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PHYSICAL REVIEW B, VOLUME 64, 174105

Influence of bulk composition on grain boundary segregation inB2 Fe-Al:An atomic-scale simulation study

Remy Besson* and Alexandre LegrisLaboratoire de Me´tallurgie Physique et Ge´nie des Mate´riaux, C.N.R.S. U.M.R. 8517, Universite´ des Sciences et Technologies de Lille

Batiment C6, 59655 Villeneuve d’Ascq Cedex, France

Joseph Morillo†

Structure des Syste`mes de Basse Dimensionnalite´, C.N.R.S.-C.E.M.E.S., 29, Rue Jeanne Marvig, 31055 Toulouse Cedex 4, Fran~Received 10 April 2001; published 4 October 2001!

The atomic structure of the~310!@001# symmetrical tilt grain boundary~GB! in B2 Fe-Al ordered alloys wasstudied atT50 K by numerical simulations usingN-body empirical potentials. As expected from the highdegree of order ofB2 Fe-Al the coincidence site lattice~CSL! theory is found a relevant approximation for GBcrystallography, the stable GB variants being close to the usual symmetric and pseudosymmetric CSL models,with no GB vacancies. However, as regards GB chemistry, the extension to interfaces of the independent pointdefect approximation, rigorous for bulk ordered compounds, and assessment of its validity by a full treatmentof interactions between GB point defects reveal a strong dependence of GB properties on alloy off-stoichiometry: single-layer Fe segregation holds in Fe-rich Fe-Al, whereas in Al-rich alloys, a complexmultilayer Al segregation appears, with the possibility of a GB phase transition, and GB glide weakly depen-dent on GB chemistry seems easy in the@001# direction. These elements may help explain the experimentallywell-known high dependence of the mechanical properties ofB2 Fe-Al on bulk composition.

DOI: 10.1103/PhysRevB.64.174105 PACS number~s!: 68.35.Md, 61.72.Mm, 68.35.Dv

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I. INTRODUCTION

In spite of promising properties~high specific stiffnessand excellent high-temperature corrosion resistance—foreview, see Ref. 1!, bcc-based ordered iron aluminides~withB2 and DO3 crystallographic structures! have found onlylimited industrial applications up to now, because they sufrom a strong intergranular brittleness. This weaknesslong believed to be an intrinsic property, until a very prnounced embrittlement by extrinsic factors~environment,dopants! was evidenced, a detrimental effect partially reedied by the addition of small amounts of boron, whichanges the fracture mode from intergranular decohesiocleavage. The precise mode of action of boron is not wunderstood yet, but whatever its influence or that of otextrinsic factors, grain boundaries~GB’s! are key parameterdetermining the macroscopic mechanical properties of ialuminides, and must therefore be characterized as regtheir intrinsic structural properties.

To this purpose, experimental techniques—higresolution transmission electron microscopy~HRTEM! andAuger electron spectroscopy, among others—are particulwell adapted to yield structural information. However, Fe-samples~high-purity bicrystals prepared in well-controlleconditions, for instance!, because of their extreme sensitivito impurities, are very difficult to obtain and manipulatwhich certainly contributes to limit experiments. Complmentarily, atomic-scale simulations offer a valuable wayinvestigating both the structure and thermodynamicsmodel interfaces and yield matter for comparison with avable or future experimental results.

In performing atomic-scale simulations, special careto be taken of the choice of the potential-energy model.Abinitio methods provide the reputedly most accurate state

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the-art potentials but require a high computational powcompared to other semi-empirical~e.g., tight-binding! or em-pirical ~e.g., embedded atom method, EAM! models. A sur-vey of the literature about the atomic modeling of GBshows that studies usingab initio methods are scarce anoften deal with very specific points.2 In spite of the continu-ous enhancement of the available computer power, the hcomputational cost ofab initio calculations limits the size otractable systems to about 50 transition-metal atoms, hining comprehensive GB studies that have to include podefect thermodynamics, chemical and segregation effectaddition to more common studies generally limited to a fspecific GB variants. In particular, in order to determine tground-state properties of a given GB, the configuratioatomic phase space that has to be investigated must incin-plane rigid-body translations~RBT’s! and a local compo-sition that can be different from the bulk one~if segregationoccurs!. Owing to the difficulty of this task, there is no example ofab initio study embracing the full problem, including both the chemical and translational degrees of freedomdeficiency that still makes relevant the use of empirical ptentials.

Although alloy interfacial segregation is a well-knowphenomenon, deeper insight into it was recently gained frthe joint use of analytic modeling and simulation with EAMlike potentials, showing in particular the existence ofmultilayer GB phase transition in Cu containing a few pecent of Ag in solid solution.3 Concerning the atomic simulation of GB’s in intermetallics, mainly Ni-Al alloys have beestudied up to now, using empirical potentials either relyion the EAM or on the second moment approximation,majority of these studies being performed atT50 K. InB2 Ni-Al, crystallographically similar to Fe-Al, simple GB’shave been studied using atomic scale simulation4–7 and ex-

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REMY BESSON, ALEXANDRE LEGRIS, AND JOSEPH MORILLO PHYSICAL REVIEW B64 174105

perimental tools~HRTEM!.8–10 By contrast, due to the lackof reliable energetic description, Fe-Al alloys have remainalmost unstudied by atomic-scale simulations. This dciency is reputedly to be put down to the strong bond dirtionality in these compounds, responsible~among others! forthe negative value of theC122C44 Cauchy discrepancy between the elastic moduli, which rules out all commonly usN-body potentials. The potential used in this work, mixiEAM and angular interactions, partially overcomes thproblem and was specifically developed11,12 to study bulkand point defect properties of bcc-based Fe-Al alloys.spite of the well-known uncertainty concerning the transfability of any empirical energetic description, this potentoffers reasonable guarantees for the present GB study, init successfully reproduces the elastic constants as webasic properties of point and extended~antiphase boundarynoted APB hereafter! bulk defects inB2 Fe-Al. Moreover, itwas designed to reproduce the properties of pure Fe anand describes rather wellDO3 ordered Fe3Al at T50 K,which is an essential point when dealing with segregatphenomena. We refer the reader to Ref. 11 for a detadescription of this energy model.

In a previous paper,13 several simple GB and free-surfacproperties were investigated inB2 FeAl andDO3 Fe3Al atT50 K, pointing out that these interfaces undergo Al seggation ~except GB’s inB2 FeAl that show Fe enrichment!,and that the interfaces in theDO3 compound display ahigher variant multiplicity than those in theB2 one. Thiswork, however, was restricted to the ideally stoichiometB2 andDO3 compounds.

