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Thermodynamic Assessment of the Ti-Al-Nb, Ti-Al-Cr, and Ti-Al-Mo Systems

Permanent Link: http://ufdc.ufl.edu/UFE0024931/00001

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Title: Thermodynamic Assessment of the Ti-Al-Nb, Ti-Al-Cr, and Ti-Al-Mo Systems
Physical Description: 1 online resource (233 p.)
Language: english
Creator: Cupid, Damian
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: aluminum, assessment, calphad, chromium, molybdenum, niobium, thermodynamics, titanium
Materials Science and Engineering -- Dissertations, Academic -- UF
Genre: Materials Science and Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Two-phase alloys based on a microstructure of disconnected sigma Nb2Al precipitates within a gamma TiAl matrix are promising materials for gas turbine blades because of their expected creep resistance and fracture toughness properties. To optimize alloy microstructures, respective alloys should solidify as single phase beta, and, on aging transform to the two phase microstructures. To extend the high temperature single phase beta field to optimal compositions, beta stabilizers such as Cr and Mo may be used. The CALculation of PHAse Diagrams (CALPHAD) method is a powerful tool that can be used to guide materials design through the application of computational thermodynamics to the calculation of phase diagrams in multi-component systems. Existing thermodynamic descriptions for the Ti-Al-Nb, Ti-Al-Cr, and Ti-Al-Mo systems could not reproduce experimentally determined phase diagrams; therefore, CALPHAD based re-optimizations of the thermodynamic parameters of the phases in the multi-component system descriptions were required. The re-optimized description for the Ti-Al-Nb system calculates the experimentally observed extension of the primary crystallization field of the beta phase, the existence of the single phase beta field at sub-solidus temperatures, and the solid state phase transformations and phase transformation temperatures of two experimentally investigated alloys. Calculations using the new description of the Ti-Al-Cr system are able to reproduce the ternary extension of the Laves phases based on TiCr2, the stoichiometric Tau phase at composition Ti-67 atomic \%Al-8 atomic \%Cr, the critically evaluated liquidus surface, and isothermal sections at 1073 K and 1273 K. The continuity of the beta phase to the Al-Mo binary in the Ti-Al-Mo system could only be reproduced through re-optimization of the thermodynamic parameters for the Al-Mo binary sub-system. The new Al-Mo binary description calculates the congruent melting of the beta phase at 50 atomic %Al, and the new Ti-Al-Mo description is in excellent agreement with the extension of the single phase beta field to the Al-Mo binary and the invariant reaction between beta, delta, Al8Mo3, and eta phases at 1540 K.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Damian Cupid.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Ebrahimi, Fereshteh.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024931:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024931/00001

Material Information

Title: Thermodynamic Assessment of the Ti-Al-Nb, Ti-Al-Cr, and Ti-Al-Mo Systems
Physical Description: 1 online resource (233 p.)
Language: english
Creator: Cupid, Damian
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: aluminum, assessment, calphad, chromium, molybdenum, niobium, thermodynamics, titanium
Materials Science and Engineering -- Dissertations, Academic -- UF
Genre: Materials Science and Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Two-phase alloys based on a microstructure of disconnected sigma Nb2Al precipitates within a gamma TiAl matrix are promising materials for gas turbine blades because of their expected creep resistance and fracture toughness properties. To optimize alloy microstructures, respective alloys should solidify as single phase beta, and, on aging transform to the two phase microstructures. To extend the high temperature single phase beta field to optimal compositions, beta stabilizers such as Cr and Mo may be used. The CALculation of PHAse Diagrams (CALPHAD) method is a powerful tool that can be used to guide materials design through the application of computational thermodynamics to the calculation of phase diagrams in multi-component systems. Existing thermodynamic descriptions for the Ti-Al-Nb, Ti-Al-Cr, and Ti-Al-Mo systems could not reproduce experimentally determined phase diagrams; therefore, CALPHAD based re-optimizations of the thermodynamic parameters of the phases in the multi-component system descriptions were required. The re-optimized description for the Ti-Al-Nb system calculates the experimentally observed extension of the primary crystallization field of the beta phase, the existence of the single phase beta field at sub-solidus temperatures, and the solid state phase transformations and phase transformation temperatures of two experimentally investigated alloys. Calculations using the new description of the Ti-Al-Cr system are able to reproduce the ternary extension of the Laves phases based on TiCr2, the stoichiometric Tau phase at composition Ti-67 atomic \%Al-8 atomic \%Cr, the critically evaluated liquidus surface, and isothermal sections at 1073 K and 1273 K. The continuity of the beta phase to the Al-Mo binary in the Ti-Al-Mo system could only be reproduced through re-optimization of the thermodynamic parameters for the Al-Mo binary sub-system. The new Al-Mo binary description calculates the congruent melting of the beta phase at 50 atomic %Al, and the new Ti-Al-Mo description is in excellent agreement with the extension of the single phase beta field to the Al-Mo binary and the invariant reaction between beta, delta, Al8Mo3, and eta phases at 1540 K.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Damian Cupid.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Ebrahimi, Fereshteh.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024931:00001


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Iwouldrstliketoacknowledgemyadvisors,Prof.Hans-JurgenSeifertandProf.FereshtehEbrahimi.Withouttheirhelp,advice,andencouragement,thisworkwouldnotbepossible.AspecialthanksisextendedtoDr.OlgaFabrichnaya,whohastaughtmeeverythingIknowaboutthermodynamicoptimization.Ithankalsomypastandpresentcommitteemembers:Prof.Phillpot,Prof.Sigmund,Prof.Sinnott,andProf.Lear.IacknowledgeallmembersoftheresearchgroupinFlorida:OrlandoRios,MikeKessler,andSonalikaGoyel.Theirexperimentalworkwastremendouslyhelpful.Ialsoacknowledgetheundergraduatestudents:JonahKlemm-Toole,DanielHeinz,andTabeaWilk,whoparticipatedatvariousstagesinthisproject.MembersoftheresearchgroupatFreibergUniversityofMiningandTechnologyalsorequirespecialacknowledgement:FrauGalinaSavinykhandMarioKriegelandDmytroPavlyuchkov.MarioKriegeldeservesspecialattentionasheworkedintensivelyontheoptimizationoftheTi{Al{Crsystem.IamgratefultomyfriendsinFloridaandinGermany.TheyhavemademystayonbothsidesoftheAtlanticmemorable.Last,Iwouldliketodeeplyacknowledgemyparents.Withouttheirsupportalongeverystepoftheway,thiswouldbeimpossible.ThisworkwassupportedbytheUniversityofFloridaCollegeofEngineeringAlumniFellowshipandbytheNSF/AFOSRundergrantnumberDMR-0605702. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 10 ABSTRACT ........................................ 19 CHAPTER 1INTRODUCTION .................................. 21 2BACKGROUNDANDTHEORY .......................... 26 2.1ThermodynamicModeling ........................... 27 2.1.1ModelingofthePureElements ..................... 28 2.1.2ModelingofStoichiometricPhases ................... 31 2.1.3ModelingofSubstitutionSolutions ................... 31 2.1.3.1Extrapolationmethods .................... 33 2.1.4ModelingofOrderedPhases{TheCompoundEnergyFormalism 35 2.1.5PhaseswithOrder{DisorderTransformations ............. 37 2.2ThermodynamicOptimization ......................... 38 2.2.1ExperimentalData ........................... 39 2.2.2TheoreticalData ............................ 39 2.2.3GibbsFreeEnergyMinimization .................... 40 2.2.4OptimizationMethod:TheLeastSquaresMethod .......... 41 3REVIEWOFTHETi{Al{NbSYSTEM ...................... 48 3.1PhasesintheTi{Al{NbSystem ........................ 48 3.2ExperimentallyDeterminedPhaseEquilibriaintheTi{Al{NbSystem ... 50 3.2.1TheLiquidusSurface .......................... 50 3.2.2IsothermalSections ........................... 51 3.2.3VerticalSections ............................. 52 3.3ThermodynamicDescriptionsoftheTi{Al{NbSystem ........... 52 3.3.1ThermodynamicDescriptionofKattnerandBoettinger ....... 53 3.3.2ThermodynamicDescriptionofServantandAnsara ......... 54 3.3.2.1Reasonsforre-optimization ................. 54 3.3.3ThermodynamicDescriptionofWitusiewiczetal. .......... 56 3.3.3.1ThermodynamicdescriptionoftheTi{Alsystem ..... 56 3.3.3.2ThermodynamicdescriptionoftheAl{Nbsystem ..... 58 3.3.3.3ThermodynamicDescriptionoftheTi{Al{Nbsystem ... 59 5

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.............. 76 4.1TieLineData .................................. 76 4.2Alloy11 ..................................... 77 4.2.1DTAPeakAnalysisforAlloy11 .................... 78 4.3Alloy12 ..................................... 80 4.3.1DTAPeakAnalysisforAlloy12 .................... 80 5RE-OPTIMIZATIONOFTHETi{Al{NbSYSTEM ................ 97 5.1UnaryDataandBinarySub-Sections ..................... 97 5.2ThermodynamicOptimization ......................... 97 5.2.1OptimizationStrategy ......................... 97 5.3SelectionofThermodynamicParameters ................... 99 5.3.1TheSigmaphase ............................ 99 5.3.2TheDeltaPhase ............................. 100 5.3.3TheBetaPhase ............................. 101 5.3.4TheGammaPhase ........................... 102 5.3.5TheOrderedBetaPhase ........................ 103 5.3.6TheDisorderedAlphaandOrderedAlpha{2Phases ......... 104 5.3.7TheLiquidPhase ............................ 105 6RESULTSOFTHERE-OPTIMIZATION ..................... 112 6.1LiquidusandSolidusProjections ....................... 112 6.2IsothermalSections ............................... 112 6.3VerticalSections ................................ 113 6.4PhaseFractionDiagrams ............................ 114 7THETi-CrSYSTEM ................................. 125 7.1PhaseEquilibriaintheTi{CrSystem ..................... 125 7.2ThermodynamicDescriptionsoftheTi{CrSystem .............. 127 7.2.1LatticeStabilityofPureCr ...................... 127 7.2.2NumberofLavesPhasesModeled ................... 128 7.2.3ModelingoftheLavesPhases ..................... 128 7.2.4TheHomogeneityRangesoftheLavesPhases ............ 129 7.2.5TheActivityofCrintheBetaPhase ................. 130 7.3OptimizationoftheParametersfortheBinaryTi{CrSystem ........ 131 7.3.1SelectionofThermodynamicParametersfortheLavesPhases ... 132 7.3.2OptimizationStrategy ......................... 134 7.3.3OptimizedResults ............................ 135 8THETi-Al-CrSYSTEM ............................... 145 8.1PhasesintheTi{Al{CrSystem ........................ 145 8.2ExperimentallyDeterminedPhaseEquilibria ................. 146 8.2.1IsothermalSections ........................... 146 6

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.......................... 147 8.3CriticalAssessmentsoftheTi{Al{CrSystem ................. 147 8.4ThermodynamicDescriptionsoftheTi{Al{CrSystem ............ 148 8.4.1ThermodynamicDescriptionofSaunders ............... 148 8.4.1.1Reasonsforre-optimization ................. 150 8.5Re-optimizationoftheTi{Al{CrSystem ................... 151 8.5.1OptimizationStrategy ......................... 151 8.5.1.1Thealpha2phase ....................... 151 8.5.1.2TheternaryTi(Al,Cr)2lavesphase ............. 152 8.5.1.3Theternarytauphase .................... 153 8.5.1.4Thebetaphase ........................ 154 8.5.1.5Theorderedbetaphase ................... 154 8.5.1.6TheTiCr2lavesphases .................... 155 8.6ResultsoftheRe-optimization ......................... 155 9REVIEWOFTHETI-AL-MOSYSTEM ...................... 164 9.1BinarySubsystems ............................... 164 9.2PhasesintheTi{Al{MoSystem ........................ 165 9.3ReviewofCriticalAssessmentsintheRegionfrom0to20at.%Ti ..... 167 9.4ThermodynamicDescriptionsoftheTi{Al{MoSystem ........... 170 10RE-OPTIMIZATIONOFTHEAL-MOANDTI-AL-MOSYSTEMS ...... 181 10.1ReviewoftheAl{MoSystem ......................... 181 10.2ThermodynamicDescriptionsfortheAl{MoSystem ............. 186 10.3Re-optimizationoftheAl{MoSystem ..................... 186 10.3.1OptimizationStrategy ......................... 187 10.3.2SelectionofAdjustableParameters .................. 187 10.3.2.1Thebetaphase ........................ 187 10.3.2.2Theliquidphase ....................... 188 10.3.2.3TheAl-richintermetallicphases ............... 188 10.3.2.4TheAlMo3phase ....................... 189 10.3.2.5The(Al)phase ........................ 190 10.3.3TheResults ............................... 190 10.4Re-optimizationoftheTi{Al{MoSystem ................... 191 10.4.1OptimizationStrategy ......................... 191 10.4.2SelectionofParameters ......................... 192 10.4.2.1Theetaphase ......................... 192 10.4.2.2Thebetaphase ........................ 193 10.4.2.3Thedeltaphase ........................ 193 10.4.3TheResults ............................... 194 11CONCLUSIONSANDSUGGESTEDFUTUREWORK ............. 204 11.1Conclusions ................................... 204 11.2FutureWork ................................... 207 7

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..... 209 BTHERMODYNAMICPARAMETERSFORTHETI-AL-CRSYSTEM ..... 214 CTHERMODYNAMICPARAMETERSFORTHETI-AL-MOSYSTEM .... 217 REFERENCES ....................................... 220 BIOGRAPHICALSKETCH ................................ 233 8

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Table page 3-1PhasesintheTi{Al{Nbsystem. ........................... 61 3-2ThermodynamicmodelingofthephasesusedinthedescriptionofKattnerandBoettinger[ 9 ]. ..................................... 63 3-3ThermodynamicmodelingofthephasesintheTi{Al{NbsystemofServantandAnsara[ 10 ]. ...................................... 66 3-4ThermodynamicmodelingofthephasesintheTi{Al{NbsystemofWitusiewiczetal.[ 17 ]. ....................................... 74 4-1KeyexperimentalworksintheTi{Al{Nbsystem ................. 83 4-2TielineandtietriangledataforalloysA2,A3,andA133thatwereusedfortheoptimization. ................................... 87 4-3Transitiontemperaturesforalloy11andalloy12. ................. 96 5-1Crystalstructureofthephase[ 135 ] ........................ 106 7-1PhasesintheTi{Crsystem. ............................. 137 7-2HomogeneityrangeoftheLavesphasesdeterminedbyChenetal.[ 148 ]. .... 137 7-3Crystalstructureofthe{TiCr2Lavesphase. ................... 138 7-4Crystalstructureofthe{TiCr2Lavesphase. ................... 139 7-5Crystalstructureofthe{TiCr2Lavesphase. ................... 140 7-6DataontheLavesphasesthatwereusedfortheoptimization. .......... 141 8-1StablesolidphasesintheTi{Al{Crsystem. .................... 156 9-1StablesolidphasesintheTi{Al{Mosystem. .................... 175 10-1InvariantreactionsintheAl{Mosystemwhichwereacceptedfortheoptimizationbasedonacriticalevaluationoftheavailableliterature. .............. 196 A-1ThermodynamicDescriptionfortheTi{Al{NbSystem .............. 209 B-1ThermodynamicDescriptionfortheTi{Al{CrSystem ............... 214 C-1ThermodynamicDescriptionfortheTi{Al{MoSystem .............. 217 9

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Figure page 2-1SchematicoftheCALPHADmethod[ 62 ]. ..................... 44 2-2ThecontributiontotheexcessGibbsfreeenergyofmixingfromtherstfourtermsintheRedlich-Kisterpolynomial. ....................... 45 2-3GraphicalrepresentationoftheA)Kohler,B)Colinet,andC)MuggianuandD)Toopternaryextrapolationmethods.Inalldiagrams,theopencirclerepresentsapointofternarycomposition(xi,xj,xk).Thelledsquaresshowthepointsonthei{jbinarywhichwillmakeacontributiontotheGibbsfreeenergyoftheternarycomposition.TheKohlerandMuggianuextrapolationmethodsuseonlyonepointalongthei{jbinarywhereastheColinetextrapolationmethodusestwopointsalongthei{jbinary.Onepointgivesthemolefractionofiandtheotherpointgivesthemolefractionofjthatwillbeused.Therefore,intheColinetmethod,xbini+xbinj6=1.InToopextrapolation,thepointsalongthei{kandj{kbinariesarechosenatconstantxk,butthepointalongthei{jbinaryischosenusinginthesamewayastheKohlermethod. .............. 46 2-4Thesurfaceofreferenceforahypotheticalcompound(A;B)p(C;D)qplottedabovethecompositionsquare. ............................ 47 3-1ThecalculatedA)Ti{Al,B)Nb{Al,andC)Ti{NbconstituentbinarysystemsusedinthethermodynamicdatasetofKattnerandBoettinger.Figurestakenfrom[ 9 ]. ....................................... 62 3-2CalculatedliquidususingthethermodynamicdescriptionofKattnerandBoettinger.Figuretakenfrom[ 9 ]. ................................ 64 3-3ThecalculatedA)Ti{Al,B)Nb{Al,andC)Ti{NbconstituentbinarysystemsusedinthethermodynamicdatasetofServantandAnsara[ 10 ]. ......... 65 3-4Isothermalsectionat1273KcalculatedusingthethermodynamicdescriptionofServantandAnsara[ 10 ].Theexperimentaldataidentiedas[1992Men]and[1998Hel]refertotheworksofMenonetal.[ 109 ]andHellwigetal.[ 101 ]respectively. 66 3-5Isothermalsectionat1373KcalculatedusingthethermodynamicdescriptionofServantandAnsara[ 10 ].Theexperimentaldataidentiedas[1999Eck]and[2002Leo]refertotheworksofEckertetal.[ 108 ]andLeonardetal.[ 107 ]respectively. 67 3-6Isothermalsectionat1473KcalculatedusingthethermodynamicdescriptionofServantandAnsara[ 10 ].Theexperimentaldataidentiedas[1992Men],[1995Zdz],and[1998Hel]refertotheworksofMenonetal.[ 109 ],Zdziobeketal.[ 13 ],andHellwigetal.[ 101 ]respectively. ........................... 67 10

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10 ].Theexperimentaldataidentiedas[1998Wan]refertotheworkofWangetal.[ 99 ].Althoughthereseemstobequitealargedisagreementbetweenthecalculatedisothermalsectionandthesuperimposedexperimentaldata,itispossiblethatWangetal.didnotheattreattheiralloysforsucientlylongperiodsoftimeat1673Ktoachievetheequilibriummicrostructures. ... 68 3-8Isothermalsectionat1813KcalculatedusingthethermodynamicdescriptionofServantandAnsara[ 10 ].Theexperimentaldataidentiedas[2000Leo]referstodataobtainedfromLeonardandVasudevan[ 14 ].Inthiswork,analloyofcompositionTi{25at.%Al{60at.%Nbwhichwasheattreatedat1813Kandquenchedwasshowntobeinthethreephaseregion++However,calculationswiththisdatasetindicatethatanalloyofthiscomposition,shownbythebluetriangle,isinthe+two-phaseregion. ...................... 68 3-9Isothermalsectionat1923KcalculatedusingthethermodynamicdescriptionofServantandAnsara[ 10 ].Theexperimentaldataidentiedas[1992Men]referstotheworkofMenonetal.[ 109 ]. .......................... 69 3-10LiquidussurfacecalculatedusingthethermodynamicdescriptionofServantandAnsara[ 10 ].Theexperimentaldataidentiedas[1992Fen],[1996Ara],and[2000Leo]refertotheworksofFengetal.[ 98 ],Leonardetal.[ 15 ],andd'Arag~aoandEbrahimi[ 4 ]respectively.Thepositionsofalloy11withnominalcompositionTi{45at.%Al{18at.%Nbandalloy12withnominalcompositionTi{45at.%Al{27at.%Nbarealsoindicated.Thesealloyshavealsobeenshowntosolidifyrstas{phasefromtheliquid.Althoughthesealloyswillbediscussedlaterinthiswork,theyareindicatedhereforconvenience. ......................... 69 3-11ThecalculatedTi{AlphasediagramusingthedescriptionsofA)Witusiewiczetal.[ 17 ].ThecalculatedTi{AlphasediagramofSaunders[ 18 ](B)isrepeatedhereforcomparison. ................................. 70 3-12ComparisonbetweencalculatedandexperimentallydeterminedstandardenthalpiesofformationofalloysintheNb{Alsystemat298K.ThereferencestateforAlisthefcc{phaseandthereferencestateforNbisthebcc{phase.ThedashedlineshowstheresultsofthecalculationusingthedatasetofServantandAnsara[ 104 ]andthesolidlineshowstheresultsofthecalculationusingthedatasetofWitusiewiczetal.[ 17 ].ExperimentalpointsweretakenfromGeorgeetal.[ 117 ]. 71 3-13ComparisonbetweencalculatedandexperimentallydeterminedenthalpiesofformationofalloysintheNb{Alsystem.ThereferencestateforAlistheliquidandthereferencestateforNbisthebcc{phase.ThedashedlineshowstheresultsofthecalculationusingthedatasetofServantandAnsara[ 104 ]andthesolidlineshowstheresultsofthecalculationusingthedatasetofWitusiewiczetal.[ 17 ].Bothcalculationswereperformedat1669K.ExperimentalpointsweretakenfromMahdouketal.[ 116 ]. .......................... 72 11

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17 ]andServantandAnsara[ 104 ].Themaindierencesbetweenthephasediagramsarethecongruentmeltingtemperatureofthe{phaseandtheeutecticreactiontemperaturebetweentheliquid,,andphases.ServantandAnsaraacceptedthelevitationthermalanalysismeasurementsofJordaetal.[ 118 ],whileWitusiewiczetal.determinedthesetemperaturesusingopticalpyrometryanddierentialthermalanalysis. .................... 73 3-15Isothermalsectionat1813KcalculatedusingthethermodynamicdescriptionofWitusiewiczetal.[ 17 ].Theexperimentaldataidentiedas[2000Leo]referstodataobtainedfromLeonardandVasudevan[ 14 ].Inthiswork,analloyofcompositionTi{25at.%Al{60at.%Nbwhichwasheattreatedat1813Kandquenchedwasshowntobeinthethreephaseregion++However,calculationswiththisdatasetindicatethatanalloyofthiscomposition,shownbythebluetriangle,isinthe+two-phaseregion. ...................... 74 3-16LiquidussurfaceintheTi{Al{NbsystemcalculatedusingthethermodynamicdescriptionofWitusiewiczetal.[ 17 ].Allalloysindicatedonthediagramshouldbeintheprimarycrystallizationeldofthe{phase.Thissuggeststhattheprimarycrystallizationeldsoftheandphasesextendtoomuchintotheternary. ........................................ 75 3-17SolidussurfaceintheTi{Al{NbsystemcalculatedusingthethermodynamicdescriptionofWitusiewiczetal.[ 17 ].Foranalloytosolidifyassinglephase,notonlyshoulditbelocatedintheprimarycrystallizationeldof,butitshouldalsobelocatedtotherightofthesolidus. ................ 75 4-1DTAcurveforalloy11measuredatheatingandcoolingratesof10oC/min. ... 88 4-2GibbstriangleshowingthepositionofAlloy11aswellasthepositionofthethreephasetriangle++measuredbyHellwigetal.[ 101 ].ThisthreephasetrianglewasdeterminedbyheattreatinganalloyofcompositionTi{40.5at.%Al{24.8at.%Nbfor48hoursat1200oCandthenwaterquenching.Thealloywasshowntocontainthe,,andphases,thecompositionsofwhichweremeasuredusingEPMAandareshownasopentriangles.Thealloyisindicatedasthelledredcircle.Accordingtotheseresults,Alloy11shouldalreadybeinthethreephaseregion++at1200oC. .................... 89 4-3PeakseparationofthetwooverlappedpeaksusingVoigtfunctions.Thepeakrepresentingthe+!++transitionisshowninredandthepeakrepresentingthe++!+transitionisshowningreen. ......... 90 4-4Peakanalysisforthe+!++transformation.Therstderivativeisshowninblue.Thetransitiontemperaturewasselectedbasedonthepointatwhichthevalueoftherstderivativedeviatedfromzero. ............. 90 12

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............. 91 4-6DTAcurveforalloy12measuredatheatingandcoolingratesof10oC/min. ... 92 4-7DTAcurveforalloy12showingtheoverlappedpeaksonheating.Becauseofthepresenceoftheshallowpeak,shownbythebluerectangle,manypeakseparationmethodsmustbeevaluated. ............................. 93 4-8Theresultsofpeakseparationusingthreepeaks.Inthismethod,allpeakparameterswererened.Theresultsshowthatalthoughtwophasetransformationpeaksarefairlywellmodeled,athirdpeak,showninred,appearsatanimprobablepositionbetweenthetwophasetransformationpeaks. ............... 93 4-9Theresultsofpeakseparationusingthreepeaks.Inthisapproach,thecentroidofonepeakwasxedat1273oCinanattempttoxthepositionoftherstpeak.AlthoughthecumulativepeakingoodagreementwiththeoriginalDTAsignal,thephasetransitiontemperaturesassociatedwiththersttwopeaksareapproximatelyequal.Thisisalsoahighlyimprobablesituation. ......... 94 4-10Theresultsofpeakseparationusingonlytwopeaks.ThecumulativepeakisingoodagreementwiththeoriginalDTAsignal.However,inanattempttotakeintoaccounttheshallowpeak,theparametersoftherstpeakarerenedtoproduceapeakthatismuchwiderthanexpected.Thisintroducesquitealargeerrorintheanalysisofthephasetransitiontemperatureusingthemethodofrstdeviationfromthebaselineusingtherstderivative. ............. 95 4-11Theoverlappedpeaksonheatingforalloy12indicatingthechoiceoftemperaturesforthecorrespondingphasetransformations.1269oCisselectedasthestarttemperatureofthe+!++transitionand1323oCisselectedasthestarttemperatureofthe++!+transition. ........................ 95 5-1CalculatedbinaryTi{NbdiagramusingthedescriptionofZhangetal.[ 129 ]. .. 106 5-2Unitcellofthephase.TheAl1andAl2atomsarelocatedatthe2aand8i2Wyckopositionsrespectively.Sincethe2aand8i2positionseachhavethesamecoordinationnumber,theatomsinthesepositionsoccupythesamesublattice.TheNb2andNb3atoms,whicharelocatedatthe8i2and8jpositions,occupythesamesublatticeastheyeachhaveacoordinationnumberof14andtheNb1atominthe4fpositionoccupiesitsownsublattice.Thisresultsinthemodel(Nb)16(Al)10(Nb)4whenthereisnomixingoneitherofthesublattices. ................... 107 5-3Theendmembersofthephase.ThedashedlinefromNb:Al:NbtoTi:Al:Nbschematicallyillustratestheinuenceofthe0LNb,Ti:Al:Nbmixingparameter. ... 108 5-4Theendmembersofthephase.ThedashedlinefromNb:AltoTi:Alschematicallyillustratestheinuenceofthe0LNb,Ti:Alparameter. ................ 108 13

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.................................. 109 5-6Theendmembersofthephase.Thedashedlinesshowtheinuenceofthesixmixingparametersoftype0Li;j:k,whichrepresenttheconditionofmixingoftwodierentcomponentsononesublatticewithasecondsublatticesinglyoccupiedbyathirdcomponent. ........................... 110 5-7Thevaluesofthe0LAl:Nb,Tiat1923K,1683K,and1473K.Expressingthisparameterasalinearfunctionoftemperatureonlywillnotresultinagoodttotheexperimentalphaseequilibria.Therefore,aquadraticvariationwithtemperaturewaschosen. 111 6-1CalculatedliquidussurfacefortheTi{Al{Nbsystem.Theexperimentaldataidentiedas[1992Fen],[1996Ara],and[2000Leo]refertotheworksofFengetal.[ 98 ],Leonardetal.[ 15 ],andd'Arag~aoandEbrahimi[ 4 ]respectively.Thecalculatedliquidussurfaceisingoodagreementwiththeliteratureshowingthatallalloysindicatedsolidifyasphase. ....................... 115 6-2CalculatedsolidussurfaceintheTi{Al{Nbsystem.ThepositionofthesolidusshowsthatallalloyssolidifyassinglephaseexceptforthealloyofcompositionTi{48at.%Al{25at.%NbfromFengetal.[ 98 ]andalloy12.Aftersolidication,thesealloysexistastwophase+. ....................... 115 6-3ScheilreactionschemeforequilibriawiththeliquidintheTi{Al{Nbsystem.Thereareveinvariantreactions:threetransitionreactions,oneeutecticreaction,andoneperitecticreaction. ............................. 116 6-4Calculatedisothermalsectionat1923K.Theexperimentaldataidentiedas[1992Men]referstotheworkofMenonetal.[ 109 ]. ................ 117 6-5Calculatedisothermalsectionat1813K.Theexperimentaldataidentiedas[2000Leo]referstodataobtainedfromLeonardandVasudevan[ 14 ].Inthiswork,analloyofcompositionTi{25at.%Al{60at.%Nbwhichwasheattreatedat1813Kandquenchedwasshowntobeinthethreephaseregion++. ..... 117 6-6Calculatedisothermalsectionat1783K.TielineandtietriangledataforalloysA2,A3,andA133areincluded.Thetielinesshowninblueweredeterminedafterthedescriptionwasre-optimized.Therefore,thecurrentdescriptionisalsoabletopredictthephaseequilibriaat1783K. ................... 118 6-7Calculatedisothermalsectionat1683K.TielineandtietriangledataforalloysA2,A3,andA133areincluded. .......................... 118 14

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99 ]issuperimposed.Whilethedatasetisabletocalculatephaseequilibriabetweenand,and,and,and,andandphases,otherphaseequilibriacannotbecalculated.However,Wangetal.[ 99 ]mayhaveinsucientlyheattreatedtheiralloysat1673K.Therefore,theymaynothavemeasuredtheequilibriumtielines. ....................................... 119 6-9Calculatedisothermalsectionat1613K.Thereisgoodagreementwiththe{tielineforalloy12. ................................. 119 6-10Calculatedisothermalsectionat1513K.Thereisgoodagreemtnwiththe{tielineforalloy11. ................................. 120 6-11Calculatedisothermalsectionat1473K.Theexperimentaldataidentiedas[1992Men],[1995Zdz],and[1998Hel]refertotheworksofMenonetal.[ 109 ],Zdziobeketal.[ 13 ],andHellwigetal.[ 101 ]respectively. ............ 120 6-12Calculatedisothermalsectionat1373K.Theexperimentaldataidentiedas[1999Eck]and[2002Leo]refertotheworksofEckertetal.[ 108 ]andLeonardetal.[ 107 ]respectively. ................................ 121 6-13Calculatedisothermalsectionat1273K.Theexperimentaldataidentiedas[1992Men]and[1998Hel]refertotheworksofMenonetal.[ 109 ]andHellwigetal.[ 101 ]respectively. ................................ 121 6-14Calculatedisopleththroughthenominalcompositionsofalloy11andalloy12.Thereisgoodagreementbetweenthecalculationsandthesolidstatetransformationsofalloys11and12measuredusingDTA.Alloy11solidiesassinglephase.However,thereisnosinglephaseeldforalloy12.Instead,directlyaftersolidication,andshouldappear. ............................... 122 6-15Calculatedisoplethat40at.%Al.TransformationtemperaturesmeasuredusingDTAfromLeonardetal.[ 15 ]areincluded. .................... 123 6-16Calculatedphasefractiondiagramforalloy11. .................. 124 6-17Calculatedphasefractiondiagramforalloy12. .................. 124 7-1Unitcellofthe{TiCr2Lavesphase.AtwosublatticemodelwithoneTi-richandoneCr-richsublatticeisused. .......................... 138 7-2Unitcellofthe{TiCr2Lavesphase.TheTi1andTi2atoms,whichoccupythe4eand4fpositionsrespectively,occupythesamesublatticebecausetheyhavethesamecoordinationnumbers,andtheCr1,Cr2,andCr3atoms,whichoccupythe4f',6g,and6hpositions,occupythesamesublatticeastheyhavethesamecoordinationnumbers. ........................... 139 15

