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Water incorporation in oxides: A moving boundary problem
Solid State Ionics 181 (2010) 154–162
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Solid State Ionics j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / s s i
Water incorporation in oxides: A moving boundary problem Ji-Haeng Yu a,c, Jong-Sook Lee a,b,⁎, Joachim Maier a a b c
Max-Planck-Institut für Festkörperforschung, 70569 Stuttgart, Germany School of Materials Science and Engineering, Chonnam National University, Gwangju 500-757, Republic of Korea Korea Institute of Energy Research, Daejeon 305-343, Republic of Korea
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 25 July 2008 Received in revised form 17 September 2009 Accepted 1 November 2009
We previously reported a peculiar non-monotonic water incorporation kinetics in a model oxide system of Fe-doped SrTiO3 [Angew. Chem. Int. Ed. 46 (2007) 8992]. In-situ optical absorption spectroscopy unambiguously indicated that the oxide was first strongly reduced and then oxidized to the final equilibrium by the decoupled water transport, in which hydrogen fast diffuses into the oxide and oxygen lags behind. Electronic carriers constitute the ambipolar diffusion couples for the respective transports. Here we performed a defect-chemical thermodynamic analysis with the frozen-in oxygen vacancies, which successfully represent the overshooting state caused by the sluggish oxygen diffusion. The quantitative investigation of the spatially resolved optical absorption images based on the thermodynamic calculation revealed a hitherto unknown novel kinetic aspect. The absorption effects cannot be explained by the simple superposition of hydrogenation and oxygenation as assumed in the integral analysis. The diffusioncontrolled hydrogenation appears to proceed in the region narrowed by the moving oxidation fronts. The oxidation front in the oxide, which is presently equivalent to the hydration front, not the sample surface in contact with water vapor, behaves as the source planes for water incorporation. The hydrogen and oxygen transport are not completely decoupled in this sense but constitute a correlated moving boundary diffusion problem. The process can be quantitatively described by modeling of the diffusion coefficients which are exponentially increasing with water concentration for hydrogenation and oxygenation, respectively. The overall oxidation process of the later stage is non-trivial but apparently surface reaction controlled. © 2009 Elsevier B.V. All rights reserved.
Keywords: Water incorporation SrTiO3 Optical absorption spectroscopy Non-monotonic relaxation Frozen-in oxygen vacancies Moving boundary problem Concentration-dependent diffusivity
1. Introduction In spite of the active research for the last decades, the kinetic details of water incorporation in proton conducting perovskites are largely unknown. The water incorporation into these vacancy-dominated oxides introduces protons as hydroxide defects (OH⋅O) as follows [1]: ::
:
H2 O + VO + OO ⇌2OHO KH2 O =
½OH:O 2 aH2 O ½VO::
ð1Þ
δ
jw j s = −kw δcw j s
One water molecule produces two protons by consuming one oxygen vacancy. The conduction mechanism of the incorporated protons has been elucidated in the last two decades [2,3]. It is essentially a phononassisted transport of interstitial protons that can be described by Eq. (2). :
:
:
OHO + OO ⇄OO ⋅⋅⋅H ⋅⋅⋅OO ⇄OO + OHO
ð2Þ
⁎ Corresponding author. School of Materials Science and Engineering, Chonnam National University, Gwangju 500-757, Republic of Korea. Tel.: +82 62 530 1701; fax: + 82 62 530 1699. E-mail address: [email protected] (J.-S. Lee). 0167-2738/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2009.11.003
The preceding kinetic steps and hence the mechanism of water incorporation into oxides are, however, barely understood. In analogy to well-established oxygen incorporation kinetics [4], one may suppose a surface reaction step which involves physi-/chemisorption, charge transfer, and dissociation of water to H+ and O2− ions. To the authors' knowledge no systematic investigation of this aspect has been reported. One may define a surface reaction constant kδw satisfying the relation ð3Þ
in which the subscript s in Eq. (3) stands for the surface and δcw|s the concentration difference from the equilibrium value at the surface. The diffusion of H+ and O2− further into oxides to hydrate the oxide should be coupled to keep electroneutrality. This may be considered as ‘water’ motion into oxides even though no molecular water is involved. The atomistic mechanism comprises the transport of protons (OH⋅O) and the counter-transport of oxygen vacancies [5] as schematically shown in Fig. 1. Thus Fick's 1st law of diffusion can be written as δ
jw = −Dw
∂ c ∂x w
ð4Þ
J.-H. Yu et al. / Solid State Ionics 181 (2010) 154–162
Fig. 1. Water incorporation through a surface reaction step and diffusion by coupled transport of 2H+ and O2− ions. Proton motion occurs according to Eq. (1), that is, by forming OH⋅O at the expense of OO. Motion of the oxygen ions occurs by occupation of oxygen vacancies V⋅⋅ O and formation of OO. According to Ref. [8].
in which Dδw is the Nernst–Planck type chemical diffusion coefficient represented by the transference numbers and the self-diffusivity of the ambipolar diffusion couples multiplied by the thermodynamic factor (see e.g. [6,7]), δ
Dw = ðtHþ DO2− + tO2− DHþ Þ
∂aw : ∂cw
ð5Þ
From the equality of the two fluxes of Eqs. (3) and (4) at the surface, the one-dimensional solution of Fick's 2nd law in the case of constant Dδw, ∂cw ∂ δ ∂ = Dw cw ∂t ∂x ∂x
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Fig. 2. Water incorporation through a surface reaction step and diffusion by a decoupled transport of 2H+ and O2− ions. The charge neutrality of the respective transports is maintained by the electronic carriers.
Ambipolar diffusivities DδH and DδO may be defined in a similar fashion as in Eq. (5). This mechanism thus requires a sufficient electronic conductivity, which is the case for many proton conducting perovskites as well as for Fe–SrTiO3 of the present study. With surface reaction rate constants kδH and kδO for hydrogenation and oxygenation, respectively, two sets of kinetic equations appear to be required to describe water incorporation, viz. for hydrogenation, δ
δ
∂ ∂cH ∂ δ ∂ = cH ; DH cH ∂t ∂x ∂x ∂x
ð8Þ
δ
∂ ∂cO ∂ δ ∂ = cO ; DO cO ; ∂t ∂x ∂x ∂x
ð9Þ
jH j s = −kH δcH j s ; jH = −DH and for oxygenation,
ð6Þ
δ
jO j s = −kO δcO j s ; jO = −DO
follows as [9]: ∞ 2Lw cosðαw;n x = dÞexpð−α2w;n Dδw t = d2 Þ cw ðx; tÞ−cw ð0Þ = 1− ∑ ð7Þ cw ð∞Þ−cw ð0Þ ðα2w;n + L2w + Lw Þcosαw;n n=1
where Lw = kδwd/Dδw, αw,n are the positive roots of αwtanαw = Lw for hydrogen and oxygen diffusion, respectively, and d is the half of the sample dimension. The kinetic parameters, kδw and Dδw can be then obtained from the experimentally obtained cw(x, t). The concentration profiles of quenched specimens (with a given t from the annealing treatment) can be analysed e.g. by SIMS (Secondary Ion Mass Spectroscopy) after isotopic exchange. In-situ techniques typically exploit the integrated solution which is a function of t only. Most popular has been following weight changes. Any other properties reflecting nonstoichiometry such as EPR [10] or NMR signals can be used. When the oxygen incorporation in nonstoichiometric oxides leads to the conductivity change, the relaxation of the overall conductivity can be monitored in-situ with a much less experimental complexity [11]. Spatially resolved optical absorption spectroscopy developed in the MPI in Stuttgart, Germany [12,13] uses the fact that the redox state of Fe4+/Fe3+ can be followed in-situ as a function of space (x) and time (t). The absorption at λ = 590 nm of Fe4+ ions is directly related to the local oxygen content and the kinetics of oxygen incorporation have been precisely and unambiguously determined. Recently, we applied this technique to the water incorporation in SrTiO3 [8]. Peculiar non-monotonic kinetics of water incorporation was observed in the optical absorption as well as overall bulk resistance of oxides distinguished by impedance spectroscopy. Based on our observations, the naively expected water incorporation kinetic scheme, represented in Fig. 1, is to be modified into the one given in Fig. 2. After a certain surface reaction step involving water dissociation, the hydrogen component (H) diffuses fast into the oxide reducing the oxide strongly and the oxygen component (O) follows at a slower rate. For hydrogenation, protons and electrons have to move in an ambipolar fashion. For oxygenation, motion of oxygen ions (O2−) and the electrons constitute the coupled fluxes.
the solutions of which have been previously used for the relaxation of bulk conductivity [8]. Non-monotonic conductivity relaxations have been meanwhile reported in other proton conducting perovskite oxides [14,15], supporting that the mechanism we are addressing can be a general one. Two-fold but monotonic relaxations were also observed in oxygen incorporation when cation as well as anion species are comparably mobile in the presence of the sufficient electronic conductivity [16,17]. Optical absorption relaxation in this work clearly indicates that the oxide is over-reduced first and then slowly reoxidized to the final state, while the resistance change is the combined effects of electron holes, oxygen vacancies, and protons [18]. The latter may not be straightforwardly interpreted. Spatially resolved insitu absorption images further revealed the hitherto unknown novel features and allowed the development of an in-depth mechanistic model for the decoupled transport of hydrogen and oxygen. It will be for the first time shown that a characteristic moving boundary diffusion problem is involved in water incorporation kinetics. 2. Experimental The change in the optical absorption coefficient of Fe4+ ions (λ = 590 nm) in the glass-sealed large surface of a 6 × 6 × 1 mm3 SrTiO3 single crystal plate doped with 5 × 1018 cm− 3 Fe according to an ICP (induction coupled plasma spectroscopy) analysis was in-situ monitored at 650 °C by a CCD camera upon a sudden water vapor pressure increase (see the images of Fig. 3). The setup was essentially the same as described previously [12,13] except that a new design of specimen holders was employed, which allowed the observation of the absorption effects up to the very specimen edges. Previously [13,19] the region of ca. 0.5 mm width from the edge of the specimen was blocked by a frame holding the specimen, sealing glass and sapphire plates together. The water vapor pressure was varied from 1.9 kPa using the wet oxygen bubbled through water at 17 °C, to 0.4 kPa set by mixing dry and wet oxygen. All the side faces as well as the sealed large surfaces were fine-polished. Two opposite small faces
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Fig. 3. Normalized integral absorption (filled squares) of the optical absorption images and the bulk resistance by impedance spectroscopy (open squares) upon PH2O change from 0.4 kPa to 1.9 kPa at 650 °C in a Fe-doped SrTiO3 single crystal. The insets in the bottom part show the impedance spectra of the initial, overshooting and final state. Bulk and electrode responses can be separated. The overall response can be represented by the superposition of the hydrogenation and oxidation processes. See the text for details.
