The effect of segregation and partial order on the thermodynamics of (1 1 1) antiphase boundaries in Ni3Al

Computational Materials Science 14 (1999) 283±290

The e€ect of segregation and partial order on the thermodynamics of (1 1 1) antiphase boundaries in Ni3Al Marcel Sluiter *, Y. Hashi, Yoshiyuki Kawazoe Institute for Materials Research, Tohoku University, 980-77 Sendai, Japan

Abstract Thermodynamic properties of thermal and a-thermal (1 1 1) antiphase boundaries (APB) in Ni3 Al are computed from ®rst-principles. The e€ect of o€-stoichiometry, partial disordering and segregation are evaluated and a rough estimate of the vibrational contribution to the antiphase boundary energy is given. Although the vibrational e€ect is found to be small, the con®gurational e€ects are large, so that at non-zero temperature and in o€-stoichiometric Ni3 Al the antiphase boundary energy (APBE) may be only half of that in perfectly ordered stoichiometric Ni3 Al at zero temperature. This result points to a discrepancy between electronic structure calculations and experimental measurements. Ó 1999 Elsevier Science B.V All rights reserved.

1. Introduction Antiphase boundaries (APBs) in Ni3 Al are among the most studied planar defects, both from the experimental and theoretical aspects [1]. NI3 Al has the Ll2 structure (Struktur Bericht notation, Cu3 Au prototype [2]), which derives from the fcc structure by ordering. When a dislocation with a Burgers vector which is a translation vector of the disordered (fcc), but not of the ordered (Ll2 ) structure moves through an ordered crystal, an APB is formed. The energy required to form APBs is an important factor in a material's resistance to deformation, and also a€ects the way deformation occurs [3]. Therefore, aside from the theoretical interest, there is a practical reason to study such

* Corresponding author. Tel.: +81 22 215 2481; fax: +81 22 215 2052; e-mail: [email protected].

defects in detail because Ni3 Al precipitates are used to strengthen high-performance alloys. The APB energy (APBE) has been determined by measuring the separation distance between the partial dislocations that form the boundaries of APBs [1]. The two bounding partial dislocations exert a repulsive force on each other due to their strain ®elds. This repulsive force decays as the reciprocal of the separation distance, and at in®nite separation the repulsive force vanishes. However, increasing the separation distance increases the surface area of the APB. The energy-cost associated with enlarging the APB provides an attractive counter force which is not a function of the separation distance, and at a particular separation a balance of repulsive and attractive forces occurs. Elasticity and dislocation theory provide an expression of the separation distance in terms of the APBE, so that by measuring this distance the APBE can be determined. This is the basis for the experimental determination of the APBE.

0927-0256/99/$ ± see front matter Ó 1999 Elsevier Science B.V All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 9 8 ) 0 0 1 2 0 - 7

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On the theoretical side, many studies have been performed to date. Possibly the earliest theoretical description by means of the Bragg±Williams method was performed by Brown in 1959 [4]. About two decades later, Kikuchi and Cahn [5] applied the more rigorous cluster variation method (CVM) [6,7] using empirical parameters that represented Cu±Au alloys. A similar CVM calculation was performed for Ni3 Al by Wu et al. [8]. More recently, wetting phenomena at APB were studied by Finel and his collaborators using both CVM and Monte Carlo approaches [9,10]. Here, we build on the work by Kikuchi and Cahn [5] and Wu et al. [8]. However, here ®rstprinciples e€ective interatomic interactions have been used, which was not possible at the time Kikuchi and Cahn did their study. Moreover, we use the tetrahedron±octahedron (TO) maximal cluster approximation rather than the tetrahedron (T) maximal cluster approximation in the CVM because Finel [9] has shown that in contrast to the T approximation, the TO approximation faithfully reproduces the features of the in principle exact Monte Carlo method. Here we examine two types of APBs, in keeping with general terminology in the literature [1] we shall call them: (1) ``thermal'' APBs and, (2) ``athermal'' APBs. The term ``thermal'' APBs shall refer to APBs which have had sucient time to develop a composition pro®le. As is well known from the Gibbs absorption equation [11], defects can attract or repel atomic species, and for a planar defect, such as an APB, the concentration is in general a function of the distance to the APB plane and the ``bulk'' concentration is reached asymptotically. ``A-thermal'' APBs shall refer to those cases where the di€usion is too slow for a longer ranged composition pro®le to develop. This occurs at low temperature and with rapid dislocation movement. We shall represent these APBs computationally by imposing that the composition at every distance from the APB plane is equal to the bulk composition. As there is much controversy regarding the effect of bulk composition on the (1 1 1) APBE in Ni3 Al [12], we shall examine in detail the e€ect of composition and temperature on the APBE.

