- Email: [email protected]
Theoretical study of the Mo–Ru sigma phase
Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 171–176 www.elsevier.com/locate/calphad
Theoretical study of the Mo–Ru sigma phase O. Gr˚an¨as a,∗ , P.A. Korzhavyi b , A.E. Kissavos c , I.A. Abrikosov c a Department of Physics, Condensed Matter Theory Group, Box 530, SE-751 21 Uppsala, Sweden b Royal Institute of Technology, Applied Material Science, Brinellv¨agen 23, SE-100 44 Stockholm, Sweden c Department of Physics, Chemistry and Biology (IFM), Univerity of Link¨oping, SE-581 83 Link¨oping, Sweden
Received 9 November 2006; received in revised form 31 May 2007; accepted 5 June 2007 Available online 17 July 2007
Abstract The thermodynamic properties of the Mo–Ru binary σ -phase are investigated using a combination of ab initio calculations and CALPHAD modeling. Total energy calculations have been performed for the complete set of 32 end-member compounds of a 5-sublattice compound energy model. The internal crystallographic parameters for each end-member compound have been determined by minimising the total energy. A simpler, 3-sublattice model of the Mo–Ru σ -phase is formulated on the basis of calculated total energies. The site occupancy is acquired by minimising the free energy given by the compound energy model. A strong preference of Mo and Ru towards high-coordination sites and icosahedral sites in the Mo–Ru σ -phase is found and analysed in terms of the electronic structure. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Ab initio calculations; Sigma phase; Mo–Ru; Thermodynamic properties; Site occupancy
1. Introduction The σ -phase occurs in transition metal alloys with a total number of valence electrons between 6.2 and 7.4 per atom. The structure type is complex tetragonal, with space group P42 /mnm [1]. The effects of this very hard and brittle phase on the mechanical properties are mainly deleterious, therefore the mechanisms that govern the σ -phase formation are important for the metallurgy of steels and superalloys. Usually, the stability of the σ -phase is discussed in terms of electronic structure and atomic size factors. Recently it was shown that the thermodynamic functions of σ -CrFe, as well as those of other σ -phases containing 3d transition elements, contain sizeable magnetic contributions [2,3]. The case of the Mo–Ru σ -phase is of particular interest in connection with the development of next generation superalloys (containing platinum group metals such as Ru, Rh, Pd, Pt and Ir) [4], but also because its constituents are non-magnetic and isoelectronic with those of the magnetic σ -CrFe. Also, the atomic radii of the components are more similar in Mo–Ru than in the other non-magnetic isoelectronic σ -phases such as ∗ Corresponding author. Tel.: +46 18 4715867; fax: +46 18 511784.
E-mail address: [email protected] (O. Gr˚an¨as). c 2007 Elsevier Ltd. All rights reserved. 0364-5916/$ - see front matter doi:10.1016/j.calphad.2007.06.001
Cr–Ru or Cr–Os. Therefore, the stability of the Mo–Ru σ -phase appears to be governed mainly by the electronic structure factor. Experimental studies of the Mo–Ru σ -phase are complicated because of the refractory nature of this material. Ab initio calculations may be used as a complementary source of data for thermodynamic modeling [5,6]. They also allow one to investigate the relationship between the electronic and atomic structure of the σ -phase. Several successful studies have been conducted in which ab initio calculations are combined with the CALPHAD modeling [3,7] or the Connolly–Williams cluster expansion approach [8–10], aiming to analyse the site occupancy in the σ -phase as a function of composition and temperature. The purpose of the present study is to calculate the electronic structure and ground-state properties of the Mo–Ru σ -phase, as well as to develop a model for the thermodynamic description for this phase using first-principles calculations as the only input. It is worth noting that ab initio and CALPHAD methods have also been used to improve the description of the σ -phase in the phase diagram of Re–W [7], Co–Mo and Fe–Mo [11]. There are five crystallographically inequivalent sites in the crystal structure of the σ -phase; see Table 1. These sites can be grouped into two larger categories according to the coordination number Z , icosahedrally coordinated (IC) sites 1 and 4 (for
172
O. Gr˚an¨as et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 171–176
Table 1 Inequivalent sites in the unit cell of the σ -phase Site
1
2
3
4
5
Multiplicity (Wyckoff index) Coordination number
2(a) 12
4(f) 15
8(i) 14
8(i) 12
8(j) 14
The reference states of Mo and Ru were chosen to be their respective ground states. The structural parameters were first optimised, followed by an accurate calculation of the total energy. The k-points were distributed according to a Monkhorst-Pack mesh consisting of 744 and 432 and k-points in the irreducible wedge of the Brillouin zone for bcc Mo and hcp Ru respectively. The volume was equilibrated for every end-member compound, assuming a zero-pressure environment. The enthalpy of formation for each end-member compound i jlkm, where (i, . . . , m) = {Mo, Ru}, was then calculated from the total energies, E, as:
which Z = 12) and high-coordination number (high-CN) sites 2, 3 and 5 (for which Z = 15, 14 and 14, respectively). If all five sites are treated as independent sublattices; the site occupancy in the Mo–Ru case may be expressed by a general formula (Mo, Ru)2 (Mo, Ru)4 (Mo, Ru)8 (Mo, Ru)8 (Mo, Ru)8 . Here (Mo, Ru) denotes a mixture of Mo and Ru on a certain sublattice, and the subscript gives the number of atoms belonging to that sublattice in the unit cell of the σ -phase (the unit cell contains 30 atoms in total). The 25 = 32 cases of integral site occupancy (where each site is completely occupied by either Mo or Ru) have special meaning within the compound energy formalism (CEF) [12] as so-called end-member compounds, the parameters of a fivesublattice model. Hereafter we use a short notation for the endmember compounds, e.g. we write RuMoMoRuMo instead of writing the full formula Ru2 Mo4 Mo8 Ru8 Mo8 . In this study we calculate the complete set of 32 total energies for the endmember compounds, whose energies are then used in order to model atomic ordering in the Mo–Ru σ -phase as well as to estimate its thermodynamic properties.