in 6 × 1 mm2 were sputter-coated by Pt, while the other sides remained bare (see Fig. 3). The optical image processing and the numerical simulation of the absorption profiles were performed using IDL (Interactive Data Language) programs (RSI, USA). The impedance change of the same specimen with the glass sealing intact was monitored by an LCR meter (HP4284A, USA) using the Pt coating as electrodes under the same condition as in the optical experiment. Thermodynamic calculations were numerically performed using Matlab (Mathworks, USA).
the redox change of Fe ions. Although the water incorporation mechanism represented by Eq. (1) is a pure acid–base reaction, the Fe4+/Fe3+ ratio change is consequent upon water uptake by the mass action equilibria as previously shown [18]. Fig. 4(a) displays the numerically calculated defect concentrations at 650 °C as a function of water partial pressure in oxygen. In the present calculation the total pressure, PH2O + PO2 is fixed to 1 atm in accord with the usual experimental condition, while in the previous calculation [18] PO2 is fixed to 1 atm and PH2O is varied independently. With small
3. Results and discussion 3.1. Non-monotonic relaxation Fig. 3 shows the integral absorption over the 6 × 6 mm2 area of the specimen as a function of logarithmic time. The optical images of the sample indicate the spatially resolved absorption by an artificial color scale. Note that the imprints at the corners are from the specimen holder. The resistance relaxation was obtained from the real part of the impedance at 100 kHz, which roughly corresponds to the bulk resistance as indicated in the impedance spectra of the insets of Fig. 3. In both measurements consistently an overshooting was observed, in contrast to the usual monotonic relaxation. The line curves show that the time response of the resistance and absorption can be described by the superposition of the two relaxation processes in the opposite direction at distinctly different rates, which will be detailed in Section 3.3. 3.2. Thermodynamic consideration To apply the optical absorption spectroscopy for the water incorporation kinetics, water vapor pressure change should lead to
Fig. 4. (a) Calculated defect concentrations of SrTiO3 doped with 5 × 1018 cm− 3 Fe as a function of PH2O at 650 °C buffered with oxygen. (b) Calculated defect concentrations with the ‘frozen-in’ oxygen vacancies from the dry state (in solid lines) indicated by asterisks (*) compared with the equilibrium concentrations (in dotted lines).
J.-H. Yu et al. / Solid State Ionics 181 (2010) 154–162
differences in the numerical estimates the features remain the same for PH2O b 0.1 atm or 104 Pa as shown in Fig. 4. Fig. 4 shows that only 68% of Fe dopants (3.4 × 1018 cm− 3) are effective acceptors, FeTi, in dry condition i.e., PH2O ≅ 1 Pa. Upon increasing PH2O oxygen vacancy concentration, [V⋅⋅ O ], decreases due to the occupation by hydroxide ions or protons (OH⋅O), which leads to a decrease in hole concentrations by oxygen incorporation reaction equilibrium. (The decrease may not be well visible in the scale adopted in Fig. 4(a).) This in turn leads to decrease in Fe4+ ions by intrinsic redox equilibrium and thus in Fe4+ absorption. Note that, with the negligible contribution of Δ[h⋅] as well as of Δ[n′], defect concentrations in Fig. 4(a) satisfy the charge neutrality condition, ::
:
′ ≈2Δ½VO + Δ½OHO : Δ½FeTi
ð10Þ
⋅ Since Δ[Fe′Ti] = Δ[Fe3+] = − Δ[Fe4+], Δ[V⋅⋅ O ] = − Δ[O] and Δ[OHO] = Δ[H], the redox effects can also be expressed in terms of component hydrogen and oxygen mass balance and eventually represented by the net hydrogen amount for redox effects:
4+
−Δ½Fe
3+
≈Δ½Fe
≈Δ½H−2Δ½O = Δ½H−Δ½Hacid−base = Δ½Hredox : ð11Þ
This thermodynamic consideration can be extended to describe the strongly reduced intermediate state of the peculiar non-monotonic relaxation. According to our model in Fig. 2 the overshooting represents the state of the specimen in which the hydrogen has diffused-in but the sluggish oxygen could not catch up. In the defect-chemical analysis, the situation can be described by the ‘frozen-in’ oxygen lattice or a fixed ⁎ oxygen vacancy concentration of the dry state, i.e. Δ[V⋅⋅ O] = − Δ[O]⁎ = 0, where asterisks (⁎) stand for the non-equilibrium situation. The results are displayed in Fig. 4(b) (in solid lines) compared with the case of equilibrium (in dashed lines) as of Fig. 4(a). Under this condition the hydrogen is incorporated only by the redox mechanism
157
and the amount is much larger than the net hydrogen for the redox effect in the final equilibrium, i.e. 4+ ⁎
−Δ½Fe
⁎
≈Δ½Hredox N Δ½Hredox :
ð12Þ
The quasi-thermodynamic calculation straightforwardly illustrates that the origin of the strongly reduced transient state observed by optical spectroscopy can be the sluggish diffusion of oxygen modeled as the fixed oxygen vacancy concentration in the defect-chemical calculation. Note that the hydrogen incorporated with the ‘frozen-in’ oxygen in Fig. 4(b) is still less than the total hydrogen amount in final equilibrium, i.e. ⁎
Δ½H = Δ½Hacid−base + Δ½Hredox N Δ½Hredox N Δ½Hredox :
ð13Þ
To reach the final equilibrium state from such an overshooting state not only oxygen but also some additional hydrogen should be further incorporated. For the illustration purpose a quantitative analysis is performed in the following specific to the current experimental condition, i.e. PH2O change from 0.4 kPa to 1.9 kPa. The diagrams of the upper row in Fig. 5 schematically represent the thermodynamic mass balance. The effects are normalized by the redox variation in the final equilibrium −Δ[Fe4+] ≈ Δ[Fe3+] which amounts to ca. 0.1 × 1018 cm− 3 (Fig. 4). The redox variation upon water uptake is less than 10%. The incorporated hydrogen amount in the final state would lead to 5.6 units of reducing effect (vertical hatched area in the diagram left), if alone, but 4.6 units thereof is compensated by oxygen incorporation (horizontal hatched area in the diagram middle). Note that the redox effect by the doubly charged oxygen is twice the concentration. The compensated redox effects then represent the incorporation of the ‘water’ or the hydration. The incorporated water concentration can be estimated as Δcw = 1/2ΔcH(acid–base) = ΔcO = 4.6/ 2 × 0.1 × 1018 cm− 3 = 0.23 × 1018 cm− 3. Under the condition of the fixed oxygen vacancy concentration the hydrogen incorporation reaches 3.2 units, i.e. ca. 61% of a total of 5.6 units of hydrogen amount
Fig. 5. Upper row diagrams represent the thermodynamic mass balance of hydrogen and oxygen upon PH2O change from 0.4 kPa to 1.9 kPa in terms of redox effects. The scale is normalized by − Δ[Fe4+] ≈ Δ[Fe3+] ≈ 0.1 × 1018 cm− 3. Lower row diagrams represent the mass balance associated with the kinetic processes. The first diagram indicates the thermodynamically estimated hydrogen incorporation with frozen-in oxygen defects which is responsible for strong reduction (Stage I). The mass balance to the final equilibrium should be realized by the later process(es) (Stage II).
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as shown in the lower left diagram in Fig. 5. Stage I thus represents the fast reduction process from the initial state of ‘1’ passing the final equilibrium value of ‘0’ and approaching the quasi-equilibrium state of ‘−2.2.’ To meet the mass balance of the final equilibrium (right diagram) the re-oxidation process (Stage II) shown in the middle diagram involves additional hydrogen incorporation of 2.4 (=5.6 − 3.2) unit, as well as the oxygen incorporation corresponding to 4.6 units of oxidation effect.
