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Theoretical approach of phase selection in refractory metals and alloys
Journal of Alloys and Compounds 334 (2002) 27–33
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Theoretical approach of phase selection in refractory metals and alloys a b c, C. Berne , M. Sluiter , A. Pasturel * a
´ Commissariat a` l’ Energie Atomique, Direction de la Recherche Technologique, Departement des Techniques des Energies Nouvelles, 17 Rue des Martyrs, 38054 Grenoble-Cedex 09, France b Institute for Materials Research, Tohoku University, Sendai 980 -8577, Japan c ´ ´ , Maison des Magisteres ` Laboratoire de Physique et Modelisation des Milieux Condenses BP 166 CNRS, 38042 Grenoble-Cedex 09, France Received 17 June 2001; accepted 13 July 2001
Abstract First-principles total energy calculations are used to examine the phase stability of the Frank–Kasper (FK) structures. The sequence of FK phases observed as a function of composition in many transition metal binary systems is shown to have a band-filling origin. We show that the alloying effects can be treated using a first-principles statistical thermodynamics approach and first results are given for Re–Ta and Re–W sigma alloys. 2002 Elsevier Science B.V. All rights reserved. Keywords: Transition metal compounds; Crystal structure; Thermodynamic modelling
1. Introduction To study the problem of phase formation in alloys, one has to understand what kind of crystalline structure and atomic configuration an alloy may probably exhibit at a given composition and temperature. This is the purpose of theoretical studies which deal with the prediction of alloy phase diagrams. Until recently, most theories were based on phenomenological models. The stability of many structures has been discussed using geometrical arguments based on space filling and size effects. Chemical or electronic effects have been also used. For instance, the Hume–Rothery rules relate the stability of several structures to the number of valence electrons per atom. Finally, some electronegativity scales have been introduced. To study the effects of composition and temperature, most approaches have been based on phenomenological thermodynamic models using regular solution models or various improvements on them. Recent research shows that it is now possible to deal with these problems from a microscopic theory of the cohesive properties of alloys. This microscopic theory is based on first-principles total energy calculations and on the development of efficient thermodynamic tools [1,2]. One can easily understand that the use of methods starting ¨ from the Schrodinger equation and from general thermo*Corresponding author. E-mail address: [email protected] (A. Pasturel).
dynamics depends on the complexity of alloys systems; this explains why alloys based upon single underlying crystalline structures such as bcc and fcc, for which a large variety of ordered states can exist, have received much attention in recent years. On the contrary few studies have been undertaken on fundamental grounds to treat alloys based on complex crystallographic arrangements as in the case of the socalled Frank–Kasper phases (FK) [3]. For transition metal based alloys, the starting point is the phenomenological approach formulated by Watson and Bennett [4]; in this approach, it is shown that the average filling of the d band is important in establishing which alloys form FK phases except the Laves phases for which the high degree of atomic order is usually the result of a dominant size factor. Previous theoretical works based on tight-binding arguments [5] suggest that electronic effects and more particularly the d-band filling are crucial to explain the stability of the A15-based structure. To study ordering effects in A15-based alloys, Turchi and Finel [6] propose to couple tight-binding calculations in the context of the generalized perturbation method (GPM) [7] with a thermodynamic variational method such as the cluster variation method (CVM) [8,9]. To address the problem of FK phases stability using a first-principles statistical thermodynamics approach, we propose a twofold contribution. In a first step, total energy calculations of several transition metals are performed to analyze how different
0925-8388 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0925-8388( 01 )01773-X
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C. Berne et al. / Journal of Alloys and Compounds 334 (2002) 27 – 33
FK structures, namely A15, sigma and chi, compete with simple structures such as bcc, fcc and hcp as a function of the band filling; more particularly, these calculations allow us to provide an energetic scale or a degree of metastability for these structures which are not observed in elemental metals but occur frequently in refractory transition metal based alloys [10]. In a second step, the alloying effects are discussed in terms of the sequence of occurrence of these FK phases as a function of composition. We will also show how we can use total energy calculations to study ordering–disordering phenomena; we will focus our attention on the sigma phase which displays different composition limits according to the studied systems. In the following pages, we briefly present the firstprinciples based method (Sec. I) used to determine the structural stability in pure transition metals. Then in Sec. II, we discuss the phase stability properties of pure transition metals for A15, sigma, chi, fcc, bcc and hcp structures. Sec. III is devoted to alloying effects and more particularly we will emphasize how first-principles calculations can be used to tackle ordering–disordering phenomena in complex structures.