In the compound energy formalism, which is a generalised Bragg–Williams approximation, the components are assumed to be randomly distributed within each crystallographic site [7, 12]. The Gibbs free energy may be written as
2. Methodology
G = H srf − T S conf ,
2.1. Ab initio calculations
where the enthalpy of the σ -phase is represented in configurational space by H srf , the so-called surface of reference [5]. Within the 5-sublattice model, where the five crystallographically inequivalent sites of the σ -phase are treated as independent sublattices, the H srf reads as: X (1) (2) (3) (4) H srf = yi y j yk yl ym(5) Hi jklm . (3)
The ab initio calculations were performed using Density Functional Theory (DFT) [13,14] as implemented in the Projector Augmented Wave (PAW) code Vienna Ab initio Simulation Package (VASP) [15–17]. The energy cut-off for the PAWs was set to 300 eV, the exchange correlation was treated within the generalised gradient approximation (GGA) as parametrised by Perdew, Burke and Ernzerhof [18]. A Monkhorst-Pack mesh with 112 k-points in the irreducible wedge of the Brillouin zone was used [19]. The c/a ratio was optimised for the RuMoMoRuMo (most stable) configuration; the optimum value was found to be c/a = 0.514. The increase in energy for a deviation of 1% in the c/a ratio (shifted to 0.509 instead of 0.514) was about 0.2 kJ/mole; this dependence was regarded as weak. Therefore, the calculated optimum value c/a = 0.514 was used for all the other compounds. To confirm the validity of this assumption for other compounds, we carried out several additional tests. For instance, for the configuration RuRuRuRuMo, which was representative of Rurich phases, the energy was lowered by 0.1 kJ/mole as a result of the relaxation of the c/a ratio. The change is small enough, and does not have any impact on subsequent analysis. The equilibrium volume for each compound was calculated, followed by full relaxation of the internal atomic positions. Relaxation energies were found to be as high as a few kJ/mole for some compounds.
Hi jklm = E i jklm − cRu E Ru − (1 − cRu )E Mo .
(1)
Here cRu is the overall concentration (mole fraction) of Ru in the end-member compound. The procedure of finding the equilibrium volume also gave us access to the energy versus volume relation, hence, the ground-state properties such as the lattice parameters and the bulk modulus were directly available. 2.2. Free energy estimation
(2)
i jklm
Here the Hi jklm is the ab initio enthalpy of formation for every end-member compound calculated at zero temperature (vibrational contribution is neglected here and thereafter) and (s) yi is the mole fraction of component i on the sublattice s. It is worth noting that in principle the mixing enthalpies of compounds with partial disorder at sublattices can be calculated directly from first principles using the so-called coherent potential approximation (CPA) [20] or using the effective cluster interactions determined from the CPA calculations, as was recently demonstrated for the hcp Ru-rich MoRu alloys [21]. However, for the Mo–Ru σ -phase the optimisation of the internal positions within the unit cell turns out to be important. For example, the energy gain due to such an optimisation for the most stable compound RuMoMoRuMo is of the order of 0.8 kJ/mole, to be compared with the enthalpy for this compound of 1.8052 kJ/mole. Unfortunately, the CPA does not allow one to perform the above-mentioned optimisation within the present-day implementations.
173
O. Gr˚an¨as et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 171–176 Table 2 Calculated enthalpies of formation, Hi jklm [kJ/mol], for the end-member compounds Index
Configuration
cRu
Hi jklm
Index
Configuration
cRu
Hi jklm
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
MoMoMoMoMo RuMoMoMoMo MoRuMoMoMo RuRuMoMoMo MoMoMoRuMo MoMoRuMoMo MoMoMoMoRu RuMoMoRuMo RuMoRuMoMo RuMoMoMoRu MoRuMoRuMo MoRuRuMoMo MoRuMoMoRu RuRuMoRuMo RuRuRuMoMo RuRuMoMoRu
0.000 0.067 0.133 0.200 0.267 0.267 0.267 0.333 0.333 0.333 0.400 0.400 0.400 0.467 0.467 0.467
16.3831 13.2151 19.8267 17.2526 4.4621 12.9547 20.1042 1.8052 10.3510 17.3479 5.9659 18.9270 20.8151 5.0607 16.2802 19.3056
17 18 19 20 21 22 23 24 25 26 27 27’ 28 29 30 31 32
MoMoRuRuMo MoMoMoRuRu MoMoRuMoRu RuMoRuRuMo RuMoMoRuRu RuMoRuMoRu MoRuMoRuRu MoRuRuRuMo MoRuRuMoRu RuRuMoRuRu RuRuRuRuMo RuRuRuRuMo RuRuRuMoRu MoMoRuRuRu RuMoRuRuRu MoRuRuRuRu RuRuRuRuRu
0.533 0.533 0.533 0.600 0.600 0.600 0.667 0.667 0.667 0.733 0.733 0.733 0.733 0.800 0.867 0.933 1.000
10.7209 10.7560 16.1144 8.7544 11.3953 13.4947 12.3769 15.2643 21.4020 14.2456 14.6906 11.9606 18.9946 20.6765 20.6323 27.3309 30.6530
Fig. 1. Calculated enthalpies of formation for the end-member compounds of the 5-sublattice model (open circles) and the 3- sublattice model (filled circles). Free energies obtained within the models for three different temperatures are also displayed. Indexing corresponds to that of Table 2.
Within the compound energy formalism the entropy contribution in Eq. (2) is calculated as: X X (s) (s) S conf = −kB a (s) yi ln(yi ). (4) s
i=Mo,Ru
Here kB is the Boltzmann constant and a (s) is the multiplicity of site/sublattice s (see Table 1). The free energy can now be minimised for specific composition and temperature to acquire the optimum sublattice concentrations. 3. Results and discussion 3.1. Ground-state properties The calculated enthalpies for all 32 end-member compounds are displayed in Fig. 1. The enthalpy of formation for the Mo–Ru σ -phase is a minimum for configuration 8 (see Table 2), in which all the 10 IC sites are occupied by Ru and all the 20 high-CN sites are occupied by Mo, thus the overall
composition is Mo2 Ru. Even for this most stable end-member compound the calculated enthalpy value is positive, indicating that the σ -phase in this binary system is not stable at 0 K relative to Mo and Ru in their respective ground states, in agreement with the experimental phase diagram by Kleykamp et al. [22]. The end-member compounds with the highest formation enthalpies are most probably dynamically unstable. As was discussed in the literature [6,23–25], this can be a problem when ab initio results are used to calculate lattice stabilities. However, these compounds have low participation ratios in the configuration that minimises the Gibbs free energy of the σ -phase in the composition and temperature range of its stability. Correspondingly, these end-member compounds have no significant effect on the results for site occupancy and ordering. The composition of the most stable end-member compound is consistent with the fact [1] that all non-magnetic isoelectronic σ -phases tend to form near the composition A2 B, where A = {Cr, Mo} and B = {Ru, Os}. Ab initio calculations show that even in the case of σ -CrFe the minimum of the enthalpy of mixing occurs at the Cr2 Fe composition, whereas the equiatomic σ -CrFe is stabilised due to magnetic entropy [3]. The stoichiometry of the most stable σ -Mo2 Ru end-member compound is a consequence of the strong tendency of Mo to occupy high-CN sites and that of Ru to occupy IC sites. The energy associated with this site preference can be estimated using the data of Table 2 as a difference in the mixing enthalpy between configuration 8 (RuMoMoRuMo) and configuration 25 (MoRuRuMoRu), in the latter configuration the occupancy of all sites is reversed. In temperature units, this energy amounts to some 2000 degrees Kelvin, implying that the Mo–Ru σ -phase should remain strongly ordered even at high temperatures. A similar site-occupancy trend has been observed experimentally in the σ -Mo2 Os phase [1]. Surprisingly, no atomic ordering was detected in the σ -Cr2 Os and σ -Cr2 Ru phases [1], although their stoichiometry implies the same kind of site occupancy as in the other isoelectronic σ -phases.
