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IET Science, Measurement & Technology Research Article Thermodynamic calculations of the Mn–Sn, Mn–Sr and Mg–Mn–{Sn, Sr} systems ISSN 1751-8822 Received on 8th December 2013 Accepted on 29th December 2014 doi: 10.1049/iet-smt.2013.0267 www.ietdl.org Mohammad Aljarrah 1 ✉, Suleiman Obeidat 1, Rami Hikmat Fouad 1, Mahmoud Rababah 2, Ahmad Almagableh 2, Awni Itradat 3 1 Industrial Engineering Department, Faculty of Engineering, The Hashemite University, P.O. Box 150459, Zarqa 13115, Jordan Mechanical Engineering Department, Faculty of Engineering, The Hashemite University, Zarqa 13115, Jordan 3 Department of Computer Engineering, Faculty of Engineering, The Hashemite University, Zarqa 13115, Jordan ✉ E-mail: maljarrah@hu.edu.jo 2 Abstract: Thermodynamic modelling of Mn–Sn and Mn–Sr binary systems is carried out using the reliable data from the literature. Thermodynamic properties of the binary liquid solutions are described using the modified quasi-chemical model. The calculated phase diagrams and the thermodynamic properties are found to be in good agreement with the experimental data from the literature. A self-consistent thermodynamic database for the Mg–Mn–{Sn, Sr} systems is constructed by combining the thermodynamic descriptions of their constituent binaries. The constructed database is used to calculate and predict liquidus projection and invariant reactions of these ternary systems. The Mg–Mn–Sr system has nine ternary eutectic reactions, two saddle points and eleven crystallisation fields. Mg–Mn–Sn has four saddle points, two quasi-peritectic and six ternary eutectic reactions. 1 Introduction Automobile manufacturers are continually searching for means to reduce vehicle weight in order to increase fuel economy. With their low density, high stiffness/weight ratio and high damping capacity, magnesium alloys have the potential to make a significant contribution to this weight reduction. Wrought magnesium alloys hold great promise for use in structural applications [1]. However, the possible application of wrought magnesium alloys is strongly reduced by the low formability and the mechanical anisotropy of this material [2]. This is mainly caused by its hexagonal crystal structure which offers limited slip systems at room temperature [3]. There are several attempts to improve formability of magnesium alloys in which alloying elements were found to play a key role [4–25]. Among them, rare earth (RE) elements, manganese and strontium additions improve formability of wrought magnesium alloys through grain size refinement, activation of non-basal slip and/or weakening the texture [16–25]. Li and RE additions to Mg are likely to bring numerous liabilities including high cost, increased flammability during processing and decreased corrosion resistance [1, 6, 19]. Mg–Mn-based alloys have some popularity because of their good weldability [26]. They provide medium strength in wrought product form such as extrusion, rolling, sheet and plate [26]. Sr provides good grain refinement in Mg-alloys and thereby enhances mechanical properties [23, 24, 27]. Alloying Mg with Sn is promising for high-temperature applications. Addition of Sn to Mg-based alloys leads to precipitations of thermally stable compound (Mg2Sn) which provides stable microstructure and enhances mechanical properties at 150°C [28, 29]. Thermodynamic descriptions of Mg–Mn–Sr and Mg–Mn–Sn systems are very important for further improvement in alloy design through providing invaluable information such as solubility of alloying addition in Mg-matrix, heat treatment temperature, amount of secondary phase precipitates and solidus and liquidus temperatures. In this paper, thermodynamic modelling of Mg–Mn–{Sn, Sr} systems are carried out for the first time according to authors knowledge. Among the constituent binaries, Mn–Sn and Mn–Sr systems were re-optimised using the modified quasi-chemical model (MQM) for the liquid solution. On the basis of calculation of phase IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692 & The Institution of Engineering and Technology 2015 diagrams (CALPHAD) method [30], all thermodynamic information such as experimental phase diagram data and thermodynamic properties data are critically evaluated and optimised simultaneously. Thermodynamic optimisation and calculations were performed in this paper using FactSage program [31]. 2 2.1 Experimental data Mn–Sn system Watchel and Ulrich [32] and Watchel et al. [33] investigated the phase relationships of the Mn–Sn system for a composition range of (0–80 at.% Mn) by metallography, magnetic susceptibility and polarisation. Their samples were prepared from 99.999 wt.% Sn and 99.99 wt.