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................................ 140 7-4TheCalculatedpartialTi{Crphasediagramfrom60at.%Crto70at.%CrusingthedescriptionofSaunders[ 18 ].Onlythelowtemperature{TiCr2andahightemperature{TiCr2Lavesphasesaremodeled.Therehomogeneityrangeofthe{TiCr2phaseisingoodagreementwiththeexperimentalworkofChenetal.[ 148 149 ].However,theagreementofthehomogeneityrangeofthe{TiCr2phasewiththeworkofChenetal.[ 148 149 ]isnotsogood. ........... 141 7-5GraphshowingthedependenceoftheGTiCr2Cr:Tiparameterforthe{TiCr2phaseasthesolidline.ThelineardependenceoftheGTiCr2Cr:Tiparameterisalsoincludedasthedashedline. .................................. 142 7-6CalculatedTi{Crphasediagramusingthenewdescription.Theexperimentaldatafromtheliteratureissuperimposed. ...................... 142 7-7CalculatedpartialTi{Crphasediagramfrom60at.%Crto70at.%Cr.Thereisgoodagreementofthehomogeneityrangeofthe{TiCr2,{TiCr2and{TiCr2LavesphaseswiththeexperimentalworkofChenetal.[ 148 149 ]. ........ 143 7-8CalculatedactivityofCrinthephasecomparedtotheexperimentaldataofPooletal.[ 152 ].SincethenewdatasetusesthesamethermodynamicparametersforthephaseasinSaunders[ 18 ],thereisnodierenceintheactivitycurvesforCrbetweenthenewdatasetandthedatasetofSaunders[ 18 ]. ........ 144 8-1CalculatedbinaryAl{CrphasediagramusingthedescriptionofSaunders[ 18 ]. 157 8-2CalculatedbinaryAl{CrphasediagramfromthedescriptionofLiangetal.[ 193 ]. 157 8-3ComparisonbetweentheA)isothermalsectionat1073KcalculatedusingthedatasetofSaunders[ 18 ]andtheB)assessed1073KisothermalsectionofBochvaretal.[ 19 ]. ....................................... 158 8-4ComparisonbetweentheA)isothermalsectionat1273KcalculatedusingthedatasetofSaunders[ 18 ]andtheB)assessed1273KisothermalsectionofBochvaretal.[ 19 ]. ....................................... 158 8-5ComparisonbetweentheA)calculated[ 18 ]andB)assessed[ 19 ]liquidussurface. 159 8-6Theendmembersoftheternary{Ti(Al,Cr)2phase.ThedashedlinefromAl:TitoCr:Tischematicallyillustratestheinuenceofthe0LAl,Cr:Tiparameter. .... 160 8-71273KisothermalsectioncalculatedusingthenewdescriptionfortheTi{Al{Crsystem.ExperimentaldataofJewettetal.[ 97 177 179 ]aresuperimposed. .. 161 16

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97 177 179 ]aresuperimposed. .. 161 8-9LiquidussurfacecalculatedusingthenewdescriptionoftheTi{Al{Crsystem. 162 8-10PartialliquidussurfacecalculatedusingthenewdescriptionoftheTi{Al{Crsystem. ........................................ 162 8-11ScheilreactionschemefortheTi{Al{Crsystem. .................. 163 9-1AssessedTi{MophasediagramaccordingtoMassalski[ 201 ]. ........... 173 9-2Ti{MophasediagramcalculatedusingthedatasetofSaunders[ 18 ]. ....... 173 9-3AssessedAl{MophasediagramaccordingtotheSchuster[ 210 ]. ......... 174 9-4CalculatedAl{MophasediagramfromthedescriptionofSaunders[ 18 ]. ..... 174 9-5Assessed1873KisothermalsectionfromtheworkofTretyachenkoetal.[ 226 ].ThealloyofcompositionTi{52at.%Al{45at.%Mo,whichhasbeenshowntosolidifyassinglephase[ 27 ],isintheAlMo+liquidtwophaseeld.Therefore,thisalloysolidiesastheAlMophase. ....................... 176 9-6AssessedpartialsolidusintheTi{Al{MosystemfromtheworkofTretyachenkoetal.[ 226 ].ThealloyofcompositionTi{52at.%Al{45at.%Mo,whichshouldsolidifyassinglephase[ 27 ],isinsteadinthe+AlMo+Al63Mo37threephaseeld.Thegroup2andgroup3alloysofNinoetal.[ 27 ]wereshowntosolidifyassinglephase. ................................... 177 9-7Calculated1773KisothermalsectionusingthedescriptionofSaunders[ 18 ].ThealloyofcompositionTi{52at.%Al{45at.%Mo,whichshouldbesinglephaseaccordingtotheworkofNinoetal.[ 27 ],isintheliquid++Al63Mo37threephaseeld. ...................................... 178 9-8Phaseequilibriaintheregionfrom0at.%Tito20at.%TifromNinoetal.[ 27 ].ThealloyofcompositionTi{52at.%Al{45at.%Moisindicated. ......... 178 9-9Calculated1540KisothermalsectionsusingthedescriptionofSaunders[ 18 ].SincethephasedoesnotextendtohighenoughAlcompositions,theinvariantreaction+Al8Mo3!+cannotbecalculated. ................ 179 9-10Phaseequilibriaintheregionfrom0at.%Tito20at.%TifromNinoetal.[ 27 ].Theinvariantreaction+Al8Mo3!+,aswellasthepositionofthephaseboundaryat1540K,isindicated. ...................... 179 17

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18 ].ThealloyofcompositionTi{52at.%Al{45at.%Moisintheprimarycrystallizationeldofthephase.However,Ninoetal.[ 27 ]showedthatthisalloyshouldsolidifyassinglephase.Additionally,thegroup3alloysinvestigatedbyNinoetal.[ 27 ]alsosolidifyassinglephase.However,thecalculatedsolidusdoesnotextendtohighenoughAlcompositionsforthistobereproduced. 180 10-1PartialAl{Mophasediagramfrom70at.%Alto100at.%AlconstructedbyEumannetal.[ 251 ]. ...................................... 196 10-2Al{Mophasediagramcalculatedusingthenewdescription.Theexperimentaldatafromtheliteraturearesuperimposed. ..................... 197 10-3Al{Mopartialphasediagramcalculatedusingthenewdescription.Theexperimentaldatafromtheliteraturearesuperimposed. ..................... 197 10-4CalculatedandexperimentallydeterminedenthalpiesofformationofthephasesintheAl{Mosystem.[1982Shi]referstotheworkofShiloandFranzen[ 249 ]and[1993Mes]referstotheworkofMeschelandKleppa[ 252 ]. .......... 198 10-5Endmembersforthephasewhichisdescribedusingthecompoundenergyformalismas(Al*,Mo,Ti)0:75(Al,Mo*,Ti)0:25wheretheasteriskidentiesthemajorspeciesoneachsublattice.Theendmembersforthephasewithmodel(Al,Mo*,Ti)0:75(Al*,Mo,Ti)0:25arethesame.Theinteractionparameter0LMo,Ti:Al,shownasthedashedline,inuencedtheextensionofthephaseintotheternary. 199 10-6LiquidussurfacecalculatedwiththenewdescriptionfortheTi{Al{Mosystem.AllalloysindicatedwereshowntosolidifyassinglephaseintheworkofNinoetal.[ 27 ],withwhichthenewliquidussurfaceisinverygoodagreement. ... 200 10-7Isothermalsectionat1773KcalculatedusingthenewdescriptionfortheTi{Al{Mosystem.[2003Nin]referstotheworkofNinoetal.[ 27 ]. .............. 201 10-8Isothermalsectionat1673KcalculatedusingthenewdescriptionfortheTi{Al{Mosystem.[2003Nin]referstotheworkofNinoetal.[ 27 ]. .............. 201 10-9Isothermalsectionat1540KcalculatedusingthenewdescriptionfortheTi{Al{Mosystem.[2003Nin]referstotheworkofNinoetal.[ 27 ]. .............. 202 10-10Isoplethcalculatedthrough50at.%Al.ThesolidstatetransformationtemperaturesmeasuredusingthermalanalysisfromNinoetal.[ 27 ]areindicated.Alloysidentiedbylledsquaresweresinglephaseattherespectivetemperatures. ....... 203 18

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19

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20

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1 ]whereastheCMSX-10Ni-basedsuperalloyhasadensityof9.05g/cm3[ 2 ].BasedonpreviousworkonTi{Al{Nballoys[ 3 4 ],two-phasealloyswithamicrostructureconsistingofa{TiAl(Nb)matrixreinforcedbysubmicron-sized,disconnected{Nb2Alprecipitatesarealsopromising{TiAlbasedalloysbecauseoftheiranticipatedhightemperaturecreepproperties.Furthermore,additionsofrefractoryelementssuchasNbhavebeenshowntoimprovethecreepresistanceaswellasroomtemperatureductilityof{TiAl[ 5 { 8 ].Severalbasicrequirementsmustbemetfordesignofthe{TiAl+{Nb2Alalloys.First,the{TiAl+-Nb2Almicrostructuresshouldbestableatoperatingtemperaturesi.e.,thereshouldbenophasetransformationsuponcooling.Second,thealloysshouldcontainatleast40at.%Altoprovideoxidationresistance.Thirdandmostimportantformicrostructuraldevelopment,the{bccsinglephasesolidsolutionshouldexistasasingle-phaseathightemperaturesandthatcanbequenchedandmaintainedatroomtemperaturesothatthe{TiAl+-Nb2Almicrostructurescanbeproducedonaging.KnowledgeofthephaseequilibriaintheTi{Al{Nbsystemcansignicantlyreducealloydevelopmenttime.Asanexample,theliquidusandsolidussurfacesindicatewhichalloysshouldsolidifyassinglephase,andisothermalsectionsshowtheequilibriumphasesinanalloyatagiventemperature.Phasediagramscaneitherbedeterminedexperimentallyorcalculatedusingthermodynamicdescriptionsforthemulti-componentsystems.Althoughthereismuchexperimentalinformationavailableintheliterature 21

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9 ]andServantandAnsara[ 10 ]areinconsistentwiththisdata.Forexample,eventhoughthesinglephaseeldintheTi{Al{Nbsystemhasbeenshowntoextendtoapproximately40at.%Alat30at.%TiintheexperimentalworksofKaltenbachetal.[ 11 ],Perepezkoetal.[ 12 ],andZkziobeketal.[ 13 ]andto40at.%Alat20at.%TiinthemorerecentworksofLeonardetal.[ 14 15 ],neitherofthethermodynamicdescriptionsreproduceaccuratelytheextensionofthesinglephaseeld.Additionally,thedescriptionsdonotreproducethepresenceofthesinglephaseeldforanalloyofnominalcompositionTi{44.5at.%Al{18.5at.%NbasdeterminedbyRiosetal.[ 16 ].AnotherinconsistencyisthatalthoughanalloyofnominalcompositionTi{25at.%Al{60at.%Nb,whichwasheattreatedat1813Kandquenched,hasbeenshowntocontain++inthemicrostructure[ 14 ],thisalloyislocatedinthe+two-phaseeldat1813KusingthedatasetofServantandAnsara[ 10 ].SincethereismuchinterestindeterminingphaseequilibriaintheTi{Al{Nbsystemforalloydesign,recentlyandparalleltothiswork,anewdescriptionfortheTi{Al{NbsystemwasdevelopedbyWitusiewiczetal.[ 17 ],whichwaspublishedin2009.AlthoughthisdescriptionreproducesthesinglephaseeldforanalloyofcompositionTi{44.5at.%Al{18.5at.%Nb[ 16 ],thisdatasetstillcannotcalculatetheextensionofthesinglephaseeldto20at.%Tiat40at.%Alorthe++threephaseeldforanalloyofcompositionTi{25at.%Al{60at.%Nbat1813K[ 14 ].Fromthebasicrequirementsforalloydesign,theoptimaltwophase{TiAl+{Nb2Almicrostructuresshouldexistatcompositionswithmorethan40at.%Altoprovideoxidationresistance.ThismeansthateveniftheextensionofthephasefromLeonardetal.[ 14 15 ]istakenintoaccount,thephasemay,onthersthand,stillnotbestableenoughtoberetainedonquenching,and,onthesecondhand,stillmaynotexistathighenoughAlcompositionstoensurealowvolumefractionof{Nb2Alphaseon 22

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18 ],thesedatasetswerecreatedbycombiningtheconstituentbinariesandextrapolatingtotheternary;CALPHADbasedassessmentsforthesesystemswerenotperformed.CalculationsintheTi{Al{CrsystemusingthedatasetofSaunders[ 18 ]couldnotreproduceeithertheliquidussurfacefromthecriticalevaluationofBochvaretal.[ 19 ]ortheextensionofthe{Ti(Al,Cr)2Lavesphaseintotheternary.Inthisdataset,thereisnodescriptionfortheternaryphase,whichwasfoundinnumerousworksintheliterature[ 20 { 26 ],andtheternaryextensionofthe{Ti(Al,Cr)2Lavesphasealongapproximately33at.%Tiisnotmodeled.Similarly,calculationsintheTi{Al{MosystemusingthedatasetofSaunders[ 18 ]couldnotreproducethephaseequilibriaintheregionfrom0to20at.%TifromtheexperimentalworkofNinoetal.[ 27 ].Forexample,althoughNinoetal.[ 27 ]showedthecontinuityofthephasetotheAl{Mobinaryat1773Kandtheinvariantreaction+Al8Mo3!+at1540K,thesefeaturescouldnotbereproducedusingthedatasetofSaunders[ 18 ]asthecalculatedsinglephaseelddoesnotextendtohighenoughAlcompositions.Basedontheaboveinformation,theobjectiveofthisdissertationistousetheCALPHADmethodtodevelopthermodynamicdescriptionsfortheTi{Al{Nb,Ti{Al{Cr,andTi{Al{Mosystemsthatareabletoreproduceexperimentalresults.UsingtheCALPHADapproach,thermodynamicmodelsforthephasesinthemulti-component 23

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28 ]moduleofTHERMO-CALC[ 29 ]wasusedtooptimizethethermodynamicend-memberandinteractionparametersofthephasedescriptions,andbothTHERMO-CALC[ 29 ]andPANDAT[ 30 ]softwarepackageswereusedtocalculatethevariouskindsofphasediagrams.Thestructureofthisdissertationisasfollows.InChapter2,theCALPHADmethodisintroduced,thethermodynamicmodelingofstoichiometricphases,solutionphases,andorderedphasesusingthecompoundenergyformalismisdeveloped,anddetailsofGibbsfreeenergyminimizationandthermodynamicoptimizationusingtheleastsquaresmethodaregiven.AreviewofexperimentalworkandthermodynamiccalculationsintheTi{Al{NbsystemisgiveninChapter3.CalculatedliquidusprojectionsandisothermalsectionsfromthedatasetsServantandAnsara[ 10 ]andWitusiewiczetal.[ 17 ]arecomparedtotheavailableexperimentaldata,discrepanciesbetweenthecalculationsandtheexperimentaldataarehighlighted,andreasonsforre-optimizationaregiven.Chapter4givesamoredetailedlookatthekeyexperimentaldatausedfortheoptimizationwithastrongemphasisplacedontheevaluationofthermalanalysisdataforAlloy11andAlloy12.ThespecicsofthethermodynamicoptimizationoftheparametersfortheTi{Al{NbsystemaregiveninChapter5,andtheresultsoftheoptimizationaregiveninChapter6.TheassessmentoftheternaryTi{Al{CrsystemstartsinChapter7withaliteraturereviewofthebinaryTi{Crsystem,reasonswhytheTi{CrdatasetofSaunders[ 18 ]shouldbemodiedtoincludeadescriptionforthehightemperature{TiCr2Lavesphase,there-optimizationofthissystem,andthenalresults.TheintroductionofthenewdescriptionfortheTi{Crbinary,there-optimizationoftheternaryparametersofsomeof 24

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18 ]arecomparedtotheexperimentalresultsintheregionfrom0to20at.%TifromNinoetal.[ 27 ].Chapter9concludeswiththestatementthattheonlywaytoreproducetheexperimentaldataintheTi{Al{Mosystemiswithacompletere-optimizationofthebinaryAl{Mosystem.InChapter10,thedetailsofthere-optimizationoftheAl{MobinarysystemandofsomeoftheternaryparametersofsomeofthephasesintheTi{Al{Mosystemaregivenandtheresultsofbothre-optimizationsarecomparedtotheexperimentaldata.ConclusionsandsuggestionsforfutureworkaregiveninChapter11. 25

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31 ]summarizesthevariousmethodsforthepresentationofphasediagramsinmulti-componentsystems.Phasediagramscanbedeterminedexperimentally,butthisisatimeconsumingprocesswithacomplexitythatdrasticallyincreaseswhenadditionalelementsareincluded.Forexample,althoughthephasediagraminabinarysystemcanbedeterminedusingcombinedmetallography,thermalanalysis,anddiractionmethods,theadditionofathirdelementintheternarysystemintroducesanextradegreeoffreedomincomposition,whichphenomenallyincreasestheamountofsamplesthatmustbepreparedtofullyunderstandandcharacterizethephaseequilibriainthesystem.ThisiswheretheCALculationofPHAseDiagrams(CALPHAD)methodplaysagreatrole.ThephilosophybehindtheCALPHADmethodisthatphasediagramscannotonlybeexperimentallydeterminedbutalsocomputedusingthermodynamicdatasets.TheadvantageoftheCALPHADmethodisthat,ontheonehand,onecouldpredictphaseequilibriainregionsofthephasediagramthathavenotyetexperimentallybeendetermined.Forexample,onecouldpredicttheisothermalsectionatagiventemperatureinaternarysystembasedonexperimentallydeterminedphaseequilibriaatothertemperatures.Ontheotherhand,onecouldpredictthephaseequilibriainamulti-componentsystembasedonanextrapolationofthethermodynamicdescriptionof 26

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1. Theselectionofthermodynamicmodelstorepresentthevariousphaseswithinagivensystem. 2. Theexpressionofthethermodynamicmodelsasanalyticfunctionsoftemperature,pressure,andcomposition. 3. Theoptimizationofthethermodynamicparametersusingallavailableexperimentalandtheoreticaldata. 4. Thestorageoftheoptimizedparametersincomputer-readablethermodynamicdatasets. 5. Thecalculationofphasediagramsandvariousphaseequilibriausingthethermodynamicdatasets.Points1and2abovewillbediscussedinSection 2.1 andpoints3,4,and5willbediscussedinSection 2.2 .AschematicoftheCALPHADmethodisshowninFigure 2-1 32 ].IntheCALPHADmethod,theGibbsenergyofthephasesinamulti-componentsystemismathematicallyformulatedusingaspecicmodelwhichtakesintoaccountsomecriticalfeatureofthephase.Forexample,intheidealgas,thereisnointeractionbetweenatoms;therefore,theidealgasmodelexpressestheGibbsfreeenergyoftheidealgasonlyasafunctionof 27

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33 ].Muchinteresthasbeendirectedtothemodelingoforderedphases.SincetheworkofTammannin1919[ 34 ]whichpostulatedthatatomsoccupyspeciclatticesiteswithinastructure,theoreticaltreatmentsfortheformationoforderedstructureshavebeenproposedbyBoreliusetal.[ 35 ],Gorsky[ 36 ]andBraggandWilliams[ 37 ].Bond-energymodelstakingintoaccounttheinteractionandexchangebetweenlikeandunlikeatomswithinalatticewerepresentedbyBethe[ 38 ]andBraggandWilliams[ 39 ].Takingtheseearlybond-energyformulationsintoaccount,stoichiometricphaseswereregardedasbeingcomposedofmultiplesublatticeswitheachsublatticesinglyoccupiedbyaspecicelement.IntheworkofHillertandStaansson[ 40 ],thisformulationwasextendedtomodelaclassofsolutionphasesconsistingoftwosublatticeswitheachsublatticeoccupiedbytwodierentelements.Inthiswork,theterm\surfaceofreference"andtheideaofsublatticesitefractionswereintroduced.Additionally,usagewasmadeofthemodelofTemkin[ 41 ],whichproposedthattheidealentropyofmixingcouldbecalculatedassumingrandommixingoneachsublattice.ThismathematicalformalismwasgeneralizedtocaseswithmorethantwoelementsoneachsublatticebyHarvig[ 42 ]andSundmanandArgen[ 43 ].AlthoughtheworkofHillertandStaansson[ 40 ]canalreadybeconsideredasthebeginningofthecompoundenergyformalism,thetermwasonlycoinedin1986byAnderssonetal.[ 44 ]. 45 ]: 28

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S=S(T)S(T=T0)=b0clnT2dT3eT2+fT2(2{3)InEquation 2{3 ,b0resultsfromthesubstitutionofthelowerintegrationlimitT0afterintegration.TheenthalpycanalsobecalculatedusingtherelationCP=(dH=dT)Pwhichgivesthedierentialequation: H=H(T)H(T=T0)=acTdT22eT3+2fT1+:::(2{5)wherearesultsfromsubstitutionofthelowerintegrationlimitT0afterintegration.TheGibbsenergyisthencalculatedusingtheequation G=HTS(2{6)Thisyieldsthesolution G=G(T)G(T=T0)=a+bT+cTlnT+dT2+eT3+fT1+:::(2{7)whereb=b0c.InEquations 2{3 2{7 and 2{5 ,attentionmustbegiventothetermsS(T=T0)andG(T=T0)andH(T=T0)respectively.Inallcases,sincetheanalyticdescriptionofCPgiveninEquation 2{1 isvalidonlyattemperaturesabove298.15K,thelowertemperaturelimitT0iscustomarilychosenas298.15K.Usingthethirdlawofthermodynamics,S(T=T0),whichisnowtheentropyofthepureelementat298.15K, 29

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2{7 canbere-writtenas: GHSERi=0Gi(T)0Hi(298:15K)=a+bT+cTlnT+dT2+eT3+fT1+:::(2{9)whereGHSERiistheGibbsenergyofthepureelementireferredtotheenthalpyofitsstablestateat298.15K,denotedas0Hi(298:15K),andEquation 2{5 canbewrittenas: HSERi=0Hi(T)0Hi(298:15K)=acTdT22eT3+2fT1+:::(2{10)TheGibbsenergyofmetastablestates'canalsobereferredtotheenthalpyofthestablestateusingtheequation: 2{11 isexactlyequaltoGHSERi,theequationcanbesimpliedtotheform: 30

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2{9 ,therighthandtermsofEquation 2{13 canbeexpressedas: whereGAaBb(T)istheGibbsenergyofformationofthecompoundAaBbattemperatureT.Themodelingofstoichiometricphasescanbeextendedtocompoundswithmorethantwoelementsusingtheexpression: 31

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2{16 ,G;idm,isthecontributiontothemolarGibbsfreeenergyresultingfromarandom,idealmixingofthecomponents.Thephysicalbasisforrandommixingistheconditionwherethereisnoenergyofinteractionbetweenthecomponents.ThecontributiontothemolarGibbsenergyresultingfromrandommixing,therefore,comesfromthecongurationalentropyofmixing,andisdenedas: 2{16 ,G;Em,istheexcessGibbsfreeenergyofmixing.ItisthedierencebetweentheGibbsfreeenergyoftheidealsolutionandtheGibbsfreeenergyoftheactualsolution,andisameasureofthedeviationoftheactualsolutionfromidealsolutionbehavior.Therearemanywaystomodelthedeviationoftheactualsolutionfromtheidealsolution.Onesuchmodelwasadvancedin1929byHildebrand[ 33 ]andcalledtheregularsolutionmodel.Theregularsolutionisdenedas\oneinvolvingnoentropychangewhenasmallamountofoneofitscomponentsistransferredtoitfromanidealsolutionofthesamecomposition,thetotalvolumeremainingunchanged"[ 33 ].Inthiscase,theenthalpyofmixingofthesolutionisnotzero(asisthecasefortheidealsolution),butequaltotheexcessGibbsfreeenergyofmixing,whichdependsonlyonthecompositionofthesolution.Forabinarysolution,thesimplestregularsolutionmodelisobtainedwhenG;Em=!x1x2where!isaconstantandx1andx2arethemolefractionsofcomponents1and2insolutionrespectively. 32

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46 ].Theseareexpressedas: 2-2 47 ],Colinet[ 48 ],Muggianu[ 49 ],andToop[ 50 ]methodsarevariousnumericalmethodswhichchoosethecompositiononthebinarysidewhichcontributestotheternary.ThesenumericalmethodsaregraphicallyillustratedinFigure 2-3 .Theextrapolationmethodsmaybeexpressedinthegeneralform: 33

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1xiG;Emfxi;1xig+xi=2 1xjG;Emf1xj;xjg(2{23)andtheMuggianumethodusesthenumericalexpression: 2{22 )isevaluated,oneobtainstheexpression: 2{23 )isevaluated,oneobtainstheexpression: 2{19 ))andthereforeistherecommendedmethodfortheextrapolationofternarysolutionbehaviorbasedonlyonananalyticexpressionforthebinaries.Unliketheothermethods,theToopextrapolationmethod[ 50 ]isanon-symmetricmethod.Thenumericalexpressionisgivenas: Fromthisexpression,itisclearthatthepointsalongthei{kandj{kbinariesarechoseninasimilarfashion,butthepointalongthei{jbinaryischosendierently.Therefore, 34

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wherethersttermrepresentsallbinaryinteractionsdeterminedusinganappliedextrapolationmethodandthesecondtermrepresentstheternaryinteractions.ThecompositiondependenceoftheternaryLijkparameterisexpressedas[ 51 ]: (2{30) Theexpressionsfori,j,andkareequaltoxi,xj,andxkrespectivelyintheternarysystem,withasumthatisalwaysunityinhigherordersystems. 52 ]wasconstructedtothermodynamicallymodelthebehaviorofsolutionphaseswithtwoormoresublatticesinwhichthecompositionofatleastoneoftheconstituentsublatticesisallowedtovary.Anexampleofsuchasolutioncanbegivenas: (A;B)p(C;D)q(2{31) 35

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2{31 .Sinceanycomponentcanoccupyanysublattice,and,dependingontheparticularcrystallographicinformation,thesamecomponentcanoccupymultiplesublattices,themolefractionofcomponentiisdenedas: 2{31 ,thestoichiometriccompoundsareApCq,ApDq,BpCq,andBpDq.Usingtheseprecepts,themodelofthecompoundenergyformalismisexpressedas: 36

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2{31 ,thesurfaceofreferencetermis: 2-4 .TheGibbsfreeenergytermrelatedtorandommixingoneachsublatticeis: 2{19 butmodiedtoincorporatethesublatticemodelingas: 37

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53 54 ]andextendedtotheternaryAl{Cr{NisystemintheworkofDupinetal[ 55 ].TheGibbsenergyoftheorderedanddisorderedphasescanbedescribedusingthesameGibbsenergyfunctionGmwhichisexpressedas: Gordm=Gordm(ysi)Gordm(ysi=xi)(2{40)whereGordm(ysi)istheGibbsenergyoftheorderedphase,expressedusingthecompoundenergyformalism,withthesitefractionsysiofcomponentionsublattices,andGordm(ysi=xi)istheGibbsfreeenergycontributionofthedisorderedphasetotheorderedphase,calculatedwhenthesitefractionsofeachcomponentysioneachsublatticeisthesameandequaltotheoverallcompositionofthephase.Inordertoensurethestabilityofthedisorderedphase,Gmmustalwayshaveanextremumwhenyi=xi.Whenthedisorderedphaseisstable,thisextremummustbeaminimum.Totakeintoaccountthisadditionalstipulationofthethermodynamicmodel,theconditiondGm 38

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2{2 willgivetheentropyofthatsubstance.DropcalorimetrycanalsobeusedtomeasurethestandardenthalpyofformationH298Kandenthalpyincrementofasubstance. 56 ],suchastheenthalpiesofformationandenthalpiesoftranformationforrealorhypotheticalcompounds.Manyoftheseworksusedensityfunctionaltheory(DFT)inthegeneralizedgradientapproximation(GGA)tocalculatethegroundstateenergiesoftheelementsand 39

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57 ]calculatedthelatticestabilitiesof78ofthemostimportantelementsandcomparedtheresultstotheSGTElatticestabilitiesofDinsdale[ 58 ].TheabinitioresultswerefoundcomparabletotheSGTElatticestabilitydataexceptforthetransitionmetals.AsanexampleofacombinedCALPHAD+abinitioapproach,Abeetal.[ 59 ]usedabinitiocalculationsoftheenthalpiesofformationofsolidphases,clustervariationmethodcalculationstoestimatethefreeenergiesatnitetemperatures,andavailableexperimentaldatainaCALPHADtypeassessmentoftheIr{Nisystem.AgoodreviewoftheapplicationofabinitioelectronicstructurecalculationsintheconstructionofphasediagramsofmetallicsystemswithcomplexphasesisgivenbySobetal.[ 60 ]. 2.1 ,theanalyticexpressionsfortheGibbsfreeenergiesofthepureelements,stoichiometricphases,solutionphases,andphasesmodeledusingthecompoundenergyformalismweredeveloped.ThetotalGibbsfreeenergyofasystem,summedoverallphases,isgivenas: 2{41 shouldbeminimizedundertheconditionsthat 2{42 meansthatthesumofthesitefractionsyiofcomponentionsublatticesofphaseisoneandEquation 2{43 calculatesthetotalnumberofmolesNofcomponentiwithinthesystem.Thisproblemcanbesolvedthoughtheintroductionof 40

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@y;si=m@Gm @m=GmXiiXsnsy;si=0(2{46)Equations 2{42 2{43 2{45 ,and 2{46 arethenfournon-linearequationswithasetofunknownsm,y;si,i,andwhichmustbesolvediterativelytoyieldthesolutionfortheGibbsfreeenergyminimumatagiventemperatureandoverallcomposition.Intherststep,aninitialguessismadeofthesitefractionsofthecomponentsonthesublatticesofthephases.Then,thechemicalpotentialsofallcomponentsiarecalculatedusingthisinitialguessofthesitefractionsforthephasesandsubstitutedintoEquation 2{45 .TheNewton-RaphsoniterativemethodisthenusedtocalculaterenedcompositionsofthephasesusingEquation 2{46 .Thesearethecompositionsofthej+1iterativestep,whicharethenusedtocalculatethechemicalpotentialsoftheelementsinthedierentphases,andsubstitutedintoEquation 2{45 again.Thisiscontinueduntilthewholecalculationconvergessuciently[ 61 ]. 41