174
O. Gr˚an¨as et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 171–176 Table 3 Experimentally reported and ab initio calculated lattice parameters for the Mo–Ru σ -phase Ru mole fraction
˚ a [A]
˚ c [A]
c/a
Reference
0.4 0.3749 0.3650 0.3515 0.3333
9.55 9.5575 9.5569 9.5573 9.653
4.95 4.9345 4.9332 4.9325 4.961
0.518 0.5163 0.5162 0.5161 0.514
[29] [30] [30] [30] This work
Fig. 2. Total and s + p density of states (DOS) for MoRu σ -phase with sublattices occupations RuMoMoRuMo (configuration 8 in Table 2, solid and dot-dashed lines, respectively) and RuMoMoMoRu (configuration 10 in Table 2, dashed and dottedlines, respectively) as a function of energy E (relative to Fermi energy E F ). The difference between the total DOS and s + p partial DOS gives the contribution due to d-electrons, which clearly dominates the spectra at the Fermi energy. See text for more discussion.
In order to investigate the relationship between the atomic configuration and the electronic structure of the σ -phase, we compare in Fig. 2 the density of states (DOS) for configurations 8 and 10, which have the same stoichiometry but a reversed occupancy of sites 4 and 5. As one can see, in configuration 8 there are three bonding peaks below the Fermi level (centred around −4.6, −3 and −1.5 eV), whereas in configuration 10 there are only two peaks (centred around −4.3 and −3 eV). This indicates that the d-states in configuration 10 are unable to hybridise to the same extent as those in configuration 8. As compared to that of configuration 10, the DOS in configuration 8 is pushed from the band centre towards the band edges. As a result, the separation between the bonding and antibonding peaks increases and the DOS near the Fermi level is depleted. This implies stronger covalency of the interatomic bonding in configuration 8 relative to configuration 10, which accounts for the clear preference by Mo for high-CN sites and by Ru for IC sites. The calculated lattice parameters of the most stable endmember compound, configuration 8, are compared with experimental data on the Mo–Ru σ -phase in Table 3. The calculated values are within the scatter of experimental data. The calculated concentration dependence of the lattice parameter a (lattice parameter for the lowest energy endmembers as a function of composition) is presented in Fig. 3. As might be expected, the lattice parameter decreases with increasing the Ru content. A more interesting feature is, however, the negative deviation of the lattice parameter from Vegard’s law. This behaviour is a result of a stronger binding (and a shorter bond length) when Mo atoms preferentially occupy the high-CN sites and Ru atoms preferentially occupy the IC sites. The bulk modulus for the most stable end-member compound (configuration 8, composition Mo2 Ru) is found by
Fig. 3. Calculated lattice parameter (solid line) and lattice parameter predicted by Vegard’s law (dashed line) for the MoRu σ -phase as a function of the Ru mole fraction.
fitting the calculated volume dependence of total energy using the Murnaghan equation of state [26]. The obtained value, 274 GPa, is slightly higher than 267 GPa, which is the concentration average of the calculated (using the same procedure) bulk moduli of bcc Mo and hcp Ru, 251 GPa and 298 GPa, respectively. The latter two values compare reasonably well with the experimental room-temperature bulk moduli of bcc Mo, 263 GPa [27], and hcp Ru, 313 GPa [28]. The high value of the bulk modulus of configuration 8 is indicative of a high hardness of the σ -phase and is consistent with the strong covalent component of interatomic bonding for this endmember compound. 3.2. Free-energy models and atomic order at high temperatures Distribution of the alloy constituents among the five sublattices at a finite temperature for the Mo–Ru σ -phase can be obtained by minimisation of the free energy given by the five-sublattice model, Eqs. (2)–(4). The results of such modeling at 700 K are shown in the upper panel of Fig. 4. Note that this temperature is much below the stability range of the σ -phase in the Mo–Ru system. At the Mo2 Ru composition the atomic order is nearly perfect. Upon a deviation from this “ideal” stoichiometry towards Mo-rich compositions, the excess Mo atoms replace Ru atoms evenly on sites 1 and 4, while the high-CN sites 2, 3 and 5 remain fully occupied by Mo. On the other side, upon small deviations from the
O. Gr˚an¨as et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 171–176
175
Fig. 5. Site occupancy for the 5-sublattice model at 37.5% Ru for the stable temperature range.
Fig. 4. Site occupancy at 700 K obtained using the 5-sublattice (upper panel) and the 3-sublattice (lower panel) CEF models.
ideal stoichiometry towards Ru-rich compositions, the IC sites remain fully occupied by Ru, the excess Ru atoms substitute preferentially on the high-CN sites 2 and 3. At compositions between 33 at.% and 50 at.% Ru the excess Ru atoms are distributed evenly between sites 2 and 3, whereas the concentration of Ru on site 5 is much smaller. For a greater Ru content this tendency changes and site 2 becomes less occupied by Ru than sites 3 and 5, so that the occupancy of all the highCN sites by Ru is different for the Ru-rich compositions of the σ -phase. According to experiments [22] the σ -phase occurs in a narrow composition range around Mo0.63 Ru0.37 at temperatures between 1416 K and 2188 K. Our calculations performed using the 5-sublattice model for a fixed composition of Mo0.625 Ru0.375 show that the σ -phase remains highly ordered in the whole temperature range of its stability; see Fig. 5. Since the Mo–Ru σ -phase exists in a very narrow composition interval, a very detailed description of this phase at large deviations from this composition seems to be redundant. Therefore, we simplify the 5-sublattice model to a 3-sublattice model by merging sites 1 and 4 into one sublattice, site 2 and 3 into another sublattice, and by treating site 5 as the third sublattice. We note that the present 3-sublattice model is different from the model by Kasper and Waterstrat [31], in which sites 3 and 5 are merged because these sites have
