CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry

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1 CALPHAD: Computer Couplng of Phase Dagrams and Thermochemstry 46 (2014) Contents lsts avalable at ScenceDrect CALPHAD: Computer Couplng of Phase Dagrams and Thermochemstry journal homepage: Soluton-based thermodynamc modelng of the system usng frst-prncples calculatons S.H. Zhou a,b,n, Y. Wang a, L.-Q. Chen a, Z.-K. Lu a, R.E. apoltano b,c a Department of Materals Scence and Engneerng, The Pennsylvana State Unversty, Unversty Park, PA 16802, USA b Materals & Engneerng Physcs Program, Ames Laboratory, USDOE, USA c Department of Materals Scence and Engneerng, Iowa State Unversty, Ames, IA 50011, USA artcle nfo Artcle hstory: Receved 31 October 2013 Receved n revsed form 9 January 2014 Accepted 6 March 2014 Avalable onlne 22 March 2014 Keywords: CALPHAD Frst-prncples calculaton phase dagram ATAT abstract A soluton-based thermodynamc descrpton of the ternary system s developed here, ncorporatng frst-prncples calculatons and reported modelng of the bnary, and systems. To search for the confguratons wth the lowest energes of the phase, the loy Theoretc Automated Toolkt (ATAT) was employed and combned wth VASP. The lqud, bcc and γ-fcc phases are modeled as random atomc solutons, and the γʹ- 3 phase s modeled by descrbng the orderng wthn the fcc structure usng two sublattces, summarzed as (,,) 0.75 (,,) Thus, γ-fcc and γʹ- 3 are modeled wth a sngle Gbbs free energy functon wth approprate treatment of the chemcal orderng contrbuton. In addton, notable mprovements are the followng: frst, the ternary effects of and n the B2- and D0 a - 3 phases, respectvely, are consdered; second, the phase s descrbed as a sold soluton usng a three-sublattce model; thrd, the X phase s treated as a stochometrc compound. del parameters are evaluated usng frst-prncples calculatons of zero-kelvn formaton enthalpes and reported expermental data. In comparson wth the enthalpes of formaton for the compounds ψ-, θ- 8 3 and B2-, the frst-prncples results ndcate that the phase, whch s stable at hgh temperatures, decomposes nto other phases at low temperature. Resultng phase equlbra are summarzed n the form of sothermal sectons and lqudus projectons. To clearly dentfy the relatonshp between the γ-fcc and γʹ- 3 phases n the ternary system, the specfc γ-fcc and γʹ- 3 phase felds are plotted n x() x() T space for a temperature range K. & 2014 Elsever Ltd. l rghts reserved. 1. Introducton The hgh temperature performance of superalloys based on the bnary system s a crtcal factor n current and future power generaton and transportaton technologes. reover, the refractory metal s well establshed as a key contrbutor to the hgh temperature stablty of many multcomponent -base superalloys [1 6]. Despte ts wdespread use, quanttatve gudelnes for future alloy development and processng desgn call for more complete descrptons of the effects of addtons on the phase stablty and phase transformaton n the system. Ths s the focus of the present work. Thermodynamc modelng for the ternary system was frst performed by Kaufman and esor [7], n whch all compounds were descrbed as stochometrc compounds. Subsequent to treatments of the [8], [9], and [10] systems, Lu et al. [11] reassessed the ternary system n Correspondng author. whch represented a great mprovement over the modelng by Kaufman and esor [7], and, despte several lmtatons, has been used wth reasonable success for qute some tme. As ncreased demands for hgh temperature materals drve the exploraton of new multcomponent alloy regmes, however, these lmtatons become problematc. Frst, the bnary phase dagram resultng from the model by Lu et al. [11] s dfferent from those by Frsk [10] n Fg. 1(a) although Lu et al. [11] use the database developed by Frsk [10]. Lu s model (dashed curves) ndcates that () the B2 phase becomes stable around 1900 K ndcatng the B2 phase can be a prmary phase n a ternary system and () the γʹ phase becomes stable at low temperature, whch does not agree wth Frsk s model [10] and the reported experments [12 14]. In addton, Lu s model was found to be nconsstent wth expermental phase equlbrum data [15 18], lmtng ts utlty n phase-feld modelng [19]. These problems arse from napproprate descrptons of the thermodynamc effects of n the ternary system [11], and several mportant mprovements are called here for. Frst, recent advances n modelng the [20] and [21] bnary systems should be ncorporated nto the ternary /& 2014 Elsever Ltd. l rghts reserved.