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(FiLi)wi=vi(2{47)Theaimofthermodynamicoptimizationistoreducetheerrortoaminimumvalue.Theconditionofbesttisobtainedwhenthesumofthesquaresoftheerrorsisaminimum.Thatmeansthat: @CjnXi=1v2i=2nXi=1vi@vi 2{50 issubstitutedintoEquation 2{49 ,aseriesofmlinearequationsisgenerated: 42

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meansquareerror=nXi=1v2i 43

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SchematicoftheCALPHADmethod[ 62 ]. 44

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ThecontributiontotheexcessGibbsfreeenergyofmixingfromtherstfourtermsintheRedlich-Kisterpolynomial. 45

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BColinet CMuggianu DToopFigure2-3. GraphicalrepresentationoftheA)Kohler,B)Colinet,andC)MuggianuandD)Toopternaryextrapolationmethods.Inalldiagrams,theopencirclerepresentsapointofternarycomposition(xi,xj,xk).Thelledsquaresshowthepointsonthei{jbinarywhichwillmakeacontributiontotheGibbsfreeenergyoftheternarycomposition.TheKohlerandMuggianuextrapolationmethodsuseonlyonepointalongthei{jbinarywhereastheColinetextrapolationmethodusestwopointsalongthei{jbinary.Onepointgivesthemolefractionofiandtheotherpointgivesthemolefractionofjthatwillbeused.Therefore,intheColinetmethod,xbini+xbinj6=1.InToopextrapolation,thepointsalongthei{kandj{kbinariesarechosenatconstantxk,butthepointalongthei{jbinaryischosenusinginthesamewayastheKohlermethod. 46

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Thesurfaceofreferenceforahypotheticalcompound(A;B)p(C;D)qplottedabovethecompositionsquare. 47

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3-1 .Theandphases,whichoriginateintheNb-Albinary,showlargesolubilitiesforTiandthephaseisacontinuoussolidsolutionbetweenNbAl3andTiAl3.Theandandsolidsolutionphasesexhibitdisorder-ordertransformationstothe0and2phasesrespectively.Latticesiteoccupationsofthe0,2,andphaseshavebeenstudiedintheliterature.Banerjeeetal.[ 63 ]investigatedatwo-phase0+2alloyofcompositionTi{25.6at.%Al{10.1at.%NbandshowedthatTiatomsoccupyonesublatticeandAlatomswithsomeNbandTiatomsoccupythesecondsublatticeinthe0{phase.OrderingseparationofTiandAlhasalsobeenobservedinthe0phasebyHouandFraser[ 64 ],Chaumatetal.[ 65 ],andLeonardandVijay[ 66 ],butforthehighNbcontaining0alloyofcompositionTi{15at.%Al{68at.%Nb,orderingseparationtoformNbandAlrichsublatticesisalsopossible[ 66 ].Inthe2andphases,NbatomssubstituteontheTisites[ 67 ].Thereareadditionallytwoacceptedternaryphases.TheorthorhombicO{phase[ 68 { 83 ]withCmcmsymmetry,basedonthecompositionTi2AlNb,isformedfromternaryorderingofthe2{Ti3Allatticesites.TiwithsomeNboccupiesthe8gsite,NbwithsomeTioccupiesthe4c2site,andAloccupiesthe4c1site[ 70 ].TransformationtotheO{phasecanbeeitherfromthehighertemperature{phasethroughintermediatehexagonal(and2)phasesorfromthehighertemperature0{phasethroughamartensitictypetransformation[ 76 77 ].Thereactionkineticsofthe0toO{phasetransformationwerestudiedusingdierentialthermalanalysis[ 80 ].TheO{phaseundergoesacontinuousorderingtransformation[ 78 82 84 ]fromthedisorderedstate,whereNbandTirandomlyoccupythe8gand4c2latticesites,totheorderedstate.TheO{phaseformsgenerallyattemperatureslowerthanapproximately1273K[ 72 78 79 81 81 ]. 48

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85 { 87 ]formsfromthehighertemperatureB2phasebydisplacivetransitionsandchemicalorderingthroughthemetastable!"{phasetothestable!phase[ 87 88 ].Sadietal.[ 81 ]investigatedthelongtermheat-treatmentandcontinuouscoolingbehaviorofTi{29.7at.%Al{21.8at.%NbandTi{23.4at.%Al{31.7at.%Nballoys.After1500hoursat923K,!{phasewasdetectedintheTi{29.7at.%Al{21.8at.%Nballoy.Phasetransformationstothemetastable!'and!"andstable!{phaseswereobservedinbothalloysusingdilatometryandusingdierentialthermalanalysisforalloyTi{29.7at.%Al{21.8at.%Nb[ 81 ].Otherternaryphaseshavebeenfoundbuthaveeitherlaterbeendisprovedortheirexistenceshowntobeuncertain.AnisolatedT2phasewithbcc-typestructurewaspresentedinanisothermalsectionat1473K[ 12 ],buttheformationoftheT2phasewaslaterinvestigatedbyJacksonandLee[ 89 ]andshownnottoexist.Theordered1{phasehasbeenpresentedinvariousworks[ 90 { 93 ].ThisphaseissuggestedtoformthroughcontinuousorderingofNbatomsonTisitesinthe{phaseuntilatsucientlylargeNbconcentrations(>18at.%),Nbatomsoccupyaspeciclatticesiteinthe1{phase[ 90 ].Thestoichiometricformulaof1,Ti4Nb3Al9,aswellasthespacegroupandWyckopositionsofthestructurearegivenintheworkofChenetal.[ 94 ].AlthoughLiuetal.[ 95 ]producedthestructurefromannealinganalloyofcompositionTi{48at.%Al{10at.%Nbatat1073Kfor34hoursandhaveusedinvariantlinetheorytopredictthecrystallographicfeaturesandmorphologyofthe1precipitate[ 96 ],theexistenceornon-existenceofthe1{phaseisstillnotclearas,inanindependentwork,Jewett[ 97 ]couldnotndthe1{phaseinanalloyofcompositionTi{51at.%Al{23at.%Nbheattreatedat1473Kfor180hours. 49

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3.2.1TheLiquidusSurfaceSeveralliquidussurfacesconstructedusingtheanalysisofas-castmicrostructuresexistintheliterature.In1989,Kaltenbachetal.[ 11 ]presentedoneoftherstliquidusprojections.Thekeyfeaturesofthisliquidussurfacepresentedinclude: 1. Arelativelylargeextensionoftheprimarycrystallizationeldforthe{phase. 2. Atransitionreactionliquid+!+at2073K. 3. Atransitionreactionliquid+!+at1623K. 4. Atransitionreactionliquid+!+at1573K. 5. Aneutecticreactionbetweenliquid,,,andat1523K.Inthiswork,however,noindicationwasmadeofaprimarycrystallizationeldforthephase.Kaltenbachetal.[ 11 ]alsopresentedaScheilreactionschemefortheTi{Al{Nbsystem.Alsoin1989,Perepezkoetal.presentedaliquidusprojectionfortheTi{Al{Nbsystem[ 12 ].Inadditiontomicrostructuralobservationofascastalloys,Perepezkoetal.alsousedinformationfromdierentialthermalanalysisofselectedalloys.Inthiswork: 1. Atransitionreactionliquid+!+wasproposedbetweentheliquid,,,andphasesinsteadoftheeutecticreactionsuggestedbyKaltenbachetal.[ 11 ]. 2. Amaximumwasindicatedintheliquid++univariantline. 3. Thecharacterofthetransitionreactionbetweentheliquid,,,andwaschangedtoliquid+!+. 4. Aprimarycrystallizationeldforthephasewaspresented.Nosignicantchangesweremadetotheextensionoftheprimarycrystallizationeldofthephase,althoughtheinvariantreactionbetweenliquid,,,andphaseswasshowntotakeplaceatlowerTicontents.In1995,Zdziobeketal.constructedanewliquidussurfacewhichwasalsobasedonthemicrostructuralanalysisofascastalloys[ 13 ].Inthiscontribution,thecharacterof 50

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12 ].However,amuchsmallerextensionoftheprimarycrystallizationofthe{phasewasgiven.Therearesomeindicationsintheliteraturethattheprimarycrystallizationofthe{eldshouldextendtoevenhigherAlcontentsthanisshownintheworksofKaltenbachetal.[ 11 ],Perepezkoetal.[ 12 ],andZdziobeketal.[ 13 ].Forexample,d'Arag~aoandEbrahimireportedthatthe{transusforanalloyofcompositionTi{40at.%Al{27at.%Nbis1723K[ 4 ].WhilethisalloyisintheprimarycrystallizationeldintheliquidusprojectionsofKaltenbachetal.[ 11 ]andPerepezkoetal.[ 12 ],thisalloyisclosetotheliquid++andliquid++univariantlinesintheliquidusprojectionofZdziobeketal[ 13 ].Additionally,analloyofcompositionTi{48at.%Al{25at.%Nbwasclaimedtohavebeenheattreatedinthesinglephaseregion[ 98 ].However,thisalloyliesinthecrystallizationeldintheliquidusprojectionsofPerepzkoetal.[ 12 ]andZdziobeketal.[ 13 ]andintheprimarycrystallizationeldintheliquidussurfaceconstructedbyKaltenbachetal.[ 11 ].Asaresultoftheseinconsistencies,Leonardetal.publishedtwoworkstoclarifytheextensionoftheprimarycrystallizationofthe{phase[ 14 15 ].BasedontheascastmicrostructuresofthreealloysofnominalcompositionTi{40at.%Al{30at.%Nb,Ti{40at.%Al{35at.%Nb,andTi{40at.%Al{40at.%Nb,theywereabletoconrmthattheprimarycrystallizationofthe{phasedidindeedextendtomuchhigherAlcompositionssinceallofthesealloyssolidiedassinglephase.TheysuggestedarevisiontotheliquidussurfaceofZdziobeketal.[ 13 ]withamuchlargerextensionofthe{phaseprimarycrystallizationregion. 91 99 ],1473K[ 11 { 13 100 101 ],1423K[ 91 93 ],and1273K[ 101 ]basedonexperimentaldataareavailableintheliterature.The1673K,1473K,and1423KisothermalsectionsofChenetal.[ 91 ]andDingetal.[ 93 ] 51

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91 99 ]indicatelargesolubilitiesofTiintheandphasesandNbinthephase.Forexample,thesolubilityofTiinthephaseis28at.%Tiat17at.%Al,thesolubilityofTiinthephaseis37at.%Tiat22at.%Al,andthesolubilityofNbinthephaseis23at.%Nbat54at.%Ti.Thesehighsolubilitieswouldcorrespondtoquitelargeextensionsofthe,,andphasesintheliquidussurface,whichwouldcontradicttheworksofLeonardetal.[ 14 15 ].Oneexplanationforthesereportedlargesolubilitiesmaycomefromtheheattreatmenthistoryofthealloysunderinvestigation.Thesampleswereheattreatedat1473Kfor160hoursfollowedbyfurnacecooling,andwerethenannealedat1673Kfor4.5hours.Itisquitepossiblethattheannealingtimeat1673Kwasinsucientforequilibriumtooccur,sothattheextensionsofthe,,,andphasesmayresultfromcompositionsofthephasestakenfrominsucientlyequilibriatedmicrostructures.Themicrostructuresmayinfactrepresentmorecloselythepossibleequilibiraat1473Kthantheequilibriaat1673K. 78 ],Ti{22at.%AltoNb[ 79 102 ],andTiAltoTiNbsections[ 72 79 ].ThesesectionswereconstructedtoshowthedevelopmentofphaseequilibriaregardingformationoftheO{phase.TheverticalsectionfromAl3TitoNb3Tiisalsoreported[ 12 ].Witusiewiczetal.[ 17 ]presenteddierentialthermalanalysisdataonverticalsectionsat45at.%Al,47at.%Al,8at.%Nb,throughTi87:2Nb12:8andNb2:2Al97:8,throughTi72:8Nb27:2andTi31:6Al68:4,andthroughTi70Al30andTi51Nb49andalsoshowedthecalculatedverticalsectionsTi{27.5at.%AltoNbandAl3TitoNb3Ti. 9 ],ServantandAnsarain1998[ 10 ],andWitusiewiczetal.in2009[ 17 ].Inallthreeworks,anoverviewofthe 52

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9 ],newthermodynamicdescriptionsfortheTi{AlandAl{NbconstituentbinariesweredevelopedbutthedescriptionoftheTi{NbsystemwastakenfromKaufman[ 103 ].AlthoughthisworkwaspublishedaftertheavailabilityoftheSGTEdataforthepureelements[ 58 ],thereisnoindicationofifthisdatawereusedtorepresenttheGibbsenergiesandlatticestabilitiesofthepureelements.ThecalculatedconstituentbinariesareshowninFigure 3-1 .Intheternarysystem,theliquid,({Ti,Nb),({Ti),and(Al)phasesweremodeledassubstitutionalsolutions,theTi2Al5,TiAl2,andTi4Al3Nbphasesweremodeledasstoichiometriccompounds,andtheNb3Al,Nb2Al,(Ti,Nb)Al3,TiAl,Ti3Al,andTi2AlNbphasesweremodeledusingthecompoundenergyformalism.ThemodelingofthephasesisgiveninTable 3-2 wherethenomenclatureforthephasesistakenfromKattnerandBoettinger[ 9 ].Althoughthisdatasetisabletopredictkeyexperimentalphaseequilibriaat1473Kand1373Kthatwereavailableatthattime,thecalculatedliquidussurface,showninFigure 3-2 ,indicatestoolargeextensionsofprimarycrystallizationoftheNb3AlandNb2Alphases,and,consequently,aninsucientextensionoftheprimarycrystallizationofthe({Ti,Nb)phase.However,thecalculatedliquidussurfaceisingoodqualitativeagreementwiththeliquidussurfaceofKaltenbachetal.[ 11 ]i.e.,thecalculatedliquidussurfacepredictscorrectlythephaseswhichoccurattheinvariantreactions.Insomecases,however,thecharacteroftheinvariantreactionsaredierent.Ofnoteisthatthisdatasetcalculatesaeutecticreactionbetweentheliquid,Nb2Al,(Ti,Nb)Al3,andTiAlphases. 53

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58 ]andtheparametersfortheTi{Al,Al{Nb,andTi{Nbconstituentbinarysub-systemsweretakenfromSaunders[ 18 ],ServantandAnsara[ 104 ],andHariKumar[ 105 ]respectively.AlthoughSaundersdidnotoriginallymodelthe!2order{disordertransformationintheTi{Alsystem,adescriptionforthistransformationwasincludedinthedatasetfortheconstituentTi{AlbinarywhichwasusedintheTi{Al{NbternaryofServantandAnsarawithoutchangingthephasediagram[ 10 ].ThecalculatedconstituentbinariesareshowninFigure 3-3 .Intheternarysystem,theliquid,,,and(Al)phasesweremodeledassubstitutionalsolutions,theTi5Al11,TiAl2,andTi4Al3Nbphasesweremodeledasstoichiometriccompounds,andthe{Nb3Al,{Nb2Al,0,{(Ti,Nb)Al3,{TiAl,2{Ti3Al,O1{Ti2AlNbandO2{Ti2AlNbphasesweremodeledusingthecompoundenergyformalism.Phaseswithorder{disordertransformations(disorderedto0anddisorderedto2{Ti3Al)weremodeledusingthesameGibbsenergyfunction.Althoughthereisanorder{disordertransformationfromthedisorderedO1{phasetotheorderedO2{phase,thisorder{disordertransformationwasnotmodeled,andbothphasesweretreatedasseparatephasesinthedataset.Inalaterwork,ServantandAnsaramodeledtheorder{disordertransitionoftheO{phaseusingthesameGibbsenergydescriptionforbothphases[ 106 ].ThethermodynamicmodelingofthephasesusedinthisdescriptionisgiveninTable 3-3 10 ]toassessitssuitabilityforuseinpredictingphaseequilibriaintheTi{Al{Nbsystem.Thecalculatedisothermalsectionsat1273K(Figure 3-4 ),1373K(Figure 3-5 ),1473K(Figure 3-6 ),1673K(Figure 3-7 ),1813K(Figure 3-8 ),and1923K(Figure 3-9 )arepresented.The 54

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107 ]andthe++phaseequilibriafromEckertetal.[ 108 ]at1373Kbecameavailableonlyafterthedatasetwaspublished,thecalculatedisothermalsectionat1373K(Figure 3-5 )showsthatthedatasetpredictssuchphaseequilibria.However,thedatabasecouldnotcorrectlypredictallexperimentallyobservedphaseequilibriaathighertemperatures.Forexample,althoughthe+tie-lineat1923KfromMenonetal.[ 109 ]couldbecalculated(Figure 3-9 ),the++threephaseequilibriaat1813KobservedinanalloyofcompositionTi{25at.%Al{60at.%Nb,whichwasheattreatedatthistemperatureandthenquenchedbyLeonardetal.[ 14 ],couldnot.Instead,analloyofthiscompositionisinthe+twophaseeldasisindicatedinthecalculatedisothermalsectionat1813K(Figure 3-8 ).Acomparisonofthecalculatedisothermalsectionat1673KwiththephaseequilibriadatadeterminedbyWangetal.[ 99 ]indicatesremarkableinconsistencies.OnepossiblereasonfortheinconsistenciescouldbethatWangetal.[ 99 ]didnotheattreatthealloysforalongenoughtimeat1673Ktoyieldtheequilibriummicrostructures.Therefore,thecalculatedisothermalsectionwithexperimentaldataat1673KinFigure 3-7 isincludedonlytoillustratethedierencebetweenthecalculationsandtheexperimentaldata.Whenonecomparesthecalculatedliquidussurface(Figure 3-10 )withexperimentaldata,moreinconsistenciesappear.Forexample,thealloysofcompositionTi{40at.%Al{30at.%Nb,Ti{40at.%Al{35at.%Nb,andTi{40at.%Al{40at.%Nb,whichhavebeenshowntosolidifyassinglephasebyLeonardetal.[ 14 15 ],areintheprimarycrystallizationeldofthephase.Additionally,thealloyofcompositionTi{48at.%Al{25at.%Nb,whichwasheattreatedbyFengetal.inthesinglephase{eld[ 98 ],isintheprimarycrystallizationeldofthe{phase.Aswell,thealloyofcompositionTi{40at.%Al{27at.%Nb,whichhasbeenshowntohavea{transustemperatureof1723K 55

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4 ],isintheprimarycrystallizationeldofthe{phase.Allinconsistenciesbetweenthecalculatedliquidussurfaceandtheexperimentaldataindicatethatthecalculatedprimarycrystallizationeldofthe{phasedoesnotextendtohighenoughAlcompositions,andthattheprimarycrystallizationeldsofthe,,andphasesextendtoomuchintotheternary.Takingintoaccounttheinconsistenciesbetweenthecalculatedliquidussurfaceandexperimentalwork,aswellastheinconsistencyofthecalculatedisothermalsectionat1813K,thethermodynamicdescriptionoftheTi{Al{NbsystemdevelopedbyServantetal.[ 10 ]shouldbere-optimized.Theprimarycrystallizationeldofthephaseshouldbeextendedtobeinagreementwiththeliterature,butthenewdatasetshouldstillbeabletopredict,withingoodagreement,theexperimentalphaseequilibriaatlowertemperatures. 17 ]foruseastheconstituentternaryinthethermodynamicdescriptionofthequaternaryAl{B{Nb-Tisystem.Inthiswork,theSGTEdataforthepureelementsAl,Nb,andTiweretakenfromDinsdale[ 58 ]andthedescriptionoftheTi{NbsystemwastakenfromHariKumar[ 105 ].However,newdescriptionsforboththeTi{Al[ 110 ]andAl{Nb[ 17 ]binariesweredeveloped. 111 ],highlightinginconsistenciesbetweenvariousrepresentationsoftheTi{Alphasediagramandtheavailableexperimentaldata.Thekeypointsofinterestare: 1. Thepresenceofameltingpointmaximumforthephaseat8.5at.%AlbasedontheworkofOgdenetal.[ 112 ],whichwasacceptedintheevaluationofSchusterandPalm[ 111 ].Themeltingpointmaximumforthephaseiscalculatedat20at.%AlinthedescriptionofSaunders[ 18 ]. 56

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Thepresenceof!0orderingintheTi{richregion.OrderingofthephasewasinvestigatedintheworkofOhnumaetal.[ 113 ].ThiswasnotacceptedinthecriticalevaluationofSchusterandPalm[ 111 ]andnotmodeledinthethermodynamicassessmentofSaunders[ 18 ]. 3. Theperitectoidformationofthe2{phaseaccordingtothereaction+!2orthecongruentformationofthe2{phasefromthe{phase.Theperitectoidformationof2wasacceptedintheevaluationofSchusterandPalm[ 111 ]butnotreproducedusingthethermodynamicdescriptionofSaunders[ 18 ]. 4. TheexistenceandstabilityoftheTi3Al5{phase.ThisphasewasconsideredtobemetastableintheworkofSchusterandPalm[ 111 ]andwasnotincludedinthedescriptionofSaunders[ 18 ]. 5. ThepresenceoftwophasesTi1xAl1+xandTi2+xAl5x[ 114 115 ]oroneone-dimensionalanti-phasestructure(1d-APS)[ 111 ]intheregionbetweentheAl{richboundaryofthe{TiAlphaseandtheTi{richboundaryofthe{TiAl3phaseattemperaturesbetween1450Kand1720K.ThethermodynamicdescriptionofSaundersincludesonlythestoichiometricTi5Al11phaseinthisregion[ 18 ].In2008,Witusiewiczetal.[ 110 ]optimizedathermodynamicdescriptionfortheTi{AlsystemusinginformationfromthecriticalassessmentofSchusterandPalm.Thenewdescription: 1. Includesameltingpointmaximumofthe{phaseat9.6at.%Aland1963K. 2. Modelsthesecondorderorder{disorder!0transformationusingthesameGibbsenergyfunctionforthedisorderedandorderedphases. 3. Calculatestheperitectoidformationof2accordingtothereaction+!2at1432K. 4. ModelstheTi3Al5{phaseasastoichiometricphasewhichformsat1083K. 5. Modelsthe1d-APSas{Ti2+xAl5x.Witusiewiczetal.didnotmodeltheorder{disordertransformationofthedisorderedphasetotheordered2phaseusingthesameGibbsfreeenergyfunction.ThecalculatedbinaryTi{AlphasediagramusingthisnewdescriptionisshowninFigure 3-11 alongwiththecalculatedTi{AlbinaryfromSaundersforcomparison. 57

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17 ].Theoptimizationofthissystemtookintoaccountdataontheenthalpiesofformationofthe{Nb3Al,{Nb2Aland{NbAl3phasesfromMahdouketal.[ 116 ]andGeorgeetal.[ 117 ]andofthe{phaseat3and6at.%AlwhichwasalsomeasuredbyGeorgeetal.ThesedatawerenotavailabletoServantandAnsaraatthetimeoftheirNb{Alassessment[ 104 ].Thecomparisonbetweenthecalculated[ 17 104 ]andexperimentallydeterminedenthalpiesofformationat298KmeasuredbyGeorgeetal.[ 117 ]usingtheemfmethodareshowninFigure 3-12 andthecomparisonbetweenthecalculatedandexperimentallydeterminedenthalpiesofformationofthealloysat1699Kand1533KcomparedtotheresultsofMahdouketal.[ 116 ]measuredusingsolutioncalorimeteryareshowninFigure 3-13 .ThecalculationsshowthatthedatasetofWitusiewiczetal.[ 17 ]isinbetteragreementwiththeenthalpyofformationdataofthephasesintheAl{NbsystemthanthedatasetofServantandAnsara[ 104 ].ThecalculatedAl{NbphasediagramsfromtheassessmentsofWitusiewiczetal.[ 17 ]andServantandAnsara[ 104 ]areshowninFigure 3-14 forcomparison.Themaindierencesbetweenthecalculatedphasediagramsarethetemperatureofcongruentmeltingofthe{phaseandtheeutecticreactiontemperaturebetweentheliquid,,andphases.Servantetal.[ 104 ]tookintoaccounttheworkofJordaetal.[ 118 ],who,usinglevitationthermalanalysis,measuredthecongruentmeltingpointofthe{phaseas19535Kandtheeutecticreactiontemperaturebetweentheliquid,,andphasesas18635K.Therefore,thecalculationoftheeutecticreactiontemperaturebetweentheliquid,,andphasesiswithintheexperimentalerrorofJordaetal.[ 118 ],andthecongruentmeltingofthe{phaseisinbetteragreementwiththeworkofJordaetal.[ 118 ]thanisthecongruentmeltingofthe{phasecalculatedusingthedatasetofWitusiewiczetal.[ 17 ].Witusiewiczetal.[ 17 ]usedhigh-temperaturepyrometrytodeterminethecongruentmeltingofthe{phaseanddierentialthermalanalysistomeasuretheeutectic 58

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17 ]aregiveninTable 3-4 .Inthiswork,theliquid,,andphasesweremodeledassubstitutionalsolutions,theTi4Al3NbandTi3Al5phasesweremodeledasstoichiometriccompounds,andthe{Nb3Al,{Nb2Al,{(Ti1xNbx)Al3,{TiAl,0,2,{Ti2+xAl5x,TiAl2,O1{Ti2AlNb,andO2{Ti2AlNbphasesweretreatedusingthecompoundenergyformalism.Althoughthereisanorder{disordertransformationforthe(!2),(!0),andO(O1!O2)phases,onlytheand0phasesweremodeledusingthesameGibbsenergyfunction.CalculationswerealsoperformedwiththisdatasettoevaluateitsabilitytocalculatecorrectlythehightemperaturephaseequilibriaintheTi{Al{Nbsystem.Thecalculationsindicatethattherearestillsomeinconsistencieswiththeliterature.Forexample,thecalculated1813Kisotherm(Figure 3-15 )showsthatthisdatasetstillcannotpredictthe++three-phaseeldforanalloyofcompositionTi{25at.%Al{60at.%NbwhichwasdeterminedusingheattreatmentandquenchexperimentsbyLeonardandVasudevan[ 14 ].Thecalculatedliquidussurface(Figure 3-16 )alsoindicatesthattheprimarycrystallizationeldofthe{phasedoesnotextendtohighenoughAlcompositionssincesomeofthealloyswhichhavebeenshowntosolidifyas{phaseareintheprimarycrystallizationeldsofand.Thesolidussurface(Figure 3-17 )showsthatnotonlydoestheprimarycrystallizationeldofthephasenotextendtohighenoughAlcompositions,butalsothesolidus,shownastheredlineinFigure 3-17 ,doesnotextendtohighenoughAlcompositions.Allalloystotheleftofthesoliduswillsolidifyassinglephasefromtheliquid,andalloystotheimmediaterightwill,oncooling,passthroughathreephaseeldliquid++xwherexcanbeeitherthe,the,the,or 59

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60

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PhasesintheTi{Al{Nbsystem. PhasePrototypePearsonsymbolSpacegroupStructurereport

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BAl{Nb CTi{NbFigure3-1. ThecalculatedA)Ti{Al,B)Nb{Al,andC)Ti{NbconstituentbinarysystemsusedinthethermodynamicdatasetofKattnerandBoettinger.Figurestakenfrom[ 9 ]. 62

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ThermodynamicmodelingofthephasesusedinthedescriptionofKattnerandBoettinger[ 9 ]. PhaseThermodynamicModel Liquid(Al,Nb,Ti)({Ti,Nb)(Al,Nb,Ti)({Ti)(Al,Nb,Ti)(Al)(Al,Nb,Ti)Ti2Al5(Ti)2(Al)5TiAl2(Ti)(Al)2Ti4Al3Nb(Ti)4(Al)3NbNb3Al(Al,Nb,Ti)0:75(Al,Nb,Ti)0:25Nb2Al(Al,Nb,Ti)0:533(Al,Nb,Ti)0:333(Nb)0:134(Ti,Nb)Al3(Al,Nb,Ti)0:25(Al,Nb,Ti)0:75TiAl(Al,Nb,Ti)0:5(Al,Nb,Ti)0:5Ti3Al(Al,Nb,Ti)0:75(Al,Nb,Ti)0:25Ti2AlNb(Nb,Ti)0:5(Nb,Ti)0:25(Al)0:25

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CalculatedliquidususingthethermodynamicdescriptionofKattnerandBoettinger.Figuretakenfrom[ 9 ]. 64

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BAl{Nb CTi{NbFigure3-3. ThecalculatedA)Ti{Al,B)Nb{Al,andC)Ti{NbconstituentbinarysystemsusedinthethermodynamicdatasetofServantandAnsara[ 10 ]. 65

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ThermodynamicmodelingofthephasesintheTi{Al{NbsystemofServantandAnsara[ 10 ]. PhaseThermodynamicModel Liquid(Al,Nb,Ti)()(Al,Nb,Ti)()(Al,Nb,Ti)(Al)(Al,Nb,Ti)Ti2Al5(Ti)2(Al)5TiAl2(Ti)(Al)2Ti4Al3Nb(Ti)4(Al)3Nb{Nb3Al(Al,Nb,Ti)0:75(Al,Nb,Ti)0:25{Nb2Al(Al,Nb,Ti)0:533(Al,Nb,Ti)0:333(Nb)0:134{(Ti,Nb)Al3(Al,Nb,Ti)0:25(Al,Nb,Ti)0:75TiAl(Al,Nb,Ti)0:5(Al,Nb,Ti)0:50(Al,Nb,Ti)0:5(Al,Nb,Ti)0:52{Ti3Al(Al,Nb,Ti)0:75(Al,Nb,Ti)0:25O1{Ti2AlNb(Al,Nb,Ti)0:75(Al,Nb,Ti)0:25O2{Ti2AlNb(Al,Nb,Ti)0:50(Al,Nb,Ti)0:25(Al,Nb,Ti)0:25 Isothermalsectionat1273KcalculatedusingthethermodynamicdescriptionofServantandAnsara[ 10 ].Theexperimentaldataidentiedas[1992Men]and[1998Hel]refertotheworksofMenonetal.[ 109 ]andHellwigetal.[ 101 ]respectively. 66

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Isothermalsectionat1373KcalculatedusingthethermodynamicdescriptionofServantandAnsara[ 10 ].Theexperimentaldataidentiedas[1999Eck]and[2002Leo]refertotheworksofEckertetal.[ 108 ]andLeonardetal.[ 107 ]respectively. Figure3-6. Isothermalsectionat1473KcalculatedusingthethermodynamicdescriptionofServantandAnsara[ 10 ].Theexperimentaldataidentiedas[1992Men],[1995Zdz],and[1998Hel]refertotheworksofMenonetal.[ 109 ],Zdziobeketal.[ 13 ],andHellwigetal.[ 101 ]respectively. 67