When dealing with an ordered alloy, the stoichiometcomposition represents a simple and convenient starpoint, because evaluating the excess free energy of an iface in such a compound does not require the knowledgchemical potentials, provided the interface is also locally sichiometric. Nevertheless, as soon as one wants to addbulk as well as GB off-stoichiometry, the chemical potentiare required. These quantities, calculated from bulk athermodynamics, show a strong variation around stoichioetry, pointing out that the stoichiometric compound is only‘‘singular point’’ from the point of view of bulk as well asinterface properties, and is therefore not representativereal alloys, hence the necessity of GB studies in ostoichiometric compounds.

In this frame, we present here a detailed analysis ofequilibrium structure of theS55 (310)@001# symmetric tiltgrain boundary. The choice of this interface is relevantcause~i! of its short period,~ii ! of the presence of a singltype of ~310! planes in the stoichiometricB2 alloy ~two fac-tors making it quite easy to study by numerical simulation!,and ~iii ! it is reasonable to suppose that~due to its highdegree of coincidence! this GB is representative of a widclass of GB’s in Fe-Al alloys. Restricting toT50 K, weshow a strong tendency to GB segregation of the bulk mjority element in slightlyB2 off-stoichiometric Fe-Al alloys.In Fe-rich alloys, Fe segregation, concentrated in the cvicinity of the GB plane, is simple to picture. In Al-richalloys, on the contrary, Al segregation is more complex,tending over several layers around the GB plane. We a

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investigate GB glide, showing a high anisotropy of this phnomenon.

The present paper is organized as follows. In Sec. II,recall the methods and procedures used throughout. InIII a short presentation of the geometric properties of theof interest is given. Focusing then on GB chemistry, wevestigate in Sec. IV the influence of departures from stoicometry in the framework of the independent point defeapproximation~IPDA, a brief description of which is pro-vided!, while Sec. V is devoted to the study of segregatibeyond IPDA. Finally, in Sec. VI, the results are discussand the conclusions and perspectives are given.

II. METHODS AND COMPUTATIONAL PROCEDURES

A. Choice of methodology

Interfacial segregation is concerned with local compotion inhomogeneities in a system. This kind of phenomenbeing conveniently studied in an ‘‘open subsystem’’ descrtion ~the subsystem being the interface zone!, a straightfor-ward investigation by atomic-scale simulation can be pformed, in principle, by Monte Carlo calculations in th(m iVT) grand canonical statistical ensemble, imposing givchemical potentials and temperature to the subsystem ofume V. Such an approach was found fruitful in describinGB segregation in the Cu-Ag disordered alloy.3 However,experiments14 as well as empirical simulations12 show thatFe-Al alloys are probably characterized by large vibratiofree energies and atomic displacements from the perfecttice positions. Thus, for Monte Carlo methods to be vawhen dealing with such alloys, large simulation cells mustused and atomic displacements have to be added to the uchemical ‘‘transmutations,’’ which leads to prohibitive computation times. A probably more fundamental criticisagainst this method is that it yields only global informatiofrom which it is not easy to isolate elementary mechanismWe thus prefer to adopt here a simpler but more enlightenapproach, based on IPDA thermodynamics and coupsimulations and analytic calculations, initially developed fbulk systems.15–17 In spite of a very similar formalism,18 itsapplication to interfaces is much more scarce up to nowdeficiency probably explained by the large number of podefects of different types that have to be calculated, entaby the complexity of GB structures in ordered alloys. A simlar detailed analysis, investigating ‘‘site by site’’ GB propeties was performed to study GB diffusion in NiO.19 In thepresent study, a systematic nomenclature for GB sitesthus adopted, which is displayed in Fig. 1 for the two crytallographic basic GB variants referred to throughonamely the symmetric~S! and the pseudosymmmetric~PS!variants obtained in the framework of coincidence site latt~CSL! theory~see Sec. III!. Each plane is labeled by an indereferring to its distance from the GB. In the PS variasingle and double primes are used to label sites belonginthe nonequivalent grains. LettersA andF refer to point de-fects on GB sublattices ‘‘normally’’~i.e., in the GB varianttaken as reference! occupied by Al or Fe atoms, respectivelMoreover, we use the following convention for the GB d

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INFLUENCE OF BULK COMPOSITION ON GRAIN . . . PHYSICAL REVIEW B64 174105

scription: in the GB plane, the two directions are labeledOx~@310#! andOz ~@001# tilt axis!, and the direction normal tothe GB plane (@130#) Oy.

B. Theoretical background

1. GB thermodynamics

From thermodynamics, the equilibrium of a bicrystal cotaining an interface of fixed areaA corresponds to the minimum of the so-called ‘‘excess free energy,’’ defined as~Ref.20!:

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for the closedsystem~entire bicrystal with constant particlnumbersNi! submitted to the external pressureP and tem-

FIG. 1. The two CSL geometrical models for theS55~310!@001# symmetrical tilt grain boundary in aB2 ordered alloy:~a! S ~symmetrical! model and~b! PS ~pseudosymmetrical! model~lengths alongOx and Oy are measured in Å!. Throughout thiswork, the Al atoms are represented by full squares, and theatoms by open squares.

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peratureT. We insist that, in this expression,E2TS1PVcorresponds to the free energy of the bicrystal, whileS im iNiis the free energy of the reference bulk system containingsame numbers of atoms and submitted to the sameP andT.Considering now anopensubsystem containing an areaA ofGB, an analogous quantitys ~without thePV term! is stillthe relevant thermodynamic potential~namely that which isminimized at equilibrium!, provided that the particle numbers Ni are now free to vary and them i are now controlparameters fixed by the surrounding bulk system. Foropen system to remain defined, its volumeV has to be speci-fied, hence an implicit dependence ofs on V. In this grandcanonical approach, since the specified volume value dnot necessarily correspond to mechanical equilibrium~bic-rystal under given external pressure!, this condition has to beensured by minimizings with respect toV. Furthermore, theequilibrium values of internal variablesyk describing thissubsystem are then those minimizing the partial thermonamic potentials(V,yk) with respect toyk and V ~through-out this paper, the symbol; represents a partial thermodynamic quantity, i.e., restricted to constant values of givinternal variables—hereyk represent the point defects numbers!:

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At zero T andP and for a fixed GB area, GB equilibrium isimply given by the minimum of the GB excess energy punit area:

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m i Ni~V,yk!G . ~3!

Since we are concerned with an open system~subsystemcontaining a GB area!, we implicitly work in the grand ca-nonical ensemble and the chemical potentials in expres~3! are to be considered as fixed parameters, fully determiby the bulk composition of the alloy for fixedP andT.

For a given set ofm i , in studying GB equilibrium, thefollowing internal variables were taken into account:

~i! the rigid-body translations of one grain relative to tother,

~ii ! the atomic relaxations around the geometrical sites~iii ! the proportions~local composition! and positions of

the various species in the GB.In the absence of atomic vibrations (T50 K), this consti-

tutes an exhaustive set of GB internal variables.

