Toward a Quantitative Understanding of the Electric Field in Thermal

Letter pubs.acs.org/JPCL

Toward a Quantitative Understanding of the Electric Field in Thermal Metal Oxidation and a Self-Consistent Wagner Theory Tian-Le Cheng* and You-Hai Wen* National Energy Technology Laboratory, 1450 Queen Avenue Southwest, Albany, Oregon 97321, United States S Supporting Information *

ABSTRACT: The electric field in the growing oxide film is important to the kinetics and mechanism of metal oxidation. However, understanding of the essential characteristics of the electric field during oxidation remains insufficient. A specialcase analytical model is presented that provides a unified understanding for the electric field from the viewpoints of kinetics and thermodynamics. More general cases are studied by computer simulations that show similar characteristics in the electric field. In particular, simulations indicate that in many situations, the electrostatic potential drop across the bulk oxide is limited to ∼kBT/e, which means that the total electrostatic potential drop across the oxide film, if on the order of 1 V by rough estimation, should have contributions mostly from the electrified interfaces. Finally, regarding the Gibbs−Duhem relation, the commonly used isobaric assumption for the diffusing species is refuted. The results contained herein also provide a self-consistent understanding of Wagner’s oxidation theory. SECTION: Physical Processes in Nanomaterials and Nanostructures

R

these assumptions is, in general, consistent with the physical situation. On the other hand, Fromhold derived and compared the electric field from the coupled-currents condition with that from the Gibbs−Duhem relation assuming local equilibrium, and the two fields significantly disagree with each other. Fromhold then denied even the local equilibrium assumption and doubted the self-consistency of Wagner theory.8,9 Quite frequently, the electric field is not explicitly considered. Instead, an “effective ion diffusivity”10 is used that, however, cannot represent all electric field effects. For example, when the characteristic size of the system is on the order of the Debye length, there could be various space charge effects.11−13 In particular, the electric field can modify the electron band profile and the defect structure of the oxide.14−17 The resultant effect has been explored for the role of a “reactive element” in the protective character of Al2O3 films, which is important for design of nickel- and iron-based superalloys.18 In this work, we aim to provide a self-consistent and quantitative understanding for the electric field in the growing oxide film during oxidation with minimal assumptions. On the basis of a prototype oxidation model, we clarify the physical situation by analytically deriving the electric field from kinetic equations, which approaches the appropriate thermodynamic limit. Subsequently, representative realistic cases are studied by computer simulations based on a mesoscopic diffuse-interface oxidation model,13 showing similar characteristics of the electric field, and, particularly, with the electrostatic potential drop

eactive metals may survive oxidizing gases for a practical life cycle due to formation of a protective thermally grown compound film.1−3 If the compound film, such as an oxide layer, remains intact and continuous, the overall oxidation process typically includes ionization of the oxidant at the gas− oxide interface and the metal at the oxide−metal interface and transport of the ions through the oxide. The latter is mostly the rate-limiting process especially in the thick film stage. Mainly by assuming an electric field that regulates the transport of charged species with different mobilities in order to maintain zero net electric current through the film (i.e., the “coupled-currents” condition) and some local equilibrium based conditions, Wagner developed a parabolic growth law for oxidation kinetics4 that is fundamental to high-temperature metal oxidation research.1,3 Unfortunately, the electric field is not solved in Wagner’s theory. Extensive studies on the electric field in oxidation were first performed by Fromhold et al.5,6 The fact that there is an electrostatic potential drop across the oxide film has been widely accepted; however, quantitative understanding of the electric field remains insufficient. In the literature, and due to the complexity of the problem involved, various assumptions have been invoked to account for the electric field, with two representative ones apparently contradictory, (1) assuming a homogeneous electric field throughout the oxide5 and (2) presolving the electric field by using the Poisson−Boltzmann equation (PBE), which assumes the system is at or near thermodynamic equilibrium.7 The linearization of PBE leads to a Debye−Hückel-type solution where the electric field away from the surface/interface decays to zero exponentially, which is in contrast to assumption (1). It will be shown that neither of © 2014 American Chemical Society

