Phase stability in the ternary Re–W–Zr system

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ScienceDirect Acta Materialia 70 (2014) 56–65 www.elsevier.com/locate/actamat

Phase stability in the ternary Re–W–Zr system Jean-Marc Joubert ⇑, Jean-Claude Crivello, Mohamed Andasmas, Pierre Joubert Chimie Me´tallurgique des Terres Rares (CMTR), Institut de Chimie et des Mate´riaux Paris-Est (ICMPE), CNRS, Universite´ Paris-Est, 2-8 rue Henri Dunant 94320 Thiais Cedex, France Received 20 August 2013; received in revised form 19 December 2013; accepted 5 February 2014 Available online 13 March 2014

Abstract The irregular shape of the v phase in the ternary system Re–W–Zr has been investigated. Both experimental measurements (joint Rietveld refinement of neutron and X-ray diffraction data) and density functional theory (DFT) calculations were used in order to obtain the site occupancies and to determine the reason why this phase is so deeply stabilized in the ternary system. In particular, the DFT calculations of all the system compounds allowed us to assess the relative phase stability and to reproduce well the experimental phase diagram isothermal section without any adjustable parameter, including the C14-to-C15 transition. In the course of this study a partial reinvestigation of the binary Re–Zr system has been performed that concluded on the absence in the phase diagram of the previously reported Zr2Re phase. Instead, the phase Zr21Re25 was proven. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Density functional theory (DFT); Joint Rietveld refinement; Neutron diffraction; Phase diagram; Phase diagram calculation

1. Introduction Four isoelectronic ternary systems (Zr–Mo–Re, Zr–W–Re [1], Hf–Mo–Re [2] and Hf–W–Re [3]) share a common specific characteristic. They all possess a v phase (aMn, Ti5Re24 or Mg17Al12 prototype, space group I43m) not only extending continuously between the two binary systems in which it exists, but also having a large, unusual, ternary extension towards the third binary system in which it does not. To illustrate this feature, an isothermal section of the Re–W–Zr system is presented in Fig. 1. The reason for this particular shape has not been discussed so far and is still unknown. However, it is interesting to note that the lines defining the homogeneity domain originating from Zr5Re24 are parallel to either the W–Zr or Re–W border, suggesting that both Zr-by-W and Re-by-W substitution ⇑ Corresponding author. Tel.: +33 1 49 78 13 44; fax: +33 1 49 78 12 03.

E-mail address: [email protected] (J.-M. Joubert).

schemes could occur. This demonstrates the interest in measuring the site occupancies in the ternary phase in order to better understand which ordered phase this homogeneity domain points towards. As three elements possibly sharing four different sites are involved, this has been done by joint Rietveld refinement of X-ray diffraction (XRD) and neutron powder diffraction (ND) data. In addition, to obtain more precise data on the phase stability, formation enthalpy calculations were carried out in the frame of density functional theory (DFT). The stability of every ordered configuration of each phase obtained by generating all the possible distributions of the three elements on the different sites has been compared. Solid solutions have also been studied by the special quasi-random structure (SQS) technique [4]. Finally, as uncertainties can be guessed from contradictory reports on the binary Re–Zr system (see detailed bibliographic review in Section 4.2), a partial reinvestigation of this system was conducted in order to clarify experimentally the phase existence.

http://dx.doi.org/10.1016/j.actamat.2014.02.010 1359-6454/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

J.-M. Joubert et al. / Acta Materialia 70 (2014) 56–65

Fig. 1. Isothermal section at 1350 °C of the Re–W–Zr system (reproduced after Ref. [1]). The nominal (closed symbols) and analyzed (open symbols) compositions of the ternary samples are shown. The line drawn between the analyzed compositions corresponds to the section studied in Fig. 9.

2. Experimental techniques Two ternary Re–W–Zr and two binary Re–Zr alloys were synthesized from the pure elements (Re, Alfa Aesar, <44 lm, 99.99%; W, Alfa Aesar, <44 lm, 99.9%; Zr, Stream Chemicals, <44 lm, 99.99%), which were carefully weighed to obtain the compositions of the required stoichiometry. The powders were thoroughly mixed using an agate mortar and a uniaxial pressure was applied to the powders in order to obtain a cylindrical compact which was then melted in an arc furnace under argon atmosphere. The alloys were remelted several times and rotated upside down between each melting to achieve better homogenization. For the sake of reaching equilibrium, the alloys were annealed as mentioned in Table 1. The annealing treatment

