Non-equilibrium point defect model for time-dependent passivation of metal surfaces

Transcription

1 Electrochimica Acta 46 (2001) Non-equilibrium point defect model for time-dependent passivation of metal surfaces Balaji Krishnamurthy, Ralph E. White, Harry J. Ploehn * Department of Chemical Engineering, Uniersity of South Carolina, Swearingen Engineering Center, Columbia, SC 29208, USA Received 21 January 2001; received in revised form 9 May 2001 Abstract This work presents an improved point defect model for the time-dependent formation of passive oxide films on metal surfaces. Like previous point-defect models, the present model assumes that charged defects, or vacancies, carry current across the growing oxide film. However, we treat the vacancies explicitly as material species which participate in oxide formation and dissolution reactions formulated for arbitrary oxide stoichiometry. The model includes boundary conditions, based on jump mass balances from formal continuum mechanics, that relate vacancy fluxes to the interfacial reaction rates as well as the motion of the film boundaries. Thus, unlike previous models, this model treats the film growth process formally as a moving boundary problem. Casting the equations in dimensionless form yields the key dimensionless groups. The dependence of the film growth rate on these groups can be rationalized in simple physical terms. The predicted trends in film growth rate and current density agree qualitatively with experimental data for nickel passivation, although the model parameters have not been optimized to achieve good agreement with current density data. The model provides a starting point for incorporating better descriptions of interfacial reaction kinetics within boundary conditions based on rigorous continuum mechanics Elsevier Science Ltd. All rights reserved. Keywords: Ionic defects; Passive film; Time dependence; Moving boundary; Film growth rate 1. Introduction Understanding the formation of passive films on metal surfaces is critically important for using metals in structural and electrochemical applications. In this context, theoretical models are needed to rationalize the observed rate of film thickness growth and the dependence of steady-state thickness (if one exists) on applied potential and solution ph. Early phenomenological models, including variants of the high-field model [1 3], proposed different rate-limiting mechanisms that yielded exponential field dependence of the film growth * Corresponding author. Tel.: ; fax: address: ploehn@sc.edu (H.J. Ploehn). rate, leading to logarithmic or inverse logarithmic variation of film thickness with time. However, Battaglia and Newman [1], echoing others [3,4], point out that none of [these] models is completely consistent with the experimentally observed temperature and potential dependences. Various models based on the existence of defects within the film have also been proposed. For example, MacDougall [4,5] argues that a continually decreasing level of film defects can explain the observed logarithmic growth law. Data from more recent experiments [6] seem to support this argument. MacDougall s defect model, albeit qualitative, introduces the concept that film growth may be related to evolution of the distribution of film defects. Macdonald and coworkers (see Refs. [2,7] and references cited therein) carry this idea forward in their development of the point defect model (PDM). In the /01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S (01)

2 3388 B. Krishnamurthy et al. / Electrochimica Acta 46 (2001) PDM, charged defects, or vacancies, carry current through the film. Unlike the high-field model, the PDM assumes that the electric field in the film does not vary with applied potential or film thickness. By postulating equilibrium conditions at the metal/film and film/solution interfaces, the PDM establishes boundary conditions that set the interfacial concentrations of defects as functions of applied potential, solution ph, and film thickness. This leads to analytical solutions for the distribution of cationic and anionic vacancies across the film as well as the steady-state film thickness. In addition, Macdonald et al. have emphasized the development of diagnostic criteria for identifying the dominant charge carrier and reaction mechanism. The time-dependence of film thickness or current density has not been addressed by the PDM until recently [2], yet this publication does not explore time-dependent phenomena in detail. Overall, the