Nonstoichiometry, Defect Chemistry and Oxygen Transport in Fe-Doped Layered Double Perovskite Cobaltite PrBaCo2−xFexO6−δ (x = 0–0.6) Membrane Materials

Mixed conducting cobaltites PrBaCo2−xFexO6−δ (x = 0–0.6) with a double perovskite structure are promising materials for ceramic semi-permeable membranes for oxygen separation and purification due to their fast oxygen exchange and diffusion capability. Here, we report the results of the detailed study of an interplay between the defect chemistry, oxygen nonstoichiometry and oxygen transport in these materials as a function of iron doping. We show that doping leads to a systematic variation of both the thermodynamics of defect formation reactions and oxygen transport properties. Thus, iron doping can be used to optimize the performance of mixed conducting oxygen-permeable double perovskite membrane materials.


Introduction
Since the discovery of fast oxide ion transport in the A-site-ordered layered double perovskite cobaltites REBaCo 2 O 6−δ (RE = rare-earth element) [1], they have received a lot of attention as suitable materials for various electrochemical devices, including oxygensemi-permeable membranes [2][3][4]. In addition, high mixed oxide-ion and electronic conductivity coupled with the catalytic activity towards the oxygen reduction reaction suggested REBaCo 2 O 6−δ cobaltites as promising cathode materials for both oxide-ion and protonconducting solid oxide fuel cells [2][3][4][5]. As a result, a number of studies describing various aspects of practical applications of double perovskites have been produced, but the understanding of the oxygen transport in these oxides, despite being directly responsible for their performance, remains underdeveloped. There is some scatter in the reported activation energies, values of the oxygen self-diffusion coefficient and surface exchange coefficient even for the undoped cobaltites [5][6][7][8][9][10][11][12][13], whereas for the doped ones, such measurements are exceptionally rare [7]. Furthermore, although the mechanism of oxygen diffusion in the A-site ordered layered double perovskites is known [6,[14][15][16][17][18][19][20][21][22] and may be described as the oxygen diffusion within the CoO 2− layers through oxygen vacancies coming from the rare-earth layers, the detailed understanding of the dopant-induced changes is also lacking. The same scarcity of systematic reports can be noted about the defect chemistry which, with the rare exceptions [23][24][25][26][27][28], has not been studied yet for doped double perovskites.
Substituting iron for cobalt in various perovskite-type cobaltites is a widely used strategy for lowering their thermal expansion coefficient (TEC) [29,30] and increasing their thermodynamic stability [31][32][33]. While there are data indicating that moderate iron content in REBaCo 2−x Fe x O 6−δ does not lower their TEC significantly [30,34], their stability seems to increase [31][32][33]. In addition, some studies report that iron doping enhances the electrochemical performance of double perovskite cobaltites as cathodes in solid oxide fuel cells [30] and oxygen evolution reaction catalysts [33]. the logarithm with base 10). The 3-way valves in the mixer effectively switch the fixed-rate gas flows, directing them either to the bypass or to the measurement cell. This eliminates the transient effects occurring upon changing the setpoints in the mass-flow controllers, which would not be possible in a simpler gas mixer setup. The volume of the sample chamber is about 20 cm 3 , whereas the gas flow rate used in this work was 150 cm 3 ·min −1 , which means the pO 2 around the sample can be changed during the experiment in a matter of a few seconds. The dry air and nitrogen supplied to the gas mixing unit were produced on site using specialized dry-air compressor (KSV-6/500, NPP Himelektronika, Moscow, Russia) and nitrogen generator (GChA-21D, NPP Himelektronika, Russia), respectively.
homemade computer-controlled 4-channel gas mixing unit (see Figure 1c) employing the RRG12 (Eltochpribor, Russia) mass flow controllers and 3-way-electrical valves (SMC, Tokyo, Japan) allows maintaining and rapidly switching the pO2 around the sample to induce the conductivity relaxation. The following stepwise changes of log(pO2/atm) were implemented: −0.678 → −0.88 → −1.14 → −1.37 → −1.55 → −1.87 → −2.20 → −2.84 → −3.25 (log is the logarithm with base 10). The 3-way valves in the mixer effectively switch the fixed-rate gas flows, directing them either to the bypass or to the measurement cell. This eliminates the transient effects occurring upon changing the setpoints in the mass-flow controllers, which would not be possible in a simpler gas mixer setup. The volume of the sample chamber is about 20 cm 3 , whereas the gas flow rate used in this work was 150 cm 3 ·min −1 , which means the pO2 around the sample can be changed during the experiment in a matter of a few seconds. The dry air and nitrogen supplied to the gas mixing unit were produced on site using specialized dry-air compressor (KSV-6/500, NPP Himelektronika, Moscow, Russia) and nitrogen generator (GChA-21D, NPP Himelektronika, Russia), respectively. The sample's electrical conductivity measured after each pO2 change was normalized according to Equation (1): where σt, σ0 and σ∞ are the conductivity at time t, the equilibrium conductivity before the pO2 change and the equilibrium conductivity after the pO2 change, respectively. The examples of the time-dependent normalized conductivity calculated with Equation (1) from the relaxation measurement results are shown in the Supplementary, Figures S3 and S4.
