Oxygen exchange at gas/oxide interfaces: how the apparent activation energy of the surface exchange coefficient depends on the kinetic regime

Abstract

In the dedicated literature the oxygen surface exchange coefficient K_O and the equilibrium oxygen exchange rate [Fraktur R]⁰_O are considered to be directly proportional to each other regardless of the experimental circumstances. Recent experimental observations, however, contradict the consequences of this assumption. Most surprising is the finding that the apparent activation energy of K_O depends dramatically on the kinetic regime in which it has been determined, i.e. surface exchange controlled vs. mixed or diffusion controlled. This work demonstrates how the diffusion boundary condition at the gas/solid interface inevitably entails a correlation between the oxygen surface exchange coefficient K_O and the oxygen self-diffusion coefficient D_O in the bulk (“on top” of the correlation between K_O and [Fraktur R]⁰_O for the pure surface exchange regime). The model can thus quantitatively explain the range of apparent activation energies measured in the different regimes: in the surface exchange regime the apparent activation energy only contains the contribution of the equilibrium exchange rate, whereas in the mixed or in the diffusion controlled regime the contribution of the oxygen self-diffusivity has also to be taken into account, which may yield significantly higher apparent activation energies and simultaneously quantifies the correlation K_O ∝ D_O^1/2 observed for a large number of oxides in the mixed or diffusion controlled regime, respectively.

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As early as 1996 Kilner et al.¹ showed that there is, for a large number of different materials, a quite obvious correlation K_O ∝ D_O^1/2 between the oxygen surface exchange coefficient K_O and the oxygen diffusion coefficient D_O (over almost 10 (!) orders of magnitude for D_O) for a number of perovskite compositions (with predominant electronic conductivity). Despite this empirical evidence it is assumed until to date,^2–4 however, that K_O and D_O are independent parameters, each of which reflects an independent microscopic mechanism. A major reason for this picture can be identified in the methodology of standard oxygen exchange experiments:⁵ Here the oxygen flux j^K_O through a surface element subject to a concentration difference Δc_O is expressed as follows

j^K_O(0) = −K_OΔc_O(0) (1)

The phenomenological eqn (1) defines K_O without recurring to any microscopic mechanism. As eqn (1) is only valid for surface reactions at x = 0 it seemed plausible that K_O depends exclusively on the defect chemistry of the surface. In 1998 Maier⁶ proposed a relation between K_O and the equilibrium oxygen exchange rate [Fraktur R]⁰_O at the gas/solid interface (s. Maier,⁶ p. 222, eqn (149))

where c_O(0) is the oxygen equilibrium concentration near the surface, w_O(0) is the thermodynamic factor near the surface, R and T have the usual meaning, and μ_O is the chemical potential of oxygen. [Fraktur R]⁰_O is the oxygen exchange rate of the rate determining step in a consecutive reaction sequence.^7,8 Without violating Maier's universally valid original approach, in the literature^4,9–14 also an effective oxygen exchange rate (, n = number of consecutive reactions) is used to describe so-called consecutive “heterogeneous” reactions. The quantitative measure of the parameter Δx (with the dimension of a length) having been left undefined, Maier's suggestion essentially means that K_O is proportional to [Fraktur R]⁰_O.¹⁵

For the oxygen isotope exchange the thermodynamic factor is unity and eqn (2) yields K̃_O = w(0)K_O*. (Throughout the text an asterisk characterizes a parameter determined in an isotope exchange experiment, whereas a tilde refers to a parameter determined in a chemical exchange experiment.) From D̃_O = w(0)D_O = w(0)D_O*/f_O, where D_O is the self-diffusion coefficient and D_O* is the tracer diffusion coefficient of oxygen, respectively, follows with the correlation factor f_O ≃ 1 a simple theoretical correlation.¹⁵

In 2006 De Souza⁷ explicitly pointed out that there is a generally observed discrepancy between experimentally determined K_O/D_O ratios and the theoretically expected ratios according to eqn (3). Instead of fulfilling eqn (3) the experimental data yield K_O*/D_O* > K̃_O/D̃_O. This was a first significant experimental indication that eqn (2) might not fully take into account all aspects of the problem.