57

of the samples was performed in a high-frequency induction furnace in a water-cooled copper crucible under an argon atmosphere, and the sample temperature was measured with the help of an optical pyrometer. After annealing, the samples were quenched by turning off the induction heating and rapid cooling of the sample took place using the water-cooled copper crucible. Thus, the transition from the annealing temperature to the ambient one proceeds in a few seconds. Prior to the structural characterization, part of each sample was polished and investigated with the help of electron probe microanalysis (EPMA; CAMECA SX100) using pure elements as standards. Large numbers of data points were measured to determine the phase composition and homogeneity. The phase composition was averaged from the point measurements performed on that phase, and the calculated standard deviation of these measurements represented the phase homogeneity. For the structural characterization, XRD and ND measurements were performed by using the powder method. All samples, by virtue of their high brittleness, were easily reduced to a fine powder (<63 lm). Powder XRD measurements were performed at room temperature with Cu Ka radiation on a Bruker D8 Advance diffractometer (Bragg–Brentano geometry) equipped with a rear graphite monochromator between 5° and 120° (2h) with a step size of 0.04°, and a total counting time up to 63 h to obtain high data statistics. The ternary samples were also measured by ND performed at the Laboratoire Le´on Brillouin (LLB; common laboratory CEA–CNRS), Saclay, France, using a 3T2 instrument. The sample in powder form (6 g, <63 lm) was introduced into a sample holder consisting of a vanadium cylinder (diameter 6 mm). The measurement was performed at room temperature with the Debye– Scherrer geometry in the presence of five detectors. For

Table 1 Thermal treatment and characterization of the samples (phase nature, lattice parameters and phase amount by XRD, phase composition by EPMA). Nominal composition

Thermal treatment (T (°C), time (h))

Present phases

Lattice ˚) parameters (A

Phase amount (wt.%)

Composition (at.%)

Re66W17Zr17

1500, 8

v C14 (ZrRe2)

a = 9.742 a = 5.277 c = 8.628 –

94 5

Re67.8(3)W18.0(1)Zr14.1(4) Re60W10Zr30

ZrO2

1

Re47W39Zr14

1500, 8

v

a = 9.789

100

Re48.5(8)W39.5(3)Zr12(1)

Re45Zr55

1500, 4.5

Zr21Re25

a = 25.820 c = 8.757 a = 3.254 c = 5.183 a = 5.280 c = 8.643

84

Re51.0(4)Zr49.0(4)

12

Re1.0(2)Zr99.0(2)

3

Re63.2(3)Zr36.8(3)

a = 25.819 c = 8.757 a = 3.254 c = 5.180 a = 5.283 c = 8.650

82

Re50.9(6)Zr49.1(6)

14

Re4(1)Zr96(1)

4

Re62.3(2)Zr37.7(2)

hcp (aZr) C14 (ZrRe2) Re33.3Zr66.6 (Zr2Re)

1500, 4.5

Zr21Re25 hcp (aZr) C14 (ZrRe2) Other phase

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each pattern, data were recorded in the angular range 5° 6 2h 6 121° with a step size of 0.05° by using neutrons ˚. of wavelength 1.2251 A To determine site occupancies, the combined Rietveld analysis of XRD and ND data was conducted using the FULLPROF program in the multi-pattern mode [5]. For each pattern, the background was interpolated between the peaks and the pseudo-Voigt function was used to define the peak shape. Because Re presents the non-negligible absorption coefficient for neutrons, an absorption correction was also applied by calculating the values of the absorption coefficient for each composition. To calculate the absorption coefficient, the density of non-compacted powders was estimated to be 40% of the density of the material. In order to get the same displacement parameters from both diffraction sets, the micro-absorption parameters corresponding to the XRD were refined. The refinement was conducted using exactly the same technique as used for the Mo–Ni–Re r phase [6]. The composition was kept constrained to the composition measured with the EPMA, allowing the refinement of the site occupancies of the three elements in each of the four sites of the crystal structure of the v phase (see the procedure in Refs. [7,8]). 3. Computational details All the calculations were performed in the frame of the DFT, using the VASP package [9,10]. The generalized gradient approximation is used with the Perdew–Burke– Ernzerfhof exchange and correlation energy functional [11,12]. A cut-off energy of 400 eV was used for the plane-wave basis set. The SQS structures corresponding to three compositions (0.25, 0.5, 0.75) of the three binary solutions have been taken from the literature with the 16-atom description for face-centered cubic (fcc) [13], body-centered cubic (bcc) [14] and hexagonal close packed (hcp) [15]. They are considered structural templates with correlation functions close to those of completely random structures with a limited number of atoms. In addition, infinite dilution is modeled by the calculation at 1/16 and 15/16 compositions for all solid solutions using a supercell. From these results, a regular interaction parameter has been calculated to match the mixing enthalpy (corresponding to the fully relaxed structure) at x = 0.5, and taken into account for the phase diagram calculation. The detailed procedure for calculation of the v and r phases can be found in other papers [16,17]. A similar method was used for the Laves C14 and C15 with 16  16  10 and 16  16  16 k-meshing, respectively. By using the ZenGen script [18], the generation and calculation of all the ordered configurations of the Re–W–Zr ternary phases have been carried out, i.e. 32 = 9 C15 (space group Fd 3m, Wyckoff positions 8a, 16d), 33 = 27 C14 (P63/mmc, 2a, 4f, 6h), 34 = 81 v (I 43m, 2a, 8c, 24g, 24g), 35 = 243 r (P42/mnm, 2a, 4f, 8i, 8i, 8j) compounds.

