Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 470–477 Contents lists available at ScienceDirect Computer Coupling of Phase Diagrams and Thermochemistry journal homepage: www.elsevier.com/locate/calphad Thermodynamic assessment of the Si–Zn, Mn–Si, Mg–Si–Zn and Mg–Mn–Si systems Adarsh Shukla ∗ , Youn-Bae Kang, Arthur D. Pelton Centre de Recherche en Calcul Thermochimique, Département de Génie Chimique, Ecole Polytechnique, Montréal, Québec, Canada article info Article history: Received 28 April 2008 Received in revised form 1 July 2008 Accepted 2 July 2008 Available online 25 July 2008 Keywords: Silicon Zinc Manganese Magnesium Phase diagrams a b s t r a c t The binary Si–Zn and Mn–Si systems have been critically evaluated based upon available phase equilibrium and thermodynamic data, and optimized model parameters have been obtained giving the Gibbs energies of all phases as functions of temperature and composition. The liquid solution has been modeled with the Modified Quasichemical Model (MQM) to account for the short-range-ordering. The results have been combined with those of our previous optimizations of the Mg–Si, Mg–Zn and Mg–Mn systems to predict the phase diagrams of the Mg–Si–Zn and Mg–Mn–Si systems. The predictions have been compared with available data. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction Although magnesium-based materials have a long history of important commercial applications, including automotive, there remains much to be learned about the basic properties of the metal and its alloys. With the recent renewed interest in lightweight wrought materials, including both sheet and tube applications, there has been an increased focus on developing a better understanding of novel magnesium alloys, including those that incorporate additions of such elements as Si, Mn and Zn. These alloy systems, along with other potential candidates, are being actively pursued as possible routes to develop magnesium materials with improved ductility, or even practical room temperature formability. The properties of cast or wrought material depend first and foremost upon the phases and microstructural constituents (eutectics, precipitates, solid solutions, etc.) which are present. In an alloy with several alloying elements, the phase relationships are very complex. In order to effectively investigate and understand these complex phase relationships, it is very useful to develop thermodynamic databases containing model parameters giving the thermodynamic properties of all phases as functions of temperature and composition. Using Gibbs free energy minimization software such as FactSage [1,2], the automotive and aeronautical industries and their suppliers will be able to access the databases to calculate the amounts and compositions of all phases at equilibrium at any ∗ Corresponding author. E-mail address: adarsh.shukla@polymtl.ca (A. Shukla). 0364-5916/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2008.07.002 temperature and composition in multicomponent alloys, to follow the course of equilibrium or non-equilibrium cooling, to calculate corresponding heat effects, etc. As part of a broader research project to develop a thermodynamic database for Mg alloys with 25 potential alloying metals, the present study reports on the evaluation and optimization of the Si–Zn, Mn–Si, Mg–Si–Zn and Mg–Mn–Si systems. Previous optimizations of the Mn–Si system in the framework of COST 507 [3] and by Chevalier et al. [4] were based upon a Bragg–Williams (BW) random-mixing model for the liquid phase. The liquid in this binary system is expected to exhibit considerable short-range-ordering (SRO) as evidenced by the very negative enthalpy of mixing curve. As has been shown by the present authors [5], the use of a BW model in liquids with a high degree of SRO generally results in unsatisfactory results and in poor predictions of ternary properties from binary model parameters. In the present work, the Modified Quasichemcial Model (MQM) has been used to account for the SRO in the liquids. As well, there are vapor pressure measurements [6,7] of Mn–Si alloys and measurement of the enthalpy of formation [8] of compounds which were not taken into account in previous optimizations. The liquid phase in the Si–Zn system shows slight positive deviations from ideality. This system was optimized previously [9] using a BW random-mixing model. The MQM, which takes clustering into account, was used in the present work in order to obtain a better description and prediction in the Mg–Si–Zn system, and for consistency with the fact that the MQM is used for the other binary subsystems in this ternary system. Hence the Si–Zn and Mn–Si systems have been re-optimized in the present study. A. Shukla et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 470–477 Fig. 1. Optimized phase diagram of the Si–Zn system. 