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Isothermalsectionat1673KcalculatedusingthethermodynamicdescriptionofServantandAnsara[ 10 ].Theexperimentaldataidentiedas[1998Wan]refertotheworkofWangetal.[ 99 ].Althoughthereseemstobequitealargedisagreementbetweenthecalculatedisothermalsectionandthesuperimposedexperimentaldata,itispossiblethatWangetal.didnotheattreattheiralloysforsucientlylongperiodsoftimeat1673Ktoachievetheequilibriummicrostructures. Figure3-8. Isothermalsectionat1813KcalculatedusingthethermodynamicdescriptionofServantandAnsara[ 10 ].Theexperimentaldataidentiedas[2000Leo]referstodataobtainedfromLeonardandVasudevan[ 14 ].Inthiswork,analloyofcompositionTi{25at.%Al{60at.%Nbwhichwasheattreatedat1813Kandquenchedwasshowntobeinthethreephaseregion++However,calculationswiththisdatasetindicatethatanalloyofthiscomposition,shownbythebluetriangle,isinthe+two-phaseregion. 68

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Isothermalsectionat1923KcalculatedusingthethermodynamicdescriptionofServantandAnsara[ 10 ].Theexperimentaldataidentiedas[1992Men]referstotheworkofMenonetal.[ 109 ]. Figure3-10. LiquidussurfacecalculatedusingthethermodynamicdescriptionofServantandAnsara[ 10 ].Theexperimentaldataidentiedas[1992Fen],[1996Ara],and[2000Leo]refertotheworksofFengetal.[ 98 ],Leonardetal.[ 15 ],andd'Arag~aoandEbrahimi[ 4 ]respectively.Thepositionsofalloy11withnominalcompositionTi{45at.%Al{18at.%Nbandalloy12withnominalcompositionTi{45at.%Al{27at.%Nbarealsoindicated.Thesealloyshavealsobeenshowntosolidifyrstas{phasefromtheliquid.Althoughthesealloyswillbediscussedlaterinthiswork,theyareindicatedhereforconvenience. 69

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BFigure3-11. ThecalculatedTi{AlphasediagramusingthedescriptionsofA)Witusiewiczetal.[ 17 ].ThecalculatedTi{AlphasediagramofSaunders[ 18 ](B)isrepeatedhereforcomparison. 70

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ComparisonbetweencalculatedandexperimentallydeterminedstandardenthalpiesofformationofalloysintheNb{Alsystemat298K.ThereferencestateforAlisthefcc{phaseandthereferencestateforNbisthebcc{phase.ThedashedlineshowstheresultsofthecalculationusingthedatasetofServantandAnsara[ 104 ]andthesolidlineshowstheresultsofthecalculationusingthedatasetofWitusiewiczetal.[ 17 ].ExperimentalpointsweretakenfromGeorgeetal.[ 117 ]. 71

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ComparisonbetweencalculatedandexperimentallydeterminedenthalpiesofformationofalloysintheNb{Alsystem.ThereferencestateforAlistheliquidandthereferencestateforNbisthebcc{phase.ThedashedlineshowstheresultsofthecalculationusingthedatasetofServantandAnsara[ 104 ]andthesolidlineshowstheresultsofthecalculationusingthedatasetofWitusiewiczetal.[ 17 ].Bothcalculationswereperformedat1669K.ExperimentalpointsweretakenfromMahdouketal.[ 116 ]. 72

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ThecalculatedNb{AlphasediagramsusingthethermodynamicdescriptionsofWitusiewiczetal.[ 17 ]andServantandAnsara[ 104 ].Themaindierencesbetweenthephasediagramsarethecongruentmeltingtemperatureofthe{phaseandtheeutecticreactiontemperaturebetweentheliquid,,andphases.ServantandAnsaraacceptedthelevitationthermalanalysismeasurementsofJordaetal.[ 118 ],whileWitusiewiczetal.determinedthesetemperaturesusingopticalpyrometryanddierentialthermalanalysis. 73

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ThermodynamicmodelingofthephasesintheTi{Al{NbsystemofWitusiewiczetal.[ 17 ]. PhaseThermodynamicModel Liquid(Al,Nb,Ti)()(Al,Nb,Ti)1(Va)3()(Al,Nb,Ti)1(Va)0:5(Al)(Al,Ti)1(Va)1Ti4Al3Nb(Ti)4(Al)3NbTi3Al5(Ti)3Al5{Nb3Al(Al,Nb,Ti)0:75(Al,Nb,Ti)0:25{Nb2Al(Al,Nb,Ti)0:533(Al,Nb,Ti)0:333(Nb,Ti)0:134{(Ti1xNbx)Al3(Al,Nb,Ti)1(Al,Nb,Ti)3{TiAl(Al,Nb,Ti)1(Al,Nb,Ti)10(Al,Nb,Ti)0:5(Al,Nb,Ti)0:5(Va)32{Ti3Al(Al,Nb,Ti)3(Al,Nb,Ti)1{Ti2+xAl5x(Al,Nb,Ti)2(Al,Nb,Ti)5TiAl2(Al,Nb,Ti)1(Al,Nb,Ti)2O1{Ti2AlNb(Al,Nb,Ti)0:75(Al,Nb,Ti)0:25O2{Ti2AlNb(Al,Nb,Ti)0:50(Al,Nb,Ti)0:25(Al,Nb,Ti)0:25 Isothermalsectionat1813KcalculatedusingthethermodynamicdescriptionofWitusiewiczetal.[ 17 ].Theexperimentaldataidentiedas[2000Leo]referstodataobtainedfromLeonardandVasudevan[ 14 ].Inthiswork,analloyofcompositionTi{25at.%Al{60at.%Nbwhichwasheattreatedat1813Kandquenchedwasshowntobeinthethreephaseregion++However,calculationswiththisdatasetindicatethatanalloyofthiscomposition,shownbythebluetriangle,isinthe+two-phaseregion. 74

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LiquidussurfaceintheTi{Al{NbsystemcalculatedusingthethermodynamicdescriptionofWitusiewiczetal.[ 17 ].Allalloysindicatedonthediagramshouldbeintheprimarycrystallizationeldofthe{phase.Thissuggeststhattheprimarycrystallizationeldsoftheandphasesextendtoomuchintotheternary. Figure3-17. SolidussurfaceintheTi{Al{NbsystemcalculatedusingthethermodynamicdescriptionofWitusiewiczetal.[ 17 ].Foranalloytosolidifyassinglephase,notonlyshoulditbelocatedintheprimarycrystallizationeldof,butitshouldalsobelocatedtotherightofthesolidus. 75

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4-1 .Theseworksclarifythecrystallographyofstableandmetastablephases,presentequilibriumtielineandtietriangledata,andsuggestthetopographyoftheliquidussurface.Therefore,althoughitisnotpossiblethatallworksaredirectlytakenintoaccountintheoptimization,allworksareimportantforafullunderstandingofthephaseequilibriaintheTi{Al{Nbsystem.Theworkswhichwereusedinthecurrentoptimizationareidentiedwithanasterisk.SeveralalloyswereinvestigatedexperimentallyusingthetechniquesofXRDandHTXRDforphaseidentication,SEMandTEMformicrostructureanalysis,EPMAforphasecompositiondetermination,andDTAforphasetransformationtemperatures.AllexperimentalworkwasperformedattheUniversityofFloridabyOrlandoRios,MichaelS.Kesler,andSonalikaGoyelunderthesupervisionandadviceofProf.FereshtehEbrahimi.ThespecicdetailsonalloypreparationandanalysisaregiveninthePh.D.thesisofRios[ 119 ].OnlytheresultsofthisworkwhichweredirectlyusedintheoptimizationoftheparametersfortheTi{Al{Nbsystemwillbediscussedhere.Asthissectionfocusesontheanalysisoforiginalexperimentaldata,whichweregivenintemperatureunitsofoC,alltemperatureunitsinthissectionwillalsobegiveninoCforconvenience. 119 ]weretakenintoaccountintheoptimization.ThebulkcompositionsofthealloysandthecompositionoftheequilibriumphasesaregiveninTable 4-2 .AlloysA2andA3werethreephase++at1510oCand1410oCwhereasalloyA133wastwophase+at1520oCand1420oC. 76

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4-1 .TheDTAcurveshowstwoexothermicpeaksoncoolingandtwooverlappedpeaksonheatingfollowedbyanextendedpeakandareturntothebaseline.ThroughHTXRD,thedetailsofwhichcanbefoundin[ 119 ],combinedwiththeanalysisoftheas-castmicrostructureandtheanalysisofasamplewhichwasheatedat1500oCfor1hour,cooledto1343oCat12oC/min,heldat1343oCfor2hoursandthenquenched,thephasesbeforeandaftereachpeakonheatingwereidentied.At1100oC,onlytheandphasesarepresent.Asthealloyisheated,phaseformsaccordingtothereaction+!++toproducethethreephaseeldmarkedas++.Astemperatureincreases,thephasedisappearsaccordingtothereaction++!+leavingonlytheandphases.Astemperatureisfurtherincreased,theamountofphasedecreasesandtheamountofphaseincreasesuntilthetransusisreached.Atthistemperature,thereaction+!takesplacesothatathighertemperatures,thealloyisinthesinglephaseregion.Sincethetransitiontemperaturesofalloy11measuredbyDTAwillbeusedforoptimization,anassessmentoftheliteraturemustbemadetoevaluatetheagreementbetweentheliteratureandtheexperimentaldata.Hellwigetal.[ 101 ]heattreatedan 77

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4-2 .AlsoincludedinFigure 4-2 isthecompositionofalloy11.Alloy11isshowntoexistinthe++threephaseregionat1200oCmeasuredbyHellwigetal.[ 101 ].Thisgureindicates,then,thatacorrectinterpretationoftheDTAcurveforalloy11shouldyieldaresultthatthetransformation+!++takesplaceatatemperaturelowerthan1200oC.Also,sincealloy11islocatedclosetothe+boundaryofthe++threephasetriangle,thetransition++!+shouldtakeplaceatatemperatureminimallyhigherthan1200oC.Ifthesearenotthecases,theneitherthemeasuredtransformationtemperaturesofalloy11ortheexperimentalworkofHellwigetal.[ 101 ]wouldhavetoberejectedintheassessment. 120 ].OnepeakfunctionwhichcanbeusedistheVoigtfunction,whichisacombinationoftheLorentzianandGaussianpeakfunctions.Itisexpressedas: 78

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4-3 .ThecalculationwasperformedusingtheprogramMicrocalOrigin.Theseparatedpeakrepresentingthe+!++transitionisshowninredandtheseparatedpeakrepresentingthe++!+transitionisshowningreen.Thecumulativepeakisshowninblue.ThepeakseparationresultsindicatethatthereisagoodagreementbetweentheanalyticallydeterminedoverlappedDTApeakusingthetwoVoigtfunctionsandthemeasuredoverlappedpeak.Eachpeakwasseparatelyanalyzedtodeterminetheonsettemperaturesfortherespectivephasetransformations(Figure 4-4 and 4-5 ).Thephasetransformationtemperaturewaschosenasthepointatwhichtherstderivativecurvedeviatedfromzero.Usingthisanalysis,thetransformationtemperatureforthereaction+!++wasselectedas1196oCandthetransformationtemperatureforthereaction++!+wasselectedas1241oC.ThesevaluesareingoodagreementwiththeexperimentaldataofHellwigetal.[ 101 ],sincethe+!++transformationoccursatatemperaturelowerthan1200oCandthe+!++transformationtakesplaceat1241oC,whichisnotmuchhigherthan1200oC,indicatinggoodagreementwiththepositionofalloy11beingclosetothe+boundaryofthe++threephasetriangleofHellwigetal.[ 101 ].Onecleardisadvantageofthismethodisthatonlysymmetricpeakscanbesimulated.Inreality,however,thepeakforaphasetransformationmeasuredusingDTAistypicallyasymmetric.Therefore,someerrorisintroducedintotheevaluationoftheonsettemperatures.Theotherpossibilitywhichcouldbeusedtoseparatetheoverlappedpeaksintotwodistinctpeaksistousealowerheatingrate.However,thismethodistime 79

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120 ]. 4-6 .TheDTAcurveshowstwoexothermicpeaksoncoolingandtwoorthreeoverlappedendothermicpeaksonheating.Throughanalysisoftheas-caststructureforalloy12,aswellastheanalysisofmicrostructuresfromheattreatingandquenchingexperimentsat1343oC,thephaseequilibriaindicatedonFigure 4-6 wasconrmed. 4-7 ,multiplemethodspeakseparationmethodswereevaluated.Intherstapproach,theoverlappedpeakswerettothreeVoigtpeakfunctionsandalladjustableparametersfortheVoigtfunctions(veparametersperpeak,thereforefteenparametersinall)wererened.ThisrstapproachwasperformedtotakeintoaccountthepossibilityoftheshallowpeakshoulderalsobeingavalidDTAheatowresponsetoanactualphasetransformation.TheresultofthistisshowninFigure 4-8 .Theresultindicatesthattwophasetransformations,oneshownbythebluepeakandtheothershownbythegreenpeak,arefairlywellmodeled.However,theshallowpeak 80

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4-9 .ThistshowsthatthereisalsoareasonablygoodagreementbetweentherenedcumulativesignalandtheoriginalDTAsignal.Usingthistprocedure,however,therstandsecondpeaks(shownastheredandbluepeaksrespectively)arereproducedwiththesametransitiontemperaturewhenthetransitiontemperatureisdeterminedusingthemethodofrstdeviationfromthebaselinecalculatedusingtherstderivativeofthecurves.Thisisalsoahighlyimprobablesolutionasitwouldsuggestthatthealloyisinthe++threephaseregionforonlyamaximumof5oC.Inthelastapproach,onlytwoVoigtpeaksweremodeledandallparameterswererened.TheresultofthistisshowninFigure 4-10 .ThistshowsthatthereisverygoodagreementbetweentherenedcumulativesignalandtheoriginalDTAsignal.Usingthistprocedure,theshallowpeakisnotreproduced.However,therstpeak,showningreen,isquiteextendedi.e.,thefullwidthathalfmaximumismorethancanbeexpected.Thereasonforthisextensionisthattheparametersfortherstpeakarerenedtotakeintoaccounttheshallowpeak.Thisresultingpeakextensionintroducesquitealargeerrorinthedeterminationofthetransitiontemperatureforthispeak.Therefore,althoughthissolutionisnotimprobable,theerrorinthetransitiontemperatureissimplytoolargeforthistemperaturetobeusedinanyoptimization.Sincenoneoftheabovepeakttingmethodscanbeusedwithaccuracy,thetransitiontemperatureswerechosenas1269oCforthe+!++transformationand1323oCforthe++!+phasetransformation.Thetemperatureofthe+!++transformationcorrespondstotheonsetpointoftherst 81

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4-11 .Someerrorisnecessarilyintroducedwiththesetransitiontemperaturechoices.Forexample,althoughthemaximumischosenasthestarttemperatureforthe+!++reaction,itisclearthatanextrapolationofthesecondpeaktolowertemperatures,intheabsenceoftherstpeak,wouldgivethestarttemperatureforthereactionatalowertemperature.Thephasetransformationtemperaturesforalloy11andalloy12whichwereusedforoptimizationaresummarizedinTable 4-3 82

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ReferenceExperimentalMethodsPhaseEquilibriaTemperature[K] [ 63 ]Metallography(TEMwithEDX)Siteoccupationsin0Temperaturenotindicated[ 64 ]*TEMSublatticeoccupationof0forselectedalloys[ 65 ]NeutrondiractionSublatticeoccupationof0forselectedalloys.!0transformationtemperaturesforselectedalloys1873,1573,1273[ 66 ]*Metallography(TEMwithEDS)XRDALCHEMISiteoccupationsintheand0phasesSamplesheattreatedat1273K[ 68 ]*Metallography(OM,TEM)IdenticationoftheOphase973K[ 69 ]Metallography(OM,SEM,TEM)XRD2{01273,1173,1073[ 70 ]Metallography(OM,TEM)NeutrondiractionStructurerenementoftheOphase973[ 71 ]Metallography(SEM,TEM)XRDTemperatureofformationoftheOphaseinaTi{24.4at.%{15.5at.%Nballoy1373{1173[ 72 ]Metallography(OM,SEM,TEM)XRD0{O973[ 73 ]Metallography(TEM)!{O(notie-linedata)1173,1123,1073,1023,973[ 74 ]Metallography(TEM)SiteoccupationsoftheOphaseformedinthealloyTi{24at.%Al{27at.%Nb[ 75 ]Metallography(OM,SEM,TEM)EectofoxygencontaminationonphaseequilibriainalloysTi{22at.%Al{23at.%NbandTi{22at.%Al{27at.%Nb1173,1073[ 77 ]Metallography(TEM)TransformationpathstotheOphase1673,1373,973[ 78 ]Metallography(SEM,TEM)2{0,0{OPeritectoidformationoftheOphase:0+2!O(1273K)1473,1423,1398,1373,1323,1273,1223,1173,1073,973 83

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ReferenceExperimentalMethodsPhaseEquilibriaTemperature[K] [ 81 ]Metallography(OM,SEM,TEM)XRD,DTA,DilatometryContinuouscoolingtrasformationsofTi{27.9at.%Al{21.8at.%NbandTi{23.4at.%Al{31.7at.%Nballoys1573,1533,1173,973[ 83 ]Metallography(OM,SEM,TEM)XRD,Dilatometry0{OPhasetransformationtemperaturesforthe0,2,andOphases1623,1173,1073,973[ 85 ]Metallography(TEM)Formationofthe!phaseinalloyswith25at.%Aland5-14at.%Nb773,673[ 86 ]Metallography(SEM,EPMA,TEM)XRDIdenticationofthemetastable!"and!phases0{{2and!{{21673,1373,973[ 86 ]Metallography(OM,TEM)Derivativesofthe!phase1673,1373,973[ 87 ]XRDCrystallographyofthe!phaseinanalloyTi3Al2:25Nb0:75[ 12 ]*Metallography(OM,SEM,EPMA,TEM)XRD,DTALiquidussurface[ 98 ]*Metallography(TEM)DeterminationofasinglephasephaseeldforanalloyofcompositionTi{48at.%Al{26at.%Nb1673[ 89 ]*Metallography(OM,SEM,TEM,EPMA)DTA{1548,1473[ 90 ]Metallography(SEM,TEM,HRTEM,EDS)Identicationof11523followedbyfurnacecooling[ 91 ]*Metallography(SEM,EPMA)XRD,DSC,DTA{,{,{1,{,{,{2,{,-,{1,{1,1{,2{,{TiAl21673,1323,1273[ 92 ]Metallography(SEM,EDX,TEM)XRDExistenceofthe1phase1423[ 93 ]Metallography(OM,EPMA)XRD{1,{,{{114231273 84

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ReferenceExperimentalMethodsPhaseEquilibriaTemperature[K] [ 94 ]Metallography(TEM)XRDCrystallographyofthe1phase1673[ 95 ]Metallography(TEM)CRDCrystallographyof1precipitatesinthealloyTi{48at.%Al{10at.%Nb1073[ 97 ]*Metallography(SEM,EDX),XRDNonexistenceofthe1phase{1423[ 11 ]*Metallography(EPMA)XRDLiquidussurface{{and{{1473[ 13 ]*Metallography(SEM,EPMA,TEM),XRDPrimarycrystallizationeldsontheliquidussurface{,{,{,{1473[ 99 ]Metallography(OM,SEM,EDS),XRD,DTACompleteisothermalsection1673[ 100 ]*Metallography(OM,SEM,TEM,EPMA)Isothermalsection1473[ 101 ]*Metallography(OM,EPMA,SEM,TEM),XRD{{,{,{{,0{,0{{,0{,0{2,0{{,{,{TiAl2,{liquid14731273[ 121 ]*Metallography(SEM,TEM,EPMA),XRDSinglephase0,2{01473[ 122 ]Metallography(OM,SEM,TEM,EPMA),XRDVolumefractionof2andinaTi{47.2at.&Al{2.14at.%NballoySamplesannealedat1323[ 109 ]*Metallography(OM,SEM,EPMA,TEM),XRD{1923,1473,1273[ 123 ]*Metallography(SEM,EPMA)0{{1473[ 102 ]Metallography(OM,SEM,TEM,EDSinTEM,EPMA),XRDTheTi{22at.%Al{Nbverticalsection1311,1273,1173,1088,1033,923[ 124 ]*Metallography(OM,SEM,TEM,EPMA),XRD{!0temperatureforaTi{13.25at.%Al{63.35at.%Nballoy1923,1473,1273[ 108 ]*VaporPressureMeasurements(Knudseneusionspectroscopy){{,{{,0{{,0{,{,0{{2,0{2,{1473,1373,1273 85

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ReferenceExperimentalMethodsPhaseEquilibriaTemperature[K] [ 125 ]EDX,TEMTioccupationofNbsitesin1173[ 126 ]*Metallpgraphy(EPMA){,{,{,2{,{{1573,1473,1273[ 5 ]*Metallography(OM,SEM,TEM,EPMA)XRDCompositionofthe(2){(0){tietriangle1473,1423,1373,1323,1273[ 15 ]*Metallography(OM,EPMA,SEM,TEM,)microhardnessXRDDTAPrimarycystallizationeldsontheliquidussurface[ 14 ]*Metallography(OM,SEM,TEM,EPMA)XRDTransformationpaths,Phaseequilibria(phasecompositionsnotincluded)1373,1173,973[ 107 ]*Metallography(OM,EPMA,SEM,TEM),XRD{,{,{,{,{{1373[ 127 ]Metallography(OM,SEM,EBSD),DSCTemperatureofthereactions+!and2+!2++foralloyTi{45at.%Al{5at.%Nb[ 128 ]Metallography(OM,TEM),XRD,DTA2{0{O,0{O,0{2,2{OLiquidustemperatureofselectedalloys1423,1373,1273,973[ 17 ]*Metallography(OM,SEM)XRD,DTASeveralsolidstatetransitionandmeltingtemperaturesarepresentedforalloyswithupto17.1at.%Nb 86

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TielineandtietriangledataforalloysA2,A3,andA133thatwereusedfortheoptimization. AlloyBulk[at.%][at.%][at.%][at.%]AlNbTiAlNbTiAlNbTiAlNbTi A2at151057.435.96.742.250.37.569.926.93.247.840.012.2A2at141057.235.77.142.647.99.569.925.64.554.334.111.5A3at151050.840.88.442.350.37.470.027.03.047.341.411.3A3at141051.439.88.841.749.29.169.726.63.754.633.312.1A133at152042.642.714.138.250.411.4{{{43.740.116.2A133at142042.642.714.139.746.314.0{{{50.633.216.2 87

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DTAcurveforalloy11measuredatheatingandcoolingratesof10oC/min. 88

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GibbstriangleshowingthepositionofAlloy11aswellasthepositionofthethreephasetriangle++measuredbyHellwigetal.[ 101 ].ThisthreephasetrianglewasdeterminedbyheattreatinganalloyofcompositionTi{40.5at.%Al{24.8at.%Nbfor48hoursat1200oCandthenwaterquenching.Thealloywasshowntocontainthe,,andphases,thecompositionsofwhichweremeasuredusingEPMAandareshownasopentriangles.Thealloyisindicatedasthelledredcircle.Accordingtotheseresults,Alloy11shouldalreadybeinthethreephaseregion++at1200oC. 89

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PeakseparationofthetwooverlappedpeaksusingVoigtfunctions.Thepeakrepresentingthe+!++transitionisshowninredandthepeakrepresentingthe++!+transitionisshowningreen. Figure4-4. Peakanalysisforthe+!++transformation.Therstderivativeisshowninblue.Thetransitiontemperaturewasselectedbasedonthepointatwhichthevalueoftherstderivativedeviatedfromzero. 90

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Peakanalysisforthe++!+transformations.Therstderivativeisshowninblue.Thetransitiontemperaturewasselectedbasedonthepointatwhichthevalueoftherstderivativedeviatedfromzero. 91

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DTAcurveforalloy12measuredatheatingandcoolingratesof10oC/min. 92

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DTAcurveforalloy12showingtheoverlappedpeaksonheating.Becauseofthepresenceoftheshallowpeak,shownbythebluerectangle,manypeakseparationmethodsmustbeevaluated. Figure4-8. Theresultsofpeakseparationusingthreepeaks.Inthismethod,allpeakparameterswererened.Theresultsshowthatalthoughtwophasetransformationpeaksarefairlywellmodeled,athirdpeak,showninred,appearsatanimprobablepositionbetweenthetwophasetransformationpeaks. 93

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Theresultsofpeakseparationusingthreepeaks.Inthisapproach,thecentroidofonepeakwasxedat1273oCinanattempttoxthepositionoftherstpeak.AlthoughthecumulativepeakingoodagreementwiththeoriginalDTAsignal,thephasetransitiontemperaturesassociatedwiththersttwopeaksareapproximatelyequal.Thisisalsoahighlyimprobablesituation. 94

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Theresultsofpeakseparationusingonlytwopeaks.ThecumulativepeakisingoodagreementwiththeoriginalDTAsignal.However,inanattempttotakeintoaccounttheshallowpeak,theparametersoftherstpeakarerenedtoproduceapeakthatismuchwiderthanexpected.Thisintroducesquitealargeerrorintheanalysisofthephasetransitiontemperatureusingthemethodofrstdeviationfromthebaselineusingtherstderivative. Figure4-11. Theoverlappedpeaksonheatingforalloy12indicatingthechoiceoftemperaturesforthecorrespondingphasetransformations.1269oCisselectedasthestarttemperatureofthe+!++transitionand1323oCisselectedasthestarttemperatureofthe++!+transition. 95

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Transitiontemperaturesforalloy11andalloy12. Alloy+!++++!++! 96

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58 ].ThedescriptionfortheAl{Tibinarysub-systemwastakenfromSaunders[ 18 ]andisthesamedescriptionusedbyServantandAnsara[ 10 ].TheAl{TibinaryofSaunders[ 18 ]wasmodiedtomodeltheand2phases,whichundergoanorder{disordertransformation,usingasingleGibbsenergyexpression[ 10 ].AlthoughthereisanewdescriptionfortheAl{TisystemfromWitusiewiczetal.[ 110 ],thisdescriptionwasnotusedastheand2phasesaremodeledasseparatephases(seeSection 3.3.3.1 ).TheAl{NbsystemwastakenfromtheworkofWitusiewiczetal.[ 17 ]asthisdatasetreproducesbettertheenthalpyofformationdataforthephasesintheAl{Nbsystem(seeSection 3.3.3.2 ).TheTi{NbsystemwastakenfromZhangetal.[ 129 ].ThisdatasetisrecommendedasitcalculatesthemetastablemiscibilitygapinthephasebasedondatafromHickmanetal.[ 130 ]andMoatandLarbalestier[ 131 132 ].Additionally,thecalculated/T0curveisabovethemeasuredtemperaturesforthemartensitictransformationfromthephasetothehexagonal'martensitequotedinBrownetal.[ 133 ].ThecalculatedbinaryTi{NbsystemisshowninFigure 5-1 28 ]moduleofTHERMO-CALC[ 29 ]. 14 ].Next,theternarymixingparametersofthephasewerere-optimized.Asthereisnoexperimentaldataontheenthalpyofmixingofthephase,thestartingvaluesoftheternarymixingparameters 97

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5.3.3 .Oncetheparametersforthe,,andphaseswerere-assessed,theparametersrepresentingbinarymixingoftwocomponentsononesublatticewithasinglyoccupiedsecondsublattice,aswellastheternarymixingparametersofthephasewerere-optimized.ThedetailsofthismethodaregiveninSection 5.3.4 .Aftertheparametersofthephasewerere-optimized,ternarymixingparametersofthephasewereintroducedtoimprovethetofthehomogeneityrangeofthephasetotheavailableexperimentaldata.TEMinvestigationsonthesinglephaseregionhaveshownthatthe{0boundaryisbetweenTi{25at.%Al{30at.%NbandTi{25at.%Al{20at.%Nbat1473K[ 68 101 ],atAlcontentslargerthan25at.%at1373K[ 68 ],andatAlcontentslargerthan23at.%at1273K[ 101 ].Additionally,analloyofcompositionTi{27.9at.%Al{21.8at.%Nbisshowntobe0at1573K[ 81 ].Therefore,inthesecompositionranges,theparametersofthe0phasewereoptimizedsothatthe0phaseismorestablethanthephase.Theparametersforthe0phaseweremodeledaslinearfunctionsoftemperature,thedetailsofwhicharegiveninSection 5.3.5 .Next,theparametersfortheand2phaseswerere-optimizedtodecreasetheprimarycrystallizationeldofthephaseontheliquidussurfaceandtocalculatetheintersectionofthe2+0and+0phaseeldsat1273K[ 101 ].Theparametersfortheliquidphasewereoptimizedlast.Thetemperature,composition,andeutecticcharacterofreactionbetweentheliquid,,,andphaseswereacceptedfromtheworkofRios[ 119 ]andthetemperatureofthetransitionreactionliquid+!+wereacceptedfromtheworkofPavlovandZakharov[ 134 ].Sincethereissomeuncertaintyinthecharactersandtemperaturesofallotherinvariantreactions,theternarymixingparametersfortheliquidphasewereoptimizedsothatthefouralloysofcomposition40at.%Aland36,30,24,and20at.%TirespectivelyarelocatedintheprimarycrystallizationeldofthephasetobeinagreementwiththeworkofLeonardetal.[ 15 ]. 98

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5.3.1TheSigmaphaseThephasehasthetP30{CrFecrystalstructure.Theatoms,Wyckopositions,andcoordinationnumbersaregiveninTable 5-1 andtheunitcellisshowninFigure 5-2 .FromtherecommendationsofFerroandCacciamani[ 135 ]andAnsaraetal.[ 54 ]threesublatticesshouldbeusedwitheachsublatticecontainingatomshavingthesamecoordinationnumber.Intheabovecase,theNbatomsonthe8i1and8jsitesoccupyonesublatticewithtotalmultiplicity16,theAlatomsonthe2aand8i2sitesoccupyasecondsublatticewithmultiplicity10,andtheNbatomonthe4fsiteoccupiesitsownsublatticewithmultiplicity4.Thismodelingresultsinthedescription(Nb)16(Al)10(Nb)4.AllowingmixingofAlandTiontherstsublattice,mixingofNbandTionthesecondsublattice,andnomixingonthethirdsublattice,themodelforthephaseis(Al,Nb*,Ti)16(Al*,Nb,Ti)10(Nb)4whichreducesto: (Al,Nb*,Ti)0:533(Al*,Nb,Ti)0:333(Nb)0:134(5{1)whenonemoleofformulaunitsisusedandwheretheasteriskidentiesthemajorspeciesoneachsublattice.ThelocationsoftheendmembersofthisdescriptionareshownontheGibbstriangleinFigure 5-3 .AllthermodynamicdescriptionsfortheendmembersweretakenfromWitusiewiczetal.[ 17 ]excepttheendmembersGTi:Al:NbandGTi:Ti:Nb,whichweretakenfromthedescriptionofServantandAnsara[ 10 ].ServantandAnsaraalsoincludedsixteenmoreparameterstomodelthephase.Thesearethe0LAl,Ti:i:Nb,0LNb,Ti:i:Nb,0Li:Al,Nb:Nb,0Li:Al,Ti:Nb,and0Li:Nb,Ti:Nbparameters,wherei=Al,Nb,Ti,aswellasthe0LAl,Nb:Ti:Nbparameter[ 10 ].Thisstudyrevealedthattheseparameterswereshowntohavenoinuenceontheternaryphaseequilibriawiththephase.Additionally,theremovaloftheseparametersdidnotaecttheTi{AlorTi{Nbconstituentbinaries,i.e.,thephasewasnotstabilizedineithertheTi{AlortheTi{Nbconstituentbinary.Infact,the0LNb,Ti:i:Nband0Li:Nb,Ti:Nbparametersfori=NbandTiwereaddedbyServant 99