As in previous articles, the enthalpy of formation DHfor of an ordered configuration is obtained by the difference between the total energy and the composition weighted energies of pure elements in their stable reference structure. For example, for a ternary Re–W–Zr compound with configuration ijkl in the structure of the v phase, DHfor can be written: hcp hcp v bcc DH for;v ijkl ¼ E ijkl  cRe  ERe  cW  E W  cZr  EZr

ð1Þ

where cX is the mole fraction of the element X. Based on the compound energy formalism (CEF) [19], the four sublattice model (i)2(j)8(k)24(l)24 is used to describe the v phase, in agreement with the site multiplicities in the v crystal structure. The Gibbs energy Gv({xi}, T) is calculated at a finite temperature T using the Bragg–Williams (BW) approximation (both interactions within sublattices and non-configurational entropies were neglected) [20]. In our example: X ð2aÞ ð24gÞ ð24gÞ Gv ðfvi g; T Þ ¼ ½y i y 8c y l   DH for;v ijkl j yk i;j;k;l¼Re;W;Zr

þ RT

X aðsÞ  s

X

ðsÞ

ðsÞ

ð2Þ

y i ln y i

i¼Re;W;Zr ðsÞ

R is the gas constant, a(s) the site multiplicity and y i the site fraction of the element i on site s. The Calphad lattice stabilities were not used, only the DFT values. An entropy term has been added to bcc Zr in order to have the phase stabilized at high temperature. The site occupancies and phase diagram were computed using Thermo-Calc software [21]. 4. Results 4.1. Re–W–Zr samples The structure of the v phase, as other Frank–Kasper phases, is characterized by extensive non-stoichiometry accommodated by a substitution mechanism (for a complete review on the v phase, see Ref. [22]). In principle, each of the four sites of the crystal structure may be shared by the three elements. In order to solve the problem of having two occupancy unknowns per site to be determined, combined diffraction analysis should be carried out. More details about this can be found in Ref. [7]. The procedure adopted for the structural investigation was identical to that used for the r Mo–Ni–Re phase [6], i.e. joint Rietveld refinement of XRD and ND powder data. Among the four systems presenting the unusual shape of the v phase mentioned in the introduction, the Re–W–Zr system was specifically chosen since it offers the best diffraction contrast by XRD and ND. Expressed in a simple way, since Re and W are close neighbors in the Periodic Table, the Zr vs. (Re + W) site occupancies may be determined by XRD while the Re-to-W ratio on each site can be obtained from ND, Zr content being constrained by the XRD, owing to the good contrast (bRe = 9.2 fm, bW = 4.9 fm). Of course,