471 Fig. 2. Optimized phase diagram of the Mn–Si system. The binary model parameters have then been combined with those from our previous optimizations of the Mg–Zn [10], Mg–Mn [11] and Mg–Si [12] systems in order to predict thermodynamic properties and phase equilibria in the Mg–Si–Zn and Mg–Mn–Si ternary systems. 2. Modified quasichemical model (MQM) The Modified Quasichemical Model (MQM) in the pair approximation [13] was used to model the liquid alloys. A description of the MQM and its associated notation is given by Pelton et al. [13]. The same notation is used in the present paper. The composition of maximum SRO is determined by the ratio of the coordination numj bers Ziji /Zij . The values of the coordination numbers selected in the present study are listed in Table 1. All the binary subsystems of the Mg–Si–Zn and Mg–Mn–Si systems exhibit SRO near the equimolar j composition; hence Ziji = Zij in all cases. From the MQM model parameters of the binary liquid phases, the thermodynamic properties of a ternary liquid phase may be estimated as discussed previously [14]. If ternary experimental data are available, additional ternary model parameters may be added. 3. Binary systems All calculations and optimizations in the present study were performed with the FactSage thermochemical software [1,2]. The optimized model parameters of all phases obtained in the present study are listed in Table 1. The crystallographic data [15] of all phases in the Si–Zn and Mn–Si systems are listed in Table 2. The Gibbs energies of all stable and metastable condensed phases of the elements were taken from Dinsdale [16], while the Gibbs energies of the gaseous elements were taken from the JANAF Tables [17]. 3.1. The Si–Zn system The optimized phase diagram of the system is shown in Fig. 1. It may be noted that no temperature dependent terms were required in the optimized parameters (Table 1) of the liquid Si–Zn solution. This system was optimized by Jacobs and Spencer [9] who used a BW random-mixing model for the liquid phase. The present calculated eutectic temperature and composition are 419.2 ◦ C and XZn = 0.999. The only data available are the coordinates of the liquidus. No thermodynamic property data were found. Girault [18], Thurmond and Kowalchik [19] and Moissan and Siemens [20] determined the liquidus in the range 0.85 to 5.3 at.% Zn by a gravimetric method. John et al. [21] and Schneidner and Krumnacker [22] measured Fig. 3. Optimized phase diagram of the Mn–Si system for the region XSi = 0.0 to 0.5. the liquidus by DTA in the range from 1 to 55 at.% Zn. The solid solubility of Zn in Si was determined by diffusion investigations in the temperature ranges from 820 to 1076 ◦ C [23], 1040 to 1200 ◦ C [24] and 900 to 1360 ◦ C [25]. All these investigations reported negligible solubility of Zn in Si. In the absence of data for the solubility of Si in solid Zn, negligible solubility was assumed. The experimental liquidus points are fitted equally well in the present and previous assessment [9]. In the present work only two fitting parameter have been used, whereas the previous optimization used three parameters. 