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Acta Materialia 56 (2008) 1890–1896

Analytical treatment of diffusion during precipitate growthin multicomponent systems

Qing Chen a,*, Johan Jeppsson b, John Agren b

a Thermo-Calc Software AB, Bjornnasvagen 21, 113 47 Stockhlom, Swedenb Department of Materials Science and Engineering, KTH, 100 44 Stockholm, Sweden

Received 17 October 2007; received in revised form 19 December 2007; accepted 19 December 2007Available online 4 March 2008

Abstract

We propose an approximate growth rate equation that takes into account both cross-diffusion and high supersaturations for modelingprecipitation in multicomponent systems. We then apply it to an Fe-alloy in which interstitial C atoms diffuse much faster than substi-tutional solutes, and predict a spontaneous transition from slow growth under ortho-equilibrium to fast growth under the non-partition-ing local equilibrium condition. The transition is caused by the decrease in the Gibbs–Thomson effect as the growing particle becomeslarger. The results agree with DICTRA simulations where full diffusion fields are calculated.� 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Precipitation; Diffusion; Kinetics; Thermodynamics; NPLE

1. Introduction

Precipitation processes may be modeled in detail by dif-fuse interface methods such as phase-field [1] or sharpinterface methods like DICTRA [2]. These simulationsare based on numerical calculations of full diffusion fieldsand may be quite time consuming. Consequently, only asmall number of particles may reasonably be handled. Inmany applications one is less interested in fine details butmore interested in handling complex alloys with more than10 elements and in the size distributions of several precipi-tate phases. It is then necessary to take a simpler approach.Usually one avoids solving the full diffusion problem andrather applies analytical expressions for the diffusive fluxesof the solute elements. Such expressions may be obtainedfrom the steady-state field approximation or similarapproaches [3–6]. These solutions only use the precipitate,the interface and the matrix average compositions to repre-sent the diffusion field [3–5]. In some approaches not eventhe interface compositions are used [6].

1359-6454/$34.00 � 2008 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2007.12.037

* Corresponding author. Tel.: +46 8 54595939; fax: +46 8 6733718.E-mail address: qing@thermocalc.se (Q. Chen).

In general, for multicomponent alloys the operating tie-line at the phase interface has to be found numerically byseeking a common growth rate for all independent compo-nents. In previous studies the growth rate was formulatedby either neglecting the cross-terms in the diffusivity matrixin order to consider high supersaturations [4], or by includ-ing cross-diffusion terms but restricting the calculations tosmall supersaturations [3]. Sometimes, even for small super-saturations, the cross-diffusion was ignored [5]. In Ref. [4], itwas felt impossible to use the analytic binary growth rateequation for multicomponent systems, and the consequenceof neglecting the cross-diffusion terms was severe deviationof the modeled results from experimental observations sothat an ad hoc growth rate had to be found. The approachin Ref. [3] was not aimed specifically at high supersatura-tions. However, for a system with significantly differentatomic mobilities, large supersaturations can occur forslow-moving elements even though the overall supersatura-tion of the system is very small. In the extreme case of a non-partitioning local equilibrium (NPLE) growth, the super-saturations of substitutional elements approach unity. Theapproach is then not applicable. Therefore, it seems neces-sary to find a multicomponent growth rate equation that

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Q. Chen et al. / Acta Materialia 56 (2008) 1890–1896 1891

can take both cross-diffusion terms and high supersatura-tions into account at the same time. In this work, we presentsuch an equation and apply it to a steel to predict a sponta-neous transition from slow growth to fast growth due to theGibbs–Thomson effect.

2. Model

For the sake of simplicity, we treat a spherical particle ofstoichiometric composition or with negligible atomic diffu-sivity growing under the diffusion-controlled condition. Westart from the exact solution to binary systems and refor-mulate it by introducing an effective diffusion distance fac-tor, and then derive a general multicomponent growth rateequation on the basis of the local equilibrium assumptionand the flux balance equations that depends on individualeffective diffusion distances, which in turn varies withchanging supersaturations of individual components.Finally, we break up diffusivities into mobilities and ther-modynamic factors, and present an equation to treatcross-diffusion directly by working with mobilities andchemical potential gradients.