J.-M. Joubert et al. / Acta Materialia 70 (2014) 56–65

in practice, the refinement is combined and both patterns are analyzed simultaneously. The studied compositions were chosen in order to be representative of the different states of order to allow clarification of the complex substitution mechanism. They should also correspond to the stable homogeneity domain as reported in the only available isothermal section [1]. Therefore, the compositions were chosen to be Re66W17Zr17 and Re47W39Zr14, inside the homogeneity domain, and they were annealed at 1500 °C, i.e. slightly above the temperature of the isothermal section (1350 °C). The primary characterization by XRD and EPMA (see Table 1) revealed that one of the samples was single phase while the other sample contained a small amount of additional C14 secondary phase and a negligible quantity of zirconium oxide. It is not known if the presence of this phase is due to insufficient annealing conditions preventing complete thermodynamic equilibrium being reached, to the uncertainties in the phase diagram determination (the composition has been chosen quite close to the reported border of the homogeneity domain and C14 phase is the phase in equilibrium with the v phase in this region of the phase diagram) or to slight changes of the isothermal section between 1350 °C and 1500 °C. However, in both samples the v phase composition was homogeneous and corresponded approximately to the nominal one. The presence of the secondary phase does not preclude further analysis and it was taken into account in the Rietveld refinement. At the end of the unconstrained refinement, occupancy parameters for some atoms were found to be slightly negative or above full occupancy. These were constrained to be 0 and 1, respectively, in the final refinement. The refined XRD and ND patterns are shown in Fig. 2 for the sample Re66W17Zr17. The results, including site occupancy measurements, are reported in Table 2. The preference of Zr for high coordination number (CN) sites 2a and 8c (CN16) can be noted. Re and W share the two sites of lower coordination. However, when present simultaneously, they are not distributed equally since a preference of W for the site of CN13 and of Re for the site of CN12 is observed. This is in agreement with the size criterion which states that the larger atoms prefer the sites with higher CN ˚ > RW = 1.41 A ˚ > RRe = 1.37 A ˚ ). (RZr = 1.60 A 4.2. Re–Zr samples Prior to the DFT computation of all phases of the ternary system, we had to know which phase had to be taken into account. However, contradictory results exist in the literature concerning the Re–Zr system, one of the constituting binary systems. A study was therefore undertaken in order to solve the discrepancies observed in the literature, in particular between the reported phase diagram, the composition of one of the phases and the stoichiometry calculated from it crystal structure.

59

a

b

Fig. 2. Rietveld plots corresponding to the XRD (a) and ND (b) patterns issued from the joint refinement of the structure of sample Re66W17Zr17. The measured (points) and calculated (line) patterns and difference curve (line below) are shown. The markers indicate the Bragg line positions of the three indexed phases: v, C14 and ZrO2 (top to bottom).

In the earlier studies of the Re–Zr phase diagram, a phase of composition Zr2Re was reported without clear structural determination [23]. This is the phase present in the isothermal section of Fig. 1. In a latter report of a complete investigation of the phase diagram [24], this phase was indicated to have the r phase structure. However, the reported lattice parameters are quite unrealistic (they are much larger than those of the r phase in any other system [25]). The other phases of the system are more clearly identified: the v phase around the composition Zr5Re24 and a C14 phase at the composition ZrRe2 (for a review, see Ref. [26]). In a more recent work, another crystal structure was identified from a single crystal investigation with a new, rather complex, structure type [27]. The composition deduced from the structural analysis, i.e. considering a fully ordered distribution of the atoms on the different sites (stoichiometric composition) was Zr21Re25. No chemical analysis supported this composition and since the analysis was made on a single crystal, the nominal composition was not helpful to estimate the composition of the phase. This structure type was also reported in the systems Mn–Ti and

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J.-M. Joubert et al. / Acta Materialia 70 (2014) 56–65

Table 2 Results of the joint refinement of ND and XRD data of the v phase (space group I 43m). Nominal composition v phase composition Lattice parameter a ˚) (A

Re66W17Zr17

Re47W39Zr14

Re67.8W18.0Zr14.2 9.7427(1)

Re48.5W39.5Zr12 9.7892(2)

Site positions and occupancies 2a 0, 0, 0 Re 0.06(10) W 0.27(10) Zr 0.67(10) 8c xxx x = 0.3161(2) Re 0 W 0.23(3) Zr 0.77(3) 24g1 xxz Re W Zr 24g2 xxz Re W Zr

x = 0.3166(3) 0 0.38 0.62

x = 0.3593(1) z = 0.0385(2) 0.69(1) 0.28(1) 0.03(1)

x = 0.3595(1) z = 0.0402(2) 0.39(1) 0.61(1) 0

x = 0.0919(1) z = 0.2851(1) 0.95(1) 0.05(1) 0

x = 0.0924(2) z = 0.2847(2) 0.78(1) 0.22(1) 0

Displacement parameters ˚ 2) 0.31(5) B2a = B8c (A ˚ 2) B24g1 = B24g2 (A 0.24(1) RB ND (%) RB XRD (%)

0 0 1

10.1 2.4

0.48(5) 0.30(2) 10.6 3.4

Hf–Re and it is shown that in these systems the phase appears in the phase diagram at a slightly different composition from that deduced from the structural analysis (50 at.% Ti or Hf instead of 45.6 at.%). In the recent assessments of the system, an intermetallic compound was systematically placed at the composition Zr2Re. In Ref. [28], the name Zr21Re25 was conserved for this compound but this implies a huge discrepancy between the composition issued from the crystal structure (45.6 at.% Zr) and the one reported in the phase diagram (66.7 at.% Zr). In Ref. [29], the composition Zr2Re is also reported but mention of the structure of Zr21Re25 disappeared. Finally, the enthalpies of formation of different phases of the Re–Zr system, including the phase of composition Zr21Re25 corresponding to the structure of Ref. [27], were calculated by Levy et al. [30] and more recently by Wang et al. [31] using the DFT. This phase appears to be present on the ground state of the Re–Zr system, which is an indication, though not a proof, of its stability. In conclusion, considering the whole knowledge of this system, several hypotheses may be formulated:

