8
Anomalous diffusion in dilute solid solutions R. Krishnamurthy a,b, * , D.J. Srolovitz a,c , M.I. Mendelev d a Department of Mechanical Engineering, Princeton Institute for the Science and Technology of Materials, Princeton University, NJ 08542, USA b Technology and Solutions Division, Caterpillar Inc., Mossville, IL 61552, USA c Yeshiva College, Yeshiva University, New York, NY 10033, USA d Ames Laboratory, Ames, IA 500014, USA Received 5 March 2007; received in revised form 1 June 2007; accepted 5 June 2007 Available online 24 July 2007 Abstract Diffusion in metals and alloys encapsulates many different physical phenomena and a range of time and length scales, and conse- quently, a hierarchical combination of simulation methods is required to study diffusion. We develop such methods to study the role interaction among defects and diffusants, and local association effects, play in diffusion in metals. We use Fe impurity diffusion in Al as an example. Using recently developed, accurate, interatomic potentials for the Fe–Al system, we calculate migration energies for atom–vacancy exchange in a variety of local atomic configurations, using lattice statics methods. These are used in a kinetic Monte-Carlo framework to calculate diffusivities. Two different activation regimes are observed at temperatures above and below 550 K. We explain this anomalous, non-Arrhenius behavior of the diffusion activation energy in terms of the interaction among vacancies and Fe atoms, and local association/ordering effects. Ó 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Diffusion; MC simulation; Molecular statics simulation; Impurities; Short-range ordering 1. Introduction Diffusion in crystalline, metallic, solid solutions is rich with a variety of interesting physical phenomena. For example, the activation energy for diffusion can be different from that for an individual atom exchanging with a vacancy or an interstitial atom. The reason for this is the following. Since many different local arrangements of vacancies and chemical species, and associated energies, are possible in a solid solution, multiple jump frequencies are associated with diffusion in a solid solution. Conse- quently, the activation energy associated with macroscopic diffusivity, which is characterized by multiple atom hops across many different atom configurations, will be different from that associated with any individual atom hop. Fur- thermore, both substitutional diffusion (typically mediated by vacancies) and interstitial diffusion differ considerably from a true random walk process. The difference is due to correlation effects. Correlation effects can be attributed to either the mechanism (vacancy or interstitial atom med- iated) by which diffusion occurs, or to the many different jump frequencies that characterize diffusion in solid solu- tions. For example, in vacancy-mediated self-diffusion, the diffusing atom is more likely to jump back to the vacant site it occupied prior to occupying its current position than hop to one of the other neighboring sites. This is because any of the other neighboring sites can become vacant only after the vacancy makes the series of jumps necessary for it to get to that site. Clearly, diffusion in this case is not a ran- dom walk, but is negatively correlated, as the jump proba- bility for the atom returning to its earlier position is much 1359-6454/$30.00 Ó 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2007.06.001 * Corresponding author. Address: Materials Modeling Group, TC-K, Technical Center, Caterpillar Inc., PO Box 1875, Mossville, IL 61552, USA. Tel.: +1 309 578 5186; fax: +1 309 578 3322. E-mail addresses: [email protected] (R. Krishnamurthy), [email protected] (D.J. Srolovitz), [email protected] (M.I. Mendelev). www.elsevier.com/locate/actamat Acta Materialia 55 (2007) 5289–5296

Anomalous diffusion in dilute solid solutions

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www.elsevier.com/locate/actamat

Acta Materialia 55 (2007) 5289–5296

Anomalous diffusion in dilute solid solutions

R. Krishnamurthy a,b,*, D.J. Srolovitz a,c, M.I. Mendelev d

a Department of Mechanical Engineering, Princeton Institute for the Science and Technology of Materials, Princeton University, NJ 08542, USAb Technology and Solutions Division, Caterpillar Inc., Mossville, IL 61552, USA

c Yeshiva College, Yeshiva University, New York, NY 10033, USAd Ames Laboratory, Ames, IA 500014, USA

Received 5 March 2007; received in revised form 1 June 2007; accepted 5 June 2007Available online 24 July 2007

Abstract

Diffusion in metals and alloys encapsulates many different physical phenomena and a range of time and length scales, and conse-quently, a hierarchical combination of simulation methods is required to study diffusion. We develop such methods to study the roleinteraction among defects and diffusants, and local association effects, play in diffusion in metals. We use Fe impurity diffusion in Alas an example. Using recently developed, accurate, interatomic potentials for the Fe–Al system, we calculate migration energies foratom–vacancy exchange in a variety of local atomic configurations, using lattice statics methods. These are used in a kinetic Monte-Carloframework to calculate diffusivities. Two different activation regimes are observed at temperatures above and below 550 K. We explainthis anomalous, non-Arrhenius behavior of the diffusion activation energy in terms of the interaction among vacancies and Fe atoms,and local association/ordering effects.� 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Diffusion; MC simulation; Molecular statics simulation; Impurities; Short-range ordering