The surface exchange coefficient, kchem, and the bulk oxygen chemical diffusion coefficient, Dchem, were estimated by fitting the solution of Fick's second law [35][36][37], Equation (2), to the resulting relaxation curves σnorm vs. time. The sample's electrical conductivity measured after each pO 2 change was normalized according to Equation (1): where σ t , σ 0 and σ ∞ are the conductivity at time t, the equilibrium conductivity before the pO 2 change and the equilibrium conductivity after the pO 2 change, respectively. The examples of the time-dependent normalized conductivity calculated with Equation (1) from the relaxation measurement results are shown in the Supplementary, Figures S3 and S4. The surface exchange coefficient, k chem , and the bulk oxygen chemical diffusion coefficient, D chem , were estimated by fitting the solution of Fick's second law [35][36][37], Equation (2), to the resulting relaxation curves σ norm vs. time.
where t is time; 2x, 2y, 2z are the dimensions of the sample, and β m , γ n , θ p are the positive, non-zero roots of the following equations: The values of β m , γ n , and θ p are computed iteratively as described in [37] during the fitting after each optimization step. The oxygen self-diffusion coefficient, D O , and the surface exchange coefficient, k, were calculated from the as-determined chemical ones, D chem and k chem , according to Equations (5) and (6): and is the thermodynamic enhancement factor and c O is the concentration of oxygen in the corresponding oxide. The values of γ were found from the isothermal pO 2 dependencies of the oxygen content measured in this work for all the studied samples of double perovskites. The ECR measurements were carried out both in the direction of decrease and increase in pO 2 . The resulting parameters D chem and k chem estimated as mentioned above did not show any dependence on the direction of pO 2 change and were averaged.
In order to obtain the pO 2 -T-δ data necessary for estimation of the thermodynamic enhancement factor, γ, and analysis of the defect structure of the double perovskites PrBaCo 2−x Fe x O 6−δ , the relative change in their oxygen nonstoichiometry (∆δ) was measured in the wide ranges of T and pO 2 by two independent techniques: coulometric titration (CT) and thermogravimetry (TG), described in detail elsewhere [38]. The CT measurements were carried out in a home-made setup [38]. The TG measurements were performed using a RuboTHERM Dyntherm LP-ST thermobalance (TA Instruments, New Castle, DE, USA). Powder samples weighing~0.5 g were used for both TG and CT measurements.
As the instantaneous equilibration criteria during CT procedure the slope of the log(pO 2 , atm) = f (time, s) curve was used. It was monitored during each titration step. The state of the titration cell was regarded as "stationary" if this slope did not exceed 10 −6 for a few hours, which gives negligible pO 2 change speed.
In the TG experiment the equilibrium state of the sample was assumed when its weight ceased to change. In addition, all the TG and CT measurements were performed so as to ensure that the results are reproducible and correspond to the true equilibrium state, the main sign of which is the reversibility, i.e., the independence of the results on the direction of the pO 2 change. To check the reproducibility, the measurements were carried out in the directions of both the increase and the decrease in the pO 2 . The relative nonstoichiometry obtained did not depend on the mode of the experiment and, hence, was assumed to correspond to the true equilibrium state of the sample.
The relative change in the oxygen nonstoichiometry obtained by TG or CT was then recalculated to the absolute scale (δ) using the absolute oxygen content, determined by means of direct sample reduction in the flow of pure hydrogen at 1200 • C in the STA 409 PC (Netzsch, Selb, Germany) thermobalance [38].

Phase Composition and Crystal Structure of the Double Perovskites PBCF
The XRD patterns of the as-prepared, slowly (2 • C·min −1 ) cooled in air, samples of PBCF are shown in Figure 2. All the samples were confirmed to be single-phase in agreement with the previous studies [4,7,39,40]. Their XRD patterns were indexed using the P4/mmm space group. The lattice parameters refined by the Rietveld method are given in Table S1. As seen, they are in good agreement with those reported earlier [4,7,39,40]. At the same time, in contrast to [40] where PBCF samples with x = 0.4 and 0.6 were found to possess cubic 'simple' perovskite crystal structure, in the present study all the PBCF samples in the range of 0 ≤ x ≤ 0.6 were obtained as double perovskites with tetragonal crystal structure. The main reason behind this discrepancy seems to be the thermal treatment of the samples during their preparation. Indeed, the highest calcination temperature used for the synthesis in [40] was 950 • C, as compared with 1100 • C employed by us. Grimaud et al. [7] also came to the same conclusions. ment with the previous studies [4,7,39,40]. Their XRD patterns were indexed using the P4/mmm space group. The lattice parameters refined by the Rietveld method are given in Table S1. As seen, they are in good agreement with those reported earlier [4,7,39,40]. At the same time, in contrast to [40] where PBCF samples with x=0.4 and 0.6 were found to possess cubic 'simple' perovskite crystal structure, in the present study all the PBCF samples in the range of 0 ≤ x ≤ 0.6 were obtained as double perovskites with tetragonal crystal structure. The main reason behind this discrepancy seems to be the thermal treatment of the samples during their preparation. Indeed, the highest calcination temperature used for the synthesis in [40] was 950 °C, as compared with 1100 °C employed by us. Grimaud et al. [7] also came to the same conclusions. In situ HT XRD was used to study the phase composition and crystal structure of the PBCF samples in the wide range of T and pO 2 employed for ECR measurements. As a result (see Supplementary, Figures S5-S8) tetragonal double perovskite structure was confirmed for all the studied samples in the ranges 550 ≤ T/ • C ≤ 1000 and 10 −3 ≤ pO 2 /atm ≤ 0.21, in full agreement with the previous reports [15,22,40]. However, at temperatures lower than 550 • C and pO 2 ≤ 10 −3 atm the formation of orthorhombic crystal structure with the Pmmm space group was observed. Nevertheless, the range of T and pO 2 corresponding to the tetragonal crystal structure is typical of the operation of oxygen separating semi-permeable membranes [2]. For this reason, all the ECR and nonstoichiometry measurements in this work were carried out under these conditions on the PBCF double perovskite samples belonging to the P4/mmm space group.