– The phase with the reported structure of Zr21Re25 exists but is strongly non-stoichiometric and has the composition of Zr2Re. – The phase with the reported structure of Zr21Re25 exists at its stoichiometric composition and Zr2Re does not exist. – Both Zr21Re25 and Zr2Re intermetallic phases exist in this region of the phase diagram, the crystal structure of the latter being unknown. In order to acquire a decisive understanding about this point, two samples were synthesized and annealed at 1500 °C during 4.5 h. The characterization of both samples is presented in Table 1. In the sample of composition Re45Zr55 two phases were evidenced and are clearly identified by the Rietveld analysis: one has the structure of the phase reported as Zr21Re25 [27] with similar lattice parameters, the other is the hcp phase corresponding to aZr. A third phase is also identified as being the C14 Laves phase corresponding to ZrRe2. This latter phase is probably not an equilibrium phase but rather results from an incomplete peritectic reaction. The XRD pattern of this sample with the obtained Rietveld calculated curve is presented in Fig. 3. It is worth noting also that the bZr, high temperature allotropic form of zirconium is not retained by quenching. It is found to transform into the a form. Unexpectedly, significantly larger lattice parameters are observed for this phase than for pure Zr, which can hardly be attributed to Re substitution because only minimal solubility is found and because Re atomic radius is smaller than that of Zr. Effects resulting from the martensitic transformation between bZr and aZr could perhaps explain this phenomenon. From the EPMA results, it is found that the crystal structure not only of Zr21Re25 but also a composition for this phase close to the stoichiometric composition (45.6 at.% Zr) is observed. Slight over-stoichiometry in Zr (49 at.% Zr) is noticed is but far from reaching the composition of the so-called Zr2Re phase (66.6 at.% Zr).

Fig. 3. Rietveld plot corresponding to the sample Zr55Re45. The measured (points) and calculated (line) patterns and difference curve (line below) are shown. The markers indicate the Bragg line positions of the three indexed phases: Zr21Re25, aZr and C14 (top to bottom).

J.-M. Joubert et al. / Acta Materialia 70 (2014) 56–65

61

Fig. 4. Calculated heat of formation of all the ordered compounds (C15, C14, v and r phases) in the three binary systems of Re–W–Zr, and of the single Zr21Re25 compound. A solid line links most stable configurations of each ordered phases. Dashed straight lines link the enthalpies of the pure element in fcc, bcc and hcp, which illustrate their lattice stability in the different structures.

A further confirmation of the composition of the phase was obtained from the Rietveld analysis. The possibility of a wrong atom assignment or mixed occupancy that would possibly explain a phase composition different from that defined by the structure was studied. The site occupancies in each position were refined, site per site. Neither a large extent of atom mixing was refined nor a significant improvement of the refinement observed, indicating that non-stoichiometry for this compound is probably minor. Finally, the measured phase amounts are in pretty good agreement with the quantities anticipated from the lever rule. Recalculating the composition of the phase Zr21Re25 from the nominal composition of the alloy, the phase amounts obtained by XRD and the EPMA composition of the minor phases (aZr and C14) gives Zr47Re53 close to the stoichiometric composition and in very good agreement with the measured composition by EPMA. A second sample was synthesized at the composition of the reported compound Zr2Re. The analysis of this sample confirmed to a large extent the results obtained with the first sample: no EPMA point over 200 measurements indicated a composition corresponding to Zr2Re. The sample is also composed of Zr21Re25 and a significant quantity of ZrRe2. However, the analysis of the XRD pattern is made complex by the fact that bZr is present in much larger amounts at the equilibrium in this sample and it may have unknown – possibly irreversible – decomposition products during quenching. As in the other sample aZr is identified. However, other peaks are present, among which remaining bcc was tentatively assigned but is not certain due to the complexity of the XRD pattern caused by the presence of Zr21Re25. We can conclude that, at least for this temperature (T = 1500 °C), the phase Zr2Re does not exist, but rather

