CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133 Contents lists available at ScienceDirect CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry journal homepage: www.elsevier.com/locate/calphad Solution-based thermodynamic modeling of the Ni–Al–Mo system using first-principles calculations S.H. Zhou a,b,n, Y. Wang a, L.-Q. Chen a, Z.-K. Liu a, R.E. Napolitano b,c a Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA Materials & Engineering Physics Program, Ames Laboratory, USDOE, USA c Department of Materials Science and Engineering, Iowa State University, Ames, IA 50011, USA b art ic l e i nf o a b s t r a c t Article history: Received 31 October 2013 Received in revised form 9 January 2014 Accepted 6 March 2014 Available online 22 March 2014 A solution-based thermodynamic description of the ternary Ni–Al–Mo system is developed here, incorporating first-principles calculations and reported modeling of the binary Ni–Al, Ni–Mo and Al–Mo systems. To search for the configurations with the lowest energies of the N phase, the Alloy Theoretic Automated Toolkit (ATAT) was employed and combined with VASP. The liquid, bcc and γ-fcc phases are modeled as random atomic solutions, and the γʹ-Ni3Al phase is modeled by describing the ordering within the fcc structure using two sublattices, summarized as (Al,Mo,Ni)0.75(Al,Mo,Ni)0.25. Thus, γ-fcc and γʹ-Ni3Al are modeled with a single Gibbs free energy function with appropriate treatment of the chemical ordering contribution. In addition, notable improvements are the following: first, the ternary effects of Mo and Al in the B2-NiAl and D0a-Ni3Mo phases, respectively, are considered; second, the N-NiAl8Mo3 phase is described as a solid solution using a three-sublattice model; third, the X-Ni14Al75Mo11 phase is treated as a stoichiometric compound. Model parameters are evaluated using first-principles calculations of zero-Kelvin formation enthalpies and reported experimental data. In comparison with the enthalpies of formation for the compounds ψ-AlMo, θ-Al8Mo3 and B2-NiAl, the first-principles results indicate that the N-NiAl8Mo3 phase, which is stable at high temperatures, decomposes into other phases at low temperature. Resulting phase equilibria are summarized in the form of isothermal sections and liquidus projections. To clearly identify the relationship between the γ-fcc and γʹ-Ni3Al phases in the ternary Ni–Al–Mo system, the specific γ-fcc and γʹ-Ni3Al phase fields are plotted in x(Al)–x(Mo)–T space for a temperature range 1200–1800 K. & 2014 Elsevier Ltd. All rights reserved. Keywords: CALPHAD First-principles calculation Ni–Al–Mo phase diagram ATAT 1. Introduction The high temperature performance of superalloys based on the Ni–Al binary system is a critical factor in current and future power generation and transportation technologies. Moreover, the refractory metal Mo is well established as a key contributor to the high temperature stability of many multicomponent Ni-base superalloys [1–6]. Despite its widespread use, quantitative guidelines for future alloy development and processing design call for more complete descriptions of the effects of Mo additions on the phase stability and phase transformation in the Ni–Al–Mo system. This is the focus of the present work. Thermodynamic modeling for the ternary Ni–Al–Mo system was first performed by Kaufman and Nesor [7], in which all compounds were described as stoichiometric compounds. Subsequent to treatments of the Ni–Al [8], Al–Mo [9], and Ni–Mo [10] systems, Lu et al. [11] reassessed the ternary Ni–Al–Mo system n Corresponding author. http://dx.doi.org/10.1016/j.calphad.2014.03.002 0364-5916/& 2014 Elsevier Ltd. All rights reserved. which represented a great improvement over the modeling by Kaufman and Nesor [7], and, despite several limitations, has been used with reasonable success for quite some time. As increased demands for high temperature materials drive the exploration of new multicomponent alloy regimes, however, these limitations become problematic. First, the binary Ni–Mo phase diagram resulting from the model by Lu et al. [11] is different from those by Frisk [10] in Fig. 1(a) although Lu et al. [11] use the database developed by Frisk [10]. Lu’s model (dashed curves) indicates that (i) the B2 phase becomes stable around 1900 K indicating the B2 phase can be a primary phase in a ternary Ni–Al–Mo system and (ii) the γʹ phase becomes stable at low temperature, which does not agree with Frisk’s model [10] and the reported experiments [12–14]. In addition, Lu’s model was found to be inconsistent with experimental phase equilibrium data [15–18], limiting its utility in phase-field modeling [19]. These