3.3. Diffusion modeling Fig. 6 represents the absorption profiles across the width (2d= 0.6 cm) of the specimen at t=120, 240, 360, 480, 600, 720, 1200, and 1560 s (upper graphs) and at t=1560, 1800, 2460, 3660, 6060, 14,460, and 34,860 s (lower graphs). Upper graphs are for the fast reduction period and the lower graphs are for the later oxidation period. The profiles were obtained by integrating the intensity of the central slabs of 50 pixel thickness from the 510×510 pixel images. Fig. 6(a) represents the horizontal profiles with Pt-coated surface as boundaries while Fig. 6(b) is from the vertical lines bounded by the well-polished bare surfaces (see Fig. 3). It should be noted that the profiles are normalized with respect to the initial state of Fe4+ absorption as ‘1’ and the final state as ‘0’ as in the schematic diagrams of Fig. 5. The in-situ profiles from the optical images shown in Fig. 6 suggest that the hydrogenation incorporation can be modeled as predominantly diffusion-controlled one-dimensional problem with the Ptcoated surface boundary. On the other hand, the re-oxidation proceeds with more or less flat profiles over most of the sample area. Little difference was observed between the Pt-coated and the bare surface. The process thus constitutes an apparently surface reaction controlled two-dimensional problem. Based on these features we first tried to analyze the integral effects shown in Fig. 3. As illustrated in the mass balance diagrams in Fig. 5 the analysis of non-monotonic optical relaxation for water incorporation kinetics requires i) a proper deconvolution of the optical effects into hydrogen and oxygen mass balance and ii) apostulation of the decoupled quasithermodynamic driving forces for the hydrogen and oxygen transport, respectively. Fig. 3 shows that the hydrogenation process of Eq. (14) multiplied by an ‘overshooting factor’ P⁎(N1) with respect to the equilibrium change relaxing from the initial state to an overshooting state (the red curves) and the oxygenation of Eq. (15) proceeding from the overshooting to the final state, thus of the magnitude (P⁎ − 1) in the opposite redox effects (green curves) can successfully
describe the experimental relaxation as shown by the black solid lines of the composite effects. The modeling details are in the following. We found that purely diffusion-controlled kinetics for hydrogenation did not satisfactorily describe the short-time behavior. The discrepancy in the short-time behavior can be easily seen in the logarithmic time representation of the integral curves as in Fig. 3. The spatially resolved absorption profiles in Fig. 6 (top left) indicate directly that the surface does not instantaneously equilibrate but approaches to the saturation point over a certain period (ca. 100 s for the current experiment). The behavior may appear to suggest the limited surface reaction kinetics modeled by Eq. (8) but the consequent analysis was not quite satisfactory either. Although the effect of the limited surface reaction rate or the dead volume effect may not be completely excluded, the non-instantaneous equilibration of the surface concentration was found to describe the behavior best. The treatment of the realistic limitation on the stepwise change of the atmosphere can be found in the monograph by Crank [9]. (The situation should be distinguished from the case of the limited surface reaction rate, in which the concentration beneath the surface in the bulk is different from the very surface concentration which is assumed to be in an instantaneous equilibrium with the atmosphere.) For an exponential saturation of the surface concentration, 1 − exp(−γt), the solution follows as [9]: 1 1 MH ðtÞ−MH ð0Þ δ 2 2 δ = 1−expð−γtÞðDH = γd Þ2 tanðγd = DH Þ2 MH ðt = ∞Þ−MH ðt = 0Þ
−
8 ∞ expð−ð2n + 1Þ2 π2 DδH t = 4d2 Þ : ∑ π2 n = 0 ð2n + 1Þ2 ½1−ð2n + 1Þ2 ðDδH π2 = ð4γd2 ÞÞ
ð14Þ On the other hand, the re-oxidation process (Eq. (9)) is assumed to occur by a surface reaction controlled two-dimensional incorporation: ! δ MO ðtÞ−MO ð0Þ 2kO t = 1−exp − ; MO ðt = ∞Þ−MO ðt = 0Þ d
ð15Þ
although the process is non-trivial as will be discussed in the following section. It should be mentioned that the integral response of the superposed relaxations may not unambiguously show the limiting process. The exponential relaxation could be easily taken as the diffusion-limited process [14,15]. The parameters used for the simulated curves are listed in Table 1. In simulating the optical relaxation curve we fixed the ‘overshooting
Fig. 6. (a) Horizontal line profiles for the optical images as shown in Fig. 3 for the reduction period (t = 120, 240, 360, 480, 600, 720, 1200, and 1560 s) (Top) and for the slow oxidation period beyond the overshooting (t = 1560, 1800, 2460, 3660, 6060, 14,460, and 34,860 s); (b) vertical line profiles corresponding to the profiles of (a). The profiles were obtained by integrating the intensity of the central slabs of 50 pixel thickness of 510 × 510 pixel images.
J.-H. Yu et al. / Solid State Ionics 181 (2010) 154–162 Table 1 Parameters for the simulated curves shown in Fig. 3.
Overshooting factor Hydrogenation (Stage I) Oxygenation (Stage II)
Parameters
Optical relaxation
Resistance relaxation
P* DδH/cm2 s− 1 γ/s− 1 kδO/cm s− 1
3.2 8 × 10− 5 0.0065 2.8 × 10− 5
4.0 8 × 10− 5 0.0065 8.5 × 10− 5
factor’ P⁎ to the maximum overshooting value ‘3.2’ observed by insitu optical images and then determined DδH, γ, and kδO values of Eqs. (14) and (15) for the best ‘eye fitting’. The hydrogenation parameters from the optical relaxation, DδH and γ, were found to be satisfactorily applied to the resistance relaxation also. For the oxidation process, however, the surface reaction rate constant kδO from the optical relaxation is significantly smaller (factor of ca. a third) than that from the resistance relaxation. This can be ascribed to the spatial inhomogeneity in Δ[Fe4+] (see the profiles of the lower graphs in Fig. 6), which is not considered in the purely surface reaction controlled oxidation model, Eq. (15), and to the consequent short-circuiting effect along the edge region. While the overshooting factor P⁎ for the absorption effect can be represented by Δ[Fe4+right]⁎ (see Fig. 4(b)), and thus directly compared to the spatially resolved absorption, the factor for the resistance effect, be it related to Δ[h⋅]⁎ in the first place, is more involved, since this is an integral quantity. The latter can be also affected by the mixed conduction of protons and oxygen ions. 3.4. Profile analysis Fig. 7 shows the simulation for the Stage I process shown in the upper graphs of Fig. 6 as one-dimensional symmetrical case. The kinetic parameters used in the profile simulation are shown in Fig. 8.
159
The simulation was performed numerically by a finite difference method [9] using IDL programs as in the previous work [12,19]. The appropriateness of the modeling and the goodness of the parameters can be more clearly seen in the profile simulation, since the sequential development of the multiple simulated profiles as functions of x and t is more sensitive to the input models and parameters than the single integral relaxation curves as a function of t only. More detailed and certainly more definite information was thus obtained as described below. 3.4.1. Novel features The profiles for the Stage I process (Fig. 6, upper) are indeed seen to be the composites of the reduction profiles from the hydrogenation thus with an upward curvature and the oxidation profiles slowly developing from the surface with a downward curvature. However, the kinetic mechanism developed for the profile analysis appears to be much more involved than those considered for the integral analysis in the previous section (Section 3.3). Thus far considered models cannot describe the following novel features of the spatially resolved absorption profiles: (i) Not in accord with the predominantly surface reaction controlled oxidation process in Stage II, clearly diffusion-controlled oxidation front was developed from the surface during Stage I. Thus a new parameter, DδO, which was not considered in the integral analysis, is introduced for the Stage I profile analysis (see Fig. 8). The diffusion-controlled oxidation is even more distinctly observed for the bare surface boundaries, Fig. 6(b), due to rather featureless hydrogenation profiles along this direction. (ii) The oxidation magnitude of such a diffusion-controlled process turned out to be the magnitude ‘O’ indicated in Fig. 7, which amounts to only 16% of the magnitude (P⁎ − 1) assumed in the previous analysis (see Section 3.3). The nature of such an overreduced quasi-equilibrium surface state should be further clarified.
Fig. 7. Simulated line profiles (upper) obtained by combining the hydrogenation profiles and oxygenation profiles (lower) for the horizontal profiles with Pt-coated surface (a) and for the vertical profiles with bare surface boundaries (b) for the simulation time tsim = 30, 150, 270, 300, 390, 510, 630 and 1110 s corresponding to the profiles of Fig. 6 with t =tsim + 90 s. The difference of 90 s represents the delay in the specimen response from the recording time since the atmosphere change. The parameters for the simulation are shown in Fig. 8. In the lower graph of (a) the oxygenation profile at tsim = 1100 s is shown in magnification.
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Fig. 8. (Left graph) kinetic parameters used in the profile simulation of Stage I process in Fig. 7: DδH and DδO (exponentially increasing with ΔXH2O); kδH,Pt, kδH,bare, kδO (regardless of the surface condition). kδO(II): from the local absorption relaxation. The surface reaction constants are indicated at ΔXH2O = 0.16 for the wet surface. The average kinetic parameter from integral analysis (Table 1) and those from the oxygen incorporation for the same material [19] are shown for comparison.
It may be ascribed to the preferential hydrogen adsorption on the oxide surface or a hydrogen ‘spill-over’ [20] during the hydrogen incorporation. (iii) The hydrogenation profiles can be satisfactorily described only when the effective sample dimension pffiffi is reduced from the surface by the distance proportional to DδO t . The oxygen diffusion fronts described above were found to serve as dynamic boundaries for the fast hydrogenation incorporation into the inner region of oxide. The modeling of this moving boundary feature in water incorporation will be detailed in the next section. (iv) It has been also noted that the advancing oxidation fronts halted or approached to a steady state after some time (ca. 300 s for the current experiment), while the hydrogen diffusion profile appears to evolve continuously with the oxidation fronts as dynamic or static boundaries during Stage I. (v) According to the modeling in Section 3.3 the broad overshooting maximum in Fig. 3 is the overall compensation effect of the time-varying spatially inhomogeneous redox processes. The spatially resolved absorption profiles however indicate the states are indeed a time-independent steady state for a certain period (i.e. for ca. 1000 s from 1000 s to 2000 s as shown in the inset of Fig. 9). (vi) Persistent, only slowly decaying, near-surface diffusion profiles during Stage II as evidenced in the lower graphs of Fig. 6 are apparently inconsistent with a much flatter middle portion strongly indicating a surface reaction controlled process. The local absorption as well as the integral absorption relaxation, shown in Fig. 9, exponentially relaxed with the apparently similar time constant which is represented by kδO(II) in Fig. 8.
single phase also leads to the development of the concentration profiles with the inflection points. It should be noted that the diffusion pffiffi is nonetheless Fickian i.e. the boundary x moves linearly with t , as long as diffusivity depends on concentration only, not on time, concentration gradient, etc. [9]. The present situation is not a simple moving boundary problem, however. The moving oxygen diffusion front serves as the source plane for the hydrogen incorporation into the inner region of the specimen. As discussed above (Section 3.2) the oxidation of the specimen in the overall reduced specimen by fast hydrogen incorporation is equivalent to hydration of the specimen, i.e. ΔcO = ΔcH2O when ΔcH N 2ΔcO. The surface oxidation layer can be thus considered as the surface hydration layer. It thus makes a physical sense then that the moving front of the hydration layer, not the very surface of the specimen in contact with water vapor, becomes the source plane for further ‘decoupled’ water incorporation into the inner region of the specimen. It is remarkable that, after all, the hydrogenation and oxygenation process are not completely split as depicted in Fig. 2 but ‘coupled’ in a correlated moving boundary diffusion problem. Even with so much understanding of the physico-chemical nature of the phenomena, the modeling presented in Fig. 7 has been achieved after a considerable maneuvering. The bottom line is that the two diffusivities, DδH and DδO should be functionally correlated. Separate adjustment of the two diffusion solutions as was done in the integral analysis (Section 3.3, Fig. 3) never succeeds in describing the two dynamics correlated in space and time. It appears also that the moving boundary aspect should be described by a diffusivity or diffusivities strongly concentrationdependent. Then the ‘water’ concentration, ΔcH2O, would be a natural choice for the common variable for both diffusivities. Two diffusivities modeled to be exponentially increasing with water concentration as shown in Fig. 8 have been found to successfully describe the experimentally observed Stage I profile evolution: Fig. 6 (upper) vs. Fig. 7. Here is a brief description on technical aspects of how we introduced the water concentration dependence of the two diffusivities for the successful simulation of the correlated moving boundary diffusion feature. We calculated the oxygenation profile by a finite difference method according to the standard moving boundary problem [9,21] with the diffusivity exponentially dependent on the concentration of the ‘own’ diffusing species, i.e. oxygen. It should be noted again that the oxygen concentration is equivalent to the water concentration in the presence of sufficient hydrogen concentration. In the magnified profile at tsim = 1110 s one may recognize an inflection point characteristic of the moving boundary problem. The
Although the exact mechanistic model is still unclear, the non-trivial features described in (iv), (v) and (vi) are suggested to indicate the presence of the ‘coupled’ water motion as represented in Fig. 1 without net redox effects, mixed with the decoupled fluxes with redox effects. The ‘coupled’ water motion after some initial period can be consequent upon the slowed-down local flux of hydrogen with the progress. 3.4.2. Concentration-dependent diffusivity A moving boundary diffusion problem can be formally described in terms of the diffusivity with a strong positive concentration dependence [9]. The extreme and representative example is the growth of tarnishing layer or a layer of a second phase between the two neighboring solid phases in which Dδ changes discontinuously at the interface. A strongly concentration-dependent diffusivity in a
Fig. 9. The normalized intensity variation in linear time scale of the entire area of the specimen (total), in the center of the specimen (center), near the Pt-coated surface and near the bare surface. The inset is the magnification near the maximum overshooting point.