similar coordination numbers. The present 3-sublattice model takes into account the site occupancy behaviour that is obtained using the full 5-sublattice model for the concentration interval of interest. Unfortunately, there are no experimental data on the site occupancy for the Mo–Ru σ -phase, although such data are available for other isoelectronic compounds [1,32]. As parameters of the 3-sublattice model we take the calculated enthalpies of formation for a subset of end-member compounds that are indicated in Table 2 using a bold font (these enthalpies are shown in Fig. 1 as filled circles). However, the energies of end-member compounds 14 and 20, which determine the slope of the enthalpy upon the Ru-rich deviations away from configuration 8, are no longer contained in the 3-sublattice model. In order to achieve good agreement between the free energy predictions by the 5-sublattice and the 3-sublattice models, the enthalpy of configuration 27 should be adjusted in such a way that the corresponding point (thereafter denoted as 27’) in Fig. 1 lies approximately on the same line with the points representing configurations 8, 14 and 20. The enthalpies of formation evaluated within the 5-sublattice and 3-sublattice models for three temperature values are displayed in Fig. 1. Our compound energy models predict that at high temperatures the free energy minimum for the σ -phase is shifted from Mo2 Ru towards more Ru-rich compositions, which agrees well with experimental findings. The free energy estimates of the two models agree very well with each other; however, the 3-sublattice model seems slightly to underestimate the entropy content. This may be a result of the fact that the 3-sublattice model predicts a smaller concentration of Ru on site 5 as compared to the predictions of the 5-sublattice model; see Fig. 4. 3.3. Summary and conclusion We present a thermodynamic description of the Mo–Ru σ -phase based on a combination of ab initio and CALPHAD methods. Lattice parameters, bulk moduli, enthalpies of formation as well as site occupancies are presented. By analysing the density of states we have found an increased
176
O. Gr˚an¨as et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 171–176
hybridisation among the d-orbitals when Mo occupies highcoordination-number sites and Ru occupies icosahedrally coordinated sites. This results in a strong tendency towards ordering in this compound. Acknowledgements The Swedish National Allocation Committee (SNAC) is acknowledged for supercomputer resources at the National Supercomputer Centre (NSC). The authors acknowledge financial support by the Swedish Foundation for Strategic Research and by the Royal Swedish Academy of Sciences. References [1] E. Hall, S. Algie, The sigma phase, Metall. Rev. 11 (1966) 61–88. [2] D. Downie, J. Martin, Heat-capacities of transition-metal alloys. II. Alpha-phase and sigma-phase Fe0.525 Cr0.475 and Fe0.566 Cr0.434 , J. Chem. Thermodyn. 16 (8) (1984) 743–752. [3] P. Korzhavyi, B. Sundman, M. Selleby, B. Johansson, Atomic, electronic, and magnetic structure of iron-based sigma-phases, Mater. Res. Soc. Symp. Proc. 842 (2005) S4.10.1. [4] J.-C. Zhao, J. Westbrook, Ultrahigh-temperature materials for jet engines, MRS Bull. 28 (9) (2003) 622–630. [5] I. Ansara, T.G. Chart, A.F. Guillermet, F.H. Hayes, U.R. Kattner, D.G. Pettifor, N. Saunders, K. Zeng, Thermodynamic modelling of solutions and alloys — Schloss Ringberg, March 10 to March 16, 1996 — Group 2: Alloy system I — Thermodynamic modelling of selected topologically close-packed intermetallic compounds, CALPHAD 21 (2) (1997) 171–218. [6] P.E.A. Turchi, I.A. Abrikosov, B. Burton, S.G. Fries, G. Grimvall, L. Kaufman, P.A. Korzhavyi, V. Rao Manga, M. Ohno, A. Pisch, A. Scott, W. Zhang, CALPHAD 31 (1) (2007) 4–27. [7] S. Fries, B. Sundman, Using Re–W sigma-phase first-principles results in the Bragg–Williams approximation to calculate finite-temperature thermodynamic properties, Phys. Rev. B 66 (1) (2002) 012203. [8] M.H.F. Sluiter, K. Esfarjani, Y. Kawazoe, Phys. Rev. Lett. 75 (17) (1995) 3142–3145. [9] C. Berne, M. Sluiter, Y. Kawazoe, A. Pasturel, Ordering effects in the Re–W and Re–Ta sigma phases, J. Phys. - Condens. Matter 13 (42) (2001) 9433–9443. [10] C. Berne, M. Sluiter, Y. Kawazoe, T. Hansen, A. Pasturel, Site occupancy in the Re–W sigma phase, Phys. Rev. B 64 (14) (2001) 144103. [11] J. Houserova, J. Vrestal, M. Sob, Phase diagram calculations in the Co–Mo and Fe–Mo systems using first-principles results for the sigma phase, CALPHAD 29 (2) (2005) 133–139.
[12] M. Hillert, The compound energy formalism, J. Alloys Compounds 320 (2) (2001) 161–176. [13] P. Hohenberg, W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136 (3B) (1964) 864–871. [14] W. Kohn, L. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140 (4A) (1965) 1133–1138. [15] G. Kresse, D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59 (3) (1999) 1758–1775. [16] G. Kresse, J. Furthm¨uller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6 (1996) 15–50. [17] P. Bl¨ochl, Projector augmented-wave method, Phys. Rev. B 50 (24) (1994) 17953–17979. [18] J. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77 (18) (1996) 3865–3868. [19] H.J. Monkhorst, J.D. Pack, Special points for brillouin-zone integrations, Phys. Rev. B 13 (12) (1976) 5188–5192. [20] A.V. Ruban, I.A. Abrikosov, H.L. Skriver, Electronic structure and thermodynamicproperties of ordered, partially ordered and random Cu–Au and Ni–Pt alloys, Phys. Rev. B 51 (19) (1995) 12958–12968. [21] A.E. Kissavos, S. Shallcross, L. Kaufman, O. Gr˚an¨as, A.V. Ruban, I.A. Abrikosov, Thermodynamics of ordered and disordered phases in the binary MoRu system, Phys. Rev. B 75 (18) (2007) 184203(8). [22] H. Kleykamp, Constitution and thermodynamics of the Mo–Ru, Mo–Pd, Ru–Pd and Mo–Ru–Pd systems, J. Nucl. Mater. 167 (1989) 49–63. [23] G. Grimvall, M. Thiessen, A.F. Guillermet, Phys. Rev. B 36 (15) (1987) 7816–7826. [24] K. Persson, M. Ekman, G. Grimvall, Phys. Rev. B 60 (14) (1999) 9999–10007. [25] A. Kissavos, S. Shallcross, V. Meded, L. Kaufman, I. Abrikosov, A critical test of ab initio and calphad methods: The structural energy difference between bcc and hcp molubdenum, CALPHAD 29 (1) (2005) 17–23. [26] F.D. Murnaghan, Amer. J. Math. 59 (2) (1937) 235–260. [27] K.V. Katahara, M.H. Manghnani, E.S. Fisher, Pressure derivatives of the elastic moduli of bcc Ti–V–Cr, Nb–Mo and Ta–W alloys, J. Phys. F. Met. Phys. 9 (55) (1979) 773–790. [28] E.S. Fisher, D. Dever, Temperature dependence of elastic moduli of ruthenium, rhenium, cobalt, dysprosium, and erbium; a study of the elastic anisotropy-phase transformation relation, Trans. Metall. Soc. AIME 239 (1) (1967) 48–57. [29] E. Bucher, F. Heiniger, J. M¨uller, Helv. Phys. Acta 34 (1961) 843–858. [30] E. Anderson, W. Hume-Rothery, J. Less-Common Met. 2 (1960) 443–450. [31] J. Kasper, R. Waterstrat, Ordering of atoms in the sigma-phase, Acta Crystallogr. 9 (1955) 289. [32] H. Yakel, Atom distributions in sigma phases. I. Fe and Cr atom distributions in a binary sigma phase equilibrated at 1063, 1013 and 923 k, Acta Crystallogr. B 39 (1) (1983) 20–28.
