Nucleation and Growth of Oxide Layers on Stainless Steels (FeCr

Oct 28, 2010 - For low chromium content, these cavities coalesce, leading to passivity breakdown and pit initiation. View: ACS ActiveView PDF | PDF | ...
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J. Phys. Chem. C 2010, 114, 19299–19307

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Nucleation and Growth of Oxide Layers on Stainless Steels (FeCr) Using a Virtual Oxide Layer Model Boubakar Diawara,* Yves-Alain Beh, and Philippe Marcus* Laboratoire de Physico-Chimie des Surfaces, CNRS/ENSCP (UMR 7045), Ecole Nationale Supe´rieure de Chimie de Paris (Chimie ParisTech), 11 rue Pierre et Marie Curie, 75005 Paris, France ReceiVed: October 1, 2009; ReVised Manuscript ReceiVed: September 11, 2010

To simulate the passivation of FeCr alloys, we have developed a new model involving an explicit 3D model of the oxide layer that allows us to simulate the nucleation and the growth of the passive film. In the new model, the alloy is immersed in a virtual empty oxide lattice oriented with a given epitaxy. During the dynamic evolution, the metal cations generated by the oxidation of the alloy elements are injected into the virtual lattice where they are associated with oxygen ions coming from the solution, leading to formation of oxide nuclei, lateral growth of oxide islands, and an increase of the layer thickness. The dynamic evolution is based on the kinetic Monte Carlo (KMC) method. The KMC simulation takes into account the fundamental processes involved in the passivation mechanism: metal diffusion in the bulk and at the surface of the alloy, formation of metal cations and their injection in the oxide, nucleation and growth of the oxide layer, mass transfer through the oxide, and oxide dissolution at the oxide-solution interface. The activation energies related to the various processes are calculated using the modified embedded atom method potential or derived from experimental data. The electric field in the oxide film, considered in the new model, decreases or increases the activation energies depending on the positive or negative charge of the ions. The local value of the electric field is considered as inversely proportional to the local thickness of the oxide film. The simulations were carried out with a model of 25 Å × 25 Å × 25 Å (11 atomic planes). The results reproduce qualitatively well the experimental data. For low concentrations in Cr (16%), the oxide layer grows, covers the whole surface, and reaches a stationary thickness of the order of 9 Å. In the intermediate zone (14-16%), the transition from incomplete or no passivation to complete passivation is continuous. The passive film is enriched with chromium. For alloys with low Cr content, extensive iron dissolution is required to obtain passivation. This leads to increased surface roughness. The oxidation process produces vacancies in the alloy that may form cavities at the oxide-metal interface or in the bulk of the alloy. For low chromium content, these cavities coalesce, leading to passivity breakdown and pit initiation. 1. Introduction Surface oxide layers (passive films) play a major role in the protection of metals and alloys against corrosion. The experimental approach of the passivation of alloys, including Fe-Cr alloys, is well-documented,1-11 whereas the modeling of the growth of oxide layers, a key issue for the prediction of the behavior of metallic materials in corrosive environments, is much less developed. Different macroscopic models of the oxidation kinetic are available (Wagner,12 Cabrera-Mott,13 Fehlner-Mot,14 Macdonald15). In the HFM (high field model) of Mott and Cabrera,13 the migration of interstitial cations ensures the film growth and, depending on the magnitude of the field, the limiting step can be the transport of cations through the film (weak field) or their injection at the metal-oxide interface (strong field). Depending on the assumptions of the model, a parabolic or logarithmic time dependence of the film thickness is derived. In the point defect model (PDM) of Macdonald,15 the growth of the oxide film involves the migration of anion vacancies and the limiting step is the injection of anion vacancies at the metal-oxide interface. * To whom correspondence should be addressed. E-mail: bob-diawara@ chimie-paritech.fr (B.D.), [email protected] (P.M.).

