MgSnCa

Phase equilibria,thermodynamics and solidification microstructures of Mg e Sn e Ca alloys,Part 1:Experimental investigation and thermodynamic modeling of the ternary Mg e Sn e Ca systemA.Kozlov a ,M.Ohno a ,1,R.Arroyave b ,2,Z.K.Liu b ,R.Schmid-Fetzer a ,*aInstitute of Metallurgy,Clausthal University of Technology,Robert-Koch-Str.42,D-38678Clausthal-Zellerfeld,GermanybThe Pennsylvania State University,University Park,PA 16801,USAReceived 6July 2007;received in revised form 4October 2007;accepted 30October 2007Available online 21December 2007AbstractThe phase equilibria of the Mg e Sn e Ca system for the entire composition and temperature ranges have been clarified based on the Calphad method.To obtain a reliable thermodynamic description,we performed key experiments for the phase boundary data and also utilized the first-principle results of the finite-temperature properties for the binary and ternary compounds.Experimental works for the phase equilibria,which consist of thermal,crystallographic and microstructural analyses,and the thermodynamic modeling combined with finite-temperature first-principle calculations are reported.The satisfying agreements between the experimental and calculated results support the reliability of the proposed thermodynamic description.The phase diagram for overall composition and temperature ranges of the ternary system based on the thermodynamic calculations is presented.In a second study this result is applied to obtain details of the phase formation during solidification for practically important Mg-rich as-cast alloys.Ó2007Elsevier Ltd.All rights reserved.Keywords:A.Ternary alloy systems;B.Phase diagrams;B.Thermodynamic and thermochemical properties;E.Phase diagram,prediction;F.Electron microscopy,scanning1.IntroductionIn recent years,magnesium alloys have become a center of interest among material engineers and scientists.The combi-nation of lightweight,high specific strength and good castabil-ity makes Mg alloys a promising engineering material of the automotive and aviation industries [1,2].Most commercial Mg alloys are Mg e Al based (AZ and AM series)with addi-tions of Zn,Mn and/or Ca,Sr,Si,RE (RE:rare earth mischme-tal),which demonstrate good room-temperature strength andductility with satisfying salt-spray corrosion resistance and ex-cellent castability [3].On the other hand,Al-free Mg alloys are currently attracting attention as a next generation of com-mercial Mg alloys,in the light of the fact that grain refinement can effectively be realized by the addition of Zr,which is not possible in the Al-containing AZ and AM series because of detrimental Al e Zr compound formation.Mg e Sn e Ca (TX series)as well as Mg e Zn e Zr (ZK series)is among the most important systems for the Al-free Mg alloys.It has been sug-gested in Ref.[4]that the addition of Sn improves strength at the expense of ductility in Mg alloys.Some precipitates in-volving Ca could be considered relevant to this behavior.The effects of additional elements on the phase equilibria in alloys are of great importance in material design and manufac-ture.The focused optimization of alloy composition and heat treatment conditions is virtually impossible without the knowl-edge of pertinent phase equilibria.Therefore phase diagrams of*Corresponding author.Tel.:þ495323722150;fax:þ495323723120.E-mail address:schmid-fetzer@tu-clausthal.de (R.Schmid-Fetzer).1Present address:Research Center for Integrated Quantum Electronics (RCIQE),Hokkaido University,North 13West 8,Sapporo 060-8628,Japan.2Present address:ENPH 119,Texas A&M University,College Station,TX 77843-3123,USA.0966-9795/$-see front matter Ó2007Elsevier Ltd.All rights reserved.doi:10.1016/j.intermet.2007.10.010Available online at Intermetallics 16(2008)299e315multi-component Mg-systems are indispensable information. Despite its importance mentioned above,the phase diagram of the ternary Mg e Sn e Ca system has not been established yet.Even isolated equilibrium phase boundary data could not be found in the literature.The aim of this work is to clarify the phase equilibria of this important ternary system over the entire composition and temperature ranges,for thefirst time, based on a combination of the