2.1. Binary systems

Given an alloy of concentration cM, the concentration inthe matrix at the phase interface cI and that in the precip-itate cP can be uniquely determined by the phase equilib-rium tie-line under the isothermal local equilibriumassumption. With these well-defined interface conditions,the diffusion-controlled growth of an isolated precipitatein an infinite matrix can be solved exactly [7,8] and thegrowth rate t can be written as

t ¼ 2Dk2

R; ð1Þ

where D is the diffusivity in the matrix, R is the radius andk is given by

2k2 � 2k3 ffiffiffipp

expðk2ÞerfcðkÞ ¼ X; ð2Þwhere X ¼ ðcM � cIÞ=ðcP � cIÞ is the so-called dimension-less supersaturation. Eqs. (1) and (2) are actually obtainedfrom the flux balance equation:

tðcP � cIÞ ¼ Drc; ð3Þwhere rc is the concentration gradient close to the phaseinterface. Let us introduce an effective diffusion distanced and rewrite the above equation as

tðcP � cIÞ ¼ DðcM � cIÞ=d: ð4ÞComparing this result with Eq. (1), we get:

d ¼ X

2k2R ¼ nR; ð5Þ

where n ¼ X=2k2 is a factor for adjusting the effective diffu-sion distance from the radius as the supersaturation varies.Inserting the above expression into Eq. (4), we get:

tðcP � cIÞ ¼ DðcM � cIÞ=nR: ð6ÞIf the supersaturation is very small, i.e. X! 0, one obtainsk ¼

ffiffiffiffiffiffiffiffiffiX=2

pand thus n ¼ 1, which recovers the well-known

steady-state equation used, for example, in coarsening the-ory [9]:

t ¼ DðcM � cIÞRðcP � cIÞ : ð7Þ

If the supersaturation is very large, i.e. X! 1, one obtains

k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3=2ð1� XÞ

pand thus n ¼ Xð1� XÞ=3! 0, i.e. the

effective diffusion distance becomes very small and thegrowth rate very large.

2.2. Multicomponent systems

In an n-component alloy of concentration cMi , generally

speaking, the interface concentrations cIi and cP

i in thematrix and precipitate phases are determined simulta-neously [10] with the interface velocity from the flux bal-ance equation for each element:

t cPi � cI

i

� �¼Xn�1

j¼1

Dijrcj ð8Þ

where Dij is the chemical diffusivity in the matrix phase andrcj is the concentration gradient of element j at the inter-face. Both i and j go from 1 to n � 1 because there are n � 1independent concentration variables in an n-componentsystem. So far we have only n � 1 equations for 2n � 1unknowns. Invoking the local equilibrium assumption,we obtain n more necessary equations:

lPi ¼ lI

i ; ð9Þwhere lP

i and lIi are chemical potentials of element i in the

precipitate and matrix at the interface, respectively, and arefunctions of cP

i and cIi . Now we have to define rcj in order

to solve 2n � 1 (Eqs. (8) and (9)). Supposing that the effec-tive diffusion distance dj or the factor nj for each element j

depends only on its own supersaturation Xj, we can obtaina general multicomponent growth rate equation directlyfrom Eq. (6):

t cPi � cI

i

� �¼Xn�1

j¼1

Dij cMj � cI

j

� �.njR: ð10Þ

During the coarsening stage of a precipitate, Xj ! 0 andthus nj ! 1, and we obtain the same multicomponentcoarsening rate equation as that of Morral and Purdy [11].

2.3. Direct treatment of cross-diffusion with mobilites and

chemical potentials

The off-diagonal terms in the diffusivity matrix aremostly related to thermodynamic effects but also dependon the choice of frame of reference. The first effect comesfrom expressing fluxes as functions of concentration gradi-ents. The driving force for diffusion is essentially a gradient

Fig. 1. The calculated growth rates of metastable M23C6 particles ofdifferent sizes in supersaturated ferrite alloys A (Fe–2 wt.% Cr–0.2 wt.%C) and B (Fe–2 wt.% Cr–0.05 wt.% C).