1. Introduction

Diffusion in crystalline, metallic, solid solutions is richwith a variety of interesting physical phenomena. Forexample, the activation energy for diffusion can be differentfrom that for an individual atom exchanging with avacancy or an interstitial atom. The reason for this is thefollowing. Since many different local arrangements ofvacancies and chemical species, and associated energies,are possible in a solid solution, multiple jump frequenciesare associated with diffusion in a solid solution. Conse-quently, the activation energy associated with macroscopic

1359-6454/$30.00 � 2007 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2007.06.001

* Corresponding author. Address: Materials Modeling Group, TC-K,Technical Center, Caterpillar Inc., PO Box 1875, Mossville, IL 61552,USA. Tel.: +1 309 578 5186; fax: +1 309 578 3322.

E-mail addresses: [email protected] (R. Krishnamurthy),[email protected] (D.J. Srolovitz), [email protected] (M.I. Mendelev).

diffusivity, which is characterized by multiple atom hopsacross many different atom configurations, will be differentfrom that associated with any individual atom hop. Fur-thermore, both substitutional diffusion (typically mediatedby vacancies) and interstitial diffusion differ considerablyfrom a true random walk process. The difference is dueto correlation effects. Correlation effects can be attributedto either the mechanism (vacancy or interstitial atom med-iated) by which diffusion occurs, or to the many differentjump frequencies that characterize diffusion in solid solu-tions. For example, in vacancy-mediated self-diffusion,the diffusing atom is more likely to jump back to the vacantsite it occupied prior to occupying its current position thanhop to one of the other neighboring sites. This is becauseany of the other neighboring sites can become vacant onlyafter the vacancy makes the series of jumps necessary for itto get to that site. Clearly, diffusion in this case is not a ran-dom walk, but is negatively correlated, as the jump proba-bility for the atom returning to its earlier position is much

rights reserved.

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higher than that for the atom to jump to another neighbor-ing site. In solid solutions, this picture is complicated bythe presence of many different local configurations, andhence, many different jump frequencies. Correlation effectsof this type were first discussed in the classic paper byBardeen and Herring [1]. A comprehensive, but datedaccount of correlation effects can be found in the mono-graph authored by Manning [2].

Diffusivities in solid solutions also vary widely, depend-ing upon whether the alloy is disordered or ordered. In adisordered alloy, the diffusivity is directly proportional tothe weighted jump frequency for the diffusant exchangingwith a neighboring vacancy; the weights correspond tothe relative probabilities for occurrence of each of the indi-vidual atom hops that contribute to diffusion. In contrast,the diffusivity may bear no relation to the weighted jumpfrequency in an ordered alloy. In fact, exchanges with near-est neighbor vacancies may result in a destruction of localorder in ordered alloys. Consequently, the effective jumpevent characterizing diffusion in these materials comprisesa sequence of consecutive jumps that collectively conserveorder [3]. Such behavior is commonly observed in orderedintermetallics, such as TiAl and FeAl [4,5]. The activationenergy for diffusion in metals and alloys can also be verydifferent in different allotropes of the same system. Forexample, the activation energy for Ti self-diffusion andthe diffusion of impurities (Al, Ga, In in Ti) changes dra-matically at the a-Ti to b-Ti allotropic transition tempera-ture [4]. Diffusivities also depend upon the nature of thepoint defects in solid solutions. In some alloys, associateddefects (like di-vacancies, di- and tri-interstitials) may bepredominant, and diffusion in these alloys requires eitherthe migration of these associated defects or their disassoci-ation. Consequently, the activation energy for atom hop-ping may be different from that for a single vacancy(interstitial)–atom exchange. Such association effects arebelieved to be important in semiconductor systems (e.g.the diffusion of quenched-in defects in Si [6]). Associationeffects play a very important role in diffusion in ionic sys-tems [7].

The discussion above illustrates the rich variety of phys-ical effects that determine diffusion in solid solutions. Aprecise determination of diffusivities in solid solutionsrequires that these many physical effects be captured accu-rately. To do this, we require a hierarchical combination ofmethods. To calculate the energies corresponding to indi-vidual atom hops in different local environments accu-rately, density functional theory methods, or molecularlattice statics using accurate inter-atomic potentials, whenthey are available, are necessary. However, an accuratedescription of the macroscopic diffusivity requires averag-ing over many diffusant atoms and over many hops (acrossthe entire energy landscape) for each of these diffusants.This necessarily involves calculations over millions ofatoms, and on time scales many decades larger than atom-istic time scales. To adequately address this, mesoscale sim-ulation techniques, such as kinetic Monte-Carlo methods,

