Embedded atom method study of the interactions between point defects in iron aluminides: Double defects

Intermetallics 13 (2005) 1233–1244 www.elsevier.com/locate/intermet

Embedded atom method study of the interactions between point defects in iron aluminides: Double defects Renata N. Nogueira, Cla´udio G. Scho¨n* Computational Materials Science Laboratory, Department of Metallurgical and Materials Engineering, Escola Polite´cnica da Universidade de Sa˜o Paulo, Av. Prof. Mello Moraes, 2463-CEP 05508-900 Sa˜o Paulo-SP, Brazil Received 30 August 2004; received in revised form 23 March 2005; accepted 11 April 2005 Available online 20 June 2005

Abstract In this work (part 1 of 2), we used the embedded atom method (EAM) to perform atomistic simulations in the molecular statics framework, aiming to investigate the interactions between point defects (both vacancies and antisite atoms) in Fe–Al alloys. This method is particularly useful in obtaining the self-energies of crystal defects characterized by strong core relaxation strains, which generate long-range elastic fields. The following cases are considered: divacancies, vacancy–antisite and antisite–antisite atom pairs in Fe (A2), FeAl (B2) and Fe3Al (D03) compounds. In each case the most stable configuration has been found and the dependence of the interactions of these pairs of defects on their separation distance has been investigated. Particular care was taken to ensure that all simulations for a given compound are performed in the canonical ensemble. q 2005 Elsevier Ltd. All rights reserved. Keywords: A. Iron aluminides; B. Bonding; D. Defects: point defects; E. Simulations atomistic; E. Defects: theory

1. Introduction Iron aluminides represent an intriguing class of new materials: they offer a good combination of mechanical properties, specific weight/strength ratio, corrosion (and oxidation) resistance and low raw material cost [1], which make them potential candidates for the substitution of stainless steel in applications at moderate to high temperature. The extensive technological application of iron aluminides, however, is impaired by their low room temperature tensile ductility. This is attributed to extrinsic (environmental embrittlement) or intrinsic (low grain boundary cohesion) mechanisms, with the dominant mechanism depending on the aluminium content of the alloy [2]. The development of new, more ductile, Fe–Al alloys depends on a thorough understanding of their properties, implicating a better comprehension of the * Corresponding author. Tel.: C55 11 3091 5726; fax: C55 11 3091 5243. E-mail addresses: [email protected] (R.N. Nogueira), [email protected] (C.G. Scho¨n).

0966-9795/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.intermet.2005.04.007

properties and behaviour of defects in these materials. Experimental [3–5] as well as theoretical studies [6–11] suggest that iron aluminides present complex point defects, especially double defects. In addition, it has been demonstrated that ‘quenched-in’ vacancies are responsible for a large strengthening effect both on B2 FeAl [12] and on disordered (A2) Fe–Al alloys [13,14]. Nevertheless, there are no conclusive descriptions of the arrangement and interactions that cause these defects, and the interest on this subject remains, despite the long history of investigation concerning these alloys. In atomistic modelling of point defects, a given simulation is performed in the canonical ensemble due to the characteristics of the method (the total number of atoms in the ‘computer crystal’ is kept constant in the course of the simulation). The reference state for calculations is the defect free crystal, and the formation energy of the defect is obtained introducing the defect into the atom block and computing the energy difference of the two blocks. However, as the numbers of atoms of the reference and defective blocks are not the same, they are not elements of the same canonical ensemble. As Mayer et al. [9] pointed out, this procedure involves a variation in the chemical potential of the block, since the simulations should be done

