A. Empirical relation involving ionic conductivity

The first step in understanding the mechanistic details of a reaction is to identify those species which are important for the rds and hence crucial for the overall rate (nevertheless, parameters of fast preceding/succeeding steps can also affect the rate). Thus, in the case of a gas–solid surface reaction, correlations with the nature and concentrations of surface species (e.g., adsorbates) and surface defects (e.g., oxygen vacancies) come to the fore. Unfortunately, in the window of relevant conditions (300–800 °C, pO₂ ≈ 0.2 bar), the surface defect concentrations are experimentally accessible in the fewest cases. We are aware of the fact that defect energetics and thus absolute concentrations in the surface layer differ from the bulk. Nevertheless, as long as the same reactions determine the equilibration with the gas phase and the majority defects remain the same, the pO₂ dependence of defect concentrations in bulk and surface layer will be similar (this at least holds for an electroneutral model). In addition to the different surface energetics, surface space charge zones may modify the kinetics, e.g., for slightly Fe-doped SrTiO₃.Reference De Souza and Martin¹² In the materials under discussion, however, such space charge zones can be assumed to be negligible (the blocking character of the grain boundaries in SrTi_1−xFe_xO_3−d vanishes with increasing Fe content, i.e., increasing carrier density and concentration of cations with variable valence, at ≥7 mol% of FeReference Jurado, Colomer and Frade¹³).

Figure 1(a) shows effective rate constants k ^q of the oxygen surface reaction (determined from impedance measurements of perovskite films on YSZ substrates) versus electronic conductivity. No clear trend can be recognized—for instance, Ba_0.5Sr_0.5Co_0.8Fe_0.2O_3−δ with the fastest exchange rate has a lower electronic conductivity than La_0.74Sr_0.18MnO_3−δ with the lowest rate. Although electrons must be involved in the oxygen exchange reaction, obviously, the supply of electronic carriers is not rate limiting for the perovskites considered here.

FIG. 1.

(a) Effective oxygen exchange rate constants k ^q of mixed conducting perovskites at 750 °C and pO₂ = 1 bar plotted versus electronic conductivity. k ^q from Refs. Reference Baumann, Fleig, Cristiani, Stuhlhofer, Habermeier and Maier7, Reference Wang, Merkle and Maier8, and Reference Fleig, Kim, Jamnik and Maier14, σ_eon from Refs. Reference Mizusaki, Yonemura, Kamata, Ohyama, Mori, Takai, Tagawa, Dokiya, Naraya, Sasamoto, Inaba and Hashimoto15–Reference Kozhevnikov, Leonidov, Patrakeev, Mitberg and Poeppelmeier19. (b) Rate constants at 750 °C, pO₂ = 1 bar for bulk samples [circles: k*, triangles: k* = k ^chem/w _O from chemical relaxation experiments, ] and dense PLD films (diamonds: k ^q) plotted versus ionic conductivity (in most cases calculated from diffusivity via Nernst–Einstein relation). For the interpretation of the dashed lines, see Sec. III. C. Data sources: a,Reference de Souza and Kilner²⁰ b,Reference Fullarton, Jacobs, van Benthem, Kilner, Brongersma, Scanlon and Steele²¹ c,Reference Berenov, Atkinson, Kilner, Bucher and Sitte²² d,Reference Ishigaki, Yamauchi, Kishio, Mizusaki and Fueki²³ e,Reference Geffroy, Bassat, Vivet, Fourcade, Chartier, Del Gallo and Richet²⁴ f,Reference Fleig, Kim, Jamnik and Maier¹⁴ g,Reference Baumann, Fleig, Cristiani, Stuhlhofer, Habermeier and Maier⁷ h,Reference Wang, Merkle and Maier⁸ i: k ^q from,Reference Jung and Tuller³ σ_ion from,Reference Rothschild, Menesklou, Tuller and Ivers-Tiffee²⁵ jReference van der Haar, den Otter, Morskate, Bouwmeester and Verweij²⁶ (with w_O taken from Ref. Reference Mizusaki, Mima, Yamauchi, Fueki and Tagawa27) k,Reference Yoo, Verma, Wang and Jacobson²⁸ l,Reference ten Elshof, Lankhorst and Bouwmeester²⁹ m.Reference Chen and Shao³⁰ If necessary, data were interpolated/extrapolated: pO₂ dependences from the references were used where available. Otherwise, D* which typically has a less pronounced dependence was not adjusted, while k* was extrapolated to 1 bar with the approximate relation , which is reasonably close to many experimentally found exponents.