% Mn and melted in high-frequency induction furnace under argon. Homogenisation was carried out by annealing the specimen in quartz tube under argon atmosphere. Their alloys were heat treated from 200°C for 40 days to up to 800°C for 10 h. They [32] reported that Mn2Sn phase melts incongruently at 883°C which is comparable with 897°C, the calculated value of Singh et al. [34] and William [35]. Watchel and Ulrich [33] indicated that MnSn2 melts incongruently at 549°C, compared with Singh et al. [34] and William [35] results as 548°C. Their data [32] indicates that Mn3Sn is a solid solution in the composition range of 76.4–77.6 at.% Mn at 700°C. Singh et al. [34] investigated Mn–Sn phase diagram below 50 at.% Sn using optical microscopy and X-ray diffraction using high purity Mn (99.9 at.%) and Sn (99.999 at.%) starting materials. Their samples were cast in recrystallised alumina crucible using high-frequency induction furnace under protective gas (argon). Moreover, their studied samples were wrapped in molybdenum foil and sealed in evacuated fused silica capsules, then annealed for 15 days at 500°C to 10 h at 1000°C. Singh et al. [34] reported low solubility of Sn in αMn phase which is in agreement with Nail’s [36] observation. Nowotny and Schubert [37] and Zwicker [38] determined the solubility of Sn in βMn phase and reported value of 8.7 at.% Sn at 600°C, whereas Singh et al. [34] indicated this solubility to be about 10 at.% Sn. The experimental phase diagram data from [32–34] are self-consistent and considered as 681 the most reliable data, therefore these data will be incorporated in the current optimisation of Mn–Sn system. Earlier investigations differ on the Mn3Sn phase region. According to Nail [36], the Mn3Sn phase exists between 23 and 24.5 at.% Sn, whereas Guillaud [39] indicated the Mn3Sn phase region between 20 and 22 at.% Sn. On the other hand, Nowotny and Schubert [37] reported the Mn3Sn phase at 21.43 at.% Sn. The structure type of Mn3Sn has been suggested to be Ni3Sn [34, 36, 37]. Elding-Pontén et al. [40] investigated NiAs–Ni2In structure types in the Mn–Sn system using X-ray diffraction and scanning electron microscopy/ energy dispersive X-ray spectroscopy (SEM/EDX). Three different phases were detected such as high temperature phase (HTP)1 (Mn17Sn7), HTP2 (Mn8Sn5) and Mn3Sn2 phases. They reported that there were difficulties in preparing the samples since traces of tin were detected on the outside of the ingot. Besides, different phases were not distinguishable because of having very similar compositions. The chemical compositions and crystal structures of HTP1 and HTP2 phases detected in this paper of Elding-Pontén et al. [40] were not reported or verified in the literature. Therefore these phases will not be incorporated in the current paper. Stange et al. [41] studied the crystal structure of binary phases in the Mn–Sn system using X-ray diffraction, neutron diffraction and differential thermal analysis. They revealed the uncertainty on the stability and composition of Mn2Sn. According to their work, Mn2Sn consists of a high-temperature phase Mn(2−x)Sn and a low-temperature phase Mn3Sn2 rather than Mn2Sn or HPT2. Moreover, Mn(2−x)Sn is stable between 480 and 884°C and the homogeneity range of this phase at 800°C is 0.18 ≤ x ≤ 0.23 and 0.28 ≤ x ≤ 0.34 at 600°C. Therefore in this paper Mn(2−x)Sn will be treated as solid solution and Mn3Sn2 will be modelled as a stoichiometric compound. The phase diagram of Mn–Sn system was assessed by Massalaski et al. [42] and Stange et al. [41]. According to these works, the composition ranges of (αMn) and (βMn) were 0.0–1.0 and 5.5–10 at.% Sn at 400°C, respectively. Miettinen [43] calculated Mn–Sn phase diagram using the experimental data from [32–34, 44–47]. He [43] also described the liquid phase using the random solution model. According to Miettinen [43], the stable phases in the Mn–Sn system were (αMn), (βMn), (γMn), (δMn), MnSn2, Mn2Sn and Mn19Sn6 (Mn3Sn). In his paper [43], Mn–Sn phase diagram was reported in two figures; Mn3Sn phase was treated as stoichiometric and in the other figure Mn3Sn modelled as a solid solution. Moreover, the calculated activity at 1000°C of Miettinen [43] could not produce good fit with experimental data of Eremenko et al. [47]. According to Okamoto’s review [48] on the Mn–Sn phase diagram, the stable phases in this system are; (αMn), (βMn), (γMn), (δMn), MnSn2, (Mn(2−x)Sn), Mn3Sn2, (Mn3Sn) and β Sn. In this paper, the stable phases adopted by Okamoto will be considered and re-optimised. 2.2 Mn–Sr system The work of Obinata et al. [49] is the only experimental data that could be found in the literature. According to their work, this system forms a miscibility gap through the interval 0.74–96.5 at.