Received: May 1, 2014 Accepted: June 16, 2014 Published: June 16, 2014 2289

dx.doi.org/10.1021/jz5008627 | J. Phys. Chem. Lett. 2014, 5, 2289−2294

The Journal of Physical Chemistry Letters

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across the bulk oxide being limited to ∼kBT/e. Finally, the longstanding puzzle raised by Fromhold regarding the local equilibrium assumption is explained. The total flux of species i is generally considered to consist of a diffusion term and an electric field drifting term, given by1 Ji = −Di∇ci + ϖiEci

i = 1, ..., n

hand, at thermodynamic equilibrium, assuming that the mobile species obey the Maxwell−Boltzmann distribution, a Debye− Hückel-type solution can be derived as E = Ae±x/lD′, where the Debye length is lD′ = [εkBT/NAe2(c1̅ + c2̅ )]1/2 (the overbars denote the mean concentration). Clearly, this solution (i.e., complete Coulomb screening) conforms to eqs 6 and 7 when E∞ = 0 as c ̅ = c1̅ + c2̅ is restricted by charge neutrality. Equation 7a clarifies that it is the difference of the intrinsic diffusivity of the charged defects that causes the nonzero bulk electric field, and this field does not vanish anywhere unless D1 = D2 or when the mass fluxes become zero (at thermodynamic equilibrium). Now returning to eq 5, the condition for neglecting the quartic term is |E(dE/dx)| ≪ (kBT/e)|d2E/dx2|, which together with eqs 6 and 7 gives

(1)

where Di is the diffusivity, ϖi the electromobility, and E the macroscopic electric field. In this work, each diffusing species i is assumed to be dissociated point defects with effective charge of zie. By the Nernst−Einstein relation,1,19 ϖi = zieDi/kBT, where kB is the Boltzmann constant and T is Kelvin temperature. When the oxide has small nonstoichiometry, the frequently used steady-state approximation can be justified.13,20 Subsequently, by the continuity condition, ∂ci/∂t = −∇·Ji = 0, meaning each flux is position-independent, that is, Ji(x,t) = Ji0(t). Meantime, the coupled-currents condition (∑i zieJi = 0) has to be satisfied because otherwise the surface charge would continuously accumulate. In addition, by Gauss’s law ε∇·E = ρ =

∑ zieNAci

kBT (8) e This condition is analogous to the small electric potential assumption in the Debye−Hückel approximation. Thus, the solution of eq 6 holds except in the close vicinity of the surface/ interface where there could be high electric field and different atomic structure from the bulk (Figure 1). The above solution |E − E∞|lD ≪

(2)

i

where NA is Avogadro’s constant, ε the dielectric constant of the oxide, and ρ the charge density. Equation 2 can be combined with eq 1 to yield −

ε d2E = NAe dx 2

⎛ zi

∑⎜ i

⎝ Di

Ji0 −

e 2 ⎞ zi ciE⎟ kBT ⎠

(3)

To obtain a clear picture of the nature of the electric field, we first conceive of a special case where z1 = z2 = −1, that is, there are two negatively charged transporting species (such as electrons and interstitial anions).13 For charge compensation, a background positive charge field (z3 = 1) is assumed. This field is uniform with a fixed density c3 ≡ c.̅ Here, c3 can be considered as the effective charge field of intentional/ unintentional impurities (namely, the oxide is doped n-type). Because J3 = 0 (as D3 = 0), the coupled-currents condition leads to J1 = −J2 = J0, and the summation in eq 3 is taken over two species, giving −

e ⎛ 1 ⎞ 0 ε d2E ε dE ⎞ ⎛ 1 E⎜ c ̅ − + − ⎟J ⎟=⎜ 2 NAe dx kBT ⎝ NAe dx ⎠ ⎝ D2 D1 ⎠

Figure 1. Schematic of the electric field profile near an interface in quasi-steady-state ionic diffusion (with zero net electric current). The field generally includes a decaying term plus a permanent bulk term.