1892 Q. Chen et al. / Acta Materialia 56 (2008) 1890–1896

in chemical potential and it can therefore be more suitableto work directly with mobilities rather than with chemicaldiffusivities. The diffusivity matrix can be divided into amobility M and a thermodynamic factor ol/oc, and thelarge off-diagonal elements in the matrix come from thethermodynamic factor:

Dik ¼Xn�1

j¼1

cjMijolj

ock: ð11Þ

The diffusional flux of element i can then be expressed as

J i ¼ �Xn�1

k¼1

Dikrck ¼ �Xn�1

j¼1

cjMij

Xn�1

k¼1

olj

ockrck

¼ �Xn�1

j¼1

cjMijrlj: ð12Þ

The off-diagonal terms in the last part of Eq. (12) are onlyrelated to the frame of reference; they are zero in the latticefixed frame of reference and non-zero in any other frame ofreference due to the Kirkendall effect. The effect is, how-ever, small compared to the approximations introducedearlier with the modified steady-state equation. Neglectingthese cross-terms, Eq. (10) can be expressed with mobilitiesand chemical potential gradients as

t cPi � cI

i

� �¼ cI

i MiðlMi � lI

i Þ=niR: ð13Þ

In the above equation, the cross-diffusion is accounted fordirectly by the dependence of chemical potentials onconcentrations.

3. Results and discussion

We now apply our model to calculate the growth rate ofa M23C6 particle metastably formed from a supersaturatedferrite alloy A (Fe–2 wt.% Cr–0.2 wt.% C) at 1053 K. Allnecessary thermodynamic and kinetic data, i.e. the temper-ature- and composition-dependent Gibbs free energy andatomic mobility, were taken from the well-assessed dat-abases TCFE4 [12] and MOB2 [13]. When mobility varieswith composition, an average value was used. TheGibbs–Thomson effect was taken into account during thecalculation of chemical potentials by adding a pressure dif-ference term, 2rV m=R, to the Gibbs energy of the M23C6

phase. The interfacial energy r and the molar volumeV m, based on substitutional elements, were assumed con-stant and equal to 0.4 J m�2 and 6 � 10�6 m3 mol�1,respectively. For each given particle radius R, the growthrate t was obtained together with the interface concentra-tions cP

i and cIi (i = C, Cr) by numerically solving five non-

linear equations that consist of Eqs. (9) and (13).The calculated results are shown as curve A in Fig. 1. It

is interesting to see that the growth rate increases dramat-ically from zero as the particle size increases from the crit-ical radius, 0.78 nm, and then reaches almost a plateau,and then shoots up again and reaches a maximum, andfinally decreases gradually. This is obviously different from

a normal t vs. R curve, such as curve B, where the growthrate of a M23C6 particle from another supersaturated fer-rite alloy B (Fe–2 wt.% Cr–0.05 wt.% C) at the same tem-perature is shown to increase sharply to its maximum ataround twice the critical nucleus size and then decreasesslowly.

We now examine the other results obtained at the sametime during the simulation, i.e. the dependence of interfaceconcentrations on the particle size for alloys A and B. Theresults are depicted in Figs. 2 and 3, where u-fractions oratomic fractions per mole of substitutional elements areused. The C and Cr contents in the body-centered cubic(bcc) matrix phase far away from the interface are shownas dotted lines, which coincide with the left end points ofthose curves for the bcc phase, i.e. concentrations at theinterface in the bcc phase for a particle of the criticalnucleus size. As can clearly be seen, for alloy A at a particlesize of about 2.1 nm, abrupt changes occur for the Cr con-tent in both bcc and M23C6 phases and for the C content inbcc. For particles larger than 2.1 nm in radius, their Crcontent at the interface is almost the same as that in thematrix. In contrast, no sudden changes exist for the varia-tions of interface concentrations in alloy B, and the redis-tributions of solutes in the matrix and precipitate phasesare continuous and very smooth.

