Calculation of the thermodynamic properties of the Ga Sb Ti liquid alloys

Calculation of the thermodynamic properties of the Ga Sb Ti liquid alloys合金热力学计算

J.Serb.Chem.Soc.70(1)9–20(2005)

JSCS–3243UDC546.681'.683':86–034:536.7.001.2OriginalscientificpaperCalculationofthethermodynamicpropertiesoftheGa–Sb–Tlliquidalloys

DRAGANMANASIJEVI]*1,DRAGANA IVKOVI]1,IWAOKATAYAMA2and IVAN

IVKOVI]1

1TechnicalFaculty,UniversityofBelgrade,VJ12,19210Bor,SerbiaandMontenegro

(e-mail:dmanasijevic@tf.bor.ac.yu)and2OsakaUniversity,GraduateSchoolofEngineering,Department

ofMaterialsScienceandProcessing,2-1Yamadaoka,Suita,Osaka565-0871,Japan

(Received21April,revised27May2004)

Abstract:Theresultsofthecalculationofthethermodynamicpropertiesforliquid

Ga–Sb–Tlalloysatthetemperature1073Karepresentedinthispaper.Initially,the

mostappropriatethermodynamicmodelfortheinvestigatedsystemwasselected.

Basedonacomparisonofthevaluescalculatedbydifferentgeometricmodels

(Kohler,Muggianu,Toop,Hillert,Chou)withtheexistingexperimentalbaseddata,

asymmetricmodelsofcalculationweredeterminedtogivethebestresults.The

asymmetricnatureoftheinvestigatedternarysystemwasadditionallyconfirmedby

theChousimilaritycoefficientconcept.Forthesereasons,furthercompletethermo-

dynamiccalculationswereperformedaccordingtotheHillertmodelinfivesections

oftheternaryGa–Sb–Tlsystemfromeachcornerwiththemoleratioofothertwo

componentsbeing9:1;7:3;5:5;3:7and1:9.Theobtainedresultsincludeintegral

excessGibbsenergydependencesoncompositionforalltheinvestigatedsections.

Thecalculatedactivityvaluesat1073Kforallcomponentsaregivenintheformof

parisonbetweenthecalculatedandexperimentallyob-

tainedgalliumactivitiesshowsgoodagreement.

Keywords:thermodynamicsofalloys,Ga–Sb–Tlsystem,geometricmodels,inte-

gralexcessGibbsenergy,activities.

INTRODUCTION

TheGa–Sbbinarysystemisasubsystemofvariouscomplexelectronicmate-rials.Forthisreason,manythermodynamicstudieshavebeencarriedoutonalloysystemscontainingtheGaSbsemiconductingcompound.ThepresentstudywasperformedinordertocreateasetofthermodynamicdataforliquidGa–Sb–Tlter-naryalloyswhichmaybeofuseforfurtherassessmentofthissystemandtoinves-tigatethemutualagreementamongtheresultsobtainedbysomeofthemostfre-quentlyusedmethodsforthepredictionofthermodynamicpropertiesofternarysystemsbasedontheknowndatafortheconstitutivebinaries.

Phone/Fax:++38130424547.

9

Calculation of the thermodynamic properties of the Ga Sb Ti liquid alloys合金热力学计算

10MANASIJEVI]etal.

Literaturethermodynamicdatafortheconstitutivebinarysystemsarenumer-ous.DirectcalorimetricmeasurementsoftheGa–Sbbinarysystemhavebeenper-formedbyYazawaetal.,1PredelandStein2andGambinoandBros.3ActivitiesoftheliquidGa–SballoyshavebeenmeasuredbyelectrochemicaltechniquesbyGerasimenkoetal.,4DanilinandYatsenko,5Andersonetal.6andKatayamaetal.,7andvaporpressuremeasurementsbyHsi-Hsiungetal.8andBergmanetal..9

DanilinandYatsenko10usedtheelectromotive-force(EMF)measurementsonfusedsaltstoobtaintheactivityofTlinliquidGa–Tlalloysinthetemperaturerange695–1024Koverthewholecompositionrange.PredelandStein11measuredtheenthalpyofmixingofliquidGa–Tlalloysusinghigh-temperaturecalorimetry,anddeterminedtheentropyofmixingbasedontheresultsofthesolubilityequilib-riaevaluation.Katayamaetal.12performedEMFmeasurementswithzirconiaassolidelectrolyteovertheentirecompositionrangeofliquidGa–Tlalloysinthetemperaturerange973–1273Ktoobtaintheactivityofgallium.

