CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 550–556

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CALPHAD: Computer Coupling of Phase Diagrams andThermochemistry

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Thermodynamic modeling of the Si–Sr systemAndres Garay a, Gerardo Trápaga a, Zi-Kui Liu b, Raymundo Arróyave c,∗a CINVESTAV-Unidad Querétaro, Querétaro, Mexicob The Pennsylvania State University, University Park, PA, United Statesc Texas A&M University, College Station, TX, United States

a r t i c l e i n f o

Article history:Received 15 December 2008Received in revised form28 May 2009Accepted 8 July 2009Available online 21 July 2009

Keywords:Thermodynamic modelingIntermetallicsFirst-principles calculationsCALPHAD

a b s t r a c t

Thermodynamicmodeling of the Si–Sr systemwas carried out using the CALPHADmethod. In this system,three of the intermetallic compounds, Sr2Si, Sr5Si3, SrSi, were considered to be stoichiometric and SrSi2was modeled as a non-stoichiometric phase. For the liquid phase, two different models were used. In onecase, Sr and Siwere assumed to constitute a random solution,with its excess Gibbs energy being describedthrough the Redlich–Kister polynomial. In the second case, the liquid phase was described through theassociate model, considering the species Si, Sr and SrSi. The terminal solid solutions, (βSr), (αSr) and (Si),are treated with negligible solubilities. A set of self-consistent thermodynamic parameters is obtainedby considering experimental phase equilibria, measurements of the mixing enthalpy in the liquid phase,as well as measured and calculated – via first-principles methods based on density functional theory –formation enthalpies of the intermetallic compounds.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Strontium is becoming an important modifier of Al–Si alloys,being increasingly preferred over sodium and antimony since it iseasier to handle and has less detrimental environmental effects.Additionally, strontium is easy to add to aluminum-based alloysas this system presents several intermetallic phases, facilitatingthe fabrication of master alloys [1]. When Sr is added in smallquantities to Al–Si alloys, their eutectic structure changes fromthick flakes to fine interconnected fibers, consequently improvingtheir mechanical properties [2,3]. The modifying effect on theAl–Si eutectic microstructures was firstly studied for the case ofNa by Edwards and Archer [4], while Lu and Hellawell [5] andNajafabadi et al. [6] studied Al–Si eutectic modification by Sr. It isgenerally accepted that segregation of modifiers to the interfacebetween the liquid and Si crystallites in the eutectic phase leadsto a modification of the growth pattern of the Si phase, from non-faceted to faceted. The change in solidification pattern results in amuch finer solidification structure with better and more uniformmechanical properties.In order to gain a better understanding of the modifying ef-

fects of Sr on Al–Si alloys, it is important to study the thermo-dynamic behavior of the Al–Si–Sr ternary system. It is thereforenecessary to have a self-consistent thermodynamic description for

∗ Corresponding address: ENPH 199, Mail Stop 3123, Texas A&M University,College Station, TX 77843-3123, United States. Tel.: +1 979 845 5416.E-mail address: rarroyave@tamu.edu (R. Arróyave).

0364-5916/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.calphad.2009.07.001

the Si–Sr system. In this work, the thermodynamic model for theSi–Sr system was obtained through the CALPHAD approach [7].This method consists of the description of the Gibbs energies of allthe phases that are likely to take part in equilibria through empiri-calmodelswhose parameters are evaluated based on experimental– and more recently first-principles – information. The thermody-namicmodels can then be used to calculate the stable (metastable)state of the system and allow the extrapolation to regions of thecomposition-temperature space not accessed/accessible throughexperiments.In this work, the available experimental data of the Si–Sr

system is critically assessed and used to evaluate the modelparameters defining theGibbs energies of the phases in the system.To supplement the experimental information, first-principlescalculations of the formation enthalpies of the intermetalliccompounds in this binary system are also included. A consistentdescription of the Al–Si–Sr system can be obtained by combiningthe description form this work with existing assessments of theAl–Sr [8,9] and Al–Si [10] binaries which will allow a betterunderstanding of the thermodynamic factors behind the structuremodification effects of Sr on Al–Si eutectics.