2. General method
2.1. Frank–Kasper structures It is well known that the FK phases are built entirely out of tetrahedral packing units [3]. In contrast, the fcc structure consists of both tetrahedral and octahedral units while the bcc structure is an intermediate case with the octahedra containing second-neighbor bonds. The tetrahedral packing in FK phases leads to characteristic polyhedra which are labeled CN12, CN14, CN15, and CN16, where the numbers refer to the coordination number of the atom centering the polyhedron. The three A15, sigma and chi structures contain all the characteristic polyhedra of the FK phases. The simplest example of FK phases is the cubic A15 structure since it has eight atoms per unit cell. Two of these are surrounded by CN12 polyhedra, which are icosahedra, and the remaining 6 by CN14 polyhedra. The sigma phase is more complex since it contains 30 atoms and in this tetragonal structure, there are 5 crystallographically different sites, commonly designated by A, B, C, D, and E. The coordination shells around the various sites are either icosahedral (around A and D sites) or are of CN14 (around C and E sites) or CN15 (around B site) type. Finally the chi structure contains 58 atoms distributed on four crystallographically different sites designated by A, B, C and D. The coordination shell around the D site is icosahedral whereas the A and B sites are surrounded by CN16 polyhedra; the remaining C sites have a coordination shell made up of 13 neighbors which are not tetrahedrally
coordinated. Strictly speaking, therefore, this structure should not be considered as a member of the FK structures. However, studies on alloy chemistry and ordering indicate that the same general principles hold for chi- and sigmarelated structures.
2.2. Total energy calculations To compute total energies of several 4d and 5d transition metals, we used the Vienna ab initio simulation program (VASP) [11] which is based on the following principles. A finite-temperature density-functional approximation is used to solve the generalized Kohn–Sham equations with an efficient iterative matrix diagonalization scheme based on a conjugate gradient technique and optimized density mixing routines. Within this framework, the free energy is the variational functional and a fractional occupancy of the eigenstates is allowed which eliminates all instabilities resulting from level crossing and quasidegeneracies in the vicinity of the Fermi level in metallic systems. The exchange–correlation functional given by Ceperley and Alder is used in the parametrization of Perdew and Zunger [12]. Non-local exchange-correlation effects are considered in the form of generalized-gradient corrections (GGCs). We use the Perdew–Wang functional [13]. The GGCs are applied self-consistently in the construction of the pseudopotentials as well as in the calculations of the Kohn–Sham ground state. The calculations are performed using fully non-local optimized ultrasoft pseudopotentials [14] allowing small energy cut-offs. In this work, particular care was taken for the total-energy convergence with regard to the k-space integration. The Methfessel– Paxton k points technique [15] with a smearing parameter of 0.10 eV was used. The number of irreducible k points was 182, 190, 150, 56, 18 and 20 for fcc, bcc, hcp, A15, sigma and chi respectively. All the parameters of the different studied unit cells as well as the possible cell internal coordinates were optimized using a conjugate gradient method [11].
3. Stability in pure transition metals The total energies were calculated for bcc, fcc, hcp, A15, sigma and chi structures and for Nb, Mo, Tc, Ru, Ta, W, Re, Os metals. As already discussed, the three A15, sigma and chi structures contain all the characteristic polyhedra of the tetrahedral packing with a percentage of icosahedral sites which is 25, 33, and 41% for A15, sigma and chi respectively. Results are illustrated in Fig. 1. The first spectacular result is the relative stability of the FK phases for valence electron per atom ratios (e /a) ranging from 4.5 to 6.5. More particularly, the A15 structure is found to strongly compete with the bcc structure for Nb and Mo metals as
C. Berne et al. / Journal of Alloys and Compounds 334 (2002) 27 – 33
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deviations from such symmetry are small enough to expect that to a first approximation the bonding of CN12 atoms is the same as the one which would exist under exact symmetry of the icosahedron point group. Thus it appears that the icosahedral sites show a predominance for atoms with filled or nearly filled, or else empty d shells. Atoms with d shells close to half filled are mostly found in coordinations of lower symmetry where the d orbitals may hybridize to form directed bonds. Therefore transition metals like Nb, Mo, Ta or W prefer occupying CN15 and CN16 positions which are of lower symmetry whereas metals like Re or Os tend to occupy the CN12 position. A mixture of both elements may occur on CN14 (CN13) positions and this will be discussed in Sec. III.
4. Alloying effects in Frank–Kasper phases
4.1. Sequence of the Frank–Kasper phases
Fig. 1. Structural energy differences for (a) 4d metals and (b) for 5d metals.