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in the grand-canonical ensemble instead. As far as defects in pure metals (for example, a single vacancy) are concerned, this procedure can be justified due to Raoult’s law, which states that the solvent’s activity1 in dilute alloys (e.g. ironC vacancies) asymptotically converges to unity at the limit of infinite dilution [15]. For more complex defects (introducing a solute atom in a crystal, for example), this change of chemical potential is no longer negligible, and it has to be evaluated in order to obtain the correct formation energy of the defect. Meyer et al. [9] suggested one way of evaluating the change of chemical potential in the case of defects in superlattices, but it has to be noticed that the variation of chemical potential particularly near to a stoichiometric point of a superlattice is strongly temperature and concentration dependent [16]. Thus, any ‘correction’ that is to be applied may introduce systematic errors in the evaluation of the formation energy of the defects, in comparison, for example, with experimental data. The situation is even more critical for the case of complex defects, since the reference and defective blocks differ by a larger amount of atoms or even by different species. The ideal situation would be to perform simulations in which the reference and the defective blocks contain the same amount and the same kind of atoms, comparing directly the internal energies of the two blocks. The embedded atom method (EAM) is capable of simulating very large atom blocks, and this property of the method is used here to achieve this goal. In the present paper, the first part of a two-part investigation, a systematic study regarding the interactions between pairs of point defects as a function of the distance between them is performed. Three different classes of point defects have been considered: two antisite atoms, two vacancies, or one antisite atom and a vacancy. The simulations have been performed in pure Fe (A2), in B2 FeAl and in D03 Fe3Al compounds, which differ from stoichiometry only by the existence of the point defects. The approach used here (to be described later in the text) allows for the realization of the simulations in the canonical ensemble for all defects in a given compound. Therefore, comparison of different classes of pairs of defects in terms of energy differences in a given compound is possible, and some general trends for the interactions can be inferred. The equally important case of triple defects will be dealt with in the second part [17].

2. Methodology

for Fe–Al alloys to calculate equilibrium spatial configurations and their energies. A detailed description of this method, as well as of the particular procedure for determination of the potential, can be found elsewhere [18–23]. In the EAM the energy of the computer crystal is written as: EZ

X 1X Vij ðrab Þ C Fi ðrea Þ 2 ab a

(1)

where the first sum runs over all pairs of atoms in the atom block (the factor 1/2 corrects, as usual, for the double counting of the pairs in the sum), Vij ðrijab Þ represents a pair potential, which is characteristic of species i and j, which occupy, respectively, block positions (hlattice sites) a and b.2 These potentials are functions of the instantaneous distance between both sites in the atom block (rab) and vanish for long distances.3 The use of central-force potentials for modelling Fe–Al alloys has been criticized [6] due to evidences of a strong covalent contribution to the bonding [24]. This leads to angular-dependent contributions to the energy, which cannot be accounted for with central-force potentials. The use of these potentials in the present work, however, is justified since they have been used in modelling many different properties of ordered Fe–Al alloys, like dislocation core structures [23,25], grain boundary [26] and surface structures [27], and crack growth [28]. Recently, centralforce potentials have even been applied to the case of M–C interactions (MZFe, Al, Nb and Ti), surely characterized by stronger covalent contributions [29]. The second sum runs over all atoms in the atom block, and Fi ðrea Þ, stands for the so called ‘embedding functional’ which is characteristic of species i occupying lattice site a and is a function of the electron density on lattice site a, rea , which is generated by all other atoms of the computer crystal. The embedding function depends on the atom positions in the block through the dependence of the electron density on the distance, i.e. rea Z rea ðfrijab gÞ, where frijab g represents the set of all pair types and distances in the atom block. A typical EAM simulation starts with a given atom configuration (i.e. specifying an initial set frijab g0 ) and then Eq. (1) is minimized with respect to the atom positions in the multi-dimensional space spanned by the set frijab g, the minimum being a stable configuration (possibly degenerate) of the atom block. The pair potentials V FeFe , VFeAl and V AlAl , the corresponding embedding functionals FFe and FAl, as well

2.1. Fundamentals 2

The atomistic simulations are performed using embedded atom method (EAM) parameterized potentials 1 At the reference state of the stable solvent’s phase at the temperature of the simualtion.