problems arise from inappropriate descriptions of the thermodynamic effects of Mo in the ternary system [11], and several important improvements are called here for. First, recent advances in modeling the Ni–Al [20] and Ni–Mo [21] binary systems should be incorporated into the ternary S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133 125 Fig. 1. The binary phase diagrams of the (a) Ni–Mo system calculated using the parameters by Frist [10] in solid-line and Lu et al. [11] in dash-line; (b) Ni–Mo system calculated using the parameters by Zhou et al. [21] in solid-line and Lu et al. [11] in dash-line; (c) Al–Mo system [9] and (d) Al–Ni system [8,20]. Table 1 A listing of phases modeled in the current treatment. Formula unit Sym. Phases extending from the Ni, Ta, Mo – Ni γ Al γ Mo – Phases extending from the Al3Ni – Al3Ni2 – AlNi – Al3Ni5 ε AlNi3 γ’ Phases extending from the NiMo δ Ni2Mo ρ Ni3Mo – Ni4Mo – Ni8Mo ζ Phases extending from the Al12Mo ϕ Al5Mo η Al4Mo φ Al8Mo3 s Al63Mo37 τ AlMo ψ AlMo3 – Ternary compounds NiAl8Mo3 N Ni14Al75Mo11 X Lattice pure component states: – Cub Cub Cub Ni–Al binary system: Ort Tri Cub Ort Cub Ni–Mo binary system: Orth Tet Tet Tet Tet Al–Mo binary system: Cub Hex Mon Mon – Cub Cub Tet – Struk. des. Prototype – A1 A1 A2 Liquid fcc fcc bcc D011 D513 B2 – L12 Fe3C Al3Ni2 ClCs Ga5Pt3 AuCu3 – – D0a D1a – NiMo Pt2Mo Cu3Ti Ni4Mo Ni8Nb – – – – – A2 A15 Al12W Al5W Al4W Al8W3 – W Cr3Si D022 – Al3Ti – description. Second, the ternary N and X phases listed in Table 1, which were not described in previous reports [7,11], should be considered. In the present work, the thermodynamic descriptions of the binary Ni–Al [8,20], Al–Mo [9], and Ni–Mo [21] systems with the associated phase diagrams shown in Fig. 1(b–d) are employed within a general solution-based approach, using first-principles calculations and experimental data [11,15–18,22–39], which are appropriate to determine the ternary model parameters. The ternary N and X phases are included and the full Ni–Al–Mo phase diagram is computed. The equilibrium phase diagram is presented with a series of isothermal sections and a 3D plot for the γ and γʹ phase fields. The resulting phase diagrams of the ternary Ni–Al–Mo system are compared with the experimental data [11,15–18,22–39] and the prior report by Lu [11]. 2. First-principles calculations To determinate the Gibbs free energy of formation for the intermediate phases, the zero-Kelvin energies of the N phases and the unstable end-members (described in Section 3) for the B2 and D0aNi3Mo phases as well as fcc-Al, fcc-Ni, bcc-Mo were calculated by means of VASP [40] employing Vanderbilt ultrasoft pseudopotential [41] and the generalized gradient approximation (GGA) [42] with the high precision choice. Monkhost 15 15 15 k points were used for the pure elements Ni, Al and Mo, 11 11 11 k points for the compounds B2 and D0a-Ni3Mo, and 6 6 6 k points for the compound N. For the stable structures, we fully relaxed the volume, the 126 S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133 cell shape, and the internal atomic coordinates while for the unstable structures of the end-members we only relaxed the cell volume to maintain the unit cell structures to their stable phases. The enthalpy of formation, ΔH Φ f , for a given compound Φ is calculated as the difference between the energy EΦ TOT of the compound and the linear combination of the pure element reference state energies, EfAlcc , EfNicc and Ebcc Mo Φ Φ f cc Φ f cc Φ bcc ΔH Φ f ¼ ETOT xNi ENi xAl EAl xMo E Mo ð1Þ where xΦ i is the mole fraction of component i in Φ. The calculated enthalpies of formation of the compounds are listed in Table 2a. The compositions of the N phase were reported as Ni12.5Al62.5Mo25 by Raman and Schubert [23,24] and as Ni3Al72Mo25 by Markiv et al. [22] as the dashed-lines in Fig. 2, respectively. To search for the configurations with the lowest energies, ATAT developed by van de Walle [43] was employed and combined with VASP. The starting geometry was limited to the D022 structure with the fixed composition (25 at% Mo) and Mo atom positions. Gamma centered k points were used and more than 65 configurations were studied. The calculated enthalpies of formation for some of the structures found by ATAT are plotted in Fig. 2 indicating that the structure at the lowest point along the concave curve is at composition NiAl8Mo3 and gives a formation energy of 38.17 kJ/mol, which is lower than the those values at both Ni12.5Al62.5Mo25 and Ni3Al72Mo25 by Raman and Schubert [23,24] and Markiv et al. [22], respectively. This ternary compound is a tripled superstructure of that of DO22 along b direction, resulting in a body