NEW COMPUTER MODEL FOR INTERNAL OXIDATION

Jürgen GegnerSKF GmbH, Material Physics, Ernst-Sachs-Str. 5, D-97424 Schweinfurt, Germany

ABSTRACT

Internal oxidation is of great technological significance as effective dispersion-hardening technique, e.g. for Ag, Cu or Ni alloys, and undesirable side effect to many heat treatment processings of steels. Reaction kinetics is usually described by Wag-ner's theory for dilute binary alloys. However, this approach comprises three practi-cally important restrictions: homogeneous concentration distribution of the less noble component M within the matrix metal is assumed, only isobaric and isothermal pro-cess conditions are considered. In order to overcome these limitations, a computer model based on iterative solution of Fick's law for the inward diffusion of oxygen with arbitrary initial distribution of immobile M atoms is proposed in the present paper. The solubility product and the stoichiometry factor of the precipitating oxide define the mass balance relationship. For homogeneously dissolved M atoms and isothermal-isobaric reaction conditions, it is shown that the computed concentration profiles and time curves of the penetration depth agree excellently with the prediction of Wagner's theory. The new simulation technique is also validated by comparison with experimental data. Heterogeneous M distributions as well as nonisobaric and/or nonisothermal internal oxidation are accessorily covered by the presented computer model. Illustrative processes are calculated and the results are discussed in detail.

SHORT LOOK BACK INTO HISTORY OF COMPUTER SIMULATIONS

In 1946, the first electronic tube mainframe ENIAC was commissioned at LANL (Los Alamos National Laboratory). A historical U. S. Army photograph is shown in Fig. 1. Operating of the machine in those days was obviously a time-consuming task. Program implementation required a lot of manual labor. Up to the seventies of the past century, further development was slowed down by insufficient calculation capacities. Since the last thirty years, however, computer simulations have been evolved into the third main pillar of research. Nowadays, they represent the crucial connecting link between experiment and analytical theory.

One of the first exercises of the ENIAC mainframe and his successor at LANL, the MANIAC computer, was the solution of advanced diffusion problems, which possess

Figure 1. ENIAC computer in use.

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high economical relevance to many industrial surface refinement and undesirable edge zone processes, like coating, case hardening or decarburization. The present pa-per deals with another technologically important and scientifically interesting repre-sentative of this complex reaction class: internal oxidation of less noble components within a metallic matrix. INTRODUCTION

Common dispersion-hardening procedures like mechanical alloying and high energy milling, applied e.g. to magnesium materials with admixed alumina (1), follow the rather expensive powder metallurgy route and usually yield coarser particle distribu-tions compared to precipitation strengthening. Internal oxidation offers a cost-saving alternative technique suitable for compatible base metals, such as palladium, silver, copper or nickel (2). Applying appropriate element combinations (e.g. Ag or Pd with Mg), this process produces very fine dispersoids with uniform particle arrangement (3-5). Internal oxidation thus represents the most widely used dispersion-hardening method for copper materials. It is also important to technical silver alloys as the effect exceeds solid solution strengthening (3). Moreover, internal oxidation provides a powerful experimental tool for the determination of oxygen permeabilities from me-tallographic cross-sections and the preparation of perfectly flat (at an atomic level) and low-energy metal-oxide phase boundaries, on or close to thermodynamic equilib-rium (5-7). Applications are available in the literature (5-8).

Internal oxidation of iron-based materials does not provide an efficient dispersion-hardening technique due to the formation of rather large, coarsely dispersed oxide particles within the Fe matrix (9, 10). It is, however, of even increasing technological importance because of its relevance, as undesirable side effect, to many common heat treatment processings of steels, like austenitizing and carburization. With the intention of cost reduction, in the field of case hardening, fast CO-H2 containing gas mixtures at boost temperatures above 1300 K also become widespread. During these (chemico-) thermal treatments, internal oxidation of different alloying and accompanying el-ements of the steel (e.g. chromium, silicon, manganese, aluminum) may occur re-sulting in a loss of hardenability in the affected rim zone and a lower fatigue strength.