Calphad methodology,first-principle calculations and own key experiments.The Calphad method has been recognized as a powerful method to provide the phase equilibria and thermodynamic quantities in multi-component systems with high accuracy [5,6].Within this approach,the Gibbs free energies for all the phases involved in the system of interest are quantified with relatively simpler mathematical expression and/or statis-tical mechanics formula based on various kinds of experimen-tal data.Hence,the reliability of the Calphad method is largely dependent on the available experimental data.As mentioned above,however,little has been clarified for the Mg e Sn e Ca phase equilibria.Therefore,we performed our own painstak-ing experiments for the determination of the phase equilibria by means of X-ray diffraction(XRD),scanning electron mi-croscopy with energy-dispersive X-ray analysis(SEM/EDX), differential thermal analysis(DTA)and differential scanning calorimetry(DSC).These experimental results are then fully employed in the Calphad modeling.In order to obtain a highly reliable thermodynamic description,furthermore,we utilize thefinite-temperaturefirst-principle results as detailed below.Alloys with considerable content of Ca often involve the difficulty to realize true equilibrium,due to the high reactivity and high vapor pressure and also the low purity of commonly used Ca.A recent work on the Calphad modeling of the Ca e Sn system by the present authors[7]has clarified that the forma-tion enthalpy of compounds from Ca-rich part experimentally determined in the early work[8]is unrealistic.In Ref.[7]the authors performed two types of Calphad modeling,one is based only on the experimental data and the other is based on thefirst-principle output offinite-temperature properties of compounds and,then,the detailed comparison between them indicates that the latter description is more realistic than the former one.In this regard,careful attention should be paid to the present ternary system where,in addition to the Ca e Sn binary compounds,a Mg e Ca binary compound (Mg2Ca)and two ternary compounds exist,as described in the next section.Thefinite-temperature properties of these compounds should be subjected to thefirst-principle investiga-tion.Especially,there have been no thermodynamic data avail-able for two reported ternary compounds.In the present work, therefore,we extend the combined approach of the Calphad andfinite-temperaturefirst-principle methods recently attemp-ted in Ref.[7]to the present ternary system.The details of the methods for the presentfirst-principle calculations are found in Refs.[9,10]and also in Ref.[7]for the Ca e Sn binary phases, and the main concern of this study is the thermodynamic mod-eling based on these outputs.As mentioned above,this is thefirst investigation of phase equilibria of the Mg e Sn e Ca ternary system over a wide range of composition and temperature,consisting of the detailed experiments on phase equilibria and the Calphad modeling combined withfinite-temperaturefirst-principle calculations and,furthermore,the solidification process for the Mg-rich alloys.We report our extensive work on Mg e Sn e Ca alloys in a series of two publications.In this paper(part I of this series), the Calphad modeling combined withfinite-temperaturefirst-principle calculations and the experimental work on the phase equilibria are presented,using ternary key samples selected at strategic intermediate compositions(16e67at.%Mg).This is followed by the comparison between the calculated and ex-perimental results concerning liquidus surface,isothermal sections at various temperatures,invariant reactions and se-quence of phase formation during solidification process to demonstrate the consistency of the constructed thermody-namic description.The second report[11](denoted‘‘part II’’here)is devoted to an application to practically important Mg-rich(92e 96.5wt.