Using also a macroscopic approach, Keddam et al16,17 and Laurent et al3 have proposed an electrochemical model of the anodic dissolution of alloys. Macroscopic models lead to analytical formulas that are easy to use and require small computer resources. Their main limits are that they assume perfect uniformity of the oxide films (structure, chemical composition, and electronic properties) and do not include the case of alloys (with corrosion properties depending strongly on their composition). Atomistic modeling is an interesting alternative, providing access to the effect of the local chemistry and structure of the film and allowing a fine-tuning of the interplay of the elementary processes involved in the oxidation process. Despite these advantages, atomistic modeling is much less developed due to the complexity of the computer codes and the computer time required. In their pioneering work on the atomistic simulation of the selective dissolution of FeCr alloys, Sieradski et al.18 have proposed a model based on the percolation theory, which relies on the idea that long-range connectivity of atoms is the determining factor. A 2D model with a square lattice was first used.19 A transition assigned to a percolation threshold was obtained with different Fe and Cr fractions, in which the rules prescribed for dissolution included complete selectivity in the

10.1021/jp909445x  2010 American Chemical Society Published on Web 10/28/2010

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dissolution; that is, the dissolution probability of Fe atoms was 1 and the one of Cr atoms was 0. Three-dimensional (3D) models have also been used,20 but a screw dislocation was present, which simplified the model to a quasi 2D model. Taking into account some of the limitations of the preceding models, we have developed a 3D model21 for modeling the selective dissolution and passivation of alloys. The real structure of the alloy was taken into account (bcc in the present case), as well as the structure of the initial surface. The passivation was modeled in considering the formation of “oxide” nuclei, resulting from the presence of local chromium-rich clusters. During the dynamic evolution of the model, based on the Monte Carlo method, surface diffusion and dissolution of atoms occurred according to probabilities dependent on the nature of the atom (Cr or Fe) and on its chemical environment. The probabilities of dissolution used in the first model were set empirically as a function of the number of first-nearest neighbors. This model was upgraded by including a new method for the calculation of the probability of dissolution, based on results of quantum chemistry calculations, while the diffusion probabilities remained set empirically.22 Atomic simulation based on kinetic Monte Carlo has been applied to the dealloying of Ag-Au alloys, including diffusion of silver and gold and dissolution of silver.23 A bond-breaking model was used for diffusion and for dissolution. Another approach based on graph theory has been proposed to explain the existence of critical alloy compositions for passivity of binary alloys.24-26 The main limit of our previous model and of all the preceding atomistic models is the absence of a truly 3D model of the oxide layer. Indeed, in all these models, the oxide layer was modeled by a monolayer of metal atoms considered as blocked. They do not allow simulating a major feature of passivation, that is, the nucleation and 3D growth with solid-state transport properties controlling the growth process. We present here a new model involving, for the first time, a 3D model of the oxide interacting with a 3D model of the alloy. The dynamic evolution of the system takes into account the main processes within the alloy, the oxide layer, and at the metal-oxide interface and the effect of physicochemical parameters, such as temperature and potential. 2. The Virtual Oxide Layer (VOL) Model 2.1. Structural Model. Numerous experiments have evidenced the duplex structure of the passive film formed on metals with an inner layer of oxide and an outer layer of hydroxide. As in the previous macroscopic models (Cabrera-Mott, PDM), the presence of a hydroxide layer is not taken into account in our atomistic model, which considers the growth of the barrier layer, essential for the passivity. Previous experimental studies,4,5 on the passivation of the FeCr alloys using scanning tunelling microscopy, have identified the presence of chromium oxide (R-Cr2O3) nanostructures, giving evidence of a nucleation of chromium oxide with subsequent coalescence to give larger single-crystal areas of Cr2O3. Insofar as the crystalline character of the oxide layer is now established, we retained the principle of a crystalline oxide layer that grows directly on the alloy surface. From a structural point of view, we have considered a virtual chromium oxide lattice embedding the alloy lattice (Figure 1). The alloy lattice is oriented along the [110] direction and filled randomly to a prescribed fractional occupancy of the alloy elements. The cationic and anionic sites of the virtual oxide lattice are filled gradually during the nucleation and growth of the oxide. A

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Figure 1. VOL model (alloy lattice embedded in a virtual oxide lattice).

Figure 2. Epitaxy relationship between the oxide lattice and the alloy lattice (from ref 5).

realistic model must take into account the roughness, as STM data5 have shown that, during the early stage of passivation of the Fe-Cr alloys, the roughness increases with surface height variations (∆z) going from 4 to 10 Å. A simple superposition of the two lattices would be valid only for smooth metal-oxide interfaces. To take into account roughening during the simulation, we extended the virtual oxide network to embed completely the alloy lattice (Figure 1). Thus, whatever the depth of dissolution of the metal, each metal site will continue to be surrounded by the virtual oxide lattice, allowing the local nucleation and growth of the oxide. The STM studies5 carried out on the structure of the oxide layer formed on the Fe-22Cr alloys showed that the basal plane of the oxide is parallel to the surface of the substrate ((0001) R-Cr2O3//Fe-22Cr (110)) with, depending on the crystalline area, a rotation of the two lattices corresponding to the alignment of the dense rows of both lattices. In our model, we have considered an R-Cr2O3 [1230]//Fe-Cr[001] alignment (Figure 2).