2. Experimental data

The experimental work on the Si–Sr system is rather scarce. Infact, it was not until the recent work by Palenzona and Pani [11]that a more complete set of intermetallic compounds formingin the central regions of the phase diagram was determined.Furthermore, the different experimental investigations of the

A. Garay et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 550–556 551

phase equilibria in this system at times contradict each other. Thevalidity of mutually excluding experimental results can be sortedout through the CALPHAD method since its application permitsthe identification of the most thermodynamically consistentexperimental data sets. In this section, the existing experimentaldata for this system will be reviewed.

2.1. Phase diagram data

The Si–Sr system has several intermetallic compounds [12],with no significant solubility detected. Until recently, the acceptedSi–Sr phase diagram [12] was based primarily on the experimentalinvestigations by Obinata et al. [13]. In that work, the authors onlyused nine alloy compositions to investigate the full Si–Sr phase dia-gram and only three intermetallic compounds – Sr2Si (PbCl2-type),SrSi (CrB-type) and SrSi2 (cubic SrSi2-type) – were identified. Ina later work, Alcock and Itkin [14] assessed the Si–Sr phase dia-gram, including the Sr2Si intermetallic phase. In the work reportedby Massalski et al. [12] the possible existence of other phases wasmentioned, although the existing evidence at the timewas not suf-ficient to determine whether these phases were metastable or ifthey really belonged to the equilibrium Si–Sr phase diagram.Recently, Palenzona and Pani [11] investigated the Si–Sr system

over the entire composition-temperature space using x-ray diffrac-tion, metallography and differential thermal analysis (DTA). Thiscomprehensive study resulted in more complete liquidus and im-portant information regarding the identity and crystal structures ofthe stable intermetallic phases in the system. In the case of Sr2Si, itwas confirmed that it has the orthorhombic PbCl2-type structure.Sr5Si3 was confirmed to be of the tetragonal Cr5B3-type and wasfound tomelt congruently at 1150 ◦C, just below themelting pointof SrSi (1195 ◦C) and above that of the β-SrSi2 phase (1120 ◦C). Theequiatomic compound SrSi was found to have the orthorhombicCrB-type, confirming previous reports [13]. At the 66.66% compo-sition, Palenzona and Pani found two different polymorphs: a hightemperature tetragonal α-ThSi2-type phase β-SrSi2, which meltscongruently at ∼1100 ◦C and a cubic structure α-SrSi2, stable atlower temperatures. It is important to note that in other investiga-tions, it had been suggested that the tetragonal isomorph becamestabilized only at high pressures [15]. Palenzona and Pani foundinstead that this compound is stable and can even be quencheddown to room temperature. The β → α transition temperaturecould not be located through DTA analysis, suggesting that the en-thalpy differences between these two structures must be rathersmall. In light of this, Palenzona and Pani did not assign a trans-formation temperature in their assessed phase diagram, but rathersuggested a range of temperatures within which this transforma-tion was likely to occur. Further analysis of the structural parame-ters of the β-SrSi2 phase indicated that it has a solution range from62% to 66.7% Si.

2.2. Thermo-chemical data

While experimental data on the phase diagram of Si–Sr isscarce, this problem is significantly greater for the case of thethermochemical data for this system. For the liquid phase, the onlyreliable data set is the one obtained by Esin and Kolesnikov [16],who measured the thermochemical properties of liquid Si–Sralloys at 1493 ◦C in a high-temperature vacuum calorimeter underan excess pressure of∼0.5 atm. Through these calorimetry studies,Esin and Kolesnikov determined the concentration dependencesof partial and the integral enthalpies of mixing. Values of liquidintegral enthalpies of mixing in the range from 0 to 50 at.% (Sr)were used for the present assessment and these values wereinterpolated in steps of 5 at.%.