provides a clear and physically consistent understanding for the electric field in metal oxidation, namely, a screening term plus a remnant term of pure kinetic origin. For the same case, our model and simulation results recover Wagner’s theory at the thick film stage and meanwhile accurately capture the Debye decay length and the uniform bulk electric field,13 both in agreement with this analytical solution. In eq 7a, J0 can be eff estimated by J0 = −Deff 1 c/L, ̅ where D1 ∞is roughly of the same order as D1 when D1 ≪ D2; therefore, E can be estimated to a first order as E∞ ≈ −kBT/eL. That is, the bulk electric field has to attenuate with increasing film thickness with the potential drop across the bulk oxide limited to ∼kBT/e. This result, if proved to be of generality, would be a distinctive feature of the electric field in metal oxidation. In general cases, quite commonly oppositely charged ionic and electronic defects are simultaneously generated at one of the gas−oxide−metal interfaces, and thus, the major diffusing species may have different valences (i.e., z1 ≠ z2) and different mobilities. Even without impurities/dopants, a “pure” metal oxide can still be n-type or p-type with a certain degree of nonstoichiometry. For example, nickel oxide is often slightly metal-deficient such as Ni1−xO. During high-temperature

(4)

Suppose that in the bulk oxide, the electric field is small so that in eq 4, the quartic term regarding the electric field can be neglected. Equation 4 then reduces to −

D − D2 0 ε d2E ec ̅ + E= 1 J 2 NAe dx kBT D1D2

(5)

This second-order ordinary differential equation has a general solution of the form E = E∞ + Ae±x / lD

(6)

where parameter A depends on the boundary condition and E∞ =

kBT D1 − D2 0 J ec ̅ D1D2

(7a)

lD =

εkBT /NAe 2 c ̅

(7b)

Equation 6 delineates incomplete Coulomb screening of the electric field in quasi-steady-state ionic diffusion. On the other 2290



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oxidation of nickel, both holes and metal vacancies are produced at the oxide surface according to 1/2O2 = VNi″ + 2h• + OXO, or 1/2O2 = VNi′ + h• + OXO (Kröger−Vink notation) under different oxygen partial pressures.1,21 Similar mechanisms were also reported for chromium, cobalt, and so forth.1 Zinc oxide (Zn1+yO) is usually metal excess n-type, and oxidation of zinc can be dominated by outward diffusion of interstitial zinc ions.3 Inward anion diffusion is also a possible mechanism.3 For a general situation where z1 ≠ z2, the electric field is difficult to solve analytically. Unfortunately, the electric field is also difficult to measure;22 therefore, computer simulation is probably the most logical solution choice. Numerical simulations are performed via a multiscale electrochemistrybased diffuse-interface oxidation model. Special care is taken to avoid numerical drifting during the long simulation of diffusion. The main algorithms have been verified and documented,13 and in this work, a permanent surface charge term at the gas−oxide interface is added to study the surface potential effect (see the Supporting Information for more technical details). The major transport species are assumed to be doubly charged metal ions and electrons (such as during the growth of ZnO). In this situation, local charge neutrality would mean 2c1 − c2 ≈ 0 in the bulk oxide as c3 = 0 is assumed. Systems with negatively charged metal vacancies and holes are parallel to this case. For compounds with different stoichiometric ratios, the situation is also expected to be similar. Both the gas−oxide and oxide− metal interfacial reactions are assumed to be close to equilibrium. Simulations show that quasi-neutrality is spontaneously approached in the bulk oxide as the system approaches steady state, but the bulk electric field yet has finite magnitude as long as the transporting ionic and electronic defects have different mobilities. The magnitude of the bulk electric potential drop, I II which is defined by |Δbϕ| = |∫ xII xI − E dx|, where x and x position the two ends of the quasi-neutral zone near the gas− oxide and oxide−metal interfaces, is limited to ∼kBT/e. Δbϕ is equal to zero when D1 = D2. All of these characteristics conform to the prototype model predictions. From eq 3, the second-order derivative of the electric field in the bulk oxide can be neglected in view of the smoothly changing field; therefore, E ≈ (kBT/e)∑i (zi/Di)J0i /∑i z2i ci. Therefore, when the total defect concentration has a gradient, the bulk electric field becomes inhomogeneous. Simulation results show that despite the electric field, the concentration profiles do not deviate significantly from linearity in the bulk oxide (Figure 2a), where roughly ci(x) ≈ cIi + (cIIi − cIi )(x − xI)/ (xII − xI). Here, cIi and cIIi are defect concentrations at xI and xII; therefore, J0i = −Deff(cIIi − cIi )/(xII − xI), where Deff is estimated as, by assuming the coupled-currents condition and local charge neutrality, Deff = (z2 − z1)D1D2/(z2D2 − z1D1) (the general ambipolar diffusion coefficient). The bulk electric field can thus be fit by Efit(x) = (kBT/e)[(D1 − D2)/(z1D1 − z2D2)][(cIIi − cIi )/(cIIi (x − xI) + cIi (xII − x))].Integrating this equation for Δbφ then gives Δbϕ = ϕII − ϕI ≈