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Figure 3. Elementary processes considered in the VOL model.

In this first version of the VOL model, the aqueous solution is not considered explicitly but the dissolution is included and a potential can be applied. 2.2. Nucleation, Growth, and Dissolution of the Oxide Layer. Oxide films can form on metal/alloy surfaces by outward cation diffusion or inward anion diffusion, and sometimes, both occur simultaneously. In this paper, the outward diffusion of cations is considered, which is physically reasonable. ln the PDM model, the cations going across the film are all dissolved. ln our model, they contribute to the oxide growth, and dissolution takes place at the oxide/solution interface. During the early stage of the simulation, cations formed by oxidation of metal atoms can be dissolved or remain at the surface of the metal where they diffuse. We have considered that an oxide nucleus is formed when a Cr3+ cation has another Cr3+ or Cr atom located in its neighborhood (i.e., in a sphere of 4.1 Å). The second case corresponds to the oxidation of a Cr atom promoted by the presence of a Cr3+ ion. In this case, the Cr atom is oxidized and incorporated directly in the oxide film. When one of the two preceding conditions is satisfied, that is, the formation of a couple of (Cr, Cr3+) or (Cr3+, Cr3+), the anion sites of the virtual oxide lattice in the first-neighbor position with respect to each species of the couple (Cr, Cr3+) or (Cr3+, Cr3+) are “activated”, that is, filled with oxygen. After the formation of the first oxide layer, the cations formed at the metal-oxide interface migrate through the oxide film. Cations that reach the oxide-solution interface can either be dissolved or remain on the surface. Nondissolved cations contribute to the growth of the oxide layer if they have another cation within a sphere with a radius of 5 Å, corresponding to the radius of the third coordination sphere of a cation in the oxide lattice. Accordingly, the required number of oxygen anions of the virtual oxide lattice is activated to form a Cr2O3 nucleus, ensuring the overall stoichiometry of the oxide film during all the simulation. Neighbors limited to the first coordination sphere have been considered in preliminary tests, and it was found that the probability of meeting of two cations in the first coordination sphere was too weak to ensure the growth of the oxide. If, after the dissolution of cations, an anion has no cation in the sphere of reference of 5 Å, it is dissolved. This mechanism controls the local dissolution of the oxide layer. 2.3. Dynamic Evolution and Elementary Processes. Our previous model used a classical Metropolis Monte Carlo algorithm.21,22 The dynamic evolution of the VOL model is

based on the kinetic Monte Carlo (KMC)27- 32 method, allowing us to simulate the realistic kinetic evolution of the system, with simulated times related to the real time. In the KMC approach, the dynamic evolution of the system is the result of n processes, each process i being characterized by a rate Ri ) V0e(-Ei)/(kT) and a probability pi ) Ri/∑nj)1Rj (were Ei is the activation energy of process i and V0 a preexponential factor). The stochastic evolution of the system is simulated by drawing a random s pj g r1. The number r1 and selecting the process for which ∑j)1 preexponential factor is set to the same value ν0 ) 1013 for all the processes. The elementary processes involved in the VOL model belong to three groups: processes (i) in the alloy and at the metal-oxide interface, (ii) in the oxide, and (iii) at the oxide-solution interface (Figure 3). The processes in the alloy and at the metal-oxide interface include surface diffusion and oxidation of the metal atoms, dissolution, or injection in the oxide film of the cations formed. For metal oxidation, the associated cathodic reaction is implicit and thus not explicitly simulated. The vacancies resulting from the injection of cations in the oxide can migrate toward the bulk of the metal by a vacancy-mediated migration mechanism. Once the first oxide layer is formed, the ions can migrate in the film. The macroscopic growth models differ on the nature of the ionic species contributing to the growth of the barrier layer: cation outward diffusion (Mott-Cabrera) or anion inward diffusion (Fehlner-Mott, Macdonald). The VOL model can manage the transport of both anions and cations. However, in this work, only cation diffusion in the film via a vacancymediated migration mechanism was considered; thus, there is no need of generating oxygen vacancies at the metal/film interface. At the oxide-solution interface, the processes are the dissolution of cations, the formation of oxide nuclei by anion deposition, and the dissolution of the film by removal of anions being in excess due to cation dissolution. 2.4. Evaluation of the Activation Energy of the Elementary Processes. The energy barriers related to the various processes are calculated or derived from experimental data. At the alloy surface, the evaluation of the diffusion and dissolution activation energy requires the calculation of the cohesive energy of clusters representing local areas of the alloy surface. For FeCr alloys, being chemically disordered, it is necessary to consider a large number of configurations in order