Theoretical study of the Mo–Ru sigma phase O. Gr˚an¨as a,∗ , P.A. Korzhavyi b , A.E. Kissavos c , I.A. Abrikosov c a Department of Physics, Condensed Matter Theory Group, Box 530, SE-751 21 Uppsala, Sweden b Royal Institute of Technology, Applied Material Science, Brinellv¨agen 23, SE-100 44 Stockholm, Sweden c Department of Physics, Chemistry and Biology (IFM), Univerity of Link¨oping, SE-581 83 Link¨oping, Sweden
Received 9 November 2006; received in revised form 31 May 2007; accepted 5 June 2007 Available online 17 July 2007
Abstract The thermodynamic properties of the Mo–Ru binary σ -phase are investigated using a combination of ab initio calculations and CALPHAD modeling. Total energy calculations have been performed for the complete set of 32 end-member compounds of a 5-sublattice compound energy model. The internal crystallographic parameters for each end-member compound have been determined by minimising the total energy. A simpler, 3-sublattice model of the Mo–Ru σ -phase is formulated on the basis of calculated total energies. The site occupancy is acquired by minimising the free energy given by the compound energy model. A strong preference of Mo and Ru towards high-coordination sites and icosahedral sites in the Mo–Ru σ -phase is found and analysed in terms of the electronic structure. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Ab initio calculations; Sigma phase; Mo–Ru; Thermodynamic properties; Site occupancy
1. Introduction The σ -phase occurs in transition metal alloys with a total number of valence electrons between 6.2 and 7.4 per atom. The structure type is complex tetragonal, with space group P42 /mnm [1]. The effects of this very hard and brittle phase on the mechanical properties are mainly deleterious, therefore the mechanisms that govern the σ -phase formation are important for the metallurgy of steels and superalloys. Usually, the stability of the σ -phase is discussed in terms of electronic structure and atomic size factors. Recently it was shown that the thermodynamic functions of σ -CrFe, as well as those of other σ -phases containing 3d transition elements, contain sizeable magnetic contributions [2,3]. The case of the Mo–Ru σ -phase is of particular interest in connection with the development of next generation superalloys (containing platinum group metals such as Ru, Rh, Pd, Pt and Ir) [4], but also because its constituents are non-magnetic and isoelectronic with those of the magnetic σ -CrFe. Also, the atomic radii of the components are more similar in Mo–Ru than in the other non-magnetic isoelectronic σ -phases such as ∗ Corresponding author. Tel.: +46 18 4715867; fax: +46 18 511784.
E-mail address: [email protected] (O. Gr˚an¨as). c 2007 Elsevier Ltd. All rights reserved. 0364-5916/$ - see front matter doi:10.1016/j.calphad.2007.06.001
Cr–Ru or Cr–Os. Therefore, the stability of the Mo–Ru σ -phase appears to be governed mainly by the electronic structure factor. Experimental studies of the Mo–Ru σ -phase are complicated because of the refractory nature of this material. Ab initio calculations may be used as a complementary source of data for thermodynamic modeling [5,6]. They also allow one to investigate the relationship between the electronic and atomic structure of the σ -phase. Several successful studies have been conducted in which ab initio calculations are combined with the CALPHAD modeling [3,7] or the Connolly–Williams cluster expansion approach [8–10], aiming to analyse the site occupancy in the σ -phase as a function of composition and temperature. The purpose of the present study is to calculate the electronic structure and ground-state properties of the Mo–Ru σ -phase, as well as to develop a model for the thermodynamic description for this phase using first-principles calculations as the only input. It is worth noting that ab initio and CALPHAD methods have also been used to improve the description of the σ -phase in the phase diagram of Re–W [7], Co–Mo and Fe–Mo [11]. There are five crystallographically inequivalent sites in the crystal structure of the σ -phase; see Table 1. These sites can be grouped into two larger categories according to the coordination number Z , icosahedrally coordinated (IC) sites 1 and 4 (for
172
O. Gr˚an¨as et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 171–176
Table 1 Inequivalent sites in the unit cell of the σ -phase Site
1
2
3
4
5
Multiplicity (Wyckoff index) Coordination number
2(a) 12
4(f) 15
8(i) 14
8(i) 12
8(j) 14
The reference states of Mo and Ru were chosen to be their respective ground states. The structural parameters were first optimised, followed by an accurate calculation of the total energy. The k-points were distributed according to a Monkhorst-Pack mesh consisting of 744 and 432 and k-points in the irreducible wedge of the Brillouin zone for bcc Mo and hcp Ru respectively. The volume was equilibrated for every end-member compound, assuming a zero-pressure environment. The enthalpy of formation for each end-member compound i jlkm, where (i, . . . , m) = {Mo, Ru}, was then calculated from the total energies, E, as:
which Z = 12) and high-coordination number (high-CN) sites 2, 3 and 5 (for which Z = 15, 14 and 14, respectively). If all five sites are treated as independent sublattices; the site occupancy in the Mo–Ru case may be expressed by a general formula (Mo, Ru)2 (Mo, Ru)4 (Mo, Ru)8 (Mo, Ru)8 (Mo, Ru)8 . Here (Mo, Ru) denotes a mixture of Mo and Ru on a certain sublattice, and the subscript gives the number of atoms belonging to that sublattice in the unit cell of the σ -phase (the unit cell contains 30 atoms in total). The 25 = 32 cases of integral site occupancy (where each site is completely occupied by either Mo or Ru) have special meaning within the compound energy formalism (CEF) [12] as so-called end-member compounds, the parameters of a fivesublattice model. Hereafter we use a short notation for the endmember compounds, e.g. we write RuMoMoRuMo instead of writing the full formula Ru2 Mo4 Mo8 Ru8 Mo8 . In this study we calculate the complete set of 32 total energies for the endmember compounds, whose energies are then used in order to model atomic ordering in the Mo–Ru σ -phase as well as to estimate its thermodynamic properties.
In the compound energy formalism, which is a generalised Bragg–Williams approximation, the components are assumed to be randomly distributed within each crystallographic site [7, 12]. The Gibbs free energy may be written as
2. Methodology
G = H srf − T S conf ,
2.1. Ab initio calculations
where the enthalpy of the σ -phase is represented in configurational space by H srf , the so-called surface of reference [5]. Within the 5-sublattice model, where the five crystallographically inequivalent sites of the σ -phase are treated as independent sublattices, the H srf reads as: X (1) (2) (3) (4) H srf = yi y j yk yl ym(5) Hi jklm . (3)
The ab initio calculations were performed using Density Functional Theory (DFT) [13,14] as implemented in the Projector Augmented Wave (PAW) code Vienna Ab initio Simulation Package (VASP) [15–17]. The energy cut-off for the PAWs was set to 300 eV, the exchange correlation was treated within the generalised gradient approximation (GGA) as parametrised by Perdew, Burke and Ernzerhof [18]. A Monkhorst-Pack mesh with 112 k-points in the irreducible wedge of the Brillouin zone was used [19]. The c/a ratio was optimised for the RuMoMoRuMo (most stable) configuration; the optimum value was found to be c/a = 0.514. The increase in energy for a deviation of 1% in the c/a ratio (shifted to 0.509 instead of 0.514) was about 0.2 kJ/mole; this dependence was regarded as weak. Therefore, the calculated optimum value c/a = 0.514 was used for all the other compounds. To confirm the validity of this assumption for other compounds, we carried out several additional tests. For instance, for the configuration RuRuRuRuMo, which was representative of Rurich phases, the energy was lowered by 0.1 kJ/mole as a result of the relaxation of the c/a ratio. The change is small enough, and does not have any impact on subsequent analysis. The equilibrium volume for each compound was calculated, followed by full relaxation of the internal atomic positions. Relaxation energies were found to be as high as a few kJ/mole for some compounds.
