Computer Coupling of Phase Diagrams and Thermochemistry

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634 S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 631–641 Table 2 The thermodynamic parameters for pure elements [30]. Mo phases 0 G liq Mo T min 298 2896 298 2896 298 T max 2896 5000 2896 3200 5000 0 G ref 0 G bcc Mo – – – G bcc Mo a 0 41831.347 3538.963 −7746.302 −30556.41 15200 a 1 −14.694912 271.6697 131.9197 283.559746 0.63 a 2 – −42.63829 −23.56414 −42.63829 – a 3 – – −0.003443396 – – a 4 – – 5.66283 × 10 −7 – – a 5 – – −1.30927 × 10 −10 – – a 6 – – 65812.39 – – a 7 – – – −4.849315 × 10 33 – a 8 4.24519 × 10 −22 – – – – Ni phases 0 G liq Ni 0 G bcc Mo 0 G bcc Ni T min 298 1728 298 298 1728 T max 1728 6000 6000 1728 6000 0 G ref 0 G fcc Ni – G fcc Ni – – a 0 16414.686 −9549.775 8715.084 −5179.159 −27840.655 a 1 −9.397 268.598 −3.556 117.854 279.135 a 2 – −43.1 – −22.096 −43.1 a 3 – – – −0.0048407 – a 4 – – – – – a 5 – – – – – a 6 – – – – – a 7 – – – – 1.12754 × 10 31 a 8 −3.82318 × 10 −21 – – – – T c – – 575 633 – β – – 0.85 0.52 – Ta phases 0 G liq Ta 0 G fcc Ta 0 G bcc Ta T min , K 298 3290 298 298 1300 2500 3290 T max ,K 3290 6000 6000 1300 2500 3290 6000 0 G ref 0 G bcc Ta – G bcc Ta – – – – a 0 29160.975 43884.339 16000 −7285.889 −22389.955 229382.886 −1042384.01 a 1 −7.578729 −61.981795 1.7 119.139857 243.88676 −722.59722 2985.49125 a 2 – 0.0279523 – −23.7592624 −41.137088 78.5244752 −362.159132 a 3 – −0.01233 – −0.002623033 0.00616757 −0.0179833 0.043117795 a 4 – 6.1459 × 10 −7 – 1.70109 × 10 −7 −6.551 × 10 −7 1.9503 × 10 −7 −1.055148 × 10 −6 a 5 – – – – – – – a 6 – −3523338 – −3293 2429586 −93813648 5.54714342 × 10 8 a 7 – – – – – – – a 8 – – – – – – – Note: 0 G θ i = a 0 + a 1 T + a 2 T ln T + a 3 T 2 + a 4 T 3 + a 5 T 4 + a 6 T −1 + a 7 T −9 + a 8 T 7 and T c (K) is the Curie temperature and β is the average magnetic moment per atom (Bohr magnetons). 0 G fcc Ni 0 G fcc Mo parameters G µ Ni:Ta:Ta:Ni (at Ni 7Ta 6 ) and G µ Ta:Ta:Ta:Ni (at Ni 6Ta 7 ) are selected first. These two parameters, however, are not sufficient to describe the composition range of the µ-NiTa phase. To extend the composition range to left side, the parameter G µ Ta:Ta:Ta:Ta as well as the associated interaction parameter 0 L µ Ta:Ta:Ta:Ni,Ta between end members Ta 1 Ta 4 Ta 2 Ni 6 and Ta 1 Ta 4 Ta 2 Ta 6 is included. Using the same arguments, the parameters G µ Ni:Ta:Ni:Ni , Gµ Ta:Ta:Ni:Ni , and 0 L µ Ta:Ta:Ni,Ta:Ni are employed to extend composition range to right side. To accommodate the small solubility of Mo in the µ-NiTa phase in ternary system, we modify this formulation to (Ni, Ta) 1 (Ta) 4 (Mo, Ni, Ta) 2 (Ni, Ta) 6 , permitting Mo on the third sublattice, and introduce the ternary parameters G µ Ta:Ta:Mo:Ni and 0 L µ Ta:Ta:Mo,Ta:Ni . These parameters are sufficient for describing the µ- NiTa phase, and the contributions from the remaining end members (i.e G µ Ni:Ta:Ta:Ta , Gµ Ta:Ta:Ni:Ta , Gµ Ni:Ta:Ni:Ta , Gµ Ta:Ta:Mo:Ta , Gµ Ni:Ta:Mo:Ni , and G µ Ni:Ta:Mo:Ta ) listed in Table 3 are assumed to be zero. The atoms in the κ-Ni 3 (Mo, Ta), C16-NiTa 2 and C11 b -Ni 2 Ta phases are distributed in two sublattices. The experimental data [6,7,10–12,15,16,28] reveal a homogeneous composition range for these phases, and each is described here with a two-sublattice model, as shown in Table 3, where a Φ i:j , bΦ i:j , 0 L Φ i,l:j and 0 L Φ j:i,l are the parameters to be evaluated. Due to a lack of experimental data for D0 22 -Ni 3 Ta, ζ -Ni 8 Ta, and ζ -Ni 8 Mo, these phases are treated as stoichiometric compounds in their respective binary systems. In the ternary Ni–Mo–Ta system, the experimental investigations by Chakravortye et al. [28] indicated a small solubility of Mo in ζ -Ni 8 Ta. Accordingly, the ζ - Ni 8 Ta and ζ -Ni 8 Mo phases are described here using a (Ni) 8 (Mo, Ta) formulation, and the corresponding Gibbs free energy function is given in Table 3. The D0 a -Ni 3 Mo and δ-NiMo phases are treated with a threesublattice model using the formulation (Mo, Ni) 4 (Ni) 2 (Mo, Ni) 2 and (Ni) 24 (Mo, Ni) 20 (Mo) 12 as described in Ref. [8]. The experimental data by Chakravorty and West [28] and Cui et al. [9] revealed a small solubility of Ta in D0 a -Ni 3 Mo and δ-NiMo. These phases, denoted by D0 a -Ni 3 (Mo, Ta) and δ-NiMo, are modeled here by considering the substitution of Ta for Mo, i.e. (Mo, Ni, Ta) 4 (Ni) 2 (Mo, Ni, Ta) 2 and (Ni) 24 (Mo, Ni, Ta) 20 (Mo, Ta) 12 , respectively, with the Gibbs free energy functions given in
S.H. Zhou et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 631–641 635 Table 3 Summary of the thermodynamic models used for the Ni–Mo–Ta ternary system with the total Gibbs free energy G Φ m = ref G Φ m + id G Φ m + xs G Φ m . Phase Sublattice formulation Model Liquid fcc (A1) bcc (A2) (Mo, Ni, Ta) 1 ref G Φ = m x Ni 0 G Φ + Ni x0 Mo GΦ + Mo x0 Ta GΦ id Ta G Φ = m RT(x Ni ln x Ni + x Mo ln x Mo + x Ta ln x Ta ) xs G Φ = ∑ m x n ∑ Mox j Ni L Φ j=0 Mo,Ni (x Mo − x Ni ) j n + x Mo x j Ta L Φ j=0 Mo,Ta (x Mo − x Ta ) j ∑ n + x Ni x j Ta L Φ j=0 Ni,Ta (x Ni − x Ta ) j + x Mo x Ni x Ta L Φ Mo,Ni,Ta µ (Ni, Ta) 1 (Ta) 4 (Mo, Ni, Ta) 2 (Ni, Ta) 6 Reference States: Group 1: Ni:Ta:Ni:Ni Ni:Ta:Ta:Ni Ta:Ta:Ni:Ni Ta:Ta:Ta:Ni Ta:Ta:Ta:Ta Ta:Ta:Mo:Ni Group 2: Ni:Ta:Ta:Ta Ta:Ta:Ni:Ta Ni:Ta:Ni:Ta Ta:Ta:Mo:Ta Ni:Ta:Mo:Ni Ni:Ta:Mo:Ta ∑ ref G µ = ∑ m i=Ni,Ta yI i 0 G µ = 0 i:Ta:j:k G fcc i + 4 0 G bcc Ta = 0(Group 2) 0 G µ i:Ta:j:k j=Mo,Ni,Ta yIII j + 20 G bcc j ∑ k=Ni,Ta yIV k 0 G µ i:Ta:j:k + 6 0 G fcc k + G µ i:Ta:j:k (Group 1) ) id G µ = m (∑i=Ni,Ta RT (yI i ln yI + i 2yIII i ln y III i + 6y IV i ln y IV i ) xs G µ = ∑ ∑ ∑ m yI Ni yI k Ta L µ i=Mo,Ni,Ta j=Ni,Ta k=0 Ni,Ta:Ta:i:j (yI − Ni yI Ta )k + ∑ ∑ ∑ ∑ ∑ l=Mo,Ni p=Ni,Ta yIII l y III p ∑ ∑ ∑i=Ni,Ta j=Ni,Ta k=0 + y IV Ni yIV k Ta L µ i:Ta:j:Ni,Ta (yIV − Ni yIV Ta )k i=Ni,Ta j=Mo,Ni,Ta k=0 k L µ i:Ta:l,p:j (yIII l − y III p )k (Mo, Ni, Ta) 4 (Ni) 2 (Mo, Ni, Ta) 2 Reference states: Group 1: ref G D0a = ∑ ∑ Mo:Ni:Mo m i=Mo,Ni,Ta yI i j=Mo,Ni,Ta yIII 0 j G D0a i:Ni:j Ni:Ni:Mo 0 G D0a = i:Ni:j 40 G fcc i + 2 0 G fcc Ni + 2 0 G bcc j + G D0a i:Ni:j (Group 1) Ta:Ni:Ta 0 G D0a i:Ni:j = (Group 2) D0 a I Ni:Ni:Ta id G D0a = ∑ m RT i=Mo,Ni,Ta (4yI i ln yI + i 2III i ln y III i ) Ni:Ni:Ni xs G D0a = ∑ ∑ ∑ ∑ Group 2: m i k>i yI i yI n k L D0a j n=0 i,k:Ni:j (yI − i yI k )n + ∑ ∑ ∑ ∑ (Mo, Ta):Ni:Ni j i k>i yIII i y III k n=0 LD0a l:Ni:i,k (yIII i − y III k )n Mo:Ni:Ta Ta:Ni:Mo κ (Ni, Mo, Ta) 3 (Ni, Mo, Ta) ref 1 G Φ = ∑ ∑ m i=Mo,Ni,Ta yI i j=Mo,Ni,Ta yII 0 j G Φ i:j D0 a II Note: 0 G D0aII = 0 Mo:Ta G D0aII = Ta:Mo 0 0 G Φ = i:j p0 G ref i + q 0 G ref j + G Φ = i:j p0 G ref i + q 0 G ref j + a Φ + i:j bΦ i:j T C16 (Ni, Ta) id G Φ = m ∑i=Mo,Ni,Ta RT (pyI i ln yI + i qyII i ln y II 1 (Ni, Ta) 2 ) i xs G Φ = ∑ ∑ ∑ ∑ C11 m i l>i yI i yI k l L Φ b (Ni, Ta) 2 (Ni, Ta) 1 j k=0 i,l:j (yI − i yI l )k + ∑ ∑ ∑ ∑ ζ (Ni) 8 (Mo, Ta) j i l>i yII i yII k l L Φ k=0 j:i,l (yII i − y II l )k (p and q are the subscript numbers of sublattices, respectively) ∑j=Mo,Ta yIII 0 j G δ Ni:i:j ref G µ m = ∑ i=Mo,Ni,Ta yII i 0 G δ = Ni:i:j 240 G fcc Ni + 20 0 G bcc i + 12 0 G bcc j + G δ Ni:i:j δ (Ni) 24 (Mo, Ni, Ta) 20 (Mo, Ta) id 12 G δ = (∑ m RT i=Ni,Ta (20yII i ln y II i + 12y III i ln y III i ) ) xs G δ = ∑ ∑ ∑ m i=Mo,Ni ∑j=Ni,Ta yII i yII j l=Mo,Ta k=0 + ∑ ∑ i=Mo,Ni,Ta yIII Mo yIII k Ta L µ Ni:i:Mo,Ta (yIII − Mo yIII Ta )k k=0 k L δ Ni:i,j:l (yII i − y II j )k Ni 2 Mo (Ni) 2 (Mo) 1 G Φ = m GΦ + Ni:Ta p0 G fcc Ni + q 0 G bcc i(i=Mo,Ta) Ni 4 Mo (Ni) 4 (Mo) 1 = a Φ + b Φ T + p 0 G fcc Ni + q 0 G bcc i(i=Mo,Ta) D0 22 (Ni) 3 (Ta) 1 (p and q are the subscript numbers of sublattices, respectively) Table 3. D0 a -Ni 3 Mo was also described with the two-sublattice model using the (Mo, Ni) 3 (Mo, Ni) 1 formulation in Ref. [8]. To be compatible with this treatment in the ternary system, another set of parameters is developed for the two-sublattice model, i.e. (Mo, Ni, Ta) 3 (Mo, Ni, Ta) 1 , denoted by D0 a II–Ni 3 (Mo, Ta). In this case, the Gibbs free energy functions can be written in the same form as those for κ-Ni 3 (Mo, Ta) in Table 3. 3. Determination of the thermodynamic model parameters In the determination of the model parameters for the ternary Ni–Mo–Ta system, the Ni–Ta binary is evaluated first and then integrated with reported models for the Ni–Mo [8] in Fig. 1(d) and Mo–Ta [9] in Fig. 1(e) systems to build the thermodynamic description. With the thermodynamic models described in the preceding section, we employ a total of twenty-four Gibbs free energy of formation parameters and fourteen interaction parameters for the binary Ni–Ta system, as listed in Tables 4 and 5. The evaluation process, utilizing both experimental data and first principles results for the Ni–Ta system, is discussed in this section. Using VASP [38] with the Vanderbilt ultrasoft pseudopotential [39] within the generalized gradient approximation (GGA) [40], the total energy of the fcc-Ni, bcc-Mo, bcc-Ta, and compounds are calculated. Monkhorst 15 × 15 × 15 k points are used for the pure elements Ni, Mo and Ta, 11×11×11 k points for the end-members of the ζ -Ni 8 Ta, D0 22 -Ni 3 Ta, C11 b -Ni 2 Ta, D0 a -Ni 3 Mo and C16-NiTa 2 phases, 9×9×9 k points for the end-members of the µ-NiTa phase and 4 × 4 × 4 k points for the end-members of the κ-Ni 3 Ta phase. To ensure that the unit cell corresponds to a stable structure, we fully relax the cell shape and the internal atomic coordinates for the stable end-members and relax only the cell volume for the unstable/metastable end-members.
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