kBT D2 − D1 c II ln 1 I e z1D1 − z 2D2 c1

Figure 2. Representative simulation result of quasi-steady-state ionic diffusion in an oxide film. The length unit is l0 = (ε0kBT/NAe2c0)1/2, where c0 is a reference concentration. The defect diffusivity ratio is DM/De = 1:10. (a) Defect concentration profiles (solid lines) in the thickness direction; (b) charge and electrostatic potential distribution corresponding to (a); and (c) charge distribution near the gas−oxide interface for a special case without permanent surface charge. [Other conditions are the same as thosei n (a) and (b). (Inset) The detailed defect density distribution with coordinates is the same as that in (a).]

is verified by the exact numerical solutions to be a good estimate for the quasi-neutral bulk zone. When D1 ≪ D2, Δbϕ ≈ (kBT/e) ln(cIIi /cIi ). cII1 should be close to the maximum ionic defect concentration in the oxide. The question now is, how small can cI1 be? In reality, a certain quantity of surface charge generally exists on both oxide surfaces;1,6,14 therefore, the linear concentration approximation is only valid at a characteristic distance from the interface (roughly this distance is on the order of a “local” Debye length). Therefore, cI1 is normally of finite value even when the environmental oxidant activity is high (see Figure 2b). From eq 9, Δbϕ should be independent of the dielectric constant of the oxide, which is also computationally verified. Interestingly, Δbϕ is also insensitive to the surface charge/ potential at the interface of the defect sink (here the G−O interface) (Table 1). Even if this interface is free of surface charge, negative space charge spontaneously aggregates therein to create the required bulk field. As a result, an apparent double layer is formed solely by space charge with a potential drop of

(9)

The end locations of the quasi-neutral zone (xI and xII) are defined by a critical space charge density, ρ*. The choice of ρ* is somewhat arbitrary, but in rough magnitude, it should be larger than εdEfit/dx, which is the space charge of kinetic origin (see the Supporting Information for more details). Equation 9 2291



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Table 1. Simulated Δbϕ Dependence on Increasing Surface Potential ΔϕOG = ϕI − ϕG, with a Change in the Amount of Prescribed Surface Chargea ΔϕOG/(kBT/e) Δbϕ/(kBT/e)

0.98 1.13

2.02 1.22

4.00 1.21

5.96 1.22

8.02 1.23

The exact value of Δbϕ could be slightly different if ρ* is chosen differently (in this work, ρ* = 0.005NAec0). ΔϕOG* = 0.98 actually corresponds to the situation where the prescribed surface charge is zero (cf. Figure 2c). a

approximately kBT/e (Figure 2c), and Δbϕ is still ∼kBT/e. The maximum defect concentration affects both cI1 and cII1 and therefore makes minor changes to Δbϕ. It is found that Δbϕ weakly depends on the film thickness but tends to saturate with increasing film thickness (Table 2). These results indicate that Table 2. Simulated Dependence of Δbϕ on the Total Oxide Film Thicknessa L/lD* Δbϕ/(kBT/e)