J.-H. Yu et al. / Solid State Ionics 181 (2010) 154–162
hydrogenation profiles were simultaneously calculated at each time sequence using the diffusivity as another exponential function of the momentary oxygen concentrations. The modeling resulted in the hydrogenation profiles flatter near the surface region with high oxygen concentration continued by portion of a normal diffusion behavior in the inner region with frozen oxygen concentration (see the lower graph of Fig. 7(a)). The upper graph of Fig. 7(a) shows that the combination of the two simulations reproduces the correlated moving boundary feature remarkably well. In Fig. 8 the water concentration is represented as the hydration degree ΔXH2O normalized with respect to the total water incorporation amount upon PH2O change from 0.4 kPa to 1.9 kPa. As described by the feature (ii) in the previous section ΔXH2O varies from 0 (at the moving ‘water’ front) to 0.16 at the surface in the quasi-equilibrium state during Stage I. The water concentration change in the final equilibrium, ΔcH2O, is 0.23 × 1018 cm− 3 (see Section 3.2) and thus the maximum hydration degree of ΔXH2O = 0.16 during Stage I corresponds to 0.37 × 1017 cm− 3 of ΔcH2O. It should be noted thus in Fig. 8 that the variation of the diffusivity by the factor of 6 or 8 over the concentration variation ΔcH2O of ca. 0.37 × 1017 cm− 3 is in fact a huge effect since the oxygen concentration variation from the water incorporation is less than one part per million (ppm) of the lattice oxygen in concentration of 5 × 1022 cm− 3. At such a dilute limit one might usually assume almost concentration-independent diffusivity. Concentration-dependent diffusivities are known to be characteristic of the gas or vapour diffusion in glassy polymers or in the sorption kinetics in porous materials even in small concentrations of diffusants [9,22,23], however. In these systems the diffusants are usually extrinsic elements, not even related to the host material structure. Diffusion of ‘water’ or hydrogen in oxides such as SrTiO3 appears to bear a similarity in this respect, since the diffusants, water or hydrogen, are not the regular constituents of oxides. The strong concentration dependency in the chemical diffusivity can be phenomenologically attributed to the thermodynamic factor with the self-diffusivity or the mobility terms being concentrationindependent. Thermodynamic factors can be obtained from the sorption/intercalation isotherms [22,23] or from the defect chemistry for materials with defects in low concentrations [12,16,17,21]. Similar to the factor ∂aw/∂cw in Eq. (5), for the decoupled hydrogen and oxygen flux, assuming a sufficiently high electronic conductivity and a concentration-independent self-diffusivity of species DH(O), the concentration dependency in the chemical diffusivity DδH(O) in Fig. 8 may be attributed to the thermodynamic factor: Γ HðOÞ =
∂ ln aHðOÞ ∂ ln cHðOÞ
ð16Þ
Since thus far powerful and versatile local equilibrium assumption relating μO, μH and μH2O is clearly invalid in the decoupled transport of hydrogen and oxygen in water incorporation, it is not immediately clear how the relevant thermodynamic factors can be exactly defined and estimated. Furthermore, the spatially resolved in-situ optical spectroscopy directly indicated an over-reduced quasi-equilibrium surface state, rather than the one in equilibrium with the ambient water atmosphere, PH2O, during Stage I process of the present concern, so that the application of the ambient atmosphere for the thermodynamic variable seems not guaranteed. From the defect-chemical calculation of Fig. 4 thermodynamic factors which relate the defect concentrations [OH⋅O] and [V⋅⋅ O ], respectively, with the water activity aw can be estimated, i.e. Γ wH =
∂ ln aw ∂ ln aw ; Γ wO = − ; ∂ ln ½V::O ∂ ln ½OH:O 2
ð17Þ
which are represented as a function of the respective defect concentrations in Fig. 10. It is shown that the thermodynamic factor
161
Fig. 10. Thermodynamic factors ΓwH = ∂lnaw/∂ln[OH⋅O]2 and ΓwO = ∂lnaw/∂ln[V⋅⋅ O] estimated from Fig. 4. They are represented as a function of the respective defect ⋅ concentrations in which [OHO]* is for the frozen-in oxygen vacancies from Fig. 4.
for the equilibrium concentration [OH⋅O], ΓwH is close to one over the concentration range concerned as expected for the dilute defect concentration. On the other hand ΓwH⁎ for [OH⋅O]⁎ in the case of the frozen oxygen lattice distinctly deviates from the ideal solution behavior with increasing concentration. Although not as strong as in DδH shown in Fig. 8 an exponential dependence is clearly indicated. The analysis thus suggests that the strong concentration dependence in the chemical diffusivity should be originated from and thus characteristic of the split non-equilibrium water diffusion. Much stronger non-ideality is shown in the case of the oxygen vacancies. Note that ΓwO⁎ is shown to increase with the oxygen vacancy concentration, which corresponds to the decreasing water activity (Fig. 4). The behavior is apparently contradictory to the strongly δ increasing DO with water concentration (see Fig. 8) and the implication is not immediately clear. It should be noted that ΓwO⁎ ⁎ values near frozen-in situation i.e., Δ[V⋅⋅ O ] ≈ 0 could be extremely large. 3.4.3. Surface reaction constant In the profile analysis we also take the finite surface reaction rate constants kδH and kδO into consideration and the values used in the simulation are indicated in Fig. 8. The surface reaction rates for Ptcoated boundaries, kH,Pt, is shown to be sufficiently large justifying the predominantly diffusion-controlled hydrogenation kinetics. The vertical profiles bounded by the bare surface boundaries can be successfully described, as shown in Fig. 6(b) by the same DδH(cH2O) or DδH(cO) but by a much smaller surface reaction constant for hydrogen incorporation, i.e. kδH,Pt ≫ kδH,bare. It should be noted that the surface reaction rate constant kδH,bare used for the simulation of the vertical profile is an apparent parameter to simulate the hydrogen flux through the Pt-coated surface, i.e., DδH ≈ kδH,bared, in the present onedimensional modeling for each direction. The true surface reaction constant for the bare surface is thus supposed to be even smaller. On the other hand, kδO for the oxygen split from water is found to be sufficiently high regardless of the surface treatment. The situation is in a great contrast to the incorporation of oxygen upon a stepwise change of oxygen partial pressure in which the surface kinetics is greatly influenced by the Pt- or other metal catalytic layers [19], as indicated in Fig. 8(right). 4. Conclusion A mechanistic model for the peculiar non-monotonic water incorporation has been presented. In the presence of a sufficiently
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high electronic conductivity, water incorporation into oxides occurs via an electrically decoupled transport of 2H+ and O2− ions. The charge neutrality of the respective transports is fulfilled by the electronic carriers. The strongly over-reduced transient state is due to the fast diffusion of hydrogen while the sluggish oxygen is lagging behind, and the situation can be described by a defect-chemical calculation assuming the ‘frozen-in’ oxygen vacancies. In-situ monitored spatially resolved optical absorption spectroscopy reveals the associated redox effects unambiguously. The decoupled hydrogen and oxygen transport turned out to constitute a correlated moving boundary problem, which can be quantitatively described by chemical diffusivities exponentially increasing with the hydration. The hydrogen incorporation thus appears to occur from the advancing ‘wet’ fronts established by the oxygen incorporation following behind not from the sample surface in contact with water vapor. While the Pt coating is shown to be catalytic for the hydrogen incorporation reaction, the oxygen incorporation from the split water appears similarly facile in Pt-coated and bare surface. Acknowledgment J.-S. Lee acknowledges the support by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2007-412-J02002).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
S. Stotz, C. Wagner, Ber. Bunsenges. Phys. Chem. 70 (1966) 781. W. Münch, K.-D. Kreuer, G. Seifert, J. Maier, Solid State Ionics 136–137 (2000) 183. M.S. Islam, R.A. Davies, J.D. Gale, Chem. Commun. (2001) 661. R. Merkle, J. Maier, Angew. Chem., Int. Ed. 47 (2008) 3874. K.D. Kreuer, E. Schönherr, J. Maier, Solid State Ionics 70–71 (1994) 278. C. Wagner, Prog. Solid State Chem. 10 (1975) 3. H. Schmalzried, Solid State Reactions, 2nd edVerlag Chemie Weinheim, 1981. J.H. Yu, J.-S. Lee, J. Maier, Angew. Chem., Int. Ed. 46 (2007) 8992. J. Crank, The Mathematics of Diffusion, Clarendon Press, Oxford, 1975. K. Sasaki, J. Maier, Phys. Chem. Chem. Phys. 2 (2000) 3055. J.B. Price, J.B. Wagner Jr., Z. Phys. Chem. NF 49 (1966) 257. I. Denk, W. Münch, J. Maier, J. Am. Ceram. Soc. 80 (1997) 279. M. Leonhardt, J. Jamnik, J. Maier, Electrochem. Solid-State Lett. 2 (1999) 333. H.-I. Yoo, J.-Y. Yoon, J.-S. Ha, C.-E. Lee, Phys. Chem. Chem. Phys. 10 (2008) 974. H.-I. Yoo, J.-K. Kim, C.-E. Lee, J. Electrochem. Soc. 156 (2009) B66. D.-K. Lee, H.-I. Yoo, Solid State Ionics 177 (2006) 1. H.-I. Yoo, C.-E. Lee, J. Am. Ceram. Soc. 88 (2005) 617. J.H. Yu, J.-S. Lee, J. Maier, Phys. Chem. Chem. Phys. 7 (2005) 3560. M. Leonhardt, R.A. De Souza, J. Claus, J. Maier, J. Electrochem. Soc. 149 (2002) J19. J.W.C. Conner, J.L. Falconer, Chem. Rev. 95 (1995) 759. R. Merkle, J. Maier, K.-D. Becker, M. Kreye, Phys. Chem. Chem. Phys. 6 (2004) 3633. H.B. Hopfenberg, V. Stannett, The Diffusion and Sorption of Gases and Vapours in Glassy Polymers, Applied Science Publishers, London, 1973, p. 504. [23] G.S. Park, The Glassy State and Slow Process Anomalies, Academic Press, 1968, p. 141.