well as for Ta and W metals. Let us mention that this peculiar behavior of the A15 phase has been used to interpret the occurrence of a metastable phase in Ta metal brought on by drop-tube undercooling [16]. The electron concentration found in our calculations is very similar to the one obtained by Phillips and Carlsson [17] or by Watson and Bennett [4]. However, our study emphasizes that, in this range, the FK structures strongly compete with the bcc and not with the fcc as assumed by Phillips and Carlsson [17]. The second important result of the present calculations is that all of the lines in Fig. 1 intersect at one particular band filling, namely e /a56.7. As a consequence, the sequence A15-sigma-chi also depends on the band filling. As these three FK phases display a different percentage of icosahedral sites, i.e. CN12, it is tempting to relate their relative structural stability with this percentage. Indeed perturbations conforming to the point group of the icosahedron do not affect the degeneracy of the atomic d levels whereas for all of the other coordination symmetries the d-level degeneracy is partly broken. Actually, the surroundings of CN12 atoms do not exactly conform to the point group of the icosahedron. However, the effective
Results presented in Sec. II can be used to understand the sequence of FK phases in transition metal based alloys. For instance, if a chi phase exists at a given composition in an alloy, a composition change which reduces the e /a ratio could possibly give rise to a new phase such as a sigma phase, an A15 phase, or a bcc solid solution, but not an hcp phase. Reciprocally, if the e /a ratio increases an hcp phase may occur. If the same arguments are applied to a sigma phase a decreasing e /a may lead to the occurrence of A15 or bcc phases but not to chi or hcp phases. Thus, with increasing the electron concentration, the following succession of phases may exist: bcc-A15-sigma-chi-hcpfcc. They are many binary alloys of 4d and 5d transition elements in which these sequences can be checked [10]. For instance, the sequence bcc-sigma-chi-hcp can be found in the Ta–Os, Ta–Re, W–Re, Nb–Re, and Mo–Re systems. The sequence bcc-A15-sigma-hcp occurs in the Mo– Os, Mo–Tc, and Nb–Rh systems. In the Nb–Os system the complete sequence bcc-A15-sigma-chi-hcp occurs. However, if this approach provides a nice interpretation of the sequence of the FK phases as a function of electron concentration, it does not predict their actual occurrence for a given alloy. To try to answer this question, it is necessary to treat alloying effects in these structures. Another difficulty arises from the fact that some FK phases like sigma or chi do not have a definite stoichiometry. Hence their occurrence varies with composition and thus with electron concentration. We present now a first-principles based treatment of the alloying effects in the sigma phase.
4.2. Alloying mechanism in the sigma phase When dealing with the general problem of phase stability in an alloy from a microscopic description, one has to
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combine electronic structure calculations and statistical studies. This key may be played by an Ising Hamiltonian for which the energetics associated with changing local atomic configurations is parametrized by a set of effective cluster interactions (ECI’s). Total energies of a large number of configurations are used to extract these effective cluster interactions following the Connolly–Williams method (CWM) [18]. The ECI Vi for a cluster i is calculated with:
Ov SE a
a
OV j D 5 min n
a form
2
2
i
a i
(1)
i 51
where j is the cluster correlation function as defined in Eq. (10) of Ref. [19]; v represents the weights assigned to each structure. Formation energies Eform are extracted from the total energies Etot by substracting the concentration weighted total energies of pure constituents with the lattice structure. Eq. (1) is solved using a singular value decomposition (SVD) procedure. As in the spirit of Ref. [20], the sets of equations corresponding to thermodynamically stable structure are given more weight than the equations pertaining to unstable structures. The weighting in the SVD procedure provides an alternative to the linear optimization technique for ensuring that the cluster expansion reproduces the proper sequence of structures energy. An additional benefit of weighting the lowest energy states more heavily in the CWM method is that it makes the CE convergence more rapid. Let us consider the case of the sigma phase. For a AB-sigma alloy, the five inequivalent sites of the sigma phase make 2 5 532 possible distributions of A and B atoms. These 32 configurations are not superstructures of the sigma phase because they all have the same space group symmetry. In addition, 10 sigma superstructures were considered that allowed us to determine the pair interactions between sites of the same type (i.e. B–B, C–C, D–D, and E–E type pairs). The formation energies of these 42 configurations are computed with a full geometry relaxation including both the optimization of the a and c parameters of the tetragonal unit cell and the optimization of the internal atomic coordinates. The SVD algorithm is used to extract effective interactions [21]. A set of 29 and 27 interactions was shown to reproduce the formation energies with a root mean square error of 0.06 kJ / atom for systems like the Re–W and the Re–Ta respectively. This error is well within the precision of the total energy calculations and is much smaller than the smallest difference between the formation energies of any of the 32 configurations used in the CWM (see Figs. 2–3). The interactions are used to compute the configurational entropy and the associated Helmholtz free energy as a function of composition and temperature within the CVM. The site occupancy of the 5 inequivalent sites are also determined in these calculations. In the present case, the
Fig. 2. Formation energy (in kJ / atom) of Re–W relaxed atomic configurations with the sigma structure: as computed with VASP (1); as computed from the effective interactions (1). Circles emphasize the 10 configurations that are sigma superstructures. The most stable configurations have been connected with a solid line and the occupancy of the A, B, C, D, and E sites have been indicated.
tetrahedron approximation is a natural choice since the sigma phase is tetrahedrally close packed. For this structure, the tetrahedron approximation leads to 17 tetrahedron maximal clusters and a total of 71 correlation functions [22] Figs. 4–5 show the site preference for two systems, namely Re–Ta and Re–W systems, at T5500 K. The first comment is that the site occupancy as a function of W or Ta composition has the same sequence for both systems. First the B site happens to be filled with W (or Ta) and takes most W (or Ta) content; then come sites C and E, and finally the A and D sites are filled. Results of Sec. II allow us to understand the clear preference for W or Ta to
Fig. 3. Formation energy (in kJ / atom) of Re–Ta relaxed atomic configurations with the sigma structure: as computed with VASP (1); as computed from the effective interactions (1). Circles emphasize the 10 configurations that are sigma superstructures. The most stable configurations have been connected with a solid line and the occupancy of the A, B, C, D, and E sites have been indicated.