centered orthorhombic structure with the lattice parameters being a ¼3.6904 Å, b¼11.1050 Å, and Table 2a A summary of results from the first-principles calculations. Phase Ni Al Mo Symbol Formula ΔH, kJ/mol of atoms Ni Al Mo NiMo AlMo Ni3Al Ni3Mo Al2NiMo Al8NiMo3 Al3Mo 0 0 0 27.84 5.88 38.12 6.15 28.01 38.17 27.19 A2 NiAl B2 CsCl Ni3Mo D0a Cu3Ti NiAl8Mo3 N Al3Ti Position type Atomic position Al1 Al1 Al2 Mo Ni1 Ni1 Ni2 Ni2 Ni3 Ni3 Ni4 Ni5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0 0.5 0.9938 0.4938 0.0194 0.7525 0.4821 0.9821 0.2486 0.7486 0.7503 0.2503 0.5210 0.2575 0.3284 0.1716 0 0 0.3399 0.1601 0.1704 0.3296 0.1518 0.3482 0 0 c¼8.5710 Å. The primitive unit cell of this ternary compound contains 12 atoms which are arranged in the symmetry. The atomic positions of this ternary compound are listed in Table 2b. In this work, we designated this ternary compound as N-NiAl8Mo3 and describe it as a solution phase with small homogeneous composition by considering the experimental observations by Raman and Schubert [23,24] and Markiv et al. [22]. As listed in Table 2a, the first-principles calculated enthalpy of formation for the N-NiAl8Mo3 phase is 38.17 kJ/mol, which is less negative than the enthalpy of mixing, i.e. 40.19 kJ/mol, for the ψ-AlMo, θ- and B2-NiAl phases from the binary systems, indicating that the N phase decomposes into other phases at low temperature. Due to the lack of structural information, the firstprinciples calculation was not performed for the compound X. 3. Thermodynamic models Prototype fcc fcc bcc γ Table 2b The crystal structure of the lowest energy structure at composition NiAl8Mo3. The thermodynamic properties of pure Al, Mo, and Ni in various structures were computed using the parameters from the SGTE database [44] as listed in Table 3. In addition to the compounds in the Ni–Al [8,20], Al–Mo [9] and Ni–Mo [21] systems, the liquid, bcc, γ-fcc, B2-NiAl, γʹ-Ni3Al, D0a-Ni3Mo, and N were modeled as ternary solution phases and X phase as a stoichiometric compound. The associated Gibbs free energy functions are defined in Table 4, where the total Gibbs free energy for the phase, Φ, is generally given by the sum of three contributions: ref Φ xs Φ Gm þ id GΦ GΦ m ¼ m þ Gm ; ð2Þ where the subscript m denotes that all terms are molar quantities. The first term in Eq. (2) is the sum of occupancy-weighted sublattice end-member contributions. The second and third terms are the ideal and excess parts of the Gibbs free energy of mixing, respectively. The excess Gibbs free energy in all Gibbs free energy equations is expressed in terms of the Redlich–Kister polynomial [45]. The specific treatment of each phase is discussed briefly here. The liquid, bcc and γ-fcc are described using the model given in Table 4, where 1GΦ i is the molar Gibbs free energy of the pure element i with the structure Φ (Φ ¼ liquid, bcc or γ-fcc) as listed in Table 3 and xi is the mole fraction of the indicated component Φ i. j LΦ i;k is the binary interaction parameter, and LAl;Mo;Ni is a composition dependent ternary interaction parameter evaluated with the experimental data. Magnetic property of the Ni is considered for its fcc structure. mg GΦ m is the magnetic contribution [46] to Gibbs free energy in solution phases, and expressed as the following: mg Fig. 2. Enthalpy of formation for the N phase calculated using ATAT and VASP with the fixed Mo atomic positions xMo ¼ 0.25 and current thermodynamic parameters in Tables 6 and 7. Reference states: fcc-Ni, fcc-Al and bcc-Mo. Φ Φ GΦ m ¼ RT lnðβ þ1Þf ðτ Þ; Φ ð3Þ where β is the quantity related to the total magnetic entropy and is set equal to the Bohr magnetic moment per mole of atom. τΦ is S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133 127 Table 3 Thermodynamic parameters for pure elements [44]. Al phases 0 Gliq Al 0 GfAlcc 0 Gbcc Al Tmin, K Tmax, K 298 933 933 6000 298 700 700 933 933 6000 298 3200 0 0 f cc GAl 11,005.029 11.841867 – – – – 7.934 10 20 – 0 f cc GAl 10,482.282 11.253974 – – – – – 1.231 1028 – – – 7976.15 137.093038 24.3671976 1.884662 10 3 74,092 8.77664 10 7 – – 11,276.24 223.048446 38.5844296 0.018531982 74,092 5.764227 10 6 – – 11,278.378 188.684153 31.748192 0 f cc GAl 10,083 4.813 – – – – – – GREF A B C D E F G H 1.234 1028 Mo phases 0 Gliq Mo 0 cc GfMo Tmin Tmax 298 2896 2896 6000 298 6000 0 0 0 0 bcc GMo 15,200 0.63 – – – – – – – Gbcc Mo 41,831.347 14.694912 – – – – 4.24519 10 22 – – GREF A B C D E F G H I Gbcc Mo 34,095.373 11.890046 – – – – 4.849315 1033 – – 0 Gbcc Mo 298 2896 0 7746.302 131.9197 23.56414 3.443396 10 3 65,812 5.66283 10 7 – – 1.30927 10 10 2896 3000 0 30,556.41 283.559746 42.63829 – – – – 4.849315 1033 – Ni phases 