%Mg)as-cast alloys,demonstrating the satisfying consistency between the phase formation obtained by compu-tational thermochemistry using the constructed description and the solidification behavior revealed in an experimental investigation.2.Literature data2.1.Binary subsystemsSuccessful Calphad modeling for each of the constituent binaries Mg e Ca,Mg e Sn and Ca e Sn has been already per-formed.In the following,we briefly refer to the Calphad mod-eling relevant to the present work and demonstrate the calculated phase diagrams for the discussion of ternary phase equilibria given in Section6.The Calphad modeling of the Mg e Sn system was reported in Ref.[12]and subsequently corrected in Ref.[13].In the Mg e Sn system,the existing phases are liquid,hcp-(Mg),bct-a Sn,diamond-b Sn,and the intermediate compound Mg2Sn. No solubility of Mg in b Sn has been experimentally observed, but there is a small amount of solubility of Sn in(Mg).There are two eutectic reactions:one is L4(Mg)þMg2Sn at 563 C on the Mg-rich side and the other is L4b SnþMg2Sn at204 C on the Sn-rich side.The congruent melting of Mg2Sn occurs at774 C.Fig.1shows the calculated phase diagram for the Mg e Sn system based on the corrected version[13].The thermodynamic modeling of the Ca e Sn system has been recently reported by the present authors[7].In that work,due to a discrepancy between the experimental and first-principle data of formation enthalpy of compounds,two types of thermodynamic modeling were attempted:one is based on thefirst-principle output(model I)and the other based on the experimental data(model II).The detailed com-parison between the several types of experimental data and the corresponding calculated results revealed that model I is supe-rior to model II.Shown in Fig.2is the phase diagram calcu-lated by model I.One sees that there are seven compounds in300 A.Kozlov et al./Intermetallics16(2008)299e315this system.Among them,three compounds,Ca 2Sn,CaSn and CaSn 3melt congruently,while the other four compounds melt through peritectic reactions.The thermodynamic modeling of the Mg e Ca system has been reported in Refs.[14,15].The Mg e Ca system exhibits a simple phase diagram consisting of the five phases,liquid,hcp-(Mg),fcc-a Ca,bcc-b Ca,and Mg 2Ca.There are three in-variant reactions:one is on the Mg-rich side,L 4(Mg)þMg 2Ca at 517 C,the second one on Ca-rich side L 4b Ca þMg 2Ca at 446 C and the third one is congruent melting of Mg 2Ca compound L 4Mg 2Ca at 710 C.The dashed line in Fig.3represents the phase diagram calculated from the parameters of Ref.[14].This line is almost completely superimposed on the solid line which is obtained from the present thermodynamic modeling as discussed in Section 3.2.2.Ternary systemFor the Mg e Sn e Ca system,neither phase diagram nor thermodynamic data have been reported in the literature.Only information about one ternary Ca 6.2Mg 3.8Sn 7compound and one ternary solid solution originating from the binary Ca 2Sn compound are available in the literature [16e 19].The existence of ‘‘ternary compound’’CaMgSn has been reported by Eisenmann et al.[16,17].They synthesized the composition CaMgSn and determined its crystal structure.It was found that CaMgSn crystallizes as the ordered anti-PbCl 2-type (orthorhombic Co 2Si-type)structure and therefore this was expected to be related to the binary compound Ca 2Sn,which forms the anti-PbCl 2-type too.Ganguli et al.[19]confirmed the existence of CaMgSn ‘‘ter-nary compound’’and found that this phase actually is the ter-nary solid solution of Ca 2Sn,which is denoted (modeled)as Ca 2Àx Mg x Sn in the present study,and terminates at the compo-sition x ¼1,i.e.CaMgSn.The alloys were prepared by melting elements in Ta crucibles at 1030 C for 2days,quickly cooled,and then investigated by X-ray powder diffraction analysis.All powder patterns over this region indicate that there are not only the orthorhombic Ca 2Sn and CaMgSn or their solid solution,but additional Mg 2Sn was also present for x >1.Ganguli et al.[18]synthesized the ternary compound Ca 6.2Mg 3.8Sn 7and determined its crystal structure.The alloys with various compositions in the ratio (Ca,Mg)5Sn 3were pre-pared by melting elements in Ta crucibles.The sample was heated at 925 C for 6h,then cooled to 600 C and annealed for 6days,which yields 85%of the subject phase.X-ray powder diffraction patterns were obtained from samples mounted on a frame between pieces of cellophane tape in the glove box to protect them from the atmosphere.An Enraf-Nonius Guinier camera with Cu K a radiation and reference Si as an internal standard were employed to obtain powder diffractiondata.Fig.1.Mg e Sn binary phase diagram calculated from the data set given in Ref.