PDM s development of mass/ mole transport equations for non-material vacancy species is problematic in the context of continuum mechanics. Battaglia and Newman [1] (BN) have developed a comprehensive time-dependent model for oxide film growth based, at least in part, on the PDM. The BN model accounts for transport of vacancies, interstitials, and electrons within the film. By viewing the diffusion and migration of various species in terms of sequences of reactions, BN obtain a high-field expression (exponential dependence of current density on applied potential) for species transport rate. The BN model also accounts for space charge effects by including the Poisson equation, although this (as well as the high-field rate expression) is not considered for the specific case of iron/iron oxide films. The boundary conditions correctly recognize the relationship among species fluxes, interfacial reaction rates, and a characteristic lattice velocity. This provides a rigorous, general treatment of interfacial kinetics and film growth rate, although BN do not explore these areas in detail. Fig. 1. Schematic diagram showing the transport processes in a barrier film that separates metal and electrolyte phases. Interfacial reactions are given in the text. For a variety of reasons, we decided to take a fresh look at time-dependent formation of passive oxide films on metal surfaces, viewing the situation explicitly as a moving boundary problem. Starting with the PDM and BN models as our foundation, we formulate the timedependent mole balances for cationic and anionic vacancy species through a growing oxide film. However, the present approach differs from that of the PDM in several respects. First, we view the defects as material species, including units of oxide lacking either a metal cation (the negatively charged cationic vacancy species ) or an oxygen anion (the positively charged anionic vacancy species ). Unlike the PDM, we assume that the electric field within the film varies inversely with film thickness, holding the applied potential constant. Building upon the BN model, we formulate rigorous jump mass balances, producing boundary conditions that relate species fluxes to the rates of interfacial reactions and the velocities of the moving film boundaries. Moreover, casting the model in dimensionless form reduces the number of independent model parameters and identifies relevant dimensionless groups that control the passivation process. After performing a limited parametric study, we compare the predictions of the model with experimental data for the growth of oxide films on nickel. 2. Continuum transport model 2.1. Assumptions The PDM [2,7] and BN [1] models provide the foundation for the development of a rigorous continuum model for transport of vacancy species through a barrier oxide film that separates a metal from an electrolyte solution (Fig. 1). To simplify the problem, we invoke several assumptions. (1) The domains of the metal, film, and electrolyte phases (Fig. 1) have planar symmetry with no variations in the lateral dimensions. Transport is one-dimensional (coordinate z) with the z=0 located at the film/electrolyte (FE) interface. (2) The model employs a frame of reference that makes the FE interface stationary. Thus the film and metal phases have finite, non-zero velocities. (3) The film phase consists primarily of a metal oxide with the general stoichiometry M x O xy/2 where y is the metal valence and x=mod(y,2) +1. The molar concentration of oxide, c o, is constant within the film. (4) The oxide film contains various defects [1,2,7], primarily cationic and anionic vacancy species, (M x 1 O xy/2 ) y and (M x O xy/2 1 ) 2+, denoted more compactly in Kroger Vink notation as V M y and V O 2+, respectively. We assume that the molar concentrations of vacancy species, c c and c a, are small compared to c o.