Figure 1(b) shows effective rate constants k determined by isotope exchange (k*) as well as k ^q (k* and k ^q are related by the same proportionality factor to the exchange rate of the oxygen incorporation reaction and thus refer to the same quantityReference Maier³¹; in some places, we will drop the superscript for simplicity) from electrochemical measurements on pore-free thin films plotted versus bulk ionic conductivity. For a given material, the scatter in the bulk property σ_ion is smaller compared to the surface rate constant k, which is more sensitive to sample preparation, aging, etc., thus preferably, data from the same source should be used for more detailed comparisons. One clear trend is observed; typically, the samples with a higher ionic conductivity also exhibit a faster surface reaction. The fact that a higher oxygen vacancy concentration typically leads to faster surface exchange has been observed and discussed already in Refs. Reference de Souza and Kilner20, Reference Yoo, Verma, Wang and Jacobson28, Reference ten Elshof, Lankhorst and Bouwmeester29, Reference Kilner, De Souza and Fullarton32, and Reference Mosleh, Sogaard and Hendriksen33. Based on results for BSCF perovskites,Reference Wang, Merkle and Maier⁸ it was further concluded that not only concentration but also mobility can significantly affect the surface reaction rate. To understand these correlations, reaction mechanisms for some perovskites will be discussed in more detail in the next section.

The importance of ionic conductivity for SOFC cathode performance is well established as a certain σ_ion is required to allow for oxygen incorporation on the whole surface of the cathode material (“bulk path”Reference Fleig and Maier³⁴) instead of being limited to the extremely narrow zone of the three-phase boundary. With its correlation to surface exchange kinetics as discussed here, σ_ion must be regarded as even more important since it increases not only the electrocatalytically active part of an electrode layer but also the actual surface reaction rate.

B. Detailed mechanistic considerations for LSM, LSCF, and BSCF

For LSM (the first perovskite widely applied as SOFC cathode material), extensive ab initio calculations are available, which allow detailed atomistic insight into the reaction mechanism that could not be extracted solely from experimentsReference Mastrikov, Merkle, Heifets, Kotomin and Maier^10, Reference Choi, Lynch, Lin and Liu³⁵ (unfortunately, due to the very small supercell used, the mutual approach of adsorbed oxygen species and is not dealt with in Ref. Reference Choi, Lynch, Lin and Liu35). As a first step, the most stable surface terminations need to be identified as they are expected to be the most exposed ones in the actual experiments with typically polycrystalline samples. For LaMnO₃, under relevant SOFC conditions (high T, high pO₂), the MnO₂(001) termination is found most stable,Reference Mastrikov, Merkle, Heifets, Kotomin and Maier¹⁰ while for Sr-doped LSM, the (La,Sr)O(001) termination can become comparably stable or even slightly lower in energy but only at low pO₂.Reference Mastrikov, Merkle, Heifets, Kotomin and Maier^10, Reference Piskunov, Heifets, Jacob, Kotomin, Ellis and Spohr^36, Reference Mastrikov, Heifets, Kotomin and Maier³⁷ Nevertheless, the formation of oxygen vacancies as an essential ingredient for oxygen incorporation is unfavorable by about 2 eV on (La,Sr)O(001) compared to MnO₂(001). At 1000 K, this corresponds to a decrease in concentration by more than 9 orders of magnitude, which will decelerate any mechanism accordingly (directly if the rds already contains ; or by making the last step requiring , see Fig. 2, rate limiting due to their low concentration). Thus, even if the (La,Sr)O(001) termination would cover 99% of the total exposed surface area, its contribution to the overall oxygen exchange rate will be negligible.

FIG. 2.

Schematic network of possible reaction steps for the oxygen incorporation into mixed conducting perovskites. (a) Predominant mechanism for perovskites with very low oxygen vacancy concentration such as LSM, =M2 in Fig. 3; (b) for perovskites with high vacancy concentration such as LSCF or BSCF, =M3 in Fig. 3; (c) hypothetical mechanism starting with adsorption into vacancy, =M1 in Fig. 3.