3.2. The Mn–Si system The optimized phase diagram of the Mn–Si system is shown in Fig. 2 and is compared with the experimental data in Figs. 3 and 4. Various calculated thermodynamic properties of the system are compared with experimental data and previous optimizations in Figs. 5–13. A complete literature review of the Mn–Si system up to 1990 was reported by Gokhale and Abbaschian [26]. COST 507 [3] and Chevalier et al. [4] performed thermodynamic optimizations of the system, using a BW random-mixing model for the liquid phase. Du et al. [27] slightly modified the optimized parameters of COST 507 [3] for the CUB and Mn6 Si. Chakraborti and Lukas [28] optimized the phase diagram data without taking account of any thermodynamic data. Kanibolotskii and Lesnyak [29] fitted the thermodynamic properties of the system to polynomial equations, but performed no optimization. The optimized Mn-rich side of the phase diagram (X ≤ 0.5) is compared with the experimental data in Fig. 3. Gokhale and A. Shukla et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 470–477 472 Table 1 Optimized parameters of the Si–Zn and Mn–Si systems from the present work, and of the liquid phase in the Mg–Zn system from Spencer [10] Liquid: (notations as described previously [13]) Co-ordination numbers: Si–Zn: Mn–Si: Mg–Zn: Si Zn Zn ZSi SiSi = ZZnZn = ZSiZn = ZZnSi = 6 Si Mn Mn ZSi SiSi = ZMnMn = ZSiMn = ZMnSi = 6 Mg Mg ZMgMg = ZZn = Z = ZZn ZnMg = 6 ZnZn MgZn Optimized values for ∆gAB , in Joules: ∆gSi–Zn : ∆gMn–Si : ∆gMg–Zn : 2299 + 1946XZnZn (−33 054 + 6.694 T) + (−20 920)XMnMn + (1.674 T)XSiSi + (11 715 − 4.184 T)X2MnMn (−6778 + 3.128 T) + (−1996 + 2.008 T)XMgMg + (−2975 + 1.674)XZnZn Solid solutions: Excess Gibbs energy (Joules/mol of atoms): CUB: CBCC: BCC: FCC: XSi XMn [(−157 737 + 26.778 T) + (−41 003 + 21.338 T)(XSi –XMn )] + XSi (5.021 T) XSi XMn [(−151 042 + 26.778 T) + (−32217 − 8.368 T)(XSi –XMn )] XSi XMn [−83 680 + 26 778(XSi –XMn )] XSi XMn [−101 253 + 17 991(XSi –XMn )] Stoichiometric compounds: Compounds 0 a ∆H298 (J/(mol of atoms)) 0 b S298 (J/[(mol of atoms) K]) 0 a ∆S298 (J/[(mol of atoms) K]) Cp (J/[(mol of atoms) K]) Mn6 Si Mn9 Si2 Mn3 Si Mn5 Si3 MnSi Mn11 Si19 −18 094 −23 091 −27 900 −35 250 −38 000 −32 333 29.471 27.745 27.000 25.562 20.800 17.267 −0.834 −2.038 −1.868 −1.630 −4.716 −6.460 0.857 Cp (Mn, CBCC) + 0.143Cp (Si, Dia.) 0.818 Cp (Mn, CBCC) + 0.181Cp (Si, Dia.) 0.375 Cp (Mn, CBCC) + 0.125 Cp (Si, Dia) 0.625 Cp (Mn, CBCC) + 0.375 Cp (Si, Dia.) 0.500 Cp (Mn, CBCC) + 0.500 Cp (Si, Dia.) 0.366 Cp (Mn, CBCC) + 0.633 Cp (Si, Dia.) a b Enthalpy and entropy of formation from the elements at 298.15 K. Absolute Third-Law entropy at 298.15 K. Table 2 Crystallographic data [15] of all phases in the Si–Zn, Mn–Si, Mg–Si–Zn and Mg–Mn–Si systems considered in the present optimization Phase Struktur-bericht Prototype Pearson symbol Space group Liquid FCC BCC CUB CBCC HCP Si (Dia.) Mg51 Zn20 Mg12 Zn13 Mg2 Zn3 MgZn2 Mg2 Zn11 Mg2 Si Mn6 Si Mn9 Si2 Mn3 Si Mn3 Si Mn5 Si3 MnSi Mn11 Si19 – A1 A2 A13 A12 A3 A4 – – – C14 D8c C1 – – D03 – D88 B20 – – Cu W Mn Mn Mg C (Dia.) Mg51 Zn20 – – MgZn2 Mg2 Zn11 CaF2 – Si2 U3 BiF3 – Mn5 Si3 FeSi – – cF 4 cI2 cP20 cI58 hP2 cF 8 oI158 – mC 110 hP12 cP39 cF 12 hR53 tP10 cF 16 – hP16 cP8 tP120 – Fm3m Im3m P41 32 I43m P63 /mmc Fd3m Immm – B2/m P63 /mmc Pm3 Fm3m R3 P4/mbm Fm3m – P63 /mcm P21 3 P4n2 Abbaschian [26] reported speculative phase boundaries between the FCC, BCC and liquid phases upon which the calculations in Fig. 3 were based. Wieser and Forgeng [30] reported phase equilibria in the region from 2 to 24 at.% Si by metallography and XRD. These authors reported a phase ε with a homogeneity range from 12 to 15 at.% Si and a phase ξ with a homogeneity range from 16 to 18 at.% Si (Mn6 Si and Mn9 Si2 respectively in Fig. 3). Kuz’ma and Gladyshevskii [31] and Gupta [32] confirmed the existence of the ε and ξ phases respectively by XRD. Later, Gokhale and Abbaschian [26] referred to these compounds as R and υ and suggested their approximate stoichiometries as Mn6 Si and Mn9 Si2 respectively. Boren [33] from XRD, and Vogel and Bedarff [34] from thermal and metallography, reported the existence of Mn3 Si. Later, Letun et al. [35] reported an allotropic transformation of this phase Comments Mn is stable phase Mn is stable phase Mn is stable phase Mn is stable phase Mg and Zn are stable phases High temperature form Low temperature form at 677 ◦ C. The Gibbs energy change of this transformation was assumed to be zero in the present work. The congruently melting compounds Mn5 Si3 and MnSi were first reported by Doerinckel [36] from thermal and metallographic analysis, and later by Vogel and Bedarff [34] who used the same technique. The optimized silicon-rich side of the phase diagram is compared with the experimental data in Fig. 4. Different authors [33, 37–41] disagree on the exact designation, the width of the singlephase region (0.3–0.5 at.