3 B. Krishnamurthy et al. / Electrochimica Acta 46 (2001) y 2+ (5) Only V M and V O are transported through the oxide, presumably through sequences of reactions [1,2,7] that recombine metal and oxygen atoms in the lattice. Rather than considering the details of such reactions, vacancy transport is treated as a diffusion/ migration process with constitutive equations for vacancy flux given by dilute solution theory [2,7,8]. Furthermore, we follow Macdonald et al. [2,7] in assuming that the mobility and diffusivity of each vacancy species are constant and related by the Nernst Einstein equation [8]. (6) The model does not consider the transport of interstitial species through the oxide [2,7]. This assumption has a firm physical basis [2,7] for systems such as Ni/NiO, but must be seriously questioned [1] for others such as Fe/Fe 2 O 3.Sørenson [12] provides a comprehensive introduction to defect structure in non-stoichiometric oxides. (7) The thickness of the film l varies with time due to various reactions that produce or consume the oxide. The reaction scheme is a generalization of the one proposed and physically justified in the work of Macdonald et al. (see Refs. [2,7] and references cites therein). At the metal/film (MF) interface, M atoms combine with cationic vacancy species to produce oxide according to y M+V M k1 M x O xy/2 +ye (1) In parallel, M atoms combine with the oxide lattice to produce anionic vacancy species: 2 M+ 1 2 y xy Mx O xy/2 k2 2+ V O +2e (2) At the FE interface, dissolution of the oxide produces cationic vacancy species: M x O xy/2 k3 y V M +M y+ (s) (3) Water in the electrolyte reduces the anionic vacancy species at the FE interface, 2+ V O +H 2 O k4 M x O xy/2 +2H + (4) producing hydrogen ions that lead to oxide dissolution via M x O xy/2 +xyh + k5 xm y+ (s)+ xy 2 H 2O (5) Eqs. (1) (5) represent elementary reactions with rate expressions implied by the given stoichiometry. Reverse reactions are not considered at this time. (8) The model neglects mass transfer limitations in the electrolyte phase and assumes constant concentrations of H 2 OandH + at the FE interface. This assumption could be relaxed by specifying appropriate hydrodynamic conditions (e.g. rotating disk hydrodynamics [8]), mass transfer coefficients in the context of Newton s law of mass transfer [9], or incorporating general solutions for momentum and mass transfer in the electrolyte phase. (9) No homogeneous reactions occur in the oxide film [2,7]. This is closely related to assumptions (5) (7). (10) The model ignores the initial stage of oxide film formation that presumably occurs by nucleation and growth of oxide clusters on a bare metal surface in contact with electrolyte. Instead, following BN [1], we assume that the initial state consists of a thin layer of oxide of molecular dimensions. The initial film thickness is l i. The initial concentrations of cationic and anionic vacancy species, c ci and c ai, are assumed to be constant and known. (11) The potential applied to the metal does not vary with time and decreases linearly across the film to a value of zero at the FE interface. Consequently, the electric field does not depend on position. The physical basis for this assumption has been discussed extensively by Macdonald et al. [2,7]. This assumption probably fails if the film becomes too thick (e.g. 10 nm). Unlike the PDM, we assume that the applied potential remains constant as the film thickness increases, so the electric field within the film decreases accordingly. As pointed out by BN [1], the correct approach to this issue introduces the Poisson equation to relate the potential distribution to the distributions of ionic species that comprise the free space charge. For the specific case of iron/iron oxide films, BN [1] distributed the potential drop so as to satisfy Poisson s equation at the interfaces, applied a small gradient approximation to linearize the field effects, and ignored space charge effects on the potential distribution in the bulk oxide. Overall, this is equivalent to our assumption (11). The non-linearity introduced by the Poisson equation presents a challenging numerical problem [1] that we, too, will not address here. We recognize that the assumption of a constant electric field strength in the film is most significant and problematic. Nevertheless, it provides a tractable starting point for the description of the processes happening within an oxide film. Forthcoming work will address this issue in more detail Transport equations The governing transport equations include differential mass balances and mass flux expressions for metal oxide, cationic vacancies, and anionic vacancies in the domain 0zl(t). Appendix A introduces these equations in their most general form and simplifies them in accord with the assumptions stated previously. In particular, we show how the boundary conditions at phase interfaces are derived from jump mass balances [9] as required by continuum mechanics. The jump mass balances relate the species fluxes to interfacial reaction rates and the motion of the film boundaries. This enables us to treat the film growth process as an explicit moving boundary problem.