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Solid State Ionics j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / s s i
Water incorporation in oxides: A moving boundary problem Ji-Haeng Yu a,c, Jong-Sook Lee a,b,⁎, Joachim Maier a a b c
Max-Planck-Institut für Festkörperforschung, 70569 Stuttgart, Germany School of Materials Science and Engineering, Chonnam National University, Gwangju 500-757, Republic of Korea Korea Institute of Energy Research, Daejeon 305-343, Republic of Korea
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 25 July 2008 Received in revised form 17 September 2009 Accepted 1 November 2009
We previously reported a peculiar non-monotonic water incorporation kinetics in a model oxide system of Fe-doped SrTiO3 [Angew. Chem. Int. Ed. 46 (2007) 8992]. In-situ optical absorption spectroscopy unambiguously indicated that the oxide was first strongly reduced and then oxidized to the final equilibrium by the decoupled water transport, in which hydrogen fast diffuses into the oxide and oxygen lags behind. Electronic carriers constitute the ambipolar diffusion couples for the respective transports. Here we performed a defect-chemical thermodynamic analysis with the frozen-in oxygen vacancies, which successfully represent the overshooting state caused by the sluggish oxygen diffusion. The quantitative investigation of the spatially resolved optical absorption images based on the thermodynamic calculation revealed a hitherto unknown novel kinetic aspect. The absorption effects cannot be explained by the simple superposition of hydrogenation and oxygenation as assumed in the integral analysis. The diffusioncontrolled hydrogenation appears to proceed in the region narrowed by the moving oxidation fronts. The oxidation front in the oxide, which is presently equivalent to the hydration front, not the sample surface in contact with water vapor, behaves as the source planes for water incorporation. The hydrogen and oxygen transport are not completely decoupled in this sense but constitute a correlated moving boundary diffusion problem. The process can be quantitatively described by modeling of the diffusion coefficients which are exponentially increasing with water concentration for hydrogenation and oxygenation, respectively. The overall oxidation process of the later stage is non-trivial but apparently surface reaction controlled. © 2009 Elsevier B.V. All rights reserved.
Keywords: Water incorporation SrTiO3 Optical absorption spectroscopy Non-monotonic relaxation Frozen-in oxygen vacancies Moving boundary problem Concentration-dependent diffusivity
1. Introduction In spite of the active research for the last decades, the kinetic details of water incorporation in proton conducting perovskites are largely unknown. The water incorporation into these vacancy-dominated oxides introduces protons as hydroxide defects (OH⋅O) as follows [1]: ::
:
H2 O + VO + OO ⇌2OHO KH2 O =
½OH:O 2 aH2 O ½VO::
ð1Þ
δ
jw j s = −kw δcw j s
One water molecule produces two protons by consuming one oxygen vacancy. The conduction mechanism of the incorporated protons has been elucidated in the last two decades [2,3]. It is essentially a phononassisted transport of interstitial protons that can be described by Eq. (2). :
:
:
OHO + OO ⇄OO ⋅⋅⋅H ⋅⋅⋅OO ⇄OO + OHO
ð2Þ
⁎ Corresponding author. School of Materials Science and Engineering, Chonnam National University, Gwangju 500-757, Republic of Korea. Tel.: +82 62 530 1701; fax: + 82 62 530 1699. E-mail address: [email protected] (J.-S. Lee). 0167-2738/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2009.11.003
The preceding kinetic steps and hence the mechanism of water incorporation into oxides are, however, barely understood. In analogy to well-established oxygen incorporation kinetics [4], one may suppose a surface reaction step which involves physi-/chemisorption, charge transfer, and dissociation of water to H+ and O2− ions. To the authors' knowledge no systematic investigation of this aspect has been reported. One may define a surface reaction constant kδw satisfying the relation ð3Þ
in which the subscript s in Eq. (3) stands for the surface and δcw|s the concentration difference from the equilibrium value at the surface. The diffusion of H+ and O2− further into oxides to hydrate the oxide should be coupled to keep electroneutrality. This may be considered as ‘water’ motion into oxides even though no molecular water is involved. The atomistic mechanism comprises the transport of protons (OH⋅O) and the counter-transport of oxygen vacancies [5] as schematically shown in Fig. 1. Thus Fick's 1st law of diffusion can be written as δ
jw = −Dw
∂ c ∂x w
ð4Þ
J.-H. Yu et al. / Solid State Ionics 181 (2010) 154–162
Fig. 1. Water incorporation through a surface reaction step and diffusion by coupled transport of 2H+ and O2− ions. Proton motion occurs according to Eq. (1), that is, by forming OH⋅O at the expense of OO. Motion of the oxygen ions occurs by occupation of oxygen vacancies V⋅⋅ O and formation of OO. According to Ref. [8].
in which Dδw is the Nernst–Planck type chemical diffusion coefficient represented by the transference numbers and the self-diffusivity of the ambipolar diffusion couples multiplied by the thermodynamic factor (see e.g. [6,7]), δ
Dw = ðtHþ DO2− + tO2− DHþ Þ
∂aw : ∂cw
ð5Þ
From the equality of the two fluxes of Eqs. (3) and (4) at the surface, the one-dimensional solution of Fick's 2nd law in the case of constant Dδw, ∂cw ∂ δ ∂ = Dw cw ∂t ∂x ∂x
155
Fig. 2. Water incorporation through a surface reaction step and diffusion by a decoupled transport of 2H+ and O2− ions. The charge neutrality of the respective transports is maintained by the electronic carriers.
Ambipolar diffusivities DδH and DδO may be defined in a similar fashion as in Eq. (5). This mechanism thus requires a sufficient electronic conductivity, which is the case for many proton conducting perovskites as well as for Fe–SrTiO3 of the present study. With surface reaction rate constants kδH and kδO for hydrogenation and oxygenation, respectively, two sets of kinetic equations appear to be required to describe water incorporation, viz. for hydrogenation, δ
δ
∂ ∂cH ∂ δ ∂ = cH ; DH cH ∂t ∂x ∂x ∂x
ð8Þ
δ
∂ ∂cO ∂ δ ∂ = cO ; DO cO ; ∂t ∂x ∂x ∂x
ð9Þ
jH j s = −kH δcH j s ; jH = −DH and for oxygenation,
ð6Þ
δ
jO j s = −kO δcO j s ; jO = −DO
follows as [9]: ∞ 2Lw cosðαw;n x = dÞexpð−α2w;n Dδw t = d2 Þ cw ðx; tÞ−cw ð0Þ = 1− ∑ ð7Þ cw ð∞Þ−cw ð0Þ ðα2w;n + L2w + Lw Þcosαw;n n=1
where Lw = kδwd/Dδw, αw,n are the positive roots of αwtanαw = Lw for hydrogen and oxygen diffusion, respectively, and d is the half of the sample dimension. The kinetic parameters, kδw and Dδw can be then obtained from the experimentally obtained cw(x, t). The concentration profiles of quenched specimens (with a given t from the annealing treatment) can be analysed e.g. by SIMS (Secondary Ion Mass Spectroscopy) after isotopic exchange. In-situ techniques typically exploit the integrated solution which is a function of t only. Most popular has been following weight changes. Any other properties reflecting nonstoichiometry such as EPR [10] or NMR signals can be used. When the oxygen incorporation in nonstoichiometric oxides leads to the conductivity change, the relaxation of the overall conductivity can be monitored in-situ with a much less experimental complexity [11]. Spatially resolved optical absorption spectroscopy developed in the MPI in Stuttgart, Germany [12,13] uses the fact that the redox state of Fe4+/Fe3+ can be followed in-situ as a function of space (x) and time (t). The absorption at λ = 590 nm of Fe4+ ions is directly related to the local oxygen content and the kinetics of oxygen incorporation have been precisely and unambiguously determined. Recently, we applied this technique to the water incorporation in SrTiO3 [8]. Peculiar non-monotonic kinetics of water incorporation was observed in the optical absorption as well as overall bulk resistance of oxides distinguished by impedance spectroscopy. Based on our observations, the naively expected water incorporation kinetic scheme, represented in Fig. 1, is to be modified into the one given in Fig. 2. After a certain surface reaction step involving water dissociation, the hydrogen component (H) diffuses fast into the oxide reducing the oxide strongly and the oxygen component (O) follows at a slower rate. For hydrogenation, protons and electrons have to move in an ambipolar fashion. For oxygenation, motion of oxygen ions (O2−) and the electrons constitute the coupled fluxes.