Oxygen Content in PBCF and Their Defect Chemistry
The measured oxygen content, 6−δ, in the double perovskites PBCFs is shown as a function of pO 2 and temperature in Figure 3. As seen, the oxygen content varies in the wide range from~5 to~5.8. It changes smoothly as a function of both pO 2 and T, indicating the absence of phase transitions in the studied double perovskites, in accordance with the results of in situ HT XRD discussed above. The oxygen nonstoichiometry grows with increasing temperature and decreasing pO 2 . It also follows from Figure 3 that oxygen content values determined by coulometric titration and thermogravimetry coincide with each other quite well. Such coincidence indicates in favor of the reliability and reversibility of the obtained oxygen nonstoichiometry data.  The defect structure of PBCFs was analyzed, first, using the quasichemical model successfully employed previously for a number of undoped double perovskites RE-BaCo2O6−δ (RE = La, Pr, Nd, Sm, Eu, Gd, Y) [32,41]. It is based on the three quasichemical reactions, which are presented in Table 1 (reactions (1)-(3)), describing the oxygen exchange with surrounding atmosphere, the preferential localization of oxygen vacancies in the rare-earth-element layers and the cobalt disproportionation. However, as the cobalt sublattice of PBCF contains potentially redox-active iron cations, the model should be modified. Two additional processes may be considered, namely, the iron disproportionation: and the electron exchange between the iron and cobalt ions: where Co × , Co ′ , Fe × , Fe • and Fe ′ are Co 3+ , Co 2+ , Fe 3+ , Fe 4+ and Fe 2+ -cations in the The defect structure of PBCFs was analyzed, first, using the quasichemical model successfully employed previously for a number of undoped double perovskites REBaCo 2 O 6−δ (RE = La, Pr, Nd, Sm, Eu, Gd, Y) [32,41]. It is based on the three quasichemical reactions, which are presented in Table 1 (reactions (1)-(3)), describing the oxygen exchange with surrounding atmosphere, the preferential localization of oxygen vacancies in the rare-earthelement layers and the cobalt disproportionation. However, as the cobalt sublattice of PBCF contains potentially redox-active iron cations, the model should be modified. Two additional processes may be considered, namely, the iron disproportionation: Membranes 2022, 12, 1200 7 of 19 and the electron exchange between the iron and cobalt ions: where Co × Co , Co Co , Fe × Co , Fe • Co and Fe Co are Co 3+ , Co 2+ , Fe 3+ , Fe 4+ and Fe 2+ -cations in the Co-sublattice, respectively (please note that hereinafter Kröger-Vink notation is employed for point defects designation and, as in our previous works [32,41], RECoO 3 was used as a reference crystal). Table 1. Results of fitting model 1 to the measured pO 2 -T-δ-datasets of PBCFs.
Taking into account that Fe 3+ disproportionation is characterized by a much larger positive enthalpy value (>100 kJ·mol −1 [42,43]) as compared to the disproportionation of Co 3+ (enthalpy~30-40 kJ·mol −1 [41]), it would be safe to neglect the contribution of Equation (7) to the equilibrium of point defects in the PBCF studied in this work. As for the reaction Equation (8), its equilibrium-due to the higher electronegativity of Co as compared to Fe-is expected to be shifted to Fe × Co and Co Co . However, since the thermodynamics of the reaction Equation (8) is unknown a priori, it is difficult to estimate the extent of this shift. Therefore, the quasichemical reaction described by Equation (8) was included in the defect structure model. The equilibrium constants of these reactions together with the charge balance and mass balance conditions form the set of equations (see, for example, [41]) which was solved with respect to the concentrations of the defect species involved. The model function pO 2 = f (T, δ, ∆H i , ∆S i ) was obtained as a result, where ∆H i and ∆S i are the standard enthalpies and entropies of the quasichemical reactions. Fitting the model function to the pO 2 -T-δ-datasets measured for all the PBCFs yields large (~100 kJ·mol −1 ) enthalpy and small entropy around 0 J·mol −1 ·K −1 for the reaction Equation (8). This results in the very low value of the equilibrium constant K 8 << 1 and, hence, the equilibrium of Equation (8) is almost entirely shifted to Fe × Co and Co Co . Therefore, similarly to the iron disproportionation, Equation (7), the electron exchange between iron and cobalt ions, Equation (8), almost does not influence the defect chemistry of PBCFs and may be neglected.