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Figure 4. Evolution of the calculated extraction energy with the topology and the chemical environment of Fe and Cr surface atoms.

to cover the diversity of structural and chemical environments. In a previous approach, we used quantum chemistry calculations for the evaluation of the extraction energy of atoms.33 The cost of these calculations limits the size of the configuration space to be explored. An alternative is the use of the modified embedded atom method (MEAM) potential.34 Originally limited to pure metals, the MEAM has been extended to alloys, and parameters for FeCr alloys have been evaluated by Lee et al.35 The energy of extraction has been calculated as the difference between the sum of the energy of the cluster after the extraction of an atom and of the energy of the free atom, and the energy of the full cluster prior to the atom extraction:

Eextrac ) (Ecluster n-1 atoms + E1 atom) Ecluster n atoms (with E1 atom ) 0 in the MEAN method) For the extraction of Cr from pure Cr clusters or an Fe atom from pure Fe clusters, the calculated energies are in the range of 0.5-6.0 eV and do not vary beyond the sixth neighbor. Thus, the size of the cluster was limited to the sixth neighbors. For the alloy, the topology and the chemical environment of the extracted atoms are taken into consideration (Figure 4). The surface diffusion barrier was evaluated by considering a cluster containing atoms up to the sixth neighbors for the initial position and the final position of the diffusing atom. The energy of the initial configuration (E1) and the final configuration (E2) are then calculated using the MEAM potential. The transition state is calculated by a conjugated gradient algorithm. As for the extraction energy, the diffusion barrier was calculated for various topologies and chemical environments. The calculated values are in the range of 1.4-2.7 eV for the diffusion barrier of iron atoms and in the range of 1.84-2.88 eV for the diffusion of chromium atoms. An examination of the results shows that, for iron atoms, the diffusion barrier height increases with increasing Cr content (Figure 5), causing blocking of iron in Cr-rich areas. The same blocking behavior is observed for the diffusion of Cr in Cr-rich areas. A preferential diffusion of Cr atoms toward Cr-rich areas is observed (lowering the energy of the system). In contrast, it is found that the diffusion of Fe atoms toward Cr-rich areas can lower or not the energy of the system, indicating that there is no preferential diffusion of Fe atoms toward Cr-rich areas. In the presence of an oxide layer, a discontinuity appears in the MEAM calculations at the metal-oxide interface as the neighbors of a metal atom involve ions (anions or cations) that are not taken into account in MEAM. To overcome this limitation, we made the following choices: (i) For surface and bulk diffusion of the metal atoms (Fe and Cr), we used the mean values obtained by MEAM calculations. (ii) For the surface diffusion of the cations (Fe3+ and Cr3+), we used the activation energy values obtained for Fe and Cr by

Figure 5. Evolution of the diffusion barrier for Fe with the local chromium content.

TABLE 1: Input Values for the Activation Energies location alloy alloy surface

alloy-oxide interface oxide oxide-solution interface

process

mean activation energy (eV)

Fe diffusion Cr diffusion Fe diffusion Cr diffusion Fe3+ diffusion Cr3+ diffusion Fe3+ dissolution Cr3+ dissolution Fe oxydation Cr oxydation Fe3+ injection Cr3+ injection Fe3+ diffusion Cr3+ diffusion Fe3+ diffusion Cr3+ diffusion Fe3+ dissolution