Armstrong et al.^12,13 used in 2011 and 2013 a novel experimental approach, called isothermal isotope exchange (IIE), to extract accurate K_O* values in a surface exchange controlled regime. This technique relies on the exchange of isotopically labelled oxygen for lattice oxygen in a similar way to isotope exchange depth profiling (IEDP) with secondary ion mass spectrometry (SIMS), but is conducted in situ on powder samples (particle size 200–300 nm). In this way the exchange kinetics can be expected to be clearly in the surface exchange regime. Very unexpectedly Armstrong et al.^12,13 found apparent activation energies of K_O* with negative (!) or very small positive values for different lanthanum based MIEC perovskite oxides like LC, LF, LSM, LSF, LSC or LSCF, but also for the standard oxygen ion conductors YSZ and GDC, which have fluorite structure. The reported range of apparent activation energies of the oxygen tracer surface exchange coefficient is −97 kJ mol⁻¹ to +12 kJ mol⁻¹ and is in strong contrast to the reported range from +60 kJ mol⁻¹ to +333 kJ mol⁻¹ measured by conventional methods generally using macroscopic sample geometries, i.e. IEDP/SIMS and electrical conductivity relaxation (ECR) (s. Armstrong et al.,¹³ Table 5), but also measured by other powder-based methods like pulsed isotope exchange (PIE)^9–11 and long-term gas phase exchange/analysis GPE/GPA.^4,14,16 These experimental results suggest that the measured oxygen surface exchange coefficient K_O, if measured in the mixed regime and in the diffusion controlled regime, may depend not only on the equilibrium oxygen exchange rate [Fraktur R]⁰_O at the gas/solid interface but also on the oxygen diffusion coefficient D_O in the bulk. Such a behaviour would clearly contradict eqn (2) which postulates that K_O depends on [Fraktur R]⁰_O only in all kinetic regimes.

In 2015 Knoblauch et al.¹⁷ performed thermogravimetric relaxation experiments on sintered CeO₂ pellets (1 mm thickness, 93% of theoretical density) with a grain size of about 20 μm. Because of the strongly enhanced chemical diffusivity¹⁸ during the oxygen reduction step of nominally undoped CeO₂ Knoblauch et al.¹⁷ concluded that their reduction experiments had been performed in the surface exchange regime. They also reported an apparent negative activation energy of −64 kJ mol⁻¹ for K̃_O. That is, the work of Knoblauch et al.¹⁷ showed that the apparent activation energy of the chemical oxygen surface exchange coefficient K̃_O can also have negative values if it is measured in the surface exchange controlled regime.

The experimental contradictions to the general validity (i.e. in all kinetic regimes) of eqn (2) necessitate a rediscussion of the meaning of the parameter K_O. In order to facilitate a fundamental theoretical discussion the experimental situation of the oxygen exchange experiments is idealized as far as possible. In analogy to Maier⁶ jump relaxation effects, space charge effects, structural changes and trapping effects are considered secondary. Likewise the mathematical treatment is simplified by considering a homogeneous semi-infinite solid oxide (s. Fig. 1). In this case one has to consider only one gas/oxide interface at x = 0 where defect chemical reactions at the interface are quantified by the equilibrium oxygen exchange rate [Fraktur R]⁰_O. The diffusion process of oxygen ions in the solid oxide is quantified by the oxygen self-diffusion coefficient D_O. The oxygen partial pressure in the gas phase is given by p_O2. That is, the considered oxide is phenomenologically described by two independent material parameters