2. IPDA for bulk and grain boundaries

According to formula~3!, GB properties depend on twterms:~i! a purely local term (E) describing the properties othe GB itself, ~ii ! a ‘‘mixed’’ term (m i Ni) involving local~GB particle numbersNi! and global properties~chemicalpotentialsm i!. This last term, reflecting the equilibrium between the interface and the surrounding bulk material,plies a realistic knowledge of the chemical potentials afunction of the bulk composition. For an ordered phase, afor small bulk departures from stoichiometry, this can

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reasonably achieved in the IPDA.15–17 In this approach, atnonzero temperatures, thermodynamic properties, functof microscopic features~point defects!, are most conve-niently obtained taking (m iVT) as variables, leading to a seof equations that must be solved numerically, whereas aT50 K, thermodynamic properties can be expressed anacally taking (Ni PT) as variables. In particular, theT50 Kchemical potential values are then derived analytically, afunction of the alloy composition, by minimizing the themodynamic potential~which simply amounts to the potentiaenergy, at zeroT and P! with respect to the point defecnumbers.12 In the IPDA, the free enthalpy being a lineafunction of these variables, the chemical potentials are cposition independent~stepwise functions! at zeroT and P,and have a unique value for each chemical species onside of stoichiometry. However, at stoichiometry, this linedescription of the free enthalpy leads to undefinedT50 Kchemical potentials, and these quantities must thereforedefined as the temperature limitT→0 K of chemical poten-tials. Concerning IPDA thermodynamics of bulkB2 Fe-Al,we merely recall here those features obtained in Ref. 17directly useful to the present study, namely the valueschemical potentials atT50 K for stoichiometric, Al-rich andFe-rich alloys~Table I!. Note that, whatever the bulk composition, theT50 K chemical potential summAl1mFe isconstant, which is a straightforward consequence ofIPDA. As mentioned above, since stoichiometry is an ‘‘insbility point’’ for many low-temperature bulk properties~forexample, chemical potentials, as Table I shows! of an or-dered compound, calculations for the stoichiometric copound can by no means be compared with experimentasults, concerning real alloys the composition of whichalways ill defined, to a certain extent. This justifies ourcusing on GB’s in two ‘‘limit’’ cases ofB2 Fe-Al, namely aunique Al-rich one and a unique Fe-rich one. The compotion range of validity of the results~corresponding to therange of validity of the IPDA! is not known precisely but canbe estimated to a few percent.

In the IPDA framework, the key quantities requiredorder to assess GB thermodynamics~for example composi-tion! are the formation~free! energies of isolated GB poindefects. These quantities are defined as a straightforwardtension of the same quantities used in IPDA for bulk phasnamely, atT50 K:

Ef~Vac,pF!5]E

]n~Vac,pF!1mFe, ~4a!

Ef~Al, pF!5]E

]n~Al, pF!1mFe2mAl ~4b!

TABLE I. Chemical potentials~eV! for the stoichiometric, Fe-rich, and Al-rich alloys atT50 K ~from Ref. 17!.

Fe-rich Stoichiometric Al-rich

mAl 23.95 23.57 23.11mFe 24.20 24.58 25.04

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for vacancies and Al antisite atoms in the GB sublatticetypepF, respectively~similar expressions hold for defects opA sublattices!. The partial derivatives~sometimes called‘‘raw formation energies’’ of point defects! are associated tothe creation of a point defect of a given type in the GB@opensystem,~m i ,V,T50! ensemble#. These quantities~which donot maintain the numbers of particles constant! are calcu-lated by subtracting to the energy of the simulation cell cotaining the defect that of the reference, undefected cell.

C. Simulation details

The GB simulations were performed with a cell containg 2160 atoms@corresponding to 33636 CSL unit cellsbuilt in the @310#, @130#, and @001# directions,respectively—see Fig. 1~a!#. Periodic boundary conditionswere applied, thus systematically generating two interfainitially identical through a 180° rotation aroundOy andallowing free relative movements of the two grains in tgrain-boundary plane directions~in-plane RBT’s! in theneighborhood of local energy minima. The identity of thetwo interfaces was always carefully checked after equirium was reached.

The individual relaxations of atoms~either after RBT’s orcompositional changes! were carried out using moleculardynamics simulations in the (NiVT) ensemble. The cell sizewas chosen in order to preserve a large amount of percrystal between the two interfaces~Oy direction! and to havea large number of equivalent sites~18 Al and 18 Fe sites! ineach (xOz) plane, parallel to the GB.

To explore the configuration space described by po~i!–~iii ! above, we proceeded practically as follows.

~i! Starting from the geometric GB model~henceforthnoted S! deduced from the CSL analysis for the~310!@001#symmetrical tilt GB inB2 compounds21 ~Fig. 1!, applicationof in-plane RBT’s (Tx ,Tz) and minimization ofEx(V) withrespect to the cell volume~i.e.,Ty! without individual atomicrelaxations yielded, through the GBg surface~detailed be-low!, the unrelaxed local GB energy minima with respect(Tx ,Tz).

~ii ! From these unrelaxed energy minima, energy minimzations were performed at constant volume~no Ty translationallowed!, by use of a well-known quasidynamic ‘‘dampingalgorithm. The value ofTy ~measured with respect to thCSL models! was adjusted in order to get the lowest bicrysrelaxed energy~corresponding to zero external pressure,mentioned above!.

~iii ! The constraint of constant cell composition was threleased, by investigating the properties of GB point defeIn the IPDA frame, these point defects have to be definwith respect to reference GB structures, initially taken tothe S and PS CSL models, and actualized if necessary~point~iv! below!. Each site of a given type~A for Al, F for Fe—see Fig. 1! in a given GB plane belonging to a new sublatice, a large number of GB point defects have to be disguished~antisite defects and vacancies!. In this IPDA studyof GB point defects~calculation of theirT50 K formationenergies!, those point defects with negative formation engies ~so-called ‘‘constitutional’’ defects! have a special im-

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portance: they are stable atT50 K, since a decrease of thGB excess energy is implied by their formation, This leadsnew GB variants, obtained by performing a systematicenrichment with such defects. The next step consisted theassessing, beyond IPDA, the excess energies of thesevariants by taking into account the actual interactionstween the newly created GB point defects. For the sakeclarity, a separate section is devoted to each analysis, inbeyond IPDA, in the following.

~iv! Iteration of the procedure@points ~ii ! and ~iii !# wasperformed if points defects with negative formation energwere found in the new GB variants generated. If no sudefects were found, the equilibrium GB variant was consered to be reached.

III. GRAIN-BOUNDARY CSL MODELS

In the case of an alloy withB2 structure, there are twoCSL variants for the~310! @001# symmetrical tilt grainboundary,21 called the S~symmetrical! and PS~pseudosym-metrical! models thereafter. These models, displayed on F1, can be deduced from each other by applying to thvariant RBT’s equal to~with $u% orthonormal basis!:

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wherea is the lattice parameter,d3105aA10/10 is the~310!interplanar distance and~Px510d310,Pz5a! are the GB pe-riods alongOx andOz ~throughout this work, the referencstate for rigid-body translations is chosen to be the S mod!.