% Sr at 1240°C and no intermetallic compound was observed in their work. Peng et al. [50] modelled all phases in the Mn–Sr binary system as completely disordered solutions. In their work, Gibbs energy is described by Redlich–Kister polynomial and model parameters were evaluated using Thermo-Calc, whereas Janz [51] calculated liquidus projection of Mg–Mn–Sr ternary system. However, he did not describe thermodynamic modelling of the Mn–Sr binary system and the optimised parameters were not reported. In the current paper, the liquid phase of the Mn–Sr system will be modelled using MQM to have consistent thermodynamic description with Mg–Sn, Mg–Mn and Mg–Sr systems. work of Ghosh et al. [52], Ghosh and Medraj [53] and Aljarrah and Medraj [54], respectively. The calculated phase diagrams of the Mg–Sn, Mg–Mn and Mg–Sr phase diagrams are in reasonable agreement with all reported data in the literature. Liquid phase was optimised using the MQM. Therefore the works of Ghosh et al. [52], Ghosh and Medraj [53] and Aljarrah and Medraj [54] were considered to construct a self-consistent thermodynamic database of the Mg–Mn–{Sn, Sr} ternary systems. 2.4 Mg–Mn–Sn ternary system In 1969, Kopetskii and Semenova [55] experimentally identified phases in the Mg-rich region of the Mg–Mn–Sn system and draw isothermal sections at 500 and 400°C. The identified phases in their isothermal section will be compared with the current calculations. They also reported a ternary eutectic at 554°C where L ↔ (Mg) + αMn + Mg2Sn. The work of Kopetskii and Semenova [55] is the only experimental data of Mg–Mn–Sn system that could be found in the literature. 2.5 Mg–Mn–Sr ternary system Celinkin et al. [27] investigated microstructure and creep behaviour of eight Mg-{0.75 − 2} wt.% Mn-{3 − 5} wt.% Sr alloys in the Mg-rich region. In their works, samples were heat treated at 225°C for 150 h and 300°C for 96 h. They measured solid solubility of Mn in Mg-matrix using TEM/EDS. These solubilities will be compared with the current thermodynamic calculation. Three phases were positively identified in the heat treated alloys, namely; Mg17Sr2, (Mg) and α-Mn. XRD analysis in the heat treated alloys show no change in phase constitution compared with the as-cast alloys. In Janz’s doctoral thesis [51], liquidus projection and isothermal sections at 500 and 400°C of the Mg–Mn–Sr ternary system were calculated. In the current paper, a comparison between the current liquidus projection and Janz’s work will be discussed. Mg–Mn–Sn and Mg–Mn–Sr systems are subsystems of multi-components Mg-alloys. The purpose of this paper is to provide a comprehensive thermodynamic description of the Mg– Mn–Sn and Mg–Mn–Sr systems which is the backbone for understanding solidification and phase equilibria in the Mg– Mn-based alloys. 3 Thermodynamic models 3.1 Pure elements The Gibbs free energy of a pure element with a certain structure f is described as a function of temperature as ◦ GAf (T ) = a + bT + cT ln T + dT 2 + eT 3 + fT −1 + gT 7 + hT −9 (1) The parameters a through h are taken from the SGTE compilation by Dinsdale [56]. 3.2 Stoichiometric compounds Intermetallic compounds in the Mn–Sn phase diagram such as Mn3Sn2 and MnSn2 are considered stoichiometric and the Gibbs free energy of these compounds is described by the following equation Gphase,f = xi ◦ Gif + xj ◦ Gjf + DGf 2.3 Mg–Sn, Mg–Mn and Mg–Sr systems Optimised Gibbs energy parameters of the constituent binary systems, Mg–Sn, Mg–Mn and Mg–Sr will be adopted from the 682 (2) where ◦ Gif and ◦ Gjf denote Gibbs free energy of elements i and j in their standard state and ΔGf = a + bT is the Gibbs energy of formation of the stoichiometric compound, where a and b are the IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692 & The Institution of Engineering and Technology 2015 Table 1 Crystal structure and lattice parameters of Mn3Sn phase [65] Crystal data Atoms WP1 CN2 PS3 Atomic position X structure type Pearson symbol space group space group number lattice parameter, nm Ni3Sn CP2 P63/mmc 194 Mn Sn 6h 2d mm2 −6m2 6 6 0.16667 1/3 Y Z 0.3333 2/3 1/4 3/4 a = b = 0.56834 and c = 0.45335, angles: α = 90, β = 90, γ = 120 WP1 Wyckoff position, CN2 coordination number and PS3 point symmetry model parameters to be optimised based on experimental data of phase equilibria and thermodynamic properties. 