[Fraktur R]⁰_O = [Fraktur R]⁰_O(T,p_O2), D_O = D_O(T,p_O2) (4)

which are functions of the temperature T and the oxygen partial pressure p_O2, which control (together with unintentionally introduced impurities and/or deliberately added dopants) the concentration of oxygen vacancies or interstitials (and of the other ionic and electronic defects). The oxygen surface exchange coefficient K_O, however, is a phenomenological parameter which, a priori, is not yet related to a specific microscopic process viaeqn (1). In view of the experimental findings outlined above it can be tentatively suggested that, at least for IEDP/SIMS and ECR experiments, but most probably also for PIE^9–11 and GPE/GPA^4,14,16 experiments, K_O should depend on the two independent parameters D_O and [Fraktur R]⁰_O as follows

K_O = K_O(D_O, [Fraktur R]⁰_O) (5)

Fig. 1 Schematic representation of a gas/oxide interface of a homogeneous semi-infinite solid oxide. D is the self-diffusion coefficient of oxygen ions (D_O) in the solid oxide, p is the oxygen partial gas pressure (p_O2) in the oxygen gas phase and [Fraktur R] is the equilibrium oxygen exchange rate ([Fraktur R]⁰_O) at the interface. At t = 0 a small step-like change δμ(0) of the chemical potential of oxygen (δμ_O(x = 0)) is experimentally induced at the gas/oxide interface at x = 0.

In order to derive the hypothetical function (5) from fundamental principal relations it is appropriate to adopt Maier's⁶ notation and his very general procedure: the oxygen flux j^K_O(0) through the surface (at x = 0) which is induced by a small change of the chemical potential of oxygen δμ_O(0) becomes⁶

where Λ_O is the conductance and ∂μ_O/∂c_O the reciprocal “chemical” capacity. It is important to keep in mind that eqn (6) is an approximation near chemical equilibrium. It is very instructive to recall Maier's comment on eqn (6): “It is worth noting that these relationships [eqn (6)] are quite general in that they do not preselect a certain mechanism.”⁶ This statement implicitly comprises the fact that the parameter K_O principally can be the result of a combination of several mechanisms.

The diffusive oxygen flux¹⁹j^D_O(0) in the near-surface region of thickness δx(0), which is driven by the gradient of the chemical potential of oxygen ∂μ_O/∂x|_x=0, becomes

where D_O is the oxygen self-diffusion coefficient and c_O(0) is the oxygen concentration (at the surface).

The first step in the derivation of the hypothetical function (5) is to study the correlation between K_O and D_O which principally results from the boundary condition at the gas/solid interface

j^D_O(0) = j^K_O(0) (8)

which connects surface exchange and bulk transport and which therefore has to be respected when solving the respective diffusion equation.²⁰ With eqn (6) and (7) the boundary condition eqn (8) yields This means that for a moderate value of δμ_O(0), i.e. a near-equilibrium situation at the surface, the (standard) boundary condition eqn (8) yields the relation D_Oc_O(0)/RT = Λ_Oδx(0). With the thermodynamic factor w_O(0) (s. eqn (2)) and K_O = Λ_O∂μ_O/∂c_O|_x=0 (s. eqn (6)) one finally obtains the following general correlation between K_O and D_O which is valid in the near-equilibrium situation

δx(0)·K_O = w_O(0)·D_O (10)

In the second step the meaning of the parameter δx(0) will be clarified. For an isotope exchange experiment w_O(0) = 1 is valid and eqn (10) simply becomes δx*(0) = D_O*/K_O*, which is equivalent to the so-called characteristic length l⁰_c used in the literature to roughly estimate in which kinetic regime a given oxygen exchange experiment is performed. Adopting this notation the subsequent discussion will be facilitated for the reader as eqn (10) now becomes