2 S.H. Zhou et al. / CALPHAD: Computer Couplng of Phase Dagrams and Thermochemstry 46 (2014) Fg. 1. The bnary phase dagrams of the (a) system calculated usng the parameters by Frst [10] n sold-lne and Lu et al. [11] n dash-lne; (b) system calculated usng the parameters by Zhou et al. [21] n sold-lne and Lu et al. [11] n dash-lne; (c) system [9] and (d) system [8,20]. Table 1 A lstng of phases modeled n the current treatment. Formula unt Sym. Lattce Struk. des. Prototype Phases extendng from the pure component states:, Ta, Lqud γ Cub A1 fcc γ Cub A1 fcc Cub A2 bcc Phases extendng from the bnary system: 3 Ort D0 11 Fe 3 C 3 2 Tr D Cub B2 ClCs 3 5 ε Ort Ga 5 Pt 3 3 γ Cub L1 2 AuCu 3 Phases extendng from the bnary system: δ Orth 2 ρ Tet Pt 2 3 Tet D0 a Cu 3 T 4 Tet D1 a 4 8 ζ Tet 8 b Phases extendng from the bnary system: 12 ϕ Cub 12 W 5 η Hex 5 W 4 φ n 4 W 8 3 s n 8 W τ ψ Cub A2 W 3 Cub A15 Cr 3 S Ternary compounds 8 3 Tet D T X descrpton. Second, the ternary and X phases lsted n Table 1, whch were not descrbed n prevous reports [7,11], should be consdered. In the present work, the thermodynamc descrptons of the bnary [8,20], [9], and [21] systems wth the assocated phase dagrams shown n Fg. 1(b d) are employed wthn a general soluton-based approach, usng frst-prncples calculatons and expermental data [11,15 18,22 39], whch are approprate to determne the ternary model parameters. The ternary and X phases are ncluded and the full phase dagram s computed. The equlbrum phase dagram s presented wth a seres of sothermal sectons and a 3D plot for the γ and γʹ phase felds. The resultng phase dagrams of the ternary system are compared wth the expermental data [11,15 18,22 39] and the pror report by Lu [11]. 2. Frst-prncples calculatons To determnate the Gbbs free energy of formaton for the ntermedate phases, the zero-kelvn energes of the phases and the unstable end-members (descrbed n Secton 3) fortheb2andd0 a - 3 phases as well as fcc-, fcc-, bcc- were calculated by means of VASP [40] employng Vanderblt ultrasoft pseudopotental [41] and the generalzed gradent approxmaton (GGA) [42] wth the hgh precson choce. nkhost k ponts were used for the pure elements, and, k ponts for the compounds B2 and D0 a - 3, and k ponts for the compound. For the stable structures, we fully relaxed the volume, the

3 126 S.H. Zhou et al. / CALPHAD: Computer Couplng of Phase Dagrams and Thermochemstry 46 (2014) cell shape, and the nternal atomc coordnates whle for the unstable structures of the end-members we only relaxed the cell volume to mantan the unt cell structures to ther stable phases. The enthalpy of formaton, ΔH Φ f, for a gven compound Φ s calculated as the dfference between the energy E Φ TOT of the compound and the lnear combnaton of the pure element reference state energes, E fcc, E fcc and E bcc ΔH Φ f ¼ E Φ TOT xφ Efcc x Φ Efcc x Φ Ebcc where x Φ s the mole fracton of component n Φ. The calculated enthalpes of formaton of the compounds are lsted n Table 2a. The compostons of the phase were reported as by Raman and Schubert [23,24] and as by Markv et al. [22] as the dashed-lnes n Fg. 2, respectvely. To search for the confguratons wth the lowest energes, ATAT developed by van de Walle [43] was employed and combned wth VASP. The startng geometry was lmted to the D0 22 structure wth the fxed composton (25 at% ) and atom postons. Gamma centered k ponts were used and more than 65 confguratons were studed. The calculated enthalpes of formaton for some of the structures found by ATAT are plotted n Fg. 2 ndcatng that the structure at the lowest pont along the concave curve s at composton 8 3 and gves a formaton energy of kj/mol, whch s lower than the those values at both and by Raman and Schubert [23,24] and Markv et al. [22], respectvely. Ths ternary compound s a trpled superstructure of that of DO 22 along b drecton, resultng n a body centered orthorhombc structure wth the lattce parameters beng a¼ Å, b¼ Å, and ð1þ Table 2b The crystal structure of the lowest energy structure at composton 8 3. Poston type Atomc poston c¼ Å. The prmtve unt cell of ths