3.17 3.69 0.60 0.50 0.55 0.45 6 4 0.41 0.35 4.85 4.20 4.85 4.21 3.00 2.11 7

MEAM decreased by 0.05 eV, considering that, in the presence of solvent, surface diffusion will be easier for ions than for atoms. For cation dissolution, we took the extraction energy obtained by MEAM in the vacuum decreased by 1 eV to take into account the presence of the solvent. (iii) The activation energies for the oxidation of Fe and Cr are set empirically. (iv) For the transport of the cations in the oxide, we used the experimental values of the activation energy for Fe and Cr diffusion in Cr203.36 We assumed the equilibrium at the metal-oxide interface, which implies that the energy of injection of the cations is equal to the activation energy of diffusion in the oxide. The input values for the activation energies are summarized in Table 1. 2.5. Electric Field in the Film. The potential drop between the two interfaces of the oxide film induces an electric field

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Figure 6. Nucleation and growth mechanisms observed with the VOL model.

controlling the ionic transport inside the film. The major part of the potential drop is in the film, but part of the voltage appears across the interfaces. The different growth models differ in their assumptions regarding the effect of the electric field in the film. In the high field model (HFM) of Mott,13 the field is constant for a given thickness and varies as the inverse of the thickness. In the point defect model (PDM) of Macdonald,15 the electric

field is considered as constant in the film and assumed to be independent of film thickness. In our model, we used the former hypothesis (electric field function of the thickness). The field is considered as perpendicular to the initial exposed surface of attack ((110) in our case). During the simulation, the intensity of the local field is calculated using the local thickness of the oxide layer and then taking into account the surface roughness.

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The diffusion barrier of charged species is corrected depending b ) versus the on the orientation of their displacement vector (dr b) following the formula direction of the electric field (E

b·b Ea ) Ed - qE dr To take into account the effect of the potential on the dissolution process, we applied a variable overpotential φ, which lowers the activation energy of the dissolution process. 3. The SIMCOR Simulation Code A simulation code implementing the VOL model has been developed. SIMCOR (SIMulation of CORrosion) allows for building a structural model of any size. From a starting configuration, SIMCOR identifies the elementary processes and performs the simulation in an iterative manner until complete passivation of the model is reached or corrosion goes through all the atomic planes of the model (perforation). The model is considered as passivated if, for a fixed number of iterations, the number of dissolved cations does not exceed a fixed threshold. It is considered as perforated if there is a channel of vacancies connecting the metal-solution interface and the last atomic plane of the alloy. Starting from each vacancy present on the last plane of the alloy, a special algorithm, developed for this purpose, tests if it can be the starting point of a channel. A user-friendly dialogue box allows us to change the simulation parameters, with the possibility to disable a group of events (surface or bulk diffusion, oxidation), in order to evaluate their influence on the simulation. Graphic display allows us to monitor the simulation with the possibility to interrupt the simulation and go back to any step of the simulation and to display selectively the alloy, the alloy-oxide interface, or the oxide. 4. Results and Discussion Simulations on a 25 × 25 × 25 Å model have been performed with the new VOL model for quantitative evaluation (passivation kinetics, passivation transition threshold, thickness of the oxide layer, influence of the potential) as well as qualitative evaluation (nucleation and growth mechanism, structure of the interfaces). For the same parameters, slightly different results may be obtained because of the stochastic aspect of the simulation. The data presented below are the mean values of ten simulations for each set of parameters. For all the tests, the model was considered as passivated if there were less than two dissolution events after 2000 iterations. 4.1. Oxide Growth Mechanism Simulated by the VOL Model. The interaction, controlled by KMC dynamics, between the structural model and the different processes considered in the VOL model produces a growth mechanism with several steps. An initial surface diffusion step (Figure 6a) occurs before the formation of the first oxide layer. During this step, a preferential diffusion of the Cr atoms toward Cr neighbors leads to the formation of small Cr clusters, in agreement with the predictions of MEAM calculations. This preferential diffusion is important for the formation of the oxide nuclei, which requires the meeting of two chromium atoms, which is a rare event, particularly for alloys with low chromium content. During this initial step, the selective dissolution of Fe at the surface favors the preferential diffusion of Cr. A nucleation step (Figure 6b) follows, with the formation of oxide nuclei each time a couple (Cr, Cr3+) or (Cr3+, Cr3+) is formed in the reference coordination

Figure 7. Effect on passivation of the Cr content in the alloy.