the phase Zr21Re25. It has a composition close to that indicated by the stoichiometry or slightly richer in Zr. Given the number of peaks and distribution of the intensities, it is possible that this phase was wrongly attributed to the r phase in earlier work. The phase diagram may most probably look like that of the Hf–Re system in which the same Hf21Re25 phase is reported and placed correctly [3,28]. 4.3. DFT results DFT calculations have been performed for the complete set of ordered configurations of all the phases present in the ternary system (C15, C14, v and r phases). The phase Zr21Re25 (92 atoms distributed on 10 sites in the primitive rhombohedral cell) has only been calculated for a single configuration corresponding to its stoichiometric composition. The enthalpy of formation is found to be similar to that of Ref. [30]. The complete set of the resulting enthalpies of formation are supplied in the Supplementary materials section. The ground state corresponding to the three constituting binary systems is plotted in Fig. 4. In each case, the predicted stable phases at 0 K correspond to what is found in the binary phase diagrams at low temperature. In the Re–Zr system, one may note that the phase Zr21Re25 actually lies on the ground state, which is another confirmation of the stability of this phase. In the Re–W system, the r phase is close to the ground state and has been shown to be stabilized at high temperature by the configurational entropy [32,33]. There is always a subtle difference between the stability of the v and r phases. As expected, the v phase is stable for the two binary compositions W5Re24 and Zr5Re24. For a more detailed discussion about the

62

J.-M. Joubert et al. / Acta Materialia 70 (2014) 56–65 Re

W

40

W -10

fcc-vol fcc-full bcc-vol bcc-full hcp-vol hcp-full

20

WZrW

-15

Gibbs energy (kJ/mol)

30

Heat of mixing (kJ/mol)

Zr

10 0

C14 - T=0 K C15 C14 - T=1000 K C15 C14 - T=2000 K C15

ReZrW

-20

-25

WZrRe ReZrRe

-30

-10 -35 -20 0.0

0.25

0.50

0.75

xRe

1.0

0.25

0.50

xZr

0.75

1.0

0.25

0.50

xW

0.75

1.0

-40

0.0

ZrW 2

0.1

0.2

0.3

0.4

xRe

0.5

0.6

ZrRe2

Fig. 5. Enthalpy of mixing for the solution phases (j: fcc, d: bcc, N: hcp) in the three binary systems of Re–W–Zr calculated with SQS first principles method. Closed and open symbols represent symmetry preserved and fully relaxed calculations, respectively.

Fig. 7. Calculated heat of formation (points) of the C15 and C14 Laves phases along the ZrW2–ZrRe2 line and computed Gibbs energies at the indicated temperatures.

comparison of the stability of both phases studied by DFT in Re–X systems, see Ref. [16]. In both Re–Zr and W–Zr, the Laves phases are stable with a sharp inflection of the ground state. Very close values are obtained between the C14 and the C15 structures, as generally observed for these two closely related phases [34]. However, the correct structure is identified in both systems (C14 for Re–Zr, C15 for W–Zr). On the contrary, both phases are far from being stable in the binary Re–W system. The enthalpies of mixing for the solution phases in the three binaries calculated with the SQS method are represented in Fig. 5. They correspond to the difference between the heat of formation of the solutions and the dashed lines in Fig. 4. The SQS structure relaxation has been done in two steps. First, only the cell volume is allowed to relax with the symmetry preserved; then, the internal parameters are optimized, keeping the obtained volume fixed. The fcc structure has been considered, even if it does not appear in this system. As usually observed, fcc and hcp show a similar behavior, with the exception of the Re–W system

(probably due to the unstable lattice of fcc W [35]. The enthalpies of mixing in bcc are negative for Re–W and Re–Zr, and positive for W–Zr. This result agrees not only with the large solubility of Re in bcc W and bZr, but also with the low solubility of W and Zr in bZr and bcc W, respectively (see Fig. 1). In the three systems, mixing energies are positive for hcp structure (almost zero for Re–W), indicating a repulsive interaction between elements in this structure. By using the CEF combined with the BW approximation, the Gibbs energy surface of each phase could be estimated at a finite temperature. As an example, Fig. 6 shows a tridimensional view of the Gibbs energy surface of C15, C14, v and r phases at 100 K. The intersections of the surface with the binary borders reproduce the ground state of Fig. 4. In the ternary composition range, v presents a flat Gibbs surface close to the Re corner, as the r phase does, but with lower energies. The Laves V shape from ZrRe2 extends into the ternary field towards ZrW2. Fig. 7 shows the competition between the Laves phases along the ZrRe2–ZrW2 line. The stability inversion is expected for xRe = 1/6 because of the presence of the stabilizing ternary C14 end-member ReWZr with Re and W occupying the two different CN12 sites 2a and 6h, respectively. The C15 phase cannot be stabilized by ternary configurations because it has only two inequivalent sites. The ground state of the v phase is represented in Fig. 8. Stable configurations differing by the occupancy of only one site define single v phase domains and are connected. Most stable configurations correspond to compounds with larger atoms on the site of higher CN, as usually observed in Frank–Kasper phases [20]. Among the 15 points defining the ground state, three correspond to ternary compositions (ZrWWRe, ZrZrWRe, ZrWReRe). The triangle limited by the composition W2W8Re24Re24, Zr2Zr8Re24Re24 and Zr2Zr8W24Re24 defines a wide zone of larger stability, as indicated by the Gibbs energy surface in the same figure, with a metastable extension towards the binary Zr2Zr8W24W24.