[13].Fig.3.Mg e Ca binary phase diagram calculated from the data sets given in Ref.[14]and this work.(Difference cannot be distinguished on the graph.).301A.Kozlov et al./Intermetallics 16(2008)299e 315Alloys with composition Mg e5Sn e0.04Ca and Mg e5Sn e 0.5Ca(in wt.%)have been investigated in Ref.[20].It was found that0.04wt.%of Ca did not introduce any characteristic features into the system and the microstructure of alloy Mg e 5Sn e0.04Ca has shown only solid solution of(Mg)containing small amount of Mg2Sn.Microstructure of alloy Mg e5Sn e 0.5Ca has shown also solid solution of(Mg)containing small amount of Mg2Sn grain boundaries.In addition,the authors re-vealed Ca as the primary constituent in rod-or plate-shaped precipitates in(Mg)and denoted this phase as Mg2Ca.How-ever,more recent investigations of alloys with composition Mg e5Sn e0.5Ca[21,22]have shown that the primary constit-uent is CaMgSn instead of Mg2Ca and this result is in good agreement with present thermodynamic calculation as dis-cussed in Section4.The solid phases in the present system are listed in Table1, comparing the crystallographic data from the literature[23e25] with present results for the most important intermetallic phases.3.Modeling3.1.First-principle calculations offinite-temperature propertiesThe details of the methods used for the presentfirst-princi-ple calculations are reported in Ref.[9,10].In the following, hence,we briefly summarize the essential points of thefi-nite-temperaturefirst-principle method utilized in this work.First-principle total energy calculations of three pure ele-ments and binary and ternary compounds at the ground state were carried out based on the Projector Augmented Wave (PAW)method[26,27]within the Generalized Gradient Approximation(GGA)[28].A cutoff energy was set to be 350eV for all the calculations and the brillouin zone integra-tions were performed using a Monkhorst e Pack mesh[29] with at least5000k points per reciprocal atom.Self-consistent static calculations were performed,following the full relaxa-tion for the structure and volume based on the Methfessel e Paxton order1smearing method[30].Thefinite-temperature properties(T>0K)for the solid phases were calculated using the supercell approach.Within the quasi-harmonic approximation,the vibrational free energy was obtained by computing the force constant tensor and the corresponding phonon density of state(DOS)as a function of volume.The temperature dependence of vibrational contri-bution was determined by minimizing the vibrational free energy with respect to volume at each temperature.Further-more,the electron-excitation contribution to the free energy was evaluated through the integration of the electronic DOS at each volume.From thesefinite-temperature calculations, we obtained the free energies of the solid phases atfinite-temperatures.The present calculations are applied to three pure elements, all the binary compounds and ternary phase CaMgSn which corresponds to the end member of the ternary solid solution Ca2Àx Mg x Sn modeled in the present study.In addition to these phases,there exists a ternary compound Ca6.24Mg3.76Sn7of which no thermodynamic data have been reported.Since the large unit cell of this ternary compound requires a substantial computational time in the supercell approach,we performed only the ground state calculations and thefinite-temperature property of this phase is obtained from the compositional average which is detailed in the following section.Since the calculated results for the other binary phases are reported in detail in Ref.[10],we present the calculated results only for Mg2Ca,Ca2Àx Mg x Sn with x¼1and Ca6.24Mg3.76Sn7 phases in Table2,which are relevant to the present thermody-namic modeling of the ternary system,and compare to exper-imental literature data[31e33].It is to be noted that the tabulated data in Ref.