Here the upper case index 0 means that l⁰_c refers to a near-surface layer (at x = 0). Until now eqn (11) was only a formal mathematical rule to calculate l⁰_c and results as such directly from the boundary condition eqn (8). The physical meaning of l⁰_c will become evident if one defines with eqn (11) a characteristic time t⁰_c (valid likewise for the near-surface region at x = 0) as follows In Appendix A it is demonstrated that the characteristic time t⁰_c physically corresponds to the relaxation time of a (sufficiently small) change in the oxygen potential δμ(0) at the surface. Eqn (12) leads to the following two relations The characteristic length l⁰_c is thus proportional to the diffusion length the oxygen ions cover in the near-surface region during the characteristic time t⁰_c. The term K_O/Δx (as formulated by Maier⁶) in eqn (2) can be considered equivalent to the inverse characteristic time t⁰_c. This relation suggests to postulate Combining eqn (13) and (14) yields the missing correlation anticipated as “the hypothetical function” in eqn (5) It should be recalled that eqn (15) is a consequence of a boundary condition, and hence, is valid in all experimental situations where the considered boundary condition (eqn (8)) is valid. The upper case index h → ∞ refers to the fact that this boundary condition is often applicable to plate-like samples of large thickness h, that is, if h ≫ l⁰_c.

For an oxygen isotope exchange experiment (with w_O(0) = 1 and D_O* = f_O·D_O) eqn (15) becomes (dropping the index h → ∞ in order to simplify the notation in eqn (16)–(19))

Here (K_O*,D_O*) are data couples determined via an isotope exchange experiment. In a chemical oxygen exchange experiment (with w(0) ≫ 1^21,22 and D̃_O = w_O(0)·D_O) the data couples (K̃_O,D̃_O) are determined. In this case eqn (15) yieldsthat is, one has K̃_O = (w_O(0)/f_O)^1/2·K_O* in contrast to D̃_O = (w_O(0)/f_O)·D_O*. Comparing K_O/D_O ratios for the isotope exchange and for the chemical exchange experiment, respectively, one obtains from eqn (16) and (17) the following theoretical relationsBecause of f_O ≈ 1 and w_O(0) ≫ 1^21,22 one immediately obtains K_O*/D_O* > K̃_O/D̃_O, which is in agreement with a large number of experimental results.⁷ Thus eqn (18) supports the postulate expressed in eqn (14).

For more complicated sample geometries and correspondingly more complicated boundary conditions a more complicated correlation than eqn (15) is to be expected. The formal calculation of a characteristic length l⁰_c by eqn (11) is mathematically always possible, but the result would mask the physical meaning of l⁰_c which is explicitly expressed in eqn (13) only. Likewise the characteristic time t⁰_c has been (almost completely) neglected in the materials science literature, which focuses practically exclusively on K_O. It is, however, principally preferable for the discussion of experimental studies to consider the two characteristic parameters t⁰_c and l⁰_c compiled together with K_O in eqn (19)

where h is the thickness of a plate-like sample. The set of eqn (19) shows that the condition h ≫ l⁰_c is realised if the oxygen self-diffusion D_O is low and/or [Fraktur R]⁰_O is high and/or w(0) is small. This means that oxygen exchange occurs practically only in a near-surface region of characteristic thickness l⁰_c during the characteristic time t⁰_c. Consequently the oxygen exchange of the whole sample is hindered due to the slow oxygen self-diffusion in the bulk. This situation corresponds to the so-called diffusion controlled regime.

The very specific condition h ≈ l⁰_c means that within the characteristic time t⁰_c practically all the oxygen of the sample is exchanged. This scenario is called mixed regime as the oxygen exchange rate at the surface ([Fraktur R]⁰_O) and the self-diffusion coefficient in the bulk (D_O) hinder the overall oxygen exchange simultaneously. Note that the condition h ≈ l⁰_c is more easily reached during chemical exchange (with w(0) ≫ 1) than during isotope exchange (with w(0) = 1) because during chemical exchange oxygen self-diffusion is enhanced by the thermodynamic factor, D̃_O = w_O(0)D_O.