Therefore, the further mechanistic discussion (see Ref. Reference Mastrikov, Merkle, Heifets, Kotomin and Maier10 for more details) will focus on the MnO₂(001) termination. As basis for exploring the network of possible reaction steps (Fig. 2), energies of defects and intermediate species as well as relevant reaction and migration barriers were calculated for LaMnO₃. While the formation of adsorbed oxygen species is exothermic (by −1.1 eV for O₂⁻, −0.9 eV for O₂²⁻, −2.2 eV for 2 O⁻_ad, relative to O_2,gas), the negative entropy for the adsorption process and repulsive interaction of charged adsorbates at high coverageReference Fleig, Merkle and Maier³⁸ restrict the estimated adsorbate coverages to <0.01 for O₂⁻, O₂²⁻ and ≤0.1 for O⁻_ad. Oxygen vacancy formation (according to ) is easier by ≈1 eV in the MnO₂(001) surface layer compared to bulk, corresponding to a much larger concentration in the surface layer. The migration barrier in the surface layer is lower by 0.3 eV. Dissociation of O₂²⁻ on the perfect surface has a barrier of 0.6 eV, while after insertion of O₂²⁻ into a vacancy, dissociation occurs without barrier. Overall rates for different pathways M1–M3 in the reaction network (Fig. 2) can then be estimated as a function of surface vacancy concentration (obtained from measured bulk D*Reference De Souza and Kilner³⁹ and Reference Ishigaki, Yamauchi, Kishio, Mizusaki and Fueki²³ values for La_0.8Sr_0.2MnO_{3 ± δ}, and the decreased formation energy in the surface layer) and coverage with adsorbed O₂⁻ or O₂²⁻, as shown in Fig. 3.

FIG. 3.

(a) Reaction rates for three alternative reaction pathways of oxygen incorporation on the MnO₂(001) surface of LSM (black lines) and LSCF (red lines) estimated based on DFT results. Adapted with permission from Ref. Reference Mastrikov, Merkle, Heifets, Kotomin and Maier10, copyright (2010) American Chemical Society. (b) Two-dimensional projection for surface vacancy concentration of 10⁻⁴ (LSM) and 10⁻¹ (LSCF).

Pathway M1 [Fig. 2(c)] starting with oxygen adsorption into a surface vacancy is unfavorable under the relevant conditions due to the low vacancy concentration on the LSM surface (cyan area in Fig. 3). Alternative paths begin with adsorption on the perfect surface. Again due to the low concentration, the dissociation with assistance of a vacancy [M3, Fig. 2(b)] is unfavorable although the reaction barrier is negligibly small. Thus, path M2 [Fig. 2(a)] with dissociation without being involved provides the fastest overall rate. Within this path, either the dissociation is predicted to be rate limiting [at low adsorbate coverage, i.e., very low pO₂, pink area in Fig. 3(a)], or the final incorporation step of into a surface (violet area). Interestingly, the barrier for migration along the surface is so high (2 eV, due to an unfavorable transition state with the migrating located atop an oxide ion) that it is the surface vacancy, which approaches (the actual incorporation then occurs without barrier). For this step being rate determining, the reaction rate is proportional to the product of concentration and mobility, the latter quantity playing the role of the rate constant (the rate for an individual to meet an is proportional to D/d ² where d is the mean distance between and O⁻_ad, and D is the diffusion coefficient of in the surface layer). The proportionality of the rate to the vacancy concentration [] is straightforward, as long as preceding equilibria do not involve [].