[10]for the enthalpy of formation corre-spond to the values at0K,while thefirst-principle data for Mg2Ca and CaMgSn phases shown in Table2are theTable1Crystallographic data of solid phases occurring in the Mg e Sn e Ca systemPhase/temperature range, C Pearson symbol/prototypeLattice parameters(A˚)Referenesa b c(b Ca)/842to443cI2/W 4.480e e[23](a Ca)/<443cF4/Cu 5.5884e e Pure Ca at25 C[23] (Mg)/<650hP2/Mg 3.2094e 5.2107Pure Mg at25 C[23] (b Sn)/231.97to13tI4/b Sn 5.8318e 3.1818Pure Sn at25 C[23] (a Sn)/<13cF8/C(diamond) 6.4892e e[23]Mg2Ca/<711hP12/MgZn2 6.23,6.251(4)e10.15,10.141(7)[24][This work]Mg2Sn/<773cF12/CaF2 6.7645(3),6.760(2)e e[19][This work] CaSn3/<630cP4/AuCu3 4.741(2)e e[25]CaSn/<1010oC8/CrB 4.813(2)11.544(2) 4.351(1)[25]Ca7Sn6/<1000oP52/Ca7Sn67.867(1)23.818(2)8.465(1)[25]Ca31Sn20/<w1130tI204/Pu31Rh2012.521(2)e39.87(2)[25]Ca36Sn23/<w1130tP118/Yb36Sn2312.499(1)e22.883(5)[25]Ca5Sn3a/<1130tI32/Cr5B38.117(2)e15.429(7)[25]Ca2Sn/<1340oP12/Co2Si7.992(3),7.981(2) 5.037(3),5.043(1)9.554(5),9.566(2)[25,19]Ca2Àx Mg x Sn/(x¼0e1)7.8731(8),7.86(2),7.847(8)4.6941(3),4.66(2),4.678(7)8.7538(7),8.74(2),8.749(8)x¼1,[19],x¼1,[17],x¼1,[This work]Ca6.2Mg3.8Sn7/<918.9oP34/Ca6.2Mg3.8Sn77.794(1)25.718(4) 4.655(1)[18]a Probably stabilized by hydrogen as Ca5Sn3H x.302 A.Kozlov et al./Intermetallics16(2008)299e315formation enthalpies at 298K.The inclusion of vibrational contributions lead to slightly more negative enthalpy of forma-tion at 298K as expected.The calculated formation enthalpy and the absolute entropy of Mg 2Ca phase are in fairly good agreement with the experimental data and the assessed value of Ref.[14],which strongly supports the reliability of the thermodynamic description of Ref.[14]on one hand and the validity of the present first-principle calculations on the other hand.This point is further addressed by reassessing the Mg e Ca binary system as described below.3.2.Thermodynamic modeling of the binary systems Within the Calphad method,the degree of accuracy of thedescription of the binary subsystems plays a crucial role in ob-taining a reliable thermodynamic description of the resulting ternary system.As mentioned in Section 2,the thermody-namic assessments of the binary Ca e Sn,Mg e Sn and Mg e Ca systems have been performed in Refs.[7,13,14],showing excellent agreements with the experimental data.In the pres-ent study,we employ the thermodynamic description of Ref.[13]for the Mg e Sn system and the model I in Ref.[7]for the Ca e Sn system (first-principle based description).The thermodynamic parameters of Mg e Sn system are presented in Table 3,since these have not appeared in any report yet.As for the Mg e Ca system,we performed our own thermody-namic assessment based on the finite-temperature property.As mentioned in Section 1,the thermodynamic state of compounds containing Ca requires the first-principle investi-gation in the light of the experimental difficulty to realize the true equilibria.As shown in Table 2,the formation en-thalpy and absolute entropy calculated with the thermody-namic parameters given in Ref.[14]are consistent with our first-principle results,which indicates the reliability of the thermodynamic description of Ref.[14]and also the present first-principle calculations.In this study,as a further check of this reliability,we reassess the Mg e Ca binary system based on this first-principle output of Mg 2Ca.Since the Mg 2Ca compound possesses no substantial solubility (see Fig.3),this phase is regarded as a stoichiometric compound.Then,the Gibbs energy (per mole of atoms)of the Mg 2Ca compound is described using the following absolute reference state formula:G Mg 2Ca ðT Þ¼23G 0;hcp Mg ð298:15K Þþ13G 0;fccCað298:15K ÞþA þB ,T þC ,T ln T þD ,T 2þE ,T 3þF ,T À1ð1a Þwhere A ,B ,.,F are the optimized parameters and G 0;hcpirep-resents the Gibbs energy for pure element i (i ¼Mg and Ca)with hcp structure and this explicit formula is given later in Eq.(2).It should be noted that this model allows a quantitative description of the heat capacity.C Mg 2Ca pðT Þ¼ÀC À2D ,T À6E ,T 2À2F ,T À1ð1b ÞFirstly,the parameters C ,D ,E and F are obtained by fitting to the temperature dependence of the heat capacity calculated from the first-principle method.Shown in Fig.4is the calcu-lated result which compares perfectly with the first-principle output for T >250K.It is noted that a slight discrepancy is observed in comparison to the published Calphad description [14].The remaining parameters to be optimized in Eq.