use ab initio methods to develop a quantitative elementary reaction model of oxygen exchange in a representative cathode material, La_{0.5}Sr_{0.5}CoO_{3−δ}, and predict that under operating conditions the ratelimiting step for oxygen incorporation from O_{2} gas on the stable, (001)SrO surface is lateral (surface) diffusion of Oadatoms and oxygen surface vacancies. We predict that a high vacancy concentration on the metastable CoO_{2} termination enables a vacancyassisted O_{2} dissociation that is 10^{2}–10^{3} times faster than the rate limiting step on the Srrich (La,Sr)O termination. This result implies that dramatically enhanced oxygen exchange performance could potentially be obtained by suppressing the (La,Sr)O termination and stabilizing highly active CoO_{2} termination.
Introduction
In solid oxide fuel cell (SOFC) cathodes, which are the primary motivation for this oxygen exchange study, the forward process of oxygen exchange is oxygen reduction reaction (ORR) that takes gas phase O_{2} and transforms it to solid phase O^{2−} in the cathode. The exchange is comprised of O_{2} adsorption, dissociation, and incorporation at the surface, followed by O^{2−} diffusion in the bulk^{1,2}. SOFC cathode materials that perform these operations efficiently are almost all mixed electronic and ionic conducting complex oxides, and typically have the perovskite structure (Fig. 1) with stoichiometry ABO_{3δ} (where A and B are generally metal and transition metal elements, respectively). In spite of many experimental^{3,4,5,6,7,8,9} and modeling^{1,10,11,12,13,14,15} efforts, a quantitative molecular understanding of oxygen exchange at the surface of mixed conducting oxides remains elusive. This limited understanding means one cannot presently predict which materials or surfaces will be most active for exchange and inhibits rational design of optimal materials.
Previous ab initio studies^{14,15,16,17,18,19,20} have investigated the nature of stable surfaces and adsorbates on SOFC cathodes, but typically without developing quantitative models for oxygen exchange and usually with a limited focus on the (001) BO_{2} terminated surface (Fig. 1b). However, recent work strongly suggests that Srdoped perovskites such as La_{1−x}Sr_{x}MnO_{3−δ} (LSM)^{21}, La_{1−x}Sr_{x}CoO_{3−δ} (LSC)^{22,23,24,25,26} and La_{1−x}Sr_{x}Co_{1−y}Fe_{y}O_{3−δ} (LSCF)^{27} all have Sr enrichment in the surface, which indicate Sr segregation and/or precipitation of the second phase. Furthermore, experiments^{23,25,27,28} and ab initio studies^{29} show some specific evidence for stabilized Srrich (La,Sr)O (or AO) terminations, demonstrating the need to understand the exchange process on the AO surface. Recent work on (La,Sr)CoO_{3} cathodes in particular has also shown significant oxygen exchange rate degradation within hours of operation^{24,30}, strong Sr segregation^{22,23,26,31,32}, reversal of degradation after chemical etching^{33} in Srdoped cathode materials, and a major role for a small number of highly active Co sites in the oxygen exchange rate of the AO surface^{34}. The coupling of these chemical and performance changes cannot be understood without a detailed model for the oxygen exchange. A recent semiquantitative work by Mastrikov et al.^{15} predicts an approximate 5 orders of magnitude difference of exchange rate between MnO_{2} termination and (La,Sr)O termination in LSM, although without a comprehensive kinetic model. There is therefore a strong need to develop better understanding of the atomic scale mechanisms controlling the oxygen exchange.
Here we combine ab initio (Density Functional Theory) reaction energetics, defect chemistry and microkinetic modeling to calculate and compare absolute rates for 53 different mechanisms of oxygen exchange (see “Results” section) on both AO (Fig. 1a) and BO_{2} (Fig. 1b) surfaces of La_{0.5}Sr_{0.5}CoO_{3−δ} (LSC50), a representative transition metal perovskite cathode for SOFCs. We note that this is a simplified model that leaves out many possible complexities, including other active surface orientations, possible roles for steps/corners and other undercoordinated surface sites, and the impact of possible additional phases at the surface (e.g., SrO_{x} and Ruddlesden–Popper phases^{35}). In particular, we cannot be sure that the assumption for SrO termination will be satisfied by the materials used in the experiments^{5,7,8,9} to which we compare (see “Results” section). However, there are many experiments^{23,25,27,28} and ab initio studies^{29} that strongly support our assumption, so we think it is a reasonable one. While the present model provides valuable insights and good agreement with available experiments, we do not mean to suggest that this model provides the definitive mechanistic understanding for all possible conditions for LSC oxygen exchange.
Results
Thermodynamic and kinetic model
To quantify the activity of a surface towards oxygen exchange we will follow common practice and use the surface exchange coefficient, K_{tr}, which is defined as the proportionality between the rate of oxygen incorporation, r, and deviation of surface oxygen concentration \(\left( {\left[ {{\mathrm{O}}_{{\mathrm{O}},{\mathrm{S}}}^{{X}}} \right]} \right)\), from equilibrium \(\left( {\left[ {{\mathrm{O}}_{{\mathrm{O}},{\mathrm{S}}}^{{{x}},{\mathrm{eq}}}} \right]} \right)\), through the relationship, \(r =  K_{\mathrm{tr}}\left( {\left[ {{\mathrm{O}}_{{\mathrm{O}},{\mathrm{S}}}^{{X}}} \right]  \left[ {{\mathrm{O}}_{{\mathrm{O}},{\mathrm{S}}}^{{{x}},{\mathrm{eq}}}} \right]} \right)\). We will also make use of a more fundamental parameter, the equilibrium surface exchange rate, R_{0}, which gives the amount of oxygen per unit area and time that is exchanged between the material and gas at equilibrium and is easily related to K_{tr} (see the Rate expressions section in the “Methods” section and ref. ^{11}).
In this work, kinetic rate expressions for individual ORR mechanisms are formulated assuming a particular step of the mechanism being ratelimiting, implying that the overall driving force (which equals the total free energy change for the reaction of O_{2} getting incorporated into bulk of the cathode \({\mathrm{O}}_2 +{{2\mathrm{V}}}^{..}_{{\mathrm{O}}_{\mathrm{b}}}+ {\mathrm{4e}}^  \leftrightarrow {\mathrm{2O}}_{{\mathrm{O}}_{\mathrm{b}}}^{{X}}\),) equals the freeenergy change in the specific ratelimiting elementary step^{11}. To describe the thermodynamics of various elementary steps of ORR we use a neutral buildingunit representation of reactants and products, which facilitates the formulation of chemical potentials and description of mechanisms. These are defined in Supplementary Table 1 (SrO surface) and Supplementary Table 4 (CoO_{2} surface). For example, neutral building units ‘s’, ‘sO_{2}’, ‘CoO’ denote surface oxygen vacancy, O_{2} inserted into surface vacancy, and Oadatom on surface Co, respectively. We use Kröger–Vink notations for defects. For more details of the thermokinetic framework used in this work please refer to Adler et al.^{11}.
As a specific example, we describe one of the mechanisms on CoO_{2} termination and then formulate the rate expressions (refer to “Methods” section for the rate expression). The same framework has been used for all the other mechanisms. The following describes Mechanism B3 of Supplementary Table 6, which involves adsorption at an oxygen vacancy.