C. Berne et al. / Journal of Alloys and Compounds 334 (2002) 27 – 33
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Fig. 4. Computed occupancy of the inequivalent sites in the Re–W sigma phase at T5500 K.
occupy the lower symmetry B, C and E sites and for Re to occupy the icosahedral A and D sites. As discussed in Ref. [21], more subtle alloying effects are necessary to understand the sequence between the sites of lower symmetry, namely B, C and E sites. For instance, another E–C–B sequence was found in the Fe–Cr sigma alloy [22]. In Figs. 6–7, the site occupancy of both Re 0.6 W0.4 and Re 0.6 Ta 0.4 alloys as a function of temperature is shown. At high temperatures, the site occupancy very slowly reaches the value corresponding to the random composition. This limit occupancy being determined purely by the alloy composition, in this case the limit of each site is 60% Re. However, this limit is not obtained because no order– disorder transformation due to the removal of symmetry elements in going to the chemically randomized configuration takes place in the present calculations. The occupancy
of B, A and D sites reaches the high temperature limit more rapidly in Re–W than in Re–Ta, due to weaker alloying effects in Re–W system. In this case, the temperature effects on the ordering mechanism are more pronounced. When the temperature is lowered, the fraction of W (Ta) on B site increases to reach an occupancy of 100% around 500 K for Re–W and 2000 K for Re–Ta. For the same temperatures, the A and D sites display an opposite behavior which gives a Re occupancy of 100%. For both alloys, the C and E sites have an important mixed occupancy which is more or less independent of the temperature. These results may be used to predict composition limits for the sigma alloys. At very low temperature, we can assume that only Re occupy A and D sites and only W (or Ta) B site and a mixture of Re and W (or Ta) C and E
Fig. 5. Computed occupancy of the inequivalent sites in the Re–Ta sigma phase at T5500 K.
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C. Berne et al. / Journal of Alloys and Compounds 334 (2002) 27 – 33
Fig. 6. Computed occupancy of the inequivalent sites in the Re 0.6 W0.4 sigma phase as a function of temperature.
Fig. 7. Computed occupancy of the inequivalent sites in the Re 0.6 Ta 0.4 sigma phase as a function of temperature.
sites; thus the predicted limits are 13–66% W (or Ta) taking into account the multiplicity of the crystallographic sites. In fact this approach provides an over-limit because the true composition limits of the sigma phase are given by the competition of the other equilibrium phases and more particularly the bcc solid solution.
when increasing electron concentration which is found in the binary metallic systems based on refractory metals. The alloying effects are discussed more particularly in the sigma phase. It has been shown that the site occupation in such a complex structure displays a peculiar behavior as a function of composition and / or temperature which can be explained from the site-coordination numbers.
5. Conclusion We have shown that the structural stability among the FK phases depends on the band filling of pure transition metals. This behavior is related to the occurrence of high local coordination sites which are more favorable for metals with d-shells close to half-filled. These results are also used to interpret the sequence bcc-A15-sigma-chi-hcp
References [1] D. de Fontaine, Solid State Phys. 34 (1979) 73. [2] F. Ducastelle, in: F.R. de Boer, D.G. Pettifor (Eds.), Order and Phase Stability in Alloys, Cohesion and Structure, Vol. 3, North Holland, Amsterdam, 1991. [3] A.K. Sinha, Prog. Mater. Sci. 15 (1973) 79.
C. Berne et al. / Journal of Alloys and Compounds 334 (2002) 27 – 33 [4] [5] [6] [7] [8] [9] [10]
[11] [12] [13] [14]
R.E. Watson, L.H. Bennett, Acta Metall. 32 (1984) 477. P.E.A. Turchi, G. Treglia, F. Ducastelle, J. Phys. F 13 (1983) 2543. P.E.A. Turchi, A. Finel, Phys. Rev. B 46 (1992) 702. F. Ducastelle, F. Gautier, J. Phys. F 6 (1976) 2039. R. Kikuchi, Phys. Rev. 81 (1951) 998. J.M. Sanchez, F. Ducastelle, D. Gratias, Physica A128 (1984) 334. T.B. Massalski, H. Okamoto, P.R. Subramanian, L. Kacprzak (Eds.), Binary Alloys Phase Diagrams, American Society of Metals, Materials Park, OH, 1990. G. Kresse, G. Furthmueller, Phys. Rev. B 54 (1996) 11169. J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. J.P. Perdew, Y. Wang, Phys. Rev. B 33 (1986) 8800. G. Kresse, J. Hafner, J. Phys.: Condens. Matter 6 (1994) 8245.