Hi jklm = E i jklm − cRu E Ru − (1 − cRu )E Mo .
(1)
Here cRu is the overall concentration (mole fraction) of Ru in the end-member compound. The procedure of finding the equilibrium volume also gave us access to the energy versus volume relation, hence, the ground-state properties such as the lattice parameters and the bulk modulus were directly available. 2.2. Free energy estimation
(2)
i jklm
Here the Hi jklm is the ab initio enthalpy of formation for every end-member compound calculated at zero temperature (vibrational contribution is neglected here and thereafter) and (s) yi is the mole fraction of component i on the sublattice s. It is worth noting that in principle the mixing enthalpies of compounds with partial disorder at sublattices can be calculated directly from first principles using the so-called coherent potential approximation (CPA) [20] or using the effective cluster interactions determined from the CPA calculations, as was recently demonstrated for the hcp Ru-rich MoRu alloys [21]. However, for the Mo–Ru σ -phase the optimisation of the internal positions within the unit cell turns out to be important. For example, the energy gain due to such an optimisation for the most stable compound RuMoMoRuMo is of the order of 0.8 kJ/mole, to be compared with the enthalpy for this compound of 1.8052 kJ/mole. Unfortunately, the CPA does not allow one to perform the above-mentioned optimisation within the present-day implementations.
173
O. Gr˚an¨as et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 171–176 Table 2 Calculated enthalpies of formation, Hi jklm [kJ/mol], for the end-member compounds Index
Configuration
cRu
Hi jklm
Index
Configuration
cRu
Hi jklm
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
MoMoMoMoMo RuMoMoMoMo MoRuMoMoMo RuRuMoMoMo MoMoMoRuMo MoMoRuMoMo MoMoMoMoRu RuMoMoRuMo RuMoRuMoMo RuMoMoMoRu MoRuMoRuMo MoRuRuMoMo MoRuMoMoRu RuRuMoRuMo RuRuRuMoMo RuRuMoMoRu
0.000 0.067 0.133 0.200 0.267 0.267 0.267 0.333 0.333 0.333 0.400 0.400 0.400 0.467 0.467 0.467
16.3831 13.2151 19.8267 17.2526 4.4621 12.9547 20.1042 1.8052 10.3510 17.3479 5.9659 18.9270 20.8151 5.0607 16.2802 19.3056
17 18 19 20 21 22 23 24 25 26 27 27’ 28 29 30 31 32
MoMoRuRuMo MoMoMoRuRu MoMoRuMoRu RuMoRuRuMo RuMoMoRuRu RuMoRuMoRu MoRuMoRuRu MoRuRuRuMo MoRuRuMoRu RuRuMoRuRu RuRuRuRuMo RuRuRuRuMo RuRuRuMoRu MoMoRuRuRu RuMoRuRuRu MoRuRuRuRu RuRuRuRuRu
0.533 0.533 0.533 0.600 0.600 0.600 0.667 0.667 0.667 0.733 0.733 0.733 0.733 0.800 0.867 0.933 1.000
10.7209 10.7560 16.1144 8.7544 11.3953 13.4947 12.3769 15.2643 21.4020 14.2456 14.6906 11.9606 18.9946 20.6765 20.6323 27.3309 30.6530
Fig. 1. Calculated enthalpies of formation for the end-member compounds of the 5-sublattice model (open circles) and the 3- sublattice model (filled circles). Free energies obtained within the models for three different temperatures are also displayed. Indexing corresponds to that of Table 2.
Within the compound energy formalism the entropy contribution in Eq. (2) is calculated as: X X (s) (s) S conf = −kB a (s) yi ln(yi ). (4) s
i=Mo,Ru
Here kB is the Boltzmann constant and a (s) is the multiplicity of site/sublattice s (see Table 1). The free energy can now be minimised for specific composition and temperature to acquire the optimum sublattice concentrations. 3. Results and discussion 3.1. Ground-state properties The calculated enthalpies for all 32 end-member compounds are displayed in Fig. 1. The enthalpy of formation for the Mo–Ru σ -phase is a minimum for configuration 8 (see Table 2), in which all the 10 IC sites are occupied by Ru and all the 20 high-CN sites are occupied by Mo, thus the overall
composition is Mo2 Ru. Even for this most stable end-member compound the calculated enthalpy value is positive, indicating that the σ -phase in this binary system is not stable at 0 K relative to Mo and Ru in their respective ground states, in agreement with the experimental phase diagram by Kleykamp et al. [22]. The end-member compounds with the highest formation enthalpies are most probably dynamically unstable. As was discussed in the literature [6,23–25], this can be a problem when ab initio results are used to calculate lattice stabilities. However, these compounds have low participation ratios in the configuration that minimises the Gibbs free energy of the σ -phase in the composition and temperature range of its stability. Correspondingly, these end-member compounds have no significant effect on the results for site occupancy and ordering. The composition of the most stable end-member compound is consistent with the fact [1] that all non-magnetic isoelectronic σ -phases tend to form near the composition A2 B, where A = {Cr, Mo} and B = {Ru, Os}. Ab initio calculations show that even in the case of σ -CrFe the minimum of the enthalpy of mixing occurs at the Cr2 Fe composition, whereas the equiatomic σ -CrFe is stabilised due to magnetic entropy [3]. The stoichiometry of the most stable σ -Mo2 Ru end-member compound is a consequence of the strong tendency of Mo to occupy high-CN sites and that of Ru to occupy IC sites. The energy associated with this site preference can be estimated using the data of Table 2 as a difference in the mixing enthalpy between configuration 8 (RuMoMoRuMo) and configuration 25 (MoRuRuMoRu), in the latter configuration the occupancy of all sites is reversed. In temperature units, this energy amounts to some 2000 degrees Kelvin, implying that the Mo–Ru σ -phase should remain strongly ordered even at high temperatures. A similar site-occupancy trend has been observed experimentally in the σ -Mo2 Os phase [1]. Surprisingly, no atomic ordering was detected in the σ -Cr2 Os and σ -Cr2 Ru phases [1], although their stoichiometry implies the same kind of site occupancy as in the other isoelectronic σ -phases.