(1)are A and B which are related to the formation enthalpy and entropy of the compound.In the present modeling we obtained these parameters using the experimental data for the congruent melt-ing temperature of Mg 2Ca and the eutectic reaction tempera-ture L 4(Mg)þMg 2Ca (e 7on Fig.3),without changing the thermodynamic parameters of the other phases,such as liq-uid and hcp-solid solution given in Ref.[14].The optimized parameters are presented in Table 3and the calculated phase diagram is demonstrated by the solid lines in Fig.3.The solid lines virtually coincide with the dashed lines of Ref.[14]mak-ing them invisible on the graph.It is not surprising to see vir-tually the same phase diagram calculated from different sets of thermodynamic parameters (Ref.[14]and present).This is why it is so important to include as much thermodynamic in-formation as possible,in the present case the first-principle results.The formation enthalpy and the absolute entropy at 298K calculated by the present modeling are shown in Table 2where one can see the satisfying consistency between the Calphad modeling,the first-principle outputs and the experimental data.These agreements again prove the reliability of thermo-dynamic assessment of Ref.[14]on one hand,and the validity of the present first-principle output on the other hand.Table 2Standard enthalpy of formation,D f H 298,and absolute entropy,S 298,of compounds Compounds D f H 298(kJ/mol atom)S 298(J/mol atom/K)FP Calphad Exp FP Calphad Mg 2CaÀ12.37[tw]À12.85[14]À13.50[31]36.21[10]32.48[14]À12.78[tw]À12.70[32]32.56[tw]À13.00[33]CaMgSnÀ56.89[tw]À56.94[tw]37.83[10]37.84[tw]Ca 6.24Mg 3.76Sn 7À56.9(0K)[10]À57.16[tw]40.59(0K)[10]40.59[tw]Stable Element Reference,SER is used.Note that the tabulated data in Ref.[10]for the formation enthalpy correspond to the values at 0K,while the first-principle data for Mg 2Ca and CaMgSn phases shown below are the formation enthalpies at 298K.[tw]¼This work.303A.Kozlov et al./Intermetallics 16(2008)299e 315As shown above,the thermodynamic descriptions obtained in Ref.[14]and in this work provide almost identical results and one can proceed to the modeling of the ternary system based on either description.Although we employ the thermo-dynamic parameters obtained in this study in the following,the replacement of the description of Mg 2Ca system with the one of Ref.[14]yields no substantial change in the resul-tant phase equilibria of the ternary system.The main reason for using the present data is their consistency with the first-principle C p and D H data.3.3.Thermodynamic modeling of ternary systemThe Gibbs energy function,G 0;f i ðT Þ¼G f i ðT ÞÀH SERi ,for the element i (i ¼Ca,Mg,Sn),in f phase is given byG 0;f i ðT Þ¼a þb ,T þc ,T ln T þd ,T 2þe ,T 3þf ,TÀ1þg ,T 7þh ,T À9ð2Þwhere H SER i is the molar enthalpy of the stable element refer-ence (SER)at 298.15K and 1bar;T is the absolute tempera-ture.The Gibbs energy functions for Ca,Mg and Sn are taken from the SGTE compilation by Dinsdale [34].In the binary Mg e Sn [13]and Ca e Sn [7]systems,the liq-uid phases are modeled using the associate solution model [35,36]with Mg 2Sn and Ca 2Sn associates,respectively.Ac-cordingly,the liquid phase in the present ternary system is described using the associate solution model as expressed by G LðT ;f y i gÞ¼X 5i ¼1y i ,G 0;L i ðT ÞþRTX 5i ¼1y i ln y iþX 4i ¼1X 5j >iy i y jX n n ¼0L n ;L i ;jÀy i Ày j Ánð3Þwhere R is the gas constant,y i is the mole fraction of specie iin the liquid,i (j )represents any of the species,Ca,Mg,Sn,Ca 2Sn and Mg 2Sn.G i L refers to one mole of formula of the liq-uid and the number of moles in the liquid phase results from an internal Gibbs energy minimization.In the first term ofEq.(3),the parameters,G 0;L Ca ðT Þ;G 0;L Mg ðT Þand G 0;LSn ðT Þ,Table 3Thermodynamic parameters Liquid phase,binary Mg e Sn (J/mol)G 0;Liquid Mg 2Sn ¼À69092:9þ97:6086,T À11:0957,T ,ln ðT Þþ2,G 0;L Mg þG 0;L Sn L 0;Liquid Mg ;Mg 2Sn ¼6902:76À9:22726,T L 0;Liquid Mg 2Sn ;Sn ¼À8289:15À10:0268,T L 0;Liquid Mg ;Sn ¼À31251þ0:74703,TLiquid phase,ternary Mg e Sn e Ca (J/mol)L 0;Liquid Mg 2Sn ;Ca ¼À70461:37L 0;Liquid Mg 2Sn ;Ca 2Sn ¼À71375:70Intermetallic compounds,absolute reference state,Eq.