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5-3 .Itwasusedparticularlytottheextensionofthephaseat1813Ktobeinagreementwiththe++tietriangleofLeonardandVasudevan[ 14 ].Startingvaluesofthe0LNb,Ti:Al:Nbparameterweregeneratedbyassumingalineartemperaturedependencebetweenthevalueoftheparameterat1813Kandthevalueoftheparameterat1373Krequiredtotthe++tietriangleofLeonardandVasudevanat1813K[ 14 ]andthe++tietriangleofLeonardetal.at1373K[ 107 ]respectively.The1LAl,Nb,Ti:i:Nbparameterswereusedtoimprovethetofthehomogeneityrangeofthephasetothe++tietriangledataforalloysA2andA3at1783Kand1683K[ 119 ],tothe+tielinedataforalloyA133at1783Kand1683K[ 119 ],tothephasediagramdataat1473K[ 13 101 108 ],1373K[ 107 ],and1273K[ 101 ],andaswelltothetemperaturesofthephasetransformations+!++and++!+foralloys11and12.The0LAl,Nb,Ti:i:Nband2LAl,Nb,Ti:i:Nbparametersweresettozeroastheydidnotaectthephaseboundariesofthephase.Alloptimizedparametersweremodeledusingalineartemperaturedependence. (Al,Nb*,Ti)0:75(Al*,Nb,Ti)0:25(5{2)wheretheasteriskidentiesthemajorspeciesoneachsublattice.ThelocationoftheendmembersgeneratedusingthisdescriptionareshownontheGibbstriangleinFigure 5-4 .ThedescriptionsfortheGAl:Al,GAl:Nb,GNb:Al,andGNb:Nbendmembersweretakenfrom 100

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17 ]whereasthedescriptionsforallotherendmembersweretakenfromthedatasetofServantandAnsara[ 10 ].Themixingparameter0LNb:Al,NbwasalsotakenfromWitusiewiczetal.[ 17 ]andthe0LTi:Al,Tiand0LAl:Al,TiparameterswerekeptfromtheoriginaldescriptionofServantandAnsara[ 10 ].ServantandAnsarausedelevenmoreparameterstotthephaseequilibriawiththephase.Thesearethe0Li:Nb,Tiand0LAl,Nb:iparameterswithi=Al,Nb,Ti,the0Li:Al,Tiparameterswithi=Al,Ti,the0LNb,Ti:iparameterswithi=Nb,Ti,andthe0LNb:Al,Tiparameter.Theseparametershadnegligibleinuenceonthephaseboundariesofthephase,andwerethereforesettozero.Theonlyparameterwhichwasoptimizedinthisworkwasthe0LNb,Ti:Alparameter,whichcontrolstheextensionofthephaseintotheternaryat25at.%Al.TheinuenceofthisparameterisshownasthedashedlineinFigure 5-4 .Thisparameterwasmodeledwithalineartemperaturedependence.Thestartingvaluesofthe0LNb,Ti:Alparameterweregeneratedbyusingthevalueoftheparameterat1813Kandthevalueoftheparameterat1373Krequiredtotthe++tietriangleofLeonardandVasudevanat1813K[ 14 ]andthe++tietriangleofLeonardetal.at1373K[ 107 ]respectively. 10 ].However,duringthecourseofthisassessment,itwasfoundthattherewasnopossibilitytottheextensionofthephasetohigherAlcompositionsusingthisconstraint.Therefore,eachoftheparameterswasassessedindividually.Tochoosecorrectstartingvaluesofthe0LAl,Nb,Ti,1LAl,Nb,Ti,and2LAl,Nb,Tiparameters,thevaluesoftheparametersrequiredtoproduceagoodttothephaseequilibriaat1473K[ 13 101 ],and1783K[ 119 ],tocalculatecorrectlythe++tietriangleat1813K[ 14 ],andtoproducereasonabletielinesat2000Kand2100Kwereselected.Thevalues 101

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5-5 .Figure 5-5 showsthatthereisnopossibilitytottheequilibriadeningtheseparametersaslinearfunctionsoftemperatureonly.Therefore,the0LAl,Nb,Tiand1LAl,Nb,Tiparametersweremodeledascubicfunctionsoftemperatureandthe2LAl,Nb,Tiparameterwasmodeledasaquadraticfunctionoftemperature.Thestartingvaluesofthe0LAl,Nb,Ti,1LAl,Nb,Ti,and2LAl,Nb,Tiparameterswerethenfurtheroptimized. (Al,Nb,Ti)0:5(Al,Nb,Ti)0:5(5{3)ThedescriptionforallendmembersarethesameasusedbyServantandAnsara[ 10 ]exceptfortheGNb:TiandGTi:Tiparameters.Intheoriginaldataset,theseparameterswereexpressedas: 5-6 .Theparameterswhichwereoptimizedinthisworkaretheparametersoftype0Li;j:k,whichrepresentmixingoftwocomponentsononesublatticewithathird,dierentcomponentonasinglyoccupiedsecondsublattice,andparametersoftype0Li;j;k:i,1Li;j;k:i,and2Li;j;k:i,whichrepresentthemixingofthreecomponentsononesublatticewithasinglyoccupiedsecondsublattice.Toreducethenumberofoptimizingvariables,theconstraint0Li;j:k=0Lk:i;jwasusedforthe0Li;j:kparameters,andtheconstraintLi;j;k:i=Li:i;j;kwasusedfortheLi;j;k:iparameters.Thebinaryparameters0LAl:Al,Nb=0LNb:Al,Nb=0LAl,Nb:Al=0LAl,Nb:AlwereincludedintheoptimizationastheywererequiredtoshiftthephasetohigherNbcompositions 102

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119 ].ThephasewasnotstabilizedintheAl{Nbbinarythroughadditionoftheseparameters.Alloptimizedparametersweremodeledaslinearfunctionsoftemperatureexceptthe0LAl:Nb,Ti=0LNb,TI:Alparameter.Thisparameterisakeyparameterusedtotthecompositionofthephaseinthe+tielineat1683K[ 119 ],totthecompositionofthephaseinthe++tie-triangleat1473K[ 101 ],andtocalculateahighNbcontainingphaseat1923K.Thevaluesofthisparameterat1923K,1683K,and1473KrequiredtocalculatetheaforementionedequilibriaareshowninFigure 5-7 .Figure 5-7 showsthatalineartemperaturedependenceisinadequatetoreproducetherequiredequilibria.Forexample,ifalineartemperaturedependenceoftheparameterwasselectedusingthevaluesoftheparameterat1923Kand1473K,thenthealloyA133,whichcontainsonly+at1683K[ 119 ](seeTable 4-2 ),iscalculatedinthe++tietriangle.Additionally,ifthelineartemperaturedependenceoftheparameterischosenforvaluesoftheparameterat1923Kand1683K,thenthecalculatedisothermalsectionat1473KisindisagreementwiththetielineandtietriangledatafromHellwigetal.[ 101 ].Therefore,aquadratictemperaturedependenceforthisparameterwasselected. (Al,Nb,Ti)0:5(Al,Nb,Ti)0:5(5{5)Sinceasimilarmodelisusedforthephase(Equation 5{3 ),thelocationsoftheendmembersforthe0phasearethesameasthatforthephase,andarethereforealreadyillustratedinFigure 5-6 .Inthemodelforthe0phase,therestrictionsofsymmetryimposetheconstraint0L0i;j:k=0L0k:i;j.Theadditionalconstraint0L0i;j:A=0L0i;j:B=0L0i;j:Cwasusedtoreducethenumberofoptimizingvariables.Thethirdsimplication0G0i:j=-0L0i;j:k,whichwasused 103

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10 ],canreducethenumberofoptimizingvariablesfrom12to6.Toassesstheinuenceofthenumberofoptimizingparametersonthephaseequilibria,re-optimizationoftheparametersofthe0phasewasperformedwithandwithoutthethirdsimplication.Theresultsindicatedthattherewasnosignicantchangetothephaseequilibria.Additionally,thereisinsucientexperimentaldatatojustifytheuseof12parameters.Therefore,thesimplication0G0i:j=-0L0i;j:k,asusedbyServantandAnsara[ 10 ],wasmaintainedinthepresentre-optimization.Theoptimizedparametersforthephasewereallmodeledaslinearfunctionsoftemperature.Togeneratethestartingvaluesoftheparametersforoptimization,thevaluesoftheparametersrequiredtotthe0++threephaseequilibriaat1473KofZdziobeketal.[ 13 ]andHellwigetal.[ 101 ]andthevaluesoftheparametersrequiredtotthe+tielinedataforalloy12at1613K[ 119 ]wereusedtoconstructalineartemperaturedependence. 10 ]modeledtheternarymixingparametersforthephaseusingtheconstraint0LAl,Nb,Ti=1LAl,Nb,Ti=2LAl,Nb,Ti.However,thismodelingleadstoquitealargeextensionoftheprimarycrystallizationofthephaseontheliquidussurfacewhichisnotexperimentallyconrmed.Therefore,eachoftheternaryparameterswasassessedindividually.TheD019structureofthe2phaseisthehcpanaloguetotheL12structurefromfccorderingwithhP8{Ni3Snstructure.ConstraintsusedtomodelGordMweredevelopedbyDupinandAnsara[ 55 ]basedonthemathematicalequivalencebetweenthe2sublatticemodel(A,B,C)0:75(A,B,C)0:25andthe(A,B,...)0:25(A,B,...)0:25(A,B,...)0:25(A,B,...)0:25foursublatticemodelwhereallsublatticesareequivalent.Onlyoneparameterwasoptimized.Thisparameter,relatedtotheformationenergyGAlNbTi2ofthehypotheticalcompound 104

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GAlNbTi2=uAlNb+uAlTi+2uNbTi+u(5{6)whereuAlNb,uAlTi,anduNbTiarethebondenergiestakenfromthebinarysystemsanduisacorrectiontermthatcanbeoptimizedtottotheternaryphaseequilibria.Astheextensionofthephasewassignicantlyreduced,toensuretheintersectionofthe2+0and+0phaseeldsat1273K[ 101 ],theparameteruwasre-optimizedusingthetrial-and-errormethodtoincreasethestabilityofthe2phase. 10 ].However,itwasfoundthattherewasnopossibilitytottheextensionoftheprimarycrystallizationofthephasetohigherAlcompositionsusingthisconstraint.Therefore,theparameterswereindependentlyassessed. 105

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CalculatedbinaryTi{NbdiagramusingthedescriptionofZhangetal.[ 129 ]. Table5-1. Crystalstructureofthephase[ 135 ] SpaceGroupP42/mnmPearsonSymboltP30AtomsWyckoPositionxyzCN Al2a0.00.00.012Nb4f0.3990.3990.015Nb8i10.4640.1310.014Al8i20.7410.0660.012Nb8j0.1830.1830.25114 106

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Unitcellofthephase.TheAl1andAl2atomsarelocatedatthe2aand8i2Wyckopositionsrespectively.Sincethe2aand8i2positionseachhavethesamecoordinationnumber,theatomsinthesepositionsoccupythesamesublattice.TheNb2andNb3atoms,whicharelocatedatthe8i2and8jpositions,occupythesamesublatticeastheyeachhaveacoordinationnumberof14andtheNb1atominthe4fpositionoccupiesitsownsublattice.Thisresultsinthemodel(Nb)16(Al)10(Nb)4whenthereisnomixingoneitherofthesublattices. 107

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Theendmembersofthephase.ThedashedlinefromNb:Al:NbtoTi:Al:Nbschematicallyillustratestheinuenceofthe0LNb,Ti:Al:Nbmixingparameter. Figure5-4. Theendmembersofthephase.ThedashedlinefromNb:AltoTi:Alschematicallyillustratestheinuenceofthe0LNb,Ti:Alparameter. 108

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BThestartingvaluesofthe1LAl,Nb,Tiparameterasafunctionoftemperature CThestartingvaluesofthe2LAl,Nb,TiparameterasafunctionoftemperatureFigure5-5. ThestartingvaluesoftheA)0LAl,Nb,Ti,B)1LAl,Nb,Ti,andC)2LAl,Nb,Tiparametersasafunctionoftemperature.Therewasnopossibilitytottheseparametersaslinearfunctionsoftemperature.Therefore,the1LAl,Nb,Tiand2LAl,Nb,Tiparametersweremodeledascubicfunctionsandthe2LAl,Nb,Tiparameterwasmodeledasaquadraticfunction. 109

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Theendmembersofthephase.Thedashedlinesshowtheinuenceofthesixmixingparametersoftype0Li;j:k,whichrepresenttheconditionofmixingoftwodierentcomponentsononesublatticewithasecondsublatticesinglyoccupiedbyathirdcomponent. 110

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Thevaluesofthe0LAl:Nb,Tiat1923K,1683K,and1473K.Expressingthisparameterasalinearfunctionoftemperatureonlywillnotresultinagoodttotheexperimentalphaseequilibria.Therefore,aquadraticvariationwithtemperaturewaschosen. 111

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6-1 .Thereareveinvariantreactionswiththeliquidphase.Aneutecticreactionwascalculatedfortheinvariantreactionbetweentheliquid,,,andphases.Thisisinagreementwithotherthermodynamicassessments[ 9 10 17 ]aswellaswiththeexperimentalworkofRios[ 119 ].Thecalculatedliquidussurfaceisinverygoodagreementwithexperimentalworksindicatinganextensionoftheprimarycrystallizationeldofthephase[ 4 14 15 98 119 ]i.e.,allalloysintheliteraturewhichhavebeenshowntosolidifyasphaseareintheprimarycrystallizationeld.Thecalculatedsoildussurface(Figure 6-2 )showsthatallalloysexcepttheTi{48at.%Al{25at.%NballoyofFengetal.[ 98 ]andalloy12solidifyassinglephase.Inthiscase,itwasnotpossibletoextendthesolidustohigherAlcompositions.Alloy12islocateddirectlyonthesoliduslinebuttheTi{48at.%Al{25at.%NballoyofFengetal.[ 98 ],whichhasahigherAlcontent,solidiesastwophase+.TheScheilreactionschemeforequilibriawiththeliquidphaseisshowninFigure 6-3 6-4 ),1813K(Figure 6-5 ),1783K(Figure 6-6 ),1683K(Figure 6-7 ),1673K(Figure 6-8 ),1613K(Figure 6-9 ),1513K(Figure 6-10 ),1473K(Figure 6-11 ),1373K(Figure 6-12 ),and1273K(Figure 6-13 )withsuperimposedkeyexperimentaldataaregiven.Thecalculatedisothermalsectionsshowgoodagreementwiththeexperimentaldata.AlthoughthetielinedataofWangetal.[ 99 ]werenottakenintoaccountintheoptimization,theyarestillsuperimposedontotheisothermalsectionat1673K.Asmentionedbefore,thedataofWangetal.[ 99 ]wouldsuggestamuchsmallerprimarycrystallizationofthephase,whichisnotinagreement 112

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99 ]isparticularlyevidentinthecalculatedsingephaseeld,where{,{,and{tielinesarereported.Asthealloyswereheattreatedat1473Kfor160hours,furnacecooled,andthenannealedat1673Kfor4.5hours,itispossiblethattherewasinsucienttimeat1673Kforthephasetocompletelydissolvethe,,andphasesthatarepresentinthemicrostructurepriortoannealingat1673K. 6-14 )andtheverticalsectionat40at.%Al(Figure 6-15 )areincludedhere.Thecalculatedverticalsectionsalsoshowquiteagoodagreementwiththeliterature[ 15 119 ].Alloy11isinthesinglephaseelduntilthetransusat1750Kisreached,atwhichpointforms.However,asimilarsinglephaseeldforalloy12isnotcalculated.Instead,alloy12passesthroughaverysmallthreephaseeldliquid++(calculatedatonly0.2K),beforeenteringintothe+twophaseeld.Alsoofnote,thecalculatedsolidstatetransformationofLeonardetal.'salloy1[ 15 ]isfromto+(Figure 6-15 ).However,intheoriginalwork,thistransformationiscitedasfromto+.Intheoriginalwork,onlythermalanalysiswasperformedonthealloytodeterminesolidstatetransformationtemperatures,butnoheattreatmentandquenchexperimentswereperformedattemperaturesbeforeandafterDTApeaksonthisalloytoconrmtheresultingmicrostructures.Therefore,itisequallyprobablethatourphasetransformationpathcalculatedhereiscorrect.TheverticalsectionsTi{27.5at.%AltoNb[ 78 ],Ti{22at.%AltoNb[ 79 102 ],andTiAltoTiNb[ 72 79 ]andat45at.%Al,47at.%Al,8at.%Nb,throughTi87:2Nb12:8andNb2:2Al97:8,throughTi72:8Nb27:2andTi31:6Al68:4,andthroughTi70Al30andTi51Nb49[ 17 ]werenotaddressedinthepresentre-optimizationastheyfocusonphasetransformationswiththe,2,,andOphaseswhichdonotgreatlyimpactthephaseequilibriainthe 113

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12 ]canonlybereproducedifthephaseismodeledas(Ti,Nb)1xAl3+x,whichwouldalloweitheranexcessordeciencyofAlinthephasewithrespecttothelinecompound(Ti,Nb)Al3.However,thethermodynamicdescriptionforthephasewasnotchangedfromthatreportedintheworkofServantandAnsara[ 10 ]. 6-16 andFigure 6-17 respectively.Foralloy11,aftersolidication,asinglephaseeldisobserved.Withcontinuedcooling,phaseformsatthetransusat1750K.Theamountofphasedecreasesandtheamountofphaseincreases.At1485K,phaseforms,andtheamountofphasecontinuestoincreaseandtheamountofphasecontinuestodecreaseuntilat1463K,onlyandarepresent.Alloy12,however,doesnotsolidifyassinglephase,butpassesthroughaverysmallregionofliquid++(calculatedatonly0.2K),beforecompletelysolidifyingas+.Infact,theamountofphasedirectlyaftersolidicationiscalculatedat99.3mol%.Thiscanbedirectlyseenonthecalculatedsolidussurface(Figure 6-2 ),asalloy12isalmostdirectlyonthesolidusline.Withcontinuedcooling,theamountofphasedecreasesandtheamountofphaseincreasesuntilat1596K,phaseforms.Afterformationofphase,theamountofcontinuestoincreaseandtheamountofcontinuestodecreaseuntilat1542K,onlyandphasesarepresent. 114

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CalculatedliquidussurfacefortheTi{Al{Nbsystem.Theexperimentaldataidentiedas[1992Fen],[1996Ara],and[2000Leo]refertotheworksofFengetal.[ 98 ],Leonardetal.[ 15 ],andd'Arag~aoandEbrahimi[ 4 ]respectively.Thecalculatedliquidussurfaceisingoodagreementwiththeliteratureshowingthatallalloysindicatedsolidifyasphase. Figure6-2. CalculatedsolidussurfaceintheTi{Al{Nbsystem.ThepositionofthesolidusshowsthatallalloyssolidifyassinglephaseexceptforthealloyofcompositionTi{48at.%Al{25at.%NbfromFengetal.[ 98 ]andalloy12.Aftersolidication,thesealloysexistastwophase+. 115

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ScheilreactionschemeforequilibriawiththeliquidintheTi{Al{Nbsystem.Thereareveinvariantreactions:threetransitionreactions,oneeutecticreaction,andoneperitecticreaction. 116

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Calculatedisothermalsectionat1923K.Theexperimentaldataidentiedas[1992Men]referstotheworkofMenonetal.[ 109 ]. Figure6-5. Calculatedisothermalsectionat1813K.Theexperimentaldataidentiedas[2000Leo]referstodataobtainedfromLeonardandVasudevan[ 14 ].Inthiswork,analloyofcompositionTi{25at.%Al{60at.%Nbwhichwasheattreatedat1813Kandquenchedwasshowntobeinthethreephaseregion++. 117

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Calculatedisothermalsectionat1783K.TielineandtietriangledataforalloysA2,A3,andA133areincluded.Thetielinesshowninblueweredeterminedafterthedescriptionwasre-optimized.Therefore,thecurrentdescriptionisalsoabletopredictthephaseequilibriaat1783K. Figure6-7. Calculatedisothermalsectionat1683K.TielineandtietriangledataforalloysA2,A3,andA133areincluded. 118

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Calculatedisothermalsectionat1673K.TheexperimentaldataofWangetal.[ 99 ]issuperimposed.Whilethedatasetisabletocalculatephaseequilibriabetweenand,and,and,and,andandphases,otherphaseequilibriacannotbecalculated.However,Wangetal.[ 99 ]mayhaveinsucientlyheattreatedtheiralloysat1673K.Therefore,theymaynothavemeasuredtheequilibriumtielines. Figure6-9. Calculatedisothermalsectionat1613K.Thereisgoodagreementwiththe{tielineforalloy12. 119

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Calculatedisothermalsectionat1513K.Thereisgoodagreemtnwiththe{tielineforalloy11. Figure6-11. Calculatedisothermalsectionat1473K.Theexperimentaldataidentiedas[1992Men],[1995Zdz],and[1998Hel]refertotheworksofMenonetal.[ 109 ],Zdziobeketal.[ 13 ],andHellwigetal.[ 101 ]respectively. 120

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Calculatedisothermalsectionat1373K.Theexperimentaldataidentiedas[1999Eck]and[2002Leo]refertotheworksofEckertetal.[ 108 ]andLeonardetal.[ 107 ]respectively. Figure6-13. Calculatedisothermalsectionat1273K.Theexperimentaldataidentiedas[1992Men]and[1998Hel]refertotheworksofMenonetal.[ 109 ]andHellwigetal.[ 101 ]respectively. 121

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Calculatedisopleththroughthenominalcompositionsofalloy11andalloy12.Thereisgoodagreementbetweenthecalculationsandthesolidstatetransformationsofalloys11and12measuredusingDTA.Alloy11solidiesassinglephase.However,thereisnosinglephaseeldforalloy12.Instead,directlyaftersolidication,andshouldappear. 122

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Calculatedisoplethat40at.%Al.TransformationtemperaturesmeasuredusingDTAfromLeonardetal.[ 15 ]areincluded. 123

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Calculatedphasefractiondiagramforalloy11. Figure6-17. Calculatedphasefractiondiagramforalloy12. 124

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7-1 .SincetheearlyworksbyVogelandWenderott[ 136 ],McPhersonandFontana[ 137 ]andMcQuillan[ 138 ]measuredtheCrcontentoftheLavesphaseas60at.%,theLavesphasestoichiometrywasmisidentiedasTi2Cr3.However,DuwezandFontana[ 139 ]andCuetal.[ 140 ],usingX-raydiraction,correctedthestoichiometrytoTiCr2.TherearethreemodicationsoftheLavesphase:thelowtemperature{TiCr2phase,theintermediatetemperature{TiCr2phase,andthehightemperature{TiCr2phase.Intheearlyworks[ 140 { 143 ],thepolymorphismoftheLavesphasewasnottakenintoaccount.However,intheexperimentallydeterminedphasediagramsfortheTi{CrsystemshownbyLevinger[ 144 ],AgeevandModel[ 145 ],andFarrarandMargolin[ 146 ],boththelowtemperature{TiCr2andthehightemperature{TiCr2modicationsoftheLavesphasewereindicated.Svechnikov[ 147 ]wasthersttoincludeallthreemodicationsoftheLavesphase.Inthiswork,thephasetransformationtemperaturesoftheLavesphasesweremeasuredusingDTA[ 147 ].AlthoughthepresenceofthreemodicationsoftheLavesphasewasconrmedbyChenetal.[ 148 149 ],laterexperimentalworksofHaoandZeng[ 150 ]andSchusterandDu[ 151 ]stillcouldnotndthe{TiCr2modication.TheworksofFarrarandMargolin[ 146 ]andChenetal.[ 148 149 ]addressthehomogeneityrangeoftheLavesphases.AlthoughFarrarandMargolin[ 146 ]measuredthehomogeneityrangeofthe{TiCr2and{TiCr2Lavesphasesfrom63at.%Crto66at.%Cr,themostcomprehensiveworkdonetoclarifythehomogeneityrangesoftheLavesphaseswasperformedbyChenetal.[ 148 149 ].Inthiswork,severalalloysofnominalcompositionsbetweenTi{62at.%CrandTi{69at.%Crwereheattreatedandquenchedfrom1573K,1473K,and1273Kfortimesvaryingfrom4hoursat1573Kto500hoursat1273K.X-raydiractionandmetallographywereusedtoidentifytheLavesphases 125

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148 149 ]aregiveninTable 7-2 .OnekeyfeatureoftheTi{Crphasediagramisthecongruenttransformationofthe{TiCr2Lavesphasetothephase.McQuillanmeasuredthetransformationtemperatureat1633K[ 138 ],Cuetal.[ 140 ]andVanThyneetal.[ 142 ]measuredthetransformationtemperatureat1623K,andFarrarandMargolinmeasuredthetransformationtemperatureat1638K[ 146 ].Inasmalltemperaturerangebetweenthecongruenttransformationofthe{TiCr2phaseandtheminimumoftheliquidus,thephaseformsacontinuoussolidsolutionfromTitoCr.Theactivityofchromiuminthephaseatcompositionsbetween10and90at.%Crinthetemperaturerange1523K,1593K,1633Kand1653KwasmeasuredbyPooletal.usingtheKnudseneusionmethod[ 152 ].AnadditionalfeatureoftheTi{Crphasediagramistheeutectoidreactionbetweenthe,,and{TiCr2phases,thetemperatureofwhichwasmeasuredtobebetween933Kand958K[ 139 140 142 153 154 ],determinedusingeithermetallography[ 139 142 153 154 ]ordilatometry[ 140 ]techniques.ResistivitymeasurementswereusedbyMikheevandChernova[ 143 ]buttheeutectoidreactiontemperatureof993Kwhichtheyobtainedismuchhigherthanthatgivenintheotherliteratureandisthereforegenerallynotaccepted.VariousauthorsinvestigatedthemeltingofTi{Cralloysofvariouscompositions.TheliquiduscurvewasmeasuredbyVanThyneetal.[ 142 ],theliquidusandsoliduscurvesweremeasuredbyMikheyevandAlekasashin[ 155 ],Minaevaetal.[ 156 ],andSvechnikovandKocherzhinskii[ 157 ],andthesoliduscurvesweremeasuredbyMcQuillan[ 138 ]andRudy[ 158 ].Theresultsallindicatethepresenceofameltingpointminimumatapproximately1883Katacompositionof45at.%Cr.TheTi{CrsystemwascriticallyassessedbyMurray[ 159 160 ]. 126

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18 ],Leeetal.[ 161 ],Zhuangetal.[ 162 ],andGhosh[ 163 ]respectively.Althoughalldatasetsmodeltheliquid,,andphasesassubstitutionalsolutions,dierencesinthedatasetsexistbasedon: 1. ThelatticestabilitydescriptionofpureCr. 2. ThenumberofLavesphasesmodeled. 3. ThemodelingoftheLavesphases. 4. ThehomogeneityrangeoftheLavesphases. 5. ThecalculatedactivityofthephasecomparedtotheexperimentalworkofPooletal.[ 152 ]. 58 ]assignsatemperatureof2810KtothemeltingpointofpureCr,whichisbasedontheassessmentofGurvichetal.[ 164 ].TheTi{CrdatasetsofSaunders[ 18 ]andGhosh[ 163 ]usethethermodynamicdescriptionofpureCrfromDinsdale[ 58 ].However,theworksofHultgrenetal.[ 165 ],Murray[ 160 ],Knackeetal.[ 166 ],Young[ 167 ],Stankus[ 168 ]andDubrovinskaiaetal.[ 169 ]reportthemeltingpointofpureCrbetween2130Kand2136K.ThethermodynamicdescriptionforpureCrfromSmithetal.[ 170 ]calculatesameltingtemperatureof2130K,whichisinagreementwiththeaboveworks.ThedatasetofLeeetal.[ 161 ]thereforeusesthedescriptionofpureCrfromSmithetal.[ 170 ]andZhuangetal.[ 162 ]modiedthelatticestabilityofpureCrtocalculateameltingpointof2136K[ 162 ].IntheworkofZhuangetal.[ 162 ],however,itisnotcleariftheauthorsmodiedonlythethermodynamicdescriptionofliquidCroriftheymodiedthethermodynamicdescriptionofliquidCraswellasthedescriptionof{Cr.BecauseLeeetal.[ 161 ]andZhuangetal.[ 162 ]usedierentdescriptionsforthelatticestabilityof 127

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159 160 ]clearlyindicatedthreemodicationsoftheLavesphase,Saunders[ 18 ]onlyconsideredtwomodications:thelowtemperature{TiCr2modicationandahexagonalhightemperaturemodicationreportedas{TiCr2.ThedescriptionofSaunders[ 18 ]isinagreementwiththeexperimentalworksofHaoandZeng[ 150 ]andSchusterandDu[ 151 ],inwhichthe{TiCr2modicationoftheLavesphasecouldnotbefound.TheworksofLeeetal.[ 161 ],Zhuangetal.[ 162 ],andGhosh[ 163 ],however,modeledallthreemodicationsoftheLavesphaseinagreementwiththeexperimentalworksofSvechnikov[ 147 ]andChenetal.[ 148 149 ]. 7-3 andtheunitcellisshowninFigure 7-1 .TheMgCu2structurecontainsonlytwocrystallographicsites;therefore,inaccordancewiththerecommendationofAnsaraetal.[ 171 ]andFerroandCacciamani[ 135 ],thetwosublatticemodel (Cr*,Ti)2(Ti*,Cr)1(7{1)shouldbeusedtomodelthehomogeneityrangeofthe{TiCr2phase.Intheabovedescription,theasteriskidentiesthemajorspeciesineachsublattice.The{TiCr2LaveshasthehP24{MgNi2structure.Theatoms,Wyckopositions,andcoordinationnumbersaregiveninTable 7-4 andtheunitcellisshowninFigure 7-2 .The{TiCr2phasehasvedierentcrystallographicsites.However,usingvesublatticestomodelthe{TiCr2Lavesphasewouldresultintoomanyendmemberandinteractionparameterswhichwouldhavetobeassessed.Therefore,simplicationstakingintoaccountthecrystallographyofthestructureareusedtoreducethenumberofsublatticestobe 128