The condition h ≪ l⁰_c means for a given value of h that the oxygen exchange at the sample surface is sufficiently slow and/or oxygen bulk transport is sufficiently rapid. In this case the oxygen exchange of the whole sample is entirely dominated by the (apparent equilibrium) oxygen exchange rate [Fraktur R]⁰_O at the surface, and this corresponds to the so-called surface exchange regime. For a thin plate-like sample with thickness h and a sphere with radius r the solution of the diffusion equation for the oxygen exchange kinetics in the surface exchange regime can be approximated by

where τ_eq is the time constant to reach chemical equilibrium.¹⁷ For a plane sheet one gets τ_eq = (h/2)/K_O and for a sphere τ_eq = (r/3)/K_O.¹⁷ Recalling that K_O is a consequence of the boundary condition one concludes Comparing K^h→∞_O for a thick plane sheet in eqn (15) with K^h,r→0_O for an extremely thin plane sheet or for powder samples, respectively, shows that, in the latter case, the oxygen exchange coefficient becomes independent of D_O. This means that the oxygen exchange rate [Fraktur R]⁰_O can be directly determined from the time constant τ_eq in eqn (20), e.g. in an isothermal isotope exchange (IIE) experiment¹³ or in a thermogravimetric relaxation experiment¹⁷ carried out in the surface exchange regime. Further, eqn (21) suggests a direct experimental check of the postulate in eqn (14): if the postulate is valid the time constant τ_eq should not depend on the surface-to-volume ratio of the sample in the surface exchange regime.

The arguments displayed above offer a clue to explain the differences in the reported range of apparent activation energies (s. Armstrong et al.,¹³ Table 5). Armstrong et al.¹³ determined K_O values via IIE clearly in the surface exchange regime (spheres: 200–300 nm, w_O(0) = 1) according to eqn (21)

Other studies (s. references compiled in Table 5 of Armstrong et al.¹³) which determined K_O values via IEDP/SIMS or ECR, respectively, as well as PIE-based^9–11 and GPE/GPA-based^4,14,16 investigations often remained in the diffusion controlled regime (h ≫ l⁰_c) and therefore unintentionally tended to determine a parameter corresponding to the following expression for the surface exchange coefficient The totally different temperature dependencies which are expected (activation energy for [Fraktur R]⁰_O only in eqn (22) and the mean value of the activation energy for [Fraktur R]⁰_Oand for D_O in eqn (23)) comprise the reported range of apparent activation energies for the two entirely different experimental scenarios and further supports the postulate in eqn (14). Eqn (23) obviously allows to rationalise the K_O ∝ D_O^1/2 correlation reported in the literature.¹

In the following section it is demonstrated how the oxygen exchange rate [Fraktur R]⁰_O can also be measured via an ¹⁸O exchange experiment on a thick plate-like sample with SIMS depth profiling, that is, in an experimental situation where the oxygen exchange is clearly in the diffusion controlled regime, h ≫ l⁰_c. In 2001 Fielitz and Borchardt,²³ without, however, discussing the physical meaning of τ_eq*, pointed out that the corresponding solution of the diffusion equation as given by Crank²⁰ should be used in the following form

to fit SIMS depth profiles (s. elsewhere²³ for further details) in order to extract the tracer isotope diffusion length σ* and the time constant τ_eq* which is the time necessary to obtain isotope equilibrium at the surface (x = 0). As the boundary condition eqn (8) was used to solve the diffusion equation²⁰ one can use the set of eqn (19) (with w_O(0) = 1 for an isotope exchange experiment) This means that the oxygen exchange rate [Fraktur R]⁰_O can be directly determined by the time constant τ_eq* in eqn (24). It should be underlined that the specific form of the solution Fielitz and Borchardt²³ proposed explicitly contains the function (compare to τ_eq* in eqn (24)), whereas this fact is not evident in the original form given by Crank.²⁰