4 3390 B. Krishnamurthy et al. / Electrochimica Acta 46 (2001) Table 1 Definitions of dimensionless variables. Dimensionless variable Dimensional variable / Characteristic quantity Z z / D c /k 1 t / 2 D c /k 1 C c c c / k 3 /k 1 C a c a / k 2 /k 4 L l / D c /k 1 U u / k 1 I i / k 3 F After using Eq. (A3) to eliminate the fluxes as variables from the boundary conditions, the governing equations contain 20 variables and parameters (counting as distinct the values of current density at each interface) in three independent units (moles, length, and time). Dimensional arguments thus imply the existence of 17 dimensionless quantities; of these, eight are dimensionless variables, and nine are dimensionless groups that parameterize the model. Table 1 gives the characteristic quantities used to define the dimensionless variables. Table 2 shows our choice of the nine independent dimensionless parameters with their physical meanings. In terms of these quantities, the governing equations and boundary conditions assume the dimensionless forms shown in Appendix B Analytical steady-state solution Setting the time derivatives in Eqs. (6) and (10) equal to zero, the vacancy concentrations profiles are easily shown to be exponential functions of position, in agreement with previous steady-state point defect models [2,7]. Application of the boundary conditions with U= 0 gives C c (Z)=1+ L s yv {1 exp[yv(1 Z/L s)]} (18) and C a (Z)=1+ L s 1 {1 exp[ 2VZ/L s ]} 2V DD (19) where L s denotes the dimensionless steady-state film thickness. These expressions simplify when evaluated at Z=0 orz=l s. With additional thermodynamic information about the equilibrium concentrations of vacancy species at either interface, the value of L s may be evaluated in terms of thermodynamic parameters. This problem has been treated extensively by Macdonald et al. [2,7] and will not be discussed in great detail here. To check the consistency of this solution, Eqs. (18) and (19) may be substituted into Eq. (14), leading to L U= xy =0 (20) since C c L s =1 and C a 0 =1 at steady state. In dimensional form, Eq. (20) becomes Table 2 Definitions of dimensionless groups, typical values, and physical interpretation Symbol Definition Base value Physical meaning y +1 metal valence [x=mod(y,2)+1=2] V FV*/RT dimensionless applied voltage (V*=0.1V) DD D a /D c 1.0 ratio of anion vacancy and cation vacancy diffusivities 12 k 1 c o /k rate of metal dissolution via cationic vacancy mechanism (Eq. (1)) relative to that via anionic vacancy mechanism (Eq. (2)) 31 k 3 /k 1 c o 0.01 rate of cationic vacancy generation (Eq. (3)) relative to the rate of cationic vacancy consumption (Eq. (1)) 24 k 2 /k 4 c o 0.01 rate of anionic vacancy generation (Eq. (2)) relative to the rate of anionic vacancy consumption (Eq. (4)) 54 k 5 /k 4 c o 10 6 rate of oxide consumption via Eq. (5) relative to the rate of oxide generation via Eq. (4) C ci c ci /c o dimensionless initial cationic vacancy concentration (c ci /c o =0.001) C ai c ai /c o dimensionless initial anionic vacancy concentration (c ai /c o =0.001) l i k 1 /D c 4.0 dimensionless initial film thickness L i

5 B. Krishnamurthy et al. / Electrochimica Acta 46 (2001) Numerical solution Fig. 2. Film thickness as a function of time for various values of applied potential. l t u= 1 2 c 0 xy k 2 k 5 =0 (21) indicating that the rate of film formation via Eq. (2) must equal the rate of dissolution via Eq. (5) at steady state, as noted previously [2,7]. If the rate constants do not satisfy Eq. (21), the film will not achieve a steady state thickness. Eqs. (16) and (17) (Appendix B) for the current densities at the FS and MF interfaces clearly differ. In general, if the concentrations of vacancies in the film vary with time, there is an accumulation of net charge, and so the current densities at the two interfaces must differ. At steady state, with C c L s =1andC a 0 =1, Eqs. (16) and (17) both simplify to the same expression, I s = y (22) The steady-state current I s is constant and negative for the chosen coordinate system. In dimensional form, we have i s = F(k 3 y+2k 2 ) (23) Obviously, both cationic and anionic vacancies may carry current. One carrier will dominate if the rate constant for production of that species is large compared to the other. Fig. 3. Film thickness as a function of time for various values of DD. The general time-dependent problem involves