ThermodynamicpropertiesoftheliquidSb–TlalloyswereinvestigatedbyKameda.13BasedontheresultsofEMFmeasurementsinthetemperaturerange953–1230K,negativedeviationsfromRaoult’slawwereobtainedforbothcom-ponentsoverthewholeconcentrationranges.

RecentlyKatayamaetal.14performedEMFmeasurementswithzirconiaassolidelectrolyteintheGa–Sb–Tlsystemandderivedgalliumactivityinthreecrosssectionsfromthegalliumcorner(xSb:xTl=3:1;1:1;1:3)inthetemperaturerange973–1273K.

Asacontributiontoamorecompleteknowledgeofthethermodynamicbe-havioroftheGa–Sb–Tlsystem,theresultsofthecalculationofthermodynamicpropertiesarepresentedinthispaper.TheresultsusingtheHillert,Toop,Kohler,MuggianuandChoumodelswerecomparedtoavailableexperimentaldataandmutuallytoeachother.Aso-obtainedsetofthermodynamicdataforliquidGa–Sb–Tlalloysmaybeusefulforfurtherassessmentofthissystem.

RESULTSANDDISCUSSION

Thereareseveraltraditionalmodelsfromwhichthermodynamicpropertiesofaternarysystemcanbeextrapolatedusingthethreeconstitutivebinarysystemsasbasis.Thesemodelsareclassified,accordingtoHillert,15intotwocategories:sym-metrical(Kohler,16Muggianu17)andasymmetrical(Toop,18Hillert15).

Thebasicanalyticalinterpretationsofthesemodelsaregivenby:

–Kohlermodel16

x1x2öx2x3öEæ2DGEæç÷ç÷DGE=(x1+x2)2DG12+x)+(x;;2323ççx+xx+x÷÷+xxxx++è1è2213ø323ø(1)x1öEæx32+(x3+x1)DG31ççx+x;x+x÷÷è1313ø

Calculation of the thermodynamic properties of the Ga Sb Ti liquid alloys合金热力学计算

Ga–Sb–TlLIQUIDALLOYS11

–Muggianumodel17

DGE=4x1x2Eæ1+x1-x21+x2-x1öDG12;ç÷+(1+x1-x2)(1+x2-x1)22øè

4x2x3Eæ1+x2-x31+x3-x2ö+DG23;ç÷+(1+x2-x3)(1+x3-x2)22øè

4x3x1Eæ1+x3-x11+x1-x3ö+DG31;ç÷(1+x3-x1)(1+x1-x3)22øè

x2xEE(x1;1–x1)+3DG13(x1;1–x1)+DG121-x11-x1

x2x3öEæç+(x2+x3)2DG23;çx+xx+x÷÷è2323ø

–Hillertmodel15

DGE=x2xEE(x1;1–x1)+3DG13(x1;1–x1)DG121-x11-x1

xxE+23DG23+(n23;n32)n23n32(3)(2)–Toopmodel18DGE=(4)

wherenij=(1+xi–xj)/2.

Inallgivenequations,DGEandDGEijcorrespondtotheintegralmolarexcessGibbsenergyforternaryandbinarysystems,respectively,wherex1,x2,x3corre-spondtothemolefractionofcomponentsinaninvestigatedternarysystem.

GraphicalinterpretationofthesemodelsisshowninFig.1.