3.3 Terminal solid solutions The Gibbs energy of a disordered solid solution phase is described by the following equation G = xi ◦ Gif + xj ◦ Gjf + RT [xi ln xi + xj ln xj ] + ex Gf (3) The configurational entropy of mixing is described by the following equation DS config = −R[nA ln (xA ) + nB ln (xB )]        x xBB xAB + n + n ln ln − R nAA ln AA BB AB 2yA yB y2A y2B (7) where xA and xB are the overall mole fractions of the components A and B, respectively. Mole fraction of component A can be described as follows where f denotes the phase in question and xi, xj denote the mole fraction of components i and j, respectively. The excess Gibbs energy is represented using Redlich–Kister equation nA nA + nB (8) nAA nAA + nBB + nAB (9) XA = Pair fraction ex Gf = xi · xj n=m  n f Li, j (xi − xj )n (4) XAA = n=0 with n Lfi, j = an + bn × T (n = 0, . . . , m) where n Lfi, j is the interaction parameters and an and bn are model parameters to be optimised using experimental phase diagram and thermodynamic data. In Mn–Sn binary system, (αMn), (βMn), (γMn) and (δMn) phases are treated as disordered solution model. 3.4 And the coordination-equivalent fractions yA = Gliq = nA ◦ gAliq + nB ◦ gBliq − T DS config + nAB liq DgAB 2 (5)     1 1 2nAA 1 nAB + A = A ZA ZAA 2nAA + nAB ZAB 2nAA + nAB     1 1 2nBB 1 nAB + B = B ZB ZBB 2nBB + nAB ZBA 2nBB + nAB ◦ + DgAB = DgAB  i≥1 io i gAB XAA +  oj j gAB XBB (6) j≥1 jo ◦ io , gAB and gAB are the model parameters to be optimised where DgAB ◦ = a + bT . and can be expressed as DgAB IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692 & The Institution of Engineering and Technology 2015 (11) (12) A B where ZAA and ZBB are the values of the coordination number of the Ath atom when all nearest neighbours are A’s and B’s, respectively. The coordination number of pure elements in the metallic solution, Table 2 Crystal structure and lattice parameters of Mn(2−x)Sn phase [65] Crystal data where nA and nB are the number of moles of the components A and B, nAB is the number of moles of (A-B) pairs, ΔS config is the configurational entropy of mixing given for randomly distributing the (A-B), (B-B) and (A-B) pairs. liq Pelton et al. [57] made modification to (5) by expanding DgAB as a polynomial in terms of the pair fractions XAA and XBB as shown in (6) (10) In addition, further modification has been made to permit coordination number to vary with composition as follows Liquid phase In the current paper, the liquid phase is modelled using the MQM where the pair approximation is utilised to describe short-range ordering in the liquid. A detailed description of the MQM for binary and multi-components solutions is available elsewhere [57–59]. Only a brief description will be presented here. The molar Gibbs energy of the liquid phase, derived from the modified quasi-chemical theory [57], is described by the following equation ZA nA ZA nA + ZB nB Atoms WP1 CN2 PS3 Atomic position X structure type InNi2 Pearson Hp6 symbol space group P63/mmc space group 194 number lattice parameter, nm angles: α = 90, β = 90, γ = 120 Y Z Mn1 Mn2 2d 2a 8 6 −3 m −6m2 1/3 1/3 2/3 2/3 3/4 1/4 Sn 2c 8 −6m2 0 0 0 a = b = 0.4187 and c = 0.5132 WP1 Wyckoff position, CN2 coordination number and PS3 point symmetry 683 Table 3 Optimised parameters of Mn–Sn and Mn–Sr binary systems Phase liquid Thermodynamic parameters (J/mole K) Mn Sn = 4; ZMnSn =5 ZMnSn Mn–Sn liquid 0 DgMnSn 10 = −418.4 − 1.72T ; DgMnSn = −20292.4 + 10.9T Mn Sr ZMnSr = 6; ZMnSr =6 Mn–Sr liquid 0 DgMnSr 10 = 15 522; DgMnSr = 5326.2 αMn Sn in αMn °L = − 12 552 βMn Sn in βMn °L = 3975 − 12.18T; 1L = 35 982; 2L = − 24 686 + 37.15T γMn Sn in γMn °L = − 29 121 + 16.86T; 1L = 5188.2 δMn Sn in δMn ◦ Mn–Sn Mn3Sn °L = − 14 602.2 + 1.88T Mn3 Sn = 3G(Mn, cbcc) + G(Sn, Diamond a4) − 17 029 − 22.84T GMn:Sn ◦ Mn3 GMn:Sn = 3G(Mn, cbcc) + 15 104 ◦ Mn(2−x)Sn Mn Sn 2 = 2G(Mn, cbcc) + G(Sn, Diamonda4 ) − 8410 − 23.89T GMn:Sn ◦ ◦ MnSn GMn:Sn = G(Mn, cbcc) + G(Sn, Diamonda4 ) − 8410 − 7.53T Mn 3 GMn:Sn = 3G(Mn, cbcc) + 27 196; MnSn2 G298.15K MnSn2 ◦ Mn3Sn2 ◦ Mn3 Sn2 G298.15K ◦ Mn 2 GMn:Sn = 2G(Mn, cbcc) + 18 828 = −14 753 − 25.6T = −7731.6 + 7.5T Fig. 1 Re-optimised Mn–Sn phase diagram with the experimental data from the literature Mg Sn Sr Mn ZMgMg = ZSnSn = ZSrSr = ZMnMn , was set to be 6. Since this value gave the best possible fit for many binary systems and was also recommended by Pelton et al. [57], Pelton and Chartrand [58] and Mn Sn = 4, ZMnSn =5 Pelton and Blander [59]. The values of ZMnSn Sr Mn = ZMnSr = 6 are chosen to permit the composition of and ZMnSr maximum short-range ordering in the binary system to be consistent with the composition that corresponds to the minimum enthalpy of mixing. For binary systems with approximately the same number of model parameters, random solution model and MQM can provide very similar and good fits to binary phase diagram data. However, this is no longer true for higher-order solutions [57–60]. Consider Mg–Mn–Sn ternary system in which the liquid solution Mg–Sn exhibits a strong tendency to short-range ordering [52, 60], whereas the Mg–Mn and Mn–Sn liquid solutions are closer to ideality. Positive deviations from ideal mixing