ternary compound contans 12 atoms whch are arranged n the symmetry. The atomc postons of ths ternary compound are lsted n Table 2b. In ths work, we desgnated ths ternary compound as and descrbe t as a soluton phase wth small homogeneous composton by consderng the expermental observatons by Raman and Schubert [23,24] and Markv et al. [22]. As lsted n Table 2a, the frst-prncples calculated enthalpy of formaton for the phase s kj/mol, whch s less negatve than the enthalpy of mxng,.e kj/mol, for the ψ-, θ- and B2- phases from the bnary systems, ndcatng that the phase decomposes nto other phases at low temperature. Due to the lack of structural nformaton, the frstprncples calculaton was not performed for the compound X. Table 2a A summary of results from the frst-prncples calculatons. Phase Symbol Prototype Formula ΔH, kj/mol of atoms fcc 0 γ fcc 0 A2 bcc 0 B2 CsCl D0 a Cu 3 T T Thermodynamc models The thermodynamc propertes of pure,, and n varous structures were computed usng the parameters from the SGTE database [44] as lsted n Table 3. In addton to the compounds n the [8,20], [9] and [21] systems, the lqud, bcc, γ-fcc, B2-, γʹ- 3, D0 a - 3, and were modeled as ternary soluton phases and X phase as a stochometrc compound. The assocated Gbbs free energy functons are defned n Table 4, where the total Gbbs free energy for the phase, Φ, s generally gven by the sum of three contrbutons: G Φ m ¼ ref G Φ m þ d G Φ m þ xs G Φ m ; ð2þ where the subscrpt m denotes that all terms are molar quanttes. The frst term n Eq. (2) s the sum of occupancy-weghted sublattce end-member contrbutons. The second and thrd terms are the deal and excess parts of the Gbbs free energy of mxng, respectvely. The excess Gbbs free energy n all Gbbs free energy equatons s expressed n terms of the Redlch Kster polynomal [45]. The specfc treatment of each phase s dscussed brefly here. The lqud, bcc and γ-fcc are descrbed usng the model gven n Table 4, where 1G Φ s the molar Gbbs free energy of the pure element wth the structure Φ (Φ¼lqud, bcc or γ-fcc) as lsted n Table 3 and x s the mole fracton of the ndcated component. j L Φ ;k s the bnary nteracton parameter, and LΦ ;; s a composton dependent ternary nteracton parameter evaluated wth the expermental data. Magnetc property of the s consdered for ts fcc structure. mg G Φ m s the magnetc contrbuton [46] to Gbbs free energy n soluton phases, and expressed as the followng: mg G Φ m ¼ RT lnðβφ þ1þf ðτ Φ Þ; ð3þ Fg. 2. Enthalpy of formaton for the phase calculated usng ATAT and VASP wth the fxed atomc postons x ¼0.25 and current thermodynamc parameters n Tables 6 and 7. Reference states: fcc-, fcc- and bcc-. where β Φ s the quantty related to the total magnetc entropy and s set equal to the Bohr magnetc moment per mole of atom. τ Φ s

4 S.H. Zhou et al. / CALPHAD: Computer Couplng of Phase Dagrams and Thermochemstry 46 (2014) Table 3 Thermodynamc parameters for pure elements [44]. phases 0 G lq 0 G bcc T mn, K T max, K G REF A 11, , , , ,083 B C D E 74,092 74,092 F G H phases 0 G lq 0 G bcc T mn T max G REF 0 G bcc 0 G bcc 0 G bcc 0 0 A 41, , , , B C D E 65,812 F G H I phases 0 G lq 0 G bcc T mn T max G REF 0 0 A 16, , , B C D E F G H T c β ote: O G θ ¼ O G Ref þaþbt þct ln T þdt 2 þet 1 þft 3 þgt 7 þht 9 þt 4 (J/mol) and T c (K) s the Cure temperature and β s the average magnetc monent per atom (Bohr magnetons). defned as T=T Φ c and T Φ c s the Cure temperature. For the soluton phase, T Φ c and β Φ are descrbed as the followng: T Φ c ¼ x A 0 T Φ ca þx B 0 T Φ cb þx Ax B n j L Φ TðA;BÞ ðx A x B Þ j ð4þ β Φ ¼ x A0 β Φ A þx B 0 β Φ B þx Ax B j ¼ 0 n j ¼ 0 j L Φ βða;bþ ðx A x B Þ j where 0 T Φ c and 0 β Φ are from the pure element as lsted n Table 3. j L Φ TðA;BÞ and j L Φ βða;bþ are the magnetc nteracton parameters between the elements A and B. In ths work, the magnetc parameters from the bnary system [8] are employed as lsted n Table 7. The thermodynamc modelng of the ordered γʹ- 3 and dsordered γ-fcc phases was dscussed by Ansara et al. [8,47] and Dupn ð5þ et al. [20] n the thermodynamc evaluaton of the, and Cr systems wth a