40 0.88

80 1.21

160 1.66

320 2.07

480 2.18

Figure 3. Simulation results of different electrostatic potential profiles with different defect mobility ratios (ϕ at the middle plane of the film is matched); in the bulk oxide, the potential gradient is positive whenDM < De, zero when DM = De, and negative when DM > De. In all cases, |Δbϕ| falls within ∼kBT/e. The same amount of surface charge at the oxide surface and the same inner potential difference at the oxide− metal interface are assumed for all cases (these values can be arbitrary).

640 2.23

The film thickness is normalized by a characteristic length lD* = (εkT/NAe2c0)1/2 = (εr)1/2l0, which is close to the Debye length at the defect injection interface (in Table 1, L/lD* = 80). Here, εr = ε/ε0 = 5; ΔϕOG, and ΔϕOM ≈ 4kBT/e. a

shows that the electric field solved from Gauss’s law and kinetic equations asymptotically approaches that from thermodynamics with increasing film thickness; therefore, it would be surprising if the solution is incompatible with the local equilibrium assumption. Actually, the kinetic equation (eq 1) together with the Einstein relation could be considered as conforming to the local equilibrium assumption for each species according to the linear irreversible thermodynamics.13 Next, it is shown that the distinct derivation of electric field from the Gibbs−Duhem relation stems from the inconsistent isobaric assumption (∇p = 0). Instead of simply assuming ∇p = 0, it should be noted that when local equilibrium is approximated, the pressure variations can be derived from the composition and chemical potential terms through the Gibbs−Duhem relation. The latter is the intrinsic relationship determined by thermodynamics, even if the pressure is not explicitly defined. For example, in a recent diffuse-interface modeling work, the pressure difference across a curved interface (Laplace pressure) is derived according to the Gibbs−Duhem relation, and the result is in agreement with the Young−Laplace equation.26 In the current model, the pressure gradient can be obtained according to ∇p = ∑i ci∇μ̅i. The summation ∑i ci∇μ̅i is obviously not zero because both the metal ions and electrons migrate in the same direction; therefore, the electrochemical potential gradients have the same sign. To understand the pressure gradient of migrating point defects, it would be more straightforward to think about a single-diffusing-species situation, for example, neutral interstitial oxygen molecules diffuse in a perfect crystal. The above formula now reduces to ∇p = c∇μ. Prior to thermodynamic equilibrium, ∇μ is always nonzero and so is ∇p. For perfect Fickian diffusion, it is found that the thermodynamic pressure p just follows the ideal gas law with respect to the diffusing species. In the current model, ∇p is part of the solution depending on ci and ∇μ̅i and thus the electric field. The latter conforming to Gauss’s law automatically maintains the coupledcurrents condition at quasi-steady state.13 Therefore, Fromhold’s argument just demonstrates that the isobaric assumption

Δbϕ could in general be on the order of kBT/e as approximated by eq 9. Thus, if the total electric potential drop Δtϕ is on the order of 1 V, as commonly estimated,1,23,24 the remaining potential drop would have to come from the interfacial zones (note that Δbϕ ≈ kBT/e is a rather small value, and at 900 °C kBT/e ≈ 0.1 V). In this situation, electric double layers must be present in close proximity to one or two of the gas−oxide− metal interfaces to account for the remaining potential drop. Consequently, if the n−p electronic transition were to occur in the oxide,18 it would likely be near one of the interfaces rather than in the middle of the bulk oxide. The dependence of the electric potential profile on the defect diffusivity ratio25 is also simulated and shown in Figure 3. This figure shows the essential features of the electric field/potential by this model in which the electric field satisfies Gauss’s Law (or, equivalently, Poisson’s equation) and maintains the coupled-currents condition. This result presents a “big picture” for understanding the band profile of a growing oxide film. The electrostatic potential drop across the bulk oxide is neither negligibly small, as predicted by the solution of PBE over the entire oxide, nor is it equal to the total potential drop as implied by the homogeneous field approximation. The main assumptions used in this work can be considered as only a subset of those in Wagner’s approach, and they conform to be (a) a uniform, isothermal, and coherent oxide film, (b) the coupled-currents condition (steady state), and (c) the Nernst−Einstein relation. Nevertheless, the Gibbs−Duhem relations are not used here. In a material system with charge interactions, when local equilibrium is assumed, the electrochemical Gibbs−Duhem relation can be defined as Vdp − SdT = ∑i Nidμ̅i, where μ̅i is the electrochemical potential of species i.8 Following Wagner’s assumption of an isothermal and isobaric condition for the diffusing species, Fromhold derived the electric field according to ∑i ci∇μ̅i = 0. This electric field is generally distinct from that deduced from the coupled-currents condition and for which Fromhold doubted the validity of the local equilibrium assumption.9 Our analytical model clearly 2292