a moving boundary and must be integrated numerically. Eqs. (6) (15) are integrated numerically using an explicit finite difference technique. First, the transformation suggested by Crank [10], =Z/L(), maps the time-dependent domain 0ZL() to the fixed domain 01. The discretization employs first-order approximations for the time derivatives and second-order approximations for all spatial derivatives. Grid sizes in space and time are small enough to reduce truncation error to an acceptable level. Furthermore, the time step is chosen to ensure that the change in the film thickness over one time step is no larger than the distance between spatial grid points. We terminate the computation if the (dimensional) concentration of either vacancy species exceeds 10% of oxide concentration. 3. Results and discussion 3.1. General trends for film growth rate Numerical solutions of the model s governing equations show how the growth of the passive film depends on various dimensionless groups. The base values given in Table 2 represent a particular starting point in our exploration of the parameter space. Order-of-magnitude variations in the initial vacancy concentrations C ci and C ai produce only minor differences in C c (Z,), C a (Z,), and L() for dimensionless times less than one. On longer time scales, the solution becomes independent of C ci and C ai. Larger values of V promote higher rates of film growth (Fig. 2). Positive values of V increase the magnitude of the migration contribution to both cationic and anionic vacancy fluxes (first term, Eq. (A3)). Since the values in the base set of dimensionless parameters (Table 2) do not satisfy Eq. (20), the film thickness does not reach a steady-state value. On the other hand, when we choose parameters that do satisfy Eq. (20), we find that the film thickness does reach steady state for various values of C ci, C ai, and L i. Increasing the values of either D c or D a also produces higher rates of film growth. The definitions of the dimensionless variables in Table 1 establish the characteristic length and time as D c /k 1 and D c /k 12, respectively. Considering an increase in D c, k 1 must also increase if we keep the characteristic length constant. Thus the characteristic time decreases: in effect, the film grows to a given thickness in a shorter period of time. Since the characteristic time is proportional to D c, varying DD is equivalent to varying D a Fig. 3 shows that increasing DD produces a higher rate of film growth.

6 3392 B. Krishnamurthy et al. / Electrochimica Acta 46 (2001) Fig. 4. Film thickness as a function of time for various values of 12. Fig. 5. Film thickness as a function of time for various values of 24. Fig. 6. Anionic vacancy species concentration at the film/electrolyte interface as a function of time for various values of 24. Next, we consider the effect of variations in relative rate constants ( 12, 31, 24, and 54 ) on the growth of the passive film. Choosing the base value 12 =1.0 suggests that the two mechanisms for metal dissolution (i.e. one based on cationic vacancies, the other on anionic vacancies) proceed at comparable rates. Within each mechanism, the rate constant for vacancy generation is low compared to that of vacancy consumption ( 31 =0.01, 24 =0.01). We also presume that acidic dissolution of the oxide is slow compared to the dissociation of water to fill anionic vacancies with oxygen ( 54 =10 6 ). These conditions favor the growth of the oxide layer. The mechanism involving cationic vacancies features oxide formation (Eq. (1)) and dissolution (Eq. (3)), with a bias toward the former since 31 =0.01. On the other hand, the anionic vacancy mechanism gives greater emphasis to oxide formation (Eq. (4)). For base values of y=1 and x=2, oxide does not participate in Eq. (2), and oxide dissolution is slow for 54 =10 6. Consequently, increasing the value of 12 leads to smaller rates of oxide layer growth (Fig. 4) because of the greater relative importance of the cationic vacancy mechanism. Eq. (14) (Appendix B) shows this mathematically: since 1+2/xy=0 in this case, increasing 12 decreases L/. In the limit of large 12, the film does not grow thicker. Increasing 24, the rate of anionic vacancy generation relative to consumption, produces a larger rate of film growth (Fig. 5). The relatively larger generation rate leads to an increase in C a at the film/electrolyte