Under the assumption that the reaction and migration barriers do not change drastically, Fig. 3(a) can even be used to “extrapolate” reaction rates to closely related perovskites such as LSCF (the measured bulk vacancy migration barrier around 0.8 eV is similar for LSM and LSCFReference Ishigaki, Yamauchi, Kishio, Mizusaki and Fueki^23, Reference De Souza and Kilner³⁹). The vacancy concentration of LSCF is much higher than for LSM as a result of the less negative oxidation enthalpy.Reference Mizusaki, Mima, Yamauchi, Fueki and Tagawa^27, Reference Mizusaki, Yoshihiro, Yamauchi and Fueki⁴⁰ On the other hand, this also leads to lower adsorbate concentrations since enthalpies of adsorption and oxidation—both involving electron transfer from the transition metal to oxygen—are closely coupled (the calculated energy for molecular oxygen adsorption amounts to ≈1/4 and for two to ≈1/2 of the O₂ incorporation energyReference Choi, Lin and Liu⁴¹). This is depicted in Fig. 3 by red lines. Owing to the much higher concentration for LSCF, path M3 with vacancy-assisted dissociation of O₂⁻ or O₂²⁻ is predicted to become the most favorable, with the encounter of molecular adsorbed oxygen and before the dissociation being rate limiting as shown by the light gray area in Fig. 3(a). The encounter of and for the final incorporation is faster due to the coverage of being higher than for molecular adsorbate species. The prediction of higher reaction rates for LSCF compared to LSM in Fig. 3 is in agreement with experiments.Reference Baumann, Fleig, Cristiani, Stuhlhofer, Habermeier and Maier⁷

The investigation of BSCF perovskites also yields valuable insight into the oxygen exchange reaction mechanism. Ba_0.5Sr_0.5Co_0.8Fe_0.2O_3−δ was initially developed for oxygen permeation membranes (i.e., high electronic and ionic conductivity) but then found to exhibit also very fast surface exchange kinetics.Reference Shao and Haile^42, Reference Baumann, Fleig, Habermeier and Maier⁴³ A plot of k ^q (measured by impedance spectroscopy on pore-free films on YSZ) for different cation compositions within the Ba_1−xSr_xCo_yFe_1−yO_3−δ family versus concentration yields a clear but nonlinear correlation.Reference Wang, Merkle and Maier⁸ In contrast to the perovskites of the (La,Sr)(Mn,Fe,Co)O_3-δ family, the mobility of increases in BSCF with increasing content of Ba and Co. Only, when plotting k ^q versus , an approximately linear correlation is obtained (Fig. 4). For hexagonal Ba_0.5Sr_0.5Co_0.8Fe_0.2O_3−δ, a lower ionic conductivity than for the cubic phase is expected (based on analogy to the closely related cubic/hexagonal SrCoO_3−δReference Deng, Yang, Liu and Chen⁴⁴). Interestingly, an investigation of oxygen exchange in Ba_0.5Sr_0.5Co_0.8Fe_0.2O_3−δ during its slow transformation to the hexagonal perovskite phase also yields a linear correlation between k ^chem (= w _O^.k ^q) and D ^chem (= w _O^.D*) (open symbols in Fig. 4).Reference Yakovlev, Yoo, Fang and Bouwmeester⁴⁵ Thus, increasing vacancy concentration and vacancy mobility () enhance the reaction rate. The latter makes the stronger contribution in the series SrFeO_3−δ−Ba_0.5Sr_0.5FeO_3−δ−BSCF (increasing by a factor of 18 at 600 °C from SF to BSCF) than the vacancy concentration, which is already high for SrFeO_3−δ to (δ = 0.25) and increases only moderately for BSCF (δ = 0.53). Pronounced differences in oxygen migration barriers in the BSCF perovskites were found also in ab initio calculations.Reference Merkle, Mastrikov, Kotomin, Kuklja and Maier¹¹ Geometrical constraints such as the distance of the migrating O to neighboring cations do not differ significantly. On the other hand, the decrease of the barriers with increasing Ba and Co content—e.g., from 0.72 eV for Ba_0.5Sr_0.5FeO_3−δ to 0.47 eV for Ba_0.5Sr_0.5Co_0.8Fe_0.2O_3−δ—appears related to the energetic cost for redistributing some electron density from the migrating O to the neighboring transition metal (and thus to decrease its effective size) in the transition state.

FIG. 4.

Effective rate constant for oxygen exchange on BSCF plotted versus diffusion coefficient. Solid symbols: k ^q, from Ref. Reference Wang, Merkle and Maier8, pO₂ = 0.2 bar (k ^q), 0.5 bar (D*). Open symbols: k ^chem (=w _Ok ^q), D ^chem (=w _OD*) for a Ba_0.5Sr_0.5Co_0.8Fe_0.2O_3−δ sample, which slowly transforms from cubic to a hexagonal perovskite phase from Ref. Reference Yakovlev, Yoo, Fang and Bouwmeester45, pO₂ = 0.33 ↔ 1 bar O₂.