Applying the previously described procedure@points ~i!and ~ii !# leads to the~unrelaxed with respect to the atompositions, but relaxed with respect to the cell volume! excessenergy profileEx(Tx ,Tz), commonly called ‘‘g surface,’’22

of the GB. Thisg surface for the CSL models is shownFig. 2. Note that, because of the GB symmetry propertthe irreducible translation domain for this GB is defined(Tx ,Tz)[email protected],0.8#3@0,0.5#. The atomic relaxations usuallmodifying the energies by;10–20%, theg surface is a ze-roth order analysis of the GB energetics with respect toternal conditions~mechanical parameters!, considered as sufficient to describe trends concerning mechanical proper~GB glide, for instance!. Figure 2 shows that GB glide ihighly anisotropic: the energy barrier for@310# glide roughlyamounts to 2 J/m2, a prohibitive value probably preventinany glide in this direction, while in the@001# direction, al-most no energy barrier exists@Fig. 2~b! gives a value below100 mJ/m2#, indicating an easy glide in this direction. Gglide in Fe-Al will be further investigated in Sec. VI.

Following then point~ii ! of our procedure, atomic anRBT relaxations yield two GB variants, characterizedatomic positions very close to those of the S and PS gmetrical models~Fig. 1!, and therefore also labeled S and PFigure 3 shows these final relaxed S and PS equilibriumstructures, and Table II gives their geometrical (Ty) and en-ergetic properties. Both variants have nearly equal excenergies andOy expansions, the latter values being positibut close to zero. Finally, it is worth noticing that the cry

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tallographic symmetry of the S variant is broken by the eergy minimization@a smallTx RBT component exists—seFig. 3~a!#, and experimental analysis of this feature~by HR-TEM! would help to assess the validity of the angular partthe energy model.

More detailed structural information~particularly usefulfor future comparison with HRTEM observations! is con-tained in the relative reticular distance@~310! planes# acrossthe GB. For an ordered compound, it is useful, in a givmixed ~310! plane, to distinguish between Fe and Al atomthe different behaviors of which may lead to the well-knowphenomenon of ‘‘rippling,’’ already observed in surfaces.23,24

The corresponding profiles for each chemical species, shin Fig. 4 for both the S and PS variants, display the usoscillatory behavior, which is to be put down to the oscilltions of the electronic density~an effect similar to those occurring at free surfaces!, and the asymmetry of both CSvariants clearly appears. Rippling is also observed: asgards Fe atoms, for both variants the effect of the GBclearly repulsive, and extends onto two GB planes, wher

FIG. 2. Excess energy surface~g surface! of the CSL GB ofS/PS type,~a! as a function of (Tx ,Tz) ~parallel to the GB plane!RBT’s ~measured in GB period units!, and~b! viewed along the tiltaxis. The GB symmetry properties~one quarter of the RBT domain!are made apparent by the 0.3Tx offset. The energy path between thS and PS variants~Tz50 and Tz50.5, respectively! contains nobarrier, hence an easy glide between mixed planes~containing 50%Al !, along@001#.

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in the case of Al atoms, the behavior is not so clear~attractive behavior in two GB planes for the S variant!.

At this stage, the empirical model therefore shows anergy degeneracy and an easy athermal transition betweetwo basic CSL GB variants, but due to its limited accuracycannot remove the ambiguity concerning the most stavariant. Energy calculations thus lead to results close to thdeduced from the CSL analysis and bring only little neinformation, but this was not obvious, however, if one cosiders the very different conclusions of other authors5 con-cerning the same GB inB2 NiAl, who found a large multi-plicity of structures unequivalent and very different from tCSL models. Our results probably reflect the covalent chacter of the bonding of FeAl. The effect of this covalencreputedly described by the angular part of the potentialdifficult to assess precisely. Acting against any large distions of the angles from their values in monocrystalliB2 FeAl, it probably has an important effect in determinithe GB stable variants. However, the S and PS variants benergetically degenerate, this effect does not appear in Fso clearly as in Mo and W, where a strong influence of ncentral forces on GB stability was noticed.25

FIG. 3. T50 K atomic structure of the CSL GB variants:~a! Svariant,~b! PS variant~distances are given in Å!.

TABLE II. T50 K excess GB energies~mJ/m2! and expansionsnormal to the GB plane~Å! in the S and PS GB variants in thstoichiometric, Fe-rich, and Al-rich alloys. The variants beingcally stoichiometric, their excess energies are independent ofbulk composition~within the IPDA!.

GB variant S PS

Excess energy 958 943Expansion 0.22 0.17

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IV. GRAIN BOUNDARIES IN THE IPDA

A. Stoichiometric alloy

Following our general procedure, taking the S andvariants as references for point defect calculations, we tahere the problem of GB local composition~segregation!from the IPDA point of view, after a preliminary paragrapshowing the probable irrelevance of considering GB disder.

1. GB disorder

GB disorder is produced by the formation of GB complpoint defects through local atomic rearrangements implyno composition change in the GB zone. This propertylocal, that is, independent of the alloy bulk composition: t~free-! energy difference between a GB ordered configuratand a disordered one is the same~for given temperature andexternal pressure! whatever the alloy bulk composition~orchemical potentials!. It thus only needs to be studied in thcase of the stoichiometric alloy. Moreover, and althoughcrystalline solids, order-disorder transformations needpresence of vacancies, such defects will be shown toprobably absent from GB’s in Fe-Al~see paragraph 2 below!. Therefore we here merely deal with antisite pa@AlFe1FeAl# in the various GB sites~we restrict ourselves tothe GB plane and that immediately adjoining!. In the IPDA,the formation energy of the complex defect~independent ofthe alloy composition! is simply equal to the sum of theformation energies of the corresponding simple defe~given in Table III!. Table III shows that in both the S and Pvariants, antisite pairs have positive formation energies,most favorable configuration being the@AlFe(1F)

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FIG. 4. Relative~310! plane spacing in~a! the S and~b! the PSvariant, after energy minimization including individual and colletive atomic movements.

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TABLE III. T50 K formation energies~eV! of GB antisite defects and vacancies~S and PS variants! inthe B2 stoichiometric FeAl compound. Bold characters show values also useful in the approach bIPDA ~throughout this article, values for bulk are taken from Ref. 17!.

Antisite defects VacanciesGB plane

Site1 2 Bulk 1 2 Bulk

A 0.62 0.22 0.77 2.44 0.89 2.85S F 0.32 0.07 0.90 0.57 0.57 0.90

A8 À0.08 1.01 0.98 2.06PS A9 0.84 1.75

F8 0.11 1.18 1.37 0.32F9 0.79 0.47

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1FeAl(1A)# defect~‘‘intraplane’’ atomic disorder! in the PSvariant ~with a slightly positive 0.03 eV formation energy!.In the S variant, the favored defect is the@AlFe(2F)1FeAl(2A)# pair, also occurring in a plane parallel to thGB. Thermal activation of GB disorder should thus be eain B2 Fe-Al, a conclusion drawn for the CSL models~not themost stable ones atT50 K—see below!, but which probablyalso holds for the equilibrium GB variants. Since the IPDdoes not allow us to draw reliable conclusions aboutoccurrence of GB disorder inB2 Fe-Al, this point will befurther examined in Sec. V.