Step 1, O_{2} adsorption on a vacancy and dissociation by insertion into the vacancy, (2s + O_{2} → sO_{2} + s(far)) or,
Here, q_{ads} denotes charge transferred to oxygen from the LSC surface during adsorption, q_{diss} denotes charge transferred to oxygen from the LSC surface during insertion into nearby vacancy, regular brackets on product side indicate O–O configuration inserted into a vacancy on the CoO_{2} surface (denoted sO_{2} in neutral building unit notation),
Step 2, Diffusion of *O towards surface vacancy (sO_{2} + s(far) → CoO + s(near)) or,
Step 3, Incorporation into the surface, (CoO + s(near) → null) or,
Step 4, Incorporation into the bulk LSC50: (v → s) × 2 or,
Where q_{diffusion}, q_{surfaceincorp}, q_{bulkincorp} are charge transfer involved in diffusion, surface incorporation and bulk incorporation of oxygen respectively. The “Methods” section details the calculations of exchange rates based on DFT energetics.
DFTpredicted molecular configurations of various ORR intermediates (products of oxygen adsorption dissociation, incorporation) are depicted in Fig. 2a (SrO termination) and in Fig. 2b (CoO_{2} termination). Based on the DFTpredicted Gibb’s free energy relative to O_{2} gas, the concentrations of these species is also indicated at relevant SOFC conditions (oxygen partial pressure pO_{2} = 0.2 atm, temperature T = 650 °C) in these figures as well as Supplementary Table 8. Many distinct (surface sitespecific) reaction mechanisms may proceed through these surface intermediates, each potentially contributing to the overall ORR reaction of O_{2}(gas) incorporation into the bulk oxide. For convenience, these mechanisms are classified here by their initial adsorption step and the consequent dissociation step(s). In this way, 41 distinct mechanisms emerge on the SrO termination (see Supplementary Table 2) and 12 mechanisms on CoO_{2} termination (see Supplementary Table 5). The ORR is unlikely to proceed through the mechanisms for which the surface intermediate concentrations are predicted to be too low (such as CoO_{2} + 2s(far) intermediate, see Fig. 2b), whereas more stable surface intermediates (such as s, CoO + s(far) on CoO_{2} surface, O_{Otop} on SrO surface) would be expected to appear in the most prominent (fastest) mechanisms, since kinetic rates generally scale with concentration of (reactant) surface intermediates. Figure 3 show the DFTpredicted energy landscape of key mechanisms on SrO and CoO_{2} terminations, respectively. The yaxis of Fig. 3 is DFT corrected enthalpies (relative to O_{2} molecular and isolated vacancies on the oxide surface), where the corrections are described in the DFT Corrections section of the “Methods” section. The initial step in the energy landscape of all mechanisms is O_{2}(gas) + 2s, whereas the final step is two oxygen ions incorporated in the bulk.
Surface oxygen exchange activity
The predicted exchange rates R_{0} (given in this work in number of O_{2} · Co^{−1}·s^{−1}, where Co are top or second layer depending on the surface termination) of the fastest oxygen exchange mechanisms are shown in Fig. 4 (SrO surface) and Fig. 5 (CoO_{2} surface) at relevant SOFC conditions (pO_{2} = 0.2 atm, T = 650 °C). The calculation details are in the Methods section. Mechanisms A9, A20, A23, A26, and B3 are predicted to be the fastest, and therefore dominant, on the SrO and CoO_{2} surfaces, respectively. As shown in Fig. 4, mechanisms A9, A20, A23, A26 on the SrO termination have one common ratelimiting step, which involves O_{ads} and vacancy lateral diffusion to find each other, with a rate governed by the expression R_{0}(diffusion) = (8 × D_{V} × 2Γ_{s} × Γ_{Oads})/d^{2}. This process involves dilute species, so can be modeled accurately in terms of each species undergoing longrange diffusion and undergoing collisions at a rate governed by meanfield theory (see Supplementary Note 2). As illustrated in Fig. 5, mechanism B3 on CoO_{2} termination is limited by the initial O_{2} adsorption and incorporation step (see “Methods” section below for details on calculation of this rate). The thermodynamic stability of the SrO surface suggests that Mechanism A9, A20, A23, A26 should give the best match for experiments, and the result is very close to experimental surface exchange data at pO_{2} = 0.2 atm, T = 650 °C^{5,7,8,9} (blue line of Fig. 4) and over a range of TpO_{2} (discussed below and in Fig. 6). It should be noted that the predicted log(R_{0}) vs. log(pO_{2}) is very linear, similar to the experimental trends at 650 °C, but missing an apparent curvature observed at 800 °C in Egger’s data and identified by Adler, et al.^{11} as illustrating a quadratic dependence of R_{0} on bulk vacancy concentration with changing pO_{2}. The model trends with pO_{2} are governed by bulk oxygen vacancy x_{v} and surface oxygen concentration \({\boldsymbol{\Gamma }}_{{\mathrm{sO}}_2}\), which decrease and increase almost linearly with increasing pO_{2} in our model, respectively. The x_{v} behavior is taken directly from experimental measurement^{36}, and therefore we assume it is correct. The Γ_{Oads} behavior is expected from simple thermodynamic arguments if O adatoms are thermodynamically ideal, and they are in our model. However, it is possible that the model has missed some electronic transfer to the O adatom, which would weaken binding as the system is oxidized at higher pO_{2} and lead to curvature in the log(R_{0}) vs. log(pO_{2}) plot. Such a mechanism might be expected to be suppressed at high Sr content as the oxidized system would have reduced electron transfer to the adatom, which is consistent with what is seen in Egger’s data^{7}.
The different behavior of these surfaces can be understood in terms of their oxidation state. We focus on pO_{2} = 0.2 atm, T = 650 °C but the qualitative arguments given here are true for a wide range of relevant conditions. The SrO surface is more reduced (less oxidized) than the CoO_{2} surface^{17}, as the former has balanced nominal charges of [Sr^{2+}O^{2−}] while the latter is oxidized, with nominal charges [Co^{3+}(O^{2−})_{2}]. The different oxidation level leads to more stable oxidizing defects (e.g., O adatoms) and less stable reducing defects (e.g., oxygen vacancies) on the SrO vs. CoO_{2} surface. This effect can also be understood as due to compensating polarity of the surface^{20}. The differences in the surfaces lead to a dramatic difference in their surface oxygen vacancy concentrations, Γ_{s}(SrO_{surf}) = 6 × 10^{−5} and Γ_{s}(CoO_{2surf}) = 0.28, which difference leads to the vacancy assisted O_{2} dissociation mechanism being much more active on the latter. The 10^{2}–10^{3} times faster oxygen exchange rate of CoO_{2} surface vs. SrO surface is due to the high vacancy concentration on CoO_{2}, which assists O_{2} adsorption on CoO_{2} through a strong stabilization of O_{2} adsorbate with nearby vacancies. Also, vacancies assist O_{2} dissociation to have a much lower barrier on the CoO_{2} surface than SrO surface.
The importance of surface segregated Sr can now also be understood in terms of surface oxidation state and the active mechanisms. The fastest mechanism (A9, A20, A23, A26) is rate limited by O_{ads} and O_{v} diffusion, and the rate of this step scales with the vacancy content on the SrO surface (Fig. 2a). As the oxygen vacancy content is greatly increased by Sr (Γ_{s}(LaO_{surf}) = 10^{−16} is 10^{11} times smaller than Γ_{s}(SrO_{surf})), it is clear that Sr enrichment within the AO surface enhances activity by stabilizing surface vacancies for oxygen diffusion and incorporation. This result suggests that for the equilibrated LSC material the Sr both in the bulk and segregated to the perovskite surface is critical for good performance. However, precipitation of Sr compounds (such as oxides, hydroxides and/or carbonates) on the surface, perhaps driven in part by Sr segregation, could potentially degrade performance^{22}. Furthermore, overall, AO surface formation is expected to reduce activity vs. the CoO_{2} surface.