which use the jump frequencies (rates) calculated fromthe atomistic simulations as input, are necessary. Weemployed such a combination of methods to accuratelypredict oxygen diffusivities in yttria-stabilized zirconia, anoxide solid solution containing a large concentration ofoxygen vacancies [8]. In that case, we demonstrated theimportance of correlation effects in accurately predictingcomplex diffusion behavior. Monte-Carlo methods are wellsuited to capture correlation effects accurately, and thesemethods have also been employed to calculate correlationeffects directly [9,10]. In this paper, we examine diffusionin dilute solid solutions, with particular attention to situa-tions where defect/diffusant association effects are impor-tant. For diffusion in dilute solutions, it is especiallyimportant to use large simulation system sizes and timesto effectively model diffusivity. This ensures that an ade-quate number of carriers, which make a sufficiently largenumber of hops, are present, thus providing appropriatelyreliable averages [11].

In this paper, we study Fe impurity diffusion in Al, as anexample of diffusion in a dilute metal solid solution, whereassociation effects are important. Fe and Al are known toform a variety of stable and metastable intermetallicphases, at both the Fe-rich and the Al-rich ends of thephase diagram. Thus, this system is a good candidate forstudies of association effects. We recently developed accu-rate embedded atom (EAM) potentials suited for atomisticsimulations of Fe–Al systems [12]. In that work, the poten-tials for pure Fe and Al were fitted to perfect crystal prop-erties, crystal defect properties and melting point data andto atomic forces calculated using density functional theory(DFT) methods. The cross-potentials were fitted to the lat-tice parameters and formation energies of Al3Fe com-pounds, to the Fe–vacancy interaction energy and toatomic forces in liquid Fe–Al systems calculated usingDFT methods. The density functional theory calculationsused to fit the EAM potential are described in Refs.[12,13]. They were performed using the VASP code [14],with ultrasoft pseudopotentials [15], and using the general-ized gradient approximation (GGA) to exchange and cor-relation [16]. Supercells containing 32 or 31 atoms wereused in these calculations [12,13]. Following work byGillan [17], which illustrates the importance of k-point con-vergence in obtaining accurate defect energies in Al, a13 · 13 · 13 Monkhorst–Pack mesh [18] was employed inthe DFT calculations. Convergence in energy within a fac-tor of 0.001 eV is obtained using this method [12,13]. Weuse this potential to calculate migration energies for Aland Fe exchange with vacancies in different arrangementsof Fe and Al atoms in the neighborhood of the exchangingatom–vacancy pair. These energies are input into kineticMonte-Carlo (kMC) simulations to calculate long-time,large-scale macroscopic diffusivities.

This paper is organized as follows. In the followingsection, we describe the different local atomic configura-tions we considered in this study, the lattice statics meth-ods we employ to calculate atom–vacancy exchange

Fig. 1. Relaxed energy vs. Fe atom displacement along the migration pathfor a few select local configurations. The number of Fe atoms among theneighbors of the exchanging atom–vacancy pair is indicated in the legend.

R. Krishnamurthy et al. / Acta Materialia 55 (2007) 5289–5296 5291

energies in these configurations, and the results of thesecalculations. The kMC simulation method is describedin Section 3. Diffusivity results are presented in Section4, and a discussion of the results and conclusions arepresented in Section 5.

2. Calculation of vacancy exchange energies

The migration energies for the exchange of an Fe or Alatom with a vacancy were calculated for different distribu-tions of Fe and Al atoms in the neighborhood of theexchanging atom–vacancy pair. In order to restrict thetotal number of local configurations to a reasonable value,the local configurations were defined based only on thetype of atoms (Fe or Al) occupying the lattice sites imme-diately neighboring the exchanging atom–vacancy pair.This implies that in the face-centered cubic (fcc) lattice,only the occupancy of the 18 neighboring sites (excludingthe exchanging pair) of a atom–vacancy pair determinesthe local configurations. Note that the vacancy and theexchanging atom will share four neighbors in the fcc lattice.Furthermore, we only consider those configurations wheretwo or less Fe atoms occupy the 18 neighboring sites of theexchanging atom–vacancy pair. Configurations wheremore than two Fe atoms are found in the neighborhoodof the exchanging pair are assumed to have the samemigration energy as an equivalent distribution with onlytwo Fe atoms in the neighborhood. Based on these assump-tions, for either Fe or Al atom exchange with a vacancy,there is one configuration with no Fe atoms in the neigh-borhood, seven configurations with only one Fe atomneighboring either the vacancy or the exchanging atomand 42 configurations with two Fe atoms neighboring theexchanging atom–vacancy pair. Hence, a total number of50 configurations are considered for calculating migrationenergies for both Fe and Al atom exchange with vacancies.This restriction of the total number of configurations to amanageable value is essential for meaningful simulationsof long-time diffusivity using kMC methods.