To prevent any confusion, the following convention was adoted in the present work: lattice sites are denoted by Greek superscripts, and species are denoted by Roman subscripts. 3 To keep the computation time feasible the pair potentials used in the EAM are usually truncated at a distance slightly larger than that of nextnearest neighbours in the non-defective lattice, vanishing for larger distances [16].

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as the electron density function rea ðfrab gÞ, have been adopted from the literature [20–23]. 2.2. Procedure for the pair defect simulations The simulation blocks have been built with the same set of defects for each compound, meaning they are all elements of the same canonical ensemble. For the reference case, those defects are placed very far from each other, so the interaction is negligible. Eq. (1) is minimized for this configuration and the corresponding energy minimum is labelled E0. For each simulation, two point defects from those available in the atom block (let us say, x and z) are selected and placed near the centre of the block, forming a defect pair in a chosen neighbourhood (the additional defects remain far from the others and from the newly formed pair). Eq. (1) is minimized for this new configuration and the energy minimum is labelled Exz. The difference between the energy for this configurations and E0 is then the interaction energy for the compound defect, in other words: DExz Z Exz K E0

(2)

Positive values of DExz imply a repulsive interaction between the defects; negative values, on the other hand, imply attractive interactions. Relative to the crystal free from defects the formation energy of a pair defects is, of course, in all cases positive. Fig. 1 presents an illustrative scheme for the procedure described above. This figure shows two (001) projections of a bcc lattice containing two defects (represented by dark grey spheres). In the first case (Fig. 1(a)), these defects are placed far from each other and represent the situation for the reference case (with energy E0). In the second case (Fig. 1(b)), we represent a next-nearest defect pair (called 2nn) configuration. We must emphasize that the blocks actually used in our calculations are much larger than the ones depicted in this figure, and that the reference configurations were chosen so that the borders of the blocks were taken into account. In order to accommodate all the non-interacting defects, larger blocks than are the usual for EAM simulations had to be used in this study. The A2 Fe and FeAl B2 blocks were formed by 26,800 atoms and the D03 Fe3Al blocks have 40,320 atoms. Their dimensions were chosen big enough so as to avoid size effects in the results. The lattice parameters (a0) found for each phase were 0.287 nm for pure Fe, 0.294 nm for the stoichiometric B2–FeAl compound and 0.579 for the stoichiometric D03–Fe3Al compound. Periodic boundary conditions were applied to the crystal blocks. The defects are denoted as XY, where XZV, Fe, Al denotes the kind of defect (vacancy, V, or antisite atom) and Y refers to the original occupancy of the site in the perfect lattice. For  space group, we have two Fe3Al, which belongs to the Fm3m non-equivalent Fe sites on Wycoff positions 4b(1/2,1/2,1/2),

Fig. 1. Schematic representation of a block with two point defects (a) noninteracting and (b) at a 2nn configuration.

containing four aluminium and four iron atoms in the nearest neighbourhood (1nn) and 8c(1/4,1/4,1/4), containing eight iron atoms in 1nn. These are referred to as Fe1 and Fe2, respectively. Since the discussion of the separation distance between the defects is central in this work, this will be referred to as the neighbourhood that one of the members of the pair occupies with reference to the other, placed in the centre of the crystal block for convenience. This neighbourhood will be referred to in contracted form, as used above, for simplicity sake. For example, the second neighbourhood will be referred to as 2nn, the third as 3nn and so on. The results of the present work will be graphically expressed as a function of the neighbourhood rather than the physical distance. This allows for better visualization of

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Table 1 Alignment directions and distances for seven neighbourhoods of an atom in A2, B2 and D03 compounds, in units of the equilibrium distance of nearest neighbour Neighbourhood

Alignment direction

Distance

1nn 2nn 3nn 4nn 5nn 6nn 7nn

h111i h100i h110i h311i h111i h100i h331i

1.000 1.155 1.633 1.915 2.000 2.309 2.517

the trends observed in the different crystal structures. Table 1 shows the equivalence between neighbourhood and physical distance for the bcc lattice. The physical distance is expressed in terms of the equilibrium distance of 1nn pairs in the compounds, so that it can be converted into metrical pffiffiffi units by multiplying column d by ð 3=2Þa0 Z 0:249 nm in the pffiffifficase of A2 Fe, 0.255 nm in the case of B2 FeAl, and by ð 3=4Þa0 Z 0:253 nm in the case of D03 Fe3Al.