will be observed, centred along the Mg2Sn–Mn corner of the composition triangle (where Mg2Sn is the binary composition of maximum short ordering). This is a typical behaviour of ternary system which has 684 one binary liquid exhibiting large negative deviation from ideality compared with the other binary liquids [60]. Such positive deviations are expected because (Mg–Sn) nearest neighbour are energetically favoured, MQM predicts a tendency for the liquid phase to separate into clusters rich in (Mg–Sn) and clusters rich in Mn solution. Random solution model overestimates the positive deviations observed in such ternary system. Kang and Pelton [60] also showed that the MQM predicts better extensions of binary miscibility gaps into a ternary system. For the binary liquid lines and ternary liquidus projection, the calculated miscibility gap using MQM is flatter than that predicted by random solution model. It is worth mentioning that the random solution model does not consider short-range ordering in the liquid phase, whereas associate model takes short-range ordering in the liquid phase into account with the assumption that some molecules occupy some lattice site which is not physically sound. Moreover, random solution model and associate model require too many parameters in optimisation to fit the experimental data. For the associate model and MQM, the optimisation results for binary systems are IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692 & The Institution of Engineering and Technology 2015 Table 4 Calculated invariant points of the Mn–Sn system in comparison with the literature Reaction type Reaction Composition, at.%Sn T, °C Reference peritectic L + (β Mn) ↔ (Mn3Sn) 22.6 22.8 35 35 66.7 66.7 66.7 66.7 36.5 36.5 40 40 40 984 984 884 884 549 549 548 548 480 480 540 540 540 this work [33] this work [33] this work [33] [34] [35] this work [32] this work [32] [43] L + (Mn3Sn) ↔ (Mn(2 −x)Sn) L + (Mn(2−x)Sn) ↔ MnSn2 eutectoid peritectoid (Mn(2−x)Sn) ↔ Mn3Sn2 + (Mn3Sn) Mn(2−x)Sn) + MnSn2 ↔ Mn3Sn2 mathematically very similar. However, Kang and Pelton [60] proved that the associate model does not correctly predict the thermodynamic properties of ternary and high-ordered system. Therefore the MQM with pair approximation is used in this paper to model the liquid phase. 3.5 Intermediate solid solutions Gibbs energy of intermediate solid solution phase is described by the compound energy formalism [61] which can be expressed as Gref = G = Gref + Gideal + Gexcess (13)  (14) q◦ yli ym j , . . . , yk G (i:j:, . . . , :k) Fig. 2 Calculated activities of Mn and Sn in the Mn–Sn liquid at a 1244°C b 1000°C IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692 & The Institution of Engineering and Technology 2015 685 Fig. 3 Calculated enthalpy of mixing in Mn–Sn liquid alloys compared with the experimental data of [66] at 1397°C Fig. 4 Calculated Mn–Sr phase diagram with the experimental data of [49] Gideal = RT   yli ln yli where i, j, …, k represent components or vacancy, l, m and q represent sublattices. yli is the site fraction of component i on sublattice l. fl is the site fraction of sublattice l relative to the total lattice site. ◦ G(i:j:, ..., :k) represents the energy of a real or hypothetical compound (end member). γL(i,j ):k represents the interaction parameters between components i and j on one sublattice when the other sublattice is occupied only by k. crystallographic data and the solubility range of the phase during the optimisation of the sublattice model parameters. The crystal structure data of the Mn3Sn intermediate solid solutions is obtained by Weitzer and Rogl [64] and listed in the Pearson handbook [65] as shown in Table 1. On the basis of crystallographic data of Mn3Sn phase, there are two atoms at different sites in the unit cell with the same coordination number and different points of symmetry as shown in Table 1. To obtain an intermediate phase which has an ideal stoichiometry, two sublattices are needed and each sublattice is occupied only by one constituent species. In other words, the direct sublattice model which is composed based on the crystallographic data of Mn3Sn phase only is the following model 3.5.1 Mn3Sn phase: According to Hari Kumar et al. [62] and Hari Kumar and Wollants [63], attention should be given to the (Mn)3 : (Sn)1 fl l Gexcess = 686  yli ylj ym k g (15) i L(i, j):k × (yli − ylj )g (16) g=0 IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692 & The Institution of Engineering and Technology 2015 Table 5 Calculated reaction temperature of the Mn–Sr system in Table 6 Invariant reaction in Mg–Mn–Sr ternary system comparison with the literature Reaction Reaction T, °C Reference gas + L1 ↔ L2 1402 1402 1236 1240 1238 1163 1165 1083 