0 Gliq Ni 0 Gbcc Ni 0 GfNicc Tmin Tmax 298 1728 1728 6000 298 6000 298 1728 1728 3000 0 GREF A B C D E F G H Tc β 0 f cc GNi 16,414.686 9.397 – – – – 3.82318 10 21 – – – 0 f cc GNi 18,290.88 10.537 – – – – – 1.12754 1031 – – 0 f cc GNi 8715.084 3.556 – – – – – – 575 0.85 0 0 5179.159 117.854 22.096 4.8407 10 3 – – – – 633 0.52 27,840.655 279.135 43.1 – – – – 1.12754 1031 – – Note: O Gθi ¼ O GRef þ a þ bT þ cT ln T þ dT 2 þ eT 1 þ f T 3 þ gT 7 þ hT 9 þ iT 4 (J/mol) and Tc (K) is the Curie temperature and β is the average magnetic monent per atom (Bohr magnetons). Φ defined as T=T Φ c and T c is the Curie temperature. For the solution Φ phase, T c and β Φ are described as the following: n 0 Φ j Φ 0 Φ j TΦ c ¼ xA T cA þ xB T cB þ xA xB ∑ LTðA;BÞ ðxA xB Þ j¼0 n j Φ j 0 Φ βΦ ¼ xA 0 βΦ A þ xB β B þ xA xB ∑ Lβ ðA;BÞ ðxA xB Þ j¼0 Φ 0 Φ T ci and 0 i are from the pure j Φ j Φ 3. LTðA;BÞ and LβðA;BÞ are the magnetic where Table β ð4Þ ð5Þ et al. [20] in the thermodynamic evaluation of the Ni–Al, and Ni–Al–Cr systems with a special emphasis on the relationship among the parameters for the ordered γʹ phase. The γʹ phase in the binary Ni–Al system was modeled with the two-sublattice model (Al,Ni)0.75 (Al,Ni)0.25 [8]. Considering the thermodynamic effect of the component Mo, the two-sublattice model for γʹ was modified as (Al,Mo,Ni)0.75 (Al,Mo,Ni)0.25. Adopting the descriptions by Ansara et al. [8,47] and ' Dupin et al. [20], the Gibbs free energy, Gγm , for γʹ is expressed as the following: element i as listed in ðaÞ interaction parameters Gγm' ¼ Gγm ðxi Þ þ ΔGord m ði : jÞ; ðbÞ ord ord ΔGord m ði : jÞ ¼ Gm ði : jÞyIi ;yIIi Gm ði : jÞxi ¼ yIi ¼ yIIi ; between the elements A and B. In this work, the magnetic parameters from the binary Al–Ni system [8] are employed as listed in Table 7. The thermodynamic modeling of the ordered γʹ-Ni3Al and disordered γ-fcc phases was discussed by Ansara et al. [8,47] and Dupin ð6Þ where Gγm ðxi Þ is the Gibbs free energy of the disordered γ-fcc phase as described in Table 4. ΔGord m ði; jÞ in Eq. (6) corresponds to the ordering energy, being described with Eq. (6b) [20,48]. Gord m ði : jÞ in Eq. (6b) is 128 S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133 Table 4 ref Φ xs Φ 0 ref 0 f cc Gm þ id GΦ ¼ 0 GfAlcc , 0 Gbcc Summary of the thermodynamic models used for the Al–Mo-Ni ternary system with the total Gibbs free energy GΦ m ¼ m þ Gm ( Gi Mo or GNi ). Phase Formulation Model Liquid γ-fcc Bcc (Al,Mo,Ni)1 ref GΦ m ¼ id xs GΦ m xi 1GΦ i ∑ i ¼ Al;Mo;Ni ¼ RT ∑ xi ln xi i ¼ Al;Mo;Ni n n j¼0 j¼0 j Φ j Φ j j GΦ m ¼ xAl xMo ∑ LAl;Mo ðxAl xMo Þ þ xAl xNi ∑ LAl;Ni ðxAl xNi Þ n Φ j þ xMo xNi ∑ j LΦ Mo;Ni ðxMo xNi Þ þ xAl xMo xNi LAl;Mo;Ni ; j¼0 2 Φ 0 Φ 1 Φ LΦ Al;Mo;Ni ¼ xAl LAl;Mo;Ni þ xMo LAl;Mo;Ni þ xNi LAl;Mo;Ni ψ-AlMo B2-NiAl D0a-Ni3Mo (Al,Mo)1(Al,Mo)1 (Al,Mo,Ni)1(Al,Mo,Ni,Va)1 (Al,Mo,Ni)0.75(Al,Mo,Ni)0.25 ref D0a D0a a GD0 Al:Al ¼ GAl:Mo ¼ GAl:Ni ¼ 0 GΦ m ¼ ∑ yIi ∑ yIIj 1GΦ i:j i ¼ Al;Mo;Ni j ¼ Al;Mo;Ni ref ref Φ 1GΦ ¼ p1G þ q1G þ ΔG ¼ p1Gref i:j i:j i j i id GΦ m ¼ xs GΦ m I I k ¼ ∑ ∑ yIi yIl ∑ ∑ k LΦ i;l:j ðyi yl Þ i l4i j k¼0 RT ∑ ðpyIi i ¼ Al;Mo;Ni ln yIi þ qyIIi Φ Φ þ q1Gref j þ ai:j þ bi:j T ln yIIi Þ II II k þ ∑ ∑ ∑ yIIi yIIl ∑ k LΦ j:i;l ðyi yl Þ i l4i j k¼0 (p and q are the subscript numbers of sublattices, respectively) γ’-Ni3Al (Al,Mo,Ni)0.75(Al,Mo,Ni)0.25 0 0 ord 1 Gord i:j ¼ Gj:i ¼ 3uij 0 ord Li;j:i 1 ¼ 0 Lord i;j:j ¼ 6uij 1 ord Li;j:i 4 ¼ 1 Lord i;j:j ¼ 3uij 1 ord Li:i;j 4 ¼ 1 Lord j:i;j ¼ uij 0 ord Li;j:k ¼ 6u1ij þ ðuiijk þ ujijk Þ3=2 1 ord Li;j:k ¼ 3u4ij þ ðuiijk ujijk Þ=2 0 ord Li;j;k:i N (Al,Ni)2(Al,Ni)1(Mo)1 D513-Al3Ni2 δ-NiMo GN Ni:Al:Mo ¼ 0 (Al)3(Al,Ni)2(Ni,Va)1 (Ni)24(Mo,Ni)20(Mo)12 ref ¼ 6uiijk 3ujijk =2 3ukijk =2 N I II III GΦ m ¼ ∑ yi ∑ yj ∑ yk 1Gi:j:k i j k ref ref ref N 1GΦ i:j:k ¼ p1Gi þ q1Gj þ r1Gk þ ΔGi:j:k id I I II II III III GΦ m ¼ RT∑ðpyi ln yi þ qyi ln yi þ ryi ln yi Þ xs n Φ n Φ I I I I n II II II II n GΦ m ¼ ∑ ∑ yi yl ∑ ∑ Li;l:j:k ðyi yl Þ þ ∑ ∑ ∑ yi yl ∑ Lj:i;l:k ðyi yl Þ i i l4i j n¼0 j i l4i n¼0 n Φ III III III n þ ∑ ∑ ∑ yIII i yl ∑ Lk:j:i;l ðyi yl Þ j n¼0 i l4i ϕ-Al12Mo φ-Al4Mo η-Al5Mo τ-Al63Mo37 θ-Al8Mo3 A15–AlMo3 D011-Al3Ni ε-Al3Ni5 D1a-Ni4Mo ρ-Ni2Mo ζ-Ni8Mo (Al)12(Mo)1 (Al)4(Mo)1 (Al)5(Mo)1 (Al)63(Mo)37 (Al)8(Mo)3 (Al)0.75(Mo)0.25 (Al)0.75(Ni)0.25 (Al)0.375(Ni)0.625 (Ni)0.8(Mo)0.2 (Ni)0.67(Mo)0.33 (Ni)0.89(Mo)0.11 ref ref Φ GΦ m ¼ ΔGi:j þ p1Gi þ q1Gj X (Al)0.75(Mo)0.11(Ni)0.14 0 f cc X GXm ¼ 0:750 GfAlcc þ 0:110 Gbcc Mo þ 0:14 GNi þ a þ b T Φ ref ¼ aΦ þ b T þ p1Gref i þ q1Gj (p and q represent subscript values in formulation, respectively.) X the Gibbs free energy contribution due to ordering with xi ¼ 3yIi =4 þ yIIi =4 as described by the two-sublattice