specal emphass on the relatonshp among the parameters for the ordered γʹ phase. The γʹ phase n the bnary system was modeled wth the two-sublattce model (,) 0.75 (,) 0.25 [8]. Consderng the thermodynamc effect of the component, the two-sublattce model for γʹ was modfed as (,,) 0.75 (,,) Adoptng the descrptons by Ansara et al. [8,47] and Dupn et al. [20], the Gbbs free energy, G γ ' m,forγʹ s expressed as the followng: ðaþ ðbþ G γ' m ¼ Gγ m ðx ÞþΔG ord m ð : jþ; ΔG ord m ð : jþ¼gord m ð : jþ y I ;yii G ord m ð : jþ x ¼ y I ¼ y ; II where G γ m ðx Þ s the Gbbs free energy of the dsordered γ-fcc phase as descrbed n Table 4. ΔG ord m ð; jþ n Eq. (6) corresponds to the orderng energy, beng descrbed wth Eq. (6b) [20,48]. G ord m ð : jþ n Eq. (6b) s ð6þ

5 128 S.H. Zhou et al. / CALPHAD: Computer Couplng of Phase Dagrams and Thermochemstry 46 (2014) Table 4 Summary of the thermodynamc models used for the - ternary system wth the total Gbbs free energy G Φ m ¼ ref G Φ m þ d G Φ m þ xs G Φ m (0 G ref Phase Formulaton del ¼, 0 G bcc or ). Lqud (,,) ref 1 G Φ m ¼ x 1G Φ γ-fcc ¼ ;; Bcc d G Φ m ¼ RT x ln x ¼ ;; xs G Φ m ¼ x x n j L Φ ; ðx x Þ j þx x n j L Φ ; ðx x Þ j j ¼ 0 j ¼ 0 þx x n j L Φ ; ðx x Þ j þx x x L Φ ;; ; j ¼ 0 L Φ ;; ¼ x 0 L Φ ;; þx 1 L Φ ;; þx 2 L Φ ;; ψ- (,) 1 (,) ref 1 G Φ m ¼ y I y II j 1GΦ :j B2- (,,) 1 (,,,Va) 1 ¼ ;; j ¼ ;; D0 1G Φ :j ¼ p1gref þq1g ref j þδg Φ :j ¼ p1gref þq1g ref j þa Φ :j þbφ :j T a - 3 (,,) 0.75 (,,) 0.25 G D0a : ¼GD0a : ¼GD0a : ¼0 d G Φ m ¼ RT ðpy I ln yi þqyii ¼ ;; xs G Φ m ¼ l 4 y I yi l k L Φ ;l:j ðyi yi l Þk j k ¼ 0 γ -3 (,,) 0.75 (,,) G ord :j ¼ 0 G ord j: ¼ 3u 1 j 0 L ord ;j: ¼ 0 L ord ;j:j ¼ 6u 1 j 1 L ord ;j: ¼ 1 L ord ;j:j ¼ 3u 4 j 1 L ord :;j ¼ 1 L ord j:;j ¼ u 4 j 0 L ord ;j:k ¼ 6u1 j þðu jk þuj jk Þ3=2 ln y II Þ þ y II yii l k L Φ j:;l ðyii y II l Þk j l 4 k ¼ 0 (p and q are the subscrpt numbers of sublattces, respectvely) 1 L ord ;j:k ¼ 3u4 j þðu jk uj jk Þ=2 0 L ord ;j;k: ¼ 6u jk 3uj jk =2 3uk jk =2 (,) 2 (,) 1 () 1 G :: ¼0 D () 3 (,) 2 (,Va) 1 δ- () 24 (,) 20 () 12 ref G Φ m ¼ y I j 1G Φ :j:k ¼ p1gref d G Φ m ¼ RT y II j y III k 1G :j:k k þq1g ref j þr1g ref þδg k :j:k ðpy I ln yi þqyii ln y II þry III ln y III Þ xs G Φ m ¼ l 4 y I yi l j n ¼ 0 þ j y III l 4 y III l n L Φ ;l:j:k ðyi yi l Þn þ j n L Φ k:j:;l ðyiii y III l Þ n n ¼ 0 y II yii l n L Φ j:;l:k ðyii y II l Þn l 4 n ¼ 0 ϕ- 12 () 12 () 1 G Φ m ¼ ΔGΦ :j þp1gref ¼ a Φ þb Φ T þp1g ref φ- 4 () 4 () 1 η- 5 () 5 () 1 τ () 63 () 37 θ- 8 3 () 8 () 3 A15 3 () 0.75 () 0.25 D () 0.75 () 0.25 ε- 3 5 () () D1 a - 4 () 0.8 () 0.2 ρ- 2 () 0.67 () 0.33 ζ- 8 () 0.89 () 0.11 þq1g ref j þq1g ref j (p and q represent subscrpt values n formulaton, respectvely.) X () 0.75 () 0.11 () 0.14 G X m ¼ 0:75 þ0:11 0 G bcc þ0:14 þa X þb X T the Gbbs free energy contrbuton due to orderng wth x ¼ 3y I =4þyII =4 as descrbed by the two-sublattce model n Table 4 where y I and yii are st-fracton n frst and second sublattces, respectvely. It has three terms as shown n Eq. (2) and n Table 4 and contans mplctly a contrbuton to the dsordered state also. When y I ¼ yii ¼ x, G ord m ð : jþ represents the extraneous excess energy contrbuton to the dsorder state. Thus ΔG ord m ð; jþ s descrbed as the dfference between G ord m ð : jþ y and G ord I ;yii m ð : jþ x ¼ y I ¼ y n Eq. (6b) and II must be zero when the phase s dsordered. 0 G ord :j, k L ord ;j:l,and k L ord :j;l lsted n Table 4 are the parameters for G ord m ð : jþ. As descrbed n reports [8,20,47,49], n order to favor the stablty of the ordered phase to dsorder transt n certan temperature and composton ranges, G γ' m should always have an extremum when y I ¼ yii ¼ x. When the dsordered phase s stable, ths extremum s a mnmum. Hence, the followng equaton [8,47] has to be fulflled: ðdg γ' m Þ x ¼ s G γ' m y ðsþ! x dy ðsþ ¼ 0 ð7þ From ths equaton, the parameters 0 G ord :j, k L ord ;j:l, and k L ord :j;l can be expressed wth the parameters u 1 j, u4 j, u jk, uj jk, and uk jk (, j, or k¼,, or ) as shown n Table 4. The detal descrptons of the parameter dervaton for the ordered γʹ- 3 phase can be found n Refs. [20,49].