3. Results and discussion The results will be discussed in three parts, depending on Y: both defects in the Fe sites, both defects in the Al sites, and each defect on a different site. 3.1. Defects in Fe sites Fig. 2 shows the interactions between Al antisite atoms4 in the three compounds considered. The interaction of Al atoms in A2 Fe is repulsive for 1nn and 2nn. For longer distances the interaction becomes attractive, but loses intensity after 3nn. A preferential formation of nearest neighbour dissimilar pairs is expected due to the existence of long-range order (LRO) in the system, this should favour, therefore, repulsive interactions for 1nn and 2nn similar pairs. The observation of attractive 3nn interactions, on the other hand, is compatible with the positive effective interaction energies for third and fourth neighbour pairs (i.e. tendency to form like pairs), as derived by inverse Monte Carlo method simulations of high temperature neutron diffuse scattering data in disordered alloy. These interactions are supposed to be responsible for the stabilization of a multicritical point for the B2–A2 equilibrium in Fe–Al (see [30] and references therein). 4

Formally one cannot refer to an ‘antisite’ atom in a disordered lattice, since there are no preferred occupancies for the species in this system. This defect, in reality, corresponds to a pair of solute aluminium atoms in an Fe host. This pair of defects, however, will be referred to in this work as an ‘antisite’ pair, in order to stress the equivalences with the defects found in the ordered compounds.

Configurations containing 3nn Al pairs are found both in the B2 and in the D03 stoichiometric superlattices and it is, therefore, natural that attractive interactions are observed between these pairs in the disordered alloy due to shortrange ordering (SRO). It is interesting, however, to observe, as mentioned before, that the pair interactions used in the EAM vanish for distances slightly above 2nn [20–23]. The range of the effective pair interactions, therefore, is ‘extended’ by the interaction between the deformation fields of the isolated defects. The same antisite defects (i.e. AlFe–AlFe) in the B2 compound follow a different trend: the most stable configuration is found for 2nn and the interaction becomes repulsive for 5nn. As one can see in Table 1, a 2nn pair is located along lattice vector h001i, which is the usual configuration of Al atoms found in stoichiometric B2, so this minimum energy can be attributed to a partial restoration of the B2 order parameter due to this defect configuration. The general curve for D03 is similar to that found for A2 Fe except for defects in Fe1 sites, which present also some proximity to the B2 results. These findings can be analysed in terms of the configuration of the first coordination shell in the compound, which, for the case of the Fe1 position, contains two Al atoms. Their influence makes the interaction to resemble the behaviour of antisite atoms placed in an environment containing four Al atoms, as it is the case of iron sites in the B2 compound. The behaviour of aluminium antisite pairs in D03 shows the most stable configuration in 3nn for both Fe1 and Fe2 sites. Comparing these two configurations (see the inset in Fig. 2) it is seen that the one in which both antisite atoms are placed on Fe2 sites presents a slightly more negative binding energy in 3nn than the alternative case (observe that some combinations are not realized due to the crystal structure, for example 3nn involving Fe1 and Fe2 positions). The first situation (AlFe2–AlFe2) leads to a B2-like shortrange order (SRO) configuration, while the second (AlFe1– AlFe1) leads to a B32-like SRO configuration. This behaviour is related to the fact that the B2 phase is found stable in the system, while B32 is not. In this context it is interesting to take into consideration the observations by Beker and Schweika [31] concerning the so called ‘K’ state: a SRO state observed at low temperatures and high aluminium contents in the A2 phase, to which these authors attributed a B32-like configuration. The results of the present study seem to contradict those observations, but one must keep in mind two limitations of the present EAM calculation: 1. the potentials are fitted using experimental data, among them the experimental formation enthalpies of the D03 and B2 superlattices in the system, in other words the B2 phase has to be more stable than B32; and 2. one important degree of freedom is not considered in the calculation: the spin orientation of the iron atoms, which