% Si), and the structure of the highest silicide, Mn11 Si19 . Gokhale and Abbaschian [26] preferred the designation MnSi1.75−x with a homogeneity width of 0.4 at.% Si. Morkhovets et al. [42], from microstructural and thermal analysis, reported this compound as MnSi1.72 . Following Chakraborti and Lukas [28] and Chevalier et al. [4], this phase is modeled in the present study as stoichiometric Mn11 Si19 . There are no data on the solubility of Mn in solid Si. The slope of the liquidus curve [43] at XSi = 1.0 is consistent with the limiting A. Shukla et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 470–477 473 Fig. 4. Optimized phase diagram of the Mn–Si system for the region XSi = 0.4 to 1. van’t Hoff equation when the solid solubility is negligible. Hence, negligible solubility was assumed. All intermediate compounds have been modeled as stoichiometric, there being no data to the contrary except for Mn6 Si and Mn9 Si2 . For these compounds, it proved impossible to reproduce simultaneously all the data points of Wieser and Forgeng [30] for the boundaries of the single-phase regions of these phases and of the CUB phase with reasonable thermodynamic parameters. Since there are no corroborating data from other authors in this region of the phase diagram, it was decided to treat these phases as stoichiometric. Zaitsev et al. [6,44] reported vapor pressure measurements in two-phase regions using high-temperature mass spectrometry. The data from their first paper [6] are in good agreement with the present calculations as can be seen in Fig. 5. The data from their second paper [44] were presented only in the form of a small ambiguous figure. In a different work, Zaitsev et al. [7] reported vapor pressure measurements of monatomic Si and Mn over liquid alloys by Knudsen effusion/mass spectrometry. The data for Mn vapor pressures are well reproduced as seen in Fig. 5(a). The calculated vapor pressure of monatomic Si is shown in Fig. 6. The composition dependence of the data in Fig. 6 is well reproduced by the model, while the calculated curves and measured points differ by a nearly constant value. In view of the expanded vertical scale of Fig. 6, and the very low vapor pressure of Si, this difference is within the experimental error limits. Since the authors [7] did not report measurements of the vapor pressure of pure liquid Si, one cannot check for consistency with the data for the vapor pressure of pure Si [17] used in the present calculations. Tanaka [45] by a transportation method at 1400 ◦ C, and Ahmad and Pratt [46] by a torsion-effusion technique, measured Mn partial pressures over liquid alloys. The present calculations are compared with these vapor pressure data in Fig. 5. At high Si contents, these data are inconsistent with the vapor pressure data of Zaitsev et al. [7] in Fig. 5(a). The data of Tanaka [45] in Fig. 5(c) are inconsistent with the vapor pressure data of Zaitsev et al. [6,7] in Fig. 5(a) and with the vapor pressure of pure liquid Mn given by JANAF [17]. Batalin and Sudavtsova [47] performed EMF measurements in the range 1247–1427 ◦ C to report the partial excess Gibbs energy of Mn in the liquid phase at 1400 ◦ C. These Gibbs energies are compared with the present calculations in Fig. 7. These data are inconsistent with the Mn vapor pressure data [6,7] in Fig. 5. Previous optimizations [3,4] did not take into account the vapor pressure data. In the present work, more weight was given to the vapor pressure data [6,7] than to the EMF measurements. Zaitsev et al. [48] derived standard enthalpies of formation of intermediate compounds from vapor pressure measurements Fig. 5. Optimized vapor pressure of monatomic Mn over liquid Mn–Si alloys. using high-temperature mass spectroscopy. Their vapor pressure data points were presented only in the form an ambiguous figure. Meschel and Kleppa [8] measured the standard enthalpies of formation of Mn5 Si3 and MnSi by direct synthesis calorimetery. Zaitsev et al. [48] also cite data from other sources [49–53]. All these data are shown on Fig. 8. In the present