Migration energies for atom–vacancy exchange werecalculated using static relaxation methods. The exchangingatom was advanced incrementally along the Æ110æ direction(the migration path), towards the neighboring vacant site,and the total energy of the simulation cell was minimizedwith respect to the displacements of all other atoms. Dur-ing the relaxation process, the jumping atom was con-strained from moving along the migration path, though itwas allowed to move in directions normal to this direction.This minimum energy was obtained as a function of atomicdisplacement along the migration path. The energy barrierfor atom–vacancy exchange is the difference between themaximum value of this relaxed energy (saddle-pointenergy) and the value corresponding to the energy of theequilibrium configuration. The energy barrier was calcu-lated with respect to both the end-point equilibrium posi-tions, in order to account for the effect of asymmetricconfiguration energies on diffusion.

The relaxed energy as a function of atomic displacementof the Fe atom as it is advanced along the migration pathtowards the vacancy is shown for a few representative con-figurations in Fig. 1. Also included in Fig. 1 is the migrationplot for the exchange of an isolated Al atom with a vacancy.From Fig. 1, we see that the activation energy for theexchange of an isolated Fe atom with a vacancy is nearlythe same as that for Al self-diffusion in pure Al. This migra-tion energy generally increases as the number of Fe atoms inthe neighborhood of the exchanging atom–vacancy pair isincreased. In configurations where the Fe atoms are asym-metrically located with respect to the atom–vacancy pair,a configuration energy change is also involved, as the bar-rier is asymmetric with respect to the two end-point equilib-rium configurations. This asymmetry is such that it is easierfor the exchange to occur if the Fe atoms neighbor thevacancy than the migrating atom; a trend that can be attrib-uted to the larger size of the Fe atom. One important trendnot captured in Fig. 1 is that configurations where two Featoms are located symmetrically with respect to theexchanging atom–vacancy pair exhibit the lowest barrierfor atom migration. This suggests that such configurationscan serve as atom traps, with the atom jumping back andforth between these configurations, rather than migratingto other configurations. Unlike its effect on Fe migration,an asymmetric distribution of Fe atoms with respect to anexchanging Al atom/vacancy pair does not induce signifi-cant asymmetry in the Al atom–vacancy exchange energy.However, as seen for Fe atom migration, the lowest energyfor Al atom–vacancy exchange is observed for the configu-ration where two Fe atoms are located symmetrically withrespect to the exchanging atom–vacancy pair. The actualvalues for the Al atom–vacancy exchange energies are typ-ically smaller in value compared with their Fe atom–vacancy counterparts.

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Vacancy migration barriers could, in principle, havebeen calculated directly using DFT methods. However,performing fully relaxed DFT calculations, particularlyfor 100 different local configurations (50 each for Al atomand Fe atom exchange with vacancies), would be computa-tionally expensive. Moreover, the semi-empirical potentialsin Ref. [12] are fitted to DFT computed values and litera-ture data pertaining to several different parameters relatedto diffusion. Consequently, we performed comparativelyinexpensive lattice statics calculations using the EAMpotential to calculate the vacancy migration energies,instead of obtaining them directly from DFT calculations.Semi-empirical potentials have been utilized successfullyover the past five decades to obtain insights that haveguided our understanding of several important thermody-namic and kinetic phenomena and, in some cases, to makeaccurate quantitative predictions.

3. Kinetic MC simulations

We model the Al(Fe) lattice using a cubic simulation cellof linear dimension L = Na, where a is the lattice parame-ter of the fcc unit cell and N = 200. This corresponds to asimulation cell containing 3.2 · 107 lattice sites. Fe atoms,of a number corresponding to a concentration of 10 ppmand vacancies of the same number were substituted at ran-dom for Al atoms in the lattice. Al and Fe atoms were bothassumed to exchange with vacancies in neighboring posi-tions along Æ110æ directions with rates calculated usingthe Boltzmann relationship:

mC ¼ m0 exp�EC

kBT

� �; ð1Þ

where the migration energies, EC, for atom–vacancy ex-change in a given configuration C are obtained from themolecular statics calculations described in the previous sec-tion. The frequency factor is assumed to be constant and isfixed at m0 = 1013/s. The simulation cell is repeated period-ically in all three dimensions to simulate a lattice of effec-tively infinite extent.