It directly follows from the above considerations that iron in the cobalt sublattice of PBCF samples studied in this work is predominantly in the oxidation state +3. This conclusion is quite consistent with the results of Mössbauer spectroscopy [7,44,45] and in situ high-resolution electron energy loss spectroscopy [46]. Therefore, for further modelling, the oxidation state of iron was fixed as +3. This assumption simplified the model, since only three quasichemical reactions were retained for further analysis, and, quite predictably, did not worsen the quality of the fit. In fact, the as-obtained model 1 is almost the same as that used previously for the undoped double perovskites REBaCo 2 O 6−δ [32,41]. The only difference consists in the mass balance condition for cobalt. Its total quantity equals: , it is lower than 2. The simplified model 1 was fitted to the pO 2 -T-δ-datasets of all the PBCF. The results are summarized in Table 1, and the fitted model surfaces are shown in Figure S9.
As seen from Figure S9 and Table 1, model 1 describes the measured pO 2 -T-δ-datasets relatively well. The coefficient of determination, R 2 , is sufficiently close to 1. The enthalpy of Co 3+ disproportionation, ∆H 3 , was found to be constant, almost independent of the iron doping level and quite close to that of the undoped double cobaltites REBaCo 2 O 6−δ [41]. The enthalpy of oxygen exchange reaction, ∆H 1 , somewhat grows with the concentration of iron in Co-sublattice. Interestingly, the enthalpy of quasichemical reaction (2) ( Table 1), which describes the preferential localization of the oxygen vacancies in the rare-earth-element layers, shows the most significant variation with the iron content. It changes from a strongly negative value in the undoped PBC to zero in the PBCF6, indicating decreasing energy gain from such localization of oxygen vacancies within the Pr-layers. In this respect, the defect chemistry of the PBCF6 sample (x = 0.6) is very similar to that of LaBaCo 2 O 6−δ [41], i.e., here iron doping has the same effect on the defect structure as the increase in the radius of the A-site cation. Indeed, as discussed above (Section 3.1), PBCF samples with high doping level were occasionally obtained as cubic 'simple' perovskites [7,40], depending on the sample's preparation history. The same is true for LaBaCo 2 O 6−δ , which can also be synthesized as a cubic or a double perovskite [47]. Therefore, model 1 is in good general agreement with various structural, spectroscopic and thermodynamic experimental evidence.
However, observing closely the agreement between the model and the data in Figure S9 and considering the R 2 decreasing with x, as seen in Table 1, reveals the decreasing goodness of fit with increasing concentration of iron. This is especially true for the sample with the largest amount of dopant, PBCF6, as seen in Figure 4. In addition, contrary to the undoped PBC sample, in the iron-containing PBCF the agreement between the oxygen content measured experimentally and that calculated according to model 1 worsens significantly at lower temperatures. This indicates that some additional contribution to the defect chemical model should be identified and included in the defect structure model. in the rare-earth-element layers, shows the most significant variation with the iron content. It changes from a strongly negative value in the undoped PBC to zero in the PBCF6, indicating decreasing energy gain from such localization of oxygen vacancies within the Pr-layers. In this respect, the defect chemistry of the PBCF6 sample (x = 0.6) is very similar to that of LaBaCo2O6−δ [41], i.e., here iron doping has the same effect on the defect structure as the increase in the radius of the A-site cation. Indeed, as discussed above (Section 3.1), PBCF samples with high doping level were occasionally obtained as cubic 'simple' perovskites [7,40], depending on the sample's preparation history. The same is true for LaBaCo2O6−δ, which can also be synthesized as a cubic or a double perovskite [47]. Therefore, model 1 is in good general agreement with various structural, spectroscopic and thermodynamic experimental evidence. However, observing closely the agreement between the model and the data in Figures S9 and considering the R 2 decreasing with x, as seen in Table 1, reveals the decreasing goodness of fit with increasing concentration of iron. This is especially true for the sample with the largest amount of dopant, PBCF6, as seen in Figure 4. In addition, contrary to the undoped PBC sample, in the iron-containing PBCF the agreement between the oxygen content measured experimentally and that calculated according to model 1 worsens significantly at lower temperatures. This indicates that some additional contribution to the defect chemical model should be identified and included in the defect structure model.  There are at least two possibilities which can be discussed in this respect. First, following the authors of [48,49] one can consider the so-called hole-hole interaction, since the contribution of the unknown process to the defect equilibria should be more prominent at lower temperatures and higher oxygen content, where model 1 ceases to describe accurately the nonstoichiometry data. In this T and δ region, the concentration of electron holes also increases. The hole-hole interaction hypothesis is, however, inconsistent with the apparent absence of such an interaction in the undoped PBC, judging by the good agreement between the calculated and measured nonstoichiometry dependences in the whole temperature range studied. The same, in principle, may be argued for any other types of defect-defect interactions.
The second possibility, which, potentially, may account for the observed disagreement between the model and the experiment, is an energetic inhomogeneity of the oxygen sublattice, i.e., the presence of nonequivalent, more or less strongly bonded, oxygen There are at least two possibilities which can be discussed in this respect. First, following the authors of [48,49] one can consider the so-called hole-hole interaction, since the contribution of the unknown process to the defect equilibria should be more prominent at lower temperatures and higher oxygen content, where model 1 ceases to describe accurately the nonstoichiometry data. In this T and δ region, the concentration of electron holes also increases. The hole-hole interaction hypothesis is, however, inconsistent with the apparent absence of such an interaction in the undoped PBC, judging by the good agreement between the calculated and measured nonstoichiometry dependences in the whole temperature range studied. The same, in principle, may be argued for any other types of defect-defect interactions.