These results suggest that LSC50, and potentially many other perovskite systems, could be greatly enhanced if a metastable BO_{2} termination could be maintained. From a number of recent experiments, the formation of Srenriched phase on the surface is known to be concurrent with ORR performance degradation for various perovskite SOFC cathodes^{22,25,30,33,37,38,39}. Crystalline LSC64 thinfilms are known to degrade by one to two orders of magnitude^{24,30,33} in their ORR performance within tens of hours^{24} of preparation or annealing, and to regain performance with etching^{33}. Studies of carefully controlled surfaces and Sr and Co deposition have also suggested that Co sites are much more active that Sr^{34}. Based on our results above we propose that these changes can be explained as due to the metastable and highly active CoO_{2} surface being present in asreceived or etched materials and being rapidly replaced by a Sr rich AO surface as the system equilibrates at high temperature. The appearance of degradation at temperatures around 500–600 °C^{22,30} and its time scale of tens of hours^{24} are both consistent with the temperature and time scales associated with the cation mobility required for moving La and Sr to the surface^{40}. Many authors have suggested the Sr itself is driving the degradation^{22,24,30}, although our model suggests that it is the formation of the AO surface and not Sr itself that leads to the degradation, at least before the formation of Srbased precipitates on the surface. We note that our proposed mechanism for the degradation is quite speculative and that other mechanisms, e.g., the formation of secondary Sr rich phases and site poisoning, might play a significant or even dominant role. Further work on LSC and related systems is needed to validate these speculations.
Table 1 lists the DFTpredicted reaction kinetic barriers, overall TpO_{2} dependences, and reaction rates (R_{0}) at SOFC conditions. Figure 6a, b show the predicted TpO_{2} dependence of R_{0} of the key mechanisms of LSC50. Figure 6c, d show the corresponding plots for K_{tr}. The pO_{2} dependence of R_{0} or K_{tr} is a function of the kinetic symmetry parameter (β), which is not readily determined from the DFT calculations, and hence value β is fit to the experimental pO_{2} dependence. The experimental R_{0} is derived from the reported K_{tr} data^{5,7}, Eq. (4) in the “Methods” section, and known defect chemistry^{41}. Only the fastest mechanisms on both surfaces (BO_{2}, orange, and AO, blue dashed line) are shown for clarity, along with the initial O_{2} adsorption rate (black). The predicted slope of log_{10}(R_{0}) vs. log_{10}(pO_{2}) for SrO termination is 0.4–0.6 in Fig. 6, matching well with the range 0.4–0.8 of slope from these experiments in Table 1. The Arrhenius activation energy (describing Tdependence) of experimental K_{tr} for LSC50^{5,7} (and the closely related La_{60}Sr_{40}CoO_{3−δ}^{7}) ranges from 0.99–1.57 eV for 0.1 atm pO_{2} and 0.55–1.67 eV for 10^{−3} atm pO_{2}. The narrow range of our predicted activation energies for the SrO surface of 1.51 eV (for 0.2 atm pO_{2}) and 1.31 eV (10^{−3} atm pO_{2}), respectively (AO surface) agree well with the upper range of these values, although more experimental data is clearly needed for quantitative comparison (see Table 1). Within the uncertainties in the experiments and ab initio energetics, our results on model SrO termination agree very well with the experimental TpO_{2} dependence, providing validation for the model. The agreement between the experiment and the model for Srrich AO termination is also fully consistent with the fact that this surface is the thermodynamically stable surface of LSC^{22,23,24,26,31}. Figure 6 also shows that the oxygen exchange rates (R_{0}) and oxygen exchange rate coefficient (K_{tr}) at the CoO_{2} surface can be two to three orders of magnitude larger than SrO surface, for wide range of TpO_{2}.
Figure 6 shows the O_{2} adsorption step (black). As the first elementary step for all ORR mechanisms, this step sets an upper fundamental limitation for ORR on any cathode surface. This limit is set by the kinetic theory of gas absorption (expressed as R_{0} in Fig. 6a, b and K_{tr} in Fig. 6c, d). Details of these estimates are in the “Methods” section and Fig. 7. At relevant SOFC conditions (0.2 atm pO_{2}, 650 °C), the upperlimit exchange rate set by this step corresponds to R_{0} ~ 3.63 × 10^{5} O_{2}·Co^{−1}·s^{−1}, K_{tr} ~ 1.69 × 10^{−3} m·s^{−1}, and an exchange current density of ~170 A·cm^{−2} (taking into account the experimental lattice parameters and 4 e^{−} carried per O_{2} molecule). Thus at these conditions of TpO_{2}, we predict that these values are an absolute upperlimit achievable for any active SOFC cathode surface. The K_{tr} for the highly active, but metastable, surface provided by the CoO_{2} termination of LSC50 (Fig. 6c, d, orange line) may get close to this upperlimit, which suggests significant opportunity for improving ORR activity for complex oxides. We note that the kinetic model developed here can also be used to predict overpotential as a function of current density. This is done and the results shown to compare very favorably to experiments in “Methods” section, further supporting the validity of the model.
Discussion
In summary, understanding, controlling, and enhancing oxygen exchange on oxide surfaces is critical to the success of many technologies, especially for lowering the operating temperature of SOFCs. We have developed an ab initio based microkinetic model that provides quantitative predictive ability for oxygen exchange rates vs. temperature and pO_{2} on La_{0.5}Sr_{0.5}CoO_{3−δ} (LSC50). The model identifies surface diffusion and adsorption of O_{2} as the ratelimiting steps on Srrich (La,Sr)Oterminated and CoO_{2}terminated surfaces, respectively, and predicts exchange rates on the stable SrO surface in excellent agreement with experiments. The O_{2} reduction on the metastable CoO_{2} surface is predicted to be orders of magnitude faster than on the thermodynamically stable SrO surface, suggesting that the loss of CoO_{2} termination is most likely the reason for the widelyobserved degradation of oxygen exchange rates over tens of hours of annealing. Also, while the (La,Sr)O surface is generally predicted to be less active than the CoO_{2} surface, the segregation of Sr to form a SrO termination is shown to be critical to good performance on this termination. Finally, this work estimates a fundamental upperlimit of oxygen exchange rate set by the O_{2} adsorption step. The metastable, pure CoO_{2} surface of LSC50 is predicted close to this limit under relevant SOFC conditions, enabling a 2–3 orders of magnitude faster oxygen exchange than Sr rich AO surface. This result demonstrates an inherent potential for enormously enhanced performance in this and related materials if a high concentration of the transition metals can be permanently stabilized on the surface.