Until recently, the only data on the formation enthalpies ofthe Si–Sr intermetallic compounds corresponded to the empiricalcalculations by Gordienko [17] and the first-principles calculationsby Imai andWatanabe [18] and Brutti et al. [19]. Balducci et al. [20]recently reported the formation enthalpies of several Si–Srintermetallics obtained through vapor pressure measurements.These calculated and experimental formation enthalpies arecombined with calculations performed by the present authors andused in the assessment of the model parameters for the Si–Srintermetallic compounds.

3. First-principles methodology

The existing thermochemical information on the intermetalliccompounds available in the literature is rather uncertain, as differ-ences between estimated, measured and calculated are consider-able. Therefore, first-principles calculations were employed to tryto elucidate the validity of the data available. In particular, this ap-proach allowed the characterization of the thermochemical prop-erties of the phase Sr5Si3, which has not reported in the literature,except for recent measurements reported by Balducci et al. [20].The DFT first-principles calculations were carried out using

VASP [21,22]. Both GGA and LDA approximations were used,with Projector Augmented-Wave (PAW) pseudopotentials [22,23].For the LDA approximation, the exchange correlation functionalproposed by Ceperley and Alder [24] was used. For the GGAexchange-correlation functional, we employ the Perdew–Wangparameterization (PW91) [25]. For Si, the 3s23p2 electronic stateswere considered as valence states, while for Sr, the valenceconfiguration corresponded to 4s24p65s2. A plane-wave cut-off of350 eV was used to calculate the electronic energies of all thestructures. For all the structures, 10,000 k-points per reciprocalatom were used, arranged according to the relative dimensionseach structure’s lattice parameters. The Monkhorst–Pack schemewas used for the Brillouin-zone integrations [26]. The energy cutoffand k-point density selected was sufficient to obtain electronicenergies converged within a few meV (∼0.1 kJ) for all structuresconsidered. For each of the structures, full optimization of theirvolume, lattice parameters and ionic positions were carried outuntil residual interatomic forceswere deemed to be negligible (lessthan 0.001 eV/A2). In some cases, this constraintmade it necessaryto perform the relaxation more than once. After the structureswere fully relaxed, a self-consistent static calculation (using thetetrahedron method with Blöchl corrections) was performed.

4. Thermodynamic modeling

In the present work, we propose two different types of modelsfor the Si–Sr system. The first model assumes that the liquid phasebehaves as a random solution and the excess Gibbs energy termsare described in terms of the Redlich–Kister polynomial [27]. Thesecond (associate) model considers short-range order in the liquidphase by assuming the presence of short-lived associates withwell defined stoichiometries [7]. This kind of behavior is usuallyexpected in liquids inwhich the constituents interact (attractively)strongly, which is reflected in turn by highly negative enthalpies ofmixing, accompanied by relatively high melting points for some ofthe intermetallic compounds.

4.1. Random solutions and compounds

In this work, the solubilities of the terminal solid phases (α-Sr(fcc), β-Sr (bcc) and Si (diamond)) are considered to be negligible.When the liquid phase is described as a random solution, its Gibbsenergy is given by the following equation:

GLm = xSrGLSr + xSiG

LSi + RT (xSr ln xSr + xSi ln xSi)+

exGLm (1)

552 A. Garay et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 550–556

where GLSr and GLSi are the Gibbs energies for pure Sr and Si in the

liquid phase which are taken from SGTE database [28]. The exGLmterm corresponds to the excess Gibbs energy of liquid describedby the equation:

exGLm = xSrxSi∑k=0

kLSr,Si (xSr − xSi)k (2)

where kLSr,Si is the kth binary interaction parameter with lineartemperature dependence, kLSr,Si = kA + kBT with A and B beingmodel parameters to be evaluated.The Gibbs energy functions of stoichiometric intermetallic

compounds, i.e. SraSib, are written as:

GSraSibm = a0GfccSr + b0Gdiamond_A4Si + ASraSib + BSraSibT (3)

where GfccSr and Gdiamond_A4Si are the Gibbs energies of the fcc Sr

and diamond Si, respectively, with A and B corresponding to theformation enthalpy and entropy of the compound.