model in Table 4 where yIi and yIIi are sit-fraction in first and second sublattices, respectively. It has three terms as shown in Eq. (2) and in Table 4 and contains implicitly a contribution to the disordered state also. When yIi ¼ yIIi ¼ xi , Gord m ði : jÞ represents the extraneous excess energy contribution to the disorder state. Thus ΔGord m ði; jÞ is described as the Gord m ði Gord m ði temperature and composition ranges, Gγm' should always have an extremum when yIi ¼ yIIi ¼ xi . When the disordered phase is stable, this extremum is a minimum. Hence, the following equation [8,47] has to be fulfilled: ! ∂Gγm' γ' ðdGm Þxi ¼ ∑ ∑ dyðsÞ ð7Þ i ¼0 yðsÞ i s i x i : jÞxi ¼ yI ¼ yII in Eq. (6b) and k ord k ord From this equation, the parameters 0 Gord i:j , Li;j:l , and Li:j;l can be k ord k ord must be zero when the phase is disordered. 0 Gord i:j , Li;j:l , and Li:j;l expressed with the parameters u1ij , u4ij , uiijk , ujijk , and ukijk (i, j, or listed in Table 4 are the parameters for Gord m ði : jÞ. As described in reports [8,20,47,49], in order to favor the stability of the ordered phase to disorder transit in certain k¼ Al, Mo, or Ni) as shown in Table 4. The detail descriptions of the difference between : jÞyI ;yII and i i i i parameter derivation for the ordered γʹ-Ni3Al phase can be found in Refs. [20,49]. S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133 129 Table 5 Symbols and related references used in Figs. 2–4. Figure Figs. 3 and 5(b) Boundary Single phase Symbol Ref. ▽γ ●bcc △ γʹ ◇δ þγ * γʹ [26] Symbol Two-phase Ref. Fig. 5(d) and (f) Fig. 5(e) and (g) Symbol Ref. Symbol Ref. ◆ γ þγʹ [27] ▼γ þδ þ bcc □γ þγʹ þbcc ▲γ þγʹ þbcc [26] △fccþ bcc þliq [37] γ þγʹ þbcc ◆ bcc þ B2 þγʹ ▲ γ þ bcc þδ B2þ bcc þ l12 fccþ l12 þbcc [15] ✖ γ þγʹþ δ * bcc þ δ þγʹ ◆ bcc þ B2 þ γʹ D0a þ γʹ þδ γ þ γʹþ D0a ✡ B2 þAlMo3 þN [15] ⊖ γ þ γʹþ δ * bcc þ δþ γʹ ✶liqþ Al3Moþ N liq þ Al3Ni2 þX N þAl3Ni2 þ X ◆γ þγʹ þ D0a ✖ γ þγʹþ δ [31] ▲ bcc þliq ○ bcc þliq ■ fcc þ liq ◆ bccþ liq ◇ fccþ liq ▼ Nþ liq ▽γ △ γʹ □ B2 δ þ bcc γ ✧γʹ ◯γ ◇γʹ B2 ▽γ △ γʹ □ B2 δ γ ✧γʹ B2 δ ○γ ◇γʹ B2 ▽γ △ γʹ ○D0a □ B2 γ ✧γʹ δ D0a Y γʹ ✚γ [25] [27] Fig. 4 Fig. 5(a) and (c) Three-phase [15] ✖N ●γ N [22] [15] [23,24] [25] [34] [38] [22] ■ γ þγʹ δ þγ ▼ bcc þγʹ * bcc þ B2 [15] ■ γ þγʹ ▲ γʹþ δ δþ γ ▼ bcc þγʹ ✰γʹþ D0a ∅δ þD0a ⊖γ þ D0a [15] ◇ δþ γ ✰δ þ γʹ ∅ bcc þ γʹ ▲Al3Niþ X @ ▼ γ þ γʹ [31] [29] [29] [22] [35] [15] [11] [24] [35] [22] ● γʹ N ■X [28] [22] [32] [36] [18] [36] [32] [35] @γ þ γʹ two-phase field at 1048 K. The N-NiAl8Mo3 phase has the D022 structure with the space group I4/mmm [22,24]. Their atoms distribute in three-sublattices. According to the first-principles data in Table 2a and experimental data [22,24] discussed in Section 2, the N-NiAl8Mo3 phase is described with the three-sublattice model, (Al,Ni)2(Al,Ni)1(Mo)1, with the Gibbs free energy given in Table 4, where yI and yII are the site fractions in the first and second sublattices, respectively. The Gibbs free energy of formation ΔGN i:j:Mo of the end-member i:j:Mo is N xs N expressed as aN þ b T. G m is the excess Gibbs free energy i:j:Mo i:j:Mo 1 N with the interaction parameters 0 LN Al:Al;Ni:Mo and LAli:Al;Ni:Mo being N assumed as a constant. Both of them as well as aN i:j:Mo and bi:j:Mo are evaluated with the experimental and first-principles data. The thermodynamic models of B2-NiAl and D0a-Ni3Mo phases were described in detail in Refs. [8,21], respectively. The experimental data [11,15,35] revealed a small solubility of Mo in B2-NiAl and Al in D0a-Ni3Mo. The non-stoichiometric B2-NiAl and D0aNi3Mo phases were modified here with the model shown in Table 4. Here we only consider the unstable end-members with B2 the Gibbs free energy of formationsΔGB2 Al:Mo and ΔGMo:Ni for B2D0a NiAl and ΔGNi:Al for D0a-Ni3Mo. The X phase was treated as stochiometric compounds with the X Gibbs free energy functions as shown in Table 4, where aX and b are the model parameters to be evaluated. 4. Determination of the thermodynamic model parameters The parameters described in the preceding section for the ternary Ni–Al–Mo system were determined in the order of γ-fcc and γʹ-Ni3Al phases, B2-NiAl, D0a-Ni3Mo, liquid, N and X phases using the available experimental data in Table 5 and first-principles data in Table 2a. The evaluated parameters as well as those of the binary Ni–Al [8,20], Al–Mo [9] and Ni–Mo [21] systems are listed in Tables 6 and 7. In the present section, the methodology used for the determination of the parameters is discussed. In both binary Ni–Al and Ni–Mo systems, Al and Mo dissolve in γ-fcc(Ni). The isothermal section at 1533 K is calculated only using the parameters of the three