6 S.H. Zhou et al. / CALPHAD: Computer Couplng of Phase Dagrams and Thermochemstry 46 (2014) Table 5 Symbols and related references used n Fgs Fgure Fgs. 3 and 5(b) Boundary Sngle phase Symbol Ref. γ bcc γʹ δ þγ * γʹ [26] Symbol Two-phase Ref. Fg. 5(d) and (f) Fg. 5(e) and (g) Symbol Ref. Symbol Ref. γ þγʹ [27] γ þδ þ bcc γ þγʹ þbcc γ þγʹ þbcc [26] fccþ bcc þlq [37] γ þγʹ þbcc bcc þ B2 þγʹ γ þ bcc þδ B2þ bcc þ l12 fccþ l12 þbcc γ þγʹþ δ * bcc þ δ þγʹ bcc þ B2 þ γʹ D0a þ γʹ þδ γ þ γʹþ D0a B2 þ3 þ γ þ γʹþ δ * bcc þ δþ γʹ lqþ 3þ lq þ 32 þx þ32 þ X γ þγʹ þ D0a γ þγʹþ δ [31] bcc þlq bcc þlq fcc þ lq bccþ lq fccþ lq þ lq 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] Fg. 4 Fg. 5(a) and (c) Three-phase γ [22] [23,24] [25] [34] [38] [22] γ þγʹ δ þγ bcc þγʹ * bcc þ B2 γ þγʹ γʹþ δ δþ γ bcc þγʹ γʹþ D0a δ þd0a γ þ D0a δþ γ δ þ γʹ bcc þ γʹ 3þ γ þ γʹ [31] [29] [29] [22] [35] [11] [24] [35] [22] γʹ X [28] [22] [32] [36] [18] [36] [32] þ γʹ two-phase field at 1048 K. The -83 phase has the D022 structure wth the space group I4/mmm [22,24]. Ther atoms dstrbute n three-sublattces. Accordng to the first-prncples data n Table 2a and expermental data [22,24] dscussed n Secton 2, the -83 phase s descrbed wth the three-sublattce model, (,)2(,)1()1, wth the Gbbs free energy gven n Table 4, where yi and yii are the ste fractons n the first and second sublattces, respectvely. The Gbbs free energy of formaton ΔG :j: of the end-member :j: s xs expressed as a þ b T. G m s the excess Gbbs free energy :j: :j: 1 wth the nteracton parameters 0 L :;: and L:;: beng assumed as a constant. Both of them as well as a :j: and b:j: are evaluated wth the expermental and first-prncples data. The thermodynamc models of B2- and D0a-3 phases were descrbed n detal n Refs. [8,21], respectvely. The expermental data [11,15,35] revealed a small solublty of n B2- and n D0a-3. The non-stochometrc B2- and D0a3 phases were modfied here wth the model shown n Table 4. Here we only consder the unstable end-members wth B2 the Gbbs free energy of formatonsδgb2 : and ΔG: for B2D0a and ΔG: for D0a-3. The X phase was treated as stochometrc compounds wth the X Gbbs free energy functons as shown n Table 4, where ax and b are the model parameters to be evaluated. 4. Determnaton of the thermodynamc model parameters The parameters descrbed n the precedng secton for the ternary system were determned n the order of γ-fcc and γʹ-3 phases, B2-, D0a-3, lqud, and X phases usng the avalable expermental data n Table 5 and first-prncples data n Table 2a. The evaluated parameters as well as those of the bnary [8,20], [9] and [21] systems are lsted n Tables 6 and 7. In the present secton, the methodology used for the determnaton of the parameters s dscussed. In both bnary and systems, and dssolve n γ-fcc(). The sothermal secton at 1533 K s calculated only usng the parameters of the three bnary systems n Tables 6 and 7 and shown n Fg. 3, where the calculated γ and γʹ two-phase regon devates far from the expermental data [25 27] ndcatng that the

7 130 S.H. Zhou et al. / CALPHAD: Computer Couplng of Phase Dagrams and Thermochemstry 46 (2014) Table 6 Gbbs free energy of formaton for end-member reference states (per mole of formula unt). Phase Parameters Value, J/mol of atoms Ref. D ΔG D011 : 48,483.73þ12.299T a D ΔG D5 13 :: 39, þ7.895T ΔG D5 13 :: 427,191.9þ79.217T ΔG D5 13 ::Va 30,000 3T ΔG D513 ::Va 357,725.92þ68.322T ε- 3 5 ΔG ε : 55, þ7.265T [8] B2- ΔG B2 a : 0 ΔG B2 :, ΔGB2 a : 152,397.3þ26.406T ΔG B2 a :Va 1000 T ΔG B2 a : 0 ΔG B2 a :Va 162, T ΔG B2 :Va 150,000 [9] ΔG B2 : 150,000 ΔG B2 :, ΔGB2 : 5876 Ths work ΔG B2 :, ΔGB2 : 27,844 γʹ- 3 u 1 13, þ2.082T [8] u T u T Ths work u þ3.411T u 1 0 u 4 0 u 20, T ϕ- 12 ΔG φ : 139,100þ26.975T [9] φ- 4 ΔG ϕ : 137,570þ29.69T η- 5 ΔG η : 