Following standard procedure for kMC simulations[19], we arrange the rates of all atom–vacancy exchangesthat can occur anywhere in the simulation cell at a giveninstant of time in serial order, generate a random numbern1 2 [0,1) and execute the atom–vacancy exchange i accord-ing to the condition:

Xi�1

j¼1

mjC

C� n1 �

Xi

j¼1

mjC

C; ð2Þ

where mjC is the rate for the jth atom–vacancy exchange, C is

the sum of the rates of all possible atom–vacancyexchanges,

C ¼Xn

j¼1

mjC ð3Þ

and n is the total number of possible exchanges. Vacancy–vacancy exchanges are forbidden and whenever two vacan-

cies occupy adjacent sites, the rates for their exchange areset to zero and the corresponding events are removed fromthe serial list of rates. In this algorithm, all chosen hops areexecuted (i.e. the acceptance probability is unity) and thetime is advanced stochastically by

Dt ¼ � log n2

Cð4Þ

where n2 2 [0,1) is another random number. This idea rep-resents a rearrangement of the Metropolis algorithm [20]that exploits the fact that if the rates are chosen correctly,the processes associated with the rates are of the Poissonprocess type, and the time given by Eq. (4) is the physicaltime scale associated with the evolution of the simulatedsystem [21].

The list of possible atom–vacancy exchanges areupdated following each successful atom–vacancy exchangeand neighbor lists are maintained for each vacancy and Featom to facilitate efficient calculation. This procedure isrepeated until the target time is reached. The total simula-tion time is divided into several equal time duration bins toobtain averages of interest.

4. Diffusion results

KMC simulations were performed at several tempera-tures below the melting temperature of Al. The same num-ber of vacancies was used for each of these simulations.Consequently, vacancy formation energies are not includedin the diffusion results presented below. To compare resultsin this paper with experimentally measured values, thesevacancy formation energies must be added to the diffusionactivation energies presented in this paper. The meansquare displacement, R2 of Fe atoms varies linearly withtime, as shown in Fig. 2. Fig. 2 corresponds to diffusionat 800 K, and calculated results for other temperaturesclose to the melting point of Al show similar behavior. Incontrast, R2 does not vary linearly with time, for diffusionat temperatures lower than 550 K, as shown in Fig. 3.Rather, we observe a crossover from a ‘‘high diffusivity’’regime to a substantially ‘‘lower diffusivity’’ regime at largetimes. Note that the late-time slope in Fig. 3 is the quantitythat is directly related to the diffusivity, and the early-timeslope has no relation to the diffusivity. The relativelyextended length of time associated with the ‘‘low-diffusiv-ity’’ regime may be due to the fact that the initial state ofa fixed number of vacancies at all temperatures corre-sponds to a state where there is a significant concentrationof quenched-in vacancies at temperatures less than 550 K.This length of time is likely to be related to the time neces-sary to form associates or establish a reasonable shortrange order, characteristic of the state of the material atthe lower temperature. The Fe diffusivity, DFe, wasextracted from the linear regimes (at long times) in theseplots using the Einstein relation:

R2 ¼ 6DFet ð5Þ

Fig. 3. Mean square displacement, R2, of Fe atoms shown as a function oftime t at 475 K.

Fig. 4. Iron diffusivity, DFe, shown as a function of inverse temperature ina standard activation plot. Note the existence of two different activationenergies at temperatures above and below 550 K, respectively.

Fig. 2. Mean square displacement, R2, of Fe atoms shown as a function oftime t at 800 K.

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These simulations were performed at different temperaturesand the Fe diffusivity DFe extracted from them was plottedas a function of inverse temperature (a standard activationplot), as shown in Fig. 4. Since a fixed vacancy concentra-tion is employed in this study, this activation energy doesnot include contributions from vacancy self-energy, andshould thus be compared with experimentally measuredFe diffusion activation energies, only after the vacancy for-mation energies have been subtracted from them. It is evi-dent from this figure that there are two distinct activationregimes present in this system. At temperatures close tothe melting point of Al, Fe diffusion is fast, and the activa-tion energy for diffusion is comparable to the migration en-

ergy of an isolated Fe atom in Al. In contrast, at lowertemperatures (<550 K), Fe diffuses with an activation en-ergy (or migration energy) that is nearly twice as large.

Why do we observe such starkly different activationenergies in the two regimes? The effective diffusion activa-tion energy comprises multiple atom–vacancy exchangesof many types (associated with the different local configura-tions), each associated with its own activation energy. Con-sequently, the diffusion activation energy will be differentfrom any of the individual activation energies. However,this cannot explain a diffusion activation energy (whichrepresents a suitably weighted average over the many indi-vidual migration energies) that is larger than any of theindividual migration barriers, as seen at low temperatures.In the next section, we will argue that this behavior can beattributed to local ordering/association effects among Feand Al atoms and vacancies. Such local association/order-ing produces an additional contribution to the activationenergy for atom migration, namely, that required to over-come or accommodate this association/ordering. A relatedinteresting observation is the following. A ‘‘diffusion’’ acti-vation energy extracted from the small times, low ‘‘diffusiv-ity’’ regimes of R2 vs. time plots, such as the one shown inFig. 3, is similar in value to the diffusion activation energyassociated with the high-temperature regime in Fig. 4.Since we employ the same concentration of vacancies forall the temperatures considered in this study, the early-timedata in Fig. 3 corresponds to a situation where we havevacancies quenched-in from higher temperatures. At earlytimes, the defect types and distributions will correspondto the high temperature situation. Given sufficient timefor ‘‘annealing’’, however, defect associates can form, lead-ing to the higher activation energies seen in the low-temper-ature regime of Fig. 4. These observations are consistent