(2)J/mol-atom/(K)Temperature range (K)A *aBC DEFG 0;Ca 2Àx Mg x SnCa:Ca:Sn298<T <1660À74159.5128.5390À25.372 4.426E À4À7.414E À735284.021660<T À86420.8214.4356À36.135000G 0;Ca 2Àx Mg x Sn Mg:Ca:Sn 298<T À65595.0146.5À27.543 2.367E À3À10.07E À789823.7L 0;Ca 2Àx Mg x Sn Ca:Sn:Ca ;Mg 298<T 1262.3400000G Ca 6:24Mg 3:76Sn 7298<T 65707.06143.13À27.478 2.556E À313.376E À775502.91G Mg 2Ca 298<T À20456.3130.6760À24.04541À0.0028942À2.31E À0735456.50G Mg 2Sn 298<TÀ31024.2110.918À21.8911À0.003028À210000aA üA þp p þq G 0;fcc Ca ð298:15K Þþqp þq G 0;bct Sn ð298:15K Þ.represent the Gibbs energies of pure Ca,Mg and Sn liquids, respectively,which are taken from the SGTE compilation[34].L n;L i;j is the n th interaction parameter between the speciesi and j in the liquid phase.Actual ternary interactions L n;L i;j are only those involving all three elements in the species i and j. They are found to be necessary in this work only for the Mg2Sn e Ca and Mg2Sn e Ca2Sn interactions.All other param-eters are already given from the binary systems.The hcp-(Mg)solution phase is described by the disordered substitutional solution model as given byG hcp¼X3i¼1x i G0;hcpiþRTX3i¼1x i ln x iþX2i¼1X3j>ix i x jX nv¼0L v;hcpi;jÀx iÀx jÁvð4Þwhere x i are the molar fractions of i¼Ca,Mg and Sn.It should be noted that no ternary interaction parameters are in-troduced,which correspond to a Redlich e Kister/Muggianu type extrapolation from the binary sets.As explained in Section2,the Ca2Sn phase possesses the ternary solubility which terminates at the CaMgSn composi-tion and this solution phase is denoted as Ca2Àx Mg x Sn.In the present study,this phase is modeled with the sublattices Ca1Sn1(Ca,Mg)1and the Gibbs energy(per1mol of atoms) of this phase can be described by the following Compound-Energy FormalismG Ca2Àx Mg x Sn¼y Ca G0;Ca2Àx Mg x SnCa:Sn:Ca þy Mg G0;Ca2Àx Mg x SnCa:Sn:Mgþ13R,Tðy Ca,ln y Caþy Mg,ln y MgÞþy Ca,y Mg,L0;Ca2Àx Mg x SnCa:Mg:Ca;Mgð5Þwhere G0;Ca2Àx Mg x SnCa:Sn:Ca corresponds to the Gibbs energy of1molof atoms Ca2Sn binary phase and G0;Ca2Àx Mg x SnCa:Sn:Mg is the Gibbsenergy of formation of the terminal composition point of solidsolution,denoted as CaMgSn.L0;Ca2Àx Mg x SnCa:Sn:Ca;Mg is the interactionparameter between Ca and Mg in the third sublattice.All the binary phases are described by the absolute refer-ence state formula given in Eq.(1).For the end member CaMgSn and ternary compound Ca6.24Mg3.76Sn7,we also employ the absolute reference formula which,for the ternary phase,is rewritten asG Ca p Mg q Sn rðTÞ¼ppþqþrG0;fccCað298:15KÞþqpþqþrG0;hcpMgð298:15KÞþrpþqþrG0;bctSnð298:15KÞþAþB,TþC,T ln TþD,T2þE,T3þF,TÀ1ð6ÞAs described,we obtained thefinite-temperature property of the end member CaMgSn phase from thefirst-principle calcu-lations.Hence,the parameters C,D,E and F for this phase can be obtained byfitting to the heat capacity offirst-principle output.Shown in Fig.5is thefitting curve of C p with the first-principle data.The parameters A and B are then optimized based on the experimental data obtained in this study which are described in detail below.As for the ternary compound Ca6.24Mg3.76Sn7,the neces-sary large unit cell results in a large computational burden in obtaining thefinite-temperature properties based on super-cell approach.It is noted that the composition of ternary com-pound Ca6.24Mg3.76Sn7is located in-between the three phases CaMgSn,CaSn and CaSn3.In the present work,we obtained the C p value of this compound from the compositional average of the C p values of these three phases.In the thermodynamic modeling of the Ca e Sn system[7],it has been demonstrated that this compositional averaging procedure works very well for the binary compounds.The resultingfitting curve for the Ca6.24Mg3.76Sn7compound is shown in Fig.6with those of the CaMgSn,CaSn and CaSn3phases.In the present model-ing,we employ this heat capacity;the A and B parameters are based on the phase equilibria of Ca6.24Mg3.76Sn7.Fig.6.C p for Ca6.2Mg3.8Sn7obtained from the compositional averaging pro-cedure from the values of CaSn3,CaMgSn and CaSn phases.305A.Kozlov et al./Intermetallics16(2008)299e315。