4.2. Associate model

The Si–Sr presents very strong attractive interactions, asindicated by the highly negative mixing enthalpy in the liquidphase as well as by the relatively stable (high melting points)intermetallic compounds, especially in the central regions of thesystem. Under these conditions, it is likely that the system willexhibit short-range ordering. This behavior can be modeled ina simple way by assuming the presence of associate with well-defined stoichiometries. In principle, any number of associateswith different stoichiometries can be assumed. To constrain theparameter space of the thermodynamic optimization, it wasassumed that the dominant cluster corresponded to the SrSistoichiometry. This assumption is based on the fact that themixingenthalpy in the liquid phase has its minimum at this compositionand that the compound with the highest (congruent) meltingtemperature is SrSi.The description of the liquid phase includes the species Sr, Si

and SrSi and can be written as:

GLm =∑

i=Sr,Si,SrSi

yLi0GLi + RT

∑i=Sr,Si,SrSi

yLi ln(yLi)+exGLm (4)

where yLi corresponds to the mole fraction of constituent i in theliquid phase and 0GLi corresponds to the Gibbs energy of the pureconstituent in the liquid state.The Gibbs energy of the SrSi associate is given by:

0GLSrSi = ∆0f HLSrSi − T∆

0f SLSrSi +

0GLSr+0 GLSi. (5)

Due to the associate’s stoichiometry, the excess Gibbs energy of theliquid phase can be described by the interactions between Sr andSrSi and between Si and SrSi, resulting in the following expression:

exGLm = ySrySrSiLLSr,SrSi(y

LSrSi − y

LSr)+ ySiySrSiL

LSi,SrSi(y

LSrSi − y

LSi). (6)

4.3. The SrSi2 phase

According to the work by Palenzona and Pani [11], the SrSi2phase presents a significant degree of non-stoichiometry atelevated temperatures, when the tetragonal phase becomesstable. This non-stoichiometry has been verified through chemicalanalysis as well as lattice parameter measurements throughXRD. Non-stoichiometry in this high temperature phase could,in principle, be modeled through either vacancies or anti-siteslocated at the crystallographic positions occupied by either Sr or Si.Palenzona and Pani postulated the dominant point defect inducingthe non-stoichiometry to be vacancies in the Si sublattice, This was

confirmed by preliminary calculations of the likely defect structureof this intermetallic, to be published elsewhere [29]. Therefore,the high temperature form of the SrSi2 phase was modeledby considering mixing of Si and vacancies in the Si sublattice,corresponding to the (Sr)(Si,Va)2 sublattice description. For thisphase, the Gibbs energy is given by:

GSrSi2m = yVa0GSrSi2Sr:Va + ySi

0GSrSi2Sr:Si

+ 2RT [ySi ln (ySi)+ yVa ln (yVa)]+ yVaySiLSrSi2Sr:Si,Va (7)

where oGSrSi2Sr:Va corresponds to the Gibbs energy of formation ofa hypothetical Sr phase with the underlying structure of thetetragonal strontium disilicide, ySi and yVa correspond to the sitefractions of Si and Va in the silicon sublattice. LSrSi2Sr:Si,Va is theinteraction parameter between Si and vacancies in the siliconsublattice and can be expressed as LSrSi2Sr:Si,Va = A + BT with A and Brepresenting the enthalpic and entropic contributions to the excessGibbs energy.

4.4. Evaluation of the model parameters

The evaluation was carried out by means of the PARROTmodule of Thermo-Calc software [30]. First, the mixing enthalpyin the liquid and the liquidus in the Sr-rich corner were used toevaluate the model parameters of the liquid phase. Afterwards,the parameters for the stoichiometric compoundswere optimized,using data for their formation enthalpies [17,19,20,31], as wellexperimental measurements of phase equilibria with the liquidphase and invariant reactions [11,14]. The final values of thethermodynamic parameters were obtained with the appropriateweight factors for all selected data, which were given according tothe importance of the reactions and the published error margin ofthe data. The liquid enthalpies of mixing and compound formationenthalpies were given higher weights due to the relatively smalluncertainty in the experimental enthalpymeasurements aswell asthe relatively narrow range in the formation enthalpies calculatedfrom first-principles.