Returning to Fig. 1, we see from curve A that thegrowth rates for small particles around 1–2 nm are severalorders of magnitude slower than those for particles largerthan 2.1 nm. So the curve can be easily divided into twoparts, one corresponding to a slow growth mode and theother to a fast growth mode. In fact, from Figs. 2 and 3,we know immediately that the slow growth mode involvesredistribution of Cr in the matrix and precipitation underlocal ortho-equilibrium, and the fast growth mode involvesno such redistribution as the Cr content in the particles isthe same as that in the matrix, which corresponds to a

Fig. 2. The calculated interface compositions for the metastable growth ofM23C6 particles in supersaturated ferrite alloy A (Fe–2 wt.% Cr–0.2 wt.%C). The dotted line represents the matrix composition.

Fig. 3. The calculated interface compositions for the metastable growth ofM23C6 particles in supersaturated ferrite alloy B (Fe–2 wt.% Cr–0.05 wt.%C). The dotted line represents the matrix composition.

Q. Chen et al. / Acta Materialia 56 (2008) 1890–1896 1893

phase transformation under the so-called NPLE condition.For alloy B, the growth of the particle is governed by theslow mode and is not influenced by particle size.

To find out why the behavior for alloys A and B are sodifferent and why the growth mode can change spontane-ously as the particle grows larger, we need to examine themetastable phase equilibrium between the bcc and M23C6

phases and the influence of the Gibbs–Thomson effect onit. The isothermal section of the Fe–Cr–C system at1053 K is calculated by using Thermo-Calc software [14]and the thermodynamic database TCFE4 [12]. The result-ing Fe-rich corner is shown in Fig. 4. To calculate the meta-stable bcc and M23C6 phase equilibrium, all other phases inthe database are suspended during the calculation. Theblack solid line is the solvus of the bcc phase, and the greensolid lines are the tie-lines of the two phases being consid-ered here. For particles with a finite radius, the phase equi-librium can be calculated by adding a curvature-inducedpressure difference term, 2rV m=R, to the Gibbs energy of

the M23C6 phase, i.e. the contribution due to the so-calledGibbs–Thomson effect. Therefore, for a series of given par-ticle radius, we may draw a series of corresponding bcc sol-vus lines and the related tie-lines. For clarity, only the linesfor the particle size of 2.1 nm, where the growth modechanges in alloy A, are plotted and they have the same col-ors, but are dashed. The two circle symbols mark the con-centrations of alloys A and B. If we calculate the bcc solvuslines for the M23C6 particles of critical sizes in alloys A andB, they should certainly pass through points A and B,respectively.

By ignoring the M23C6 phase, the iso-carbon-activitylines for the two alloys can also be calculated, and theseare shown by the red lines passing through points A andB. The former intersects at C with the solvus line of onlythe 2.1 nm particles, and the latter crosses both solvus lines,especially at E for that corresponding to an infinitely largeparticle. It should be mentioned that two special tie-linesfor M23C6 containing the same amount of Cr as in the

Fig. 4. The calculated isothermal Fe–Cr–C phase diagram at 1053 K. Thesolid black line is the solvus of the bcc phase, and the solid green lines arethe conjugate tie-lines of the bcc and M23C6 phases. The dashed black andgreen lines are similar ones corresponding to a particle size of 2.1 nm. Thered lines are iso-carbon-activity lines for ferrite alloys A and B,respectively. The heavy blue lines are the loci of the compositions of thebcc phase at the interface with M23C6 particles of different sizes.

Fig. 5. The calculated growth rates of metastable M23C6 particles ofdifferent sizes in supersaturated ferrite alloys A (Fe–2 wt.% Cr–0.2 wt.%C) and B (Fe–2 wt.% Cr–0.05 wt.% C). The solid black lines indicateresults from direct treatment of cross-diffusion. The dashed lines indicateresults from indirect treatment of cross-diffusion. The dotted lines indicateresults from indirect treatment without use of cross-diffusivities. Thesquare and diamond symbols indicate results from DICTRA simulations.