2. Relaxation and vibrational e€ects Theoretically, various contributions to the APBE can be recognized: 1. most important is the con®gurational internal energy contribution from the ``wrong'' bonds associated with an APB. This contribution is reduced by several other terms corresponding to: 2. relaxation of order, i.e., partial (dis) ordering near the APB, 3. in ``thermal'' APBs only, segregation near the APB, i.e., the formation of the composition pro®le, and 4. a relaxation of atomic positions in the vicinity of the APB. 5. a vibrational free energy due to APB local modes can a€ect the APBE. The contributions from (1) and (4) have been computed with density functional electronic structure methods [13±15]. In those calculations, e€ects (2), (3) and (5) are usually neglected, and only APB at zero temperature are treated. Considering that the terms (2) and (3) reduce the APBE, it follows that upperbounds are to be expected. The e€ect from the relaxation of the atomic positions (4) can be computed exactly however, and it has been shown to reduce the APBE computed from (1) by about 10±20% in a variety of intermetallic compounds [13±16]. The e€ect of the vibrational free energy is not known. Schoeck and Korner [17] gave an expression for an estimate of the vibrational entropy in terms of a ratio of bond frequencies. Here, the vibrational free energy is computed from the vibrational density of states (VDOS) associated with a (1 1 1) APB in the harmonic approximation. The VDOS of two supercells, one with and another without an APB, have been computed using ®rst- and second-nearest-neighbor force constants, see Table 1. The supercell consists of 30 (1 1 1) planes with or without APBs on planes 15 and 30. The di€erence between p the  two supercells is just an APB surface area of 2 3a2 , and thus the VDOS of both supercells are similar, see Fig. 1. The VDOS of the perfect Ll2 supercell agrees reasonably well with the one computed by Foiles and Daw [16]. It is remarkable that the most signi®cant contributions to the APB VDOS occur at

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285

Table 1 Force constant matrices for Ni3 Al in THz2 amu, the lattice parameter is 0.36 nm Element

Vector

xx

xy

zz

yx

Ni±Ni

(1/2 1/2 0) (1 0 0)

11 371 ÿ831

11 598 0

ÿ226 388

11 598 0

Al±Ni

(1/2 1/2 0) (1 0 0)

10 228 30

10 907 0

ÿ679 755

10 907 0

Al±Al

(1/2 1/2 0) (1 0 0)

6458 891

6658 0

ÿ1574 1123

6658 0

The Ni±Ni and Al±Ni (1/2 1/2 0), and Al±Al (1 0 0) values are from Ref. [18], the Al±Al (1/2 1/2 0) value is from a pseudo-potential calculation in Ref. [19], and the Al±Ni (1 0 0) force constants are the average of the value for Al±Al and Ni±Ni.

where N is the number of atoms, 0 is the ``zeropoint'' energy, T is the temperature, V is the volume, x is the frequency and n is the VDOS. The APB vibrational free energy, given by the di€erence of the free energies of the supercell with and without. APB, is well described by the entropy term alone, F vib  ÿTS vib where S vib  ÿ0:004 mJ/ m2 K. This means that the vibrational contribution increases the (1 1 1) APBE, but, as we shall see below, it is small compared to the con®gurational contributions (1), (2), and (3). Therefore, in this paper we will focus on the con®gurational contributions to the APBE. It should be mentioned however, that the F vib is strongly dependent on the force constants for those bonds which do not occur in the perfect Ni3 Al structure, such as nearest neighbor Al±Al bonds. Fig. 1. VDOS for a supercell with (solid) and without (dashed) an (1 1 1) APB as described in the text, normalized to 180 states (60 atoms). The di€erence of the supercells gives the VDOS of a (1 1 1) APB with area a2 sqrt 3 (chain dashed). 1.3 ´ 107 eigenvalues were used for the VDOS computation.