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(Cr*,Ti)4+6+6(Ti*,Cr)4+4=(Cr*,Ti)2(Ti*,Cr)1(7{2)whichisalsorecommendedbyAnsaraetal.[ 171 ].The{TiCr2LavesphasehasthehP12{MgZn2structurewiththreecrystallographicsites.Theatoms,Wyckopositions,andcoordinationnumbersaregiveninTable 7-5 andtheunitcellisshowninFigure 7-3 .Justasinthecaseforthe{TiCr2Lavesphase,the2aand6hCrsites,whichhavethesamecoordinationnumber,canbecombined.ThissimplicationresultsalsointhemodelgiveninEquation 7{1 [ 171 ].Therecommendedtwo-sublatticemodelfortheLavesphaseswasusedbySaunders[ 18 ].Leeetal.[ 161 ]andGhosh[ 163 ]alsodescribedallthreemodicationsoftheLavesphaseusingthetwosublatticemodel.However,Zhuangetal.[ 162 ]usedathreesublatticemodeltodescribethe{TiCr2and{TiCr2Lavesphasessincetheydidnotusetheadditionalsimplicationwhichcouldbeimposedbycombiningcrystallographicsiteswiththesamecoordinationnumberintoasinglesublattice.Therefore,theymodeledthe{TiCr2and{TiCr2phasesas: (Cr)2(Cr*,Ti)4(Ti*,Cr)6(7{3)ThisisalsoasimplicationasthereisnoevidenceintheliteraturethatonlytheCratomoccupiestherstsublattice. 18 ]isshowninFigure 7-4 .TheexperimentaldataonthehomogeneityrangeoftheLavesphases 129

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148 149 ]aresuperimposed.Althoughthecalculatedhomogeneityrangeofthe{TiCr2phaseisingoodagreementwiththeworkofChenetal.[ 148 149 ],theTi-richsideofthe{TiCr2phasedoesnotextendtohighenoughTicontentsandtheCr-richsideofthe{TiCr2phasedoesnotextendtohighenoughCrcompositions.AsSaundersdidnottakeintoaccountthe{TiCr2phase,calculationsperformedusingthedatasetofSaunders[ 18 ]cannotreproducethesinglephase{TiCr2eldofChenetal.[ 148 149 ].Thehomogeneityrangeofthe{TiCr2LavesphasecalculatedusingthedatasetsofZhuangetal.[ 162 ]andGhosh[ 163 ]arenotingoodagreementwiththeexperimentaldataofChenetal.[ 148 149 ].AlthoughChen'sPh.D.thesis[ 148 ]givesthecompositionofthe{TiCr2phaseinequilibriumwiththeTi-richphaseforanalloyofcompositionTi{62at.%Crat1273K,thisdatawasprobablynottakenintoaccountintheassessmentsofZhuangetal.[ 162 ]andGhosh[ 163 ].Therefore,intheseworks,theTi-richboundaryofthe{TiCr2LavesphaseextendstohigherTicompositionsthanthosemeasuredbyChen[ 149 ].However,incomparisontothedatasetofSaunders[ 18 ],wherethehomogeneityrangeofthe{TiCr2LavesphaseisnotinagreementwiththeworkofChenetal.[ 148 149 ],thehomogeneityrangeofthe{TiCr2LavesphaseintheworksofZhuangetal.[ 162 ]andGhosh[ 163 ]isingoodagreementwiththemeasuredhomogeneityrangeofChenetal.[ 148 149 ].ThepublishedpaperofLeeetal.[ 161 ]doesnotpresentanenlargedpartoftheTi{Crphasediagramfrom60at.%Crto70at.%CrshowingdetailsofthehomogeneityrangesoftheLavesphasemodications.Therefore,theagreementofthehomogeneityrangesofthethreeLavesphaseswiththeworkofChenetal.[ 148 149 ]isunclear. 152 ]usingvaporpressuremeasurements.Themeasuredactivitiesshowquitealarge 130

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18 ]reproducestheactivityofCrinthephaseat1523Kand1593K,whereasthecalculationsofZhuangetal.[ 162 ]andGhosh[ 163 ]reproducetheactivitydataforCrat1633Kand1653K.Zhuangetal.[ 162 ]attemptedtoreproducethelargetemperaturevariationofCractivityinthephaseinaccordancewiththemeasurementsofPooletal.[ 152 ]byoptimizingthe0LCr,Tiparameter.However,thisresultedinarelativelylargetemperaturedependenttermfortheparameter,which,accordingtoOkamoto[ 172 ],maybeunrealisticinsuchabinarysystem.Ghosh[ 163 ]additionallyreportedthatabetterttotheexperimentaldataonthephaseboundarycouldbeachievedifonlythehighertemperatureactivitydataofPooletal.[ 152 ]istakenintoaccount.However,whentheactivityofCrinthephaseiscalculatedusingthedatasetofGhosh[ 163 ],thereisalargerdeviationoftheactivityofCrfromtheidealathighertemperaturesthanthedeviationoftheactivityofCrfromtheidealatlowertemperatures.ThisindicatesthatGhosh[ 163 ]modeledaphysicallyunrealisticbehavioroftheactivityofCrinthephase.ThepublicationofLeeetal.[ 161 ]givesnoindicationonthetoftheactivityofCrinthephasetotheexperimentalresultsofPooletal.[ 152 ]. 18 ],Leeetal.[ 161 ],Zhuangetal.[ 162 ],andGhosh[ 163 ]andtheexperimentaldata,theonlyclearstrategywouldbetore-optimizethethermodynamicparametersforthedescriptionoftheTi{Crsystemusingoneoftheexistingdatasetsasabasefromwhichfurtherparameterre-optimizationcouldbeperformed.ThedatasetsofZhuangetal.[ 162 ]andLeeetal.[ 161 ]arenotrecommendedsincetheyuseddierentdescriptionsforpureCrthanthatrecommendedbySGTE[ 58 ].Additionally,Zhuangetal.[ 162 ]modeledthe{TiCr2and{TiCr2LavesphasesusingthreesublatticeswhichisnotrecommendedbyAnsaraetal.[ 171 ].ItwouldhavebeenpossibletousethedatasetofGhosh[ 163 ],except 131

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18 ]usedthelatticestabilityofpureCrrecommendedbySGTE[ 58 ],modeledtheLavesphasesusingthetwosublatticemodelrecommendedbyAnsaraetal.[ 171 ],andmodeledaphysicallyrealisticbehavioroftheactivityofCrinthephase.Therefore,thedatasetofSaundersisagoodcandidatedatasetfromwhichfurtheroptimizationcouldproceedalthoughSaundersdidnottakeintoaccountallthreemodicationsoftheLavesphases.However,anadditionaljusticationforthechoiceoftheSaundersdatasetistheapplicationofthedescriptionoftheTi{CrsystemtothehigherorderTi{Al{Crsystem,whichSaundersalsomodeledusinghisdescriptionfortheTi{Crbinary[ 18 ]astheconstituentbinarydescription.Therefore,ifonlythedescriptionsforthetwoLavesphasesintheTi{CrdatasetofSaundersarere-optimizedandanadditionaldescriptionforthe{TiCr2Lavesphaseisincluded,thenitwouldstillbepossibletocalculateallotherphaseequilibriaintheTi{Al{Crsystemthatincludethe,,andliquidphasesasthedescriptionsforthesephaseswouldnotbechanged.SincethedatasetofSaunders[ 18 ]wasused,allthermodynamicdescriptionsforthepureelementsweretakenfromtheSGTErecommendations[ 58 ].Allre-optimizationwasperformedusingthePARROTmodule[ 28 ]ofTHERMO-CALC[ 29 ]. 7{1 .Suchadescriptionresultsinthefourend-membersGCr:Cr,GCr:Ti,GTi:Cr,andGTi:TiwhererepresentseitherofthethreeLavesphases.TheonlyphysicallyrealisticendmemberistheGCr:Titerm,whichcorrespondstofulloccupationoftheCrlatticesitesandTilatticesitesbyCrandTiatomsrespectively.Thisendmembertermismodeledas: 132

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7{4 ,Equation 7{5 ,andEquation 7{6 intoEquation 7{7 resultsintheexpression: 7{4 .ThetwosublatticemodelfortheLavesphasesresultsinthefourbinarymixingparameters0LCr,Ti:Ti,0LCr,Ti:Cr,0LCr:Cr,Ti,and0LTi:Cr,Ti.Toreducethenumberof 133

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7-6 .Thesevalueswerechosenbasedonacriticalandcarefulreviewofallavailableexperimentaldata.TheCPbehavioroftheLavesphaseswaschosentoobeytheNeumann-Kopprule.Intherststep,theendmemberparameterGTiCr2Cr:Tiandthe0LTiCr2Cr,Ti:*and0LTiCr2*:Cr,Timixingparametersforthe{TiCr2phasewerere-optimizedtotthehomogeneityrangeofthe{TiCr2phaseandtocalculatetheeutectoidreactiontemperaturebetweenthe{TiCr2,,and{Tiphasesat959K.Next,theGTiCr2Cr:Tiendmemberparameterandthe0LTiCr2Cr,Ti:*and0LTiCr2*:Cr,Timixingparmatersforthe{TiCr2phasewerere-optimizedsimultaneouslywiththeparametersforthe{TiCr2phasetotthetemperaturesandphasecompositionsfortheeutectoiddecompositionofthe{TiCr2phaseat1077K,theperitectoidformationofthe{TiCr2phaseat1496K,andtheeutectoidreactionbetweenthe{TiCr2,,and{Tiphasesat959K. 134

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7{4 ),valuesoftheGTiCr2Cr:TiparameterattwotemperaturesarerequiredtoestimatestartingvaluesofaandbinEquation 7{4 .TherstvalueoftheGTiCr2Cr:TiparameterisGTiCr2,1640KCr:Ti,whichisthevalueoftheGTiCr2Cr:Tiparameterat1640Krequiredtocalculatethecongruentdecompositionofthe{TiCr2phaseatapproximately1640K.Toestimateavalueforthisparameter,aslightlynegativetermofapproximately60JwasaddedtothevalueoftheGTiCr2Cr:Tiendmemberparameterat1640K,accordingtotheequation: 7-5 .Inthenextstep,theGTiCr2Cr:Ti,0LTiCr2Cr,Ti:*,and0LTiCr2*:Cr,Tiparameterswereoptimizedseparatelytotthecongruentdecompositionofthe{TiCr2at1640Kandthehomogeneityrangeofthe{TiCr2phasetobeinagreementwiththeworkofChenetal.[ 148 149 ].Inthelaststep,allendmemberandinteractionparametersofallthreemodicationsoftheLavesphaseswereoptimizedsimultaneously. 7-6 .Theexperimentaldatafromtheliteraturesuperimposed.Figure 7-6 showsthatthereisaverygoodagreementbetweenthecalculatedphase 135

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7-7 .TheexperimentaldataonthehomogeneityrangesoftheLavesphasesfromChenetal.[ 148 149 ],Minaevaetal.[ 156 ]andFarrarandMargolin[ 146 ]aresuperimposed.Thecalculationsalsoshowgoodagreementwiththeexperimentalworks[ 146 148 149 156 ].Figure 7-8 showsthecalculatedactivityofCrinthephasefrom10at.%Crto90at.%Crat1523K,1593K,1633K,and1653KalongwiththeexperimentaldatapointsofPooletal.[ 152 ].Sincetheparametersforthephasewerenotoptimizedinthiswork,thisdiagramisexactlythesameasthatwhichwouldbecalculatedusingtheoriginalTi{CrdescriptionofSaunders[ 18 ]. 136

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PhasesintheTi{Crsystem. PhasePrototypePearsonsymbolSpacegroupStructurereport Table7-2. HomogeneityrangeoftheLavesphasesdeterminedbyChenetal.[ 148 ]. TemperatureLavesPhaseTi{richboundaryCr{richboundary 1573K{TiCr263.8at.%Cr66.3at.%Cr1473K{TiCr263.6at.%Cr66.3at.%Cr1273K{TiCr264.0at.%Cr66.0at.%Cr 137

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Unitcellofthe{TiCr2Lavesphase.AtwosublatticemodelwithoneTi-richandoneCr-richsublatticeisused. Table7-3. Crystalstructureofthe{TiCr2Lavesphase. SpaceGroupFd3mPearsonSymbolcF24AtomsWyckoPositionxyzCN Ti8a0.00.00.016Cr16d0.6250.6250.62512 138

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Unitcellofthe{TiCr2Lavesphase.TheTi1andTi2atoms,whichoccupythe4eand4fpositionsrespectively,occupythesamesublatticebecausetheyhavethesamecoordinationnumbers,andtheCr1,Cr2,andCr3atoms,whichoccupythe4f',6g,and6hpositions,occupythesamesublatticeastheyhavethesamecoordinationnumbers. Table7-4. Crystalstructureofthe{TiCr2Lavesphase. SpaceGroupP63=mmcPearsonSymbolhP24AtomsWyckoPositionxyzCN Ti4e0.00.00.09416Ti4f0.3330.6670.84416Cr4f'0.3330.6670.12512Cr6g0.5000.00.012Cr6h0.1670.3340.25012 139

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Unitcellofthe{TiCr2Lavesphase.TheTiatomsoccupythe4fposition,andtheCr1andCr2atomsoccupythe2aand6hpositionsrespectively.SincetheCr1andCr2atomshavethesamecoordinationnumbers,theyareplacedonthesamesublattice. Table7-5. Crystalstructureofthe{TiCr2Lavesphase. SpaceGroupP63=mmcPearsonSymbolhP12AtomsWyckoPositionxyzCN Cr2a0.00.00.012Ti4f0.3330.6670.06316Cr6h0.8300.6600.25012 140

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TheCalculatedpartialTi{Crphasediagramfrom60at.%Crto70at.%CrusingthedescriptionofSaunders[ 18 ].Onlythelowtemperature{TiCr2andahightemperature{TiCr2Lavesphasesaremodeled.Therehomogeneityrangeofthe{TiCr2phaseisingoodagreementwiththeexperimentalworkofChenetal.[ 148 149 ].However,theagreementofthehomogeneityrangeofthe{TiCr2phasewiththeworkofChenetal.[ 148 149 ]isnotsogood. Table7-6. DataontheLavesphasesthatwereusedfortheoptimization. Reaction1+2+3Temperature[K]Composition[at.%Cr]123 141

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GraphshowingthedependenceoftheGTiCr2Cr:Tiparameterforthe{TiCr2phaseasthesolidline.ThelineardependenceoftheGTiCr2Cr:Tiparameterisalsoincludedasthedashedline. Figure7-6. CalculatedTi{Crphasediagramusingthenewdescription.Theexperimentaldatafromtheliteratureissuperimposed. 142

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CalculatedpartialTi{Crphasediagramfrom60at.%Crto70at.%Cr.Thereisgoodagreementofthehomogeneityrangeofthe{TiCr2,{TiCr2and{TiCr2LavesphaseswiththeexperimentalworkofChenetal.[ 148 149 ]. 143

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CalculatedactivityofCrinthephasecomparedtotheexperimentaldataofPooletal.[ 152 ].SincethenewdatasetusesthesamethermodynamicparametersforthephaseasinSaunders[ 18 ],thereisnodierenceintheactivitycurvesforCrbetweenthenewdatasetandthedatasetofSaunders[ 18 ]. 144

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8-1 .The{Al8Cr5andAl7CrphasesfromtheAl{CrbinaryhavebeenshowntoexhibitsomesolubilityforTi.Forexample,the{Al8Cr5dissolvesupto10at.%Tiatroomtemperature[ 22 24 173 ]andZoller[ 174 ]reportedthesolubilityofTiinAl7Cras1.03at.%whereasSokolovskayaetal.[ 175 ]reportedthissamesolubilityasupto2at.%Ti.However,thereisnoinformationonthesolubilityofTiintheotherbinaryAl{Crphasesavailableintheliterature.The{TiAl,2{Ti3Al,andTiAl2phasesextendintotheternaryTi{Al{CrsystemthroughdissolutionofCr.ThesolubilityofCrin{TiAlis2at.%Crat47at.%Aland8at.%Crat58at.%Alat1273K[ 176 ].At1073K,themaximumsolubilityofCrin{TiAlis4.5at.%Cr.ThesolubilityofCrin2{Ti3Alisabout2.5at.%Crinthetemperaturerange1073Kto1273K,andthesolubilityofCrinTiAl2is3.5at.%Crand6at.%Crat1273Kand1073Krespectively[ 176 ].ThesolubilityofAlinthe{TiCr2and{TiCr2phasesfromtheTi{Crbinaryhasbeenshowntobe1at.%Aland4at.%Alrespectively[ 177 178 ].ThreeternaryphasesareformedintheTi{Al{Crsystem.ThephaseformsthroughstabilizationofthebinarymetastableTiAl3(m)phasebyadditionofCr[ 20 { 26 ].Thisphaseexhibitsahomogeneityrangefrom7at.%Crto15at.%Crand23at.%Tito27at.%Tiat1473K[ 22 ],from6at.%Crto13at.%Crand60at.%Alto68at.%Alat1273K,andfrom6at.%Crto10at.%Crand64at.%Alto67at.%Alat1073K[ 179 ].AccordingtotheworksofMabuchietal.[ 173 ],Ichimaruetal.[ 180 ],andBarabashetal.[ 181 ],itispossiblethatthephaseformsperitecticallyfromtheliquid.Ichimaruetal.[ 180 ]gavetheformationtemperatureas1623KalthoughBarabashetal.[ 181 ]gaveaformationtemperatureof1523K. 145

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178 ],from5at.%Alto42at.%Alat1273K,andfrom5at.%Alto40at.%Alat1073K[ 177 ].ThedeviationofTicontentfromthestoichiometric33.3at.%Tiismeasuredas1at.%at5at.%Aland3at.%at35at.%Alat1273Kand3at.%at40at.%Alat1073K[ 176 177 ].Fujitaetal.[ 182 ]didnotndthe{Ti(Al,Cr)2phaseat1473Kalthoughitwasfoundat1323KintheworkofXuetal.[ 178 ];therefore,theformationofthe{Ti(Al,Cr)2phaseisassumedtotakeplaceatatemperaturebetween1323Kand1473K.Variousinvestigators[ 22 24 { 26 173 183 ]havefoundthe{TiCr2phasetobestabledowntoroomtemperature,i.e.,thereisnoevidenceofasolidstatedecompositionofthe{Ti(Al,Cr2)phaseatlowertemperatures.Orderingofthedisorderedphasetothe0phaseintheTi-richregionoftheTi{Al{Crsystemhasbeenobservedinseveralworks[ 6 126 176 184 { 186 ].ThehomogeneityrangeoftheTi-richphaseat1273KhasbeenmeasuredbyJewettetal.[ 176 177 ].TheformationoftheTi-rich0phaseisassumedtotakeplacebetween1323Kand1273KasXuetal.[ 178 ]didnotndthe0phaseinTi-richalloyswhichwereheattreatedandquenchedfrom1323K,butthephasewasfoundbyJewettetal.[ 176 ]andKainumaetal.[ 126 186 ]at1273K. 8.2.1IsothermalSectionsCompleteIsothermalsectionsat1273Kand1073KarepresentedbyJewettetal.[ 176 ]basedonmeasuredtielinedatafromheattreatmentandquenchexperimentsforvariousalloysat1273Kand1073K[ 176 177 179 ].AccordingtoJewettetal.[ 176 ],however,the+{TiAl+TiAl2and+TiAl2+Ti5Al11threephaseeldsat1273Karenotfullycharacterized.Additionally,phaseequilibriaatAl-richcompositionsabovethe+{(Ti,Cr)Al3+{Al8Cr5threephaseeldisunclear.The+{TiAl 146

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176 ]wasconrmedintheexperimentalworkofKainumaetal.[ 126 ].Kainumaetal.[ 186 ]alsopresented{2and2{tielinedataat1273Kwhichisingoodagreementwiththe1273KisothermalsectionofJewettetal.[ 176 ].Intheisothermalsectionat1073KofJewettetal.[ 176 ],phaseequilibriaintheAl-richregionatAlcompositionshigherthanthe{(Ti,Cr)Al3+{Al9Cr4twophasetielinearearemainunclear. 187 ]basedonDTAresults,andIchimaruetal.[ 180 ]presentedliquidusisothermallinesforthephaseinthetemperaturerangefrom1643Kto1523K.ShaoandTsakiropoulos[ 185 ]investigatedthesolidicationofaTi{40at.%Al{10at.%Cralloy,aTi{50at.%Al{10at.%Cralloy,andaTi{52at.%Al{20at.%Cralloy,andshowedthattherstalloysolidiedasphase,thesecondalloysolidiedasphaseandthethirdalloysolidiedastheternaryphase. 19 ],Hayes[ 188 ]andRaghavan[ 189 ]withthemostrecentassessmentperformedbyBochvaretal.[ 19 ].InthereviewsofRaghavan[ 189 ]andBochvaretal.[ 19 ],liquidussurfaces,variouscompleteandpartialisothermalsectionsattemperaturesbetween1473Kand770K,andScheilreactionschemesarepresented.ThekeydierencesbetweentheliquidussurfacesofBochvaretal.andRaghavanare: 1. Raghavan[ 189 ]includestheprimarycrystallizationeldfortheTi1xAl1+xphaseaswellasfortheTi5Al11phasewhereasBochvaretal.[ 19 ]showonlyaprimarycrystallizationeldfortheTi2Al5phase.ThereasonforthisisthatBochvaretal.[ 19 ]usetheTi{AlphasediagramcalculatedfromthedatasetofWitusiewiczetal.[ 110 ]astheconstituentbinaryfortheirassessmentwhereasRaghavan[ 189 ]useshisownassessedTi{Albinary[ 190 ]astheconstituentbinary.TheTi{Albinaryof 147

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17 ]combinestheTi1xAl1+xandTi5Al11phasesasoneTi2Al5phasewithahomogeneityrangefrom62at.%Alto68at.%Al.Therefore,onlytheTi2Al5phasecanbeindicatedintheliquidussurfaceofBovhvaretal.[ 19 ]. 2. TheassessmentofRaghavan[ 189 ]showsamuchlargerprimarycrystallizationeldfortheTi5Al11phase(shownastheTi2Al5phaseintheassessmentofBochvaretal.[ 19 ]).Asaresult,Raghavanincludestheunivariantlineliquid++Ti5Al11andinvariantreactionsbetweentheliquid,Ti5Al11,,andphasesandbetweentheliquid,Ti5Al11,{TiAl,andphases.TheassessmentofBochvaretal.[ 19 ]includesnosuchequilibriai.e.,theTi2Al5phasedoesnotextendtosucientlyhighCrcompositionssoastoexhibitaunivariantreactionwiththeliquidandphasesontheliquidussurface. 3. Thephaseformsperitecticallyaccordingtothereactionliquid+Ti5Al11!intheassessmentofRaghavan[ 189 ],butaccordingtothetransitionreactionliquid+{TiAl!+intheassessmentofBochvar[ 19 ]. 191 ],ShaoandTsakiropoulos[ 185 ],andKaufman[ 192 ].However,neitherdetailsofthethermodynamicmodelingandoptimizationstrategynorthenalthermodynamicparametersaregiven.Saunders[ 18 ]alsodevelopedathermodynamicdatasetfortheTi{Al{CrsystemaspartoftheCOST507Europeanaction.Thisthermodynamicdescriptionwasreadilyavailable,andwasthereforeusedtoperforminitialcalculationsintheTi{Al{Crsystemtoassessitsabilitytoreproducekeyphaseequilibria.ThecalculationswerecomparedtothephasediagramspublishedintheassessmentofBochvaretal.[ 19 ]asthisassessmentisthemostrecentandmostcomprehensiveassessmentavailable. 58 ].ThebinaryTi{AldescriptionofSaunders,whichisusedastheconstituentbinaryintheternaryTi{Al{Crsystem,wasdiscussedinSection 3.3.3.1 .SaundersalsousedhisdescriptionoftheTi{CrbinaryastheconstituentbinaryinthedescriptionfortheTi{Al{Crsystem.However,aswasdiscussedinChapter 7 ,onlytwo 148

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18 ]andistheSGTErecommendeddatasetalthoughanother,morerecentdatasetbyLiangetal.[ 193 ]existsintheliterature.ThecalculatedbinaryAl{CrsystemofSaunders[ 18 ]isshowninFigure 8-1 .SomediscrepanciesexistbetweenthecalculatedphasediagramofSaunders[ 18 ],thecalculatedphasediagramofLiangetal.[ 193 ](Figure 8-2 ),andthenumerousexperimentalworkperformedonthesystem.Someofthemajordiscrepanciesare: 1. Grushkoetal.[ 194 ]showedthatinthecompositionrangefrom30at.%Crto42at.%Cr,onlyonehightemperaturephasewithaperfect-brassstructureandonelowtemperaturephasewitharhomboherallydistorted{brassstructureexist.ThehomogeneityrangeofthephasewasreproducedinthemorerecentthermodynamicdescriptionofLiangetal.[ 193 ].However,Saundersmodelstwostoichiometricphases,theAl8Cr5andAl9Cr4phases,eachwithapolymorphictransformationfromthelowtemperaturemodicationtothehightemperaturemodication,withinthiscompositionrange. 2. Grushkoetal.[ 195 ]foundanewphaseofcompositionAl11Cr4whichformsbyaperitectoidreactionbetweentheAl4Crandphases.ThisnewphasewasnotincludedinthedatasetofLiangetal.[ 193 ]orinthedatasetofSaunders[ 18 ]. 3. MahdoukandGachon[ 196 ],usingheattreatingandquenchingexperiments,foundthattheAl11Cr2phaseundergoeseutectoiddecompositiontotheAl7Cr2andAl4Crphasesatatemperaturebetween1123Kand923K.However,thiswasnotconrmedintheworksofGrushkoetal.[ 194 195 197 ].TheeutectoiddecompositionoftheAl11Cr2phaseisreproducedinthedescriptionofLiangetal.[ 193 ]butnotinthedescriptionofSaunders[ 18 ]. 4. HelanderandTolochko[ 198 ]showedthatalloyscontainingbetween58.4at.%Aland64.8at.%Alheattreatedandquenchedfrom1158Kand1178KcontainedtheorderedB2structure.However,theorder{disordertransformationofthephasewasnotmodeledineitherthedatasetofLiangetal.[ 193 ]orthedatasetofSaunders[ 18 ]. 149

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18 ]toassesstheabilityofthisdatasettoreproducethephasediagramsgivenintheassessmentofBochvaretal.[ 19 ].Thecalculatedandassessedisothermalsectionsat1073Kand1273KareshowninFigure 8-3 andFigure 8-4 respectively.Themajordiscrepanciesbetweenthecalculatedandassessedphasediagramsare: 1. Theextensionoftheternary{Ti(Al,Cr)2Lavesphase,whichisindicatedinthe1073Kand1273KassessedisothermalsectionsofBochvaretal.[ 19 ],isnotreproducedinthedescriptionofSaunders[ 18 ]. 2. Saunders[ 18 ]didnotincludeadescriptionfortheternaryphase. 3. Theisolatedsinglephase0eldatapproximately18at.%Crand30at.%Alshowninthe1073KisothermalsectionassessedbyBochvaretal.[ 19 ]isnotreproducedusingthedatasetofSaunders[ 18 ]. 4. The2{Ti3AlphaseappearstobestabilizedintheTi-richcornerofthecalculated1073KisothermalsectioninthedescriptionofSaunders[ 18 ]. 5. ThecalculatedsolubilityofAlinthe{TiCr2and{TiCr2Lavesphasesismuchlowerthanthatobservedexperimentally. 6. ThereisnodescriptionforthesolubilityofTiintheAl11Cr2,Al4Cr,{Al9Cr4,{Al8Cr5,andAlCr2phases.ThecalculatedandassessedliquidussurfacesareshowninFigure 8-5 .ThemaindierencebetweentheliquidusprojectionsisthepresenceofaprimarycrystallizationeldforthephaseintheliquidussurfaceofBochvaretal.[ 19 ].SincethephaseisnotmodeledinthedatasetofSaunders[ 18 ],theliquidussurfaceofSaundersshowslargerprimarycrystallizationareasforthe{TiAl,{(Ti,Cr)Al3,and{Al8Cr5phases.ToimprovetheagreementbetweenthecalculatedphasediagramsandthephasediagramsfromtheassessmentofBochvaretal.[ 19 ],descriptionsfortheand{Ti(Al,Cr)2Lavesphasesshouldbeintroduced.Additionally,theAlsolubilityofthe{TiCr2,{TiCr2,and{TiCr2Lavesphasesshouldbemodeledforbetteragreementwiththe 150

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8.5.1OptimizationStrategyFirst,the2{Ti3AlphasewasmadelessstableintheTi{Crbinarybyadditionofapositivetemperatureindependenttermtooneofitsendmembers.Second,theparametersofthe{Ti(Al,Cr)2phasewereoptimizedtocalculatetheexperimentallyobservedextensionofthe{Ti(Al,Cr)2phaseintotheternaryTi{Al{Crsystemat33.3at.%Ti.Third,thephasewasintroducedasastoichiometricphasewithalineartemperaturedependenceoftheGibbsenergyofformationfromtheelements.Thetemperaturedependentandindependenttermswereassessedsothatthreephaseequilibriawiththephaseat1273KreportedintheworksofJewettetal.[ 176 177 179 ],aswellasthepresenceofaprimarycrystallizationeldforthephaseontheliquidusprojectioncouldbecalculated.Next,theparametersofthephasewerereoptimizedtocalculatethe/+{Ti(Al,Cr)2phaseboundariesat1073Kand1273KfromJewettetal.[ 176 177 179 ].Oncetheparametersforthedisorderedphasewerere-optimized,theparametersoftheordered0phasewerere-assessedtoreproducetheisolated0eldat1073KwhichisgivenintheworksofJewettetal.[ 176 177 ].Last,thesolubilityofAlinthe{TiCr2and{TiCr2Lavesphasesweremodeledtobeinagreementwiththe1073Kand1273KisothermalsectionsassessedbyBochvaretal.[ 19 ]. 18 ]indicatedastabilizationofthe2{Ti3AlphaseintheTi{CrsystemandTi-richcorneroftheTi{Al{Crsystem.SincethisstabilizationoccurredinTi-richcorner,apositivetermof4000Jwasaddedto 151

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7-5 .ThetwosublatticemodelgiveninEquation 7{1 ,whichwasusedtomodelthebinaryTi{CrLavesphases,wasmodiedto: (Cr*,Ti,Al)2(Ti*,Cr,Al)1(8{1)wheretheasteriskidentiesthemajorspeciesineachsublattice,toreproducethesolubilityofAlintheternary{Ti(Al,Cr)2Lavesphase.Sincethereisnoevidenceintheliteratureofaseparationoftheternary{Ti(Al,Cr)2andbinary{TiCr2phases,andsincebothphaseshavethesameMgZn2crystalstructure,theyweremodeledasasinglephase.ThelocationsoftheendmembersthatresultfromthedescriptiongiveninEquation 8{1 areindicatedontheGibbstriangleinFigure 8-6 .ThedescriptionfortheGCr:TiendmemberwastakenfromthebinaryTi{Crdatasetwhichwasassessedinthepresentwork(detailsgiveninChapter 7 )whereasthedescriptionsforallotherendmemberswerekeptfromtheoriginalTi{Al{CrdatasetofSaunders[ 18 ].The0L*:Cr,Ti,0LCr,Ti:*,and0LAl,Cr:Timixingparameterswereoptimizedinthiswork.The0L*:Cr,Tiand0LCr,Ti:*mixingparameterswereacceptedfromthebinarydescriptionoftheTi{Crsystemperformedinthepresentwork(Chapter 7 ).Theseparameterscontrolthewidthofthe{Ti(Al,Cr)2phase.The0LAl,Cr:Tiparameter,whichwasmodeledusingalineartemperaturedependence,inuencestheextensionofthe{Ti(Al,Cr)2phaseintotheternary.Togeneratestartingvaluesforthetemperatureindependentandtemperaturedependenttermsofthe0LAl,Cr:Tiparameter,twovaluesofthisparameterwereapproximated.Therstvalue,0L;1273KAl,Cr:Ti,isthevalueofthe0LAl,Cr:Tiparameterrequiredtocalculateanextensionofthe{Ti(Al,Cr)2phasetoapproximately42at.%Al 152