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(5) Fromhold, A. T.; Cook, E. L. Kinetics of Oxide Film Growth on Metal Crystals: Electronic and Ionic Diffusion in Large Surface-Charge and Space-Charge Fields. Phys. Rev. 1968, 175, 877−897. (6) Fromhold, A. T. Theory of Metal Oxidation, Fundamentals, Defects in Crystalline Solids; North-Holland: New York, 1976; Vol. 1. (7) Xu, Z. J.; Rosso, K. M.; Bruemmer, S. Metal Oxidation Kinetics and the Transition from Thin to Thick Films. Phys. Chem. Chem. Phys. 2012, 14, 14534−14539. (8) Fromhold, A. T.; Coriell, S. R.; Kruger, J. Transport and Thermodynamic Analysis of Steady-State Currents in Solids. J. Phys. Soc. Jpn. 1973, 34, 1452−1459. (9) Fromhold, A. T. Theory of Metal Oxidation, Fundamentals, Defects in Crystalline Solids; North-Holland: New York, 1976; Vol. 1, Chapter 6, pp 121−130. (10) Kröger, F. A. The Chemistry of Imperfect Crystals, 2nd ed.; NorthHolland: Amsterdam, The Netherlands, 1974; Vol. 3. (11) Maier, J. Ionic Transport in Nano-Sized Systems. Solid State Ionics 2004, 175, 7−12. (12) Zhang, Z. F.; Jung, K.; Li, L.; Yang, J. C. Kinetics Aspects of Initial Stage Thin γ-Al2O3 Film Formation on Single Crystalline βNiAl (110). J. Appl. Phys. 2012, 111, 034312. (13) Cheng, T.-L.; Wen, Y.-H.; Hawk, J. A. Diffuse-Interface Modeling and Multi-Scale-Relay Simulation of Metal Oxidation KineticsWith Revisit on Wagner’s Theory. J. Phys. Chem. C 2014, 118, 1269−1284. (14) Richter, N. A.; Sicolo, S.; Levchenko, S. V.; Sauer, J.; Scheffler, M. Concentration of Vacancies at Metal−Oxide Surfaces: Case Study of MgO(100). Phys. Rev. Lett. 2013, 111, 045502. (15) Hine, N. D. M.; Frensch, K.; Foulkes, W. M. C.; Finnis, M. W. Supercell Size Scaling of Density Functional Theory Formation Energies of Charged Defects. Phys. Rev. B 2009, 79, 024112. (16) Li, X.; Finnis, M. W.; He, J.; Behera, R. K.; Phillpot, S. R.; Sinnott, S. B.; Dickey, E. C. Energetics of Charged Point Defects in Rutile TiO2 by Density Functional Theory. Acta Mater. 2009, 57, 5882−5891. (17) Yu, J. G.; Rosso, K. M.; Bruemmer, S. M. Charge and Ion Transport in NiO and Aspects of Ni Oxidation from First Principles. J. Phys. Chem. C 2012, 116, 1948−1954. (18) Heuer, A. H.; Nakagawa, T.; Azar, M. Z.; Hovis, P. B.; Smialek, J. L.; Gleeson, B.; Hine, N. D. M.; Guhl, H.; Lee, H. S.; Tangney, P.; et al. On the Growth of Al2O3 Scales. Acta Mater. 2013, 61, 6670− 6683. (19) Weinert, U.; Mason, E. A. Generalized Nernst−Einstein Relations for Nonlinear Transport Coefficients. Phys. Rev. A 1980, 21, 681−690. (20) Fromhold, A. T. Kinetics of Oxide Film Growth on Metal Crystals. I.: Formulation and Numerical Solutions. J. Phys. Chem. Solids 1963, 24, 1081−1092. (21) Haugsrud, R. On the High-Temperature Oxidation of Nickel. Corros. Sci. 2003, 45, 211−235. (22) The voltage commonly measured by applying electrodes is the electrochemical potential difference of electrons (divided by −e), not equal to the in situ electrostatic potential drop. (23) Jeurgens, L. P. H.; Sloof, W. G.; Tichelaar, F. D.; Mittemeijer, E. J. Growth Kinetics and Mechanisms of Aluminum-Oxide Films Formed by Thermal Oxidation of Aluminum. J. Appl. Phys. 2002, 92, 1649−1656. (24) Quantitative evaluation of the total electrostatic potential difference is difficult. It is expected to be on the same order of the thinfilm stage Mott potential if virtual electronic equilibrium across the oxide is established, assuming that the electron activities at the two oxide surfaces do not significantly change from those at the thin-film stage. The Mott potential is estimated to be on the order of 1 V, as in refs 1 and 23. (25) In most metal oxides, electronic conductivity is much higher than ionic conductivity. However, there are some cases in which the ionic conductivity is considerable or even predominant; see: Kofstad, P. High Temperature Corrosion; Elsevier: London, 1988; pp 124−125