the solutions of which have been previously used for the relaxation of bulk conductivity [8]. Non-monotonic conductivity relaxations have been meanwhile reported in other proton conducting perovskite oxides [14,15], supporting that the mechanism we are addressing can be a general one. Two-fold but monotonic relaxations were also observed in oxygen incorporation when cation as well as anion species are comparably mobile in the presence of the sufficient electronic conductivity [16,17]. Optical absorption relaxation in this work clearly indicates that the oxide is over-reduced first and then slowly reoxidized to the final state, while the resistance change is the combined effects of electron holes, oxygen vacancies, and protons [18]. The latter may not be straightforwardly interpreted. Spatially resolved insitu absorption images further revealed the hitherto unknown novel features and allowed the development of an in-depth mechanistic model for the decoupled transport of hydrogen and oxygen. It will be for the first time shown that a characteristic moving boundary diffusion problem is involved in water incorporation kinetics. 2. Experimental The change in the optical absorption coefficient of Fe4+ ions (λ = 590 nm) in the glass-sealed large surface of a 6 × 6 × 1 mm3 SrTiO3 single crystal plate doped with 5 × 1018 cm− 3 Fe according to an ICP (induction coupled plasma spectroscopy) analysis was in-situ monitored at 650 °C by a CCD camera upon a sudden water vapor pressure increase (see the images of Fig. 3). The setup was essentially the same as described previously [12,13] except that a new design of specimen holders was employed, which allowed the observation of the absorption effects up to the very specimen edges. Previously [13,19] the region of ca. 0.5 mm width from the edge of the specimen was blocked by a frame holding the specimen, sealing glass and sapphire plates together. The water vapor pressure was varied from 1.9 kPa using the wet oxygen bubbled through water at 17 °C, to 0.4 kPa set by mixing dry and wet oxygen. All the side faces as well as the sealed large surfaces were fine-polished. Two opposite small faces
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Fig. 3. Normalized integral absorption (filled squares) of the optical absorption images and the bulk resistance by impedance spectroscopy (open squares) upon PH2O change from 0.4 kPa to 1.9 kPa at 650 °C in a Fe-doped SrTiO3 single crystal. The insets in the bottom part show the impedance spectra of the initial, overshooting and final state. Bulk and electrode responses can be separated. The overall response can be represented by the superposition of the hydrogenation and oxidation processes. See the text for details.
in 6 × 1 mm2 were sputter-coated by Pt, while the other sides remained bare (see Fig. 3). The optical image processing and the numerical simulation of the absorption profiles were performed using IDL (Interactive Data Language) programs (RSI, USA). The impedance change of the same specimen with the glass sealing intact was monitored by an LCR meter (HP4284A, USA) using the Pt coating as electrodes under the same condition as in the optical experiment. Thermodynamic calculations were numerically performed using Matlab (Mathworks, USA).
the redox change of Fe ions. Although the water incorporation mechanism represented by Eq. (1) is a pure acid–base reaction, the Fe4+/Fe3+ ratio change is consequent upon water uptake by the mass action equilibria as previously shown [18]. Fig. 4(a) displays the numerically calculated defect concentrations at 650 °C as a function of water partial pressure in oxygen. In the present calculation the total pressure, PH2O + PO2 is fixed to 1 atm in accord with the usual experimental condition, while in the previous calculation [18] PO2 is fixed to 1 atm and PH2O is varied independently. With small
3. Results and discussion 3.1. Non-monotonic relaxation Fig. 3 shows the integral absorption over the 6 × 6 mm2 area of the specimen as a function of logarithmic time. The optical images of the sample indicate the spatially resolved absorption by an artificial color scale. Note that the imprints at the corners are from the specimen holder. The resistance relaxation was obtained from the real part of the impedance at 100 kHz, which roughly corresponds to the bulk resistance as indicated in the impedance spectra of the insets of Fig. 3. In both measurements consistently an overshooting was observed, in contrast to the usual monotonic relaxation. The line curves show that the time response of the resistance and absorption can be described by the superposition of the two relaxation processes in the opposite direction at distinctly different rates, which will be detailed in Section 3.3. 3.2. Thermodynamic consideration To apply the optical absorption spectroscopy for the water incorporation kinetics, water vapor pressure change should lead to
Fig. 4. (a) Calculated defect concentrations of SrTiO3 doped with 5 × 1018 cm− 3 Fe as a function of PH2O at 650 °C buffered with oxygen. (b) Calculated defect concentrations with the ‘frozen-in’ oxygen vacancies from the dry state (in solid lines) indicated by asterisks (*) compared with the equilibrium concentrations (in dotted lines).
J.-H. Yu et al. / Solid State Ionics 181 (2010) 154–162
differences in the numerical estimates the features remain the same for PH2O b 0.1 atm or 104 Pa as shown in Fig. 4. Fig. 4 shows that only 68% of Fe dopants (3.4 × 1018 cm− 3) are effective acceptors, FeTi, in dry condition i.e., PH2O ≅ 1 Pa. Upon increasing PH2O oxygen vacancy concentration, [V⋅⋅ O ], decreases due to the occupation by hydroxide ions or protons (OH⋅O), which leads to a decrease in hole concentrations by oxygen incorporation reaction equilibrium. (The decrease may not be well visible in the scale adopted in Fig. 4(a).) This in turn leads to decrease in Fe4+ ions by intrinsic redox equilibrium and thus in Fe4+ absorption. Note that, with the negligible contribution of Δ[h⋅] as well as of Δ[n′], defect concentrations in Fig. 4(a) satisfy the charge neutrality condition, ::
:
′ ≈2Δ½VO + Δ½OHO : Δ½FeTi
ð10Þ
⋅ Since Δ[Fe′Ti] = Δ[Fe3+] = − Δ[Fe4+], Δ[V⋅⋅ O ] = − Δ[O] and Δ[OHO] = Δ[H], the redox effects can also be expressed in terms of component hydrogen and oxygen mass balance and eventually represented by the net hydrogen amount for redox effects:
4+
−Δ½Fe
3+
≈Δ½Fe
≈Δ½H−2Δ½O = Δ½H−Δ½Hacid−base = Δ½Hredox : ð11Þ
This thermodynamic consideration can be extended to describe the strongly reduced intermediate state of the peculiar non-monotonic relaxation. According to our model in Fig. 2 the overshooting represents the state of the specimen in which the hydrogen has diffused-in but the sluggish oxygen could not catch up. In the defect-chemical analysis, the situation can be described by the ‘frozen-in’ oxygen lattice or a fixed ⁎ oxygen vacancy concentration of the dry state, i.e. Δ[V⋅⋅ O] = − Δ[O]⁎ = 0, where asterisks (⁎) stand for the non-equilibrium situation. The results are displayed in Fig. 4(b) (in solid lines) compared with the case of equilibrium (in dashed lines) as of Fig. 4(a). Under this condition the hydrogen is incorporated only by the redox mechanism
157
and the amount is much larger than the net hydrogen for the redox effect in the final equilibrium, i.e. 4+ ⁎
−Δ½Fe
⁎
≈Δ½Hredox N Δ½Hredox :
ð12Þ
The quasi-thermodynamic calculation straightforwardly illustrates that the origin of the strongly reduced transient state observed by optical spectroscopy can be the sluggish diffusion of oxygen modeled as the fixed oxygen vacancy concentration in the defect-chemical calculation. Note that the hydrogen incorporated with the ‘frozen-in’ oxygen in Fig. 4(b) is still less than the total hydrogen amount in final equilibrium, i.e. ⁎
Δ½H = Δ½Hacid−base + Δ½Hredox N Δ½Hredox N Δ½Hredox :
ð13Þ
To reach the final equilibrium state from such an overshooting state not only oxygen but also some additional hydrogen should be further incorporated. For the illustration purpose a quantitative analysis is performed in the following specific to the current experimental condition, i.e. PH2O change from 0.4 kPa to 1.9 kPa. The diagrams of the upper row in Fig. 5 schematically represent the thermodynamic mass balance. The effects are normalized by the redox variation in the final equilibrium −Δ[Fe4+] ≈ Δ[Fe3+] which amounts to ca. 0.1 × 1018 cm− 3 (Fig. 4). The redox variation upon water uptake is less than 10%. The incorporated hydrogen amount in the final state would lead to 5.6 units of reducing effect (vertical hatched area in the diagram left), if alone, but 4.6 units thereof is compensated by oxygen incorporation (horizontal hatched area in the diagram middle). Note that the redox effect by the doubly charged oxygen is twice the concentration. The compensated redox effects then represent the incorporation of the ‘water’ or the hydration. The incorporated water concentration can be estimated as Δcw = 1/2ΔcH(acid–base) = ΔcO = 4.6/ 2 × 0.1 × 1018 cm− 3 = 0.23 × 1018 cm− 3. Under the condition of the fixed oxygen vacancy concentration the hydrogen incorporation reaches 3.2 units, i.e. ca. 61% of a total of 5.6 units of hydrogen amount
Fig. 5. Upper row diagrams represent the thermodynamic mass balance of hydrogen and oxygen upon PH2O change from 0.4 kPa to 1.9 kPa in terms of redox effects. The scale is normalized by − Δ[Fe4+] ≈ Δ[Fe3+] ≈ 0.1 × 1018 cm− 3. Lower row diagrams represent the mass balance associated with the kinetic processes. The first diagram indicates the thermodynamically estimated hydrogen incorporation with frozen-in oxygen defects which is responsible for strong reduction (Stage I). The mass balance to the final equilibrium should be realized by the later process(es) (Stage II).
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as shown in the lower left diagram in Fig. 5. Stage I thus represents the fast reduction process from the initial state of ‘1’ passing the final equilibrium value of ‘0’ and approaching the quasi-equilibrium state of ‘−2.2.’ To meet the mass balance of the final equilibrium (right diagram) the re-oxidation process (Stage II) shown in the middle diagram involves additional hydrogen incorporation of 2.4 (=5.6 − 3.2) unit, as well as the oxygen incorporation corresponding to 4.6 units of oxidation effect.