Theuseofasymmetricalmodelwhenanasymmetricalmodelismoreappro-priatecanoftengiverisetoerrors.Categorizationofaninvestigatedternarysystemintooneortheotherofthesetwocategoriesisinsomecases,especiallywhenade-quateexperimentaldataarelacking,anuncertaintask.Also,adifferentchoiceofthearrangementofthethreecomponentstothethreeapexesofthetriangleinthecaseoftheapplicationofanasymmetricmodelleadstoadifferentresultoftheter-naryGibbsenergyofmixing.Forthesereasons,Chou19proposedanewmodelbasedonthe“similaritycoefficientconcept”,theadvantageofwhichisthatitsap-plicationrequiresneitherthepredeterminationofwhetherasystemissymmetricalornot,northechoiceofthesymmetricandasymmetriccomponentsinaparticularternarysystem.Thecorrectnessofthismodelhasalreadybeenconfirmedinsomepracticalexamples(Ga–Sb–Snsystem20;Ga–Sb–Bisystem21;Ga–In–Tesys-tem22).Therefore,besidesthetraditionalmodels,thismodelisalsoapplicableforthecalculationofthethermodynamicpropertiesoftheGa–Sb–Tlternarysystem.

Calculation of the thermodynamic properties of the Ga Sb Ti liquid alloys合金热力学计算

12MANASIJEVI]etal.Fig.1.

Selectedcompositionsofthethreeconstitutivebinarysystemsforthecalculationofthein-

tegralexcessGibbsenergyofaternaryalloyaccordingtotheKohler,Muggianu,Toopand

Hillertmodel.

ThebasicequationoftheChoumodelisgivenasfollows(indetailseeRef.19):

DGE=x1x2(A012+A112(x1–x2)+A212(x1–x2)2)+x2x3(A023+A123

(x2–x3)+A223(x2–x3)2)+x3x1(A031+A131(x3–x1)+

+A231(x3–x1)2)+fx1x2x3(5)

whereDGEistheintegralexcessGibbsenergyforaternarysystem,x1,x2,x3arethemolefractionsofaternaryalloy,A0ij,A1ij,A2ijareparametersforthebinarysystem“ij”whichcanbetemperaturedependent.

Thefunctionfistheternaryinteractioncoefficientexpressedby:

f=(2x12–1){A212((2x12–1)x3+2(x1–x2))+A112}+(2x23–1)

{A223((2x23–1)x1+2(x2–x3))+A123}+(2x31–1)

{A231((2x31–1)x2+2(x3–x1))+A131}(6)

wherexijarethesimilaritycoefficientsdefinedbynicalledthedeviationsumofsquares:

Calculation of the thermodynamic properties of the Ga Sb Ti liquid alloys合金热力学计算

Ga–Sb–TlLIQUIDALLOYS13

xij=hi/(hi+hj)

where:

EE2hI=ò(DG12)dX1-DG13

EE2hII=ò(DG21)dX2-DG23

1

0011(7)hIII=ò

andE(DG31E2)-DG32dX3(8)

X1(12)=x1+x3x12

X2(23)=x2+x1x23

X3(31)=x3+x2x31(9)

Inordertoinvestigatewhichamongthechosengeometricmodelsgivethemostaccurateresults,experimentallybaseddata14oftheexcessGibbsenergyat1073KforthesectionsfromthegalliumcornerwithmoleratioofxSb:xTlequalto3:1,1:1and1:3werecomparedwiththeresultscalculatedaccordingtothefollow-ingcalculationmodels–Kohler,Muggianu,Toop,HillertandChou.

ThebasicdataforthemodelcalculationswerethevaluesoftheparametersoftheRedlich-KisterequationgivenasDGE=XiXj(å(Xi-Xj)uL(u)ij(T))at

v=0n

1073KfortheconstitutivebinarysystemstakenfromRef.7fortheSb–Gasystem,Ref.12fortheGa–TlsystemandRef.13fortheTl–SbsystemgiveninTableI.TABLEI.TheRedlich–KistercoefficientsfortheSb–Ga7;Ga–Tl12andTl–Sb13binarysystemsat1073K

Sb–Ga

L(0)SbGa

L(1)SbGa

L(2)SbGa–8341.951810.725363.025L(0)GaTlL(1)GaTlL(2)GaTlGa–Tl14210.511501740.5L(0)TlSbL(1)TlSbL(2)TlSbTl–Sb–11227.3–5197.6146.725

Thebinaryregular-solutionparametersofthebinarysystemsSb–Ga,Ga–TlandTl–SbareequaltotheRedlich–KistercoefficientsshownintheTableI,whilerelatedsimilaritycoefficientswerecalculatedaccordingtoEq.(7)andareshownintheTableII.