binary systems in Tables 6 and 7 and shown in Fig. 3, where the calculated γ and γʹ two-phase region deviates far from the experimental data [25–27] indicating that the 130 S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133 Table 6 Gibbs free energy of formation for end-member reference states (per mole of formula unit). Phase Parameters Value, J/mol of atoms D011-Al3Ni 11 ΔGD0 Al:Ni 48,483.73þ 12.299T 13 ΔGD5 Al:Al:Ni 39,465.978þ 7.895T D513-Al3Ni2a ε-Al3Ni5 B2-NiAl γʹ-Ni3Al 13 ΔGD5 Al:Ni:Ni 427,191.9þ 79.217T 13 ΔGD5 Al:Al:Va 30,000 3T 13 ΔGD5 Al:Ni:Va 357,725.92 þ 68.322T ΔGεAl:Ni 55,507.7594 þ7.265T 0 a ΔGB2 Al:Al B2 a ΔGB2 Al:Ni , ΔGNi:Al 152,397.3 þ 26.406T a ΔGB2 Al:Va 1000 T a ΔGB2 Ni:Ni 0 a ΔGB2 Ni:Va 162,397.3 27.406T ΔGB2 Mo:Va 150,000 ΔGB2 Mo:Mo 150,000 B2 ΔGB2 Al:Mo , ΔGMo:Al 5876 B2 ΔGB2 Mo:Ni , ΔGNi:Mo 27,844 u1AlNi u4AlNi u1MoNi ϕ-Al12Mo φ-Al4Mo η-Al5Mo τ-Al63Mo37 θ-Al8Mo3 ψ-AlMo A15–AlMo3 δ-NiMo ρ-Ni2Mo D0a-Ni3Mo D1a-Ni4Mo ζ-Ni8Mo N X a [9] This work 13,415.515 þ 2.082T [8] 341 3.515T This work u4MoNi 3621.1 þ3.411T u1AlMo 0 u4AlMo 0 uNi AlMoNi 20,375 12.033T ΔGφAl:Mo ΔGϕAl:Mo ΔGηAl:Mo ΔGτAl:Mo ΔGθAl:Mo ΔGψAl:Al ΔGψMo:Al ΔGψMo:Mo ΔGA15 Al:Mo ΔGδNi:Mo:Mo ΔGδNi:Ni:Mo ΔGρNi:Mo 139,100 þ 26.975T [9] 137,570 þ29.69T 139,104þ 30.156T 2,268,100 þ167.2T 412,500 þ105.05T 20,166.8 9.626aT 36,850þ 1.0T 0 89,000 þ20T 0.003T2 169,981 þ1154.981T 155.484T ln (T) [21] 154,106þ 2855.001T 94.923T ln (T) 9421 þ 49.551T 6.231T ln (T) a ΔGD0 Mo:Mo 42,650 a ΔGD0 Ni:Mo 10,131.9þ 58.132T 7.366T ln (T) a ΔGD0 Mo:Ni 17,060 a ΔGD0 Ni:Ni 2840 a ΔGD0 Ni:Al 37,910 þ5.651T This work a ΔGD1 Ni:Mo ΔGςNi:Mo ΔGN Ni:Ni:Mo ΔGN Al:Ni:Mo ΔGN Al:Al:Mo ΔGXAl:Mo:Ni 9021 þ 55.004T 7.080T ln (T) [21] 6115 þ33.258T 4.085T ln (T) 29,600þ 32.028T [21] This work 119,040 108,760 þ22.128T 39,464.2 þ6.521T 0 0 0 bcc Reference state is Al-bcc and Ni-bcc, i.e. Gref ¼ Gbcc Al or GNi . i [21], the thermodynamic properties of the γ and γʹ phases in the ternary Ni–Al–Mo system are described with the ternary interacuNi AlMoNi . [8] 7088.7363 3.687T binary interaction parameters for γ-fcc from both Ni–Al [8,20] and Ni–Mo [21] systems predict a ternary γ-fcc-field that is too expansive; hence positive ternary interaction parameters here had to be considered to reduce it. Similar behavior was also found in the ternary Ni–Al–Cr system modeled by Dupin et al. [20], in which the positive ternary interaction parameter for γ-fcc(Ni) was employed. In addition to the parameters in the Al–Ni [8,20] and Ni–Mo tion parameters Ref. γ γ 0 γ LAl;Mo;Ni , 1 LAl;Mo;Ni , 2 LAl;Mo;Ni , u1Mo;Ni , u4Mo;Ni and These parameters are evaluated with the experimental data [11,15–18,22,26–35] and listed in Tables 6 and 7. The experimental data [11,15,22,26,35] indicate little solubility of Mo in B2-NiAl. The model parameters ΔGB2 Mo:i (i ¼ Al,Ni), which are considered as a constant in Table 4, were fixed with the firstprinciples data as listed in Tables 6 and 7. Based on the thermodynamic description in Section 3, the D0a a model parameters aD0 and bNi:Al in Table 4 are used for the Ni:Al a was determined with compound D0a-Ni3Mo. The parameter aD0 Ni:Al D0 a was evaluated with the phase the first-principles data, while bNi:Al equilibrium data [15], i.e. the four-phase δ þ γ-γʹ þD0a reaction at 11637 5 K. The evaluated parameters are listed in Table 6. According to the experimental phase equilibrium data [22–24] and first-principles data in Fig. 2, the composition range of the S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133 131 Table 7 Excess Gibbs free energy interaction parameters. Phase Parameters Value, J/mol Ref. Liquid 0 Liq LAl;Ni 1 Liq LA;Ni 2 Liq LA;Ni 3 Liq LA;Ni 4 Liq LA;Ni 0 liq LAl;Mo 1 liq LAl;Mo 0 liq LMo;Ni 1 liq LMo;Ni 2 liq LMo;Ni 0 liq LAl;Mo;Ni 1 liq LAl;Mo;Ni 2 liq LAl;Mo;Ni 0 bcc LMo;Ni 1 bcc LMo;Ni 0 bcc LAl;Mo 0 γ LAl;Mo 0 γ LAl;Ni 1 γ LAl;Ni 2 γ LAl;Ni 3 γ LAl;Ni 0 γ LTðAl;NiÞ 1 γ LTðAl;NiÞ 0 γ LMo;Ni 1 γ LMo;Ni 2 γ LMo;Ni 0 γ LAl;Mo;Ni 1 γ LAl;Mo;Ni 2 γ LAl;Mo;Ni 0 D513 LAl:Al;Ni:Ni 0 D513 LAl:Al:Ni;V a 0 D513 LAl:Ni:Ni;Va 0 D513 LAl:Al;Ni:V a 0 B2 LAl;Ni:Ni 0 B2 LAl:Ni;Va 0 B2 LNi:Ni;Va 0 B2 LAl;Ni:Va 0 ψ LAl;Mo:Al 0 ψ LAl:Al;Mo 0 ψ LMo:Al;Mo 1 ψ LMo:Al;Mo 0 ψ LAl;Mo:Mo 1 ψ LAl;Mo:Mo 0 δ LNi:Mo;Ni:Mo 1 δ LNi:Mo;Ni:Mo 0 D0a LMo;Ni:Mo 0 D0a LNi:Mo;Ni 0 N LAl:Al;Ni:Mo 1 N LAl:Al;Ni:Mo 207,109þ 41.315T [8] bcc γ-Fcc D513-Al3Ni2 B2-NiAl ψ-AlMo δ-NiMo D0a-Ni3Mo N 10,186þ 5.871T 81,205 31.957T 4365 2.516T 22,101.64 þ 13.163T 100,000 þ 35T [9] 15,000 þ 6.3T 39,597þ15.935T [21] 7373þ 4.102T 12,123þ 5.551T 50,748 This work Fig. 3. Isothermal section of the Ni–Al–Mo phase diagram calculated at 1533 K using the parameters of the three binary systems in Tables 6 and 7 in comparison with experimental data with the symbols listed in Table 5. 