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Fig. 2. Interactions for two Al antisite atoms placed in Fe sites for A2 Fe, B2 FeAl alloy and D03 Fe3Al alloy (Fe1 and Fe2 sites). The lines are just guides for the eyes.

is expected to play an important role at low temperatures [32–34]. Fig. 3 shows the results for divacancies. As can be seen, the interaction of vacancies in Fe sites tends to follow a universal curve valid for the three compounds. There is an oscillatory behaviour and 1nn, 2nn and 4nn show attractive interactions, whereas the others are repulsive or neutral,

the most stable configuration being 2nn. The oscillatory behaviour clearly indicates some degree of anisotropy in the interactions. Interactions originating from central forces (as is the case of the used EAM pair potentials) are expected to depend only on the distance and not on the orientation, decreasing smoothly in the limit of infinite separation. The origin of this anisotropy in crystals with a centre of symmetry (all compounds treated here belong to the m3m

Fig. 3. Interactions for two vacancies placed in Fe sites for A2 Fe, B2 FeAl alloy and D03 Fe3Al alloy (Fe1 and Fe2 sites). The line is just a guide for the eyes.

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fewer atoms, mostly in the first neighbourhood of the complex, so the original symmetry of the system is just slightly affected. Of course these considerations are only qualitative and a proper analysis of the strain tensor field in the two configurations should be performed for demonstration, but this is outside the scope of the present work. Fig. 5 shows the results for a vacancy–antisite atom pair. In this case, the most stable configurations, the ones with the lowest energies, are 1nn. The interactions are attractive in almost all the cases studied. Two different trends can be identified, one characteristic of A2 Fe and another characteristic of B2 FeAl. One must notice that the neighbourhood of the vacancy seems to play a very important role in the definition of which trend a given defect belongs to. This can be seen from defects in the D03 compounds: the cases where the vacancy has Al atoms in the first coordination sphere belong to the B2-like trend and those cases where it has only iron atoms in the first neighbourhood belong to the A2 trend. For the A2 trend the most stable configuration was found to be 1nn, while for the B2 trend the most stable configuration is found farther, in 3nn. It is important to notice that the magnitude of the attraction for divacancies is in general larger than that obtained for the other pairs of defects. Furthermore, the lowest magnitudes of the interactions were found for vacancy–antisite atom pairs. 3.2. Defects in Al sites

Fig. 4. Schematic representations of the displacements around (a) 2nn and (b) 3nn divacancies in A2 (Fe). Projection view of the A2 lattice along [100]. The centres of the vacancies are located on the (011) plane.

point group) should be ascribed to the interaction of the strain fields generated by the pair of defects, which disturbs the local symmetry of the crystal. Fig. 4 represents the equilibrium configurations obtained in the simulations of the 2nn and 3nn divacancies. The atom displacements in this figure have been multiplied by 10, in order to allow for clear visualization of the relaxation strains around the defects. For the 2nn configuration (Fig. 4(a)), the figure clearly shows large compressive strains around the defect complex, irradiating up to a distance of two neighbourhoods away from the complex. For 3nn, instead (Fig. 4(b)), some of the atoms are attracted to the empty region and the strain perturbation is accommodated by