3.3. Diffusion modeling Fig. 6 represents the absorption profiles across the width (2d= 0.6 cm) of the specimen at t=120, 240, 360, 480, 600, 720, 1200, and 1560 s (upper graphs) and at t=1560, 1800, 2460, 3660, 6060, 14,460, and 34,860 s (lower graphs). Upper graphs are for the fast reduction period and the lower graphs are for the later oxidation period. The profiles were obtained by integrating the intensity of the central slabs of 50 pixel thickness from the 510×510 pixel images. Fig. 6(a) represents the horizontal profiles with Pt-coated surface as boundaries while Fig. 6(b) is from the vertical lines bounded by the well-polished bare surfaces (see Fig. 3). It should be noted that the profiles are normalized with respect to the initial state of Fe4+ absorption as ‘1’ and the final state as ‘0’ as in the schematic diagrams of Fig. 5. The in-situ profiles from the optical images shown in Fig. 6 suggest that the hydrogenation incorporation can be modeled as predominantly diffusion-controlled one-dimensional problem with the Ptcoated surface boundary. On the other hand, the re-oxidation proceeds with more or less flat profiles over most of the sample area. Little difference was observed between the Pt-coated and the bare surface. The process thus constitutes an apparently surface reaction controlled two-dimensional problem. Based on these features we first tried to analyze the integral effects shown in Fig. 3. As illustrated in the mass balance diagrams in Fig. 5 the analysis of non-monotonic optical relaxation for water incorporation kinetics requires i) a proper deconvolution of the optical effects into hydrogen and oxygen mass balance and ii) apostulation of the decoupled quasithermodynamic driving forces for the hydrogen and oxygen transport, respectively. Fig. 3 shows that the hydrogenation process of Eq. (14) multiplied by an ‘overshooting factor’ P⁎(N1) with respect to the equilibrium change relaxing from the initial state to an overshooting state (the red curves) and the oxygenation of Eq. (15) proceeding from the overshooting to the final state, thus of the magnitude (P⁎ − 1) in the opposite redox effects (green curves) can successfully
describe the experimental relaxation as shown by the black solid lines of the composite effects. The modeling details are in the following. We found that purely diffusion-controlled kinetics for hydrogenation did not satisfactorily describe the short-time behavior. The discrepancy in the short-time behavior can be easily seen in the logarithmic time representation of the integral curves as in Fig. 3. The spatially resolved absorption profiles in Fig. 6 (top left) indicate directly that the surface does not instantaneously equilibrate but approaches to the saturation point over a certain period (ca. 100 s for the current experiment). The behavior may appear to suggest the limited surface reaction kinetics modeled by Eq. (8) but the consequent analysis was not quite satisfactory either. Although the effect of the limited surface reaction rate or the dead volume effect may not be completely excluded, the non-instantaneous equilibration of the surface concentration was found to describe the behavior best. The treatment of the realistic limitation on the stepwise change of the atmosphere can be found in the monograph by Crank [9]. (The situation should be distinguished from the case of the limited surface reaction rate, in which the concentration beneath the surface in the bulk is different from the very surface concentration which is assumed to be in an instantaneous equilibrium with the atmosphere.) For an exponential saturation of the surface concentration, 1 − exp(−γt), the solution follows as [9]: 1 1 MH ðtÞ−MH ð0Þ δ 2 2 δ = 1−expð−γtÞðDH = γd Þ2 tanðγd = DH Þ2 MH ðt = ∞Þ−MH ðt = 0Þ
−
8 ∞ expð−ð2n + 1Þ2 π2 DδH t = 4d2 Þ : ∑ π2 n = 0 ð2n + 1Þ2 ½1−ð2n + 1Þ2 ðDδH π2 = ð4γd2 ÞÞ
ð14Þ On the other hand, the re-oxidation process (Eq. (9)) is assumed to occur by a surface reaction controlled two-dimensional incorporation: ! δ MO ðtÞ−MO ð0Þ 2kO t = 1−exp − ; MO ðt = ∞Þ−MO ðt = 0Þ d
ð15Þ
although the process is non-trivial as will be discussed in the following section. It should be mentioned that the integral response of the superposed relaxations may not unambiguously show the limiting process. The exponential relaxation could be easily taken as the diffusion-limited process [14,15]. The parameters used for the simulated curves are listed in Table 1. In simulating the optical relaxation curve we fixed the ‘overshooting
Fig. 6. (a) Horizontal line profiles for the optical images as shown in Fig. 3 for the reduction period (t = 120, 240, 360, 480, 600, 720, 1200, and 1560 s) (Top) and for the slow oxidation period beyond the overshooting (t = 1560, 1800, 2460, 3660, 6060, 14,460, and 34,860 s); (b) vertical line profiles corresponding to the profiles of (a). The profiles were obtained by integrating the intensity of the central slabs of 50 pixel thickness of 510 × 510 pixel images.
J.-H. Yu et al. / Solid State Ionics 181 (2010) 154–162 Table 1 Parameters for the simulated curves shown in Fig. 3.
Overshooting factor Hydrogenation (Stage I) Oxygenation (Stage II)
Parameters
Optical relaxation
Resistance relaxation
P* DδH/cm2 s− 1 γ/s− 1 kδO/cm s− 1
3.2 8 × 10− 5 0.0065 2.8 × 10− 5
4.0 8 × 10− 5 0.0065 8.5 × 10− 5
factor’ P⁎ to the maximum overshooting value ‘3.2’ observed by insitu optical images and then determined DδH, γ, and kδO values of Eqs. (14) and (15) for the best ‘eye fitting’. The hydrogenation parameters from the optical relaxation, DδH and γ, were found to be satisfactorily applied to the resistance relaxation also. For the oxidation process, however, the surface reaction rate constant kδO from the optical relaxation is significantly smaller (factor of ca. a third) than that from the resistance relaxation. This can be ascribed to the spatial inhomogeneity in Δ[Fe4+] (see the profiles of the lower graphs in Fig. 6), which is not considered in the purely surface reaction controlled oxidation model, Eq. (15), and to the consequent short-circuiting effect along the edge region. While the overshooting factor P⁎ for the absorption effect can be represented by Δ[Fe4+right]⁎ (see Fig. 4(b)), and thus directly compared to the spatially resolved absorption, the factor for the resistance effect, be it related to Δ[h⋅]⁎ in the first place, is more involved, since this is an integral quantity. The latter can be also affected by the mixed conduction of protons and oxygen ions. 3.4. Profile analysis Fig. 7 shows the simulation for the Stage I process shown in the upper graphs of Fig. 6 as one-dimensional symmetrical case. The kinetic parameters used in the profile simulation are shown in Fig. 8.
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The simulation was performed numerically by a finite difference method [9] using IDL programs as in the previous work [12,19]. The appropriateness of the modeling and the goodness of the parameters can be more clearly seen in the profile simulation, since the sequential development of the multiple simulated profiles as functions of x and t is more sensitive to the input models and parameters than the single integral relaxation curves as a function of t only. More detailed and certainly more definite information was thus obtained as described below. 3.4.1. Novel features The profiles for the Stage I process (Fig. 6, upper) are indeed seen to be the composites of the reduction profiles from the hydrogenation thus with an upward curvature and the oxidation profiles slowly developing from the surface with a downward curvature. However, the kinetic mechanism developed for the profile analysis appears to be much more involved than those considered for the integral analysis in the previous section (Section 3.3). Thus far considered models cannot describe the following novel features of the spatially resolved absorption profiles: (i) Not in accord with the predominantly surface reaction controlled oxidation process in Stage II, clearly diffusion-controlled oxidation front was developed from the surface during Stage I. Thus a new parameter, DδO, which was not considered in the integral analysis, is introduced for the Stage I profile analysis (see Fig. 8). The diffusion-controlled oxidation is even more distinctly observed for the bare surface boundaries, Fig. 6(b), due to rather featureless hydrogenation profiles along this direction. (ii) The oxidation magnitude of such a diffusion-controlled process turned out to be the magnitude ‘O’ indicated in Fig. 7, which amounts to only 16% of the magnitude (P⁎ − 1) assumed in the previous analysis (see Section 3.3). The nature of such an overreduced quasi-equilibrium surface state should be further clarified.
Fig. 7. Simulated line profiles (upper) obtained by combining the hydrogenation profiles and oxygenation profiles (lower) for the horizontal profiles with Pt-coated surface (a) and for the vertical profiles with bare surface boundaries (b) for the simulation time tsim = 30, 150, 270, 300, 390, 510, 630 and 1110 s corresponding to the profiles of Fig. 6 with t =tsim + 90 s. The difference of 90 s represents the delay in the specimen response from the recording time since the atmosphere change. The parameters for the simulation are shown in Fig. 8. In the lower graph of (a) the oxygenation profile at tsim = 1100 s is shown in magnification.
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Fig. 8. (Left graph) kinetic parameters used in the profile simulation of Stage I process in Fig. 7: DδH and DδO (exponentially increasing with ΔXH2O); kδH,Pt, kδH,bare, kδO (regardless of the surface condition). kδO(II): from the local absorption relaxation. The surface reaction constants are indicated at ΔXH2O = 0.16 for the wet surface. The average kinetic parameter from integral analysis (Table 1) and those from the oxygen incorporation for the same material [19] are shown for comparison.
It may be ascribed to the preferential hydrogen adsorption on the oxide surface or a hydrogen ‘spill-over’ [20] during the hydrogen incorporation. (iii) The hydrogenation profiles can be satisfactorily described only when the effective sample dimension pffiffi is reduced from the surface by the distance proportional to DδO t . The oxygen diffusion fronts described above were found to serve as dynamic boundaries for the fast hydrogenation incorporation into the inner region of oxide. The modeling of this moving boundary feature in water incorporation will be detailed in the next section. (iv) It has been also noted that the advancing oxidation fronts halted or approached to a steady state after some time (ca. 300 s for the current experiment), while the hydrogen diffusion profile appears to evolve continuously with the oxidation fronts as dynamic or static boundaries during Stage I. (v) According to the modeling in Section 3.3 the broad overshooting maximum in Fig. 3 is the overall compensation effect of the time-varying spatially inhomogeneous redox processes. The spatially resolved absorption profiles however indicate the states are indeed a time-independent steady state for a certain period (i.e. for ca. 1000 s from 1000 s to 2000 s as shown in the inset of Fig. 9). (vi) Persistent, only slowly decaying, near-surface diffusion profiles during Stage II as evidenced in the lower graphs of Fig. 6 are apparently inconsistent with a much flatter middle portion strongly indicating a surface reaction controlled process. The local absorption as well as the integral absorption relaxation, shown in Fig. 9, exponentially relaxed with the apparently similar time constant which is represented by kδO(II) in Fig. 8.