Fig. 5. (a) The probability of finding a Fe atom (solid line) and a vacancy(dotted line) as a function of distance from an average vacancy. Thesuccessive data points (circles) represent the value of this probability at thenearest neighbor position, the next nearest neighbor position and so on.These probabilities are shown at 800 and 475 K. The probability of findinga Fe atom as a function of distance from an average Fe atom is shownin (b).

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with our argument that local association effects are respon-sible for the presence of two different activation regimes.They also indicate why it is essential to employ hierarchicalmethods of the type used in this study to calculate diffusiv-ities in alloys, as large simulation times are required to ade-quately capture interaction effects at low temperatures.

While these calculations pertain to an initial random dis-tribution of Fe atoms and vacancies in the lattice, calcula-tions were also performed for initial states correspondingto random distributions of pairs and triplets of Fe atoms,arranged relative to one another according to the local con-figurations in Section 2. These initial states also producedtrends in activation energy (and absolute values) similarto those shown here.

5. Discussion and conclusion

In the previous section, we ascribed the presence of twodifferent activation energies at temperatures greater than orless than 550 K (see Fig. 4) to local association/orderingamong defects and diffusants that occurs at lower temper-atures. The fact that the high temperature activation energycompares well with the energy extracted from the smalltimes portion of the R2 vs. time relation at low tempera-tures (see Fig. 3) is another indication that such associationis taking place.

Before we produce more evidence of defect and/or Featom interactions at low temperatures, we note that activa-tion plots very similar to those shown in Fig. 4 have beenobserved in alloys where order/disorder transitions areobserved, a striking example of which is Cu diffusion inb-brass [22]. Two different diffusion activation regimeshave also been observed in Fe–Al intermetallics, where arich variety of ordered and disordered alloys is found[23]. Theoretical calculations on FeAl ordered intermetal-lics also show similar behavior, with two different activa-tion energies associated with temperatures below andabove the ordering transition temperature, respectively[24]. These examples, however, pertain to concentrated bin-ary alloys, and cannot be applied to the very dilute concen-trations considered here. Nevertheless, this comparisonindicates that atom rearrangements due to a phase transi-tion/association can lead to large changes in the diffusionactivation energy. Similar behavior has also been observedfor self- and impurity diffusion in metals that have multipleallotropes, such as in Ti [4].

An example relevant to small impurity concentrationscomes from experimental measurements and analysis ofthe resistivity of as-quenched, dilute Au–Sn and Au–Agalloys [25,26]. In these systems, a non-Arrhenius tempera-ture-dependence of the difference between the ‘‘quenched-in’’ resistivity and the ‘‘equilibrium’’ resistivity wasobserved even for small concentrations of impurities(�100 ppm Sn in Au). The activation energy for the resis-tivity at low temperatures was substantially lower than thatat high temperatures. This was explained on the basis ofthe formation of Sn–Sn associates at low temperatures

[26]. These results make an encouraging comparison withthose presented here in that they demonstrate that, evenat small impurity contents, interactions among impurityatoms and vacancies can have a significant effect on theactivation energies of transport parameters.

To further investigate possible defect association effects,we examine the averaged local concentration of Fe atomsand vacancies at lattice positions near a vacancy or Featom in the lattice. Both ensemble- and time-averagingwere employed to compute these averages. In Fig. 5a andb, we plot these averages (normalized by the far-field con-centrations of the respective species), as a function of dis-

Fig. 6. (a) The probability of finding a Fe atom (solid line) and a vacancy(dotted line) as a function of distance from an average vacancy; (b) theprobability of finding a Fe atom as a function of distance from an averageFe atom. These probabilities are extracted from the two different‘‘diffusion regimes’’ (early and late times, respectively) at 475 K.