work, the greatest weight was given to the results of Meschel and Kleppa [8]. It should be noted that the optimized entropies of formation of the intermetallic compounds from the elements at 25 ◦ C (Table 1) are all very small as is generally expected for such compounds. That is, the optimized entropies are physically reasonable. Gel’d et al. [54] and Esin et al. [55] measured partial enthalpies of mixing in Mn–Si melts at 1500 ◦ C by high-temperature isothermal calorimetry. It is unclear in the article of Esin et al. [55] whether all their data points are experimental or if some points were obtained by integration of the Gibbs-Duhem equation. These 474 A. Shukla et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 470–477 Fig. 6. Optimized vapor pressure of monatomic Si over liquid Mn–Si alloys. Fig. 9. Optimized partial enthalpies of mixing at 1500 ◦ C in liquid Mn–Si alloys. Fig. 10. Optimized integral enthalpy of mixing at 1500 ◦ C in liquid Mn–Si alloys. Fig. 7. Optimized excess Gibbs energy of Mn in liquid Mn–Si alloys at 1400 ◦ C. Fig. 11. Optimized entropy of mixing at 1500 ◦ C in liquid Mn–Si alloys. ◦ Fig. 8. Optimized standard enthalpy of formation at 25 C of the intermediate compounds in the Mn–Si system. data are compared with the present and previous optimizations in Fig. 9. Batalin et al. [56] at 1450 ◦ C, and Gorbunov et al. [57] at 1500 ◦ C, measured the integral enthalpy of mixing in liquid solutions by high-temperature isothermal calorimetry. These data are well reproduced by the present optimization as seen in Fig. 10. The shape of the partial enthalpy curves in Fig. 9, which are very negative with points of inflection and a point of intersection near the equimolar composition, are strongly indicative of a high degree of SRO about this composition. An entropy of mixing with a strong minimum near the equimolar composition as in Fig. 11 is thus expected. These shapes of the enthalpy and entropy of mixing curves are well reproduced by the MQM. 4. Ternary systems 4.1. The Mg–Si–Zn system The phase diagrams of the Mg–Zn and Mg–Si systems from previous optimizations [10,12] are shown in Figs. 12 and 13 respectively. Spencer [10] obtained the optimized phase diagram in Fig. 12 using the MQM for the liquid solution, and taking model parameters for the solid phases from the optimization of Liang A. Shukla et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 470–477 475 Fig. 12. Previously optimized phase diagram of the Mg–Zn system [10]. Fig. 14. Predicted liquidus projection of the Mg–Si–Zn system. Fig. 13. Previously optimized phase diagram of the Mg–Si system [12]. et al. [58]. His optimized model parameters for the liquid phase are reported in Table 1. These previous optimizations were combined with the present optimization of the Si–Zn system in order to predict the thermodynamic properties and phase diagram of the Mg–Si–Zn system. Mutual solubilites between binary compounds were assumed to be negligible, there being no experimental data available. All the intermetallic compounds have different crystal structures (Table 2) and different stoichiometries. These factors mitigate against there being appreciable mutual solubility of the compounds. In the absence of any evidence for ternary compounds, none were assumed. The thermodynamic properties of the ternary liquid phase were calculated by the MQM from the binary model parameters. The ‘‘symmetric approximation’’ [14,59] was used. No additional ternary parameters were included. The resultant calculated polythermal projection of the liquidus surface is shown in Fig. 14. Bollenrath [60], using cooling curves, reported a partial section along the Mg2 Si–MgZn2 join. These data are compared with the calculated section in Fig. 15. When the ‘‘asymmetric approximation’’ [14,59] was used with Si and Mg as ‘‘asymmetric component’’, the calculated liquidus deviated by over 50◦ from the measurements [60]. Hence the ‘‘symmetric approximation’’ was preferred. 