The pO₂ dependence of k ^q for BSCF as measured on dense films exhibits slopes m in in the range of 0.54 for SrFeO_3−δ to 0.82 for Ba_0.5Sr_0.5FeO_3−δ.Reference Wang, Merkle and Maier⁸ The overall pO₂ dependence results from the reaction order of adsorbed oxygen species (1 for molecular adsorbate, 1/2 for atomic O resulting from a preceding fast dissociation reaction in quasiequilibrium) and contributions from point defects involved in the rate of the rds such as or electrons. Since the exponent of the pO₂ dependence of electron and concentration is always negative (while its magnitude will depend on the actual material), the defect contributions will decrease the overall pO₂ dependence to values lower than the reaction orders of adsorbed oxygen (1 for molecular species, 1/2 for atomic species); therefore, values larger than 1/2 clearly indicate molecular oxygen species such as O₂⁻, O₂²⁻ are involved in the rds.

Figure 5 shows a mechanism, which is in accordance with the experimental findings, that migration as well as molecular oxygen species appear in the rds. Chemisorption is fast on the surface while it would be slower directly into a V_Ö (less vacancies available as adsorption site than transition metal ions, and more strict requirements on the orientation of O₂ approaching from gas phase). After fast chemisorption (I) on the surface, the rds consists of the approach of an oxygen vacancy to the adsorbed O₂²⁻ (II). In analogy to the DFT results for LSM, the following dissociation (III) is expected to occur fast with negligible barrier. The approach of another to the remaining (IV) is faster than the rds due to the larger coverage with . The equilibrium exchange rate (Reference Maier³¹) for this proposed mechanism is

(1)

FIG. 5.

Scheme of suggested reaction mechanism for BSCF perovskites.

where square brackets indicate concentrations, K is the mass action constant of the chemisorption, and represents the mobility of the approaching . In contrast to LSM, in BSCF, the bulk concentration is high and the bulk migration barrier low so that the approach to adsorbed oxygen can occur not only by diffusion in the surface layer but also from the bulk. Thus, in contrast to LSCF, it is also plausible that rather migrates toward O₂²⁻ before dissociation and not vice versa. The lowering of the overall pO₂ dependence from the upper limit of m = 1 is mainly due to the contribution of electrons in Eq. (1) since the pO₂ as well as T-dependence of [] is small. This suggests that probably, the two electrons donated to the adsorbed oxygen are not transferred from (Co,Fe)³⁺ species (the (Co,Fe)³⁺ concentration has a comparably small pO₂ and T-dependence like []) but from other electronic states (e.g., (Co,Fe)²⁺ originating from T- and pO₂-dependent disproportionation of Co³⁺Reference Jung, Misture and Edwards^46, Reference Bucher, Sitte, Caraman, Cherepanov, Aksenova and Ananyev⁴⁷ and/or O deficiency when δ > 0.5). Also the activation energy of k (1.4–1.6 eV for k ^q of Ba_0.5Sr_0.5Co_yFe_1−yO_3−δReference Wang, Merkle and Maier⁸; other literature values 1–1.4 eVReference Shao and Haile^42, Reference Burriel, Niedrig, Menesklou, Wagner, Santiso and Ivers-Tiffee^48, Reference Bouwmeester, Song, Zhu, Yi, van Sint Annaland and Boukamp⁴⁹) cannot be due to migration (0.5 eV) and oxygen adsorption (probably with an enthalpy close to zero) alone but has a certain contribution from the T-dependent electron concentration.

C. Mechanistic interpretation of the observed k–σ_ion correlation

Based on this mechanistic discussion in the previous section, the differences in the exchange rates between LSM, LSCF, and BSCF can largely be interpreted as shown in Fig. 6. The increase from LSM to LSCF is essentially due to the increased vacancy concentration since are involved in the rds. How far the actually migrate in the rds is less relevant as the vacancy mobility at a given temperature is in good approximation independent of cation composition for these perovskites.Reference Ishigaki, Yamauchi, Kishio, Mizusaki and Fueki^23, Reference De Souza and Kilner³⁹ The rate increase from LSCF to BSCF is mainly caused by the increased mobility as the relative increase in concentration is small.