2. GB segregation

Turning to GB segregation, we proceed with theT50 Kformation energies of GB antisite defects and vacancies,the various sites quoted on Fig. 1 in both the S and PSvariants. These quantities are displayed in Table III: sinceGB vacancies exhibit large positive formation energ~larger than 0.30 eV!, it is reasonable to admit that no Gvacancies exist at low temperature. We thus ignore thesefects in the subsequent considerations. On the other h~bold characters in Table III!, several kinds of antisite defectare found to have low~positive! formation energies~1A, 2F,2A, and 1F sites in the S variant, 1F sites in the PS variant!.One defect even shows a slightly negative energy~1A sitesin the PS variant,20.08 eV!, which implies a 100% occupation rate of 1A sites by Fe atoms in the PS variant. Tcorresponding excess energy lowering~with respect to thePS variant! is equal to 18 mJ/m2, a small but significantvalue leading to a GBT50 K ground state of PS type with100% Fe GB plane. With an evident notation, we label tvariant PS-1A @Fig. 5~a!#.

At this point, our conclusion relies on the negative formtion energy of Fe antisite atoms on 1A sites in the PS variantnamely that of only one particular defect with a low magtude ~0.08 eV!. Since other antisite defects do exist wipositive but very low formation energies, the question ariof whether their signs~which determine the presence of thdefects in GB’s atT50 K! would be affected by a moreprecise energy model. This point, pertaining toab initio cal-culations, will not be dealt with in this paper. On the othhand, the validity of the IPDA will be investigated in Sec.This assumption, which when applied to GB’s, describes

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segregation of energetically independent particles~impuritiesor dopants—a phenomenon generally referred to as McLsegregation26!, is usually supposed to be accurate enouwhen studying the segregation of elements present atlow levels. On the other hand, this assumption is higquestionable when the defect concentration is high, andpecially when GB segregation layers are formed, as is prably the case in Fe-Al~for 100% Fe GB enrichment, two 1AFe antisite atoms are separated by only one lattice paramroughly 3 Å!.

B. Nonstoichiometric alloys

As formula~4! shows, the point defect formation energidepend on the alloy composition, through the chemicaltentials, hence the necessity to re-calculate theT50 K for-mation energies of isolated GB antisite defects, on either sof stoichiometry for both the S and PS variants. Thesestoichiometric antisite defects formation energies are dplayed in Table IV. Note that this table, presented explici

FIG. 5. GB variants with single layer Fe segregation and lenergy in the stoichiometric and Fe-rich alloys:~a! PS-1A GB vari-ant, ~b! S-2A variant ~distances are given in Å!.

5-7

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TABLE IV. T50 K formation energies~eV! of antisite defects in the S and PS GB variants for smdepartures from bulkB2 stoichiometry. Bold characters are used for those with significantly negative v~with respect to the assumed precision of the empirical potential!.

Fe-rich alloys Al-rich alloysGB plane

Site1 2 Bulk 1 2 Bulk

A 20.15 À0.55 0.00 1.53 1.13 1.68S F 1.10 0.84 1.68 À0.58 À0.83 0.00

A8 À0.85 0.24 0.83 1.92PS A9 0.07 1.75

F8 0.88 1.96 À0.79 0.28F9 1.57 20.11

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for clarity, can be deduced from formula~4! and Table III byapplying the proper offsets due to the dependence of checal potentials on composition~the same remark will hold forTables VI and VII—see below!. According to Table IV, thesituation in off-stoichiometric alloys is more intricate thanthe stoichiometric compound, since numerous types of asite defects appear with negative formation energies~TableIV only gives values for the first two GB planes, but slightnegative values are also obtained in planes more distantthe GB!. In the frame of the IPDA, all these structural dfects should thus be considered as present atT50 K. How-ever, owing to the limited accuracy of the empirical enermodel used and the previous results in stoichiometric Fewe concentrate, as usually, on those defects having thesignificant ~lowest! formation energies. These energi~around 21 eV! are found to be one order of magnitudlarger than the single negative one found in stoichiomeFeAl ~1A antisite Fe atoms in the PS variant!, which re-moves the previous ambiguity about their sign.

In Fe-rich alloys, only Fe antisite atoms~in A sites! arepresent as structural defects at low temperature. Two typeantisite defects are clearly favored: those in 1A sites in thePS variant and those in 2A sites in the S one. Using thpreviously defined terminology to specify the 100% enricment of a GB structure in one type of defect, this corsponds to two Fe-rich GB’s, namely PS-1A and~to a second-ary extent! S-2A @Fig. 5~b!#. Inspection of the atomicpositions in these two GB variants easily shows that theybe deduced from each other by the same RBT’s that linkS and PS variants~see Fig. 1!.

In Al-rich alloys, Al antisite atoms inall but one F sitehave negative or zero formation energies~Table IV!, the onlypositive one being particularly small~0.28 eV!. Three ofthem have particularly low values~20.83 eV for 2F sites inthe S variant,20.79 eV for 1F sites in the PS variant, andto a lesser extent,20.58 eV for 1F sites in the S variant!.Considering only these three antisite defects leads to tAl-rich variants of the GB, among which S-2F and PS-1Fare displayed in Fig. 6, hence the GB segregation of the bexcess element in slightly nonstoichiometric Fe-Al.

The IPDA thus reveals clear-cut trends of GB’s inB2Fe-Al alloys at low temperature, particularly the irrelevanof considering the exactly stoichiometric compound, thesence of GB vacancies and the segregation of the bulk

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jority element. These features, which may have importconsequences in transport~atomic diffusion, electronic con-ductivity! and mechanical GB properties, must, however,checked from a more general point of view, namely takiinto account the real interactions between GB point defeThis is the purpose of Sec. V.

V. GRAIN BOUNDARIES BEYOND IPDA

A. Stoichiometric alloy

1. GB disorder

Before dealing with GB segregation, the hypothesisGB disorder must be examined again within the framewof fully interacting GB point defects. As already mentionewe assume no GB vacancies and limit ourselves to pairantisite defects. Owing to the large number of crystalgraphically different such pairs, the full calculation was pformed only in the most critical case, namely for th@AlFe(1F)1FeAl(1A)# pair in the PS variant~IPDA forma-

FIG. 6. GB variants with single layer Al segregation and loenergy in Al-rich alloys:~a! S-2F GB variant,~b! PS-1F variant~distances are given in Å!.