70,748 115,748 27,691 [21] 18,792 75,000þ 25T [9] 92,220 þ 20T 162,407.75 þ16.213T [20] 73,417.798 34.914T 33,471.014 9.837T 30,758.01 þ10.253T 1112 [8] 1745 8916 þ3.591T [21] 5469 0.249T 1549 2.741T 41,546 50.349T This work 91,047 193,074 12.948T 193,484.18 þ 131.79T [8] 22,001.7þ 7.033T 22,001.7þ 7.033T Fig. 4. Liquidus projection of the Ni–Al–Mo system calculated in the composition triangle using the parameters in Tables 6 and 7 indicating the primary phases and comparing with experimental data as the symbols in Table 5 and dash-line by Lu et al. [11]. 193,484.18 þ 131.79T 52,440.88 þ 11.3012T 64,024.38þ 26.4949T 64,024.38þ 26.494T 52,440.88 þ 11.301T 5000 [9] 5000 25,000 10,000 25,000 10,000 829,211þ 825.923T [21] 417,368 þ 326.504T three binary systems and ternary parameters, 0 Lliq , 1 Lliq Al;Mo;Ni Al;Mo;Ni 6710 and 2 Lliq . Henry [25] studied the two alloys of Ni65.7Al17.6Mo16.7 Al;Mo;Ni 1198–0.401T 140,870 0 N function of the N phase in which ΔGN LAl:Al;Ni:Mo and Al:Ni:Mo , assumed to be constant. The parameters, and aN Al:Al:Mo , were fixed by the first-principles N N data. The parameters, bNi:Ni:Mo and bAl:Al:Mo , were evaluated with the phase equilibrium data [22,24]. The interaction parameters, 0 N LAl:Al;Ni:Mo and 1 LN Al:Al;Ni:Mo , were evaluated with the first-principles data in Fig. 2. For the X phase, due to the lack of experimental data, the parameter aX in Table 4 had to be assumed by calculating the enthalpy of mixing of the compounds N, D011-Al3Ni and D513X Al3Ni2. The parameter b was evaluated with the phase equilibrium data [36]. The liquid phase was described with the parameters of the 1 N LAl:Al;Ni:Mo were N aN Ni:Ni:Mo , aAl:Ni:Mo This work 60,570 stable N phase is estimated between Ni12.5Al62.5Mo25 and Ni3Al72Mo25 at high temperatures. Fig. 2 shows the composition with the minimum enthalpy of formation for the N phase being at NiAl8Mo3. Therefore, the interaction parameters, 0 LN Al:Al;Ni:Mo and 1 N LAl:Al;Ni:Mo , and the Gibbs free energies of formation, ΔGN Ni:Ni:Mo , ΔGNAl:Ni:Mo and ΔGNAl:Al:Mo were considered for the Gibbs free energy and Ni65.6Al14.4Mo20 using the optical microscopy and observed that the primary phase of the two alloys is the Mo-bcc phase during the solidification. Svetlov et al. [34] studied the liquid þsolids (solid¼ fcc or bcc) two-phase field at the Ni-rich corner using DTA and optical microscopy. Yoshizawa et al. [37] studied the eutectic reaction (liquid-bcc þ γ) using SEM and TEM. , 1 Lliq and 2 Lliq , were evaluated The parameters, 0 Lliq Al;Mo;Ni Al;Mo;Ni Al;Mo;Ni with the primary solidification phase data [25,34,37]. The para1 bcc 2 bcc meters 0 Lbcc Al;Mo;Ni , LAl;Mo;Ni and LAl;Mo;Ni were considered to be zero. 132 S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133 5. Phase equilibrium results The Ni–Al–Mo phase diagrams computed from the Gibbs free energy functions of individual phases described above are shown in Figs. 4 and 5. Relevant Fig. 5. The isothermal section calculated at different temperatures in comparison with experimental data as the symbols in Table 5 and the calculated results in dash-line by Lu et al. [11]. (a) At 1573K, (b) at 1533K, (c1) overall composition at 1473K, (c2) Ni–rich corner at 1473K, (d1) overall composition at 1373K, (d2) Ni– corner at 1373K and at 1273K, (e) at 1273K, (f) at 1153K and (g) at 1073K. experimental data in Table 5 as well as the phase diagram calculated using the parameters by Lu et al. [11] are also shown in Fig. 1(a), Figs. 4 and 5 for comparison. The phase diagrams we propose here include several dramatic differences from previously suggested phase diagrams as described briefly here. Our model describes the ternary compound N and X phases using a threesublattice model and as a stoichiometric compound, respectively, which were ignored by Lu et al. [11]. The first-principles calculated enthalpies of formation of the N phase in Fig. 2 were used to evaluate the thermodynamic parameters of the N phase. In addition, our first-principles data were used to determine the Gibbs energy of the end-members of the B2-NiAl and D0a-Ni3Mo phases. This differs greatly from the modeling by Lu et al. [11]. The other major contributor of this work is that the recent descriptions of the binary Ni–Al [20] and Ni–Mo [21] in Fig. 1 were adopted for better description of γ and γʹ in the Ni–Al–Mo system. The current model liquidus projection of the Ni–Al–Mo system is plotted in Fig. 4, showing a good agreement with experimental data [22,25,34,37,38]. In comparison with our results, the liquidus projection calculated using parameters by