Methods
Modeling approach
Calculations were performed with the Vienna Ab initio Simulation Package^{42,43} (VASP) using Density Functional Theory (DFT) and the ProjectorAugmented planeWave (PAW)^{44} method. Exchangecorrelation was treated in the PerdewWang91 (PW91^{45}) Generalized Gradient Approximation (GGA) with electronic configurations of La (5s^{2} 5p^{6} 6s^{2} 5d^{1}), O_s (‘soft’ oxygen pseudopotential, 2s^{2} 2p^{4}), Sr (4s^{2} 4p^{6} 5s^{2}) and Co (3d^{8} 4s^{1}). In order to take into account the effect of the strongly correlated electronic systems we applied the rotationally invariant GGA+U^{46} method, with U and J entering as an effective parameter, U_{eff} = 5.6 eV, which was fit so our bulk oxygen vacancy formation energy matches the experiment data from Mizusaki et al.^{41}. We used the Uramping method^{47} to search the global minimal energy. Bulk calculations were simulated with 80atom unit cell (2a_{pc} × 2a_{pc} × 4a_{pc}); a_{pc} is the pseudocubic lattice constant of LSC). In calculating the bulk oxygen vacancy formation energy, the oxygen vacancy was placed such that two of its nearest neighbor Asites are filled with doped Sr and two with La. The surface calculations were done with an eightlayer slab (2_{ac} × 2_{ac} × 4_{ac} with 4 layers of AO, 4 layers of BO_{2}). Slabs were asymmetric (had both AO and BO_{2} terminations) and dipole corrections^{48,49} were used. In the calculations of the energies of adsorbed oxygen (*O, *O_{2}) on either (001) surfaces, we embedded a vacancy on the exposed CoO_{2} termination to represent the significant vacancy concentration expected on this surface. In case of adsorbates on the CoO_{2} termination itself, the adsorbate and surface vacancy were separated by more than 5 Å in order to reduce the effects of neighbor interactions, except in the case while treating adsorbate with a nearest neighbor vacancy itself. To study the ORR on Srrich AOtermination, SrOterminated surfaces were created with a 2 × 2 × 4 or 80atom supercell similar to the CoO_{2} slabs above, but keeping a 100% Sr concentration at the top surface. The total concentration of Sr in the slab was maintained at 50% Sr. This cell represents a model Srenriched LSC surface that is expected to be largely covered with an SrOtermination^{22,24}. All energies are free energies with vibration contributions included for gaseous, lattice and surface atoms. For the details of types of ORR intermediates, and choice of thermodynamic reference states, reader is referred to the Supplementary Note 1. The energy cutoff was set to 450 eV and the brillouin zone was sampled by a MonkhorstPack kpoint mesh of (2 × 2 × 1) for a 2 × 2 × 4 perovskite supercell. The 3delectron configuration of Co(III) was modeled with Intermediate Spin (IS) state \(\left( {t_{2g}^5e_g^1} \right)\). Using GGA+U, the intermediate spin state in LSC was found to be the most stable and to yield the most accurate oxygen vacancy formation energies^{50} among the different spins (low, intermediate, high spin), hence all the spinpolarized calculations were relaxed from an IS state^{51}. The magnetic configuration for electron spins was chosen to be Ferromagnetic (FM) after the work of Lee et al.^{17}. Use of Ferromagnetic (FM) configuration and use of pseudocubic lattice is justified as it closely simulates the highTemperature nature of LSC lattice^{17}. The bulk and slab energies were converged to within 3 meV per formula unit with respect to the kpoints and energy cutoff. Structural relaxations were converged to within 1 meV per atom. Surface structures were calculated by truncating the bulk with insertion of 10 Å vacuum above the surface under periodic conditions.
Surface exchange rates are determined by enumerating possible pathways (mechanisms) and finding the rates of each elementary step, setting the total rate to that slowest elementary step in each mechanism. For the surfaces of LSC50, we enumerate and organize different mechanistic pathways of ORR in terms of their adsorption and dissociation sites. Supplementary Tables 2 and 5 show the list mechanisms organized according to the adsorption (rows) and dissociation sites (columns) of the (La,Sr)O and CoO_{2} termination, respectively. A typical (La,Sr)O termination without a specific ordering of the Asite cations would have (m=)14 probable sites for oxygen adsorption, and (n=) 14 sites for dissociation. Only a few of the probable m × n site combinations give rise to molecular pathways that are meaningful and unique. In case of (La,Sr)O and CoO_{2} terminations respectively, we identify 41 and 12 unique molecular pathways that can describe the sitespecific adsorption and dissociation steps of ORR. Refer to Supplementary Tables 3, 6 and 7 for the details of molecular pathways, Supplementary Tables 1 and 4 for neutral building units and their chemical potentials, and Fig. 3 for the reaction energy landscape.
Rate expressions
In the following, we formulate the rate equations for individual elementary steps of any oxygen exchange mechanism. We use ratelimiting step approximation to then calculate the rate of oxygen exchange mechanisms. In other words, given the rates of all elementary steps of a specific oxygen exchange mechanism, the rate of oxygen exchange through this mechanism will be the slowest of the series of elementary steps and the rate of overall ORR, the sum of all parallel mechanisms, dominated by the fastest ORR mechanism. Applying transition state theory, the net kinetic rate for an elementary step, i of a reaction, viz. X + Y = Z can be written as^{11},
where k_{i} is a temperatureindependent preexponential factor, ΔG_{f,i} is the unperturbed freeenergy barrier for the reaction step (evaluated at ΔE_{i} = 0), Γ_{X} and Γ_{Y} are the concentrations of reactants X and Y, β is a reaction symmetry parameter analogous to that used in electrochemical kinetics, Λ_{i} is the total freeenergy driving force for the reaction step, λ_{i} is the stoichiometric coefficient of the elementary step, and ΔE_{i} is the energyshift component of the thermodynamic driving force for the i^{th} elementary step. Equation (15) gives the net forward rate (r_{i}) of a single elementary step(i) of any reaction mechanism. The driving force for step i, Λ_{i} is the difference in the free energy of reactants minus products. The driving force for overall oxygen incorporation is the deviation in the chemical potential of O_{2} due to a deviation in its external pressure from equilibrium. Under the ratelimiting step approximation, the driving force (Λ) for the total reaction is assumed to be contributed in full by the ratelimiting reaction step, i.e., (Λ_{i}), or, \({{{\boldsymbol{\Lambda}} }}_{\mathrm{i}} = {{\boldsymbol{\Lambda}}/}\lambda _{\mathrm{i}} = k_{\mathrm{B}}T\lambda _{\mathrm{i}}^{  1}{\mathrm{ln}}\left( {P_{{{\mathrm{O}}_2}}^{\mathrm{gas}}{\mathrm{/}}f_{{{\mathrm{O}}_2}}^{\mathrm{gas}}} \right)\), where λ_{i} is the stoichiometric ratio between reaction i and overall reaction under quasiequilibrium conditions, \(P_{{{\mathrm{O}}_2}}^{\mathrm{gas}}\) is the partial pressure of oxygen in the gas relative to atmospheric pressure and, \(f_{{{\mathrm{O}}_2}}^{\mathrm{gas}}\) the oxygen fugacity in the oxide. At no net driving force (Λ = 0), the forward and backward rates will be identical, represented by an equilibrium exchange rate, R_{0}. Net rate can be written in terms of R_{0} as,
The ‘chemical’ surface exchange rate coefficient K_{tr} can further be calculated in terms of R_{0}, assuming a linearity in Λdependence of total rate r (see Adler et al.^{11} page 98),
Where x_{v} is bulk oxygen vacancy site fraction (unitless), \(\gamma = \left( {\partial {\mathrm{ln}}x_{\mathrm{v}}{\mathrm{/}}\partial {\mathrm{ln}}f_{{{\mathrm{O}}_2}}^{\mathrm{gas}}} \right)_T\) is a material specific thermodynamic factor (unitless) and C_{0} (in moles of oxygen lattice sites·m^{−3}) is the oxygen site concentration in the bulk. For convenience, we stick to the discussion of R_{0} in the units of (#O_{2}·Co^{−1}·s^{−1}), although the results can be translated to the exchange rate coefficient (K_{tr}, m·s^{−1}) using Eq. (4) and appropriate unit conversions. Supplementary Table 8 lists all the key thermodynamic and kinetic parameters used in this work, and gives the parameter value, or refers to the specific tables and figures where to find the information within “Methods” section and main text.