1894 Q. Chen et al. / Acta Materialia 56 (2008) 1890–1896

matrix have also been calculated for infinitely large and2.1 nm particles. They are those lines starting from thebcc side at D and C, respectively. The loci of the composi-tions of the bcc phase at the interface shown in Figs. 2and 3 are also superimposed in the diagram as heavy bluelines. As we can see, for alloy B, the locus starts from Band follow the iso-carbon-activity line and end at E; foralloy A, the locus starts from A and follow the iso-car-bon-activity line to C, and then change its course abruptlyand move almost horizontally to D. As expected, the locuscorresponding to the slow mode coincides with the iso-car-bon-activity lines and this means that the chemical potentialgradient of carbon near the interface in the bcc phase isalmost negligible. However, despite the negligible gradient,a small finite interface velocity, which matches that of thevery slow-moving Cr atoms, can be obtained for carbondue to its extremely large atomic mobility. Hence, thegrowth of the particles is controlled by the diffusion of Crin this mode.

In the case of alloy A, when the particle size is largerthan 2.1 nm, the fast growth mode is adopted because itis now possible for the precipitates to inherit the Cr contentin the matrix, which means undergoing a NPLE duringgrowth that is fully controlled by the diffusion of carbon.In this mode, the supersaturation of Cr is almost 1, so itseffective diffusion distance factor is approaching 0, andthe gradient of chemical potential of Cr becomes tremen-dously large, which makes it possible for the slow-movingCr to acquire a very high interface velocity, the same asthat for C, whose chemical potential gradient is now finiteand whose mobility is always very large.

It is understandable that the NPLE growth mechanismis the only feasible one for particles larger than a critical

value, which was calculated to be 2.8 nm, because thereexists no cross-point for the bcc solvus and iso-carbon-activity lines above this critical value. For particles thatare smaller than 2.8 nm but larger than 2.1 nm, the bcc sol-vus and iso-carbon-activity line do intersect with eachother, but this does not mean that the slow growth modeis possible. In fact, the intersection points for particles inthis range of size would yield tie-lines suggesting that theCr content in M23C6 at the interface is lower than that inthe matrix. This is simply not viable for the growth ofM23C6 particles from the bcc matrix where the distributioncoefficient of Cr is greater than 1.

3.1. Comparison with DICTRA simulation results

The derived model is intended for multiparticle, multi-phase simulations in complex alloys and is therefore a sim-plified approach to the diffusion problem. The model hasbeen compared to solutions of the full diffusion problemto show that the introduced approximations are not severe.The software DICTRA [2] was used to simulate the growthof one spherical M23C6 particle in alloys A and B. All ther-modynamic and kinetic data were taken from the databasesTCFE4 [12] and MOB2 [13], and the Gibbs–Thomsoneffect was accounted for in the usual way.

It should be mentioned that the simulations in DICTRAwere numerically tricky to perform due to large differences inthe magnitude of both particle size and growth rate. Forexample, in the case of alloy A, concentration gradients inthe range of fractions of nanometers have to be resolvedin the matrix close to the particle, and the matrix has to beof the order of micrometers to avoid impingement. The growthrate also changes abruptly by five orders of magnitude.

Fig. 6. Same as Fig. 4 except that the dashed blue lines are the resultsfrom indirect treatment of cross-diffusion, and the dotted blue lines are theresults from indirect treatment without use of cross-diffusivities, and thesquare and diamond symbols are the results from DICTRA simulations.

Q. Chen et al. / Acta Materialia 56 (2008) 1890–1896 1895

The results from the DICTRA simulations are indicatedby square and diamond symbols in Figs. 5 and 6, togetherwith the results from our simplified approach indicated bysolid lines. For both the growth rate and composition as afunction of particle size, the agreement is very good. Thesimplified model can even in the extreme case of alloy Afully describe the switch of the growth mode from ortho-equilibrium to NPLE.