5. Results and discussion

5.1. First-principles calculations

The formation enthalpies were obtained within the LDA andGGA approximations. Table 1 shows the differences between theexperimental and calculated lattice parameters. The data indicatesthat the difference is insignificant for the GGA approximation,while the maximum difference for LDA approximation was 3.3%.For the thermodynamic model, both approximations were takeninto account, assuming that the calculations representedupper andlower bounds for the expected values for the formation enthalpiesfor the intermetallic compounds. The experimental and calculatedformation enthalpies are listed in Table 2.

5.2. Phase diagram

The model parameters are listed in Table 3 (liquid modeledas a random solution) and Table 4 (liquid described throughthe associate model). The phase diagram calculated using theseparameters is compared to the available experimental data [11]in Fig. 1. The calculated liquidus temperatures between 40 at.%and 75 at.% Si differ from those reported by Palenzano andPanni [11]. Given the limited number of experimental data pointsfor the liquidus, we judged that reproducing the data moreaccurately would result in over fitting of the model, which wouldcompromise its transferability to higher-order systems. For the

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Table 1First-principles lattice parameters calculated at 0 K within the GGA and LDA approximations.

Phase Prototype Pearson Space group Lattice parameters (Å)Exp [11] GGA LDA

Sr2Si PbCl2 oP12 Pnma(a) 8.111 8.141 (+0.37%) 7.843 (−3.30%)(b) 5.132 5.134 (+0.04%) 5.027 (−2.05%)(c) 9.545 9.611 (+0.69%) 9.379 (−1.74%)

Sr5Si3 Cr5B3 tI32 I4/mcm (a) 8.100 8.056 (−0.54%) 7.869 (−2.85%)(b) 15.728 15.890 (+1.03%) 15.454 (−1.74%)

SrSi CrB oC8 Cmcm(a) 4.814 4.816 (+0.04%) 4.699 (−2.39%)(b) 11.288 11.39 (+0.90%) 11.154 (−1.19%)(c) 4.040 4.049 (+0.22%) 3.962 (−1.93%)

β-SrSi2 A-ThSi2 tI12 I41/amd(a) 4.429 4.467 (+0.86%) 4.346 (−1.87%)(b) 13.842 13.826 (−0.10%) 13.620 (−1.59%)

α-SrSi2 SrSi2 cP12 P4332 (a) 6.536 6.536 (+0.41%) 6.436 (−1.53%)

Table 2Formation enthalpies (kJ/mol-atom).

Gordienko [17] (calc) Imai and Watanabe [18] First-principles LDA First-principles GGAa Experiments [20]

Sr2Si −66.3 −35.3 −41.3 −37.9 (−37.6) −39.7Sr5Si3 −44.3 −40.3 (−40.2) −43.8SrSi −67 −43.5 −53.1 −46.9 (−46.7) −51.7α-SrSi2 −67.3 −34.6 −42.0 −35.7 (−35.8) −42.7β-SrSi2 −44.7 −35.9 (−35.7) −40.3

Reference states: fcc Sr and diamond cubic Si.a The numbers in parentheses correspond to the calculated formation enthalpies by Brutti [19,31].

Table 3Model parameters assuming that the liquid phase is modeled as a random solution.

Liquid (Sr,Si)0L(Sr, Si) = −180,478+ 14.367 ∗ T1L(Sr, Si) = −11,506− 14.061 ∗ T2L(Sr, Si) = 6,671+ 31.293 ∗ T