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[15] M. Methfessel, A.T. Paxton, Phys. Rev. B 40 (1989) 3616. [16] L. Cortella, B. Vinet, P.J. Desre, A. Pasturel, A.T. Paxton, M. van Schilgaarde, Phy. Rev. Lett. 70 (1993) 1469. [17] R. Philipps, A.E. Carlsson, Phys. Rev. B 42 (1990) 3345. [18] J.W.D. Connolly, A.R. Williams, Phys. Rev. B 27 (1983) 5169. [19] M. Sluiter, D. de Fontaine, X.Q. Guo, R. Podloucky, A.J. Freeman, Phys. Rev. B 42 (1990) 10460. [20] M. Sluiter, Y. Watanabe, D. de Fontaine, Y. Kawazoe, Phys. Rev. B 53 (1996) 6137. [21] C. Berne, M. Sluiter, A. Pasturel, Phys. Rev. B (2001) in press. [22] M. Sluiter, K. Esfarjani, Y. Kawazoe, Phys. Rev. Lett. 75 (1995) 3142.
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www.elsevier.com / locate / jallcom
Theoretical approach of phase selection in refractory metals and alloys a b c, C. Berne , M. Sluiter , A. Pasturel * a
´ Commissariat a` l’ Energie Atomique, Direction de la Recherche Technologique, Departement des Techniques des Energies Nouvelles, 17 Rue des Martyrs, 38054 Grenoble-Cedex 09, France b Institute for Materials Research, Tohoku University, Sendai 980 -8577, Japan c ´ ´ , Maison des Magisteres ` Laboratoire de Physique et Modelisation des Milieux Condenses BP 166 CNRS, 38042 Grenoble-Cedex 09, France Received 17 June 2001; accepted 13 July 2001
Abstract First-principles total energy calculations are used to examine the phase stability of the Frank–Kasper (FK) structures. The sequence of FK phases observed as a function of composition in many transition metal binary systems is shown to have a band-filling origin. We show that the alloying effects can be treated using a first-principles statistical thermodynamics approach and first results are given for Re–Ta and Re–W sigma alloys. 2002 Elsevier Science B.V. All rights reserved. Keywords: Transition metal compounds; Crystal structure; Thermodynamic modelling
1. Introduction To study the problem of phase formation in alloys, one has to understand what kind of crystalline structure and atomic configuration an alloy may probably exhibit at a given composition and temperature. This is the purpose of theoretical studies which deal with the prediction of alloy phase diagrams. Until recently, most theories were based on phenomenological models. The stability of many structures has been discussed using geometrical arguments based on space filling and size effects. Chemical or electronic effects have been also used. For instance, the Hume–Rothery rules relate the stability of several structures to the number of valence electrons per atom. Finally, some electronegativity scales have been introduced. To study the effects of composition and temperature, most approaches have been based on phenomenological thermodynamic models using regular solution models or various improvements on them. Recent research shows that it is now possible to deal with these problems from a microscopic theory of the cohesive properties of alloys. This microscopic theory is based on first-principles total energy calculations and on the development of efficient thermodynamic tools [1,2]. One can easily understand that the use of methods starting ¨ from the Schrodinger equation and from general thermo*Corresponding author. E-mail address: [email protected] (A. Pasturel).
dynamics depends on the complexity of alloys systems; this explains why alloys based upon single underlying crystalline structures such as bcc and fcc, for which a large variety of ordered states can exist, have received much attention in recent years. On the contrary few studies have been undertaken on fundamental grounds to treat alloys based on complex crystallographic arrangements as in the case of the socalled Frank–Kasper phases (FK) [3]. For transition metal based alloys, the starting point is the phenomenological approach formulated by Watson and Bennett [4]; in this approach, it is shown that the average filling of the d band is important in establishing which alloys form FK phases except the Laves phases for which the high degree of atomic order is usually the result of a dominant size factor. Previous theoretical works based on tight-binding arguments [5] suggest that electronic effects and more particularly the d-band filling are crucial to explain the stability of the A15-based structure. To study ordering effects in A15-based alloys, Turchi and Finel [6] propose to couple tight-binding calculations in the context of the generalized perturbation method (GPM) [7] with a thermodynamic variational method such as the cluster variation method (CVM) [8,9]. To address the problem of FK phases stability using a first-principles statistical thermodynamics approach, we propose a twofold contribution. In a first step, total energy calculations of several transition metals are performed to analyze how different
0925-8388 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0925-8388( 01 )01773-X
28
C. Berne et al. / Journal of Alloys and Compounds 334 (2002) 27 – 33
FK structures, namely A15, sigma and chi, compete with simple structures such as bcc, fcc and hcp as a function of the band filling; more particularly, these calculations allow us to provide an energetic scale or a degree of metastability for these structures which are not observed in elemental metals but occur frequently in refractory transition metal based alloys [10]. In a second step, the alloying effects are discussed in terms of the sequence of occurrence of these FK phases as a function of composition. We will also show how we can use total energy calculations to study ordering–disordering phenomena; we will focus our attention on the sigma phase which displays different composition limits according to the studied systems. In the following pages, we briefly present the firstprinciples based method (Sec. I) used to determine the structural stability in pure transition metals. Then in Sec. II, we discuss the phase stability properties of pure transition metals for A15, sigma, chi, fcc, bcc and hcp structures. Sec. III is devoted to alloying effects and more particularly we will emphasize how first-principles calculations can be used to tackle ordering–disordering phenomena in complex structures.