DFT corrections
Corrections were applied to DFT energies of surface adsorbates and kinetic barriers, as explained below:
Identifying electron doping contributions: When dealing with formation and adsorption energies, it is useful to separate out the energetics of the process that is associated with electron donation/removal from the Fermi level, as this quantity depends on the overall defect chemistry and therefore T and pO_{2}. In this work, the DFT simulations of bulk and surfaces of LSC50 were performed with 2 × 2 × 4 supercells, or with 16 formula units (FUs) of LSC50, which leads to significant electron donation/removal contributions in the basic DFT energies. For example, when simulating vacancy formation energy, 1 oxygen vacancy is simulated in such a 2 × 2 × 4 supercell, which dopes 2 electrons into the supercell. Following the rigid band formalism for LSC50 from Lankhorst et al.^{52}, the two electrons will be doped at the Fermi level, and the Fermi level will be incremented by \(\frac{n}{{16 \times {g}(E_{\mathrm{F}})}}\), where n is doping concentration (per 16 FU supercell), g(E_{F}) is the density of states at the Fermi level. This density of states has the value g(E_{F}) = 4/a = 1.71 eV^{−1} FU^{−1}, where the parameter a = 2.34 eV is the vacancy interaction parameter from Mizusaki et al.^{41}. Since we calculate oxygen vacancy formation energy as a difference between a supercell with one vacancy and a perfect (no vacancy) one, an electron donation correction energy of \(E_{\mathrm{DFT}}^{\mathrm{don}} = \int_{n = 0}^{n = 2} \frac{n}{{16 \times {g}(E_{\mathrm{F}})}}\mathrm{d}n\) or \(\frac{2}{{16 \times{g}(E_{\mathrm{F}})}}\) needs to be removed (subtracted) from the vacancy formation energy to remove this energy contribution. Similarly when calculating an adsorption energy of species *O_{2}^{1−}, an energy of \(E_{\mathrm{DFT}}^{\mathrm{don}} = \int_{n = 0}^{n = 1} \frac{n}{{16 \times {g}(E_{\mathrm{F}})}}\mathrm{d}n = \frac{n}{{32 \times {g}(E_{\mathrm{F}})}}\) needs to be added to the adsorption energy. These corrections are identified by a symbol \(E_{\mathrm{DFT}}^{\mathrm{don}}\) in Eqs. (6) and (8) below.
Correction to adsorbate energy that contains O–O bond: As the GGA calculations overestimate the binding energy of oxygen molecule, a standard O_{2}overbinding correction (0.33 eV/O2 from the work of Lee et al.^{17}) is applied for the DFTenergy of O_{2}, and surface adsorbates that retain an O–O bond (species of the type *O_{2}^{0.5−}, *O_{2}^{1−}, *O_{2}^{2−}, where ‘*’ denotes an adsorbate).
Correction to kinetic barrier: The energy component of the driving force shifts the reaction barrier by (1 − β)ΔE_{i}, giving an effective reaction barrier \({\mathrm{\Delta }}G_{\mathrm{eff,i}} = {\mathrm{\Delta }}G_{\mathrm{f,i}}^0  (1  \beta ){\mathrm{\Delta }}E_{\mathrm{i}}\), where β is empirical symmetry parameter (usually ranges from 0.5 to 1). For a reaction step involving the breaking of oxygen dimmer bond (O–O dissociation step), applying the above DFT oxygen overbinding correction of +0.33 eV on the initial state of this reaction would result in increasing the driving force for forward reaction, hence decreasing the effective reaction barrier. For example, the corrected effective barrier for oxygen dissociation at divacancy of Mechanism B2 (Step 2 of Mechanism B2), is \(\Delta G_{\mathrm{eff,i}} = {\mathrm{\Delta }}G_{\mathrm{f,i}}^0  \left( {1  \beta } \right){\mathrm{\Delta }}E + (1  \beta ) \times 0.33\).
Besides these, errors remain in any DFT calculated reaction energies and reaction barriers. These errors are discussed in detail in Supplementary Note 2, where we estimate an error bar of ±1.5 order of magnitude.
Determining vibrational free energy of solid phase oxygen
In this work we use thermodynamic parameters for bulk oxygen from the work of Mizusaki et al.^{41}. In particular the chemical potential of LSC50 bulk oxygen in reference to O_{2,gas} is set using their reported parameters \({\mathrm{\Delta }}h_{\mathrm{O}}^0\), ΔS_{0} and a (Figure 13a of Mizusaki et al.^{41}). However, we need to make use of bulk defect energies from DFT and vibrational models relative to the oxygen gas (obtained from an empirical model from NIST^{53}). To assure that our DFTbased model matches the Mizusaki model here we fit an enthalpy correction and Einstein temperature for our DFTbased model. The fit matches the free energy change of bulk vacancy formation at relevant SOFC conditions between the our calculated and Mizusaki models. This approach is illustrated as follows:
Reaction: oxygen from bulk LSC goes to oxygen (1/2 O_{2}) in gas forming a bulk vacancy,
ΔG_{1} represents an ideal, nonconfigurational Gibb’s free energy change of the formation of oxygen vacancy. ΔG_{1} can be expanded as sum of DFT and Tdependent terms and a constant correction term,
Where, \(E_{\mathrm{O}_2}^{\mathrm{DFT}}\) is the DFTGGA(PW91^{45}) predicted energy of isolated O_{2} molecule (=−9.09 eV/O_{2}^{17}), term \({\mathrm{\Delta }}h_{\mathrm{O}}^0\) is the O–O overbinding correction added to the O_{2} DFT energy. Term \({\mathrm{\Delta }}h_{\mathrm{O}}^{\mathrm{corr}}\) (=0.33 eV/O_{2} from Lee et al.^{17}) is the O–O overbinding relative to lattice oxygen at standard conditions (T^{0} = 298.15 K), hence a term \(2H_{\mathrm{vib,O  solid}}^{T = T^0}\) (vibrational enthalpy of two latticeO at T^{0} = 298.15 K) is added to avoid doublecounting. Term \(E_{n_{\mathrm{b}}}^{\mathrm{DFT}}\) is the DFT energy of a bulk supercell with n atoms, \(E_{n  1_{\mathrm{b}}}^{\mathrm{DFT}}\) is the DFT energy of a bulk supercell with 1 oxygen vacancy, \(E_{\mathrm{DFT}}^{\mathrm{don}}\) is the electron donation correction energy (explained above), \(H_{\mathrm{O}_2}^{\mathrm{NIST}}\) and \(S_{\mathrm{O}_2}^{\mathrm{NIST}}\) are the enthalpy (relative to 298.15 K) and entropy (absolute scale) of oxygen gas, from NIST database^{53}, \({\mathrm{\Delta }}h_{\mathrm{O}}^{\mathrm{corr}}\) is correction term, which is explained as follows. We have used Einstein model to treat vibrational free energy of latticeoxygen, with θ_{E} as the effective Einstein temperature. A similar expression for the ideal, nonconfigurational free energy of bulk oxygen vacancy formation can be written based on parameters from Mizusaki et al.^{41}, viz. −1 × ΔG^{0} = Δh^{0} − T × ΔS^{0}, where both Δh^{0} and ΔS^{0} are enthalpy and entropy of bulk lattice oxygen (relative to gasO with an arbitrary reference) referred from Figure 13a of Mizusaki et al.^{41}, and the negative sign accounts for the vacancy formation instead of bulkO formation. As shown in Fig. 8 below, tuning ΔG_{1} with θ_{E} = 275 K matches the slope (relative to temperature) of −1 × ΔG^{0}. Any other approximations to the Einstein temperature (such as θ_{E} = 500 K shown as example), will set ΔG_{1} and −1 × ΔG^{0} on different slopes, in other words would give a wrong estimation of the relative entropy of oxygen.