139,104þ30.156T τ ΔG τ : 2,268,100þ167.2T θ- 8 3 ΔG θ : 412,500þ105.05T ψ- ΔG ψ : 20, a T ΔG ψ : 36,850þ1.0T ΔG ψ : 0 A15 3 ΔG A15 : 89,000þ20T 0.003T 2 δ- ΔG δ :: 169,981þ T T ln (T) [21] ΔG δ :: 154,106þ T T ln (T) ρ- 2 ΔG ρ : 9421þ49.551T 6.231T ln (T) D0 a - 3 ΔG D0a : 42,650 [21] ΔG D0a : 10,131.9þ58.132T 7.366T ln (T) ΔG D0a : 17,060 ΔG D0a : 2840 ΔG D0a : 37,910þ5.651T Ths work D1 a - 4 ΔG D1a : 9021þ55.004T 7.080T ln (T) [21] ζ- 8 ΔG ς : 6115þ33.258T 4.085T ln (T) ΔG :: 29,600þ32.028T Ths work ΔG :: 119,040 ΔG :: 108,760þ22.128T X ΔG X :: 39,464.2þ6.521T a Reference state s -bcc and -bcc,.e. 0 G ref ¼ 0 G bcc or 0 G bcc. bnary nteracton parameters for γ-fcc from both [8,20] and [21] systems predct a ternary γ-fcc-feld that s too expansve; hence postve ternary nteracton parameters here had to be consdered to reduce t. Smlar behavor was also found n the ternary Cr system modeled by Dupn et al. [20], nwhchthe postve ternary nteracton parameter for γ-fcc() was employed. In addton to the parameters n the [8,20] and [21], the thermodynamc propertes of the γ and γʹ phases n the ternary system are descrbed wth the ternary nteracton parameters 0 L γ ;;, 1 L γ ;;, 2 L γ ;;, u1 ;, u4 ; and u. These parameters are evaluated wth the expermental data [11,15 18,22,26 35] and lsted n Tables 6 and 7. The expermental data [11,15,22,26,35] ndcate lttle solublty of n B2-. The model parameters ΔG B2 : (¼,), whch are consdered as a constant n Table 4, were fxed wth the frstprncples data as lsted n Tables 6 and 7. Based on the thermodynamc descrpton n Secton 3, the model parameters a D0a : and b D0a : n Table 4 are used for the compound D0 a - 3. The parameter a D0a was determned wth : the frst-prncples data, whle b D0a : was evaluated wth the phase equlbrum data,.e. the four-phase δþγ-γʹþd0 a reacton at K. The evaluated parameters are lsted n Table 6. Accordng to the expermental phase equlbrum data [22 24] and frst-prncples data n Fg. 2, the composton range of the

8 S.H. Zhou et al. / CALPHAD: Computer Couplng of Phase Dagrams and Thermochemstry 46 (2014) Table 7 Excess Gbbs free energy nteracton parameters. Phase Parameters Value, J/mol Ref. Lqud bcc γ-fcc 0 L Lq ; 1 L Lq A; 2 L Lq A; 3 L Lq A; 4 L Lq A; 0 L lq ; 1 L lq ; 0 L lq ; 1 L lq ; 2 L lq ; 0 L lq ;; 1 L lq ;; 2 L lq ;; 0 L bcc ; 1 L bcc ; 0 L bcc ; 0 L γ ; 0 L γ ; 1 L γ ; 2 L γ ; 3 L γ ; 0 L γ Tð;Þ 1 L γ Tð;Þ 0 L γ ; 1 L γ ; 2 L γ ; 0 L γ ;; 1 L γ ;; 2 L γ ;; D L D5 13 :;: 0 L D513 ::;Va 0 L D513 ::;Va 0 L D5 13 :;:Va B2-0 L B2 ;: 0 L B2 :;Va 0 L B2 :;Va 0 L B2 ;:Va ψ- 0 L ψ ;: δ- D0 a L ψ :; 0 L ψ :; 1 L ψ :; 0 L ψ ;: 1 L ψ ;: 0 L δ :;: 1 L δ :;: 0 L D0a ;: 0 L D0a :; 0 L :;: 1 L :;: 207,109þ41.315T [8] 10,186þ5.871T 81, T T 22,101.64þ13.163T 100,000þ35T [9] 15,000þ6.3T 39,597þ15.935T [21] 7373þ4.102T 12,123þ5.551T 50,748 Ths work 70, ,748 27,691 [21] 18,792 75,000þ25T [9] 92,220þ20T 162,407.75þ16.213T [20] 73, T 33, T 30,758.01þ10.253T 1112 [8] þ3.591T [21] T T 41, T 91, , T 193,484.18þ131.79T [8] 22,001.7þ7.033T 22,001.7þ7.033T 193,484.18þ131.79T 52,440.88þ T 64,024.38þ T 64,024.38þ26.494T 52,440.88þ11.301T 5000 [9] ,000 10,000 25,000 10, ,211þ T [21] 417,368þ T T 140,870 60,570 Ths work Ths work stable phase s estmated between and at hgh temperatures. Fg. 2 shows the composton wth the mnmum enthalpy of formaton for the phase beng at 8 3. Therefore, the nteracton parameters, 0 L :;: and 1 L :;:, and the Gbbs free energes of formaton, ΔG ::, ΔG :: and ΔG :: were consdered for the Gbbs free energy Fg. 3. Isothermal secton of the phase dagram calculated at 1533 K usng the parameters of the three bnary systems n Tables 6 and 7 n comparson wth expermental data wth the symbols lsted n Table 5. Fg. 4. Lqudus projecton of the system calculated n the composton trangle usng the parameters n Tables 6 and 7 ndcatng the prmary phases and comparng wth expermental data as the symbols n Table 5 and dash-lne by Lu et al.