2. General method
2.1. Frank–Kasper structures It is well known that the FK phases are built entirely out of tetrahedral packing units [3]. In contrast, the fcc structure consists of both tetrahedral and octahedral units while the bcc structure is an intermediate case with the octahedra containing second-neighbor bonds. The tetrahedral packing in FK phases leads to characteristic polyhedra which are labeled CN12, CN14, CN15, and CN16, where the numbers refer to the coordination number of the atom centering the polyhedron. The three A15, sigma and chi structures contain all the characteristic polyhedra of the FK phases. The simplest example of FK phases is the cubic A15 structure since it has eight atoms per unit cell. Two of these are surrounded by CN12 polyhedra, which are icosahedra, and the remaining 6 by CN14 polyhedra. The sigma phase is more complex since it contains 30 atoms and in this tetragonal structure, there are 5 crystallographically different sites, commonly designated by A, B, C, D, and E. The coordination shells around the various sites are either icosahedral (around A and D sites) or are of CN14 (around C and E sites) or CN15 (around B site) type. Finally the chi structure contains 58 atoms distributed on four crystallographically different sites designated by A, B, C and D. The coordination shell around the D site is icosahedral whereas the A and B sites are surrounded by CN16 polyhedra; the remaining C sites have a coordination shell made up of 13 neighbors which are not tetrahedrally
coordinated. Strictly speaking, therefore, this structure should not be considered as a member of the FK structures. However, studies on alloy chemistry and ordering indicate that the same general principles hold for chi- and sigmarelated structures.
2.2. Total energy calculations To compute total energies of several 4d and 5d transition metals, we used the Vienna ab initio simulation program (VASP) [11] which is based on the following principles. A finite-temperature density-functional approximation is used to solve the generalized Kohn–Sham equations with an efficient iterative matrix diagonalization scheme based on a conjugate gradient technique and optimized density mixing routines. Within this framework, the free energy is the variational functional and a fractional occupancy of the eigenstates is allowed which eliminates all instabilities resulting from level crossing and quasidegeneracies in the vicinity of the Fermi level in metallic systems. The exchange–correlation functional given by Ceperley and Alder is used in the parametrization of Perdew and Zunger [12]. Non-local exchange-correlation effects are considered in the form of generalized-gradient corrections (GGCs). We use the Perdew–Wang functional [13]. The GGCs are applied self-consistently in the construction of the pseudopotentials as well as in the calculations of the Kohn–Sham ground state. The calculations are performed using fully non-local optimized ultrasoft pseudopotentials [14] allowing small energy cut-offs. In this work, particular care was taken for the total-energy convergence with regard to the k-space integration. The Methfessel– Paxton k points technique [15] with a smearing parameter of 0.10 eV was used. The number of irreducible k points was 182, 190, 150, 56, 18 and 20 for fcc, bcc, hcp, A15, sigma and chi respectively. All the parameters of the different studied unit cells as well as the possible cell internal coordinates were optimized using a conjugate gradient method [11].
3. Stability in pure transition metals The total energies were calculated for bcc, fcc, hcp, A15, sigma and chi structures and for Nb, Mo, Tc, Ru, Ta, W, Re, Os metals. As already discussed, the three A15, sigma and chi structures contain all the characteristic polyhedra of the tetrahedral packing with a percentage of icosahedral sites which is 25, 33, and 41% for A15, sigma and chi respectively. Results are illustrated in Fig. 1. The first spectacular result is the relative stability of the FK phases for valence electron per atom ratios (e /a) ranging from 4.5 to 6.5. More particularly, the A15 structure is found to strongly compete with the bcc structure for Nb and Mo metals as
C. Berne et al. / Journal of Alloys and Compounds 334 (2002) 27 – 33
29
deviations from such symmetry are small enough to expect that to a first approximation the bonding of CN12 atoms is the same as the one which would exist under exact symmetry of the icosahedron point group. Thus it appears that the icosahedral sites show a predominance for atoms with filled or nearly filled, or else empty d shells. Atoms with d shells close to half filled are mostly found in coordinations of lower symmetry where the d orbitals may hybridize to form directed bonds. Therefore transition metals like Nb, Mo, Ta or W prefer occupying CN15 and CN16 positions which are of lower symmetry whereas metals like Re or Os tend to occupy the CN12 position. A mixture of both elements may occur on CN14 (CN13) positions and this will be discussed in Sec. III.