The second possibility, which, potentially, may account for the observed disagreement between the model and the experiment, is an energetic inhomogeneity of the oxygen sublattice, i.e., the presence of nonequivalent, more or less strongly bonded, oxygen atoms. The origin of this nonequivalence may be related to the different coordination environments. Indeed, in the undoped PBC all the oxygen sites are coordinated by two Co ions, whereas in the Fe-doped samples three different coordination environments are possible: (1) Co-O-Co, (2) Fe-O-Fe, and (3) Fe-O-Co. This suggests that, in principle, three distinct oxidation enthalpies may be identified. However, to make the corresponding defect chemical model simpler and avoid introducing too many fitting variables, we chose to consider two different oxidation enthalpies as the simplest possible scenario. The first one corresponds to the oxygen site coordinated by iron only, Fe-O-Fe, and the second one is characteristic of all other oxygen sites, both those with only cobalt and mixed, iron-cobalt, coordination. Therefore, to modify the defect structure model, two nonequivalent oxygen positions were introduced. Their concentrations were calculated assuming the statistical distribution of iron and cobalt in the B-sublattice of the double perovskite. Then, the following quasichemical reaction were proposed: are the oxygen sites and the vacancies coordinated by either cobalt only or cobalt and iron, and the oxygen sites and the vacancies coordinated by iron only, respectively.
Since reactions (12) and (13) describe the preferential arrangement of oxygen ions in the rare-earth layers, in principle, it is possible to express them using the real crystallographic positions of various oxygen sites, as it was done in [50] for the defect structure model of Sr 3 Fe 2−x Mo x O 7−δ . However, since the ratio of rare-earth sites to oxygen sites in the rare-earth layers of PBCF is exactly 1:1, we found that it is easier to represent the vacancy ordering by the pseudocluster formation.
The equilibrium constants of reactions (9)-(13) and the conditions of mass and charge balance result in the set of equations: where α denotes the fraction of oxygen sites coordinated by iron only. Quasichemical reactions (12) and (13) were assumed, for the sake of simplicity, to possess the same thermodynamic parameters (∆S 4 = ∆S 5 , ∆H 4 = ∆H 5 ). The set of equations (14), despite its apparent complexity, can be solved symbolically. Although the resulting equations are rather long and for this reason are not given here, the concentrations of all the defect species were obtained as along with the model 2 function pO 2 = f (T, δ, ∆H i , ∆S i ) which was fitted to the experimental pO 2 -T-δ-datasets of PBCF. The results are summarized in Figures 5-8 and in Table 2.
where α denotes the fraction of oxygen sites coordinated by iron only. Quasichemical reactions (12) and (13) were assumed, for the sake of simplicity, to possess the same thermodynamic parameters (Δ 4 ○ = Δ 5 ○ , Δ 4 ○ = Δ 5 ○ ). The set of equations (14), despite its apparent complexity, can be solved symbolically. Although the resulting equations are rather long and for this reason are not given here, the concentrations of all the defect species were obtained as along with the model 2 function pO2=f′(T, δ, Δ ○ , Δ ○ ) which was fitted to the experimental pO2-T-δ-datasets of PBCF. The results are summarized in Figures 5-8 and in Table 2.      Table 2) vs. the concentration of dopant in PBCF. * Numbers following '±' represent the expanded uncertainty (~95% confidence level); ** assumed to equal 0 during the fitting procedure, for details see [41]; *** this is a combination of reactions  Table 2) vs. the concentration of dopant in PBCF. Tables 1 and 2, respectively) were found to be almost the same within the two models considered. Hence, the qualitative conclusions about the cobalt disproportionation and vacancies' interaction with the rare-earth ions are the same irrespective of the model employed. Further, the enthalpy of oxygen exchange reaction (1) in model 2 (see Table 2) which involves O × O1 -sites, i.e., those that do not have two iron ions as their nearest neighbors, is also similar to the one estimated within the model 1 (see Table 1, reaction (1)) both in terms of its values and its tendency towards the gradual increase with the amount of the dopant, as seen in Figure 8. As for the oxygen exchange involving O × O2 -sites (having only iron as their nearest neighbors)-reaction (6) in Table 2-the value of its enthalpy (see Table 2) is the same for all the PBCF, being almost two times lower than that of reaction (1) (see Table 2) and somewhat higher than the one found for PrBaFe 2 O 5+w by Karen [51]. This is quite consistent with dominating oxidation state +3 of Fe in PBCFs as discussed above.  * Numbers following '±' represent the expanded uncertainty (~95% confidence level); ** assumed to equal 0 during the fitting procedure, for details see [41]; *** this is a combination of reactions (1), (2), (4) and (5), and its thermodynamic parameters were calculated after the fitting from the ∆H i , ∆S i of those reactions; **** calculated using binomial distribution assuming statistical distribution of Co and Fe in the B-sublattice.