Sr2Si GSr2SiSr:Si = 2 ∗ GFccSr + G

Dia_A4Si − 122,330+ 5.482 ∗ T

Sr5Si3 GSr5Si3Sr:Si = 5 ∗ GFccSr + 3 ∗ G

Dia_A4Si − 334,792

SrSi GSrSiSr:Si = GFccSr + G

Dia_A4Si − 93,134

β-SrSi2 GSrSi2Sr:Si = GFCCSr + 2 ∗ G

Dia_A4Si − 114,500

GSrSi2Sr:Va = GFCCSr + 5,000 LSrSi2Sr:Si,Va = −40,000

α-SrSi2 GSrSi2Sr:Si = GFCCSr + 2 ∗ G

Dia_A4Si − 120, 000+ 6.5 ∗ T

case in which the liquid was modeled as a random solution,it was found that the second order Redlich–Kister polynomialwas needed in order to reproduce the published liquidus andinvariant reactions in an acceptable manner. Having only zero andfirst order parameters was found to be insufficient to match theliquidus and invariant reactions simultaneously, especially in thecentral region of the phase diagram. Although additional degreesof freedom could be provided by the parameters correspondingto the formation entropies of the intermetallic compounds, theglobal optimization consistently yielded a very small temperaturedependence (entropy) of the Gibbs energy of formation for most ofthese phases. The low formation entropy of the Sr–Si compoundshas already been confirmed by first-principles calculations tobe published elsewhere [29]. In those calculations, it was foundthat the formation entropy was within +/−2 J/mol-atom/K forall compounds. The negligible formation entropies of the Si–Srintermetallics meant that the liquidus and invariant reactionscould only be matched to the experimental data by using second-order Redlich–Kister polynomial to describe the random liquid’sexcess Gibbs energy.Table 5 shows the calculated invariant reactions comparedwith

those reported in the literature, where a tolerance of 12 ◦C wasaccepted for the experimental uncertainty. For the case in whichthe liquid wasmodeled as a random solution, the temperatures forthe L→ Sr5Si3 + SrSi and L→ SrSi+ SrSi2 eutectic reactions arehigher than the experimental measurements and a displacementof the eutectic composition occurs from 45.0 at.% to 42.5 at.% of

Table 4Model parameters assuming that liquid phase is describedwith the associatemodel.

Liquid (Sr,Si)0GLSrSi = G

LSr + G

LSi − 102,771

0L(SrSi,Sr) = 24,145− 47.8828 ∗ T0L(SrSi,Si) = −55,639

Sr2Si GSr2SiSr:Si = 2 ∗ GFccSr + G

Dia_A4Si − 134,099

Sr5Si3 GSr5Si3Sr:Si = 5 ∗ GFccSr + 3 ∗ G

Dia_A4Si − 390,894

SrSi GSrSiSr:Si = GFccSr + G

Dia_A4Si − 106,000

β-SrSi2 GSrSi2Sr:Si = GFCCSr + 2 ∗ G

Dia_A4Si − 129,000

GSrSi2Sr:Va = GFCCSr + 5,000 LSrSi2Sr:Si,Va = −55,000

α-SrSi2 GSrSi2Sr:Si = GFCCSr + 2 ∗ G

Dia_A4Si − 135,000+ 7 ∗ T

Si in the first reaction and from 63.0 at.% to 59.98 at.% of Si inthe second one. For the case in which the associate model wasused for the liquid phase, only the reaction L → Sr5Si3 + SrSipresented a displacement with respect to the experimental datafrom 45.0 at.% to 41.61 at.% of Si. While the liquidus is betterdescribed – particularly within the 3–38 at.% range – when theassociate model is used, the overall agreement with publishedexperimental data is found to be acceptable for both models.

5.3. Thermodynamic properties

Fig. 2 show that the formation enthalpies in the randomsolution model of liquid lie within the values obtained by LDA andGGA and are very close to the experimental results for the Sr-richregion of the composition range. On the other hand, for the casein which the liquid phase was modeled with the SrSi associate,the formation enthalpies of the Sr2Si and Sr5Si3 are slightly morenegative than even the most stable results obtained from the LDAcalculations in the Sr-rich composition range, while yielding betteragreement with experimental measurements at higher Si contents(i.e SrSi and SrSi2). Considering that the experimental error about±3.5 kJ/mol-atom, both models are considered to be acceptable.The liquid enthalpies of mixing were calculated at 1493 ◦C and

compared with the experimental measurements from Esin andKolesnikov [16] in Fig. 3. The calculations are found to be consistentwith experimental results in the region where experimental

554 A. Garay et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 550–556

Table 5Invariant reactions in the Si–Sr system.