[11]. functon of the phase n whch ΔG ::, 0 L :;: and 1 L :;: were assumed to be constant. The parameters, a ::, a :: and a ::, were fxed by the frst-prncples data. The parameters, b :: and b ::, were evaluated wth the phase equlbrum data [22,24]. The nteracton parameters, 0 L :;: and 1 L :;:, were evaluated wth the frst-prncples data n Fg. 2. For the X phase, due to the lack of expermental data, the parameter a X n Table 4 had to be assumed by calculatng the enthalpy of mxng of the compounds, D and D The parameter b X was evaluated wth the phase equlbrum data [36]. The lqud phase was descrbed wth the parameters of the three bnary systems and ternary parameters, 0 L lq ;;, 1 L lq ;; and 2 L lq ;;. Henry [25] studed the two alloys of and usng the optcal mcroscopy and observed that the prmary phase of the two alloys s the -bcc phase durng the soldfcaton. Svetlov et al. [34] studed the lqudþsolds (sold¼fcc or bcc) two-phase feld at the -rch corner usng DTA and optcal mcroscopy. Yoshzawa et al. [37] studed the eutectc reacton (lqud-bccþγ) usng SEM and TEM. The parameters, 0 L lq ;;, 1 L lq and 2 L lq, were evaluated ;; ;; wth the prmary soldfcaton phase data [25,34,37]. The parameters 0 L bcc ;;, 1 L bcc ;; and 2 L bcc ;; were consdered to be zero.

9 132 S.H. Zhou et al. / CALPHAD: Computer Couplng of Phase Dagrams and Thermochemstry 46 (2014) Phase equlbrum results The phase dagrams computed from the Gbbs free energy functons of ndvdual phases descrbed above are shown n Fgs. 4 and 5. Relevant expermental data n Table 5 as well as the phase dagram calculated usng the parameters by Lu et al. [11] are also shown n Fg. 1(a), Fgs. 4 and 5 for comparson. The phase dagrams we propose here nclude several dramatc dfferences from prevously suggested phase dagrams as descrbed brefly here. Our model descrbes the ternary compound and X phases usng a threesublattce model and as a stochometrc compound, respectvely, whch were gnored by Lu et al. [11]. The frst-prncples calculated enthalpes of formaton of the phase n Fg. 2 were used to evaluate the thermodynamc parameters of the phase. In addton, our frst-prncples data were used to determne the Gbbs energy of the end-members of the B2- and D0 a - 3 phases. Ths dffers greatly from the modelng by Lu et al. [11]. The other major contrbutor of ths work s that the recent descrptons of the bnary [20] and [21] n Fg. 1 were adopted for better descrpton of γ and γʹ n the system. The current model lqudus projecton of the system s plotted n Fg. 4, showng a good agreement wth expermental data [22,25,34,37,38]. In comparson wth our results, the lqudus projecton calculated usng parameters by Lu et al. [11] shows two sgnfcant dfferences n Fg. 4. In our calculaton, the phase s a prmary phase on rch sde whle the calculated prmary phase by Lu et al. [11] s the θ- 8 3 phase as marked wth 8 3 n gray n Fg. 4. Second, a curve as marked A-lne n Fg. 4 s calculated usng the parameters by Lu et al. [11]. Ths curve starts from the bnary system as shown n Fg. 1 (a) ndcatng that the B2 phase can be the prmary phases n the lqudus projecton. Fg. 5. The sothermal secton calculated at dfferent temperatures n comparson wth expermental data as the symbols n Table 5 and the calculated results n dash-lne by Lu et al. [11]. (a) At 1573K, (b) at 1533K, (c1) overall composton at 1473K, (c2) rch corner at 1473K, (d1) overall composton at 1373K, (d2) corner at 1373K and at 1273K, (e) at 1273K, (f) at 1153K and (g) at 1073K. Fg. 6. γ-fcc and γʹ- 3 sngle-phase domans plotted n 3D as computed usng the present model where symbol ϑ (ϑ¼lq, bcc, B2, and δ) ndcates the phase equlbrum wth γ-fcc or γʹ- 3.