1063 772 740 772 707 707 547 547 this work [50] this work [49] [50] this work [50] this work [50] this work [49] [50] this work [50] this work [50] L1 ↔ δMn + L2 δMn + L2 ↔ γMn γMn + L2 ↔ βMn βMn + L2 ↔ βSr βMn + βSr ↔ αMn αMn + βSr ↔ αSr Reaction type Mg, at. % Sr, at. % Mn, at. % T, °C E1 5.71 94.14 0.15 706.9 E2 82.38 17.26 0.36 621.2 E3 85.77 13.85 0.38 612.5 E4 86.61 13 0.39 610 S1 E5 94.77 94.18 5.21 5.33 0.02 0.49 603.4 602.6 E6 18.02 81.94 0.04 566 S2 E7 E8 19.55 19.66 38.24 80.42 80.3 61.77 0.03 0.04 0.03 549 547 518 E9 35.93 35.93 0.01 432 L#2 ↔ L#1 + αMn + Sr-bcc L#2 ↔ L#1 + αMn + Mg23Sr6 L#2 ↔ L#1 + αMn + Mg38Sr9 L#1 ↔ αMn + Mg17Sr2 + Mg38Sr9 L#1 ↔ Mg17Sr2 + (Mg) L#1 ↔ (Mg) + αMn + Mg17Sr2 L#1 ↔ L + Sr-bcc + αMn L#1 ↔ L + Sr-bcc L#1 ↔ L + Sr-fcc + αMn L#2 ↔ L#1 + αMn + Mg2Sr L#2 ↔ Sr-fcc + αMn + Mg2Sr Fig. 5 Liquidus projection of Mg–Mn–Sr system This model does not represent the homogeneity range of Mn3Sn phase which was obtained by Stange et al. [41]. To achieve the deviation from stoichiometry, it is necessary to allow mixing of atoms in one or more sublattices. For the phases which have relatively a narrow range of homogeneity such as Mn3Sn the mixing is performed by ‘defects’, which may be vacancies or anti-structure atoms (i.e. atoms at lattice sites belonging to the other kinds of atoms in the ideal structure) [62, 63]. Since the structure of Mn3Sn phase is not closed packed, vacancy is more appropriate than anti-structure atom. Therefore vacancies (Va) in Sn sublattice is the defect considered in this model. Therefore the model takes the form (Mn%)3 : (Sn%, Va)1 The range which is covered by this model is Mn3Sn to pure Mn. Therefore this satisfies the homogeneity range requirement for Mn3Sn phase which was obtained by Stange et al. [41]. Hence, the Gibbs energy per mole of formula unit of Mn3Sn is described by the compound energy formalism as shown in the following equation (see (17) at the bottom of the next page) where i is the species inside the sublattice.yIMn is the site fraction of Mn3 Sn Mn3 Sn sublattice I.yIISn , yIIVa is the site fractions of lattice II.0 GMn:Sn , 0 GMn:Va represents real or hypothetical compound (end member) energy, Mn3 Sn Mn3 Sn Mn3 Sn LMn:Sn , 0 LMn:Va , 0 LMn:Sn, Va represents the interaction parameters which describe the interaction within the sublattice. 0 3.5.2 Thermodynamic modelling of the Mn(2−x)Sn phase: The crystallographic data of the Mn(2−x)Sn phase were listed in Table 2. On the basis of the crystallographic data of Mn(2−x)Sn phase, there are three atoms at different sites in the unit cell with different points of symmetry as shown in Table 2. To obtain an IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692 & The Institution of Engineering and Technology 2015 687 Fig. 6 Amount of Mn in Mg solid solution calculated using equilibrium cooling compared with the experimental data [27] for a Mg − 3 wt.% Sr–Mn b Mg − 5 wt.% Sr–Mn To obtain a deviation from this stoichiometry, mixing of constituents is applied. Grouping was not allowed in this model because the atomic position and point symmetry for each atom is different from the other atoms as shown in Table 2. To obtain the homogeneity range, mixing of Mn anti-structure atom in the second lattice and vacancy (Va) in the first lattice are considered intermediate phase which has an ideal stoichiometry, three sublattices are required and each sublattice is occupied by only one constituent species. In other works, the direct sublattice model which is derived from the crystallographic data of Mn(2−x)Sn phase is as follows (Mn1) : (Sn) : (Mn2) GmMn3 Sn = Mn3 Sn yIMn yIISn 0 GMn:Sn + 688 + Mn3 Sn yIMn yIIVa 0 GMn:Va Mn3 Sn yIISn yIIVa yIMn 0 LMn:Sn, Va  + RT 0.75 Mn  i=Mn yIi ln yIi + 0.25 Va  i=Sn yIIi ln yIIi  Mn3 Sn Mn3 Sn + yIMn yIISn 0 LMn:Sn + yIIVa 0 LMn:Va (17) IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692 & The Institution of Engineering and Technology 2015 Fig. 7 Liquidus projection of Mg–Mn–Sn system in this model. Therefore the model takes the form Table 7 Invariant reaction in Mg–Mn–Sn ternary system (see (18)) The optimised model parameters of different phases in the Mn–Sn and Mn–Sr binary systems are summarised in Table 3. 