rather high frequency, that is, from the optical branches. The negligible contribution from the acoustic branches implies that the elastic constants are not usually a good indicator of VDOS di€erences and of vibrational free energy di€erences. The supercell VDOS were used to compute the vibrational free energy F using Z  dx log …1 ÿ eÿhx=T †n…x†x; …1† F ˆ N0 ‡ TV

3. Con®gurational e€ects The con®gurational e€ects of an APB can be computed at three levels. · Geometrical APBs [17]: The only e€ect considered is the ``incorrect'' bonds associated with an APB increase the con®gurational internal energy U conf . · A-thermal APBs: The e€ect of local disordering is considered, but the longer-ranged di€usion required for segregation in the vicinity of the APB is not. Local disordering reduces the con®gurational free energy F conf by increasing the entropy S conf while increasing the internal energy only a little.

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· Thermal APBs: The e€ect of local (dis)ordering as well as segregation near the APB is considered. This reduces the con®gurational grand potential Xconf through an entropy term as well as a chemical energy term (lAl ÿ lNi )C, where l is the chemical potential and C is the Aluminum concentration. The APBE for geometrical APBs contains internal energy contributions only and thus can be evaluated easily by counting the number of bonds, U …1† ˆ 16 APBE0 C 2 ; U …1† ˆ

0 6 C 6 0:25;

16 2 APBE0 …1 ÿ C† ; 9

0:25 6 C 6 1;

…2† …3†

where APBE0 is the APBE at stoichiometry at zero temperature. Closed form expressions for thermal and athermal APBEs are unknown, and below the methodology and basis approximations involved in the numerical calculations will be outlined. The con®gurational contributions from partial disordering and segregation are computed with the CVM in the TO approximation. The con®gurational internal energy is computed from ®rstprinciples using a modi®ed Connolly±Williams method [20,21]. Total energies of 18 ordered structures as computed with the linear mun±tin orbital method in the atomic sphere approximation (LMTO±ASA) were used to compute the 11 e€ective cluster interactions (ECI) associated with the TO approximation in the CVM, see Table 2. This cluster expansion predicted the formation energy of unknown ordered structures with an accuracy of about 6 mRyd/atom, which is less than 9% of the range of the formation energies. The expansion gives a Ni3 Al order±disorder temperature of 2200 K, in reasonable accord with the 1750 K estimated from extrapolation of ternary phase diagrams [22,23]. The (1 1 1) APBE is stoichiometric Ni3 Al at 0 K gives 300 mJ/m2 . Electronic structure calculations for positionally relaxed APBEs have given rather lower values: 141 [24], 142 [25], 156 [16], 220 [26], 175 [15], 170 [27], with as exception an unrelaxed KKR±CPA±GPM calculation: 283 [28] which compares favorably with the present result. Experimental results [29,1] are generally a bit

Table 2 Formation energies of non-magnetic Ni±Al compounds with an underlying fcc lattice parameter a ®xed at 0.362 nm as computed with the LMTO±ASA Structure

C (a/o)

Eform (mRyd/atom)

fcc C2/m L12 DO22 C11b C2/m Z3 L10 K40 L11 Z2 C11b C2/m Z3 L12 DO22 C2/m fcc

0.0 0.167 0.25 0.25 0.333 0.333 0.333 0.5 0.5 0.5 0.5 0.667 0.667 0.667 0.75 0.75 0.833 1.0

0 ÿ35.72 ÿ54.40 ÿ51.57 ÿ56.90 ÿ57.55 ÿ47.10 ÿ71.52 ÿ72.47 ÿ52.15 ÿ31.35 ÿ55.19 ÿ54.72 ÿ41.17 ÿ40.72 ÿ40.18 ÿ27.07 0

Structures are named as in Ref. [21], C is the Aluminum concentration.