A comparison of Tables 1 and 2 shows that the thermodynamic parameters of cobalt disproportionation (equations (3) within the tables) and localization of oxygen vacancies in Pr-layers (Equations (2) and (4)-(5) in
To sum up, model 2, which is a gradual improvement over model 1, is consistent with the available experimental structural, spectroscopic and thermodynamic data, and will be employed further for the analysis of oxygen transport in PBCF double perovskites.

Oxygen Diffusion and Surface Exchange
The chemical surface exchange coefficient, k chem , and the chemical diffusion coefficient, D chem , obtained as a result of ECR measurements for PBCFs double perovskites are summarized in Figures S10-S13 (see Supplementary). As seen, both k chem and D chem increase with temperature. D chem of all the studied PBCF is almost independent of pO 2 , whereas k chem of PrBaCo 2 O 6−δ is roughly proportional to p 0.5 O 2 at low pO 2 and tends to saturate at higher pO 2 . This transitory behavior is more pronounced at high temperatures. For the Fe-doped samples, k chem shows ∼ p 0.5 O 2 dependence within the whole range of studied oxygen partial pressures. D chem and k chem were recalculated to the oxygen self-diffusion coefficient, D O , and the surface exchange coefficient, k, as described in the Experimental. The as-calculated values are presented in Figures 9-12 as a function of T and pO 2 . found for PrBaFe2O5+w by Karen [51]. This is quite consistent with dominating oxidation state +3 of Fe in PBCFs as discussed above.
To sum up, model 2, which is a gradual improvement over model 1, is consistent with the available experimental structural, spectroscopic and thermodynamic data, and will be employed further for the analysis of oxygen transport in PBCF double perovskites.

Oxygen Diffusion and Surface Exchange
The chemical surface exchange coefficient, kchem, and the chemical diffusion coefficient, Dchem, obtained as a result of ECR measurements for PBCFs double perovskites are summarized in Figures S10-S13 (see Supplementary). As seen, both kchem and Dchem increase with temperature. Dchem of all the studied PBCF is almost independent of pO2, whereas kchem of PrBaCo2O6−δ is roughly proportional to O 2 0.5 at low pO2 and tends to saturate at higher pO2. This transitory behavior is more pronounced at high temperatures.   As seen, the surface exchange coefficient is thermally activated and shows pO2 dependence similar to kchem, i.e., for the undoped cobaltite k~O 2 0.5 at low pO2 and saturates at high pO2, whereas for the Fe-doped samples ~O 2 0.5 in the whole range of pO2. Similar, proportional to O 2 0.5 , dependence of k was reported for PBC by Yoo et al. [9], who also found the change in the rate determining step of the oxygen exchange from the oxygen dissociative adsorption to the incorporation of oxygen adatoms upon pO2 increase from 0.1 to 0.95 atm. Later, similar conclusions were reported for REBaCo2O6-δ (RE = Pr, Sm, As seen, the surface exchange coefficient is thermally activated and shows pO2 dependence similar to kchem, i.e., for the undoped cobaltite k~O 2 0.5 at low pO2 and saturates at high pO2, whereas for the Fe-doped samples ~O 2 0.5 in the whole range of pO2. Similar, proportional to O 2 0.5 , dependence of k was reported for PBC by Yoo et al. [9], who also found the change in the rate determining step of the oxygen exchange from the oxygen dissociative adsorption to the incorporation of oxygen adatoms upon pO2 increase from As seen, the surface exchange coefficient is thermally activated and shows pO 2 dependence similar to k chem , i.e., for the undoped cobaltite k~p 0.5 O 2 at low pO 2 and saturates at high pO 2 , whereas for the Fe-doped samples k~p 0.5 O 2 in the whole range of pO 2 . Similar, proportional to p 0.5 O 2 , dependence of k was reported for PBC by Yoo et al. [9], who also found the change in the rate determining step of the oxygen exchange from the oxygen dissociative adsorption to the incorporation of oxygen adatoms upon pO 2 increase from 0.1 to 0.95 atm. Later, similar conclusions were reported for REBaCo 2 O 6-δ (RE = Pr, Sm, Gd) [10]. This explanation is consistent with the transitory behavior of k observed in the present work. Indeed, Yoo et al. [9] found the rate of incorporation of oxygen adatoms to be almost independent of pO 2 , contrary to the oxygen dissociative adsorption which is roughly proportional to p 0.8 O 2 . In turn, the oxygen self-diffusion coefficient in PBCF, as shown in Figures 9-12, depends mainly on temperature and is practically insensitive to variations of pO 2 . Similar observations were reported in [10] for the undoped double perovskite cobaltites REBaCo 2 O 6−δ (RE = Pr, Sm, Gd). The fact that D O in double perovskites was found to be independent of pO 2 can be explained from the viewpoint of the oxygen diffusion mechanism in these oxides. As revealed by a number of studies [6,[14][15][16][17][18][19][20][21][22] using different techniques, oxygen ions diffuse in REBaCo 2 O 6−δ along the CoO 2− planes through the oxygen vacancies coming from the rare-earth-element planes. These planes, therefore, serve only as a source-sink for the oxygen vacancies and do not participate in the oxygen transfer directly. As a consequence, the amount of oxygen vacancies in the Co(Fe)O 2− planes should limit the diffusion of oxygen. In the defect chemistry modelling discussed above, V •• O1 and V •• O2 represent such diffusion-active vacancies in PBCF. The calculations according to model 2 show that, for example, at 1000 • C the pO 2 increase from 10 −3 to 0.21 atm, i.e., by almost three orders of magnitude, leads to the increase in the total concentration of such vacancies in the undoped PBC sample by 2.86 times only, or by~0.46 in the logarithmic scale. In the iron-doped samples, this increase is even lower: for example, in PBCF6 under similar conditions the total concentration of the oxygen vacancies participating in the diffusion increases by 1.433 times, or by~0.16 in the logarithmic scale. Such changes are well within the uncertainties of the diffusion coefficients determined by fitting the conductivity relaxation curves, which causes the observed D O values to appear independent of pO 2 .