In the following, a new computer model for the realistic simulation of the kinetics of internal oxidation, based on iterative analytical solution of Fick's law for the in-diffu-sion of oxygen by taking account of the simultaneous chemical reaction (oxide forma-tion), is developed. This advanced calculation scheme involves nonisobaric and non-isothermal processes, and can also be applied to heterogeneous initial distributions of the alloyed component. Such extension, although of practical importance, are rarely included in models available in the literature (11). A numerical approach, based on the finite difference method, also deals with nonisothermal internal oxidation under varying boundary condition of the first kind (12). The simulation examples presented below use palladium as noble base metal. Data on oxygen diffusivity and solubility is taken from the literature (3, 8, 13).

WAGNER'S THEORY

The description of the process kinetics usually rests on Wagner's theory of isothermal-isobaric internal oxidation of dilute homogeneous binary alloys without scaling, i.e. outer oxidation (10, 14). If oxygen and the less noble component M, having a suffi-

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ciently high affinity for oxygen, possess suitable solubility in the base metal B and their diffusivities in B, DO and DM, fulfill the condition DODM, metal M can be oxi-dized inside the B matrix according to the following reaction equation:

(1)The stoichiometry factor is the number of oxygen atoms per M atom in the precipi-tated oxide MO. Under the assumption of a bulk diffusion-controlled process, which means that no preferential interface diffusion occurs and that neither surface reactions nor oxide formation are rate-controlling steps (in particular, such a behavior is typical of systems exhibiting an endothermic or weakly exothermic solution and a rather low diffusivity of oxygen in the solvent metal, e.g. palladium), a parabolic kinetics law can be applied to describe the relationship between time t and the corresponding posi-tion of the precipitation boundary, i.e. penetration depth or subscale thickness:

(2)

Assuming a highly stable oxide MO, which is mostly the case, reaction occurs com-pletely and solely at x. Note that only uniform initial M distribution is considered by Wagner's theory. As for the alloying content of component M and the equilib-rium solid solubility of oxygen in base metal B (both expressed in mole fraction) for usual oxidizing conditions,   is generally valid, the dimensionless par-ameter fulfills the relation 1. By solving Fick's second law for one-dimensional (x denotes the distance from the surface, semi-infinite plane geometry) in-diffusion of oxygen and out-diffusion of metal M (concentrations cO and cM) for the initial condi-tion cM(x0, t0)  and the boundary conditions cO(x0, t0)  (suitable ap-proximation for small ratios DOO, O oxygen mass transfer coefficient) and cO(x  , t  0)cM(x  , t  0)  0 (zero-solubility model, i.e. extremely low solubility product of the assumed highly stable oxide MO), can be calculated if in the mass balance relationship equivalent flux at x  , i.e. jO  jM, is considered (14):

(3)

The parameter is defined as   (DODM)½. In Eq.(3), function F takes into account the counter-diffusion of component M (  ):

(4)

Here, erf stands for the Gaußian error function. The range of values of F is 0  F()  1. The dependence of the diffusivities DO and DM on concentration is neglected (small mole fractions). In almost all cases of practical interest, the outward diffusion of metal M can be neglected (  1  DMDO   1). Thus, with F()  1, substitution of Eq. (3) into Eq. (2) yields the time development of the penetration depth according to a parabolic kinetics law:

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(5)

The used approximation of totally immobile M atoms can be applied to dilute silver, copper, nickel, palladium, iron, cobalt, and chromium alloys (2). Under this condition, the concentration cM remains constant (no counter-diffusion), cM(x,t0) . The resulting oxygen depth profile can be expressed as follows (14, 15):

(6)

The concentration of M in the whole subscale (bound as MO), x  , is equal to . Strictly speaking, both solubility and diffusivity DO of oxygen refer to metal B containing MO precipitations. However, at least at small contents of component M, the difference from the corresponding data on pure B is negligible (2).

MODELING OF INTERNAL OXIDATION AND VALIDATION

As discussed above, slight outward transport of component M can generally be ne-glected since the diffusion coefficient of interstitial oxygen usually exceeds the one of the alloying elements by several orders of magnitude. Thus, the often-made assump-tion of immobile M atoms is applied (11). The used one-dimensional approach con-forms to almost all cases of practical interest.