Figure 8. Passivation probability vs Cr content in the alloy.

sphere. The nucleus is simulated by depositing anions around the two Cr of the couple. Fe3+ cations are trapped in the oxide layer during this step. The extension of the initial small nuclei and their coalescence, promoted by cation migration, takes place during the coalescence step (Figure 6c). The following step is a coVering step (Figure 6d), corresponding to the formation of the first oxide layer. The formation of multilayers starts only once the first layer is completely formed. For the Cr content considered in Figure 6 (Fe-24Cr), this first layer is relatively smooth but (see below) it can be rough for alloys with low Cr content. The final step is the growth and stabilization step of the oxide film (Figure 6e). It proceeds by formation of small islands whose height can reach several layers locally. These small islands, which grow by deposition of anions around surface cations, coalesce by a mechanism of oxide nucleation inside the interstices between them. The global mechanism induced by the VOL model is a nucleation and growth mechanism, similar to that observed by STM for the Fe-22Cr alloy5 4.2. Influence on Passivation of the Cr Content in the Alloy. For low Cr contents in the alloy, the selective dissolution of Fe dominates and the passive oxide nuclei cannot cover completely the alloy surface. The alloy is strongly corroded, leading to a very rough surface (Figure 7a). For larger chromium contents, the oxide layer grows more rapidly and reaches a stationary thickness. The alloy is passivated, and the final surface is relatively smooth (Figure 7b). Figure 8 presents the evolution of the passivation rate (% of tests leading to complete passivation for a set of 10 tests) as a function of the Cr content of alloy. A continuous transition from nonpassivated to passivated alloys is observed between 14 and 16% Cr. This transition, which is not a percolation threshold, is consistent with the view that a nucleation and growth mechanism governs the initial stages of passivation. The absolute location of this transition depends on the parameters used. The important fact here is that, with an explicit model of oxide layer and all the complex processes related to its nucleation and growth, one of the

Growth of Oxide Layers on FeCr Using a VOL Model

Figure 9. Amount of corroded material as a function of time for Fe-Cr alloys with various Cr contents (in the range of 12-22%).

fundamental aspects of the passivation of the FeCr alloys is retrieved, that is, the existence of a content threshold under which no passivation occurs.37 4.3. Influence of the Cr Content on the Dissolution Kinetics. To investigate the kinetics of passivation of Fe-Cr alloys, we have computed the amount of corroded material

J. Phys. Chem. C, Vol. 114, No. 45, 2010 19305 (number of dissolved atoms) versus time. The “kinetic” data (Figure 9) were obtained as follows: ten simulations with the same values of the parameters were carried out, and the corresponding mean value of corroded material was calculated. The kinetic curves exhibit the main features of the passivation kinetic curves of the FeCr alloys. They present an initial part of fast dissolution corresponding to the phase of nucleation and formation of the first oxide layer. The dissolution rate then decreases with the increase of the oxide layer thickness. The decreasing amount of corroded material prior to passivation with increasing Cr content in the alloy is clearly observed. For high Cr content, the number of dissolved atoms reaches asymptotically a plateau corresponding to a film thick enough to limit the cation flow through the film. For low Cr content, the oxide film does not cover completely the alloy surface and dissolution continues. 4.4. Kinetics of the Oxide Film Growth (Thickening). We have calculated the time dependence of the average film thickness for Cr contents of 12, 14, 17, 18, and 22%. A marked increase with time of the film thickness, followed by stabilization at a constant value, is observed. The final thickness increases with the Cr content, the magnitude of the change decreasing when the Cr content increases. For a content of 22%, the final thickness is 10 Å, in agreement with the experimental value of 14 Å measured by XPS for the Fe-22Cr alloy after 63 h of

Figure 10. Comparison between the VOL model (blue curve) and macroscopic models (red curve): thickness of the passive film (Å) vs time (s).

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Figure 11. Influence of the potential on the film thickness.

polarization in acidic solution (0.5 M H2SO4,).5 The increase of the film thickness with increasing Cr content in the alloy is also in agreement with the experimental observations.38 We have compared the time dependence of the film thickness obtained with the VOL model with the prediction of the kinetic laws of macroscopic models of metal oxidation. We have considered the Mott-Cabrera model for weak or strong fields13 and the PDM model that includes constant dissolution of the oxide.15 The values of the parameters of the macroscopic models were obtained by fitting the prediction of each model with the results of the VOL model using a Newton-Raphson method. The results, including the fitted parameters, are presented in Figure 10. For each model, the first figure shows kinetics on a long time scale, whereas the second is a zoom on the first 100 ms. The Mott model with a weak field leads to a marked difference with the VOL model, whereas the agreements with the Mott model with a strong field and the PDM model are good. A detailed examination of the curves for short times shows that the agreement with the PDM model is better than with the Mott model with a strong field. The better agreement with the

Figure 12. Cavity observed at the metal-oxide interface.