FIG. 6.

Comparison of effective rate constants k ^q for oxygen exchange for LSM, LSCF, and BSCF films on YSZ. Data from Refs. Reference Baumann, Fleig, Cristiani, Stuhlhofer, Habermeier and Maier7, Reference Wang, Merkle and Maier8, and Reference Fleig, Kim, Jamnik and Maier14. The vacancy concentration [] is given as per perovskite formula unit.

Despite some scatter, in the double logarithmic plot of Fig. 1(b), the data points seem to follow a correlation with a slope close to 0.5 in the left part and gather between the dashed lines with slopes close to 1 and 0.5 in the right part. These limiting values of the slopes can be understood on the basis of the underlying mechanisms. The slope close to 0.5 in the left part corresponds to the correlation between k* and D* previously found for (Ln,Sr)(Mn,Fe,Co)O_3−δ perovskites.Reference de Souza and Kilner^20, Reference Merkle, Maier and Bouwmeester⁵⁰ (As the DFT results for LSM indicate a higher surface concentration and mobility than in bulk, the LSM data should appear further right in the plot, but since k* may be underestimated due to experimental limitations,Reference De Souza⁵¹ the effects on the slope may largely cancel.) For the perovskites in the left part of Fig. 1(b), the mobility is essentially independent of cation composition.Reference Ishigaki, Yamauchi, Kishio, Mizusaki and Fueki^23, Reference De Souza and Kilner³⁹ The increase of σ_ion can be ascribed to the increased concentration and thus to the oxidation enthalpy ΔH _ox (= - formation enthalpy) decreasing from manganate to cobaltate perovskites.Reference Merkle, Maier and Bouwmeester⁵⁰ Parallel to decreasing ΔH _ox, the oxygen chemisorption enthalpy decreases,Reference Choi, Lin and Liu⁴¹ and the resulting smaller adsorbate coverages will counterbalance a part of the rate increase caused by the increased concentration.

The mechanistic interpretation of the k*–D* correlation given in Ref. Reference Merkle, Maier and Bouwmeester50 must be updated in one point: according to the present picture, the rds is not the “dropping” of into a neighboring (this was found to occur without barrier in the DFT results for LSMReference Mastrikov, Merkle, Heifets, Kotomin and Maier¹⁰) but the encounter of and adsorbed oxygen. What is in common is that are involved in the rds, and that the change in concentration with changing oxidation enthalpy is the origin of the correlation. The contribution bΔH _ox to the activation energy of k*Reference Lee, Kleis, Rossmeisl, Shao-Horn and Morgan⁵⁰ corresponds to the combined (and partly compensating) effect of ΔH _ox on concentration and adsorbate coverage, while bΔH ^≠ matches with the decreased migration barrier in the surface layer (0.5 ≤ b ≤ 0.7, ΔH ^≠ = migration enthalpy). The correlation between k and the “oxygen p-band center” from DFT suggested in Ref. Reference Lee, Kleis, Rossmeisl, Shao-Horn and Morgan52 is a largely similar finding, as the O p-band center is closely related to ΔH _ox. The attempt of rationalizing the trends in ΔH _ox in terms of d-electron configuration in Ref. Reference Pavone, Ritzmann and Carter53 seems not really appropriate since it deals with the 2+/3+ redox couple, while the relevant couple under SOFC conditions is 3+/4+ for most materials.

A slope close to 1 as observed in the right part of Fig. 1(b) corresponds to the correlation between k* and σ_ion found for (Ba,Sr)(Fe,Co)O_3−δ.Reference Wang, Merkle and Maier⁸ Such a slope will arise from the general proportionality of k to σ_ion for reaction mechanisms involving migration in the rds, when the increase of k is mainly due to an increase of mobility, i.e., the enthalpy of oxidation remains approximately constant, and thus, the opposing effect of decreasing adsorbate coverage is absent.

Since electronic charge carriers appear in the chemisorption equilibria preceding the rds, the electronic structure (position of Fermi level) can also have some influence. On the other hand, it is expected that for most of the discussed materials, such effects are less significant than the differences in σ_ion, which span a range of 9 orders of magnitude.