Fig. 6 presents the interactions regarding Fe atoms placed in the Al sites. The values are considerably smaller than those presented before (!0.05 eV) and slightly attractive for both B2 and D03 compounds. The behaviour for the case of divacancies (Fig. 7) in Al sites of B2 alloys is oscillatory and very similar to those found for the vacancies placed in Fe sites (Fig. 3). For the D03 compound, however, the interactions showed a different behaviour, and in this compound the values for the interactions are always very small in modulus. Regarding the interaction of a vacancy and one Fe antisite atom occupying the Al sites, presented in Fig. 8, their interactions are slightly attractive, with values smaller than 0.1 eV in modulus. 3.3. One defect in Fe and one in Al sites The last cases to be analysed are those which have one of the two defects placed in a Fe site and the other in an Al site. Fig. 9 presents the set of configurations involving two antisite atoms. These results show that this interaction is attractive, and vanishes after 4nn. The most stable configuration found here is 1nn. This can be rationalized by the following considerations: the environment of an antisite atom in the first coordination sphere is, by nature, poor in Al–Fe like bonds and rich in either Fe–Fe or Al–Al unlike bonds. The formation of the AlFe–FeAl nearest

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Fig. 5. Interactions for one vacancy and one antisite placed in Fe sites A2 Fe, B2 FeAl and D03 Fe3Al alloy (Fe1 and Fe2 sites). The lines are just guides for the eyes.

neighbour antisite pair restores one like bond, which contributes to the stabilization of the defect. Considering the divacancies (Fig. 10), it can be seen that they also follow the same universal trend presented for vacancies in pure Fe or Al sites. However, as some of the neighbourhoods are not possible in these cases (e.g. a 3nn VFe1–VAl), the oscillatory behaviour cannot be evidenced. The interactions are mostly attractive.

Finally, Fig. 11 presents the results for one antisite atom and one vacancy. The most striking feature of these curves is the extremely large negative values for the cases involving one Al antisite atom and one vacancy in the Al site in 1nn. These configurations are mechanically unstable and result in a reaction between the defects, characterized by the exchange of the Al antisite atom with the vacancy, producing a vacancy on Fe site and destroying the antisite

Fig. 6. Interactions for two antisite Fe atoms placed in Al sites for B2 FeAl and D03 Fe3Al compounds. The lines are just guides for the eyes.

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Fig. 7. Interactions for two vacancies placed in Al sites for B2 FeAl and D03 Fe3Al compounds. The lines are just guides for the eyes.

defect. This reaction can be schematically represented by: AlFe C VAl / VFe

(3)

Therefore, this large negative energy contains a significant contribution of the annihilation energy of the aluminium antisite defect. This reaction was observed for both B2 and D03 compounds. The significance of these values, however, must be analysed with care: in all other cases investigated in the present work the individual point

defects keep their identity during the simulation. This condition is violated by the reaction expressed in Eq. (3). These configurations are, therefore, anomalous and the energy values cannot be compared with the remaining cases. The most stable configuration represented in Fig. 10 corresponds, therefore, to the 2nn VFe2–FeAl pair of the D03 compound. This attraction is similar to the one observed for 1nn pairs, but in this case the collapse of the defect is not observed.

Fig. 8. Interactions for one vacancy and one antisite placed in Al sites for B2 FeAl and D03 Fe3Al compounds. The lines are just guides for the eyes.

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Fig. 9. Interactions for two antisite atoms, one of them placed in a Fe site and the other in an Al site for B2 FeAl and D03 Fe3Al alloy. The lines are just guides for the eyes.

It is interesting to notice that, in the diffusion literature, vacancies on aluminium sites are sometimes considered not to be energetically favoured in B2–FeAl [35], which is compatible with the mechanical instability observed for the 1nn defect pair. The reaction described above is one example of a possible recovery mechanism observed in Fe–Al alloys.