R. Krishnamurthy et al. / Acta Materialia 55 (2007) 5289–5296 5295

tance from an average Fe atom and an average vacancyrespectively, at 800 and 475 K. These averages wereobtained from the late-time regime appropriate for diffu-sion in the lower temperature case. Fig. 5a shows that thereis a pronounced increase in the concentration of Fe atomsand vacancies in the vicinity of a vacancy at 475 K, whileno such trend is seen at 800 K. In Fig. 5b, we see that thereis a depletion of Fe atoms in the vicinity of a Fe atom at475 K. Note that the average concentration of vacanciesas a function of distance from an average Fe atom andthe average concentration of Fe atoms as a function of dis-tance from an average vacancy refer to the same concentra-tion distribution. For this reason, this distribution is notshown in Fig. 5b.

It is clear from these figures that Fe atoms/vacanciesoccupy preferred positions in relation to other vacancies.This indicates strong interactions between vacancies andFe impurity atoms. These interactions contribute an addi-tional term to the diffusion barrier. The Fe atom–vacancyinteraction energy has been calculated in Ref. [12] to be0.04 eV, which agrees well with DFT calculation results,also reported in Ref. [12]. An earlier calculation reports avalue that is 7–8 times larger [27]. The difference betweenthese two values may be attributed to the different DFTtechniques employed (GGA in Ref. [12] and LDA in Ref.[27]) and the fact that the ion positions in the lattice werefixed in the calculations performed in Ref. [27], whilerelaxed configurations were considered in Ref. [12]. TheFe atom concentration is also depleted compared withthe average concentration, in the neighborhood of anotherFe atom. This implies that Fe atoms prefer neighboring Alatoms. Since the probability of finding an Al atom in latticepositions near an Fe atom is directly proportional to theshort-range order parameter, this implies the developmentof some degree of local order. Local order can constrictvacancy migration paths, which can, in turn, lead to anadditional contribution to the diffusion activation energy.The creation of favorable relative arrangements of vacan-cies and Fe atoms can thus, contribute additional termsto the diffusion activation energy. Theoretical studies ofdiffusion in Fe–Al ordered intermetallics show that astrong interaction between Fe atoms and vacancies wasessential to match quasi-elastic Mossbauer spectral datafor Fe diffusion [28]. An additional contribution to the dif-fusion barrier is attributed to this interaction. While thisstudy corresponds to a 50:50 body-centered cubic alloy, itis known that a wide range of stable and metastable inter-metallic phases exist across the entire concentration rangein Fe–Al alloys [29]. Thus, it is likely that similar Featom–vacancy interactions are also present in fcc systemsfound near the Al-end of the concentration range. Thiscompares favorably with the strong Fe–vacancy interac-tions implied by the concentration plots in Fig. 5a.

The plots in Fig. 6a and b are constructed in a mannersimilar to Fig. 5a and b, but in this case, the average con-centrations are extracted from the early time, fast ‘‘diffusiv-ity’’ regime and the late time, slow ‘‘diffusivity’’ regime at

475 K (see Fig. 4). We see that the concentration profilescorresponding to the ‘‘slow diffusivity’’ (late time) and‘‘fast diffusivity’’ (early times) regimes (Fig. 6a and b) varyfrom each other in a manner strikingly similar to the wayconcentration profiles at 475 and 800 K (Fig. 5a and b) var-ied from each other. Since we use the same vacancy concen-tration at all temperatures, defect types and distributions atearly times at 475 K correspond to a high-temperature,quenched-in distribution. After an ‘‘annealing’’ time at475 K, defects and diffusants form favorable local arrange-ments and associates, and these associates are responsiblefor the ‘‘slower diffusivity’’ regime at later times. The strik-ing similarity between Figs. 5 and 6 is a strong confirma-tion that the development of local order or thedevelopment of localized, preferred arrangements of atoms

5296 R. Krishnamurthy et al. / Acta Materialia 55 (2007) 5289–5296

and vacancies is responsible for the high activation energyregime found at low temperatures (see Fig. 4).

In summary, we employed a combination of molecularstatics/kMC simulations to study diffusion in metallic sys-tems where association effects are important, using Feimpurity diffusion in Al as an example. Using this method,we can include the effect of local distribution of Fe atomsand vacancies on Fe– and Al–vacancy exchange energetics.We have demonstrated that different activation regimescharacterize Fe diffusion at high and low temperatures.Fe atom and vacancy concentration distributions in theimmediate neighborhood of Fe and vacancy atoms clearlyindicate that impurity–vacancy and vacancy–vacancy inter-actions (local ordering) occur at low temperatures, and areresponsible for the presence of these two different regimes.The study of solid-state diffusion effects such as the forma-tion of associates requires very long simulation times, asevident from this study. Hierarchical methods of the typeemployed in this study represent the most promising wayto study diffusion in such systems.

Acknowledgements

This work was supported in part by the US Departmentof Energy through Grant DE-FG02-99ER45797 and theComputational Materials Science Network. Work at theAmes Laboratory was supported by the Department of En-ergy, Office of Basic Energy Sciences, under Contract No.DE-AC02-07CH11358.