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177 ].Thesecondvalue,0L;1073KAl,Cr:Ti,isthevalueofthe0LAl,Cr:Tiparameterrequiredtocalculatetheextensionofthe{Ti(Al,Cr)2phasetoapproximately40at.%Alat1073K,whichwasalsomeasuredbyJewettetal.[ 177 ]. 24 173 199 200 ]reportthestoichiometryofthephaseasTi25(Al67Cr8),whichisobtainedwhenx=y=0.32.Otherauthors[ 21 179 ]giveahomogeneityrangeofthephasewhichincludesthisstoichiometry.Therefore,thephaseismodeledasastoichiometricphaseusingtheequation: 8{2 .TherstvalueofGTi25(Al67Cr8)waschosenasthevalueoftheparameterrequiredtocalculatethe+{Ti(Al,Cr)2+,+{Al8Cr5+,{(Ti,Cr)Al3+{Ti5Al11+,and{TiAl+{Ti(Al,Cr)2+seriesofthreephaseequilibriaat1273KtobeinagreementwiththeexperimentalworkofJewettetal.[ 176 177 179 ].ThesecondvalueofGTi25(Al67Cr8)waschosentottheexperimentaldataontheformationofthephaseathighertemperatures.AccordingtotheworksofMabuchietal.[ 173 ],Ichimaruetal.[ 180 ],andBarabashetal.[ 181 ],itispossiblethatthephaseformsattemperaturesbetween1623K[ 180 ]and1523K[ 181 ].Takingintoaccountthe100Kdierenceinphaseformationtemperatures,thevalueoftheGTi25(Al67Cr8)parameterwaschosentottheformationofthephaseatatemperaturebetween1623Kand1523K,aswellastoreproduceprimarycrystallizationeldsof 153

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19 ].ThecalculatedliquidussurfacewasfoundtobeingoodqualitativeagreementwiththeassessedliquidusprojectionofBochvaretal.[ 19 ]whentheformationtemperatureofthephasewascalculatedat1603K.Therefore,thevaluesoftheGTi25(Al67Cr8)parameterat1273Kand1603KwereusedtogeneratestartingvaluesoftheaandbparametersinEquation 8{2 176 177 179 ]. (Al,Nb,Ti)0:5(Al,Nb,Ti)0:5(8{3)Thelocationsoftheendmembersforthe0phasearethesameasthatforthephaseintheTi{Al{Nbsystem(Section 5.3.4 ),andarethereforealreadyillustratedinFigure 5-6 .Thedescriptionsfortheendmembersofthe0phasewerekeptfromtheoriginaldescriptionofSaunders[ 18 ].Inthemodelforthe0phase,therestrictionsofsymmetryimposetheconstraint0L0i;j:k=0L0k:i;j.Onlythe0L0Al,Cr:Ti=0L0Ti:Al,Crparametersaectedtheformationoftheisolated0eld.Allotherparametersoftype0L0i;j:kwerethereforegivenavalueof0.The0L0Al,Cr:Ti=0L0Ti:Al,Crparametersweremodeledwithalineartemperaturedependence,thestartingvaluesforwhichwereselectedtotthephaseequilibriawiththe0phaseat1273Kand1073KgivenintheworksofJewettetal.[ 176 177 ]. 154

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8{1 .TheonlyparameterswhichinuencethesolubilityofAlinthe{TiCr2and{TiCr2Lavesphasesarethe0LTiCr2Al,Cr:Tiand0LTiCr2Al,Cr:Tiparametersrespectively.Therefore,theseparameterswereoptimizedtobeinagreementwiththerecommendationsofBochvaretal.[ 19 ]. 8-7 and 8-8 respectively.TheexperimentaldataofJewettetal.[ 176 177 179 ]aresuperimposed.Thecalculatedisothermalsectionsshowagoodagreementwiththeexperimentaldata.However,the2{Ti3Al+0+{TiCr2and2++{TiCr2threephaseeldssuggestedbyBochvaretal.[ 19 ]couldnotbereproduced.Instead,the2{Ti3Aland0andthe2{Ti3Alandphasesformthreephaseequilibriawiththeternary{Ti(Al,Cr)2phase.ThecalculatedliquidussurfaceandpartialliquidussurfaceareshowninFigures 8-9 and 8-10 respectively.TheScheilreactionschemeforequilibriawiththeliquidphaseisshowninFigure 8-11 155

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StablesolidphasesintheTi{Al{Crsystem. PhasePrototypePearsonSymbolSpacegroup Al7CrV7Al45mC104C2=mAl11Cr2{oC584CmcmAl4CrMnAl4hP574P63=mmc{Al9Cr4Cr4Al9hR156R3m{Al9Cr4Cu4Al9cI5I43m{Al8Cr5Cr5Al8hR78R3m{Al8Cr5Cu5Zn8cI52I43mAlCr2MoSi2tI6I4=mmm{TiCr2MgCu2cF24Fd3m{TiCr2MgNi2hP24P63=mmc{TiCr2MgZn2hP12P63=mmcWcI2Im3m0CsClcP2Pm3mMghP2P63=mmc2{Ti3AlNi3SnhP8P63=mmc{TiAlAuCutP2P4=mmm"{TiAl2Ga2HftI24I41=amd{(Ti,Cr)Al3Al3TitI8I4=mmm{Ti5Al11Al3ZrtI16I4=mmm(Al)CucF4Fm3m{(Ti1x+yCrx)Al3yAuCu3cP4Pm3m{Ti(Al,Cr)2MgZn2hP12P63=mmc

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CalculatedbinaryAl{CrphasediagramusingthedescriptionofSaunders[ 18 ]. Figure8-2. CalculatedbinaryAl{CrphasediagramfromthedescriptionofLiangetal.[ 193 ]. 157

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BFigure8-3. ComparisonbetweentheA)isothermalsectionat1073KcalculatedusingthedatasetofSaunders[ 18 ]andtheB)assessed1073KisothermalsectionofBochvaretal.[ 19 ]. BFigure8-4. ComparisonbetweentheA)isothermalsectionat1273KcalculatedusingthedatasetofSaunders[ 18 ]andtheB)assessed1273KisothermalsectionofBochvaretal.[ 19 ]. 158

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BFigure8-5. ComparisonbetweentheA)calculated[ 18 ]andB)assessed[ 19 ]liquidussurface. 159

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Theendmembersoftheternary{Ti(Al,Cr)2phase.ThedashedlinefromAl:TitoCr:Tischematicallyillustratestheinuenceofthe0LAl,Cr:Tiparameter. 160

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1273KisothermalsectioncalculatedusingthenewdescriptionfortheTi{Al{Crsystem.ExperimentaldataofJewettetal.[ 97 177 179 ]aresuperimposed. Figure8-8. 1073KisothermalsectioncalculatedusingthenewdescriptionfortheTi{Al{Crsystem.ExperimentaldataofJewettetal.[ 97 177 179 ]aresuperimposed. 161

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LiquidussurfacecalculatedusingthenewdescriptionoftheTi{Al{Crsystem. Figure8-10. PartialliquidussurfacecalculatedusingthenewdescriptionoftheTi{Al{Crsystem. 162

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ScheilreactionschemefortheTi{Al{Crsystem. 163

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3.3.3.1 .CriticalevaluationsofthephaseequilibriaintheTi{MobinarysubsystemwereperformedbyMurray[ 160 ]andMassalski[ 201 ].Thesecriticalevaluationsacceptthepresenceofamiscibilitygapinthebccphasewithamonotectoidreactionat1123KbasedonthemetallographicinvestigationandelectricalresistivitymeasurementsofseveralTi{MoalloysperformedbyTerauchietal.[ 202 ].ThecriticallyevaluatedTi{MophasediagramofMassalskiisshowninFigure 9-1 .ThermodynamicdescriptionsfortheTi{MosystemwereoriginallydevelopedbySaunders[ 18 ]andShimetal.[ 203 ].ThedescriptionofShimetal.[ 203 ]wasupdatedbyChungetal.[ 204 ]tobeabletocalculateexperimentallyobservedinvariantequilibriaintheTi{Mo{Nsystem.TheassessmentofChungetal.[ 204 ]reproducesthemiscibilitygapinthebccphasewhereastheassessmentofSaunders[ 18 ]doesnottakethismiscibilitygapintoaccount(Figure 9-2 ).AlthoughTerauchietal.[ 202 ]observedamiscibilitygapintheTi{Mosystem,whichwasacceptedintheevaluationsofMurray[ 160 ],Massalski[ 201 ],andtheassessmentofChungetal.[ 204 ],thismiscibilitygapcouldnotbeconrmedintheworksofMorniroliandGantonis[ 205 ]andDupouyandAverbach[ 206 ].BothworksusedX-raydiractiontoshowthetendencyforshortrangeordering,notphaseseparation,inthebccsolidsolution.TheresultsoftherstprinciplescalculationsofRubinandFine[ 207 ]alsoquestionthepresenceofthemiscibilitygapinthebccphase.Therefore,theTi{ModescriptionofSaunders[ 18 ]isrecommendedbySGTE.TheAl{MosystemwascriticallyassessedbyBreweretal.[ 208 ],Saunders[ 209 ],andSchuster[ 210 ].AftertheworkofSaunders[ 209 ],SchusterandIpser[ 211 ]publishedthe 164

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210 ].TheassessedAl{ModiagramfromSchuster[ 210 ]isgiveninFigure 9-3 andthecalculatedAl{ModiagramfromSaunders[ 18 ]isgiveninFigure 9-4 .Thereareobviousdierencesbetweenthediagrams.First,Saunders[ 18 ]modelsthe{AlMo3phaseasalinecompoundwhereasSchuster[ 210 ]showsthatthe{AlMo3phaseexistsoverahomogeneityrange.Second,Schuster[ 210 ]showsaneutecticreactionbetweentheliquid,{AlMo3,andAlMophaseswhereasthedatasetofSaunderscalculatesaperitecticreaction[ 18 ].Last,Schuster[ 210 ]includesmorephasesintheAl{Mo3Al8regionthanSaunders[ 18 ]. 9-1 .ThereisnoinformationonthesolubilityofTiinthetheAl12Mo,Al5Mo,Al22Mo5,Al17Mo4,Al4Mo,andAl63Mo37phasesoftheAl{Mobinarysystem,butthereissomeevidenceintheliteraturethattheAl8Mo3and{AlMo3phasesdissolveTi.Forexample,theequilibrium+Al8Mo3+{AlMo3tie-triangledataforanalloyofcompositionTi{52at.%Al{45at.%Mo,whichwasheattreatedat1673Kandquenched,showedthattheAl8Mo3phasedissolved1at.%Tiandthe{AlMo3phasedissolved2.7at.%Ti[ 27 ].Additionally,analloyofcompositionTi{24at.%Al{58at.%Mo,whichwasheattreatedat1198Kandquenched,containedonlythe{AlMo3phase[ 212 ],meaningthatthe{AlMo3phaseextendstoatleast18at.%Tiat1198K.The{(Ti,Mo)Al3phase,whichisbasedonthebinaryTiAl3phase,existsoveralargehomogeneityrangeintheternaryTi{Al{Mosystem.HansenandRaman[ 212 ]showedthatalloysofcompositionTi{64at.%Al{10at.%Mo,Ti{68at.%Al{16at.%Mo,andTi{75at.%Al{12.5at.%Mowereallsinglephase{(Ti,Mo)Al3at1198K.The{(Ti,Mo)Al3phasewasfoundtoextendtoMocompositionsashighas20at.%MoandAlcompositionsaslowas62at.%Alat1198K[ 212 ].Eremenkoetal.[ 213 ]conrmedthelargehomogeneityrangeofthe{(Ti,Mo)Al3phaseat1573K,andAbdel-Hamid[ 214 ] 165

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212 ].The{TiAlphasehasbeenshowntodissolvesomeMo.Dasetal.[ 215 ]showedthatat1448K,the{TiAlphasecontains1at.%Moat52at.%Tiand3at.%Moat48at.%TiwhereasMorrisetal.[ 216 ]measuredtheMocontentofthe{TiAlphaseinequilibriumwiththe2{Ti3Aland0phasesat1173Kand1473Kas1at.%Moat51at.%Tiand1.5at.%Moat53at.%Tirespectively.KimuraandHashimoto[ 217 ]measuredtheMocontentofthe{TiAlphaseinequilibriumwiththe2{Ti3Aland0phasesat1473Kas0.5at.%Moat50at.%Ti,whichismuchlowerthanthatmeasuredintheworkofDasetal.[ 215 ].However,Dasetal.[ 215 ]heattreatedtheiralloysfor120hourswhereasKimuraandHashimoto[ 217 ]heattreatedtheiralloysforonly4hours.Therefore,theworkofDasetal.[ 215 ]maymorecloselygivetheequilibriumcompositionofthe{TiAlphaseat1473K.AlthoughthemaximumsolubilityofMointhephasehasnotbeenmeasured,SinghandBanerjee[ 218 ]showedthatanalloyofcompositionTi{48at.%Al{2at.%Moisinthesinglephaseeldat1673K.ThesameauthorssuggestedthatthephaseeldextendstoanalloyofcompositionTi{50at.%Al{6at.%Moat1673Kalthoughthiswasnotexplicitlyproved.TheTi-richsinglephasewasalsoobservedinanalloyTi{2.5at.%Al{2.5at.%Mowhichwasheattreatedat1073Kfor222hoursandquenched[ 219 ].ThesolubilityofMointhe2{Ti3Alphaseisunclear.AlthoughBanerjeeetal.[ 220 ]publishedpartialisothermalsectionsat1673K,1573K,and1473Kshowingthe2{Ti3Alphase,themicrostructuresoftheinvestigatedalloyswereinterpretedtakingintoaccountadescriptionofthebinaryTi{AlphasediagramfromMargolin[ 221 ]whichshowedtheperitecticformationof2{Ti3Alfromtheliquidandphasesat1745K.However,thecriticallyassessedTi{AlphasediagramofSchusterandPalm[ 111 ]showstheperitectoidformationofthe2{Ti3Alphasefromtheandphasesat1473K.Therefore,itis 166

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220 ]misidentiedthephaseasthe2{Ti3Alphase.Infact,thepartialisothermalsectionsat1448KofDasetal.[ 100 ]andat1573Kand1473KofKimuraandHashimoto[ 217 ]showonlythephase.However,Morrisetal.[ 216 ]claimedtoalsohavethe2{Ti3Alphaseat1473K.Theordered0phasewasrstfoundbyBohmandLohberg[ 222 ]inalloyswithinthecompositionrangefrom60wt.%Moto30wt.%Moat50at.%Tiwhichwereheattreatedat1073Kandthenquenched.Sincethen,the0phasehasbeenfoundbymanyauthors.Particularly,Chenetal.[ 223 ]foundthe0phaseinanalloyofcompositionTi{50at.%Al{15at.%Moat1673K,1623K,1473K,1273K,and1073K.Singhetal.[ 224 ]performedRietveldrenementofX-rayandneutrondiractiondataforanalloyofcompositionTi{25at.%Al{25at.%Mowhichwashomogenizedat1273Kandfurnacecooled.Thealloywasfoundtocontainthe0phase,andRietveldrenementofthedatashowedthatTiatomsoccupyAsitesandAlandMoatomsoccupytheBsitesintheCsClstructure.ExcessMoatomscanalsooccupyAsiteswithTiatoms[ 224 ].ThereisoneternaryphaseintheTi{Al{Mosystem.HansenandRaman[ 212 ]foundtheternaryphaseinthesinglephasealloyofcompositionTi{41at.%Al{33at.%Moat1198K.ThephasewasfoundtohaveaverysmallhomogeneityrangeandthestoichiometryTi1:5Al2Mo1:5wasassigned.ThepresenceoftheternaryphasewasconrmedintheworkofEremenkoetal.[ 219 ].Thephaseformsthroughaperitectoidreactionat1523K. 225 ]basedonresultspublishedupto1990,Tretyachenko[ 226 ]basedonresultspublishedupto2003,andRaghavan[ 227 ]basedonresultspublishedupto2005.TheassessmentofTretyachenko[ 226 ]isanupdatetotheearlierassessmentofBudbergandSchmid-Fetzer[ 225 ]totakeintoaccountnewerexperimentalworksince1990.TretyachenkousedtheassessedAl{ModiagramfromSchuster[ 210 ] 167

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228 ]astheconstituentbinariesfortheassessmentwhereasRaghvan[ 227 ]usedtheAl{MofromSaunders[ 209 ],updatedtoincludetheexperimentalworkofSchusterandIpser[ 211 ]ontheAl{Mo3Al8region,andtheAl{TisystemfromRaghavan[ 190 ].BothTretyachenkoandRaghavanusetheMo{TibinaryfromMassalski[ 201 ].Ninoetal.[ 27 ]publishedthemostcomprehensiveworkonthephaseequilibriaintheTi{Al{Mosystemfrom0to20at.%Ti.SeveralsampleswithAlcontentsrangingfrom45at.%Alto55at.%AlandTicontentsrangingfrom3at.%Tito17at.%Tiwereheattreatedat1773K,1723K,and1673Kandquenched.DTAwasalsoperformedonthealloystomeasuresolidstatetransformationtemperatures.BasedontheresultsofmetallographicanalysisandDTA,partialisothermalsectionsfrom0at.%Tito20at.%Tiat1673K,1573K,1540K,1473K,and1373K,aswellasanisopleththrough50at.%Alfrom0at.%Tito20at.%Ti,wereconstructed.SincetheassessmentofTretyachenko[ 226 ]waspresumablyperformedbeforethepublicationofNinoetal.[ 27 ],Tretyachenkocouldnothavetakenthisworkintoaccountintheassessment.Raghavan[ 190 ],however,presentsisothermalsectionsat1573K,1473K,and1373KthattakeintoaccounttheworkofNinoetal.[ 27 ].OnekeyconclusionofNinoetal.[ 27 ]isthecontinuityofthephaseat1773KfromtheTi{MobinarytotheAl{Mobinary.Ninoetal.conrmedthephasecontinuitysinceanalloyofcompositionTi{52at.%Al{45at.%Mo(whichcontainsonly3at.%TiandisthereforequiteclosetotheAl{Mobinary),showedasinglephasemicrostructurefollowingheattreatmentat1773Kandquenching[ 27 ].SinceTretyachenkodoesnotincludeanisothermalsectionat1773K,nodirectcomparisoncanbemadebetweentheexperimentalresultsofNinoetal.[ 27 ]andthecriticalassessmentofTretyachenko[ 226 ].However,Tretyachenkodoesincludeanisothermalsectionat1873K,whichisshowninFigure 9-5 .WhenthealloyofcompositionTi{52at.%Al{45at.%Moissuperimposedonthe1873Kisothermalsection,however,it 168

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210 ],whichwasusedbyTretyachenko[ 226 ].However,intheoriginalworkofRexer[ 229 ],inwhichtheAlMophasewasdiscovered,thephasewasassignedtheW-A2structureafteranalysisofthex-raydiractionpattern.Infact,Schuster[ 210 ]givesnoindicationwhytheAlMophasewasassignedtheCsCl{B2structurewhiletheotherAl{MoassessmentsofSaunders[ 209 ]andBreweretal.[ 208 ]assigntheW-A2structuretotheAlMophase.Tretyachenko,however,doesindicatethephaseequilibriaat1873KclosetotheAl{Mobinaryedgeusingdashedlines,whichmeansthatthephaseequilibriainthisregionisunclear.TretyachenkoalsoincludedanassessedsolidussurfacewhichisshowninFigure 9-6 .WhenthealloyofcompositionTi{52at.%Al{45at.%Moissuperimposedonthesolidussurface,itislocatedinthe+AlMo+Al63Mo37threephaseeldwhichisformedfromtheinvariantreactionliquid+AlMo!+Al63Mo37.AccordingtothepartialScheilreactionschemeconstructedbyTretyachenko[ 226 ],thisinvariantreactionshouldtakeplaceat1823K.Thus,combiningtheinformationavailablefromtheassessed1873Kisothermalsection,theinformationavailablefromtheassessedsolidussurface,andthepartialScheilreactionscheme,theTi{52at.%Al{45at.%Moalloywouldsolidifyasliquid!liquid+AlMo!liquid+AlMo+!+AlMo+Al63Mo37andcouldneverexistassinglephase.Therefore,theevaluationofTretyachenko[ 226 ]isnotinagreementwiththeexperimentalworkofNinoetal.[ 27 ]forthisalloycomposition.Ninoetal.[ 27 ]heattreatedfouralloyswithTicompositionsrangingfrom15at.%Tito17at.%TiandAlcompositionsrangingfrom47at.%Alto50at.%Alat1723Kandquenchedthealloystoanalyzetheresultingmicrostructures.Metallographicinvestigation 169

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9-6 accordingtoTretyachenko[ 226 ],theywouldsolidifyassinglephase,whichisingoodagreementwiththisaspectoftheworkofNinoetal.[ 27 ].Asstatedearlier,Ragahavan[ 227 ]tookintoaccounttheexperimentalworkofNinoetal.[ 27 ]inhisassessment.Therefore,asonlypartialisothermalsectionsfrom0at.%Tito30at.%Tiarepresentedat1573K,1473K,and1373K,thereisnoobviouscontradictionwiththeworkofNinoetal.[ 27 ].AlthoughRaghavandoesnotpresentanycriticallyassessedliquidussurface,solidussurface,orhighertemperatureisothermalsections,hedoesmentionthatNinoetal.[ 27 ]haveruledoutthepossibilityoftheexistenceoftwoseparateb.c.cphasesinthecompositionrangethatwasinvestigated.BasedontheDTAoftheseveralalloysinvestigated,Ninoetal.[ 27 ]concludedthattheinvariantreaction+Al8Mo3!+shouldtakeplaceat1540K.AccordingtotheScheilreactionschemeofTretyachenko[ 226 ],thisreactionoccursat1598KandRaghavan[ 227 ]acceptedtheinvariantreactiontemperatureofNinoetal.[ 27 ]forhiscriticalassessment. 18 ].Calculationswereperformedwiththisdatasettoassessitsabilitytopredictthephaseequilibriaintheregionfrom0at.%Tito20at.%TithatwasinvestigatedbyNinoetal.[ 27 ].Inthisdataset,thelatticestabilitiesforthepureelementsweretakenfromSGTE[ 58 ]andthedescriptionsforallthreeconstituentbinariesweredevelopedbySaunders.Thecalculatedisothermalsectionat1773KisshowninFigure 9-7 .ThealloyofcompositionTi{52at.%Al{45at.%Mo,whichshouldbesinglephaseat1773KaccordingtotheworkofNinoetal.[ 27 ],isintheliquid++Al63Mo37threephaseeldbecauseSaunders[ 18 ]didnotmodeltheternaryphaseandtheAlMophaseasacontinuoussolidsolution.ThiswouldindicatethatthedescriptionforthebinaryAl{Mo 170

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27 ]at1673Kand1573KaregiveninFigure 9-8 ,inwhichthephaseisshowntoextendto6at.%Tiat1673Kandto10at.%Tiat1573K.Additionally,theinvariantreaction+Al8Mo3!+couldnotbecalculatedatanytemperature.Forsuchatransitionreactiontotakeplace,twothreephaseequilibria,+Al8Mo3+and+Al8Mo3+mustexistattemperaturesabovetheinvariantreactiontemperature.Thecalculatedisothermalsectionat1540Kandthepartialisothermalsectionat1540Kfrom0at.%Tito20at.%TifromNinoetal.[ 27 ]aregiveninFigures 9-9 and 9-10 respectively.ThepartialisothermalsectionofNinoetal.[ 27 ]showsaphasethatextendsto14at.%Ti.Thecalculatedisothermalsectionat1540Kshowsthatwhilethe+Al8Mo3+threephaseeldispresent,thephasedoesnotextendtolowenoughTicompositionstoformthe++Al8Mo3threephaseequilibriumatanytemperature.TheliquidussurfacealongwiththesoliduslinecalculatedusingthedescriptionoftheTi{Al{MosystemofSaundersisshowninFigure 9-11 .Accordingtothecalculations,thealloyofcompositionTi{52at.%Al{45at.%Moisintheprimarycrystallizationeldofthephase,whichisindisagreementwiththeworkofNinoetal.[ 27 ].Additionally,althoughthefouralloyswithTicompositionsrangingfrom15at.%Tito17at.%TiandAlcompositionsrangingfrom47at.%Alto50at.%Al,whichhavebeenshowntosolidifyassinglephaseat1723KbyNinoetal.[ 27 ],arelocatedintheprimarycrystallizationareaofthephase,accordingtothecalculations,thesealloysdonotsolidifyassinglephasebecausethesolidusdoesnotextendtohighenoughAlcompositions.Theresultsofthesepreliminarycalculationssuggestthefollowingpointsforre-optimization.Theseare: 1. SincetherearelargedierencesbetweentheassessedAl{MophasediagramofSchuster[ 210 ]andthecalculatedphasediagramofSaunders[ 18 ],thethermodynamic 171

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2. There-optimizedAl{MosystemshouldmodeltheMo-richphaseandtheAlMophaseasthesamephasesothattheternaryphaseiscontinuousfromtheTi{MoboundarytotheAl{Moboundaryat1773KtobeinaccordancewiththeworkofNinoetal.[ 27 ]. 3. Theternaryinteractionparametersforthephaseandtheparametersforthephaseshouldbere-optimizedsothattheinvariantequilibrium+Al8Mo3!+couldbecalculatedasastableone. 172

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AssessedTi{MophasediagramaccordingtoMassalski[ 201 ]. Figure9-2. Ti{MophasediagramcalculatedusingthedatasetofSaunders[ 18 ]. 173

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AssessedAl{MophasediagramaccordingtotheSchuster[ 210 ]. Figure9-4. CalculatedAl{MophasediagramfromthedescriptionofSaunders[ 18 ]. 174

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StablesolidphasesintheTi{Al{Mosystem. PhasePrototypePearsonSymbolSpacegroup Al12MoWAl12cI26Im3Al5Mo(r){hP36R3cAl5Mo(h1){hP60P3Al5Mo(h2)WAl5hP12P63Al22Mo5{oF216Fdd2Al17Mo4{mC84C2Al4MoWAl4mC30CmAl3+xMo1-xWO3cP8Pm3nAl3Mo(h)Cr3SimC32CmAl8Mo3{mC22c2=mAl63Mo37{{{AlMoWcI2Im3m{AlMo3Cr3SicP8Pm3nWcI2Im3m0CsClcP2Pm3mMghP2P63=mmc2{Ti3AlNi3SnhP8P63=mmc{TiAlAuCutP2P4=mmc"{TiAl2Ga2HftI24I41=amd{(Ti,Mo)Al3Al3TitI8I4=mmm{Ti5Al11Al3ZrtI16I4=mmm(Al)CucF4Fm3m{Ti1:5Al2Mo1:5-CrFetP30P42mnm

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Assessed1873KisothermalsectionfromtheworkofTretyachenkoetal.[ 226 ].ThealloyofcompositionTi{52at.%Al{45at.%Mo,whichhasbeenshowntosolidifyassinglephase[ 27 ],isintheAlMo+liquidtwophaseeld.Therefore,thisalloysolidiesastheAlMophase. 176

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AssessedpartialsolidusintheTi{Al{MosystemfromtheworkofTretyachenkoetal.[ 226 ].ThealloyofcompositionTi{52at.%Al{45at.%Mo,whichshouldsolidifyassinglephase[ 27 ],isinsteadinthe+AlMo+Al63Mo37threephaseeld.Thegroup2andgroup3alloysofNinoetal.[ 27 ]wereshowntosolidifyassinglephase. 177

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Calculated1773KisothermalsectionusingthedescriptionofSaunders[ 18 ].ThealloyofcompositionTi{52at.%Al{45at.%Mo,whichshouldbesinglephaseaccordingtotheworkofNinoetal.[ 27 ],isintheliquid++Al63Mo37threephaseeld. Figure9-8. Phaseequilibriaintheregionfrom0at.%Tito20at.%TifromNinoetal.[ 27 ].ThealloyofcompositionTi{52at.%Al{45at.%Moisindicated. 178

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Calculated1540KisothermalsectionsusingthedescriptionofSaunders[ 18 ].SincethephasedoesnotextendtohighenoughAlcompositions,theinvariantreaction+Al8Mo3!+cannotbecalculated. Figure9-10. Phaseequilibriaintheregionfrom0at.%Tito20at.%TifromNinoetal.[ 27 ].Theinvariantreaction+Al8Mo3!+,aswellasthepositionofthephaseboundaryat1540K,isindicated. 179

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LiquidussurfacecalculatedusingthedescriptionfortheTi{Al{MosystemofSaunders[ 18 ].ThealloyofcompositionTi{52at.%Al{45at.%Moisintheprimarycrystallizationeldofthephase.However,Ninoetal.[ 27 ]showedthatthisalloyshouldsolidifyassinglephase.Additionally,thegroup3alloysinvestigatedbyNinoetal.[ 27 ]alsosolidifyassinglephase.However,thecalculatedsolidusdoesnotextendtohighenoughAlcompositionsforthistobereproduced. 180

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230 ]andWalford[ 231 ].TheAl12MophasewasalsoconrmedintheworksofSperner[ 232 ],Clare[ 233 ],PotzschkeandSchubert[ 234 ],andKamei[ 235 ].TheperitecticformationofAl12Mowasmeasuredat973K[ 232 ],976K[ 233 ],and957K[ 235 ].ThehexagonalAl5MophasewasrstidentiedbyYamaguchiandSimizu[ 236 ]andlaterconrmedintheworksofSperner[ 232 ],PotzschkeandSchubert[ 234 ],andKameietal.[ 235 ].TheperitecticformationofAl5Mowasmeasuredat976K[ 236 ],1008K[ 232 ],and990K[ 235 ].ThecrystalstructureoftheAl5WprototypeoftheAl5MophasewasclariedbyAdamandRich[ 237 ].TheAl4MophasewasrstmentionedbyWohlerandMichel[ 238 ]andwasincludedinthepartialAl{MophasediagramofReimann[ 239 ],alongwiththeAl3Mophase.Themicrostructuresofancastalloywith12wt.%MoinvestigatedbyReimann[ 239 ]conrmedtheperitecticformationofAl4Mo.Usingthermalanalysis,Reimann[ 239 ]measuredtheperitecticformationtemperatureas1008K.TheAl4MophasewasconrmedbySperner[ 232 ],PotzschkeandSchubert[ 234 ],andLeake[ 240 ].IntheworkofLeake[ 240 ],thecrystalstructureofAl4Mowasgiven.PotzschkeandSchubert[ 234 ]foundtheAl4Mophasetobestableonlyattemperaturesabove973K.ThepartialAl{MophasediagramofReimenn[ 239 ]alsoindicatedtheperitecticformationofAl3Moat1403K.Sperner[ 232 ]heattreatedandquenchedalloyswithMocontentsfrom20at.%Moto95at.%Mo.Basedonmicrostructuralinvestigation,x-raydiractionresults,andthermalanalysismeasurements,therstcompleteAl{Mophasediagramwaspresented.InadditiontotheAl12Mo,Al5Mo,andAl4Mophases,theAl{MophasediagramofSperner 181