10 S.H. Zhou et al. / CALPHAD: Computer Couplng of Phase Dagrams and Thermochemstry 46 (2014) The calculated sothermal sectons at 1573, 1533 and 1473 K along wth the expermental data n Table 5 are shown n Fg. 5(a c), n whch the calculated equlbrum phases are n good agreement wth the expermental data [15,22 27,29,35]. The calculated sothermal sectons at 1473 and 1373 K plotted n Fg. 5 (c and d) show the nvarant reacton γþbcc-γʹþδ, whch occurs at 1400 K. Accordng the expermental data, the nvarant reacton γþbcc-γʹþδ temperature s between 1356 and 1420 K by Mracle et al. [26], between 1300 and 1400 K by Hong et al. [30] and at 1403 K by Wakashma et al. [31]. Ths means that our calculated nvarant reacton γþbcc-γʹþδ s n good agreement wth the expermental data [26,30,31]. Furthermore, our calculated results show an agreement wth the expermental data of the γþγʹ two-phase regon n Fg. 5(c), whle the calculated results by Lu et al. [11] are n the sngle fcc phase regon. Fg. 5(d) and (e) shows the sothermal secton calculated at 1373 and 1273 K wth expermental data [11,15,18,22,24,28,31,32,35,36]. There are two sgnfcant dfferences between Fg. 5(d) and (e). Frst, the X phase appears n Fg. 5(e). Our calculated results ndcate that the X phase forms at 1282 K, whch s n reasonable agreement wth the expermental data 1288 K measured by Grushko et al. [36]. Second, the nvarant reacton, lqþθ-þφ, occurs at 1355 K. It should be noted that the data by Grushko et al. [36] plotted n Fg. 5(e) ndcate the lqþ 3 þ three phases n equlbrum at 1273 K. Ths s nconsstent wth our calculated lqþφ- 4 þ three-phase equlbrum. Accordng to the thermodynamc descrpton for the bnary system [9] adopted n ths work wth the assocated phase dagram n Fg. 1(d), the lqud, 4, 8 3, 3 and bcc phases are stable at 1273 K, whle the 3 phase s not consdered as a stable phase. The stablty of the 3 phase needs to be confrmed n future work. Fahrmann et al. [16,17] nvestgated the and alloys. In the alloy aged at 1258 K, mnor amounts of the δ- phase was observed [16,17] whch s consstent wth our calculated result.e. the mole fracton of the δ- phase n alloy s whch s very lttle. The volume fracton of the γʹ phase for the alloy calculated at 1258 K n ths work s 0.61, whch s n good agreement wth the expermental data [16,17], whle that calculated usng parameters by Lu et al. [11] s It s clear that our model yelds better agreement wth expermental data [16,17]. Fg. 5(f) and (g) shows the sothermal sectons calculated at 1153 and 1073 K, respectvely, wth the expermental data [15,22,24,32,35,36]. Comparng wth Fg. 5 (e), the calculated four phase δþγ-γʹþd0 a reacton occurs at 1167 K, whch s n a good agreement wth the expermental phase transformaton data K. In Fg. 5(f), the expermental data, whch ndcated the γþγʹ two-phase regon, are n agreement wth our calculated results, but outsde the calculated γþγʹ twophase regon usng the parameters by Lu et al. [11]. A smlar problem s also shown n Fg. 5(g), where our calculated results shown better agreement wth the expermental γþγʹ two-phase feld data [18] than those by Lu et al. [11]. To understand the stabltes of the γ and γʹ phases, we plotted the stable γ and γʹ phase feld n the related composton trangle over the temperature range K n Fg. 6, whch s useful for -based superalloy desgn by showng the stable γ, γʹ and γþγʹ phase domans wth related composton and temperature dmensons. 6. Summary By combnng CALPHAD approach wth the frst-prncples calculatons, a thermodynamc model was developed for the ternary system. The present work exhbts several key dfferences relatve to prevously reported models: () frstprncples calculatons were used to determne some thermodynamc parameters, () both ternary compounds and X, and the frst-prncples calculatons confrm that the -compound s only margnally stable; () calculated phase dagrams exhbt mproved agreement wth the expermental data. Acknowledgment Ths work was supported by the Pennsylvana State Unversty by the SF Grants (DMR , DMR , and DMR ). Frst-prncples calculatons were carred out on the LIO clusters at the Pennsylvana State Unversty supported n part by the SF Grants (DMR , DMR , and DMR ). Appendx A. Supportng nformaton Supplementary data assocated wth ths artcle can be found n theonlneversonathttp://dx.do.org/ /j.calphad References [1] S. Myazak, Y. Murata, M. rnaga, Tetsu To Hagane J. Iron Steel Inst. Jpn. 80 (1994) 166. [2] S. Myazak, Y. Kusunok, Y. Murata, M. rnaga, Tetsu To Hagane J. Iron Steel Inst. Jpn. 81 (1995) [3] R. Hashzume, A. Yoshnar, T. Kyono, Y. Murata, M. rnaga, Development of -based sngle crystalsuperalloys for power-generaton gas turbnes, n: K.A. Green, H. Harada, T.M. Pollock, T.E. Howson, R.C. Reed (Eds.), Superalloys 2004, Mnerals, Metals & Materals Soc, Champon, PA, 2004, p. 53. [4] R. Yamamoto, Y. Kadoya, H. Kawa, R. Magosh, S. Ueta, T. oda, S. Isobe, J. Iron Steel Inst. 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