higher than the previously mentioned relaxed electronic structure results, which is surprising considering that partial disordering and segregation are neglected in the electronic structure calculations. Taking into account that positional relaxation should reduce the APBE computed here by some 20% as was mentioned earlier, it appears that our APBE result agrees favorably with experiment. The reasonable values for the predictive error, the order±disorder temperature, and the APBE at 0 K, verify that the current cluster expansion for the con®gurational energy is adequate. In order to model the spatial aspects of a (1 1 1) APB a large supercell with many parallel (1 1 1) planes is required. In many supercell calculations separate calculations are performed for the cell with and without the defect, and the in¯uence of the defect is obtained as the di€erences of the two calculations [30]. Here, a slight variation will be employed: as planes far removed from the APB strongly resemble the bulk, the bulk thermodynamic properties are extracted from those distant planes. This makes it possible to extract the APB thermodynamic excess properties from a single

M. Sluiter et al. / Computational Materials Science 14 (1999) 283±290

supercell calculation. By calculating properties per (1 1 1) plane, the convergence towards bulk values, and the excesses due to the APB are easily examined. Thermodynamic properties such as internal energy and entropy are site-wise decomposed, here for site i, as follows: X Ja na pa …i†; …4† U …i† ˆ a

S…i† ˆ ÿkB

X a

ca Sa pa …i†;

…5†

where a represents a cluster (correlation function), J is an ECI, n is a correlation function, c is a Kikuchi±Barker coecient, and Sa is the entropy contribution from cluster a. The contribution pa (i) of cluster a to site i is given by P j2a dij ; …6† pa …i† ˆ P j2a 1 where j refers to a site which is part of cluster a, and d is the Kronecker symbol. Properties per plane are obtained by summing over all sites within that plane. The excess properties DQ associated with the APB can be spatially resolved by subtracting, for each plane x in the supercell, the value of the property in the bulk,

DQ…x† ˆ Q…x† ÿ Qbulk

287

…7†

and the sum Rx DQ(x) represents the excess property DQAPB tot associated with the APB. The APBE, for example, is given by Rx DX(x) for thermal APBs, and by Rx DF(x) for a-thermal APBs. Calculations were performed with a monoclinic supercell with translation vectors á1, ÿ1, 0ñ, á1, 0, ÿ1ñ, and á20.5, 19.5, 20ñ. This cell contains 60 (1 1 1) planes, so that e€ects up to 30 planes removed from the center of the APB can be evaluated. It was found that in all cases the most distant planes were converged to the bulk values. In the TO approximation, this supercell has 3735 correlation functions (includes the empty cluster). Calculations were made feasible by taking advantage of the sparsity of the con®guration matrix [7], and the Hessian matrix [31]. Calculation for a-thermal APBs are carried out by imposing the constraint that the composition for each plane remain ®xed at the bulk value. Thermal APBs were computed by either ®xing the concentration of the total supercell, or by ®xing the chemical potential. Some representative results for an alloy with a bulk Al concentration of 24.25% at 1600 K will be discussed below. Concentration pro®les are shown

Fig. 2. Aluminum concentration per sublattice () and average Al concentration as a function of distance to the center of the APB. (a) A-thermal APB, (b) thermal APB.

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in Fig. 2, and Fig. 3 shows how the contribution to the APBE is spatially distributed for thermal and a-thermal APBs. In the lowest approximation, geometrical APBs, DUtot ˆ 282 mJ/m2 , resulting in an APBE of 282 mJ/m2 . Relaxing the local order while preserving the planar concentration, a-thermal APBs, increases the DUtot to 346 mJ/m2 , but the concomitant entropy production of DStot ˆ 0.101 mJ/m2 K more than o€sets this gain: APBE ˆ 184 mJ/m2 . Relaxation of the composition pro®le, thermal APBs, much increases DUtot to 1475 mJ/m2 , but DStot is almost doubled to 0.194 mJ/m2 K and the chemical energy term lC ˆ 1020 mJ/m2 brings the APBE to the still lower value of 145 mJ/m2 . Clearly, relaxation of the local order reduces the APBE the most. This result indicates that those electronic structure calculations that attempt to describe ®nite temperature APB energies with idealized, perfectly ordered semi-crystals, should give too large values [13,14]. Ironically, most electronic structure calculations for (1 1 1) APBEs in Ni3 Al have given rather low values: 141 [24], 142 [25], 156 [16], 220 [26], 175 [15], 170 [27], with as exception an unrelaxed KKR±CPA±GPM result 283 [28] which compares favorably with our result.