4. Alloying effects in Frank–Kasper phases
4.1. Sequence of the Frank–Kasper phases
Fig. 1. Structural energy differences for (a) 4d metals and (b) for 5d metals.
well as for Ta and W metals. Let us mention that this peculiar behavior of the A15 phase has been used to interpret the occurrence of a metastable phase in Ta metal brought on by drop-tube undercooling [16]. The electron concentration found in our calculations is very similar to the one obtained by Phillips and Carlsson [17] or by Watson and Bennett [4]. However, our study emphasizes that, in this range, the FK structures strongly compete with the bcc and not with the fcc as assumed by Phillips and Carlsson [17]. The second important result of the present calculations is that all of the lines in Fig. 1 intersect at one particular band filling, namely e /a56.7. As a consequence, the sequence A15-sigma-chi also depends on the band filling. As these three FK phases display a different percentage of icosahedral sites, i.e. CN12, it is tempting to relate their relative structural stability with this percentage. Indeed perturbations conforming to the point group of the icosahedron do not affect the degeneracy of the atomic d levels whereas for all of the other coordination symmetries the d-level degeneracy is partly broken. Actually, the surroundings of CN12 atoms do not exactly conform to the point group of the icosahedron. However, the effective
Results presented in Sec. II can be used to understand the sequence of FK phases in transition metal based alloys. For instance, if a chi phase exists at a given composition in an alloy, a composition change which reduces the e /a ratio could possibly give rise to a new phase such as a sigma phase, an A15 phase, or a bcc solid solution, but not an hcp phase. Reciprocally, if the e /a ratio increases an hcp phase may occur. If the same arguments are applied to a sigma phase a decreasing e /a may lead to the occurrence of A15 or bcc phases but not to chi or hcp phases. Thus, with increasing the electron concentration, the following succession of phases may exist: bcc-A15-sigma-chi-hcpfcc. They are many binary alloys of 4d and 5d transition elements in which these sequences can be checked [10]. For instance, the sequence bcc-sigma-chi-hcp can be found in the Ta–Os, Ta–Re, W–Re, Nb–Re, and Mo–Re systems. The sequence bcc-A15-sigma-hcp occurs in the Mo– Os, Mo–Tc, and Nb–Rh systems. In the Nb–Os system the complete sequence bcc-A15-sigma-chi-hcp occurs. However, if this approach provides a nice interpretation of the sequence of the FK phases as a function of electron concentration, it does not predict their actual occurrence for a given alloy. To try to answer this question, it is necessary to treat alloying effects in these structures. Another difficulty arises from the fact that some FK phases like sigma or chi do not have a definite stoichiometry. Hence their occurrence varies with composition and thus with electron concentration. We present now a first-principles based treatment of the alloying effects in the sigma phase.
4.2. Alloying mechanism in the sigma phase When dealing with the general problem of phase stability in an alloy from a microscopic description, one has to
C. Berne et al. / Journal of Alloys and Compounds 334 (2002) 27 – 33
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combine electronic structure calculations and statistical studies. This key may be played by an Ising Hamiltonian for which the energetics associated with changing local atomic configurations is parametrized by a set of effective cluster interactions (ECI’s). Total energies of a large number of configurations are used to extract these effective cluster interactions following the Connolly–Williams method (CWM) [18]. The ECI Vi for a cluster i is calculated with:
Ov SE a
a
OV j D 5 min n
a form
2
2
i
a i
(1)
i 51
where j is the cluster correlation function as defined in Eq. (10) of Ref. [19]; v represents the weights assigned to each structure. Formation energies Eform are extracted from the total energies Etot by substracting the concentration weighted total energies of pure constituents with the lattice structure. Eq. (1) is solved using a singular value decomposition (SVD) procedure. As in the spirit of Ref. [20], the sets of equations corresponding to thermodynamically stable structure are given more weight than the equations pertaining to unstable structures. The weighting in the SVD procedure provides an alternative to the linear optimization technique for ensuring that the cluster expansion reproduces the proper sequence of structures energy. An additional benefit of weighting the lowest energy states more heavily in the CWM method is that it makes the CE convergence more rapid. Let us consider the case of the sigma phase. For a AB-sigma alloy, the five inequivalent sites of the sigma phase make 2 5 532 possible distributions of A and B atoms. These 32 configurations are not superstructures of the sigma phase because they all have the same space group symmetry. In addition, 10 sigma superstructures were considered that allowed us to determine the pair interactions between sites of the same type (i.e. B–B, C–C, D–D, and E–E type pairs). The formation energies of these 42 configurations are computed with a full geometry relaxation including both the optimization of the a and c parameters of the tetragonal unit cell and the optimization of the internal atomic coordinates. The SVD algorithm is used to extract effective interactions [21]. A set of 29 and 27 interactions was shown to reproduce the formation energies with a root mean square error of 0.06 kJ / atom for systems like the Re–W and the Re–Ta respectively. This error is well within the precision of the total energy calculations and is much smaller than the smallest difference between the formation energies of any of the 32 configurations used in the CWM (see Figs. 2–3). The interactions are used to compute the configurational entropy and the associated Helmholtz free energy as a function of composition and temperature within the CVM. The site occupancy of the 5 inequivalent sites are also determined in these calculations. In the present case, the
Fig. 2. Formation energy (in kJ / atom) of Re–W relaxed atomic configurations with the sigma structure: as computed with VASP (1); as computed from the effective interactions (1). Circles emphasize the 10 configurations that are sigma superstructures. The most stable configurations have been connected with a solid line and the occupancy of the A, B, C, D, and E sites have been indicated.