It is also of interest to compare the temperature dependencies of the oxygen selfdiffusion coefficient and the surface exchange coefficient for the undoped PBC sample with the data available in the literature. Such a comparison is shown in Figure 13. Please note that since D O did not show the pO 2 dependence, its values, for the sake of comparison, were averaged at each T over the whole pO 2 range studied. For k, in turn, either the values at pO 2 = 0.21 atm or those averaged in the range of 0.0063-0.21 atm were plotted in Figure 13, since these pO 2 correspond to the oxygen pressures most commonly employed in the literature for isotope exchange or conductivity relaxation, respectively. A very good agreement between various datasets presented in Figure 13 is obvious, irrespective of the particular experimental technique employed. This is an additional confirmation of the validity of the results obtained in this work.
A comparison of Figure 9a, Figure 10a, Figure 11a, Figure 12a shows that iron doping almost does not influence the value of k, since all the PBCFs studied were shown to possess comparable surface exchange coefficients within the investigated ranges of T and pO 2 . Grimaud et al. [7] also showed that samples of PBC and PrBaCo 1.5 Fe 0.5 O 6−δ (PBCF5) possess almost the same surface exchange coefficient. At the same time, as clearly seen in Figure 14, the activation energy of oxygen self-diffusion decreases with x. As a result, as seen in Figure 14a, all the iron-doped PBCF samples possess D O either comparable with or higher than the D O of PBC in the range of intermediate temperatures (T < 600-700 • C). This is especially pronounced for the PBCF6 sample, which demonstrates the lowest activation energy of oxygen self-diffusion and the largest D O already at T < 900 • C and, for example, at 600 • C its D O is more than three times that of undoped PBC.
were averaged at each T over the whole pO2 range studied. For k, in turn, either the values at pO2 = 0.21 atm or those averaged in the range of 0.0063-0.21 atm were plotted in Figure  13, since these pO2 correspond to the oxygen pressures most commonly employed in the literature for isotope exchange or conductivity relaxation, respectively. A very good agreement between various datasets presented in Figure 13 is obvious, irrespective of the particular experimental technique employed. This is an additional confirmation of the validity of the results obtained in this work.
A comparison of Figures 9a-12a shows that iron doping almost does not influence the value of k, since all the PBCFs studied were shown to possess comparable surface exchange coefficients within the investigated ranges of T and pO2. Grimaud et al. [7] also showed that samples of PBC and PrBaCo1.5Fe0.5O6−δ (PBCF5) possess almost the same surface exchange coefficient. At the same time, as clearly seen in Figure 14, the activation energy of oxygen self-diffusion decreases with x. As a result, as seen in Figure 14a, all the iron-doped PBCF samples possess DO either comparable with or higher than the DO of PBC in the range of intermediate temperatures (T < 600-700 °C). This is especially pronounced for the PBCF6 sample, which demonstrates the lowest activation energy of oxygen self-diffusion and the largest DO already at T < 900 °C and, for example, at 600 °C its DO is more than three times that of undoped PBC.  Table 2).
These observations are fully in line with both the data on oxygen permeability through PBCF membranes [4] and those on polarization resistance of PBCF cathodes [40]. In both cases, it was shown that in the intermediate temperature range the iron-doped samples with x = 0.4-0.6 have either similar or better performances compared to the undoped cobaltite PBC. However, Grimaud et al. [7] came to the opposite conclusion that polarization resistance increases with iron content. It seems the discrepancy between the works [7] and [40] may be related to the electrolyte employed for the study. Zou et al. [40] used substituted ceria, whereas Grimaud et al. [7]-proton-conducting BaCe0.9Y0.1O3-δ. The latter study reported negative influence of water vapor on the polarization resistance of iron-doped samples contrary to the undoped cobaltite. Since with a proton-conducting electrolyte it is almost impossible to avoid the water vapor influence, this may be the source of the disagreement between the results of [7] and [40]. However, this is just one possibility among many, since electrode performance depends on a lot of factors, including the morphology. Therefore, it is always difficult to compare electrochemical characteristics measured in different works.  Table 2).