Lu et al. [11] shows two significant differences in Fig. 4. In our calculation, the N phase is a primary phase on Al rich side while the calculated primary phase by Lu et al. [11] is the θ-Al8Mo3 phase as marked with “Al8Mo3” in gray in Fig. 4. Second, a curve as marked “A-line” in Fig. 4 is calculated using the parameters by Lu et al. [11]. This curve starts from the binary Ni–Mo system as shown in Fig. 1 (a) indicating that the B2 phase can be the primary phases in the liquidus projection. Fig. 6. γ-fcc and γʹ-Ni3Al single-phase domains plotted in 3D as computed using the present model where symbol ϑ (ϑ ¼ liq, bcc, B2, and δ) indicates the phase equilibrium with γ-fcc or γʹ-Ni3Al. S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 46 (2014) 124–133 The calculated isothermal sections at 1573, 1533 and 1473 K along with the experimental data in Table 5 are shown in Fig. 5(a–c), in which the calculated equilibrium phases are in good agreement with the experimental data [15,22–27,29,35]. The calculated isothermal sections at 1473 and 1373 K plotted in Fig. 5 (c and d) show the invariant reaction γ þbcc-γʹ þ δ, which occurs at 1400 K. According the experimental data, the invariant reaction γ þbcc-γʹþ δ temperature is between 1356 and 1420 K by Miracle et al. [26], between 1300 and 1400 K by Hong et al. [30] and at 1403 K by Wakashima et al. [31]. This means that our calculated invariant reaction γ þ bcc-γʹþ δ is in good agreement with the experimental data [26,30,31]. Furthermore, our calculated results show an agreement with the experimental data [15] of the γ þ γʹ two-phase region in Fig. 5(c), while the calculated results by Lu et al. [11] are in the single fcc phase region. Fig. 5(d) and (e) shows the isothermal section calculated at 1373 and 1273 K with experimental data [11,15,18,22,24,28,31,32,35,36]. There are two significant differences between Fig. 5(d) and (e). First, the X phase appears in Fig. 5(e). Our calculated results indicate that the X phase forms at 1282 K, which is in reasonable agreement with the experimental data 1288 K measured by Grushko et al. [36]. Second, the invariant reaction, liqþ θ-Nþ φ, occurs at 1355 K. It should be noted that the data by Grushko et al. [36] plotted in Fig. 5(e) indicate the liqþ Al3Mo þN three phases in equilibrium at 1273 K. This is inconsistent with our calculated liq þ φ-Al4Mo þN three-phase equilibrium. According to the thermodynamic description for the binary Al–Mo system [9] adopted in this work with the associated phase diagram in Fig. 1(d), the liquid, Al4Mo, Al8Mo3, AlMo3 and bcc phases are stable at 1273 K, while the Al3Mo phase is not considered as a stable phase. The stability of the Al3Mo phase needs to be confirmed in future work. Fahrmann et al. [16,17] investigated the Ni79.2Al6.4Mo14.4 and Ni77.97Al12.9Mo9.13 alloys. In the Ni79.2Al6.4Mo14.4 alloy aged at 1258 K, minor amounts of the δ-NiMo phase was observed [16,17] which is consistent with our calculated result i.e. the mole fraction of the δ-NiMo phase in Ni79.2Al6.4Mo14.4 alloy is 0.0057 which is very little. The volume fraction of the γʹ phase for the Ni77.97Al12.9Mo9.13 alloy calculated at 1258 K in this work is 0.61, which is in good agreement with the experimental data 0.60 7 0.06 [16,17], while that calculated using parameters by Lu et al. [11] is 0.54. It is clear that our model yields better agreement with experimental data [16,17]. Fig. 5(f) and (g) shows the isothermal sections calculated at 1153 and 1073 K, respectively, with the experimental data [15,22,24,32,35,36]. Comparing with Fig. 5 (e), the calculated four phase δ þ γ-γʹþ D0a reaction occurs at 1167 K, which is in a good agreement with the experimental phase transformation data 1163 75 K [15]. In Fig. 5(f), the experimental data [15], which indicated the γ þ γʹ two-phase region, are in agreement with our calculated results, but outside the calculated γ þ γʹ twophase region using the parameters by Lu et al. [11]. A similar problem is also shown in Fig. 5(g), where our calculated results shown better agreement with the experimental γ þ γʹ two-phase field data [18] than those by Lu et al. [11]. To understand the stabilities of the γ and γʹ phases, we plotted the stable γ and γʹ phase field in the related composition triangle over the temperature range 1200– 1800 K in Fig. 6, which is useful for Ni-based superalloy design by showing the stable γ, γʹ and γ þ γʹ phase domains with related composition and temperature dimensions. 6. Summary By combining CALPHAD approach with the first-principles calculations, a thermodynamic model was developed for the ternary Ni–Al–Mo system. 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