Fig. 6. Gibbs energy surface of the C15, C14, v and r phases in Re–W–Zr system at 100 K calculated by DFT using the BW approximation.

J.-M. Joubert et al. / Acta Materialia 70 (2014) 56–65

W

verification, one may be confident that DFT results describe properly both the substitution behavior even outside the homogeneity domain and the relative phase stability in all the composition range. These two features are analyzed in the two following sections.

ZrWWW ZrZrWW

WWWRe

5.2. Substitution mechanism

ZrWWRe ZrZrZrW

Re

ZrReReRe ZrZrReRe

3e4 4e4

1e 4 2e4

3

5e4 6e4 7e4 8e4 9e4

ZrWReRe WReReRe

1e

-1 e

3

ZrZrWRe

WWReRe

63

ZrZrZrRe

Zr

Fig. 8. Ground state of the ternary v phase. The compositions of the different configurations are shown. These defining the ground state are represented as filled blue squares, the other ones as open red squares. The lines between stable configurations figure out the substitution on one site only. In addition, the Gibbs energy surface of Fig. 6 is represented here for the v phase as iso-contours (the reference states for pure elements have been changed compared to Fig. 6 for better understanding). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

5. Discussion 5.1. Comparison between experimental and calculated site occupancies The ternary site occupancies have been calculated along the composition line passing by the two ternary samples studied experimentally. They are plotted in Fig. 9 as a function of W composition in the section of the ternary system represented by a line in Fig. 1. The excellent agreement obtained between experimental and calculated results testifies the quality of both approaches. Such a good agreement has also been obtained for other Frank–Kasper phases like the r phase in other binary or ternary systems [6,33,36]. With this experimental

If we observe Fig. 9, it is clear that the site occupancies are driven mainly by the size criterion, as usual for Frank–Kasper phases. Zr, definitely the larger atom ˚ ), occupies the two sites with CN16. (RZr = 1.60 A However, a slight difference is observed between the two sites with a stronger preference for site 2a than 8c. This is the first time a difference of occupancies between these two sites is mentioned [22] but it is also the first time a ternary v phase is completely characterized from the structural point of view and it is more unlikely to occur in a binary phase. This is due to the stability of the configuration ZrWReRe in the ground state of Fig. 8. Starting from Zr5Re24, it is evident that W replaces Zr to reach W5Re24. As mentioned, the replacement occurs on both 2a and 8c sites but with a slight preference of W for 8c. The behavior along the other direction, i.e. going towards the metastable Zr5W24 is less predictable and is close to what is shown in Fig. 9 (we chose for this figure the section passing by the experimental points, but this does not make a large difference). It is evident that W replaces Re first on site 24g1 of CN13 and then on site 24g2 of CN12. This is caused by the stable configuration ZrZrWRe shown on the ground state of Fig. 8. This may also be attributed to the size effect, W being larger than ˚ , RRe = 1.37 A ˚ ). Contrary to the differRe (RW = 1.41 A ence of occupancy parameters between the two sites of high CN, that between the two sites of low CN has been abundantly commented in Ref. [22]. Finally, it is obvious that plotting the site occupancies along the line joining W5Re24 and metastable Zr5W24 would yield a complete reversal of site occupancies, W being the intermediate size atom, having to switch between the two sites of high CN in the binary system, in which it is the larger atom, to the sites of low CN in the binary system, in which it is the smaller atom. This strange behavior has already been demonstrated experimentally in e.g. the r Mo–Ni–Re phase [6]. 5.3. Calculation of the DFT phase diagram and comparison with the experimental section

Fig. 9. Experimental (points) and calculated (lines) site occupancies on the four sites of the structure of the v phase of the Re–W–Zr system along the section Re84.3Zr15.7 –W93.1Zr6.9 represented in Fig. 1.