The quantity ΔG_{1} when calculated with Einstein temperature of 275 K and corrected with \({\mathrm{\Delta }}h_{\mathrm{O}}^{\mathrm{corr}}\) (=+0.16 eV), then equals −1 × ΔG^{0} from Mizusaki et al.^{41} and sets the bulk vacancy site fraction (x_{v,bulk}) at given TpO_{2}:
2E_{F} term accounts for the increment in the Fermi level^{52,54} of LSC50 due to 2 electrons added to the Fermi level, when a vacancy is created. E_{F} = 1.5 × a × x_{v,bulk}^{11}, where parameter a is referred from Mizusaki et al.^{41}. In all the calculations of chemical potential of surface oxygen species (adsorbates, surface vacancy etc.), the bulklike modes of surface oxygen are treated with Einstein model with an Einstein temperature of 275 K.
Determining surface oxygen energetics by referencing to bulk
To minimize possible DFT errors we have taken the energies of all oxygen adsorbates and surface vacancies relative to the bulk model. As the bulk model is fit to experiments, this allows surface O energetics to be controlled by just the relative energies of surface and bulk O, which we expect to lead to significant cancellation of errors.
Consider first the example of formation of a surface vacancy,
We assume that the modes of vibration of the surfacelatticeO are identical to the modes of the bulkO. Two electrons are doped at the Fermi level in 2 × 2 × 4 simulation supercells for both surface and bulk DFT calculations, implying the terms \({{E}}_{{{\mathrm{DFT}}}}^{{{\mathrm{don}}}}\) (see section on DFT corrections above) in both ΔG_{1} and ΔG_{2} will be equal. Hence, comparing the expressions (6) and (8), difference between ΔG_{1} and ΔG_{2} is depends on only the difference in the DFT terms:
The sitefraction of surface oxygen vacancies (Γ_{s}) can be written in terms of ΔG_{2} as
Subtracting Eq. (7) from Eq. (10) and rearranging, with the use of Eq. (9) gives:
From DFT calculations, we get that vacancy formation on CoO_{2} surface of LSC50 is 0.21 to 0.8 eV/vacancy more favorable than the bulk (the range is due to the wide range of vacancy concentrations on the CoO_{2} surface and the strong concentration dependence of the vacancy energetics), while unfavorable by 0.5 eV/vacancy for SrOterminated LSC50 (relative to bulk). Based on these value, taking Mizusaki et al.^{41} experiment data for LSC50 bulk vacancy concentrations, at typical SOFC conditions (650 °C, 0.2 atm) CoO_{2} surface would have 0.28 (per Osite) vacancies, while SrOtermination of LSC50 would have 5.99 × 10^{−5} vacancies (per Cosite). Since the high oxygen vacancy concentration will have strong vacancy interaction, we write E_{Form}(V_{O}) as a linear function of oxygen vacancy concentration for BO_{2} surface, with a DFT calculated linear dependence on oxygen vacancy concentration of 5.48 eV (per unit concentration) (see Supplementary Table 8). This interaction parameter is estimated by fitting to DFT calculations at surface vacancy concentrations from 0.125 to 0.25 (per Osite), which are close to the final predicted value of 0.28. These calculations therefore should provide a good guide for BO_{2} surface oxygen vacancy interactions. Concentrations of surface species such as *O_{2} and *O can be similarly calculated, Supplementary Table 9 lists the assumptions about degrees of freedom used for calculating concentrations of each surface species.
Consider another example for the reaction of bulk vacancy and surface *O,
We use the vibration model in Supplementary Table 9 to calculate the reaction energy. The charge changing for the reaction can be calculated Fig. 2 and Supplementary Table 4.
\(E_{1,\mathrm{DFT}}^{\mathrm{don}}\) and \(E_{2,\mathrm{DFT}}^{\mathrm{don}}\) is the donation correction for two DFT calculations. The sitefraction of surface oxygen vacancies (Γ_{s}) and surface *O \(\left( {{{{\boldsymbol{\Gamma}} }}_{{\mathrm{sO}}_2}} \right)\) can be written in terms of ΔG_{3} as
Oxygen adsorption rates from kinetic theory of gases
In the following section, we describe calculations for preexponential factors (k_{i}) of chemisorption of oxygen molecules on the surfaces of a cathode, using transition state theory. Oxygen chemisorption involves a significant change in the degrees of freedom from the initial (gaseous) state to the transition state, and assumptions about the nature of transition state have been made in order to quantify the preexponential factors. Below we state the assumptions we make about the nature of chemisorbed transition state, and use them to calculate the preexponential factors for immobile (C,I) and mobile (C,M) type chemisorption. These give different preexponential factors, \(k_{\mathrm{ads}}^{\mathrm{I}}\) (immobile) and \(k_{\mathrm{ads}}^{\mathrm{M}}\) (mobile) as derived below. The surfacesitespecific choices of mobile (C,M) or immobile (C,I) transition state for O_{2} adsorption are also listed in column headers of Supplementary Table 2(mechanisms of SrO termination) and Supplementary Table 5 (mechanisms of CoO_{2} termination).
The general framework for treating chemisorption of gas molecules is taken from the books of Chorkendorff et al.^{55} (chapter 3) and Newman et al.^{56}. Consider a cathode surface with M sites, total area A (thus having N_{0}=M/A sites per unit area) in presence of an ideal O_{2} gas at a constant (T, \(P_{{{\mathrm{O}}_2}}^{\mathrm{gas}}\)). Oxygen molecules will adsorb on different surface sites, for example, the surface Co of a CoO_{2}terminated LSC cathode. For a particular type of adsorption site, out of total M such sites, let us assume M^{/} sites are free for chemisorption (or having available site fraction, Γ_{*} = M′/M). According to transition state theory, the forward rate of chemisorption can be expressed assuming a chemical equilibrium between the initial state (O_{2} in gas) and the transition state (denoted by a superscript #) for chemisorption. The forward rate (r_{f}) can be expressed as a product of an attempt frequency at the barrier and concentration of molecules in transition state:
where, ν is the attempt frequency along the reaction coordinate, N is the number of O_{2} in the transition state, M is the number of available surface sites for physisorption.
It is useful to express N in terms of the partition functions of the initial \(\left( {P_{{{\mathrm{O}}_2}}^{\mathrm{gas}}} \right)\) and transition state (q^{#}), as it can enable calculation while rigorously including all the relevant physics of the adsorption of O_{2} molecules. For a given system of N indistinguishable gas molecules (such as pure O_{2}) the total partition function of the system is given by \(Q = \frac{{q^N}}{{N!}}\), where q is the total partition function of individual molecule in transition state, expressed as the product of partition functions of constituent degrees of freedom (typically, q = q_{trans} × q_{rot} × q_{vib} × q_{electronic} where q_{trans}, q_{rot}, q_{vib}, q_{electronic} denote translational, rotational, vibrational and electronic partition functions respectively). The chemical potential of this gas can be expressed in terms of the total individual partition function q,
where, we have used Stirling’s approximation (ln(N!) = N · ln(N) − N).