Diawara et al. Macdonald model may be due to the taking into account of the dissolution of oxide by both. Indeed, for short times, oxide dissolution slows down the growth of the oxide layer, and for long times, it limits the oxide thickness. This is not the case in Mott’s model. 4.5. Influence of the Potential on the Film Thickness. In the VOL model, a potential can be applied (equivalent to an external applied potential). It creates a local electric field inversely proportional to the local thickness of the film. The final oxide thickness varies with this potential (see Figure 11). It increases with increasing potential because the migration of cations is enhanced by the electric field. For high values of the potential, the effect is reversed, which may be related to the formation of cavities that slow down the injection of cations in the oxide film. It is to be noted that, with the atomistic model, the behavior of the system for a given voltage does not depend on the results obtained below or above this voltage (it is not “historydependent”). 4.6. Formation of Cavities. In some cases, the formation of cavities under the oxide layer is observed (Figure 12). It is caused by a difference in the flux of vacancies diffusing into the bulk of the alloy and their formation at the metal-oxide interface. If the injection rate of cations in the film is lower than the bulk diffusion, the vacancies formed at the interface can diffuse toward the interior of the alloy, preventing the formation of cavities at the interface. They may then form vacancy clusters, leading to the formation of cavities in the bulk. If the injection rate is too high, the vacancies formed at the interface cannot be fully annihilated by bulk diffusion and cavities are formed at the metal-oxide interface. These cavities can slow down the growth of the oxide layer and also weaken the adherence of the protective layer. They can lead to passivity breakdown, as postulated in the PDM model.15 A recent experimental investigation by STM39 demonstrated the formation of an ordered array of nanocavities at the γ-Al2O3/TiAl interface, resulting from atomic vacancies injected at the interface during the growth of the oxide layer. A recent density functional theory calculation of the atomic structure of the Al2O3/TiAl(111)

Growth of Oxide Layers on FeCr Using a VOL Model interface confirmed that vacancies injected by the oxide film growth are stabilized in the topmost plane of the alloy and condense to form 2D clusters.40 5. Conclusions Addressing the limits of a previous atomistic model of the passivation of binary alloys,21,22 we have developed a truly 3D model of the passive film, including nucleation and growth, mass transport in the oxide and at the interfaces, and taking into account the influence of the electric field. The main basis of the new model is the concept of a virtual oxide layer (VOL) embedding the alloy, with the crystallographic features of the oxide layer previously measured by scanning tunneling microscopy. The simulation parameters (diffusion or dissolution) take into account the chemical and topological environment of the atoms. The results obtained show the ability of the VOL model to simulate the growth of the passive layer on a stainless steel (FeCr alloy). Our model accounts, in a quantitative manner, for (i) the nucleation and growth mechanisms of passivation, observed experimentally by STM; (ii) the enrichment of chromium oxide in the passive film on stainless steel, observed experimentally by, for example, XPS; (iii) the thickness of the passive film of about 1 nm, in agreement with experimental data; and (iv) the effect of the chromium content on the alloy’s ability to be passivated, with a transition in the range of 10-18%. An important feature is that, despite the great number of elementary processes taken into account and the complexity of their interplay, the stochastic evolution controlled by the KMC algorithm leads to a realistic simulation of the passivation. References and Notes (1) Pickering, H. W.; Wagner, C. J. Electrochem. Soc. 1967, 114, 698. (2) Forty, A. J.; Rowlands, G. Philos. Mag. A 1981, 43, 171. (3) Laurent, J.; Landolt, D. Electrochim. Acta 1991, 36, 49. (4) Maurice, V.; Yang, W. P.; Marcus, P. J. Electrochem. Soc. 1994, 141, 3016. (5) Maurice, V.; Yang, W.; Marcus, P. J. Electrochem. Soc. 1996, 143, 1182. (6) Schmutz, P.; Landolt, D. Corros. Sci. 1999, 41, 2143. (7) Olsson, C.-O. A.; Hamm, D.; Landolt, D. J. Electrochem. Soc. 2000, 147, 4093. (8) Kirchheim, R.; Heine, B.; Fischmeister, H.; Hofmann, S.; Knote, H.; Stolz, U. Corros. Sci. 1989, 29, 899.