4 4.1 Results and discussion Mn–Sn binary system The re-optimised Mn–Sn binary phase diagram in relation to the experimental data from the literature is shown in Fig. 1. Table 4 summarises the calculated and experimental invariant points of this system. As can be seen in the table, all the deviations between the calculated temperature for the invariant reactions and the literature are within ±4°C. Fig. 2 shows the calculated activities of Mn and Sn in the Mn–Sn liquid at 1000 and 1244°C in comparison with literature values and both show good agreement. The calculated enthalpy of mixing for liquid at 1397°C compared with the experimental data [66] is shown in Fig. 3. The reference states are liquid Mn and liquid Sn. As can be seen in Fig. 3, the presently calculated enthalpy of mixing agrees with the experimental data [66]. In the current optimisation, Mn3Sn2 phase is modelled as a stoichiometric phase. Mn3Sn2 phase was only reported by Stange et al. [41] and adopted by Okamoto [48], whereas Miettinen [43] excluded this phase from his calculations. According to the work of Stange et al. [41], Mn3Sn2 appears as low-temperature GmMn3 Sn = Mn(2−x)Sn yIMn yIISn 0 GMn:Sn:Mn + yIMn yIVa + Mn(2−x)Sn yIMn yIIMn 0 GMn:Mn:Mn + Type Sn, at.% Mg, at.% Mn, at.% T, °C L#2 ↔ L#1 + βMn L#2 ↔ L#1 + L#2 + βMn L#2 ↔ L#1 + Mn3Sn L#2 ↔ L#1 + L#2 + Mn3Sn L#2 ↔ L#1 + Mn3Sn + Mg2Sn L#2 ↔ L#1 + αMn + Mg2Sn L#2 ↔ L#1 + Mn(2−x) Sn + Mg2Sn L ↔ hcp + Mg2Sn L#2 ↔ L#1 + Mn2Sn S1 U1 38.83 18.69 56.6 3.6 4.57 77.71 1048.8 1047.6 S2 45.36 47.86 6.78 938.2 U2 29.95 6.32 63.73 938.2 E1 41.93 56.36 1.71 724 E2 21.51 78.19 0.3 687.8 E3 56.52 39.63 3.85 589.1 S3 S4 10.4 9–12 0.27 1–2.5 569.8 525 L#2 ↔ L#1 + Mg2Sn + Mn3Sn2 L#2 ↔ L#1 + Mg2Sn + MnSn2 L#2 ↔ bct + Mg2Sn + MnSn2 E4 82.17 63.71 89.33 90– 85.5 5.19 32.43 12.64 3.86 540.5 509.3 E5 68.18 28.56 3.26 460.4 E6 91.95 8.04 0.01 202.7 Reaction (Mn%, Va) : (Sn%, Mn) : (Mn%) Mn(2−x)Sn yIVa yIISn 0 GVa:Sn:Mn [55] stoichiometric phase in the Mn–Sn system. The latter precisely determined the crystal structure of Mn3Sn2 using XRD and neutron diffraction. It is worth mentioning that the synthesis of pure Mn3Sn2 phase was difficult [41] and its crystal structure was determined from two-phase region containing MnSn2 and Mn3Sn2. (Mn3Sn) melts incongruently at 883°C which is in accord with the + Mn(2−x)Sn yIVa yIIMn 0 GVa:Mn:Mn  + RT 0.667 Va  yIi i=Mn     Mn(2−x)Sn Mn(2−x)Sn Mn(2−x)Sn I 0 II II I 0 II 0 II 0 Mn(2−x)Sn ySn LMn, Va:Sn:Mn + yMn LMn, Va:Mn:Mn + ySn yMn yMn LMn:Mn:Sn, Mn + yVa LVa:Mn:Sn, Mn IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692 & The Institution of Engineering and Technology 2015 Reference ln yIi + 0.333 Mn  i=Sn yIIi ln yIIi  (18) 689 Fig. 8 Isothermal section of Mg–Mn–Sn in Mg-rich region compared with the experimental data of [55] at a 500°C b 400°C work of [41, 48], whereas Miettinen [43] reported that Mn3Sn as Mn19Sn6 that melts congruently. 4.2 Mn–Sr binary system The calculated Mn–Sr phase diagram in relation to the few available experimental data of [49] is shown in Fig. 4. Table 5 summarises the calculated and experimental invariant points of this system. In Sr-rich region, the monotectic reaction L1 ↔ L2 + δMn was observed experimentally [49] and well produced in the current calculations. The calculated monotectic reaction is 96.5 at.% Sr at 1236°C, whereas the measured one is 96.5 at.% Sr at 1240°C. 690 4.3 Mg–Mn–Sr ternary system Thermodynamic properties of the Mg–Mn–Sr liquid were estimated from the optimised binary parameters using Kohler extrapolation [67]. The projection of the liquidus surface of the Mg–Mn–Sr system is shown in Fig. 5. As can be seen in Fig. 5, the miscibility gap covers most of the composition triangle and the primary crystallisation field of (Mg) is very small. Since there is no experimental data available for the entire Mg–Mn–Sr system, it is possible that the size of the miscibility gap is over or underestimated by the extrapolation. One important clue for understanding Mg-alloy development is its narrow crystallisation field of (Mg). Since changing the alloy’s composition slightly can IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692 & The Institution of Engineering and Technology 2015 lead to precipitate secondary phase(s) that enhance mechanical properties of Mg-alloys. Invariant reactions of Mg–Mn–Sr are listed in Table 6. The calculated liquidus projection of Mg–Mn–Sr is divided into 11 narrow crystallisation fields: (Mg), αMn, βMn, γMn, δMn, Mg17Sr2, Mg38Sr9, Mg23Sr6, Mg2Sr, α Sr and β Sr. The model predicted nine ternary eutectic reactions and two saddle points. All of these reactions are in the Mg–Sr side. In the current calculations of the Mg − 3 wt.%Sr − {0 − 5}wt.% Mn and Mg − 5 wt.%Sr − {0 − 5}wt.% Mn alloys indicate that up to ≃ 1.2 wt.% Mn, αMn precipitates from Mg-matrix, whereas above ≃ 1.2 wt.% Mn, α-Mn phase formation occurs from liquid and Mg-matrix. To estimate the amount of Mn in Mg solid solution under casting conditions, equilibrium simulations are carried out and compared with the TEM/EDS data of Celinkin et al. [27] as shown in Fig. 6. Fig. 6 shows that the amount of Mn in Mg-matrix increases with Mn alloying addition up to 1.75 wt.% Mn. According to Celikin et al. [27] works, TEM and XRD analyses for heat treated alloys at 225°C (Mg–(3–5 wt.