interface (Z=0) as shown in Fig. 6. With 1+2/ xy=0 in Eq. (14), this leads directly to greater values of L/. On the other hand, increasing 31, the rate of cationic vacancy generation relative to consumption, leads to a decrease in film growth rate (not shown here). Cationic vacancies are created from the oxide (Eq. (3)), so a larger value of 31 implies a greater rate of oxide dissolution relative to formation (Eq. (1)). The value of C c at the metal film interface (Z=L) is generally less than one (Fig. 7), so increasing 31 reduces the magnitude of L/. Finally, larger values of 54, representing greater rates of acidic dissolution of oxide (Eq. (5)), produce smaller rates of film growth (not shown) Comparison with experimental data Fig. 7. Cationic vacancy species concentration at the metal/ film interface as a function of time for various values of 31. The veracity of the present model can be assessed through comparisons of theoretical predictions with experimental data. Unfortunately, there are relatively few papers in the literature that report careful measurements of the time-dependent thickness of primary passive films on metal surfaces. One example is the work of Sato and Kudo [11] who studied the passivation of nickel in alkaline, 0.15 N sodium borate buffer solu-

7 B. Krishnamurthy et al. / Electrochimica Acta 46 (2001) tions at 25 C. Fig. 8 reproduces their ellipsometric and coulometric data for the thickness of NiO passive films on Ni after 1.0 h of anodic polarization at various applied potentials. Fig. 8 also shows the model s prediction of thickness versus applied potential. The model involves many parameters for which we do not have values. To make qualitative comparisons with the data, we first adjust the model s parameters to match the predicted and experimental thicknesses at one point. Then, we vary the applied potential without any further adjustment of model parameters. To fit the ellipsometric thickness at an applied potential of 0.2 V, we used the base set of parameters (Table 2) except for 24 = (necessitating C ai =2 since the true initial vacancy fraction is held at 0.001) and D a /D c =0.8. To convert dimensionless length and time into dimensional quantities, we specify D c =10 17 cm 2 /sandk 1 =10 9 cm/s, yielding characteristic length and time of 10 8 cm and 10 s, respectively. The model predictions (Fig. 8) show an increase in thickness with increasing applied potential, in qualitative agreement with experiment, but the thickness predictions are too low at zero and negative potentials. A number of factors could be responsible for the discrepancy between theoretical and experimental film thicknesses. In particular, the model does not incorporate reverse reactions or consider the possibility that some of the reactions may be fast enough to be considered at equilibrium. The model does not account for the dependence of the reaction rate constants on potential. Finally, the model does not treat the effect of distributed space charge (carried by defects) on the potential distribution across the film. Each of these factors could be addressed through appropriate modifications of the present model. 4. Conclusions This work builds upon previous research [1,2] in the development of a continuum model for time-dependent formation of passive oxide films on metal surfaces. The present model introduces several new features. First, the model extends the steady-state analysis of the PDM [2] to explicit time dependence and treats oxide film growth much more extensively than BN [1]. Second, the model introduces boundary conditions based on jump mass balances [9] in order to describe film formation rigorously as a moving boundary problem similar to that formulated previously by BN [1]. This provides a starting point for incorporating more sophisticated treatments of interfacial reactions, including equilibrium conditions [2], finite kinetics, or a combination thereof [1]. The formulation of the model in dimensionless form identifies the key governing dimensionless parameters. The dependence of the film growth rate on these parameters can be rationalized in simple physical terms. Comparison of the model s predictions with experimental data for Ni passivation [11] shows that the model predicts the correct trends in the growth of film thickness with applied potential. Although quantitative agreement between the model experimental data is achieved at positive potentials, predicted thicknesses at negative potentials are too low. Nevertheless, we hope that this model provides suitable framework for subsequent improvements, including more realistic description of transport mechanisms and reaction kinetics for interfacial and homogeneous reactions. In particular, we are currently exploring how the assumption of constant field strength within the film affects the model predictions of time-dependent passive film thickness for a variety of metal/metal oxide systems. 