The presented revision of the principal nature of the surface exchange coefficient has been essentially triggered by the publications of Armstrong et al.^12,13 since these investigations demonstrated clearly that the experimentally determined apparent activation energies for K_O* strongly depend on the kinetic regime of the respective experiments. Armstrong et al.^12,13 studied a relatively broad spectrum of materials (various perovskite MIECs like LC, LF, LSM, LSF, LSC or LSCF as well as two widely used oxygen electrolyte materials, namely YSZ and GDC). The similarities of the regime dependency of K_O* therefore suggests that their observation does not reflect a specific property of a class of materials, but rather an underlying universal phenomenon. It should, however, be recalled that there is another method to determine K_O* on powders which does not find any dependence of the apparent activation energies on the kinetic regime. Bouwmeester et al.⁹ and Yoo et al.^10,11 used a novel pulse ¹⁸O–¹⁶O isotopic exchange (PIE) technique for the rapid determination of the oxygen surface exchange rate of oxide ion conductors. For this method powders with an average particle size in the range 0.4–5.0 μm are used. The reported apparent activation energies did not significantly differ from the values determined on bulk samples via IEDP/SIMS (or ECR, respectively). Recent GPE/GPA-based work^4,14,16 comprises similar activation energies for the surface exchange coefficient and for the oxygen diffusivity.

During the PIE experiment, a powder sample is loaded in a packed-bed micro-reactor. Maintaining conditions of chemical equilibrium, the response to an ¹⁸O-enriched pulse fed through the reactor under continuous flow conditions is measured by mass spectrometric analysis of the gas phase fractions of oxygen isotopomers (¹⁸O₂, ¹⁶O¹⁸O and ¹⁶O₂) at the exit of the reactor. The essential difference between PIE^9–11 and IIE^12,13 is the residence time between the ¹⁸O₂ atmosphere and the powder sample: for PIE the mean residence time is 5 to 40 ms,^9–11 whereas in an IIE experiment the residence time is 15 to 40 min.^12,13 In this context, it is worth recalling that the surface exchange regime is not reached before the ¹⁸O concentration gradient (in the bulk of a powder particle) practically disappears, which means that the value of D_O* has become no longer relevant for the ¹⁸O exchange process. Since principally a certain (finite) diffusion time t has to elapse before an ¹⁸O concentration gradient in a particle with nominal radius r disappears, IIE seems to be more suitable than PIE to operate in a true surface exchange regime. This can be easily demonstrated by a straightforward estimation: the condition relates the necessary diffusion length, which is required to reach the surface exchange regime, to the particle radius. Estimating an average particle radius r_PIE ≈ 3 μm^9–11 for the PIE experiment and an average particle radius r_IIE ≈ 0.3 μm^12,13 for the IIE experiment one has t_PIE/t_IIE = (r_PIE/r_IIE)² ≈ 100 as a condition to reach the surface exchange regime in the published PIE studies compared to the published IIE studies. Since, however, the experimentally realized residence times t_PIE during the PIE experiment are only in the millisecond range,^9–11 so that one can estimate the ratio t_PIE/t_IIE ≈ 40 ms/40 min = 1.7 × 10⁻⁵, it cannot be excluded that PIE operates in a transitory, i.e. mixed controlled regime, similarly to the great majority of the published IEDP/SIMS studies.

Likewise the GPE/GPA-based investigations^4,14,16 use macroscopic samples and fairly long annealing times (for details see Appendix B). This inevitably implies a shift to L values typical for a mixed controlled or diffusion controlled regime (see Table 1 and Appendix C).

Table 1 Characteristic dimensionless parameters of the kinetic regimes

Kinetic regime	L = (h/2)(KO/DO)	ϕ(β1,L)	β 1 ^2
Surface control	<0.3	1	L
Mixed regime	0.3 to 30	—	—
Diffusion regime	>30	8/π^2	π^2/4

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Summary

In 1998 Maier⁶ proposed eqn (2) according to which the exchange rate [Fraktur R]⁰_O (for oxygen) is proportional to the surface exchange coefficient K_O. Since then two well documented significant experimental findings became known which contradicted the general validity of eqn (2):

(i) Instead of the theoretically expected relation¹⁵K_O*/D_O* ≅ K̃_O/D̃_O the experiments yield⁷K_O*/D_O* > K̃_O/D̃_O.