The partition function for oxygen molecules in chemisorption transition state can be similarly calculated. If N^{#} is the number of O_{2}chemisorption species transition state, the total partition function (for N^{#} molecules) and respective chemical potential μ^{#} is,
where, we have used Eq. (15). Thus for an equilibrium between oxygen gas and oxygen molecules in transition state, we can write \(\mu _{{{\mathrm{O}}_2}}^{\mathrm{gas}} = \mu _{{{\mathrm{O}}_2}}^\#\), and it can be shown that,
Where, θ^{#} is the coverage of O_{2} species in transition state, which are likely to be small, compared to the available adsorption sites (Γ_{*} = M′/M) and can be neglected. Substituting Eq. (18) into Eq. (15) yields (assuming O_{2} as an ideal gas and using \(P_{{\mathrm{O}}_2}^{\mathrm{gas}}V = N_{{\mathrm{O}}_2}^{\mathrm{gas}}k_{\mathrm{B}}T\),)
Calculation of the partition function of oxygen gas is straightforward (expressed as \(q_{{\mathrm{O}}_2}^{\mathrm{gas}} = q_{\mathrm{trans}}^{3\mathrm{D}}q_{\mathrm{rot}}^{\mathrm{gas}}q_{\mathrm{vib}}^{\mathrm{gas}}q_{\mathrm{elec}}^{\mathrm{gas}}\)), where, \(q_{\mathrm{trans}}^{\mathrm{3D}}\), \(q_{\mathrm{rot}}^{\mathrm{gas}}\), \(q_{\mathrm{vib}}^{\mathrm{gas}}\), \(q_{\mathrm{elec}}^{\mathrm{gas}}\), denote 3D translational, rotational and vibrational and electronic partition functions of oxygen gas.
Immobile transition state for chemisorption
For the chemisorption transition state of the immobile type, we assume the following about its nature, which allows us to estimate the partition function of transition state.

1.
We assume that the transition state of O_{2} chemisorption is immobile and only retains the vibrational degree of freedom from the 3Dgas phase

2.
We assume the transition state has gained the latticevibrational degrees of freedom of the final chemisorbed state, except the ‘reaction coordinate’ degree of freedom. Thus we approximate the partition function of transition state q^{#} assuming 4 modes of vibration of lattice oxygen, 1 mode of O–O vibration retained from gas, and the reaction coordinate.
Under these assumptions, q^{#} becomes:
Thus, forward rate of oxygen chemisorption, from combining Eqs. (19) and (20) is,
The ratio of electronic partition functions of the transition and initial state \(\left( {q_{\mathrm{elec}}^\# {\mathrm{/}}q_{\mathrm{elec}}^{\mathrm{gas}}} \right)\) can be approximated to be \({\mathrm{e}}^{  {\mathrm{\Delta }}G_{\mathrm{eff}}/k_{\mathrm{B}}T}\)where, \({\mathrm{\Delta }}G_{\mathrm{eff}} = ({\mathrm{\Delta }}G_{\mathrm{f}}^0  (1  \beta ){\mathrm{\Delta }}E)\) as described above, is the difference between the electronic energy of the transition state and the initial (gas) state, estimated using DFT. For an ideal gas, the forward rate then becomes,
where, N_{0}, A are as defined above, and \(q_{\mathrm{trans}}^{2\mathrm{D}} = A(2\pi mk_{\mathrm{B}}T){\mathrm{/}}h^2\) is the partition function for 2D ideal gas, normalized per adsorption site (hence divided by M).
The forward rate of chemisorption reaction \(\left( {\mathrm{O}_2 + \ast \to \mathrm{O}_2^{\mathrm{ads}}} \right)\) via immobile transition state from the thermokinetic model is given by,
where, \(k_{\mathrm{ads}}^{\mathrm{I}}\) is the preexponential factor for oxygen chemisorption via immobile transition state. Clearly,
As discussed in “Methods” section (main text), an Einstein temperature of T_{θ} = 275 K, gives accurate expression for enthalpy and entropy of lattice oxygen in equilibrium with the gas. Using this Einstein temperature, the vibrational partition function for latticeO at T = 650 °C is approximately 3.34 (unitless). At T = 650 °C, the value of preexponential factor for oxygen chemisorption via immobile transition state, \(k_{\mathrm{ads}}^{\mathrm{I}}\) = 0.65 m s kg^{−1}. Values of various terms of above equation are in Supplementary Table 8.
Mobile transition state for chemisorption
For the chemisorption transition state of the mobile type, we assume the following about its nature, which allows us to estimate the partition function of transition state.

1.
We assume that the transition state of O_{2} chemisorption is relatively mobile, and has retained both the rotational and vibrational degrees of freedom from the 3Dgas phase, but lost all 3translational modes

2.
We assume the transition state has gained 2 latticevibrational degrees of freedom of the final chemisorbed state, and a ‘reaction coordinate’ degree of freedom. Thus we approximate the partition function of transition state q^{#} assuming 2 modes of vibration of lattice oxygen, 1 mode of O–O vibration and 2 rotational modes retained from gas, and the reaction coordinate.
Under these assumptions, similar to Eq. (22), r_{f} becomes:
The forward rate of chemisorption reaction \(\left( {\mathrm{O}_2 + \ast \to \mathrm{O}_2^{\mathrm{ads}}} \right)\) via immobile transition state from the thermokinetic model is given by,
where, \(k_{\mathrm{ads}}^{\mathrm{M}}\) is the preexponential factor for oxygen chemisorption via a mobile transition state. Thus we can write,
At T = 650 °C, the value of preexponential factor for oxygen chemisorption via mobile transition state, \(k_{\mathrm{ads}}^{\mathrm{M}}\) ~ 17.55 m·s·kg^{−1}. Values of various terms of the above equation are in Supplementary Table 8.
Adsorption limit on R _{0}, K _{tr} and oxygen exchange
Here we calculate the fundamental upper limits on K_{tr} (m·s^{−1}) as well as ORR rate (moles of O_{2}·m^{2}·s^{−1}) for any perovskite oxide cathode material. Recent work by De Souza et al.^{57} shows the k* limited by arrival from the gas phase using a simplified kinetic model that requires proposing a critical incident kinetic energy needed to incorporate an arriving O_{2}, E_{ct}. Their work predicts a k* maximum about 10^{0} cm·s^{−1}, which occurs when E_{ct} = 0. Our more detailed kinetic model does not require positing a value of E_{ct} and matches the approximate k* upper limit in De Souza et al.^{57} when their critical energy is E_{ct} = 0.3 eV, which seems a physically reasonable value and lends support to both approaches. To represent typical ITSOFC working conditions, we calculate the limiting ORR rates and upper limits on K_{tr} for a temperature range of 500–750 °C, at a 0.2 atm oxygen partial pressure. In these conditions, typical SOFC cell voltages are in the range of 1–1.5 V. Let us assume that the entire driving force for the fuel cell is represented in the deviation from equilibrium of the ORR reaction, \(\frac{1}{2}{\mathrm{O}}_2 + {\mathrm{V}}_{\mathrm{O}}^{\cdot\cdot} + 2{\mathrm{e}}^  \to {\mathrm{O}}_{{\mathrm{O}}_{\mathrm{b}}}^X\) where \({\mathrm{V}}_{\mathrm{O}}^{..}\) is oxygen vacancy and \({\mathrm{O}}_{{\mathrm{O}}_{\mathrm{b}}}^X\) is lattice oxygen in the cathode.
From Eqs. (5) and (7), we have nonconfigurational Gibbs free energy ΔG = −ΔG_{1.} ΔG_{1} represents an ideal, nonconfigurational Gibb’s free energy change of the formation of oxygen vacancy.