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TABLE V. T50 K excess energies~mJ/m2! of nonstoichiometric GB variants~one full GB plane ofpoints defects of a given type! for slightly off-stoichiometricB2 Fe-Al. The composition-independent exceenergies of the locally stoichiometric S and PS variants are also recalled. Bold characters labevariants.

Fe-richalloys

Stoichiometricalloy

Al-richalloys

S: 958 S-1F: 1118 S: 958PS: 943 S-2A: 1409 PS: 943

S-2A: 951 S-2F: 892 S-1F: 1022PS-1A: 309 PS-1A: 767 PS-1F: 332

PS-1F: 886 S-2F: 338

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tion energy equal to 0.03 eV— see Table III! in first nearest-neighbor position. It leads to a formation energy@defined inthe same way as those of simple defects, given by Eq.~4!#equal to20.32 eV for the isolated double defect. The corsponding double defect having a large positive 1.36 eVmation energy in the bulk~which simply reflects the highdegree of order of bulkB2 Fe-Al!, low temperature disorderif any, is thus probably a typical GB effect. However, theffect cannot be accounted for by applying the IPDA to Gantisite pairs, since strong repulsive interactions do existtween these antisite pairs, as shown by the calculation ofmean defect formation energy for such pairs systematicarranged~i! along the 310& Ox direction~20.32 eV, namelyalmost equal to that of the isolated pair! and ~ii ! along the@001# Oz direction~20.05 eV!. These values also show ththe mean formation energy of a defect pair strongly depeon its spatial arrangement: GB disorder is thus highly antropic and occurs preferentially in the310& direction,whereas the strong repulsive interactions between antisitefects make generalized disorder along the@001# direction un-likely. These interactions drastically limit the number of atisite pairs in the GB zone and prevent from generalizeddisorder, which thus appears to be necessarily limitedfact, the end of the study shows that this phenomenon nnot be taken into account, segregation being more favora

2. GB segregation

Turning now to GB segregation, and studying the crosinfluences of GB composition variations and GB point definteractions, let us consider two GB variants 1 and 2. Thexcess energy differences, within and beyond IPDA, futions of the alloy composition, are related by

Ex~2!2Ex~1!5ExIPDA~2!2Ex

IPDA~1!1DE~2!2DE~1!,~6!

where DE5E2EIPDA measures the compositionindependent influence of point defect interactions on theergy of each 1 or 2 GB subsystem. This relation obvioushows that for a given alloy composition the point defeinteractions may affect the relative GB stabilities, whereconversely the influence of point defect interactions issame whatever the alloy composition. To take into accothe expected interactions between GB point defects, fullergy calculations were thus performed@point ~iii ! of the pro-

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cedure described in Sec. II# for those GB variants resultingfrom the 100% enrichment of the S and PS models in ekind of point defect found relevant in the previous secti~bold values in Table III!. Table V ~middle column! displaysthe excess energies of the GB variants, showing thatinteractions between point defects indeed have drasticfects. The most critical one is clearly the opposite trendsT50 K segregation resulting from the different signs of seregation energies. More precisely, for those defects withmation energies positive and larger than about 0.20 eV, sregation is associated with an increase of the excess enwhereas for those defects with positive but low formatienergy values~less than about 0.10 eV, bold values in tmiddle column of Table V! there is largedecreaseof theexcess energy, hence a favored segregation. The seconddifference between the IPDA and full approach is the mnitude of the excess energy variations associated with segation ~with respect to the reference S and PS varian!:the most significant energy variation concerns the PS-Avariant, for which Ex(PS21A)2Ex~PS!'2180 mJ/m2

whereas, as mentioned above, in the IPDAExIPDA(PS21A)

2ExIPDA~PS!'220 mJ/m2. The IPDA is thus highly ques-

tionable when applied to GB’s in Fe-Al. Nevertheless,spite of these discrepancies, some IPDA conclusionshold: the variant with the lowest excess energy remaPS-1A, the energy difference with the next favorable o~PS-1F! being significant~roughly 100 mJ/m2!. The simplepicture of GB single layer Fe segregation in stoichiometFeAl is therefore confirmed when accounting for GB podefect interactions.

To ensure that the PS-1A Fe-rich variant is the mosstable one, we checked that the formation energies of allpoint defects in this variant are largely positive. This corsponds to perform the IPDA analysis described in Sec.but taking now the PS-1A variant as the new ground statTable VI, displaying these point defect formation energiconfirms that this is indeed the case, the smallest formaenergy~that relative to Fe vacancies on 2F sites! being equalto 0.33 eV, which proves, according to the energy moaccuracy, that no point defect is present atT50 K in thePS-1A variant at bulk stoichiometric composition. The inteplanar distance profile for this highly stable variant is shoin Fig. 7. The symmetry properties of this PS-type variastrikingly appear in this figure. The analysis of this cryst

5-9


TABLE VI. T50 K point defect formation energies~eV! in the PS-1A variant in the stoichiometric alloy.

Plane 1 2 3 Bulk

Antisite A sites 1.11 1.65 0.77atoms F sites 0.43 1.22 0.50 0.90

Vacancies A sites 2.38 2.58 2.85F sites 1.44 0.33 0.88 0.90

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lographic feature byab initio calculations and HRTEM experiments would lead to a better knowledge of the atombonding in Fe-Al compounds. Moreover, Fig. 7, displayi15 ~310! planes on each side of the GB, also indicates tthe perturbation due to the GB is more highly felt by Fe thby Al atoms: contrary to the latter, the oscillatory spacingFe atoms is not fully damped within 15 planes.

Therefore, in the stoichiometric alloy, a careful investigtion of the minimum excess energy configuration leads tsingle stable ~310!@001# GB structure ~called PS-1A,Eexcess5767 mJ/m2!, already displayed in Fig. 5. Its maifeature is its strong Fe enrichment~100% Fe in the GB central plane!, indicating that GB’s inB2 Fe-Al have probably astrong trend to Fe segregation, at least at low temperaMoreover, and although it roughly leads to the same consions as full calculations atT50 K, the McLean hypothesisof independent point defects may be hazardous to studytemperature segregation inB2 Fe-Al, since the interactionbetween intergranular point defects may reverse the relastabilities of the GB structures with any slight temperatuincrease.

B. Nonstoichiometric alloys

1. Single layer segregation

The final point that needs to be tackled concerns thepoint defect interactions in off-stoichiometric alloys. As epression~6! shows, the influence of point defect interactio~DE term! on the relative GB stabilities is the same whatevthe alloy composition and can thus be evaluated at stoiometry, knowing~i! the IPDA excess energy differences fthe given bulk composition~global term!, and ~ii ! thecomposition-independent correctionDE due to point defectinteractions~local term!. An alternative approach, yieldin

FIG. 7. Relative~310! plane spacing in the PS-1A variant~sto-ichiometric and Fe-rich alloys!, showing the restored symmetry othis PS-type variant.