(ii) If measured in the surface exchange regime (h ≪ l⁰_c) the apparent activation energies for K_O* and K̃_O are very small^13,17 whereas the apparent activation energies measured in the mixed controlled regime (h ≈ l⁰_c) or in the diffusion controlled regime (h ≫ l⁰_c) are considerably larger (s. Armstrong et al.,¹³ Table 5).

These two surprising experimental observations, but especially the latter one, raise doubtful questions about the general validity of eqn (2), which necessitate a thorough fundamental rediscussion of the surface exchange coefficient for oxygen (on oxide surfaces).

Following principally Maier's⁶ procedure the simplest possible experimental situation was chosen, i.e. an oxide surface close to chemical equilibrium (s. eqn (6)). For a complete picture, one has, however, to include the diffusive transport of oxygen (s. eqn (7)) and the fact that at the gas/solid interface there is a boundary condition to be respected. For the gas/solid interface of a homogeneous semi-infinite oxide this is eqn (8). This boundary condition inevitably leads to a mathematical correlation between K_O and D_O (s. eqn (11)), which, at first glance, seems to be only a formal recipe to calculate the so-called characteristic length l⁰_c. Through the definition of a characteristic time t⁰_c, this parameter can be attributed a physical meaning in eqn (13) as t⁰_c is the time necessary for the relaxation of a small change of the chemical potential of oxygen at the surface, δμ_O(0). Further, it becomes obvious that in eqn (2) the term K_O/Δx (as formulated by Maier⁶) is equivalent to the inverse characteristic time t⁰_c. It is therefore consistent to correlate the oxygen exchange rate [Fraktur R]⁰_O and the characteristic time t⁰_c directly, as postulated through eqn (14). This postulate and the boundary condition (eqn (8)) entail an explicit dependency between the (surface) exchange rate and the (bulk) diffusivity of oxygen K_O = K_O(D_O,[Fraktur R]⁰_O) (s. eqn (15)). At the same time such a dependency implies that for more complicated diffusion boundary conditions more complicated functions for K_O have to be expected.

The relations for K_O in the different kinetic regimes derived in the framework of the proposed novel interpretation of the standard theoretical concept⁶ for K_O finally enable to unambiguously explain the two experimental observations (i) and (ii). In the presented model eqn (2) is only valid for the special case of the true surface exchange regime (s. eqn (21)). The parameter Δx in eqn (2), however, then is no longer a constant, but corresponds to the surface-to-volume ratio of the sample. As a consequence, the time constant τ_eq to reach equilibrium (s. eqn (20) and (21)) should not depend on the surface-to-volume ratio of the sample in the surface exchange regime.

Appendix A

For a thin surface layer δx′(x = 0) the following expression can be formulated If δx′(0) is sufficiently small and/or if the oxygen flux j^K_O(0) is approximately constant within the layer of thickness δx′(0) one obtains with eqn (6) That is, the time constant t⁰_c for the exponential relaxation kinetics of a small oxygen potential change δμ_O(0) at the surface is given by Together with K_O = Λ_O∂μ_O/∂c_O|_x=0 and eqn (A1), (A3) yields eqn (12).

Appendix B

A widely applicable reaction model comprises the following consecutive steps:⁷

(1) adsorption dissociation (eq.)