3.2. Method with diffusion coefficients

In Section 2.2, we derived a method of treating multi-component particle growth including cross-diffusion withdiffusion coefficients. The method is indirect in the sensethat the cross-diffusion can be accounted for directly byusing mobilities and chemical potentials as shown in Sec-tion 2.3.

The results from the simulations for alloys A and Busing this indirect method, i.e. solving Eqs. (9) and (10),are also shown in Figs. 5 and 6. For comparison, the resultsfrom the calculations using diffusivities without cross-termshave also been added.

Starting with alloy B, it is easy to see that the cross-terms are important. Neglecting cross-diffusivities resultsin large deviations from the direct method and the DIC-TRA simulation results. The growth rates are overesti-mated because the obtained operating compositions atthe interface are wrong. Ignoring the cross-terms in Eq.(10), we see that in order to achieve a slow growth mode,the matrix carbon composition at the interface has to beabout the same as that far away from the interface. Notethat it is composition and not chemical potential or activitythat is used. This is why, without cross-terms, the operatinginterface compositions follow almost the iso-carbon-com-position line in Fig. 6. The correct interface compositionsshould follow the iso-carbon-activity line as indicated by

the direct method or DICTRA. The indirect method oftreating cross-terms gives a similar result as the directmethod and agrees more or less with the DICTRA simula-tion, which indicates that the chemical potential gradientsare described well with cross-diffusivities and compositiongradients for alloy B in our individual effective diffusiondistance approach.

In the case of alloy A, the results are different. Byneglecting the cross-diffusivities, the transition fromortho-equilibrium to NPLE is captured but the growthrates are overestimated and the transition is not at the cor-rect particle size. This is due to the same reason asdiscussed in the preceding paragraph, i.e. the wrong inter-face compositions that follow the iso-carbon-compositionline and deviate from the iso-carbon-activity line. An evenworse, and maybe surprising, result is that the indirectmethod of treating cross-terms cannot capture the transi-tion to NPLE, and the growth rate is orders of magnitudetoo low when the particle size is over 2.1 nm. In Fig. 6, wecan see that the interface compositions never reach NPLEand the growth therefore never gets fully controlled by car-bon diffusion. The indirect method of treating cross-diffu-sion estimates incorrectly the cross-term effect in thisextreme case where the diffusion distance of one compo-nent is very short and therefore the cross-terms becomevery important. The direct method of treating cross-diffu-sion estimates correctly the cross-term effect because it isaccounted for directly by the dependence of chemicalpotential on concentrations.

4. Summary and conclusions

We have proposed an approximate growth rate equationthat takes into account both cross-diffusion and high super-saturations for modeling precipitation in multicomponentsystems. Although the derivation is confined to sphericalparticles, it can be readily extended to other simple geom-etries. With this equation, different possible growth modescan be captured automatically without any ad hoc treat-ment. We have applied the equation to a Fe-alloy whereinterstitial C atoms diffuse much faster than substitutionalsolutes and found a spontaneous transition from slowgrowth under ortho-equilibrium to fast growth under theNPLE condition. From analysis of phase equilibrium andthermodynamics subjected to pressures induced by the cur-vature of particles of finite sizes, we understand that thetransition is caused by the decrease in the Gibbs–Thomsoneffect as the growing particle becomes larger. DICTRAsimulations have also been carried out and the obtainedresults corroborate that of our simplified approach. It hasalso been found that the method using mobilities insteadof diffusivities within our individual effective diffusion dis-tance approach is probably always superior due to the factthat chemical potential gradients instead of compositiongradients are then used and the cross-term effect can there-fore be accounted for directly by the composition depen-dence of chemical potentials.

1896 Q. Chen et al. / Acta Materialia 56 (2008) 1890–1896

Acknowledgments

One of the authors (J.J.) gratefully acknowledges sup-port from the Swedish foundation for strategic research(SSF) through the MATOP program.

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