is problematic, and without it, the incompatibility issue is absent. When only one ionic species is predominant, however, Wagner’s formula can be derived without using the Gibbs− Duhem relations (thus, without the need to know ∇p).10,13 The situation that one ionic species is predominant (i.e., significant inward oxidant diffusion and outward metal diffusion do not occur simultaneously), as assumed in this work, is actually quite common.1−3 For Wagner’s theory, the best quantitative verification obtained is just under this limiting situation (for the growth of CoO2,27), which also supports this analysis.

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ASSOCIATED CONTENT

S Supporting Information *

Computational details and additional simulation results, including the electric field and space charge in the bulk oxide. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (T.-L.C.). *E-mail: [email protected] (Y.-H.W.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge Drs. Jeffrey A. Hawk and Bryan D. Morreale for carefully reading the manuscript and making helpful suggestions. We would also like to thank the Strategic Center for Coal, Cross-Cutting Research, Dr. Susan Maley, Technology Manager, for supporting this ORD activity through the IPT project led by Dr. David E. Alman. T.-L.C. acknowledges the Postgraduate Research Program operated by Oak Ridge Institute for Science and Education (ORISE) and the Extreme Science and Engineering Discovery Environment (XSEDE) that is supported by National Science Foundation Grant Number OCI-1053575. This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by tradename, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

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REFERENCES

(1) Atkinson, A. Transport Processes during the Growth of OxideFilms at Elevated-Temperature. Rev. Mod. Phys. 1985, 57, 437−470. (2) Kofstad, P. High Temperature Corrosion; Elsevier: London, 1988. (3) Birks, N.; Meier, G. H.; Pettit, F. S. Introduction to the HighTemperature Oxidation of Metals; Cambridge: New York, 2006. (4) Wagner, C. Z. Phys. Chem. B 1933, 21, 25. 2293



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(26) Cheng, T.-L.; Wang, Y. U. Shape-Anisotropic Particles at Curved Fluid Interfaces and Role of Laplace Pressure: A Computational Study. J. Colloid Interface Sci. 2013, 402, 267−278. (27) Lawless, K. R. Oxidation of Metals. Rep. Prog. Phys. 1974, 37, 231−316.

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Toward a Quantitative Understanding of the Electric Field in Thermal

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