(2) charge transfer (rds) (3) incorporation (eq.): The reactions are formulated according to the Kröger–Vink notation, using square brackets for the respective concentrations, K_n for the mass action constants (n = 1, 2, 3), for the forward/backward rate constants, for the forward/backward reaction rates and [Fraktur R]_n for the net reaction rates of these reactions. The so called apparent or effective equilibrium exchange rate [Fraktur R]⁰_O of the rate determining step (charge transfer (B2) is often assumed as the rate determining step⁷) is defined by After inserting the mass action constantsinto eqn (B4) and defining notational abbreviations one finally gets if, e.g., charge transfer is rate determiningEqn (B6) demonstrates in a very convincing manner that the surface exchange coefficient will have a non-trivial temperature dependence in the surface controlled regime (K_O ∝ [Fraktur R]⁰_O, see eqn (21)). The resulting apparent activation energy should generally be less than the average energy of D_O and [Fraktur R]⁰_O. In the diffusion controlled regime (, see eqn (15)) the resulting apparent activation energy is the average activation energy of D_O and [Fraktur R]⁰_O.

We will now demonstrate that there is an equivalence of the definitions of the phenomenological parameters used in Maier's approach⁶ and the approach of Klier and Kučera.²⁴ Comparing both approaches it becomes evident that recent work,^4,14 based on the work of Klier and Kučera,²⁴ uses the same phenomenological description for their consecutive model as Maier⁶ and the present authors (neglecting, however, the charge transfer step, which seems to be a weakness of this approach). With r_H for the “heterogeneous” reaction rate, r_a for the (dissociative) “adsorption” rate, and r_i for the “incorporation” rate they obtain (see eqn (8) and (12) in Farlenkov et al.¹⁴) r_H = r_a·r_i/(r_a + r_i), i.e. they set 1/r_H = 1/r_a + 1/r_i as expected for a consecutive model. In Maier's notation this would become 1/[Fraktur R]⁰ = 1/[Fraktur R]₁ + 1/[Fraktur R]₃ if the charge transfer step (eqn (B2)) is neglected (in order to facilitate the comparison). In eqn (13) of Farlenkov et al.¹⁴ the surface exchange coefficient, k, without specifying the kinetic regime, is defined as k = const. × r_H (where the constant contains the specific data of the material, which have only a negligible temperature dependence). This corresponds formally to the relation originally proposed by Maier⁶ as well as to our result for the mere surface controlled regime, K_O = const. × [Fraktur R]⁰_O, as expressed by eqn (21). It is, however, debatable (see Appendix C) whether the pure surface controlled regime can anyhow be studied if the dense (polycrystalline and single crystalline) samples have a thickness of 5 mm and undergo annealing times of more than 5 × 10⁴ s.¹⁴

Appendix C

The most direct way to follow the kinetics of oxygen uptake or release in situ at high temperatures is a gravimetric experiment.¹⁷ For plate-like samples with thickness h the mass change, Δm(t), if normalized to the mass change at equilibrium, Δm(t_eq), is given by (Crank²⁰ p. 60) The dimensionless quantities β_n are positive roots of β_n tan β_n = L. As the infinite sum in eqn (C1) converges rapidly one has approximately¹⁷where D_O is the effective oxygen diffusion coefficient in the solid and K_O is the oxygen surface exchange coefficient defined by eqn (1). Positive roots of β₁ tan β₁ = L are plotted in Fig. 2 which allows a graphical visualisation of the kinetic regimes. The values of the resulting characteristic dimensionless parameters of the kinetic regimes are compiled in Table 1.

Fig. 2 Graphical visualisation of the kinetic regimes¹⁷ (open points are positive roots of β₁ tan β₁ = L tabulated by Carslaw and Jaeger²⁵). The diffusion regime is characterised by β₁² ≃ π²/4 and the surface exchange regime by β₁² ≃ L.

Acknowledgements

We are indebted to Martin Schmücker, Aldo Steinfeld, Robert Pitz-Paal, Clemens Feldmann and their co-workers for valuable discussions. Financial support by the Initiative and Networking Fund of the Helmholtz Association of German Research Centers within the Virtual Institute SolarSynGas as well as by Deutsche Forschungsgemeinschaft (DFG) for one of the authors (P. F.) is gratefully acknowledged.

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