Fig. 3. Plane decomposed contributions to the APBE; + athermal APB (DF), ´ thermal APB (DX).

Segregation too, lowers the APBE and this leads to a pinning e€ect [32±34]. The concentration pro®le shown in Fig. 2(b) indicates that the concentration in the center of a thermal APB can signi®cantly deviate from the bulk value, as was also found by Wu et al. [8]. The two central (1 1 1) planes have an Al concentration of 21.27% which is ®ve times as far from stoichiometry as the bulk concentration. The general result is that the concentration at the APB is enriched in the component that is in excess: an alloy with 75.75% Ni in the bulk is Ni-rich and a (1 1 1) APB attracts some of the excess Ni. The same occurs in Al-rich alloys, although slightly less pronounced. Computations reveal that in alloys with about 25.2% Al almost no segregation occurs. Away from stoichiometry up to 10 lmol/m2 of the excess Ni or Al can be absorbed at a (1 1 1) APB. The full width at half maximum of the XAPB (x) or FAPB (x) curve (Fig. 3) indicates pa width of about 6 (1 1 1) planes, which is 2 3a, or about 1.2 nm. Similarly, the width can be obtained from the planar decomposition of the excess entropy or excess internal energy. In Fig. 4 the APBE is shown for all three cases. The geometrical APBE is of course independent of temperature, but the more realistically modeled thermal and a-thermal APBs are strongly temperature dependent. Previous calculations, where thermal APBs only were considered, too revealed a strong temperature dependence [4,5,35,8], and it is experimentally well established also [1,36,37]. Clearly, the APB con®gurational entropy cannot be ignored. At higher temperatures a value of 0.2 mJ/m2 K is found which is more than an order of magnitude larger than our estimate for the APB vibrational entropy (which has a di€erent sign as well). The e€ect of o€-stoichiometry is strong, both in thermal and a-thermal APBs. Segregation in thermal APBs enhances the in¯uence of o€-stoichiometry. We ®nd that the e€ect of o€-stoichiometry is not the same for excess Ni or excess Al. Experimentally, there is a controversy regarding the e€ect of o€-stoichiometry: Douin and Veyssiere ®nd that the APBE is lowered by excess Ni, but increased by excess Al [38]; Dimiduk et al. Find that the APBE is nominally composition inde-

M. Sluiter et al. / Computational Materials Science 14 (1999) 283±290

Fig. 4. APBE as a function of Al concentration in the bulk. Geometrical APB, solid; a-thermal APB, dashed; thermal APB, chain-dashed; experiment (see text), .

pendent [12]; Yu et al. [39] ®nd that for (1 0 0) APBs only, the APBE is much lowered by excess Al, in marked contrast to Dimiduk et al. [12]. However, as is clear from compilations of APBEs [29,1] there is a large spread in the experimental values even for alloys at exactly the same composition. It should be noted that in actual alloys the composition range of the Ni3 Al phase is limited to a narrow range around stoichiometry that appears to decrease with increasing temperature. The single phase region is probably less than 1 a/o wide at elevated temperatures. This might be one reason that experimental determination of the composition dependence is non trivial. However, when experimental data from binary Ni±Al alloys [40,41,29,12] and substitutional ternary Ni±Al±X alloys (X ˆ Ti [42], Hf [43], V [40], Ta [44], Sn [40]) alloys is combined a trend emerges that has not been apparent in previous work, see the black bullets in Fig. 4. As is known from theoretical and experimental work [45] the X elements all occupy the Al sublattice in Ni3 Al, so that we summed the concentrations of Al and X for the ternary cases. Of course, this is a rather crude procedure that does not properly account for the presence of the ternary species. If two APBE values of 250 mJ/m2