Outline and discussion of the computer model

The developed calculation scheme for the simulation of internal oxidation, i.e. inward diffusion of oxygen accompanied by chemical reaction, bears on the iterative solution of Fick's law for non-uniform O profile. An existing Fortran code is upgraded that solves the plane-sheet problem for an arbitrary, discretely defined initial concentration distribution under the constant boundary condition (x0, t'  ) (15):

(7)

Due to the neglect of counter-diffusion, no center depletion of component M occurs. Thus, large thickness l is implied, i.e. original alloy always remains in the middle of the material. For the sake of simplicity, the oxygen concentration profile is linearly interpolated between the closely adjacent evaluation points (for instance,  x  xj1xj 10 nm) so that the arising integral in Eq. (7) can easily be calculated

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incrementally. The parameters i and j are naturals, n is a non-negative integer. After each iteration i (from to ) of time step t (from to t), formation of new oxide MO is considered by satisfying the mass balance equation at all xj with

(xj)  0 and cM(xj) 0, according to the stoichiometry factor and the applied zero-solubility model. Thusly, the oxygen distribution , derived from the previously calculated concentration-distance curve by evaluating Eq. (7), is locally lowered and then redefined as . Corresponding modification of the M profile, cM(xj), also occurs before the next iteration step.

The precipitation boundary proceeds with complete consumption of the M supply. Evaluation ends after itot iterations on reaching the desired oxidation time t (0 t' t, t itott). The adequate choice of t depends on x and the process conditions. Note that represents the initial oxygen concentration profile at starting time 0, i.e. 0 in accordance with the zero-solubility model.

Several alloying elements can simply be considered in the applied mass balance relationship by averaging over the different stoichiometry factors with respect to the individual contents of the added components.

Firstly, a uniform initial M distribution is considered, cM(x)  .

Figure 2. Simulated concentration profiles and definition of penetration depth .

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Figure 2 shows the concentration profiles computed for an internal oxidation of a dilute Pd alloy with 1 at.% of monovalent M at temperature T1273 K in oxygen atmosphere (partial pressure 105 Pa). Both the process conditions and the modeling parameters are given.

Owing to the free inward diffusion of oxygen for each time period t, no strictly sharp interface develops at x0x(cOcM0) with cM(xx0) but the M profile slightly smudges across the small transition zone. Thus, the penetration depth, i.e. the position of the oxidation front, is best defined at half the initial M concentration,   x ( /2). As can be seen in Fig. 2, however, the difference from x0 is small. Note that in Wagner's theory,   x0 holds. In this approach, a jump of the M concentration from 0 to occurs, cM (x  )  .Comparison with Wagner's theory of internal oxidation

Figure 3. Comparison of the computed oxygen profile with the prediction of Wagner’s theory.

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Except for the process time t, the same conditions as before are applied. Figure 3 reveals the calculated oxygen profile after internal oxidation of the Pd-1 at.% M alloy for 3 min. For comparison with Wagner's theory, the concentration-distance curve according to Eq. (6) is also shown. Both profiles virtually agree quantitatively. As often done in the literature (11), the oxygen distribution through the subscale can be assumed to be linearly decreasing. The concentration gradient as driving force of diffusion in Fick's law is then approximately given as /. Thus, the oxygen mass flux can be written as jO  DO /.

In Fig. 4, a comparison of the computed penetration depth with the prediction of Wag-ner's theory according to Eq. (5) is drawn. Both curves agree excellently. The para-bolic kinetics law characterizes the diffusion-controlled process.

Figure 4. Comparison of the computed -t curve with the prediction of Wagner’s theory.

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Experimental verification

The preeminent agreement with the predictions of Wagner's theory, which has often been proved correct over the past decades, sufficiently demonstrates the applicability of the presented computer model for internal oxidation. In addition, Fig. 5 shows the comparison with an experiment. The dilute Pd -1.99 at.% Fe alloy (M  Fe) is annealed at 1223 K in ambient air (  2.10  104 Pa) for 23 h 55 min. The stoichiometry factor  1.5 results from the identification of the formed oxide as Fe2O3 by SAD (selected area diffraction) in a TEM (transmission electron microscope). Several lattice structures (hematite, corundum, spinel) appear. The outward diffusion of iron in palladium towards the oxidation front is negligibly small (16). Determination of the penetration depth by SEM (scanning electron microscopy) yields (115.210) µm. As can be seen in Fig. 5, the prediction of the computer simulation agrees well with this experimental result.ADVANCED MODELING OF INTERNAL OXIDATION

The new iterative process simulation tool provides several extensions of Wagner's theory with high relevance to practical applications. Heterogeneous M distributions are included in the model. Furthermore, nonisobaric and/or nonisothermal internal oxidation can be simulated.