4.2. The Mg–Mn–Si system The phase diagrams of the Mg–Mn and Mg–Si systems from previous optimizations [11,12] are shown in Figs. 16 and 13 respectively. These optimizations were combined with the Fig. 15. Optimized section of the Mg–Si–Zn phase diagram along the Mg2 Si–MgZn2 join. Fig. 16. Previously optimized phase diagram of the Mg–Mn system [11]. present optimization of the Mn–Si system in order to predict the thermodynamic properties and phase diagram of the Mg–Mn–Si system. The resultant calculated polythermal projection of the liquidus surface is shown in Fig. 17. No measurements of the liquidus have been reported. As in the Mg–Si–Zn system, and for the same reasons, no mutual solubilities between binary compounds were assumed. In 476 A. Shukla et al. / Computer Coupling of Phase Diagrams and Thermochemistry 32 (2008) 470–477 This results in a better representation of the partial properties of the solutes in dilute solution in magnesium, the activities of solutes in dilute solution being of much practical importance. As shown by the present authors [5], the use of the MQM generally also results in better estimations of the properties of ternary and higher-order liquid alloys from binary model parameters. In the present work the extension of the Mg–Mn binary liquid miscibility gap into the ternary Mg–Mn–Si system is significantly smaller than would be obtained by using a Bragg–Williams model. Also, the positive deviations obtained in the present work along the Mg–MnSi join would be over-estimated by a Bragg–Williams model. These better predictions and estimations of phase equilibria will aid in the design of novel magnesium alloys. Acknowledgements Financial support from General Motors of Canada Ltd. and the Natural Sciences and Engineering Research Council of Canada through the CRD grants program is gratefully acknowledged. Fig. 17. Predicted liquidus projection of the Mg–Mn–Si system. Calculated temperatures of invariants points are shown (◦ C). the absence of any evidence for ternary compounds, none were assumed. The thermodynamic properties of the ternary liquid phase were calculated by the MQM from the binary model parameters. No additional ternary parameters were included. The ‘‘asymmetric approximation’’ [14,59] with Si as ‘‘asymmetric component’’ was used since the Mg–Mn liquid phase exhibit positive deviations from ideal solution behavior whereas the Mg–Si and Mn–Si liquid phases exhibit considerable negative deviations. The Mg–Mn binary system exhibits a large liquid miscibility gap (Fig. 16). With the addition of Si, this miscibility gap extends into the ternary system. As shown by the present authors [5], the use of the MQM generally results in better predictions of ternary miscibility gaps than when a BW random-mixing model is used. It should be noted that the consolute temperature of the binary miscibility gap calculated in our previous optimization of the Mg–Mn system [11], in which the liquid phase was modeled by the MQM, is approximately 2000◦ lower than the calculated consulate temperatures in earlier optimizations [61,62] in which the BW random-mixing model was used. This results in a much smaller extension of the calculated miscibility gap into the ternary system in the present optimization. In Fig. 17 the liquidus surface approximately along the Mg–MnSi join is very flat. This is a result of the high degree of SRO in the Mn–Si binary liquid phase which results in a tendency for Mn and Si atoms to cluster and exclude Mg. As shown by the present authors [5] such liquidus surfaces are generally predicted better by the MQM than by BW or ‘‘associate’’ models. 5. Conclusions Gibbs energy functions for all the phases in the binary Si–Zn and Mn–Si systems have been obtained. All available thermodynamic and phase equilibrium data have been critically evaluated in order to obtain one set of optimized model parameters of the Gibbs energies of all phases which can reproduce the experimental data within experimental error limits. Evaluations and tentative phase diagrams for the ternary systems Mg–Si–Zn and Mg–Mn–Si have been given. The use of the Modified Quasichemical Model results in a better fitting of the partial enthalpy of mixing in the Mn–Si system than is the case when a Bragg–Williams random-mixing model is used. References [1] C.W. Bale, P. Chartrand, S.A. Degterov, G. Eriksson, K. Hack, R. Ben Mahfoud, J. Melançon, A.D. Pelton, S. Petersen, CALPHAD 26 (2002) 189. [2] C.W. Bale, A.D. Pelton, W. 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