%) Sr–(0.75–2 wt.%) Mn) show that the stable phases are Mg17Sr2, Mg and αMn. These phases are in good agreement with the current FactSage calculations. In the composition range experimentally studied, Celikin et al. [27] observed in TEM/EDS scan that Mn dissolves in interdendritic phase Mg17Sr2. No experimental data could be found in the literature to prove the amount of Mn solubility in Mg17Sr2. This demands experimental investigation in order to verify the solid solubility of Mn in the intermetallic phases of the Mg–Mn–Sr system. Janz [51] calculated liquidus projection of Mg–Mn–Sr system as extrapolation of the binary subsystems. In his work, miscibility gap cover most of the ternary system that is in accord with the current calculations. In Mg–Sr-rich region, Janz [51] projection showed very narrow crystallisation field for all phases in the ternary system compared with the current paper. In the current calculations, liquidus projection displays smooth and gradual curving, whereas Janz [51] calculations showed curvature of γ and δMn phase boundaries in the ternary system. This particular change in the curvature might be because of not modelling liquid phase by the MQM since the MQM predicts flatter liquidus projection compared with random solution model which is in accord with Kang and Pelton [60] observations. It is worth mentioning that Janz [51] and current isothermal sections of the Mg–Mn–-Sr system at 400 and 500°C are identical. 4.4 Mg–Mn–Sn ternary system In the present paper, the constituent binaries are extrapolated according to the Kohler extrapolation model [67] to construct Mg–Mn–Sn ternary system without addition of any ternary parameters. Liquidus projection of Mg–Mn–Sn is shown in Fig. 7 and invariant reactions of this system are listed in Table 7. Ten primary crystallisation fields are predicted in Mg–Mn–Sn system: hcp, αMn, βMn, γMn, δMn, Mg2Sn, Mn3Sn, Mn3Sn2, Mn(2−x)Sn and Mn2Sn. Isothermal sections of the Mg–Mn–Sn system in the Mg-rich region were calculated at 500 and 400°C and compared with the work of Kopetskii and Semenova [55] as shown in Figs. 8a and b. Experimental data covered wider region of (Mg) + Mg2Sn and (Mg) compared with calculated isothermal section at 500°C, whereas the predicted (Mg) + Mg2Sn + αMn region is in accord with the work of Kopetskii and Semenova [55]. Wider experimental regions of (Mg) + Mg2Sn and (Mg) might belong to supersaturated solid solution of (Mg) phase. The calculated phase regions of (Mg) + Mg2Sn + αMn, (Mg) + Mg2Sn and (Mg) are in accord with the experimental data of [55]. It is worth mentioning that the work of Kopetskii and Semenova [55] is the only ternary experimental data that could be found in the literature. The differences between Mg–Mn–Sr and Mg–Mn–Sn liquidus projections drawn in Figs. 5 and 7, respectively, are: miscibility gap in Mg–Mn–Sr system is wider than that of Mg–Mn–Sn, intermetallic compounds of Mn–Sn show wide crystallisation field IET Sci. Meas. Technol., 2015, Vol. 9, Iss. 6, pp. 681–692 & The Institution of Engineering and Technology 2015 in the ternary system and the Mg–Mn–Sr system has nine ternary eutectic reactions, and two saddle points and 11 crystallisation fields. Mg–Mn–Sn has four saddle points, two quasi-peritectic and six ternary eutectic reactions. 5 Summary A self-consistent thermodynamic database has been constructed for the Mn–Sn, Mn–Sr, Mg–Mn–Sn and Mg–Mn–Sr systems using CALPHAD method. The liquid phase is modelled using the MQM to account for the short-range ordering in Mn–Sn liquid. The model parameters of the Mn–Sn and Mn–Sr systems are evaluated by incorporating all experimental data available in the literature. The phase diagrams and thermodynamic properties of the two binaries show good agreement with the experimental data. The established database for Mg–Mn–Sr system predicted two saddle points and nine ternary eutectics, whereas four saddle points, six ternary eutectic and two quasi-peritectic reactions were predicted in Mg–Mn–Sn system. This is the first attempt to construct the ternary phase diagrams of the Mg–Mn–Sn and Mg–Mn–Sr systems using the MQM for the liquid. 6 Acknowledgment Financial support from the Deanship of Scientific Research and Graduate Studies, The Hashemite University, Jordan is gratefully acknowledged. 7 References 1 Avedesian, M.M., Baker, H.: ‘Magnesium and magnesium alloys’ (ASM Specialty Handbook, Materials Park, OH, 1999), pp. 98–102 2 Eliezer, D., Aghion, E., Fores, F.H.: ‘Magnesium science, technology and applications’, Adv. Perform. Mater., 1998, 5, pp. 201–212 3 Mordike, B.L., Ebert, T.: ‘Magnesium properties – application-potential’, Mater. Sci. Eng. 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