ThevaluesofDGEwasdoneusingdatafromTablesIandIIandtheresultsforthethreesectionswithmoleratioofSb:Tlequalto3:1,1:1and1:3at1073K,areshowninFig.2,togetherwithliteraturedata14forcomparison.

Calculation of the thermodynamic properties of the Ga Sb Ti liquid alloys合金热力学计算

14MANASIJEVI]etal.

TABLE.II.BinarysolutionparametersandsimilaritycoefficientscalculatedbytheChoumodelfortheGa–Sb–Tlliquidalloysat1073K

A(0)SbGa

–8341.95

A(0)GaTl

14210.5

A(0)TlSb

–11227.3A(1)SbGa1810.725A(1)GaTl1150A(1)TlSb–5197.6A(2)SbGa363.025A(2)GaTl1740.5A(2)TlSb146.725hI517419.8hII17254719hIII22151446xSbGa0.029114xGaTl0.437869xTlSb0.977175

Further,rootmeansquaredeviationanalysiswasappliedontheDGEdataobtainedforthementionedthreesectionsinordertodetermineaccuratelywhichoftheappliedmodelswasthemostadequateoneforliquidGa–Sb–Tlalloys:

St=1/N[S(DGEexp–DGEcalc)2]1/2

where:St–rootmeansquaredeviation,N–thenumberofcountingpoints,DGEexp–literatureresults14forDGEandDGEcalc–calculatedvaluesforDGE.TheresultsofsuchananalysisarepresentedinTableIII,whichindicatesthat:a)theinvesti-gatedsystemGa–Sb–Tlshouldberegardedasanasymmetricsystemandb)theHillertmodelisthemostadequatemodelforitsthermodynamicdescription.a)b)

c)

parisonbetweencalculatedandlit-

eraturedataforDGEofliquidGa–Sb–Tlal-

loysinthesectionswithmoleratioxSb:xTl=

3:1(a);1:1(b)and1:3(c)at1073K.

Calculation of the thermodynamic properties of the Ga Sb Ti liquid alloys合金热力学计算

Ga–Sb–TlLIQUIDALLOYS15

TABLEIII.Therootmeansquaredeviationfromtheexperimentaldataforeachcalculationmodel

Section

xSb:xTl=3:1

xSb:xTl=1:1

x:x=1:3Hillert7072179Toop7674182Chou7078184Muggianu139156198Kohler174158185

TheChousimilaritycoefficientconcept19wasusedasanadditionaltoolintheprocessofselectinganappropriatemethodforthermodynamicprediction.Thisconcept,dependingonsimilaritycoefficientsvaluesforthethreebinariesintheternarysystemconsidered,indicateswhetheranasymmetricalorasymmetricalmodelistobeusedinaspecificcase.InthecaseoftheGa–Sb–Tlsystem,thecal-culatedsimilaritycoefficientsforthethreeconstitutivebinariesarepresentedinTableIIandtheobtainedrelationsbetweentheternaryandbinarycompositionac-cordingtoEq.(9)areshowninFig.3(foratemperatureof1073K).

Fig.3.Selectedbinarycompositions

forthethreebinariesintheternary

systemGa–Sb–Tlaccordingtothe

Choumodelat1073

K.

AnalyzingthecalculatedvaluesofthesimilaritycoefficientsfortheGa–Sb–Tlsys-tem(TableII)itmaybeseenthatonesimilaritycoefficient(xSbGa)isclosetozero,whileoneisclosetounity(xTlSb).TheselectedbinarycompositionsaccordingtotheChoumodelinthiscaseareveryclosetotheselectedbinarycompositionsac-cordingtotheasymmetricHillertandToopmodels(Fig.3).ItcouldbeconcludedthatthisparticularsituationreducestheChoumodeltoanasymmetricmodelverysimilartotheTooporHillertmodelwhenantimonyischosenascomponent"1"(Eqs.(3)and(4)).