J. Phys. Chem. C, Vol. 114, No. 45, 2010 19307 (9) Keddam, M. Anodic Dissolution. In Corrosion Mechanisms in Theory and Practice, 2nd ed.; Marcus, P., Ed.; Marcel Dekker Inc.: New York, 2002; pp 97-170. (10) Fehlner, F. P.; Graham, M. J. Thin Oxide Film Formation on Metals. In Corrosion Mechanisms in Theory and Practice, 2nd ed.; Marcus, P., Ed.; Marcel Dekker Inc.: New York, 2002; pp 171-188. (11) MacDougall, B.; Graham, M. J. Growth and Stability of Passive films. In Corrosion Mechanisms in Theory and Practice, 2nd ed.; Marcus, P., Ed.; Marcel Dekker Inc.: New York, 2002; pp 189-216. (12) Wagner, C. Z. Phys. Chem. 1933, B21, 25. (13) Cabrera, N.; Mott, N. F. Rep. Prog. Phys. 1948, 12, 163. (14) Fehlner, F. P.; Mott, N. F. Oxid. Met. 1970, 2, 59. (15) Macdonald, D. Pure Appl. Chem. 1999, 71, 99951. (16) Keddam, M.; Mattos, O. R.; Takenouti, H. Electrochim. Acta 1986, 31, 1147. (17) Keddam, M.; Mattos, O. R.; Takenouti, H. Electrochim. Acta 1986, 31, 1159. (18) Sieradzki, K.; Corderman, R. R.; Shukla, K.; Newman, R. C. Philos. Mag. 1989, 59, 4–713. (19) Quian, S.; Newman, R. C.; Cottis, R. A.; Sieradzki, K. J. Electrochem. Soc. 1990, 137, 435. (20) Quian, S.; Newman, R. C.; Cottis, R. A.; Sieradzki, K. Corros. Sci. 1990, 31, 621. (21) Legrand, M.; Diawara, B.; Legendre, J.-J.; Marcus, P. Corros. Sci. 2002, 44, 773. (22) Diawara, B.; Legrand, M.; Legendre, J.-J.; Marcus, P. J. Electrochem. Soc. 2004, 151, B172. (23) Erlebacher, J.; Aziz, M. J.; Karma, A.; Dimitrov, N.; Sieradski, K. Nature 2001, 410, 450. (24) McCafferty, E. Electrochem. Solid-State Lett. 2000, 3, 28. (25) McCafferty, E. Corros. Sci. 2000, 42, 1993. (26) McCafferty, E. Corros. Sci. 2002, 44, 1409. (27) Bortz, A. B.; Kalos, M. H.; Lebowitz, J. L. J. Comput. Phys. 1975, 17, 10. (28) Levy, A. C.; Kotrlaz, M. J. Phys.: Condens. Matter 1997, 9, 299. (29) Schulze, T. P. Phys. ReV. E 2002, 65, 036704. (30) Adam, E.; Billard, L.; Lancon, F. Phys. ReV. 1999, E59, 1212. (31) Henkelman, G.; Johnsson, H. J. Chem. Phys. 2001, 21, 1. (32) Frenklach, M. Pure Appl. Chem. 1998, 70, 417. (33) Legrand, M.; Diawara, B.; Legendre, J.-J.; Marcus, P. Chemom. Intell. Lab. Syst. 2002, 62, 1. (34) Lee, B.-J.; Baskes, M. I. Phys. ReV. 2000, B62, 8564. (35) Lee, B.-J.; Shim, J.-H.; Park, H. M. CALPHAD 2001, 25, 527. (36) Sabioni, A. C. S.; Huntz, A. M.; Silva, F.; Jomard, F. Mater. Sci. Eng., A 2005, 392, 254. (37) Horvath, J.; Uhlig, H. H. J. Electrochem. Soc. 1968, 115, 791. (38) Hamm, D.; Olsson, C. O. A.; Landolt, D. Corros. Sci. 2002, 44, 1009. (39) Maurice, V.; Despert, G.; Zanna, S.; Bacos, M. P.; Marcus, P. Nat. Mater. 2004, 3, 687. (40) Islam, M. M.; Diawara, B.; Maurice, V.; Marcus, P. J. Phys. Chem. C. 2009, 113, 9978.

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