These observations are fully in line with both the data on oxygen permeability through PBCF membranes [4] and those on polarization resistance of PBCF cathodes [40]. In both cases, it was shown that in the intermediate temperature range the iron-doped samples with x = 0.4-0.6 have either similar or better performances compared to the undoped cobaltite PBC. However, Grimaud et al. [7] came to the opposite conclusion that polarization resistance increases with iron content. It seems the discrepancy between the works [7] and [40] may be related to the electrolyte employed for the study. Zou et al. [40] used substituted ceria, whereas Grimaud et al. [7]-proton-conducting BaCe 0.9 Y 0.1 O 3-δ . The latter study reported negative influence of water vapor on the polarization resistance of iron-doped samples contrary to the undoped cobaltite. Since with a proton-conducting electrolyte it is almost impossible to avoid the water vapor influence, this may be the source of the disagreement between the results of [7] and [40]. However, this is just one possibility among many, since electrode performance depends on a lot of factors, including the morphology. Therefore, it is always difficult to compare electrochemical characteristics measured in different works.
At the same time, Grimaud et al. [7] also observed decreased activation energy of oxygen self-diffusion in PBCF5 as compared to PBC, which is in agreement with our results (see Figure 14). The origin of the fact that Fe doping causes the decrease in the oxygen self-diffusion activation energy in PBCF can be understood based on the diffusion mechanism discussed above and results of the defect structure modelling. Indeed, since oxygen transfer in the CoO 2− planes requires the presence of oxygen vacancies (V •• O1 and V •• O2 in the defect chemical model 2) the source of which are the PrO 1−δ -layers where vacancies are trapped in the form of quasi-complexes Pr × the dissociation enthalpy of these complexes must contribute to the activation energy of self-diffusion in double perovskites. Therefore, the higher (more positive) the enthalpy of vacancy trapping in Pr-layers (∆H 4 in the model 2) the lower should be the activation energy of oxygen self-diffusion. This type of inverse correlation between E A and ∆H 4 can be observed in Figure 14b and in Figure S14. Therefore, activation energy may be represented by the sum of, at least, two contributions: the formation enthalpy of oxygen vacancies in CoO 2− layers, equal to −∆H 4 (see Table 2), and the migration enthalpy of these vacancies in the same layers, ∆H m : Hence, the ∆H m term can be estimated as the sum E A + ∆H 4 . The as-estimated ∆H m is shown in Figure 14b as a function of iron content x in PBCF. As seen, ∆H m gradually grows from about 0.53 eV for undoped PBC to 0.66 eV for PBCF6 sample, reflecting the increasing difficulty of the vacancy migration with the increase in the dopant concentration. Therefore, the main positive implication of the Fe doping for oxygen diffusion in double perovskite cobaltites is that it seems to facilitate the formation of the mobile oxygen vacancies (i.e., those in the CoO 2− layers). However, this happens at the cost of somewhat increase in their migration enthalpy.

Conclusions
Oxygen content in the Fe-doped double perovskites PBCF was measured as a function of T and pO 2 . The as-obtained pO 2 -T-δ-datasets were successfully described using the defect chemical model based on the reactions of oxygen exchange with the surrounding atmosphere, Co 3+ disproportionation and vacancies' preferential localization in the Prlayers. The different coordination environment of oxygen in the Fe-doped samples was also taken into account in the model, since this was shown to be essential for the accurate description of the behavior of oxygen content in the PBCF, especially at lower temperatures. Fe-doping was also shown to significantly affect the enthalpy gain of oxygen vacancies' localization in the Pr-layers, decreasing the energetic cost of the diffusion-active oxygen vacancy formation in the CoO 2− layers. This leads to the decreasing activation energy of oxygen self-diffusion with increasing concentration of iron in PBCF, as verified by the results of electrical conductivity relaxation measurements. The linear correlation of the activation energy of oxygen self-diffusion with the enthalpy of formation of oxygen vacancies in CoO 2− layers also provides an independent confirmation of the oxygen selfdiffusion mechanism in the layered double perovskite cobaltites, studied earlier by neutron diffraction [15,19,22] and computations [14,[16][17][18]20,21].
We found that iron doping almost does not affect either the mechanism of surface exchange or the value of the surface exchange coefficient. The results of the present study are quite consistent with the rate-determining step of the oxygen exchange in the Fedoped samples being the oxygen dissociative adsorption in the whole pO 2 and T ranges investigated. At the same time, in the undoped PBC cobaltite at high temperatures, the rate-determining step changes from the O 2 dissociative adsorption to the incorporation of oxygen adatoms upon pO 2 increase.
In addition to retaining the already high values of the surface exchange coefficient, typical of the undoped PBC oxide, iron doping decreases the activation energy of oxygen self-diffusion and promotes the oxygen self-diffusion in PBCF in the intermediate temperature range. These findings indicate that iron doping enhances the oxygen transport properties of double perovskite membrane materials. Furthermore, the previous studies demonstrated that iron doping increases the relative thermodynamic stability of double perovskites [31], which is also beneficial for their practical applications.
The decrease in the activation energy of oxygen self-diffusion with iron concentration in PBCF, demonstrated in this work, and its interpretation through the defect structure analysis, possess twofold significance. On the one hand, they provide an important link between the particularities of the defect chemistry of double perovskites and their transport properties. On the other hand, as shown in this work, the defect chemistry guidance allows one to optimize the properties of double perovskite materials through the targeted variation of their defect energetics. This can be regarded as one of the general, not limited to double perovskites, methods for the intentional modification and development of novel materials for various practical applications, including membranes and energy conversion devices.

Informed Consent Statement: Not applicable.
Data Availability Statement: The reported data are available by a reasonable request from the corresponding authors.