174
O. Gr˚an¨as et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 171–176 Table 3 Experimentally reported and ab initio calculated lattice parameters for the Mo–Ru σ -phase Ru mole fraction
˚ a [A]
˚ c [A]
c/a
Reference
0.4 0.3749 0.3650 0.3515 0.3333
9.55 9.5575 9.5569 9.5573 9.653
4.95 4.9345 4.9332 4.9325 4.961
0.518 0.5163 0.5162 0.5161 0.514
[29] [30] [30] [30] This work
Fig. 2. Total and s + p density of states (DOS) for MoRu σ -phase with sublattices occupations RuMoMoRuMo (configuration 8 in Table 2, solid and dot-dashed lines, respectively) and RuMoMoMoRu (configuration 10 in Table 2, dashed and dottedlines, respectively) as a function of energy E (relative to Fermi energy E F ). The difference between the total DOS and s + p partial DOS gives the contribution due to d-electrons, which clearly dominates the spectra at the Fermi energy. See text for more discussion.
In order to investigate the relationship between the atomic configuration and the electronic structure of the σ -phase, we compare in Fig. 2 the density of states (DOS) for configurations 8 and 10, which have the same stoichiometry but a reversed occupancy of sites 4 and 5. As one can see, in configuration 8 there are three bonding peaks below the Fermi level (centred around −4.6, −3 and −1.5 eV), whereas in configuration 10 there are only two peaks (centred around −4.3 and −3 eV). This indicates that the d-states in configuration 10 are unable to hybridise to the same extent as those in configuration 8. As compared to that of configuration 10, the DOS in configuration 8 is pushed from the band centre towards the band edges. As a result, the separation between the bonding and antibonding peaks increases and the DOS near the Fermi level is depleted. This implies stronger covalency of the interatomic bonding in configuration 8 relative to configuration 10, which accounts for the clear preference by Mo for high-CN sites and by Ru for IC sites. The calculated lattice parameters of the most stable endmember compound, configuration 8, are compared with experimental data on the Mo–Ru σ -phase in Table 3. The calculated values are within the scatter of experimental data. The calculated concentration dependence of the lattice parameter a (lattice parameter for the lowest energy endmembers as a function of composition) is presented in Fig. 3. As might be expected, the lattice parameter decreases with increasing the Ru content. A more interesting feature is, however, the negative deviation of the lattice parameter from Vegard’s law. This behaviour is a result of a stronger binding (and a shorter bond length) when Mo atoms preferentially occupy the high-CN sites and Ru atoms preferentially occupy the IC sites. The bulk modulus for the most stable end-member compound (configuration 8, composition Mo2 Ru) is found by
Fig. 3. Calculated lattice parameter (solid line) and lattice parameter predicted by Vegard’s law (dashed line) for the MoRu σ -phase as a function of the Ru mole fraction.
fitting the calculated volume dependence of total energy using the Murnaghan equation of state [26]. The obtained value, 274 GPa, is slightly higher than 267 GPa, which is the concentration average of the calculated (using the same procedure) bulk moduli of bcc Mo and hcp Ru, 251 GPa and 298 GPa, respectively. The latter two values compare reasonably well with the experimental room-temperature bulk moduli of bcc Mo, 263 GPa [27], and hcp Ru, 313 GPa [28]. The high value of the bulk modulus of configuration 8 is indicative of a high hardness of the σ -phase and is consistent with the strong covalent component of interatomic bonding for this endmember compound. 3.2. Free-energy models and atomic order at high temperatures Distribution of the alloy constituents among the five sublattices at a finite temperature for the Mo–Ru σ -phase can be obtained by minimisation of the free energy given by the five-sublattice model, Eqs. (2)–(4). The results of such modeling at 700 K are shown in the upper panel of Fig. 4. Note that this temperature is much below the stability range of the σ -phase in the Mo–Ru system. At the Mo2 Ru composition the atomic order is nearly perfect. Upon a deviation from this “ideal” stoichiometry towards Mo-rich compositions, the excess Mo atoms replace Ru atoms evenly on sites 1 and 4, while the high-CN sites 2, 3 and 5 remain fully occupied by Mo. On the other side, upon small deviations from the
O. Gr˚an¨as et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 171–176
175
Fig. 5. Site occupancy for the 5-sublattice model at 37.5% Ru for the stable temperature range.
Fig. 4. Site occupancy at 700 K obtained using the 5-sublattice (upper panel) and the 3-sublattice (lower panel) CEF models.
ideal stoichiometry towards Ru-rich compositions, the IC sites remain fully occupied by Ru, the excess Ru atoms substitute preferentially on the high-CN sites 2 and 3. At compositions between 33 at.% and 50 at.% Ru the excess Ru atoms are distributed evenly between sites 2 and 3, whereas the concentration of Ru on site 5 is much smaller. For a greater Ru content this tendency changes and site 2 becomes less occupied by Ru than sites 3 and 5, so that the occupancy of all the highCN sites by Ru is different for the Ru-rich compositions of the σ -phase. According to experiments [22] the σ -phase occurs in a narrow composition range around Mo0.63 Ru0.37 at temperatures between 1416 K and 2188 K. Our calculations performed using the 5-sublattice model for a fixed composition of Mo0.625 Ru0.375 show that the σ -phase remains highly ordered in the whole temperature range of its stability; see Fig. 5. Since the Mo–Ru σ -phase exists in a very narrow composition interval, a very detailed description of this phase at large deviations from this composition seems to be redundant. Therefore, we simplify the 5-sublattice model to a 3-sublattice model by merging sites 1 and 4 into one sublattice, site 2 and 3 into another sublattice, and by treating site 5 as the third sublattice. We note that the present 3-sublattice model is different from the model by Kasper and Waterstrat [31], in which sites 3 and 5 are merged because these sites have