References

[1] Bardeen J, Herring C. Diffusion in alloys and the Kirkendall effect. In:Hollomon JH, editor. Atom movements. Cleveland (OH): ASM;1951. p. 87–111.

[2] Manning JR. Diffusion kinetics for atoms in crystals. Princeton(NJ): Van Nostrand; 1968.

[3] Mishin Y. Atomistic computer modeling of intermetallic alloys.Mater Sci Forum 2005;502:21–6.

[4] Mishin Y, Herzig C. Diffusion in the TiAl system. Acta Mater2000;48:589–623.

[5] Weinkamer R, Fratzl P, Sepiol PB, Vogl G. Monte-Carlo simulationof diffusion in a B2 ordered alloy. Phys Rev B 1998;58:3082–8.

[6] Lopez GM, Fiorentini V. Structure, energetics, and extrinsic levels ofsmall self-interstitial clusters in silicon. Phys Rev B 2004;69. Art No.155206.

[7] Lankhorst MHR, Bouwmeester HJM, Verweij H. Thermodynamicsand transport of ionic and electronic defects in crystalline oxides. JAm Ceram Soc 1997;80:2175–98.

[8] Krishnamurthy R, Yoon YG, Srolovitz DJ, Car R. Oxygen diffusionin yttria stabilized zirconia: a new simulation model. J Am Ceram Soc2004;87:1821–30.

[9] Belova IV, Murch GE. Tracer correlation factors in the random alloy.Phil Mag A 2000;80:1469–79.

[10] Mishin Y, Farkas D. Monte-Carlo simulation of correlation effects ina random bcc alloy. Phil Mag A 1997;75:201–9.

[11] Belova IV, Murch GE. Solvent diffusion kinetics in the dilute randomalloy. Phil Mag A 2003;83:393–9.

[12] Mendelev MI, Srolovitz DJ, Ackland GJ, Han S. Effect of Fesegregation on the migration of a non-symmetric Sigma 5 tilt grainboundary in Al. J Mater Res 2005;20:208–18.

[13] Mendelev MI, Srolovitz DJ, Ackland GJ, Han S, Atomistic simula-tion of grain boundary migration in Al, unpublished.

[14] Kresse G, Furthmuller J, VASP The Guide, http://cms.mpi.uni-vie.ac.at/vasp/vasp/vasp.html.

[15] Vanderbilt D. Soft self-consistent pseudopotentials in a generalizedeigenvalue formalism. Phys Rev B 1990;41:7892–5.

[16] Perdew JP, Chevary JA, Vosko SH, Jackson KA, Pederson MR,Singh DJ, Fiolhais C. Atoms, molecules, solids, and surfaces:applications of the generalized gradient approximation for exchangeand correlation. Phys Rev B 1992;46:6671–87.

[17] Gillan MJ. Calculation of the vacancy formation energy in alumin-ium. J Phys Condens Matter 1989;1:689–711.

[18] Monkhorst HJ, Pack JD. Special points for Brillouin-zone integra-tions. Phys Rev B 1976;13:5188–92.

[19] Battaile CC, Srolovitz DJ. Kinetic Monte-Carlo simulation ofchemical vapor deposition. Annu Rev Mater Res 2002;32:297–319.

[20] Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, TellerE. Equation of state calculations by fast computing machines. J ChemPhys 1953;21:1087–92.

[21] Fichthorn KA, Weinberg WH. Theoretical foundations of dynamicalMonte-Carlo simulations. J Chem Phys 1991;95:1090–6.

[22] Girifalco LA. Statistical physics of materials. New York: Wiley;1973.

[23] Mehrer H. Diffusion in intermetallics. Mater Trans JIM1996;37:1259–80.

[24] Weinkamer R, Fratzl P, Sepiol B, Vogl G. Monte-Carlo simulation ofdiffusion in B2 ordered model alloy. Phys Rev B 1998;58:3082–8.

[25] Bass J. Quenched resistance in dilute gold–tin gold–silver alloys. PhysRev A 1965;137:765–82.

[26] Liu GCT, Girifalco LA, Maddin R. Quenched-in electrical resistivityof dilute binary alloys. Phys Stat Sol 1969;31:303–12.

[27] Hoshino T, Zeller R, Dederichs PH. Local-density-functional calcu-lations for defect interactions in Al 1996;53:8971–4.

[28] Weinkamer R, Fratzl P, Sepiol B, Vogl G. Monte-Carlo simulation ofMossbauer spectra in diffusion investigations. Phys Rev B1999;59:8622–5.

[29] Contreras-Solorio DA, Mejia-Lira F, Moran-Lopez JL, Sanchez JM.Modeling the Fe–Al phase diagram. Phys Rev B 1998;38:11481–5.