Finally, the phase diagram has been computed from the complete set of DFT data. It is therefore a purely ab initio diagram. It is plotted in Fig. 10. One may notice very strong similarities with the experimental section. In particular, as anticipated by the observation of the relative Gibbs energies of the C14 and C15 phases, the transition between the two phases is calculated at the exact composition. Note also that all the tie-triangles of the isothermal section are

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W

C15

W5Re24

Re

Zr5Re24 C14

Zr21Re25

Zr

Fig. 10. Isothermal section at 2223 °C calculated by DFT and the CEF in the BW approximation.

correctly reproduced, thus indicating a good representation of the relative stability of the different phases, not only the binary ones, but also in the ternary field. Finally, the most striking similarity is the reproduction in the calculated section of the odd shape of the v phase. It is therefore guided by the stability of the different end-members in this region. We mentioned the flat behavior of the Gibbs energy curve in the triangle W2W8Re24Re24–Zr2Zr8Re24Re24–Zr2Zr8W24Re24. The relative stability of the ternary end-member Zr2Zr8W24Re24 and to a lesser extent of the metastable binary Zr2Zr8W24W24 is therefore responsible for the shape of the v phase together with the stabilization by the configurational entropy. This is in line with the description made in the previous paragraph about the complex substitution scheme and the possible replacement of both Zr-by-W and Re-by-W in Zr5Re24. The reason for the anomalous shape of the stable homogeneity domain of the v phase in the Re–W–Zr system is therefore elucidated: it points toward the stable state Zr2Zr8W24Re24 though, for this composition, a significant atom mixing of Re and W among the two sites of CN13 and CN12 is observed due to configurational effects. Then it extends as a metastable phase towards Zr2Zr2W24W24. The presence of the v phase in the binary W–Zr system is hindered by the presence of the much more stable Laves phase, as observed in the ground state of Fig. 4. At this point, a comparison between the Laves and the v phases can be made. The binary Laves phases need a large difference of the atomic sizes of the two constituents. For this reason, the Laves phases are far from being stable in Re–W system. As a consequence, the homogeneity domains of the C14 and C15 phases extend parallel to the binary system constituted by the two elements with rather similar sizes (Re and W). The v phase is less sensitive to this effect and has less exothermic enthalpies of formation in the systems in which it is stable but also less endothermic enthalpies in the systems in which it is not.

It has therefore a smoother ground state and a less directed ternary extension. In addition, in our case, it is stabilized by a ternary configuration at the composition Zr2Zr8W24Re24. Differences between the calculated and the isothermal sections should, however, be mentioned, like the need to compute at a higher temperature (T = 2223 °C) than the experimental section (T = 1350 °C) to develop the ternary extension of the v phase. To some extent, this may be explained by the approximations that have been used. In particular, no entropies of formation have been used, so the dependence of the relative phase stability cannot be expected to be very accurate as a function of temperature. The homogeneity domain of the Laves phases is too large. Possibly, the BW model used for the computation which neglects interactions within given sublattices and which has been shown to be very good for complex phases like r and v is not well adapted for the simple Laves phase in which a given atom has many neighbors on the same crystallographic site. The r phase does not appear at this temperature but only slightly above. When it appears the associated tie-triangles are reproduced correctly. Finally, a major difference is the position of the binary Zr2Re phase which is replaced in Fig. 10 by its correct composition Zr21Re25, as demonstrated in our experimental work. 6. Conclusion The reason for the irregular shape of the stable homogeneity domain of the v phase in the Re–W–Zr system has been elucidated, mainly thanks to DFT calculations. We should, however, mention how well these calculations are supported by our own experimental data. One may therefore conclude with the extremely interesting contribution of DFT calculations in understanding phase stability and metastability. It is also highly probable that the same behavior is responsible for the same anomaly in the homologous systems Hf–W–Re, Zr–Mo–Re and Hf–Mo–Re, though these systems are more difficult to study experimentally because of the lack of XRD and/or ND contrast. Finally, the present work represents a decisive achievement in the frame of DFT calculations of phase diagrams, showing that it is possible with certain conditions to reproduce an experimental phase diagram in a rather complex system without using any adjustable parameter. Acknowledgments Financial support from the Agence Nationale de la Recherche (ANR) (Armide 2010 BLAN 912 01) is acknowledged. This work was performed using HPC resources from GENCI-CINES/IDRIS (Grant 2012-096175). The authors are grateful to Jean-Marc Fiorani from Institut Jean Lamour for useful discussions. The authors also thank Eric Leroy for EPMA measurements and Florence Porcher for the neutron diffraction measurements.

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