Reaction Alcock and Itkin [14] Palenzona and Pani [11] Associate RandomTemp (◦C) x (Si) Temp (◦C) x (Si) Temp (◦C) x (Si) Temp (◦C)

1 L→ Sr2Si+ (βSr) 700 0.03 745 0.028 737 0.029 751 Eutectic2 L→ Sr5Si3 + SrSi 0.45 1110 0.416 1123 0.425 1125 Eutectic3 L+ Sr5Si3 → Sr2Si 0.28 1080 0.28 1077 0.29 1090 Peritectic4 L→ Sr5Si3 0.375 1150 0.375 1139 0.375 1155 Congruent5 L→ SrSi 1140 0.5 1195 0.5 1183 0.5 1183 Congruent6 L→ SrSi+ SrSi2 1045 0.63 1055 0.595 1057 0.599 1072 Eutectic7 L→ SrSi2 1100 0.667 1120 0.647 1107 0.645 1114 Congruent8 L→ Si+ SrSi2 1000 0.75 1045 0.744 1054 0.75 1044 Eutectic

a

b

Fig. 1. Calculated phase diagramwith the triangles representing the experimentaldata [11]. (a) corresponds to the case in which the liquid was modeled as a randomsolution while (b) corresponds to the case in which the associate model was used.

measurements exist, i.e. 0 to 50 at.% Si. In the figure, it can be seenthat the shape of themixing enthalpy of the liquid becomes sharperclose to the 50 at.% composition due to the presence of the SrSiassociate. However, the reduction in curvature is still consistentwith the published experimental data [16].

5.4. The SrSi2 phase

Asmentioned above, previous experimental investigations [11],as well as calculations [29] performed in our group, suggest that

0

-10

-20

-30

-40

-50

-600.0 0.2 0.4 0.6 0.8 1.0

GGALDARANASOEXP

Fig. 2. Formation enthalpies of Si–Sr intermetallics at 298 K. Comparison betweencalculations by thermodynamic models and first-principles (this work) as well asexperimental data [20]. The calculations by Brutti et al. [19,31] are not included,since they are within 1 kJ/mol-atom of current GGA calculations. Note that first-principles calculationswere done at 0 K as temperature effects arewithin numericaluncertainties. Reference states: fcc Sr and diamond cubic Si.

Fig. 3. Calculated and experimental [16] mixing enthalpy of the liquid phase. Thecalculated mixing enthalpy using the associate model is not symmetrical and has asharper curvature in the central composition range due to the presence of the SrSiassociate. Both models are consistent with the experimental data.

the SrSi2 phase has a high degree of non-stoichiometry due tothe presence of vacancies in the Si sublattice. Fig. 4 shows thecalculated phase diagramconsidering the non-stoichiometry of theSrSi2 phase. The addition of vacancies to the model is detrimentalto the accuracy of the predicted invariant reactions, mainly thereaction L→ SrSi + SrSi2 (see Table 5), where the temperature ishigher than the experimental value, while the congruent melting

A. Garay et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 550–556 555

a

b

Fig. 4. Calculated phase diagram considering the α → β transition in the SrSi2phase and non-stoichiometry in the β-SrSi2 phase. (a) corresponds to the case inwhich the liquid was modeled as a random solution while (b) corresponds to thecase in which the associate model was used.

point of SrSi2 was displaced in composition and temperature. Bothmodels (liquid as a random solution or as an associate) showthat at the SrSi + β-SrSi2 phase equilibrium, the composition ofthe tetragonal phase at 700 ◦C is SrSi1.83, while Palenzona andPani suggest a composition of SrSi1.69. This large discrepancy inthe predicted non-stoichiometry could not be resolved withoutseverely affecting the agreement with the rest of the experimentaland first-principles data used to optimize the model parameters.Although theα-SrSi2 (cubic) toβ-SrSi2 (tetragonal) transforma-