Figure 5. Comparison with experiment.

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Heterogeneous initial distribution of the alloying element

Production processes of technical alloys often yield near-surface enrichment or depletion of added components. In Fig. 6, the last-mentioned case is exemplarily con-sidered. The chosen process conditions are given. The inset shows the applied initial M distribution, . For comparison, the corresponding uniform M profile is drawn in as dashed line. Here, 1 at.% represents the mean value:

(8)

Figure 6. Edge zone depletion of element M.

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The two different slopes of the calculated time curve of the penetration depth in Fig. 6 correlate with the homogeneous parts of the initial M distribution, 0.3 at.% and

1.18 at.%. In the second simulation example, a periodic M profile is considered. Figure 7 presents the result of this computation. The applied sinusoidal initial M distribution is shown in the inset. Again, averaging of (x) follows from Eq. (8).

Nonisobaric internal oxidation

Under the assumption of fast oxygen-releasing interfacial reactions, the changing surface concentration, cO(x0), is given by the equilibrium solid solubility of the base metal, . In the following example, calculation of occurs in accordance with the varying partial pressure in the gas atmosphere. Simulation of a nonisobaric (isothermal) process demands for the consideration of a variable boundary condition of the first kind. This is included in the presented computer model by recalculating for each time step t in Eq. (7). Note that firstly, constant temperature T is

assumed. As for a nonisobaric reaction, varies with process time t, the applied oxidation conditions define the control curve.

Figure 7. Sinusoidal initial M profile.

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An example is given in Fig. 8. According to the solubility data on palladium (3, 13), the assumed linear decrease of with time from 105 to 1 Pa (see inset) leads to the shown relationship between and t. For isothermal nonisobaric internal oxidation, the diffusion coefficient DO of oxygen in palladium remains constant at the given value (3, 8).

Figure 9 reveals the result of the corresponding computer simulation for two uniform initial M distributions ( 0.5 at.%, 1 at.%) within the palladium matrix. To il-lustrate the consequence of the nonisobaric process on the time dependence of the penetration depth, the calculated parabolic -t curves for constant oxygen surface concentration consistent with the starting value of 105 Pa are drawn in as dotted lines for comparison. For t0, the simulation yields:

(0.5 at.% M) / (1 at.% M) 2

Figure 8. Control curve of the nonisobaric process.

Figure 9. Computer simulations of the process defined in Fig. 8.

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According to Eq. (5), this result agrees with the - relationship derived from Wagner’s theory, ( )½.

Nonisothermal internal oxidation

As long as the oxygen offer remains constant (e.g. no compensating control in the gas atmosphere), nonisothermal internal oxidation inevitably leads to a quasi-nonisobaric process, i.e. variable surface concentration ( data used in the following example), as treated in the previous section.

This interrelation is illustrated in Fig. 10. The inset reveals the assumed linear temperature increase with time that results in the shown DO-t curve. Internal oxidation in ambient air is assumed, i.e.  2.10104 Pa  const. The solid solubility of oxygen slightly decreases with rising temperature T (weakly exothermic solution). Therefore, simulation of nonisothermal internal oxidation by means of the presented computer model involves recalculation of both DO and for each iteration of time step t in Eq. (7). The DO - t and - t control curves result from the process conditions, which are expressed by the T-t relationship (cf. Fig. 10). Note that this computation scheme also covers general nonisothermal nonisobaric reactions, as it takes varying oxygen surface concentrations into account.

Figure 10. Control curves of the nonisothermal process.

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Figure 11 shows the calculated time dependence of the penetration depth for the example process defined in Fig. 10. Homogeneous M distribution within the palladium matrix is assumed. For comparison, two different chemical compositions of the formed oxide MO are considered with stoichiometry factors 1 and 2, respectively. The simulation provides ½ for t0, in agreement with Wagner’s theory according to Eq. (5).