5. Nomenclature Fig. 8. Comparison of theoretical and experimental film thicknesses (curves and symbols, respectively) after 1 h of passive film growth for various values of applied potential. Model parameters are given in the text. Film thickness data [11] for NiO films on Ni in borate buffer solution as measured by coulometry (diamonds) and ellipsometry (triangles). c k concentration of species k, mol/cm 3 C k concentration of species k, dimensionless D k diffusivity of species k, cm 2 /s e unit vector normal to a phase interface F Faraday s constant, C/equiv k i rate constant for reaction i, i=1 5 i current density, A/cm 2 I current density, dimensionless l film thickness, cm L film thickness, dimensionless N flux of species k, mol/cm 2 k s R universal gas constant, J/mol K

8 3394 B. Krishnamurthy et al. / Electrochimica Acta 46 (2001) R k r k t T u U u k v v k y V M V O 2+ V x y z Z Z k Greek Subscripts a c i o s rate of production of species k by homogeneous reactions, mol/cm 3 s rate of production of species k by interfacial reactions, mol/cm 2 s time, s temperature, K velocity of the metal/film interface, cm/s velocity of the metal/film interface, dimensionless mobility of species k, cm 2 mol/j s molar average velocity, cm/s velocity of species k, cm/s cationic vacancy species, (M x 1 O xy/2 ) y anionic vacancy species, (M x O xy/2 1 ) 2+ applied potential, dimensionless mod(y,2)+1 metal valence position relative to the film/electrolyte interface, cm position, dimensionless valence of species k potential gradient, V/cm transformed position variable dimensionless groups defined in Table 2 time, dimensionless anionic vacancy species cationic vacancy species initial value at time t=0 oxide steady state In the absence of homogeneous reactions, assumption (9), R k 0. Dilute solution theory specifies the mass flux N i = Z k u k Fc k D k c k +c k v (A2) as the sum of migration, diffusion, and convection contributions. The oxide has no charge and constant molar concentration c o in the film. With these assumptions, Eq. (A2) reduces to v =N o /c o for the oxide. Substituting this back into Eq. (A2) shows that the convective contribution to the mass flux of cationic and anionic vacancy species (k=c, a) is proportional to c k /c o. Assumption (4) implies that this convective contribution can be neglected. Further simplifications based on assumptions (1), (5), and (11) produce N k = D k V Zk c k l(t) +c k (A3) z for k=c, a. In accord with assumption (11), has been replaced by the dimensional potential drop divided by the film thickness; V represents the constant, dimensionless applied potential. Substitution of the mass flux expression into Eq. (A1) yields c c t =D 2 c c c z yd V c c 2 c l z and (A4) c a t =D 2 c a a z +2D V c a 2 a (A5) l z as the final equations to be solved for the concentration profiles of cationic and anionic vacancy species in the film phase. Since oxide has a constant molar concentration c o in the film, Eq. (A1) reduces to N o =0 (A6) Acknowledgements We acknowledge the financial support of this work by the US Army Research Office under grant number DAAH Appendix A. Governing equations A.1. Transport equations The differential mass balance [9] for material species k in the film phase is given by c k t = N k+r k (A1) indicating that the oxide flux is independent of position. A.2. Boundary conditions Appropriate boundary conditions must be established using jump balances valid at phase interfaces [9]. The jump mass balance for a species i across a phase interface is [c k (v k u) e]=r k (A7) where the boldface brackets signify the jump of the enclosed quantity across the phase interface. Here u denotes the vector-valued interface velocity. First, let us consider the jump mass balances for the oxide. At the MF interface, metal atoms combine with cationic vacancies to produce oxide according to Eq.