It is interesting that the annihilation of the antisite defects is followed by such a large release of energy. This suggests, together with the fact that the diffusion distance is of a very short range, that this kind of process should take place at very low temperatures, being highly exothermic. This is in agreement with experimental indications [36], which point to temperatures lower than 140 8C for the start of the first

Fig. 10. Interactions for two vacancies, one of them placed in a Fe site and the other in an Al site for B2 FeAl and D03 Fe3Al compound. The lines are just guides for the eyes.

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Fig. 11. Interactions for one vacancy and one antisite, one of them placed in a Fe site and the other in an Al site for B2 FeAl and D03 Fe3Al compound. The lines are just guides for the eyes.

stage of recovery in Fe–Al alloys (ascribed to the annihilation of defects and stress release in the sample). Other results indicate that recovery and in particular reordering take place very quickly during annealing [37]. 3.4. General analysis Fa¨hnle et al. [10] performed ab initio calculations in the framework of the density functional theory (DFT) in the local density approximation (LDA) of the interaction of point defects in 54 atom supercells of B2 FeAl, without allowing spin polarization in the calculation. Although the cases are fundamentally different due to the size difference of the modelled systems, it is instructive to compare their results with the ones obtained in the present work. In this comparison one observes that in some cases there are similar qualitative trends, e.g. VFe–VFe divacancies present a strong attractive interaction (K0.38 eV/pair) for 2nn, which vanishes for 3nn (0.00 eV/pair) and becomes repulsive for 5nn (C0.05 eV/pair) [10]. Comparing with the present results (respectively, K0.14, C0.04 and 0.01 eV/pair, Fig. 3) we observe that both trends qualitatively agree. On the other hand, the case of VFe–FeAl 1nn pairs presents contradictory results (K0.10 eV/pair for Fa¨hnle et al. [10] and C0.14 eV/pair in the present case). In all other cases investigated by Fa¨hnle et al. [10] a qualitative agreement with the present results is obtained. Usually, first principles calculations can be considered more precise than atomistic ones. However, in this case, some particular characteristics of the studied systems may lead to an inappropriate description by the ab initio approach. First of all, the maximum size of the supercell

(dictated by computer limitations) is too small to avoid the long-range interactions with defects in the neighbouring cells produced by the periodic boundary conditions. Individual EAM calculations using 54 atom cells, performed in the present work, showed variations of the order of 0.05 eV/pair for the interaction energies of similar defects in the large computer blocks. Besson and Morillo [6] used similar arguments to justify similar discrepancies with the ab initio results in their atomistics simulations (using angular dependent potentials). A second limitation is the non-consideration of spin polarization in the calculations of Fa¨hnle et al. [10]. Recent results [34,39] clearly show that spin polarization is essential when treating Fe–Al systems by ab initio methods; moreover, the application of the DFT to Fe–Al system is still a controversial issue [39–44]. It could be argued that magnetism is not included in the EAM calculations as well, but EAM potentials are effective potentials, fitted to experimental properties of the existing compounds, and therefore, this limitation is expected to play a lesser role in comparison with the ab initio case. The general results obtained in the present work indicate that the interaction between pairs of point defects is quite long-ranged, extending in some cases up to the fourth neighbourhood with appreciable magnitudes for the interaction energy. This means that strong configurational contributions to the stability of point defects should be expected to be operative in Fe–Al alloys (both ordered and disordered). As an example, considering the data presented in Fig. 3: the existence of a vacancy in the nearest neighbourhood of an antisite Al atom decreases the energy of both defects by about 0.1 eV in all investigated structures (the formation enthalpy of a vacancy in high-purity A2 Fe is