TakingintoaccountTableIIIandFig.3,itisobviousthatinthecaseoftheGa–Sb–Tlsystem,theresultsofasymmetricalmodelsaremoreprecisethanthesymmetricmethods,whiletheresultsoftheChoumodelresultsarepositionedveryclosetotheresultsoftheasymmetricmodelsforthisparticularternarysys-tem.Hence,thegraphicalrepresentationgiveninFig.3confirmsasymmetricna-

Calculation of the thermodynamic properties of the Ga Sb Ti liquid alloys合金热力学计算

16MANASIJEVI]etal.

tureoftheinvestigatedsystemandfinallyindicatestheapplicationoftheHillertmodelasthemostadequateinthethermodynamicinvestigationoftheGa–Sb–Tlliquidalloys.Forthisreason,allfurtherdetailedthermodynamicanalysiswasper-formedusingtheHillertmodelwithantimonyascomponent"1"andgalliumandthalliumascomponents"2"and"3",respectively.

FurtherthermodynamiccalculationsoftheinvestigatedternarysystemGa–Sb–TlwerepreformedaccordingtoEq.(4)infivesectionsfromanycornerwithmolera-tiosofothertwocomponentsequalto9:1;7:3;1:1;3:7and1:9atatemperature1073K.ThevaluesoftheintegralmolarGibbsexcessenergiesforthechosensec-tions,calculatedbytheHillertmodel,arepresentedinTableIV.

a)b)

c)

Fig.4.Isoactivitycurvesforgallium(a),anti-

mony(b)andthallium(c)at1073K.

UsingthecalculatedvaluesfortheintegralexcessGibbsenergiesasthestart-ingdataandtheknownrelationbetweenintegralandpartialexcessGibbsenergyEæöad(DG)iE+(1–x)÷givenbythefollowingexpression:GEi=RTlnç,the=DGiçx÷dxèiøi

activitiesofallthreecomponentsintheinvestigatedsectionsweredetermined.Theisoactivitycurvesforgallium,antimonyandthalliumat1073KaregraphicallypresentedinFig.4.

ThecalculatedisoactivitycurvesforgalliumandantimonyareverysimilartothoseshowninourpreviouspaperabouttheGa–Sb–Pbliquidalloys.23Within-

Calculation of the thermodynamic properties of the Ga Sb Ti liquid alloys合金热力学计算

Ga–Sb–TlLIQUIDALLOYS17

TABLEIV.IntegralmolarexcessGibbsenergies,DGE(inJ/mol),forthechosensectionsintheGa–Sb–Tlsystemat1073KaccordingtotheHillertmodel

a)Antimonycornerxx:x=9:1

01422

0.1205

0.2–697

0.3–1315

0.4–1678

0.5–1812

0.6–1745

0.7–1503

0.8–1112

0.9–599

xx:x=9:1

0–632

0.1–1031

0.2–1360

0.3–1606

0.4–1755

0.5–1798

0.6–1720

0.7–1512

0.8–1160

0.9–654

c)Thalliumcorner

x0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9x:x=9:1–610–1038–1488–1911–2259–2482–2530–2354–1904–1132DGE/Jmol-1x:x=7:3x:x=1:13139361015031793253411–645–576–1226–1212–1524–1538–1575–1596–1413–1429–1071–1078–586–585DGE/Jmol-1x:x=7:3x:x=1:1–1917–2803–1786–2125–1639–1518–1475–985–1297–534–1104–173–89893–681255–452305–213237DGE/Jmol-1x:x=7:3x:x=1:1–1589–2075–1558–1577–1578–1194–1620–904–1652–690–1646–533–1570–414–1395–316–1090–220–626–108x:x=3:729441159–163–1070–1612–1839–1802–1549–1130–595x:x=1:91251–323–1416–2089–2403–2415–2185–1770–1228–617x:x=3:7–2790–1685–741376401060128513081116701x:x=1:9–1380–959491763233526562712249319851178x:x=3:7–1893–968–244296666880950889710424x:x=1:9–8714041363204224552615253422251700972

Calculation of the thermodynamic properties of the Ga Sb Ti liquid alloys合金热力学计算

18MANASIJEVI]etal.

creasingcontentofantimonyinGa–Sb–Tlalloys,boththegalliumandthalliumactivitydecreasestrongly.Ontheotherhand,replacementbetweengalliumandthalliumhasonlyasmalleffectontheantimonyactivity.