9 B. Krishnamurthy et al. / Electrochimica Acta 46 (2001) (1). The involvement of oxide in the production of anionic vacancies, Eq. (2), presents a complication. For metals in low valence states (y=1 or 2), metal atoms can form anionic vacancies without the participation of the oxide (1 2/xy=0). For higher metal valences, oxide must be consumed in order to supply additional oxygen atoms. This difference affects the order of the elementary reaction, Eq. (2), and thus the jump mass balance. For this reason, the oxide jump mass balance at the MF interface is written as z=l: (N o c o u)=k 1 c c 1 2 xy k2 c o (1 2/xy) (A8) The coefficient of the oxide consumption term is zero when y=1 or 2. The constant c o can be combined with k 2 to give an effective rate constant k 2. Rearrangement of Eq. (A8) yields z=l: c o u=k 1 c c 1 2 xy k2 +N o =k 1 c c k 2 *+N o (A9) with the understanding that k 2 *=0 when y=1 or2.eq. (A9) relates the velocity of the MF interface, u, tothe oxide flux N o c o o and the net oxide production rate. At the stationary FE interface, the oxide jump mass balance reduces to z=0: xy N o = k 3 +k 4 c H2 Oc a k 5 c H+ = k 3 +k 4 c a k 5 (A10) based on the stoichiometry of Eqs. (3) (5) and the fact that the electrolyte phase contains no oxide. The second equality redefines the rate constants based on the assumption (8) of constant c H2 O and c H+ at the FE interface. Since N o does not vary with position (Eq. (A6)), combining Eqs. (A9) and (A10) yields dl dt u= 1 (k 1 c c 0 k* k 2 3 +k 4 c a l k 5 ) (A11) c o y Jump mass balances for V M at the MF and FE interfaces provide z=l: N c =c c (k 1 +u) (A12) and z=0: N c =k 3 (A13) as the necessary boundary conditions for Eq. (A4). Similarly, for V O 2+,wefind z=l: N a =c a u k 2 (A14) and z=0: N a = k 4 c a (A15) Eq. (A3) defines the fluxes in terms of the vacancy species concentration profiles. In addition, initial conditions are required. As discussed above in assumption (10), the film has a small initial thickness, l i, and constant initial concentrations of vacancy species. A.3. Current density At any position within the domain 0zl(t), the dimensional current density is i=f Z i N i =F( yn c +2N a ) (A16) i The current densities at each interface can be evaluated by substituting the appropriate values of the species fluxes from Eqs. (A12), (A13), (A14) and (A15). After defining the dimensionless current density (Table 1), expressions for this quantity at each interface can be derived readily (Appendix B). Appendix B. Dimensionless governing equations and boundary conditions B.1. Cationic acancy species C c =2 C c Z 2 yv L Z=0: C c Z (6) y V L C c C c =1 (7) Z Z=L: y V L C c C c Z =C c(u+1) (8) =0: C c =C ci (9) B.2. Anionic acancy species C a = DD 2 C a Z 2 +2V L Z=0: Z=L: DD V 2 L C a+ C a Z DD V 2 L C a+ C a Z C Z a (10) C a = (11) =Ca U (12) =0: C a =C ai (13) B.3. Film thickness L U= 31(C c L 1) Ca xy (14) =0: L=L i (15)

10 3396 B. Krishnamurthy et al. / Electrochimica Acta 46 (2001) B.4. Current densities Z=0: I= y 2C a 0 (16) Z=L: I= y(1+u)c c L References ( UC a L 1) (17) [1] V. Battaglia, J. Newman, J. Electrochem. Soc. 142 (1995) [2] L. Zhang, D.D. Macdonald, E. Sikora, J. Sikora, J. Electrochem. Soc. 145 (1998) 898. [3] C. Lukac, J.B. Lumsden, S. Smialowska, R.W. Staehle, J. Electrochem. Soc. 122 (1975) [4] B. MacDougall, J. Electrochem. Soc. 127 (1980) 789. [5] B. MacDougall, J. Electrochem. Soc. 130 (1983) 114. [6] G. Dagan, M. Tomkiewicz, J. Electrochem. Soc. 139 (1992) 461. [7] D.D. Macdonald, M. Urquidi-Macdonald, J. Electrochem. Soc. 137 (1990) D.D. Macdonald, J. Electrochem. Soc. 139 (1992) L. Zhang, D.D. Macdonald, Electrochim. Acta 43 (1998) 679. [8] J. Newman, Electrochemical Systems, Prentice-Hall, Englewood Cliffs, NJ, 2nd edn., [9] J.C. Slattery, Advanced Transport Phenomena, Cambridge University Press, Cambridge, [10] J. Crank, Free and Moving Boundary Problems, Oxford Science, Oxford, [11] N. Sato, K. Kudo, Electrochim. Acta 19 (1974) 461. [12] O.T. Sørenson, in: O.T. Sørenson (Ed.), Nonstoichiometric Oxides, Academic Press, New York, 1981, p 1..