Therefore, at the equilibrium state, we can write the cathode reaction total Gibbs free energy \(\Delta G_1^ {\ast}\) and cathode potential E_{1} as,
F is Faraday constant. Similarly, at the nonequilibrium state, we have the fixed oxygen pressure and new bulk vacancy concentration x_{v,bulk}, the cathode reaction total Gibbs free energy \(\Delta G_2^ \ast\) and cathode potential E_{2} is calculated as,
At the small overpotential η(V),
\(\left[ {{\mathrm{O}}_{{\mathrm{O}}_{\mathrm{b}}}^X} \right]\) is the concentration of latticeO ions
For LSC50 cathode, with typical SOFC cathode overpotential of η = −0.2 V, pO_{2} = 0.2 atm, the term \(\left( {\left[ {{\mathrm{O}}_{{\mathrm{O}}_{\mathrm{b}}}^X} \right]  \left[ {{\mathrm{O}}_{{\mathrm{O}}_{\mathrm{b}}}^{X,\mathrm{eq}}} \right]} \right)\) is calculated at T = 500 °C, 650 °C, 750 °C, 800 °C, respectively, to be 1168 molesO/m^{3} (0.015 sitefraction), 2231 molesO·m^{−3} (0.029 sitefraction), 2929 molesO·m^{−3} (0.038 sitefraction) and 3299 molesO·m^{−3} (0.042 sitefraction). Values of x_{v} are fitted using the parameters from Mizusaki et al.^{41} for LSC50. Figure 7 below plots the term (\(\left[ {{\mathrm{O}}_{{\mathrm{O}}_{\mathrm{b}}}^X} \right]  \left[ {{\mathrm{O}}_{{\mathrm{O}}_{\mathrm{b}}}^{X,\mathrm{eq}}} \right]\), % Osite fraction) on the xaxis and ORR rates (molesO_{2}·m^{−2}·s^{−1}) on the yaxis. Limiting rates are shown as horizontal lines of the type y = R_{0} (molesO_{2}·m^{−2}·s^{−1}), with R_{0} of the O_{2} chemisorption step that sets the fundamental upper limit to ORR. Limiting K_{tr} is represented by the slope (m) of the line of the type y = mx; in other words, a line which gives the limiting rate at the typical values of driving force \(\left( {\left[ {{\mathrm{O}}_{{\mathrm{O}}_{\mathrm{b}}}^X} \right]  \left[ {{\mathrm{O}}_{{{\mathrm{O}}_{\mathrm{b}}}}^{X,\mathrm{eq}}} \right]} \right)\). Plots at three different temperatures are made for the purpose of illustration.
Overpotential calculations
Based on our surface reaction model we can calculate the cathode overpotential vs. current density, providing another method of comparing our model to experiments. The calculation involves a few steps, which we describe below. First, we obtain R_{0} for the material under polarization as follows. From Eq. (30), we can calculate the bulk oxygen vacancy concentration x_{v} under polarization. We can then use the bulk oxygen vacancy concentration under polarization to calculate the concentration of surface oxygen adatom Γ_{Oads} and surface oxygen vacancy Γ_{s}. The equilibrium exchange rate from our rate limiting step can then be updated to include polarization effect from the equation R_{0}(diffusion) = (8 × D_{V} × 2Γ_{s} × Γ_{Oads})/d^{2}. Now we relate R_{0} to the current density and overpotential through the total reaction rate.
The total reaction rate r can be written in terms of equilibrium exchange rate R_{0} in Eq. (3a)
where λ_{i} is the stoichiometric ratio between reaction i and overall reaction under quasiequilibrium conditions, detail see Adler et al.^{11}. Λ is the driving force for overall reaction.
We can write the current density as j = 4e · r/S. The unit of r is O_{2}·Co^{−1}·s^{−1}. S is the area of one Co site. e is the charge of one electron.
Based on the Eqs. (3a), (30) and (32), we can calculate the activation potential of the overpotential curve. The total overpotential consists of activation polarization, ohmic polarization and concentration polarization.^{58,59} For a dense thin film, the concentration polarization is assumed to be negligible, and for the comparisons in this work we use experimental data from which the ohmic polarization has been subtracted.
The predicted activation polarization values are given in Fig. 9 (overpotential vs. log(I) for cathodic polarization), Fig. 10 (overpotential vs. I for cathodic polarization) and Fig. 11 (log(I) vs. overpotential for both anodic and cathodic polarization). In these figures we include a number of experimental data sets on similar materials under similar conditions. However, we were not able to find any studies under the exact conditions of our modeling. Due to the sensitivity of overpotential to exact composition and temperature, a direct quantitative comparison between our model and these studies is difficult. As our best attempt at such a comparison, we shift the La_{0.6}Sr_{0.4}Co/YSZ data from Sase and Kawada^{60} (black) by +1.1 log units in current density to obtain the La_{0.6}Sr_{0.4}CoO_{3}/YSZ “with shifting” curves (purple data). This shift is based on the k*temperature dependence and k* difference between La_{0.6}Sr_{0.4}CoO_{3} and La_{0.5}Sr_{0.5}CoO_{3} from Egger et al.^{7} Specifically, this data suggests we should expect a shift of +1.1 log units in going from La_{0.6}Sr_{0.4}CoO_{3} at 600 °C to La_{0.5}Sr_{0.5}CoO_{3} at 650 °C (which are our modeling conditions), which should lead to an equivalent shift in current density since these measurements on are surface exchange limited films. In comparing our result to that of the shifted curve from Sase and Kawada we see the similar exponential relationship of log(I) and overpotential in Fig. 9 and the same decreasing slope in Fig. 10. Given the uncertainties in the experiments and modeling, and the difference of condition, the agreement is very satisfactory. The slope of our prediction in Fig. 9 is slightly smaller than the experiment at high current density. According to the Eq. (30), the overpotential consists of one linear and one log function of a term that changes approximately proportional to oxygen vacancy concentration. At high oxygen vacancy concentration, the linear term will dominate the overpotential. The faster increase of the model as compared to experimental overpotential suggests that the linear term is changing too fast in our model, i.e., too much overpotential is required to create enough vacancies to enable the observed current density. This could be because it is too hard to create oxygen vacancies in our model, particularly at higher oxygen vacancy content. Our model follows that of Mizusaki, et al.^{41} in having oxygen vacancy formation energies increase with increasing oxygen vacancy concentration through filling of electronic states at the Fermi level, which creates a significant coupling between oxygen vacancy formation and the density of states at the Fermi level. Our density of states is from the fitting from Mizusaki et al.^{41}, and represented by the bulk vacancy interaction parameter “a” in Table I of Mizusaki et al.^{41}. However, our vacancy concentration at high current density is beyond any values explored in the experimental data that was fit by Mizusaki et al.^{41}. It is therefore possible the bulk vacancy interaction parameter will decrease at high vacancy concentration. That would lead to the decrease of overpotential at high current density, which would match the experiment data in Fig. 9. Further study of the behavior of the vacancy energetics at high vacancy concentration therefore might be of use, but it beyond the scope of this work.
Data availability
The authors declare that the main data supporting the findings of this study are available within the article and its Supplementary Information files. Extra data that support the findings of this study are available from the authors on reasonable request.
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Acknowledgements
This work was financially supported by U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences, and Engineering under award number DESC0001284. Computing resources in this work benefitted from the use of the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI1053575. We gratefully acknowledge helpful conversations on this work with Dr. YuehLin Lee, Prof. Yang ShaoHorn, and Prof. Bilge Yilidz.
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All authors provided feedback and input to the research, analysis, and manuscript. Y.C., M.J.G., and A.T.N. have equal contributions to this work. M.J.G., A.T.N., D.M., and S.B.A. conceived this project and constructed the basic kinetic theory. M.J.G. and A.T.N. performed initial ab initio calculations and theoretical analyses, with assistance from D.M. and S.B.A. Y.C. performed the final sets of ab initio calculations and theoretical analyses, with assistance from D.M. and S.B.A. All authors contributed to scientific discussions and read and commented on the manuscript. This project was supervised by D.M.
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Cao, Y., Gadre, M.J., Ngo, A.T. et al. Factors controlling surface oxygen exchange in oxides. Nat Commun 10, 1346 (2019). https://doi.org/10.1038/s41467019086744
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