ThecalculatedactivitiesofgalliumfromthisworkaregraphicallycomparedwiththeexperimentallyobtainedgalliumactivitiesinFig.5.

TheresultspresentedinFig.5showthatthereisgoodagreementbetweenex-perimentallyobtainedandcalculatedactivitiesofgalliuminthexSb:xTl=3:1and1:1sectionswhileforthesectionwithamolefractionsratioxSb:xTl=1:3,theagreement

isnotsogood.

parisonoftheexperimentally

obtainedgalliumactivityandthecalculated

valuesintheGa–Sb–Tlsystemat1073K.

TheactivityofgalliumshowsslightnegativedeviationsfromtheRaoultlawinthesectionwithxSb:xTl=3:1,moderatelypositivedeviationsinxSb:xTl=1:1sectionandlargepositivedeviationsfromtheidealthermodynamicbehaviorinthesectionwithxSb:xTl=1:3.

CONCLUSION

ThethermodynamicpropertiesoftheliquidGa–Sb–Tlsystemwereinvesti-gatedusingdifferentmodelsforthermodynamicpredictions.

ComparisonofthepredictedintegralexcessGibbsenergieswiththeexperi-mentallybasedresultsindicatedtheHillertmodeltobethemostaccuratemodelforthethermodynamicdescriptionoftheGa–Sb–Tlsystem.Theasymmetricna-tureoftheinvestigatedternarysystemwasadditionallyconfirmedbytheChousimilaritycoefficientconcept.FurtherthermodynamiccalculationusingtheHillertmodelwasdoneinfivesectionsoftheternaryGa–Sb–Tlsystemfromeachcornerwithmoleratiosofothertwocomponentsequalto9:1;7:3;5:5;3:7and1:positionforalltheinvestigatedcrosssectionsandisoactivitydiagramsforallcomponentsoftheGa–Sb–parisonbetweenthecalculatedandexperi-mentallyobtainedgalliumactivitiesshowsgoodagreementinthesectionwithxSb:xTl=3:1and1:1andslightdisagreementinthesectionwithxSb:xTl=1:3.

Calculation of the thermodynamic properties of the Ga Sb Ti liquid alloys合金热力学计算

Ga–Sb–TlLIQUIDALLOYS19

Theresultspresentedinthispaperareacontributiontoamorecompletether-modynamicdescriptionoftheGa–Sb–Tlsystemandmaybeusefulforthefurtherthermodynamicassessmentofthissystem.

IZVOD

PRORA^UNTERMODINAMI^KIHOSOBINATE^NIHGa–Sb–TlLEGURADRAGANMANASIJEVI]1,DRAGANA@IVKOVI]1,IWAOKATAYAMA2i@IVAN@IVKOVI]11Tehni~kifakultet,UniverzitetuBeogradu,VJ12,19210Bor,SrbijaiCrnaGorai2OsakaUniversity,GraduateSchoolofEngineering,DepartmentofMaterialsScienceandProcessing,2-1Yamadaoka,Suita,Osaka565-0871,

Japan

Uovomradujeizlo`enprora~untermodinami~kihveli~inaute~nimGa–Sb–Tlleguramana1073K.Naosnovupore|ewaprora~unatihintegralnihekscesnihGibsovihenergijaporazli~itimgeometrijskimmodelimatermodinami~kogpredvi|awa(Kohler,Muggianu,Toop,Hillert,Chou)sarezultatimabaziranimnaeksperimentalnimpodacimautvr|enojedaasimetri~nimodelidajuta~nijerezultate.Asimetri~nostispitivanogsistemajedodatnopotvr|enaChou-ovimkonceptomkoeficijenatasli~nosti.Izovihrazlogapotpuniprora~untermodinami~kihkarakteristikauternarnomGa–Sb–Tlsi-stemujeizvr{enHillert-ovimmodelom.RezultatiobuhvatajuzavisnostiintegralnihekscesnihGibsovihenergijaodsastavazapetnaestispitivanihpresekakaoidijagra-meizoaktivnihlinijazasvetrikomponenteispitivanogsistemana1073K.Dobijenirezultatipokazujudobroslagawesapostoje}imeksperimentalnimrezultatima.

(Primqeno21.aprila,revidirano27.maja2004)

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