289

are excluded [46,12], a weak trend appears which indicates a maximum APBE at stoichiometry, just as has been computed theoretically. Nevertheless, the decrease of the APBE with o€-stoichiometry is much less than has been computed theoretically. The theoretical values for the APBE might be reduced by 10±20% as mentioned earlier, but even without that correction, the comparison with experiment is rather good. Fig. 4 shows that the di€erence between thermal and a-thermal APBEs is an almost linear function of the deviation from stoichiometry, at stoichiometry the di€erence vanishes. Somewhat surprisingly, at a ®xed deviation from stoichiometry, i.e. composition, the di€erence is not sensitive to temperature. Clearly, it persists up to temperatures close to the order±disorder temperature. This means that the pinning e€ect is not a function of temperature, but is a sole function of composition. Of course, at high temperature a-thermal APBs rapidly transform to thermal APBs [34,36]. 4. Conclusion A calculation of the vibrational density of states of a (1 1 1) APB revealed that the most signi®cant contributions arise from the optic braches, and that the acoustic contributions are negligible. The APB vibrational free energy is well represented by the entropy term alone, but that vibrational entropy is much smaller and of opposite sign as the con®gurational entropy associated with o€-stoichiometry, partial disorder, and segregation. The con®gurational entropy is so large in fact, that zero temperature APBEs computed at stoichiometry, such as in electronic structure calculations, could be reduced to as little as 50% in actual alloys. This ®nding suggests that electronic structure calculations for perfectly ordered antiphase boundaries do not agree well with experimental data. Generally, the experimental antiphase energies are much higher than one would infer from the computed zero temperature results. The current calculations indicate that segregation at APBs can well explain the energetic di€erences between thermal and a-thermal APBs. The pinning e€ect, which results from this di€erence, is

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found to be a linear function of the deviation from stoichiometry, but is found to be temperature independent. Acknowledgements The authors acknowledge support from the sta€ at the Computer Center at IMR, Tohoku University for help with use of the Hitachi S-3800 supercomputer. References [1] Y.-Q. Sun, in: J.H. Westbrook, R.L. Fleischer (Eds.), Intermetallic Compounds: Principles and Practice, vol. 1, Wiley, New York, 1995, p. 495; and references therein. [2] J.L.C. Daams, P. Villars, J.H.N. van Vucht, Atlas of Crystal Structures for Intermetallic Phases, ASM International, Materials Park, Ohio, 1991. [3] H.P. Karnthaler, E. Muehlbacher, C. Rentenberger, Acta Mater. 44 (1996) 547. [4] N. Brown, Phil. Mag. 4 (1959) 693. [5] R. Kikuchi, J.W. Cahn, Acta Metall. 27 (1979) 1337. [6] R. Kikuchi, Phys. Rev. 81 (1951) 988. [7] D. de Fontaine, Sol. St. Phys. 47 (1994) 33; D. de Fontaine, in: J.L. Moran-Lopez, J.M. Sanchez (Eds.), Theory and Applications of the Cluster Variation and Path Probability Methods, Plenum Press, New York, 1996, p. 125. [8] Y.P. Wu, J.M. Sanchez, J.K. Tien, in: C.T. Liu et al. (Eds.), High Temperature Ordered Intermetallic Alloys III, Mater. Res. Soc. Proc. 133, 1989, p. 119. [9] A. Finel, in: P.E.A. Turchi, A. Gonis (Eds.), Statics and Dynamics of Alloy Phase Transformations, NATO ASI Series B: Physics, vol. 319, Plenum Press, New York, 1993, p. 516. [10] A. Finel, V. Mazauric, F. Ducastelle, Phys. Rev. Lett. 65 (1990) 1016. [11] A.P. Sutton, R.W. Balu, Interfaces in Crystalline Materials, Monographs on the Physics and Chemistry of Materials, Clarendon Press, Oxford, UK, 1995. [12] D.M. Dimiduk, A.W. Thompson, J.C. Williams, Phil. Mag. A 67 (1993) 675. [13] C.L. Fu, M.H. Yoo, Acta Metall. Mater 40 (1992) 703. [14] C.L. Fu, Phys. Rev. B 52 (1995) 3151. [15] C.L. Fu, Y.-Y. Ye, M.H. Yoo, in: I. Baker et al. (Eds.), High Temperature Ordered intermetallic Alloys V, Mater. Res. Soc. Proc. 288, 1993, p. 21. [16] S.M. Foiles, M.S. Daw, J. Mater. Res. 2 (1987) 5. [17] G. Schoeck, A. Korner, Phil. Mag. A 61 (1990) 909. [18] B. Fultz, L. Anthony, L.J. Nagel, R.M. Nicklow, S. Spooner, Phys. Rev. B 52 (1995) 3315.

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