single phase also leads to the development of the concentration profiles with the inflection points. It should be noted that the diffusion pffiffi is nonetheless Fickian i.e. the boundary x moves linearly with t , as long as diffusivity depends on concentration only, not on time, concentration gradient, etc. [9]. The present situation is not a simple moving boundary problem, however. The moving oxygen diffusion front serves as the source plane for the hydrogen incorporation into the inner region of the specimen. As discussed above (Section 3.2) the oxidation of the specimen in the overall reduced specimen by fast hydrogen incorporation is equivalent to hydration of the specimen, i.e. ΔcO = ΔcH2O when ΔcH N 2ΔcO. The surface oxidation layer can be thus considered as the surface hydration layer. It thus makes a physical sense then that the moving front of the hydration layer, not the very surface of the specimen in contact with water vapor, becomes the source plane for further ‘decoupled’ water incorporation into the inner region of the specimen. It is remarkable that, after all, the hydrogenation and oxygenation process are not completely split as depicted in Fig. 2 but ‘coupled’ in a correlated moving boundary diffusion problem. Even with so much understanding of the physico-chemical nature of the phenomena, the modeling presented in Fig. 7 has been achieved after a considerable maneuvering. The bottom line is that the two diffusivities, DδH and DδO should be functionally correlated. Separate adjustment of the two diffusion solutions as was done in the integral analysis (Section 3.3, Fig. 3) never succeeds in describing the two dynamics correlated in space and time. It appears also that the moving boundary aspect should be described by a diffusivity or diffusivities strongly concentrationdependent. Then the ‘water’ concentration, ΔcH2O, would be a natural choice for the common variable for both diffusivities. Two diffusivities modeled to be exponentially increasing with water concentration as shown in Fig. 8 have been found to successfully describe the experimentally observed Stage I profile evolution: Fig. 6 (upper) vs. Fig. 7. Here is a brief description on technical aspects of how we introduced the water concentration dependence of the two diffusivities for the successful simulation of the correlated moving boundary diffusion feature. We calculated the oxygenation profile by a finite difference method according to the standard moving boundary problem [9,21] with the diffusivity exponentially dependent on the concentration of the ‘own’ diffusing species, i.e. oxygen. It should be noted again that the oxygen concentration is equivalent to the water concentration in the presence of sufficient hydrogen concentration. In the magnified profile at tsim = 1110 s one may recognize an inflection point characteristic of the moving boundary problem. The
Although the exact mechanistic model is still unclear, the non-trivial features described in (iv), (v) and (vi) are suggested to indicate the presence of the ‘coupled’ water motion as represented in Fig. 1 without net redox effects, mixed with the decoupled fluxes with redox effects. The ‘coupled’ water motion after some initial period can be consequent upon the slowed-down local flux of hydrogen with the progress. 3.4.2. Concentration-dependent diffusivity A moving boundary diffusion problem can be formally described in terms of the diffusivity with a strong positive concentration dependence [9]. The extreme and representative example is the growth of tarnishing layer or a layer of a second phase between the two neighboring solid phases in which Dδ changes discontinuously at the interface. A strongly concentration-dependent diffusivity in a
Fig. 9. The normalized intensity variation in linear time scale of the entire area of the specimen (total), in the center of the specimen (center), near the Pt-coated surface and near the bare surface. The inset is the magnification near the maximum overshooting point.
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hydrogenation profiles were simultaneously calculated at each time sequence using the diffusivity as another exponential function of the momentary oxygen concentrations. The modeling resulted in the hydrogenation profiles flatter near the surface region with high oxygen concentration continued by portion of a normal diffusion behavior in the inner region with frozen oxygen concentration (see the lower graph of Fig. 7(a)). The upper graph of Fig. 7(a) shows that the combination of the two simulations reproduces the correlated moving boundary feature remarkably well. In Fig. 8 the water concentration is represented as the hydration degree ΔXH2O normalized with respect to the total water incorporation amount upon PH2O change from 0.4 kPa to 1.9 kPa. As described by the feature (ii) in the previous section ΔXH2O varies from 0 (at the moving ‘water’ front) to 0.16 at the surface in the quasi-equilibrium state during Stage I. The water concentration change in the final equilibrium, ΔcH2O, is 0.23 × 1018 cm− 3 (see Section 3.2) and thus the maximum hydration degree of ΔXH2O = 0.16 during Stage I corresponds to 0.37 × 1017 cm− 3 of ΔcH2O. It should be noted thus in Fig. 8 that the variation of the diffusivity by the factor of 6 or 8 over the concentration variation ΔcH2O of ca. 0.37 × 1017 cm− 3 is in fact a huge effect since the oxygen concentration variation from the water incorporation is less than one part per million (ppm) of the lattice oxygen in concentration of 5 × 1022 cm− 3. At such a dilute limit one might usually assume almost concentration-independent diffusivity. Concentration-dependent diffusivities are known to be characteristic of the gas or vapour diffusion in glassy polymers or in the sorption kinetics in porous materials even in small concentrations of diffusants [9,22,23], however. In these systems the diffusants are usually extrinsic elements, not even related to the host material structure. Diffusion of ‘water’ or hydrogen in oxides such as SrTiO3 appears to bear a similarity in this respect, since the diffusants, water or hydrogen, are not the regular constituents of oxides. The strong concentration dependency in the chemical diffusivity can be phenomenologically attributed to the thermodynamic factor with the self-diffusivity or the mobility terms being concentrationindependent. Thermodynamic factors can be obtained from the sorption/intercalation isotherms [22,23] or from the defect chemistry for materials with defects in low concentrations [12,16,17,21]. Similar to the factor ∂aw/∂cw in Eq. (5), for the decoupled hydrogen and oxygen flux, assuming a sufficiently high electronic conductivity and a concentration-independent self-diffusivity of species DH(O), the concentration dependency in the chemical diffusivity DδH(O) in Fig. 8 may be attributed to the thermodynamic factor: Γ HðOÞ =
∂ ln aHðOÞ ∂ ln cHðOÞ
ð16Þ
Since thus far powerful and versatile local equilibrium assumption relating μO, μH and μH2O is clearly invalid in the decoupled transport of hydrogen and oxygen in water incorporation, it is not immediately clear how the relevant thermodynamic factors can be exactly defined and estimated. Furthermore, the spatially resolved in-situ optical spectroscopy directly indicated an over-reduced quasi-equilibrium surface state, rather than the one in equilibrium with the ambient water atmosphere, PH2O, during Stage I process of the present concern, so that the application of the ambient atmosphere for the thermodynamic variable seems not guaranteed. From the defect-chemical calculation of Fig. 4 thermodynamic factors which relate the defect concentrations [OH⋅O] and [V⋅⋅ O ], respectively, with the water activity aw can be estimated, i.e. Γ wH =
∂ ln aw ∂ ln aw ; Γ wO = − ; ∂ ln ½V::O ∂ ln ½OH:O 2
ð17Þ
which are represented as a function of the respective defect concentrations in Fig. 10. It is shown that the thermodynamic factor
161
Fig. 10. Thermodynamic factors ΓwH = ∂lnaw/∂ln[OH⋅O]2 and ΓwO = ∂lnaw/∂ln[V⋅⋅ O] estimated from Fig. 4. They are represented as a function of the respective defect ⋅ concentrations in which [OHO]* is for the frozen-in oxygen vacancies from Fig. 4.
for the equilibrium concentration [OH⋅O], ΓwH is close to one over the concentration range concerned as expected for the dilute defect concentration. On the other hand ΓwH⁎ for [OH⋅O]⁎ in the case of the frozen oxygen lattice distinctly deviates from the ideal solution behavior with increasing concentration. Although not as strong as in DδH shown in Fig. 8 an exponential dependence is clearly indicated. The analysis thus suggests that the strong concentration dependence in the chemical diffusivity should be originated from and thus characteristic of the split non-equilibrium water diffusion. Much stronger non-ideality is shown in the case of the oxygen vacancies. Note that ΓwO⁎ is shown to increase with the oxygen vacancy concentration, which corresponds to the decreasing water activity (Fig. 4). The behavior is apparently contradictory to the strongly δ increasing DO with water concentration (see Fig. 8) and the implication is not immediately clear. It should be noted that ΓwO⁎ ⁎ values near frozen-in situation i.e., Δ[V⋅⋅ O ] ≈ 0 could be extremely large. 3.4.3. Surface reaction constant In the profile analysis we also take the finite surface reaction rate constants kδH and kδO into consideration and the values used in the simulation are indicated in Fig. 8. The surface reaction rates for Ptcoated boundaries, kH,Pt, is shown to be sufficiently large justifying the predominantly diffusion-controlled hydrogenation kinetics. The vertical profiles bounded by the bare surface boundaries can be successfully described, as shown in Fig. 6(b) by the same DδH(cH2O) or DδH(cO) but by a much smaller surface reaction constant for hydrogen incorporation, i.e. kδH,Pt ≫ kδH,bare. It should be noted that the surface reaction rate constant kδH,bare used for the simulation of the vertical profile is an apparent parameter to simulate the hydrogen flux through the Pt-coated surface, i.e., DδH ≈ kδH,bared, in the present onedimensional modeling for each direction. The true surface reaction constant for the bare surface is thus supposed to be even smaller. On the other hand, kδO for the oxygen split from water is found to be sufficiently high regardless of the surface treatment. The situation is in a great contrast to the incorporation of oxygen upon a stepwise change of oxygen partial pressure in which the surface kinetics is greatly influenced by the Pt- or other metal catalytic layers [19], as indicated in Fig. 8(right). 4. Conclusion A mechanistic model for the peculiar non-monotonic water incorporation has been presented. In the presence of a sufficiently
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high electronic conductivity, water incorporation into oxides occurs via an electrically decoupled transport of 2H+ and O2− ions. The charge neutrality of the respective transports is fulfilled by the electronic carriers. The strongly over-reduced transient state is due to the fast diffusion of hydrogen while the sluggish oxygen is lagging behind, and the situation can be described by a defect-chemical calculation assuming the ‘frozen-in’ oxygen vacancies. In-situ monitored spatially resolved optical absorption spectroscopy reveals the associated redox effects unambiguously. The decoupled hydrogen and oxygen transport turned out to constitute a correlated moving boundary problem, which can be quantitatively described by chemical diffusivities exponentially increasing with the hydration. The hydrogen incorporation thus appears to occur from the advancing ‘wet’ fronts established by the oxygen incorporation following behind not from the sample surface in contact with water vapor. While the Pt coating is shown to be catalytic for the hydrogen incorporation reaction, the oxygen incorporation from the split water appears similarly facile in Pt-coated and bare surface. Acknowledgment J.-S. Lee acknowledges the support by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2007-412-J02002).
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