Based on these interpretations, the question arises to what limit the surface exchange rates could further be increased. For fast exchange, (i) a high concentration combined with (ii) chemisorption enthalpies as negative as possible and (iii) a high mobility are desired. The first two points are to significant extent coupled to the oxidation enthalpy ΔH _ox, a decrease of which will increase [] at the expense of a lower adsorbate coverage. The (molecular) adsorbate coverage might be increased by tuning parameters such as the polarizability or basicity of the cations (although to some extent, they will also be coupled to the oxidation enthalpy). With respect to mobility (iii), BSCF appears to be already close to the maximum that can be achieved in perovskites, and the high mobility seems to be related to the Ba/Co size mismatch, which, on the other hand, facilitates unwanted transformation to hexagonal perovskite phases.Reference Merkle, Mastrikov, Kotomin, Kuklja and Maier^11, Reference Svarcova, Wiik, Tolchard, Bouwmeester and Grande^54, Reference Kuklja, Mastrikov, Jansang and Kotomin⁵⁵ BSCF has further shortcomings with respect to SOFC operation (see e.g., Ref. Reference Wang, Merkle and Maier8). Another approach could be a combination of two phases being intimately mixed (with grain sizes smaller than in usual perovskite/electrolyte composite cathodes), one fostering oxygen adsorption and the other supplying a high vacancy concentration and mobility for the final incorporation.

Layered double perovskites and perovskite-related Ruddlesden–Popper phases are currently under intensive investigation. Nevertheless, in the vast majority of studies, significantly higher k values than BSCF have not been achieved so far (see e.g., Refs. Reference Tarancon, Skinner, Chater, Hernendez-Ramirez and Kilner56 and Reference Burriel, Garcia, Santiso, Kilner, Chater and Skinner57). A particular feature of the Ruddlesden–Popper phases is that in the ab planes, oxygen diffusion occurs by an interstitialcy mechanism, i.e., not requiring .Reference Parfitt, Chroneos, Kilner and Grimes⁵⁸ Interestingly, exchange rates higher than that for the single phases are reported for (La,Sr)CoO_3−δ/(La,Sr)₂CoO_{4 ± δ} composites and heterostructures.Reference Sase, Yashiro, Sato, Mizusaki, Kawada, Sakai, Horita and Yokokawa^59, Reference Crumlin, Mutoro, Ahn, la O’, Leonard, Borisevich, Biegalski, Christen and Shao-Horn⁶⁰ As another means for improvement, moderate tensile strain (up to few percent) has the potential to accelerate the surface reaction by the effects of increased concentration, mobility, and adsorbate coverage,Reference la O’, Ahn, Crumlin, Orikasa, Biegalski, Christen and Shao-Horn^6, Reference Han and Yildiz⁶¹ if the strain can actually be sustained and does not relax, e.g., by formation of higher-dimensional defects.

Finally, one may ask why the surface reaction on mixed conducting perovskites—in the majority of cases discussed here limited by the migration of a (surface) oxygen vacancy—is typically so slow that the subsequent bulk oxygen transport is faster even over a distance of 100 nm (i.e., involving thousands of bulk jumps). One major reason is that the oxygen incorporation reaction has a strongly negative entropy (loss of translational and rotational degrees of freedom) of typically ΔS ⁰ ≈ −190 J/mol K per O₂ at 1000 K. This value is estimated from ΔS ⁰ for oxide formation (SrO, BaO, Mn₂O₃, Fe₂O₃, Co₃O₄), and amounts to about 3/4 of the absolute entropy S ⁰ of gaseous O₂; ΔS ⁰ for oxygen incorporation in La_1−xSr_xCoO_3−δReference Mizusaki, Yoshihiro, Yamauchi and Fueki⁴⁰ is also comparable. The respective contribution of −TΔS ⁰ ≈ 1.9 eV to the standard Gibbs enthalpy of reaction ΔG ⁰ corresponds to a factor of up to 10⁻¹⁰ appearing in the concentration of intermediate species (such as adsorbates) before the rds, which decreases the reaction rate accordingly (this aspect is discussed also in Ref. Reference Adler, Chen and Wilson62). Although chemisorption as fast preceding reaction is in quasiequilibrium, its mass action constant can affect the reaction rate by keeping the concentrations of certain intermediate species low.