similar coordination numbers. The present 3-sublattice model takes into account the site occupancy behaviour that is obtained using the full 5-sublattice model for the concentration interval of interest. Unfortunately, there are no experimental data on the site occupancy for the Mo–Ru σ -phase, although such data are available for other isoelectronic compounds [1,32]. As parameters of the 3-sublattice model we take the calculated enthalpies of formation for a subset of end-member compounds that are indicated in Table 2 using a bold font (these enthalpies are shown in Fig. 1 as filled circles). However, the energies of end-member compounds 14 and 20, which determine the slope of the enthalpy upon the Ru-rich deviations away from configuration 8, are no longer contained in the 3-sublattice model. In order to achieve good agreement between the free energy predictions by the 5-sublattice and the 3-sublattice models, the enthalpy of configuration 27 should be adjusted in such a way that the corresponding point (thereafter denoted as 27’) in Fig. 1 lies approximately on the same line with the points representing configurations 8, 14 and 20. The enthalpies of formation evaluated within the 5-sublattice and 3-sublattice models for three temperature values are displayed in Fig. 1. Our compound energy models predict that at high temperatures the free energy minimum for the σ -phase is shifted from Mo2 Ru towards more Ru-rich compositions, which agrees well with experimental findings. The free energy estimates of the two models agree very well with each other; however, the 3-sublattice model seems slightly to underestimate the entropy content. This may be a result of the fact that the 3-sublattice model predicts a smaller concentration of Ru on site 5 as compared to the predictions of the 5-sublattice model; see Fig. 4. 3.3. Summary and conclusion We present a thermodynamic description of the Mo–Ru σ -phase based on a combination of ab initio and CALPHAD methods. Lattice parameters, bulk moduli, enthalpies of formation as well as site occupancies are presented. By analysing the density of states we have found an increased
176
O. Gr˚an¨as et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 171–176
hybridisation among the d-orbitals when Mo occupies highcoordination-number sites and Ru occupies icosahedrally coordinated sites. This results in a strong tendency towards ordering in this compound. Acknowledgements The Swedish National Allocation Committee (SNAC) is acknowledged for supercomputer resources at the National Supercomputer Centre (NSC). The authors acknowledge financial support by the Swedish Foundation for Strategic Research and by the Royal Swedish Academy of Sciences. References [1] E. Hall, S. Algie, The sigma phase, Metall. Rev. 11 (1966) 61–88. [2] D. Downie, J. Martin, Heat-capacities of transition-metal alloys. II. Alpha-phase and sigma-phase Fe0.525 Cr0.475 and Fe0.566 Cr0.434 , J. Chem. Thermodyn. 16 (8) (1984) 743–752. [3] P. Korzhavyi, B. Sundman, M. Selleby, B. Johansson, Atomic, electronic, and magnetic structure of iron-based sigma-phases, Mater. Res. Soc. Symp. Proc. 842 (2005) S4.10.1. [4] J.-C. Zhao, J. Westbrook, Ultrahigh-temperature materials for jet engines, MRS Bull. 28 (9) (2003) 622–630. [5] I. Ansara, T.G. Chart, A.F. Guillermet, F.H. Hayes, U.R. Kattner, D.G. Pettifor, N. Saunders, K. Zeng, Thermodynamic modelling of solutions and alloys — Schloss Ringberg, March 10 to March 16, 1996 — Group 2: Alloy system I — Thermodynamic modelling of selected topologically close-packed intermetallic compounds, CALPHAD 21 (2) (1997) 171–218. [6] P.E.A. Turchi, I.A. Abrikosov, B. Burton, S.G. Fries, G. Grimvall, L. Kaufman, P.A. Korzhavyi, V. Rao Manga, M. Ohno, A. Pisch, A. Scott, W. Zhang, CALPHAD 31 (1) (2007) 4–27. [7] S. Fries, B. Sundman, Using Re–W sigma-phase first-principles results in the Bragg–Williams approximation to calculate finite-temperature thermodynamic properties, Phys. Rev. B 66 (1) (2002) 012203. [8] M.H.F. Sluiter, K. Esfarjani, Y. Kawazoe, Phys. Rev. Lett. 75 (17) (1995) 3142–3145. [9] C. Berne, M. Sluiter, Y. Kawazoe, A. Pasturel, Ordering effects in the Re–W and Re–Ta sigma phases, J. Phys. - Condens. Matter 13 (42) (2001) 9433–9443. [10] C. Berne, M. Sluiter, Y. Kawazoe, T. Hansen, A. Pasturel, Site occupancy in the Re–W sigma phase, Phys. Rev. B 64 (14) (2001) 144103. [11] J. Houserova, J. Vrestal, M. Sob, Phase diagram calculations in the Co–Mo and Fe–Mo systems using first-principles results for the sigma phase, CALPHAD 29 (2) (2005) 133–139.
[12] M. Hillert, The compound energy formalism, J. Alloys Compounds 320 (2) (2001) 161–176. [13] P. Hohenberg, W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136 (3B) (1964) 864–871. [14] W. Kohn, L. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140 (4A) (1965) 1133–1138. [15] G. Kresse, D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59 (3) (1999) 1758–1775. [16] G. Kresse, J. Furthm¨uller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6 (1996) 15–50. [17] P. Bl¨ochl, Projector augmented-wave method, Phys. Rev. B 50 (24) (1994) 17953–17979. [18] J. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77 (18) (1996) 3865–3868. [19] H.J. Monkhorst, J.D. Pack, Special points for brillouin-zone integrations, Phys. Rev. B 13 (12) (1976) 5188–5192. [20] A.V. Ruban, I.A. Abrikosov, H.L. Skriver, Electronic structure and thermodynamicproperties of ordered, partially ordered and random Cu–Au and Ni–Pt alloys, Phys. Rev. B 51 (19) (1995) 12958–12968. [21] A.E. Kissavos, S. Shallcross, L. Kaufman, O. Gr˚an¨as, A.V. Ruban, I.A. Abrikosov, Thermodynamics of ordered and disordered phases in the binary MoRu system, Phys. Rev. B 75 (18) (2007) 184203(8). [22] H. Kleykamp, Constitution and thermodynamics of the Mo–Ru, Mo–Pd, Ru–Pd and Mo–Ru–Pd systems, J. Nucl. Mater. 167 (1989) 49–63. [23] G. Grimvall, M. Thiessen, A.F. Guillermet, Phys. Rev. B 36 (15) (1987) 7816–7826. [24] K. Persson, M. Ekman, G. Grimvall, Phys. Rev. B 60 (14) (1999) 9999–10007. [25] A. Kissavos, S. Shallcross, V. Meded, L. Kaufman, I. Abrikosov, A critical test of ab initio and calphad methods: The structural energy difference between bcc and hcp molubdenum, CALPHAD 29 (1) (2005) 17–23. [26] F.D. Murnaghan, Amer. J. Math. 59 (2) (1937) 235–260. [27] K.V. Katahara, M.H. Manghnani, E.S. Fisher, Pressure derivatives of the elastic moduli of bcc Ti–V–Cr, Nb–Mo and Ta–W alloys, J. Phys. F. Met. Phys. 9 (55) (1979) 773–790. [28] E.S. Fisher, D. Dever, Temperature dependence of elastic moduli of ruthenium, rhenium, cobalt, dysprosium, and erbium; a study of the elastic anisotropy-phase transformation relation, Trans. Metall. Soc. AIME 239 (1) (1967) 48–57. [29] E. Bucher, F. Heiniger, J. M¨uller, Helv. Phys. Acta 34 (1961) 843–858. [30] E. Anderson, W. Hume-Rothery, J. Less-Common Met. 2 (1960) 443–450. [31] J. Kasper, R. Waterstrat, Ordering of atoms in the sigma-phase, Acta Crystallogr. 9 (1955) 289. [32] H. Yakel, Atom distributions in sigma phases. I. Fe and Cr atom distributions in a binary sigma phase equilibrated at 1063, 1013 and 923 k, Acta Crystallogr. B 39 (1) (1983) 20–28.