tion appears to lie between 450 and 650 ◦C [11,15], an accurate es-timate for the onset of this transformation has not been attained,perhaps in part due to the fact that the enthalpy difference be-tween these two structures is very small. Recently, first-principlescalculations by Brutti et al. [31] determined that the α-SrSi2 be-longs to the ground state because its energy is slightly more neg-ative that the β-SrSi2. The difference in energies, however, is lessthan 0.1 kJ/mol-atom, which is within the numerical uncertaintiesof the methods used. Our own calculations suggest that β-SrSi2 isslightly more stable (by less than 0.1 kJ/mol-atom) than α-SrSi2,While inconclusive, the calculations suggest that both polymorphsare extremely close in energy, which is consistent with the phasediagram investigations by Palenzona and Pani [11] and the vaporpressure measurements of Balducci et al. [20].

Table 6Reactions involving the α and βSrSi2 phases.

Randommodel Associate modelX (Si) T (◦C) X(Si) T (◦C)

Euctectoid: β → α + SrSi 0.657 480 0.656 475Peritectoid: β + Si→ α 0.666 566 0.666 576

In the model described in Tables 3 and 4 we did not considerthe α → β transformation and assumed that the β phase isstable through the entire temperature range. This assumptionis obviously at odds with other experimental information. Themodels described in Tables 3 and 4 were further modified byincorporating the α-SrSi2 phase, assuming that it belonged to theground state. The formation enthalpy and entropy for this phasewere adjusted to ensure that the α → β transformation occurredwithin the observed experimental ranges. Fig. 4 shows thecalculated phase diagram considering the α → β transformation,where the non-stoichiometric of the SrSi2 phase creates twoinvariant reactions (see Table 6), which incidentally lie within thetemperature range suggested for the α→ β transformation [11].

6. Conclusions

Thermodynamic models of the Si–Sr binary system weredeveloped using two different approaches: one in which the liquidwas described with the associate model and another one in whichit was described as a random solution. It was found that the formercase yielded better agreement with the experimental liquidusand invariant reactions involving the liquid phase. Both modelsreproduce the experimental and calculated thermochemical datain an acceptable manner and, in most cases, within the expecteduncertainties of the experiments/calculations.The homogeneity range of the β-SrSi2 phase was considered

to be due to the presence of Si vacancies in the Si sublattice.This defect model was based on first-principles calculationsand is supported by interpretations of experimental structuralrefinements performed by other researchers. Unfortunately, itwas found that incorporating vacancies to the Si sublattice leadsto significant discrepancies in some of the invariant reactionsinvolving the β-SrSi2 phase when compared with experiments,with the model in which the liquid was considered to be a randomsolution presenting the largest discrepancies.In accordance with experiments and calculations, the cubic α-

SrSi2 phase seems to be the ground state at the 2/3 Si compositionand it therefore must transform to the cubic variant at hightemperatures. When this transformation was incorporated, it wasfound that the cubic α-SrSi2 phase was involved in two invariantreactions: a peritectoid reaction involving β-SrSi2, α-SrSi2 and SiSrand a eutectoid reaction among β-SrSi2, α-SrSi2 and SiSr phases.The temperature interval between these two reactionsmay explainthe observed range in temperatures reported for the α → βtransition of SrSi2.

Acknowledgments

AG acknowledges CONACYT (México) for the support throughscholarship number 216341. ZKL is supported by the NationalScience Foundation (NSF) through grant DMR-0510180. Somefirst-principles calculations were carried out on the LION clustersat the Pennsylvania State University supported in part by theNSF grants (DMR-9983532, DMR-0122638 and DMR-0205232)and in part by the Materials Simulation Center and the GraduateEducation and Research Services at the Pennsylvania StateUniversity. RA and GT would like to acknowledge the supportof the TAMU-CONACyT program for partial support to carry outthis research. First-principles calculations werepartially carried

556 A. Garay et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 550–556

out in the Chemical Engineering Computer Cluster under supportfrom the Department of Mechanical Engineering at Texas A&MUniversity.M.E.Williams is acknowledged for hiswork on the first-principles calculations.

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