tetrahedron approximation is a natural choice since the sigma phase is tetrahedrally close packed. For this structure, the tetrahedron approximation leads to 17 tetrahedron maximal clusters and a total of 71 correlation functions [22] Figs. 4–5 show the site preference for two systems, namely Re–Ta and Re–W systems, at T5500 K. The first comment is that the site occupancy as a function of W or Ta composition has the same sequence for both systems. First the B site happens to be filled with W (or Ta) and takes most W (or Ta) content; then come sites C and E, and finally the A and D sites are filled. Results of Sec. II allow us to understand the clear preference for W or Ta to
Fig. 3. Formation energy (in kJ / atom) of Re–Ta relaxed atomic configurations with the sigma structure: as computed with VASP (1); as computed from the effective interactions (1). Circles emphasize the 10 configurations that are sigma superstructures. The most stable configurations have been connected with a solid line and the occupancy of the A, B, C, D, and E sites have been indicated.
C. Berne et al. / Journal of Alloys and Compounds 334 (2002) 27 – 33
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Fig. 4. Computed occupancy of the inequivalent sites in the Re–W sigma phase at T5500 K.
occupy the lower symmetry B, C and E sites and for Re to occupy the icosahedral A and D sites. As discussed in Ref. [21], more subtle alloying effects are necessary to understand the sequence between the sites of lower symmetry, namely B, C and E sites. For instance, another E–C–B sequence was found in the Fe–Cr sigma alloy [22]. In Figs. 6–7, the site occupancy of both Re 0.6 W0.4 and Re 0.6 Ta 0.4 alloys as a function of temperature is shown. At high temperatures, the site occupancy very slowly reaches the value corresponding to the random composition. This limit occupancy being determined purely by the alloy composition, in this case the limit of each site is 60% Re. However, this limit is not obtained because no order– disorder transformation due to the removal of symmetry elements in going to the chemically randomized configuration takes place in the present calculations. The occupancy
of B, A and D sites reaches the high temperature limit more rapidly in Re–W than in Re–Ta, due to weaker alloying effects in Re–W system. In this case, the temperature effects on the ordering mechanism are more pronounced. When the temperature is lowered, the fraction of W (Ta) on B site increases to reach an occupancy of 100% around 500 K for Re–W and 2000 K for Re–Ta. For the same temperatures, the A and D sites display an opposite behavior which gives a Re occupancy of 100%. For both alloys, the C and E sites have an important mixed occupancy which is more or less independent of the temperature. These results may be used to predict composition limits for the sigma alloys. At very low temperature, we can assume that only Re occupy A and D sites and only W (or Ta) B site and a mixture of Re and W (or Ta) C and E
Fig. 5. Computed occupancy of the inequivalent sites in the Re–Ta sigma phase at T5500 K.
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C. Berne et al. / Journal of Alloys and Compounds 334 (2002) 27 – 33
Fig. 6. Computed occupancy of the inequivalent sites in the Re 0.6 W0.4 sigma phase as a function of temperature.
Fig. 7. Computed occupancy of the inequivalent sites in the Re 0.6 Ta 0.4 sigma phase as a function of temperature.
sites; thus the predicted limits are 13–66% W (or Ta) taking into account the multiplicity of the crystallographic sites. In fact this approach provides an over-limit because the true composition limits of the sigma phase are given by the competition of the other equilibrium phases and more particularly the bcc solid solution.
when increasing electron concentration which is found in the binary metallic systems based on refractory metals. The alloying effects are discussed more particularly in the sigma phase. It has been shown that the site occupation in such a complex structure displays a peculiar behavior as a function of composition and / or temperature which can be explained from the site-coordination numbers.
5. Conclusion We have shown that the structural stability among the FK phases depends on the band filling of pure transition metals. This behavior is related to the occurrence of high local coordination sites which are more favorable for metals with d-shells close to half-filled. These results are also used to interpret the sequence bcc-A15-sigma-chi-hcp
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