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the absolute values of excess energies, consists, as pously, in calculating directly the GB excess energies onther side of stoichiometry. Following the latter approach,thus calculated~Table V, columns 1 and 3!, for both theFe-rich and Al-rich alloys, the excess energies of GB’s dplaying 100% enrichment in those antisite defects withlowest off-stoichiometric formation energies~bold charactersin Table IV!.

In Fe-rich compounds, the ground state is nondegeneand definitely corresponds to the Fe-rich PS-1A variant. Thissimple picture of single layer Fe segregation with no Gvariant degeneracy is therefore an essential result, sincwas found to occur in ideally stoichiometric FeAl, but alsholds in Fe-rich alloys, within the IPDA as well as includinpoint defect interactions. The other candidate predictedthe IPDA, namely the S-2A variant, although it has an energy considerably lower than that of the other structuresvestigated, remains more energetic than the PS-1A variantby a considerable amount of about 640 mJ/m2.

On the other hand, in Al-rich alloys, the same two Al-riccompeting structures~PS-1F and S-2F! emerge, the thirdone suggested by the IPDA~S-1F! being much more energetic, which again points out the limitations of the IPDA foGB’s in Fe-Al. GB structures in Al-rich alloys are thus somwhat more intricate than in Fe-rich alloys, because ofexistence of several energy minima for GB’s in Al-rich aloys. As a final structural feature, the interplanar distanprofiles in the S-2F and PS-1F variants are displayed in Fig8. Here again, segregation~i.e., local chemistry! induces ageometrically symmetric PS-type PS-1F variant, whereasthe symmetric-type S-2F is made asymmetric by the presence of an Al plane adjacent to the central plane. InPS-1F, as in the previously studied PS-1A variant, the AlGB layers contract, while the Fe layers expand.

2. Multilayer segregation

Up to now, whatever the bulk composition, 100% enricment in only one GB plane was considered, which corsponds to single layer GB segregation. Table VII indicathat in Fe-rich alloys, this description is probably valid, bcause the PS-1A variant contains no site with negative fomation energy. On the contrary, as shown in Table VIII,Al-rich alloys, several GB sites ofpF type are found to havenegative antisite defect formation energies~suggesting thatGB segregation probably spreads on several layers!, and ourprevious conclusion concerning the two PS-1F and S-2FGB’s with single-layer segregation has to be re-examinTherefore, to assess the occurrence of multilayer Al segretion in Al-rich alloys, we performed full~beyond IPDA! en-

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ergy calculations on GB’s containing an increasing numof 100% Al-rich planes around the GB plane. Figure 9 plthe GB excess energy as a function of the number ofriched planes for both the S- and PS-type variants. Theexcess energy is a decreasing function of the number oplanes, which confirms the previously obtained trendmultilayer segregation. For an Al enrichment extending omore than three planes, negative excess GB energies artained, a surprising point that needs explanation. From a geral point of view, a negative value for the excess energya subsystem containing an interface is impossible in a pelement. However, in an alloy, a simple analysis showsthis negative value is not critical. Indeed for theentire alloybicrystal which is aclosed system, and because in this casthe presence of interfaces is thermodynamically related toincrease of the Gibbs free energy, the excess free energthis bicrystal is necessarily positive. Conversely, in the cof a subsystemcontaining an interface, equilibrium is giveby the minimum of the grand potential, as described in S

FIG. 8. Relative~310! plane spacing in the GB variants witsingle layer Al segregation pertaining to Al-rich alloys:~a! S-2Fvariant,~b! PS-1F variant.

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II. Assuming zero pressure throughout the subsystem,grand potentialG(ss) is related to its free energyG(ss) bythe formula

G~ss!5G~ss!2Ni~ss!m i~ext!, ~7!

wherem i~ext! are the chemical potentials correspondingthe global atomic fractions~entire system!. If the localatomic fractions are equal to the global ones, tNi(ss)m i~ext! term is equal to the reference free energy osystem containing the same numbers of atoms as thesystem. ThusG(ss) can be interpreted as an excess freeergy, and should then be positive, following the argumabove. Nevertheless, in general, for subsystems with locompositions different from the global one, there is noquirement as regards the sign ofG(ss). The cells used in thesimulation, because of their small size, correspond tolatter case~in GB’s involved in multilayer Al segregationthe cells may contain up to 65% Al!, whereas the chemicapotentials calculated in the framework of the IPDA~andgiven in Table I! correspond to slight departures fromB2stoichiometry—of the order of 1%, which can provide aexplanation of the negative sign of the GB excess energ

In the present case, the negative excess energy valuealso to be attributed to a bulk structural transition in the bAl layer with B2 lattice parameter, switching towards a nocubic structure, constrained by the presence of the GB. Sa transition was already noted in Cu precipitates in a bccmatrix.27 In spite of the uncertainty concerning the exanature of this noncubic Al GB layer, the thickness of the G

FIG. 9. Influence of the thickness of Al segregation on the Gexcess energies in Al-rich alloys. When a sufficient thicknessreached~about four Al planes!, the bcc Al GB layer, recoveringbulk properties, is unstable and expands in the direction normathe GB.

TABLE VII. T50 K point defect formation energies~eV! in the PS-1A variant in Fe-rich alloys.

Plane 1 2 3 Bulk

Antisite A sites 0.34 0.88 0.00atoms F sites 1.21 2.00 1.28 1.68

Vacancies A sites 2.00 2.20 2.47F sites 1.82 0.71 1.26 1.28

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TABLE VIII. T50 K formation energies~eV! of antisite defects in the PS-1F and S-2F GB variants inAl-rich alloys.

PS-1F 1A 2A 3A 4A 2F 3F 4F0.88 1.29 1.52 1.72 20.31 20.16 20.18

S-2F 1A 2A81 2A82 2A9 3A8 3A9 4A8 4A91.01 0.85 0.73 0.82 0.94 1.47 1.16 1.401F 2F9 3F8 3F9 4F8 4F9

20.64 20.50 0.73 20.54 20.61 20.26

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region containing the segregated~bulk excess! element isthus probably higher in Al-rich than in Fe-rich Fe-Al. Thfree equilibrium structure of Al being fcc, this multilayer Asegregation in a strained bcc-type structure is a ratherprising result, needing experimental confirmation.

VI. DISCUSSION AND CONCLUSION

A. GB chemistry in B2 Fe-Al

To our knowledge, the only computational result conceing the atomic structure of interfaces inB2 Fe-Al alloys up

to now concerns the (110) 12 @111# APB, studied by means o

ab initio calculations.28 The authors considered two casethe stoichiometricB2 FeAl and theB2 Fe-Al alloy contain-ing 40% Al. Owing to the computational cost ofab initioenergy models, these authors do not take into accountatomic relaxations around the geometrical sites. Theyscribe the strongly nonstoichiometric alloy in a very simpway consisting in a systematic Fe enrichment of a givfraction of the~110! crystallographic planes. Despite the dferences between their work and ours~APB vs GB, large vssmall off-stoichiometry!, the general trends seem to agr~the interfaces show a strong Fe enrichment!, which givesreasonable confidence in our results.