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estimated to be in the range 1.61 eV %DHVf Fe;A2 %1.75 eV [45]). Antisite Al atoms are naturally present, for example, in overstoichiometric D03 alloys or in B2 and A2 alloys at high temperatures due to the so-called configurational and thermal disorders. Thus, these results could justify the tendency of Fe–Al alloys to present anomalously large vacancy concentrations at high temperatures, the composition dependence of this concentration, the tendency to form different kinds of defect complexes as a function of composition and temperature and the high tendency for retention of ‘quenched-in’ vacancies during cooling of specimens [4]. Vacancy clusters, similar to those predicted by the present results, have been also identified by Mo¨ssbauer spectroscopy and positron annihilation techniques after annealing of filing deformed Fe–Ti–Al samples with the D03 structure [46]. Another important feature of the interactions between point defects is their directional character, which is better appreciated in the case of divacancies in A2 (Fe): a 2nn divacancy (i.e. defects located along h100i) is more stable than 1nn (defects located nearer, but along h111i). A 3nn divacancy, on the other hand is characterized by a repulsive interaction (defects located along h110i vector). Experimental evidences for an anisotropy in the divacancy interactions have been found by Schaible et al. [47], who observed a resonance peak in internal friction experiments of aluminium-rich B2 NiAl samples, attributed to divacancy reorientation on nearest neighbourhood sites in the Ni sublattice. It should be questioned whether such longrange complexes could be experimentally evidenced by techniques like positron annihilation, which rely on the detection of the free volume space left by the vacancies in the alloy. For positron annihilation spectra a 3nn vacancy pair would probably appear just like two isolated vacancies. It has to be remembered that directional interactions are expected to be important in FeAl alloys as discussed before. The qualitative agreement between the trend obtained by the ab initio calculations of Fa¨hnle et al. [10] (which includes the covalent contribution to the energy) and that obtained in the present work for the VFe–VFe divacancies suggests that the degree of the anisotropy originated in the interaction of the strain fields is larger than the one due to the covalent contribution to the bonding. The confirmation of this hypothesis would, however, require the calculation of all cases investigated in the present work using first principles methods.

4. Conclusions The results we presented here for point defects in Fe–Al alloys show that their interaction is characterized by a high degree of anisotropy, originated from the discrete relaxation displacements at the complex core.

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Divacancies follow a general trend which is independent of the investigated compound and present attractive interactions if they are either nearest or next-nearest pairs and repulsive for 3nn. Antisite atom pairs, on the other hand, show a diverse behaviour and the interactions depend on the local configurations (at the nearest neighbour shell around each individual point defect) and two general trends can be observed: one characteristic of the A2 compound (repulsive interactions at 1nn and 2nn and attractive interactions for 3nn and beyond) and other of the B2 compound (attractive interactions for both 2nn and 3nn, with the latter being weaker), the behaviour of these defect pairs in the D03 compound follows either the B2 or the A2 trend depending, respectively, on the presence or absence of aluminium atoms in the nearest neighbour environment around the individual defects. Vacancy–antisite atom pairs are characterized by attractive interactions, but their magnitudes are smaller than those observed for vacancy pairs. Among the double defect configurations investigated in the present work, VFe–VFe (both 1nn and 2nn, in the three compounds), AlFe2–VFe2 (1nn in D03 Fe3Al), VAl–VAl (2nn, B2 FeAl), VFe2–VAl (2nn, D03–Fe3Al) and AlFe–FeAl (1nn, both in the B2 and D03–Fe3Al compounds) are potential candidates for the formation of low temperature bound states due to large (DEab!K0.15 eV) attractive interaction energies. AlFe–VAl 1nn pairs are mechanically unstable and react during the simulation, resulting in the annihilation of the antisite defect.

Acknowledgements The authors thank the Sa˜o Paulo State Research Funding Agency (FAPESP) for financial support under grants No. 99/07570-8 and 00/07299-1. The authors would like to thank Prof. Dr He´lio Goldenstein (University of Sa˜o Paulo, Sa˜o Paulo-SP, Brazil), Prof. Dr Diana Farkas (Virginia Technological Institute and State University, Blacksburg, VA, USA) and Prof. Dr Gerhard Inden (Max-Planck-Institut fu¨r Eisenforschung, Du¨sseldorf, Germany) for the helpful discussions.

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