SUMMARY AND CONCLUSIONS

In the present paper, a new iterative computer model for internal oxidation is introduced. Under the simplified conditions of Wagner’s theory, both approaches agree admirably. Applicability is also confirmed by comparison with experiment. The proposed simulation tool, based on an iterative analytical solution for the inward diffusion of oxygen with chemical reaction, additionally allow consideration of heterogeneous distributions of alloying elements within the base metal matrix, as it is usually the case with engineering materials. The implemented calculation scheme also permits simulation of nonisothermal and/or nonisobaric internal oxidation. The involved assumption of immobile alloying atoms, i.e. neglect of counter-diffusion, represents a realistic approximation that generally holds for material systems of prac-tical interest. The application potential of the developed computer model is discussed in detail by means of characteristic example processes. Both dispersion hardening procedures and undesirable internal oxidation can be predicted or mathematically ana-lyzed by the new simulation tool.

REFERENCES

1. Naser J, Riehemann W, Ferkel H: 'Dispersion hardening of metals by nanoscaled ceramic powders'. Mater. Sci. Eng. A 1997 234-236 467-69.

2. Meijering, J L: 'Internal Oxidation in Alloys'. In: Herman, H (ed.): Advanc. in Mat. Res. 1971 5 1-81, Chichester, Wiley-Interscience.

3. Gegner J: Segregation von Sauerstoff an inneren Metall/Oxid-Phasengrenzen. Thesis, University of Stuttgart, 1995.

Figure 11. Computer simulations of the process defined in Fig. 10.

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4. Gegner J, Weible A: 'Internal Oxidation of Palladium/Magnesium-Alloys'. Z. Me-tallkd. 2000 91 (12) 1026-29.

5. Pippel E, Woltersdorf J, Gegner J, Kirchheim R: 'Evidence of Oxygen Segregation at Ag/MgO-Interfaces'. Acta Mater. 2000 48 (10) 2571-78.

6. Mader W: 'Metal/Metal-Oxide Interfaces: Structure and Chemical Composition using HRTEM'. Z. Metallkd. 1992 83 (7) 478-86.

7. Gegner J: 'On the Kinetics of Oxygen Segregation at Internal Metal-Oxide Inter-faces'. 4th Int Congress Materials Week, Munich, Werkstoffwoche-Partnerschaft, CD-ROM ISBN 3-88355-302-6, 2001.

8. Gegner J, Hörz G, Kirchheim R: 'Segregation of Oxygen at Metal/Oxide-Inter-faces'. Interface Sci. 1997 5 231-43.

9. Meijering J L: 'On the Diffusion of Oxygen through Solid Iron'. Acta Metall. 1955 3 (3) 157-62.

10. Gegner J: 'Decarburization-Controlled Internal Oxidation of Rolling Bearing Steel SAE 52100 at the Dissociation Pressure of Wüstite'. Z. Metallkd. 2003 94 (1) 30-35.

11. Nesbitt J A: 'Numerical Modeling of High-Temperature Corrosion Processes'. Oxid. Met. 1995 44 (1/2) 309-38.

12. Vedula K M, Funkenbusch A W, Heckel R W: 'A Mathematical Model for Inter-nal Oxidation'. In: Rapp R A (ed.): High Temperature Corrosion. Houston/Texas, National Association of Corrosion Engineers, 1983, pp. 192-97.

13. Jehn H, Grallath E: 'Solid Solubility of Oxygen in Palladium'. In: Kudryk V, Corrigan D A, Liang W W (Eds.): Precious Metals: Mining, Extraction, and Pro-cessing. New York, TMS-AIME, Warrendale, 1984, pp. 595-602.

14. Wagner C: 'Reaktionstypen bei der Oxydation von Legierungen'. Z. Elektrochem. 1959 63 (7) 772-82.

15. Crank J: The Mathematics of Diffusion. Oxford, Clarendon Press, 1990.16. Fillon J, Calais D: 'Self-Diffusion in Concentrated Fe-Pd Alloys'. J. Phys. Chem.

Sol. 1977 38 81-89.

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