From the experimental point of view, a recent work29 car-ried out on nanocrystalline Fe-Al led to the conclusion thGB’s have a density close to that of the bulk. Although heagain comparison with our results is not straightforwagiven the sample preparation~by ball milling!, prone tomodify significantly bulk or GB structures~highly meta-stable states can be favored in such severe thermomechatreatments! and their strong off-stoichiometry~40% Al!, theabsence of GB vacancies predicted by our simulations iagreement with this experimental result.

Faced with the lack of available information about GBin iron aluminides, the present work gives better insight inthe relative roles of GB disorder and segregation inB2 Fe-Alordered alloys. In fact, in spite of a negative formation eergy for GB antisite pairs, no disorder, even limited, existsT50 K at equilibrium, the main phenomenon being GB seregation~however, GB disorder might exist as a GB responto processes creating interfaces without diffusion and hesegregation, such as purely mechanical and nontheprocesses—mechanical alloying, for instance, as in Ref.!.As regards GB segregation in Fe-Al, the proportions ofchemical elements present in the GB’s strongly dependthe bulk composition. Small departures from st

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ichiometry in the bulk are drastically amplified in GB’s. Asconsequence, from an experimental point of view, smvariations in the bulk chemical concentration probablyduce sharp modifications in the structure of GB’s. GB’sFe-rich alloys are characterized by a single layer Fe segrtion, whereas in Al-rich alloys, they undergo multilayer Asegregation. GB segregation of the excess element wasobtained by simulation inB2 NiAl, in a S53 ~111! twinboundary.30 These results are probably to be related tohighly ordered character of these alloys. The multiplicityGB stable variants in Fe-Al also depends on the bulk coposition. In Fe-rich alloys and at stoichiometry, we identifione single GB variant~PS-1A!, whereas a variety of S- anPS-type competing variants~S-2F, PS-1F and variants withAl multilayer! were found in Al-rich alloys.

B. Consequences on GB mechanical properties

The multilayer Al segregation may also have importaconsequences concerning the macroscopic mechanicahavior, since a multilayer segregation of either Al or Fshould induce a local increase of ductility by destroying tlocal order. This ductility increase should be more prnounced in the case of Al segregation~as that predicted inthe present work! because for reasons relying on the eletronic structure of each type of atom, Al stackings are ctainly more ductile than Fe stackings, whatever the stack~bcc, fcc! type. Therefore multilayer Al segregation shoufavor, for instance, GB glide with respect to intergranu

FIG. 10. Calculatedg surface of the12 @111#(110) antiphase

boundary inB2 FeAl. Contrary to the GB case, no privileged drection exists for glide~between mixed planes!.

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fracture. To make this point precise, we now examine briethe coupling between GB segregation and glide. As mtioned in Sec. III, GB glide can be investigated by meansthe g surfaces, roughly describing the GB behavior undexternal mechanical constraints. Our first calculation~Fig.2!, concerning the CSL models, pointed out glide ‘‘pathalong the@001# tilt axis, the glide occurring between twmixed planes~containing 50% Al!. Although generalizingthis crystallography-dependent feature to other GB’s ththat presently studied„~310!@001# symmetrical tilt… requiresprecaution, it is worth inquiring about its sensitivity witrespect to the precise type of interface. A common interfin B2 alloys being the1

2 @111#(110) APB, we calculated itsglide properties, which also correspond to the relative dplacement between two mixed planes connecting bulkAPB plane stackings. The~bulk-APB! g surface, shown inFig. 10, shows no high energy barrier~,500 mJ/m2! what-ever the glide direction, but indicates that no direction exfor enhanced glide~the most favorable one seems to be t@111# direction!. In particular,@001# glide is limited by anenergy barrier close to 500 mJ/m2. This points out that the@001# glide previously detected is specifically a GB featuintricately involving GB chemistry and crystallography. Thcomplexity is further demonstrated by the calculation ofg surfaces pertaining to the GB’s shown to be the most staones in the foregoing study. Without being exhaustive~inparticular, in spite of the multilayer Al segregation evidencabove in Al-rich alloys, we do not analyze glide betweenplanes or between Al planes!, we proceeded with glide in theS/PS-1A ~mixed-Fe plane glide!, S/PS-1F ~mixed-Al planeglide!, and S/PS-1A-2F ~Al-Fe plane glide! variants. Thegsurfaces of these variants, displayed in Fig. 11, are oslightly different from those of the initial CSL model. Nnoticeable influence of the GB chemistry appears,@001#glide still being strongly favored. Since a full release of texternal stress was allowed in theOy direction~GB normal!,the only degree of freedom unaccounted for is the individatomic relaxation. This degree of freedom is sometimtaken into account by allowing atomic movements onlytheOy direction, but as this procedure imposes arbitrary cstraints on the particles, we did not follow it, all the moreFe-Al alloys probably present large but unknown entroand vibrational energy contributions,14 which may affect the‘‘response’’ of the GB to the external constraint.

To conclude, the present work is a comprehensive atomscale simulation of grain boundaries inB2 Fe-Al alloys. Tomake it possible, we adopted a simplified energetic mo~thoroughly tested in the bulk! combined with a thermodynamic approach relying on two main approximations:~i! wedid not take into account the vibrational contributions, a~ii ! the GB’s were always considered as open systemthermodynamic equilibrium with the bulk~grand canonicalensemble!. Within these assumptions, we were led to tfollowing clear-cut properties of GB’s in slightly offstoichiometricB2 Fe-Al alloys:

~i! CSL models are sufficient to describe GB crystalloraphy, but strong chemical effects~segregation! existthat cannot be addressed by these models,

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~ii ! the independent point defect approximation is notliable to describe GB segregation,

~iii ! GB vacancies do probably not exist,~iv! in Fe-rich alloys single layer Fe segregation occu

whereas in Al-rich alloys there is multilayer Al segregation with possible interfacial phase transitions~to-wards noncubic bcc-type strained Al!,

FIG. 11. Influence of GB compositional changes on theg sur-face: ~a! 1A-type variants~Fe segregation, glide between Fe amixed plane!, ~b! 1F-type variants~Al segregation, glide betweenAl and mixed plane!, ~c! 1A-2F-type variants~no segregation butGB disorder, glide between Al and Fe planes!. The qualitative shapeof the GB g surface, mainly influenced by crystallography,chemistry-independent.

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~v! GB glide is anisotropic and independent of the seggating chemical species.

These results are a guideline to undertakeab initio studies ona reduced set of GB configurations, hence tractable wcomputationally so expensive methods.Ab initio calculations~to be published! of GB’s in B2 Fe-Al are currently under

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way, and preliminary results already confirm the main trenunderlined here.

ACKNOWLEDGMENT

The authors are indebted to M. Biscondi for helpful dicussions.

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An atomic-scale simulation study