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232 ]showsthetetragonalAl3Mophase,whichformsperitecticallyat1403K,aswellasahigherMo-containingphasewhichwasgiventhestoichiometryAl2Mo.ItisunclearintheworkofSperner[ 232 ]ifthe1403KperitecticformationtemperatureoftheAl3MophasewasindependentlymeasuredbythermalanalysisoracceptedfromtheearlierworkofReimann[ 239 ].TheAl2Mophasewasshowntomeltcongruentlyat2323KintheAl{MophasediagramofSperner[ 232 ].PotzschkeandSchubert[ 234 ],usingx-raydiraction,renedthestoichiometryoftheAl2MophasetoAl8Mo3andgavethecrystalstructureoftheAl8Mo3phase.Thelatticeparametersforthemonoclinicunitcellwerea=9.164A,b=3.639A,c=10.04A,and=100.50o.ForsythandGran[ 241 ]isolatedtheAl8Mo3phasetodetermineitscrystalstructureusingx-raydiraction.Inthiswork,thelatticeparametersofthemonoclinicunitcellweregivenasa=9.208A,b=3.6378A,c=10.065A,and=100.47o,whicharequitesimilartotheparametersgivenbyPotzschkeandSchubert[ 234 ].However,sinceForsythandGran[ 241 ]incorrectlyassignedahomogeneityrangeoftheAl8Mo3phasefrom25wt.%Moto31wt.%Mo,theyidentiedtheAl3MophaseofSperner[ 232 ]asAl8Mo3.Rexer[ 229 ],however,usingEPMAmeasurementsofAl{Modiusioncoupleswhichwereheattreatedat1873K,1773K,and1673Kandquenched,couldnotconrmthecompositionoftheMo-richboundaryoftheAl8Mo3phasegivenbyForsythandGran[ 241 ].PotzschkeandSchubert[ 234 ]additionallyfoundtwophasesinthecompositionrangefrom80at.%Alto83at.%Al.OnephasewasidentiedasastackingvariantoftheAl5Mophasewhereastheotherwasdescribedasa`linienreiche'phasebecauseofitscomplexdiractionpattern.VanTenderlooetal.[ 242 ],usingelectrondiractionandmicroscopy,clariedthestoichiometryofthesephasesasAl17Mo4andAl22Mo5respectively.TheAl{richliquiduscurvewasmeasuredbyYamaguchiandSimizu[ 236 ],Yeremenkoetal.[ 243 ],andMalinovskii[ 244 ]attemperaturesupto1273K.Theresultsoftheworks 182

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245 ]andVigdorovichetal.[ 246 ],whicharealsoingoodagreementwitheachother.PhaseequilibriaatMocompositionshigherthanthatofAl8Mo3werestudiedbyHam[ 247 ],HamandHerzig[ 248 ],Rexer[ 229 ],andShiloandFranzen[ 249 ].TheMo-richsolvuswasmeasuredattemperaturesbetween1477Kand2423KbyHamandHerzig[ 248 ],at1773K,1873K,and1973KbyRexer[ 229 ],andatvarioustemperaturesbetween1845Kand2021KbyShiloandFranzen[ 249 ].Theresultsareingenerallygoodagreementwitheachother.Thehomogeneityrangeofthe{AlMo3phasewasmeasuredbyRexer[ 229 ]andShiloandFranzen[ 249 ]atthesametemperatureslistedabove.HamandHerzig[ 248 ]estimatedtheperitecticformationofthe{AlMo3phaseat2423K.ThecongruentmeltingoftheAlMophase,itseutectoiddecompositionat1733KtoAl8Mo3and{AlMo3,theeutecticreactionbetweenliquid,AlMoand{AlMo3phasesat1993K,theperitecticformationandeutectoiddecompositionoftheAl63Mo37phaseat1843Kand1763Krespectively,andtheeutecticreactionbetweentheliquid,Al8Mo3,andAl63Mo37phaseswererecommendedbyRexer[ 229 ]basedonthemetallographicinvestigationofthemicrostructuresofascastandheattreatedandquenchedalloysaswellasonthermalanalysismeasurements.TheW-A2crystalstructureoftheAlMophasewasconrmedbyRexer[ 229 ]usingx-raydiraction.Basedontheaboveliterature,criticalevaluationsofthephaseequilibriaintheAl{MosystemwereperformedbyBreweretal.[ 208 ],andSaunders[ 209 ].Walford[ 250 ]presentedanevaluationofthepartialAl{Modiagramfrom0at.%Moto30at.%Mo.Someaspectsoftheevaluationsareinconsistentwitheachotherandwiththeliterature.Walford[ 250 ]modiedthepartialAl{MophasediagramofSperner[ 232 ]bychangingthestoichiometryoftheAl3MophasetoAl8Mo3inaccordancewiththerecommendationofForsythandGran[ 241 ],andaddedtheAl4MophaseofPotzschkeandSchubert[ 234 ].Walford[ 250 ]alsomodiedthecompositionoftheAl7Mophase,whichwasfoundbyClare[ 233 ]inslowcooledmeltsofAl-richalloys,toAl6Mo.However,neithertheAl7Mophasenorthe 183

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250 ]thenattributedtheperitecticformationofAl3Moat1403KfromSperner[ 232 ]totheperitecticformationoftheAl4Mophaseat1403K.AlthoughPotzschkeandSchubert[ 234 ]indicatedthattheAl4Mophaseisstableonlyabove973K,Walford[ 250 ]showstheAl4Mophaseexistingdowntoroomtemperature.TheevaluationofBreweretal.[ 208 ]acceptedtheAl-richregionfromWalford[ 250 ]butaddedtheperitecticformationoftheAl17Mo4andAl22Mo5phasesofPotzschkeandSchubert[ 234 ].TheevaluationofSaunders[ 209 ]showstheAl4Mophaseexistingdowntoroomtemperature,removestheAl6Mophase,andassignsahomogeneityrangetotheAl8Mo3phasefrom25at.%Moto31.5at.%MoinaccordancewiththeworkofForsythandGran[ 241 ].Therefore,althoughPotzschkeandSchubert[ 234 ]showedthatthemonoclinicAl8Mo3isthecorrectstoichiometryfortheAl2MophaseofSperner[ 232 ],theevaluationsofWalford[ 250 ]andBreweretal.[ 208 ],andSaunders[ 209 ]missthisandreplacethetetragonalAl3MophaseofSperner[ 232 ]withtheAl8Mo3phase.TheevaluationofBreweretal.[ 208 ]acceptedtheworkofRexer[ 229 ]ontheMo-richregion.However,Saunders[ 209 ]replacedthecongruentmeltingoftheAlMophaseat1993KwiththeperitecticformationoftheAlMophaseatthesametemperature,statingthatthecongruentmeltingoftheAlMophaseis`thermodynamicallyunlikely'becauseitisicostructuralwiththeMo-richphase.BasedontheinconsistenciesintheAl-richregionoftheAl{Mosystem,SchusterandIpser[ 211 ]completelyreinvestigatedtheareafromAltoAl8Mo3throughmicrostructuralanalysisof15alloysrangingincompositionfrom12at.%Moto28at.%Mowhichwereheattreatedattemperaturesrangingfrom873Kfor240hoursto1493Kfor3hoursandquenched.Thermalanalysiswasalsoperformedtomeasurephasetransformationtemperatures.ThepartialAl{MophasediagramofSchusterandIpser[ 211 ]showstheAl12Mophase,whichmeltsincongruentlyat985K,threemodicationsoftheAl5Mophasewiththehightemperaturemodicationmeltingincongruentlyat1119K,theAl22Mo5phasewhichformsperitecticallyat1273Kandundergoeseutectoid 184

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232 ],whichwasmisidentiedasAl8Mo3byForsythandGran[ 241 ]andacceptedintheevaluationsofWalford[ 250 ]andSaunders[ 209 ],wasconrmedbySchusterandIpser[ 211 ].Itisshowntoformperitecticallyat1495Kandundergoeutectoiddecompositionat1091K.Anewphase,identiedasAl3+xMo1xwith0x0.2,wasfoundwiththesamecrystalstructureas{AlMo3.Thisphasewasshowntoexistoverthetemperaturerangefrom1427Kto1533K.SchusterandIpsermeasuredthecongruentmeltingoftheAl8Mo3phaseas182810K.TheresultsofthisworkwereacceptedinthecriticalevaluationofSchuster[ 210 ]andareshowninFigure 9-3 .ThemostrecentexperimentalworkontheAl8Mo3toAlregionwasperformedbyEumannetal.[ 251 ].Themicrostructuresofninebinaryalloys,whichwereheattreatedatvarioustemperaturesbetween873Kand1473Kandquenched,wereanalyzed.Thermalanalysiswasalsoperformedtomeasurephasetransformationtemperatures.ThephaserelationsestablishedbySchusterandIpser[ 211 ]wereconrmed,exceptforthepresenceoftheAl3+xMo1xphase,whichwasnotdetected.TheAl22Mo5phasewasfoundtoexistdowntoroomtemperatureandtheAl3Mophasewasfoundtoextendtotemperaturesslightlylowerthanthe1091KtemperaturewhichwasoriginallygivenintheworkofSchusterandIpser[ 211 ].ThecongruentmeltingofAl8Mo3wasmeasuredas18193K,whichisingoodagreementtotheworkofSchusterandIpser[ 211 ].AccordingtoEumannetal.[ 251 ],allinvestigatedphasesexhibitedahomogeneityrangebetween0.1and0.4at.%Mo.Eumannetal.[ 251 ]alsogavetielinedataforthe{AlMo3+Al8Mo3equilibriumat1073K,1273K,and1473K.ThepartialAl{Mophasediagramfrom70at.%Alto100at.%AlisshowninFigure 10-1 185

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249 ]andMeschel[ 252 ].ShiloandFanzen[ 249 ]usedvaporpressuremeasurementsfromtheKnudseneusionmethodtocalculatetheenthalpiesofformationoftheAl8Mo3,AlMo,-AlMo3phasesandoftheMo-richsolidsolutionat89,92,96,and98at.%Mo.Meschel[ 252 ]useddirectreactioncalorimetrytomeasuretheenthalpiesofformationofthe{AlMo3andAl8Mo3phases. 18 209 ].BothdescriptionsreproducetheperitecticformationoftheAlMophasefromtheliquidand{AlMo3phasesanddonotmodeltheAl17Mo4,Al22Mo5,andAl3MophasesintheAl{Al8Mo3regioninaccordancewiththecriticalevaluationofSaunders[ 209 ].Additionally,the{AlMo3phaseismodeledasastoichiometricphasewithnohomogeneityrange.TherstdatasetofSaunders[ 209 ]didnotusetheSGTErecommendeddescriptionsforpureMoandAlfromDinsdale[ 58 ].However,theAlMoandMo-richphasesweremodeledasasinglephase.Intheseconddataset[ 18 ],theSGTEdescriptionsforthepureelements[ 58 ]wereused,buttheAlMoandMo-richphasesweremodeledasseparatephases. 10-1 .Theperitecticformationof{AlMo3wasacceptedfromHamandHervig[ 248 ],andthephaseequilibriabetween{AlMo3andAl8Mo3wereacceptedfromRexer[ 229 ].ThecongruentmeltingtemperatureofAlMowastakenfromthecriticalevaluationofSchuster[ 210 ].PhaseequilibriaintheAl{richregionwereacceptedfromtheexperimentalworkofEumann[ 251 ],inwhichtheAl{Al8Mo3regionwasre-investigatedtofurtherclarifytheexperimentalworkofSchusterandIpser[ 211 ].SincethehomogeneityrangesoftheAl{richintermetallicphasesweremeasuredasbetween0.1 186

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18 ]wasusedasthestartingpointforthere-optimization. 248 ],Rexer[ 229 ],andShiloandFranzen[ 249 ].Theparametersoftheliquidwereoptimizedtottheexperimentaldataontheenthalpiesofmixingoftheliquidat5,10,15and18at.%MofromSudavtsovaetal.[ 253 ]andtheAl-richliquidusdeterminedbyYamaguchiandSimizu[ 236 ],Yeremenkoetal.[ 243 ],andMalinovskii[ 244 ]attemperaturesupto1273K.Atthisstepoftheoptimization,onlytheAl8Mo3phaseintheAl-richregionwasconsidered.Next,theparametersofthe{AlMo3phasewereoptimizedtottheAl-richboundaryofthe{AlMo3phaseinequilibriumwiththeAl8Mo3phaseat1073K,1273K,and1473KaccordingtotheexperimentalresultsofEumannetal.[ 251 ].Inthefollowingsteps,theAl-richintermetalliclinecompoundsweresuccessivelyintroduced,startingwithAl3MoandendingwithAl12Mo,toreproducetheAl{Al8Mo3regionofthephasediagrambyEumannetal.[ 251 ].Next,theparametersofthe(Al)solidsolutionwerere-optimizedtotthedataonthe(Al)solvusfromRontgenandKoch[ 245 ]andVigdorovichetal.[ 246 ].Inthenalstep,allparametersofallphaseswerere-optimizedsimultaneouslytoreproducetheavailableexperimentaldata. 10.3.2.1ThebetaphaseThephasewithA2-Wstructureismodeledasasubstitutionalsolution(Al,Mo).Inthiswork,the0LAl,Mo,1LAl,Mo,and2LAl,Moparameterswereoptimized.The0LAl,Moparameterwasmodeledasalinearfunctionoftemperaturewhereasthe1LAl,Moand 187

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251 ],thesephaseweremodeledasstoichiometriccompounds.TheGibbsenergyGAlxMoyoftheintermetallicphaseAlxMoy,wasmodeledasalinearfunctionoftemperatureas: 249 ].ThevalueofbwasassignedtotthecongruentmeltingofAl8Mo3at1819KmeasuredbyEumannetal.[ 251 ]. 188

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251 ],thevaluesoftheaandbparameterswerechosentoreproducetheperitecticformationofAl3MofromAl8Mo3andliquidat1495KandtheeutectoiddecompositionofAl3MotoAl8Mo3andthehigherAl-containingliquidat1063K.TheAl4Mophasealsoexistsoveratemperaturerangefrom1423Kand1215K.TheparametersaandbwerechosentoreproducetheperitecticformationofAl4MofromAl3Moandliquidat1425KandtheeutectoiddecompositionofAl4MotoAl3MoandtheliquidphasewithhigherAlcontent.TheAl17Mo4,Al22Mo5,Al5Mo,andAl12Mophasesexistdowntoroomtemperature.Twostepswererequiredtogeneratestartingvaluesfortheaandbparameters.First,G(TP),whichisthevalueoftheGibbsfreeenergyofthephaserequiredtoreproduceitsperitecticformationattherespectivetemperatureTPgiveninTable 10-1 ,wasdetermined.SincetheGibbsfreeenergyismodeledasalinearfunctionoftemperature,theequations wereusedtogeneratethestartingvaluesofaandb.FromthevalueG(TP)andavalueofbchosenbythetrialanderrormethodsothatthephaseisstabledowntoroomtemperature,theGibbsenergyofthephaseasafunctionoftemperature,G(T),couldbedened. (Al,Mo*)3(Al*,Mo)(10{3)wheretheasteriskidentiesthemajorspeciesoneachsublattice.ThismodelingresultsinthefourendmembersGAl:Al,GAl:Mo,GMo:Al,andGMo:Mo.TheGMo:Alendmember,whichinuencedtheperitecticformationtemperatureofthe{AlMo3phasefromtheliquidandMo-richphases,theeutecticreactionbetweenthe{AlMo3,liquid,andphaseat50 189

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251 ],aswellasthecompositionofthe{AlMo3phaseattheinvariantreactionsfromtheevaluationofSchuster[ 210 ]. 245 ]andVigdorovichetal.[ 246 ].Thisparameterwasmodeledwithalineartemperaturedependence. 10-2 .Theexperimentaldatafromtheliteratureissuperimposed.Thenewdescriptionisinagreementwiththecongruentmeltingofthephaseat50at.%Moandtheeutecticreactionbetweentheliquid,{AlMo3andphasesofRexer[ 229 ].ThecongruentmeltingofAl8Mo3iscalculatedat1821K,whichisingoodagreementwiththethermalanalysismeasurementsofSchusterandIpser[ 211 ](182810K)andEumannetal.[ 251 ](18195K).Thepartialphasediagramfrom70at.%Alto100at.%AlisshowninFigure 10-3 .AllintermetallicphasesintheAl{Al8Mo3regionarereproduced,andthetemperaturerangesofstabilityareinverygoodagreementwiththeworkofEumannetal.[ 251 ].Figure 10-4 showsthecalculatedenthalpiesofformationofthephaseat2,4,8,11,and50at.%Al,the{AlMo3phaseat25at.%Mo,andoftheAl8Mo3phasefromthepureelements.TheexperimentaldataofShiloandFranzen[ 249 ]andMeschelandKleppa[ 252 ]aresuperimposed.WhiletheenthalpiesofformationoftheMo-rich

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249 ]givesavalueof-31.3kJ/molusingthethirdlawmethodand-27.8kJ/molusingthesecondlawmethod.However,thedescriptionofSaunders[ 209 ]givesavalueof-16.21kJ/mol,whichisalsomuchlowerthanthatreportedbyShiloandFranzen[ 249 ].TheresultsofShiloandFranzen[ 249 ],however,havenotbeenindependentlyconrmedbyotherauthorsintheliterature. 18 ]butthere-optimizeddescriptionoftheAl{Mosystemfromthepresentworkwasadopted.Onlytheternaryparametersoftheand,and{AlMo3phaseswerere-optimizedtoreproducetheequilibriainthe0to20at.%TiregionoftheTi{A{MosystemfromNinoetal.[ 27 ]. 27 ].Then,theparametersforthe,andphaseswereoptimizedsimultaneouslytoreproducetheAl8Mo3++threephasetriangleat1673Kandthecompositionofthephaseinequilibriumwiththeandphasesat1423K.Next,theparametersofthephasewerere-optimizedtoimprovethetofthecompositionofthephaseintheAl8Mo3+{AlMo3+threephasetriangleat1673K.Last,theparametersoftheandphaseswerere-optimizedsimultaneouslytocalculatethesolidstatetransformationtemperaturesofalloyswith5,7,10,13,15,and17at.%Tiand50at.%Al,whichweremeasuredusingthermalanalysisbyNinoetal.[ 27 ]. 191

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10.4.2.1TheetaphaseThephase,whichhasthetI8{Al3Tistructure,wasoriginallymodeledbySaunders[ 18 ]as: (Al*,Mo,Ti)0:75(Mo*,Ti)0:25(10{4)wheretheasteriskidentiesthemajorspeciesoneachsublattice.However,sincethephasealsoexistsintheTi{Al{Nbsystem,whereitismodeledas (Al,Nb,Ti)0:75(Al,Nb,Ti)0:25(10{5)forextrapolationtothequaternaryTi{Al{Nb{Mosystem,thephaseshouldbesimilarlymodeledas: (Al*,Mo,Ti)0:75(Al,Mo*,Ti)0:25(10{6)whereAlatomsareallowedtomixonboththerstandsecondsublattices.ThismodelingresultsinseveralendmemberswhichareindicatedontheGibbstriangleinFigure 10-5 .ThedescriptionsfortheGAl:Ti,GMo:Mo,GMo:Ti,GTi:Mo,andGTi:TiendmemberswereacceptedfromtheTi{Al{ModatasetofSaunders[ 18 ]andtheGAl:Moendmemberwastakenfromthere-optimizedparametersoftheAl{Mosysteminthepresentwork.BecauseSaundersdidnotmodelAlmixingonthesecondsublattice,theGAl:Al,GMo:Al,andGTi:AlendmemberswereoriginallynotdenedintheoriginalTi{Al{ModatasetofSaunders[ 18 ].However,inSaunders'datasetfortheTi{Alsystem[ 18 ],descriptionsfortheGAl:AlandGTi:Alendmembersofthephaseareavailable.Therefore,theseparameterswereacceptedinthecurrentwork.FortheGMo:Alparameter,apositivetermof+20000J/molwasaddded,givingthedescription: 18 ],onlythe0LAl,Mo:Mo,0LAl:Mo,Ti,and0LAl,Ti:Tiparameterswereused.However,theuseofonly 192

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212 ]andEremenkoetal.[ 213 ].Therefore,moremixingparameterswereusedinthepresentworkandtheassumptions: 18 ].The0LAl:Mo,Ti=0L*:Mo,Tiparameterwasre-optimizedtoreproducethe+Al8Mo3!+invariantequilibriumat1540KofNinoetal.[ 27 ]. 18 ]. (Al,Mo*,Ti)0:75(Al*,Mo,Ti)0:25(10{11)TheendmembersforthisdescriptionarealsoschematicallyshownontheGibbstriangleinFigure 10-5 .TheGTi:AlendmemberwasacceptedfromtheoriginaldescriptionofSaunders[ 18 ]andtheGMo:Alendmemberwastakenfromthere-optimizedparametersoftheAl{Mosysteminthepresentwork.Forallotherendmemberparameters,apositive 193

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27 ]. 10-6 .AllalloysofNinoetal.[ 27 ]whichhavebeenshowntosolidifyassinglephasearecalculatedintheprimarycrystallizationeldofthephase.Fromthepositionofthecalculatedsolidusline,itisclearthatthesealloysarecalculatedtosolidifyassinglephase.Additionally,becausetheMo-richphaseandthephaseat50at.%MoaremodeledasasinglephaseintheAl{Mobinarydescription,thephaseextendsfromtheTi{MobinarytotheAl{Mobinary.Figure 10-7 showsthecalculatedisothermalsectionat1773K.ThealloyofcompositionTi{52at.%Al{45at.%Moiscalculatedinthesinglephaseeld,whichisinexcellentagreementwiththeexperimentalworkofNinoetal.[ 27 ].Thecalculatedisothermalsectionat1673KisshowninFigure 10-8 .Theresultsindicateverygoodagreementwiththeexperimentallydetermined+Al8Mo3+threephaseeldat1673KofNinoetal.[ 27 ].Last,theisothermalsectionat1540Kshowsthatitispossibletocalculatethesolidstateinvariantreaction+Al8Mo3!+at1540K.Thecalculatedisopleththough50at.%Alfrom0to20at.%TiisshowninFigure 10-10 .ThesolidstatetransformationtemperaturesmeasuredusingthermalanalysisfromtheworkofNinoetal.[ 27 ]areincludedaslledtriangles.Alloycompositionsindicatedaslledsquareswereshowntocontainsinglephaseintheirmicrostructuresaftertheywereheattreatedattherespectivetemperaturesand 194

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27 ]. 195

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PartialAl{Mophasediagramfrom70at.%Alto100at.%AlconstructedbyEumannetal.[ 251 ]. Table10-1. InvariantreactionsintheAl{Mosystemwhichwereacceptedfortheoptimizationbasedonacriticalevaluationoftheavailableliterature. Reaction1+2+3Temperature[K]Composition[at.%Al]123 196

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Al{Mophasediagramcalculatedusingthenewdescription.Theexperimentaldatafromtheliteraturearesuperimposed. Figure10-3. Al{Mopartialphasediagramcalculatedusingthenewdescription.Theexperimentaldatafromtheliteraturearesuperimposed. 197

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CalculatedandexperimentallydeterminedenthalpiesofformationofthephasesintheAl{Mosystem.[1982Shi]referstotheworkofShiloandFranzen[ 249 ]and[1993Mes]referstotheworkofMeschelandKleppa[ 252 ]. 198

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Endmembersforthephasewhichisdescribedusingthecompoundenergyformalismas(Al*,Mo,Ti)0:75(Al,Mo*,Ti)0:25wheretheasteriskidentiesthemajorspeciesoneachsublattice.Theendmembersforthephasewithmodel(Al,Mo*,Ti)0:75(Al*,Mo,Ti)0:25arethesame.Theinteractionparameter0LMo,Ti:Al,shownasthedashedline,inuencedtheextensionofthephaseintotheternary. 199

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LiquidussurfacecalculatedwiththenewdescriptionfortheTi{Al{Mosystem.AllalloysindicatedwereshowntosolidifyassinglephaseintheworkofNinoetal.[ 27 ],withwhichthenewliquidussurfaceisinverygoodagreement. 200

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Isothermalsectionat1773KcalculatedusingthenewdescriptionfortheTi{Al{Mosystem.[2003Nin]referstotheworkofNinoetal.[ 27 ]. Figure10-8. Isothermalsectionat1673KcalculatedusingthenewdescriptionfortheTi{Al{Mosystem.[2003Nin]referstotheworkofNinoetal.[ 27 ]. 201

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Isothermalsectionat1540KcalculatedusingthenewdescriptionfortheTi{Al{Mosystem.[2003Nin]referstotheworkofNinoetal.[ 27 ]. 202

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Isoplethcalculatedthrough50at.%Al.ThesolidstatetransformationtemperaturesmeasuredusingthermalanalysisfromNinoetal.[ 27 ]areindicated.Alloysidentiedbylledsquaresweresinglephaseattherespectivetemperatures. 203

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10 ]toreproducenewexperimentalresults,and,forthersttime,togeneratethermodynamicdescriptionsforthetheTi{Al{CrandTi{Al{Mosystemstakingintoaccountavailableexperimentalphasediagramandthermodynamicdata.Throughthecalculationofvariouskindsofphasediagramsandphasefractiondiagrams,thenewthermodynamicdatasetsareexpectedtocontributetothedesignanddevelopmentofturbinebladematerialsbasedonamicrostructureofdisconnected{Nb2Alprecipitatesina{TiAlmatrixbypredictingtheequilibriumphasesundervariousconditions.AlthoughexistingdescriptionsfortheTi{Al{NbsystemfromServantandAnsara[ 10 ]andWitusiewiczetal.[ 17 ]areavailable,thesedescriptionscouldnotreproducethesolidstatetransformationtemperaturesandphaseequilibriaforalloysofcompositionTi{44.65at.%Al{18.1at.%Nb(alloy11)andTi{44.9at.%Al{28.5at.%Nb(alloy12)determinedattheDepartmentofMaterialsScienceandEngineeringattheUniversityofFlorida,andtheexperimentalworkofLeonardetal.[ 14 15 ]showingtheextensionoftheprimarysolidicationeldofthephaseontheliquidussurfaceto40at.%Alat20at.%Ti.SincethesearecriticalfeaturesofphaseequilibriaintheTi{Al{Nbsystemrelevanttoalloydesign,onegoalofthisworkwastodevelopanewthermodynamicdescriptionforthesystemabletoreproducetheexperimentaldata.ThethermodynamicdatasetofServantandAnsara[ 10 ]wasusedasthestartingpointforthere-optimization.TheTi{NbdescriptionfromZhangetal.[ 129 ]andtheAl{NbdescriptionfromWitusiewiczetal.[ 17 ]wereselectedastheconstituentbinaries,whereastheTi{AldescriptionfromSaunders[ 18 ],whichwasusedintheoriginaldescriptionofServantandAnsara[ 10 ]waskept.TheTi{NbdescriptionofZhangetal.[ 129 ]isbetter 204

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130 ],andMoatandLarbalestier[ 131 132 ]thanthedescriptionofHariKumar[ 105 ],whichwasoriginallyusedbyServantandAnsara[ 10 ],andtheAl{NbdescriptionofWitusiewiczetal.[ 17 ]isinbetteragreementwiththeenthalpyofformationdataforthe,,andphasesintheAl{NbsystemthantheoriginallyuseddescriptionofServantandAnsara[ 104 ].Next,theparametersforthe,,,,0,,0,andliquidphaseswerere-optimizedtottheexperimentalphasediagramdataavailable.Thenewdatasetnowreproducestheextensionoftheprimarysolidicationofthephaseontheliquidussurfaceto40at.%Alat20at.%TifromLeonardetal.[ 14 15 ],thesolidstatetransformationsandtransformationtemperaturesofalloy11andalloy12whichwereinvestigatedusingDTA,XRD,SEM,andTEM,andotherphaseequilibriadatawhichisreportedintheliterature.FortheTi{Al{Crsystem,calculationsperformedusingthedatasetofSaunders[ 18 ],whichwasgeneratedbycombinationandextrapolationoftheTi{Al,Ti{Cr,andAl{Crconstituentbinariesintotheternarysystem,couldnotreproducetheextensionofthe{Ti(Al,Cr)2Lavesphasealong33.3at.%Ti[ 176 { 178 ],phaseequilibriawiththephaseofstoichiometryTi25(Al67Cr8)[ 22 179 ],theisolated0eldatapproximately18at.%Crand30at.%Al[ 176 177 ],andtheliquidusprojectionofBochvaretal.[ 19 ].Additionally,although,thebinaryTi{CrdescriptionofSaunders[ 18 ]reproducedthelowtemperature{TiCr2andhightemperature{TiCr2modicationsoftheLavesphase,whichisinaccordancewiththeworksofHaoandZeng[ 150 ]andSchusterandDu[ 151 ],thiswasfoundtobeinconsistentwithotherliteratureindicatingthreemodicationsoftheLavesphase[ 147 { 149 ].Tore-assessthethermodynamicparametersoftheTi{Al{Crsystem,rst,theintermediatetemperature{TiCr2LavesphasewasincludedinthebinaryTi{CrdescriptionofSaunders[ 18 ],andthethermodynamicparametersofallthreeLavesphaseswereoptimizedtoreproducethehomogeneityrangeoftheLavesphases,thecongruent 205

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18 ],whichwasalsogeneratedbycombinationandextrapolationoftheTi{Al,Ti{Mo,andAl{Mobinariesintotheternary,couldnotreproducethephaseequilibriaintheregionfrom0to20at.%TifromtheexperimentalworkofNinoetal.[ 27 ]whichshowedthecontinuityofthephasetotheAl{Mobinaryat1773Kandtheinvariantreaction+Al8Mo3!+at1540K.Itwasdemonstratedinthepresentstudythatthesephasediagramfeaturescouldnotbereproducedwithoutacompletere-assessmentofthebinaryAl{ModescriptionofSaunders[ 18 ].AcriticalevaluationoftheAl{MoliteratureshowedthattheMo-richphaseandtheAlMophaseshouldbedescribedasasinglephase,thatthephaseat50at.%Moshouldmeltcongruentlyatapproximately2023K,andthataneutecticreactionshouldexistbetweentheliquid,AlMo,and{AlMo3phasesatapproximately1993K[ 229 ].ThesephaseequilibriawerenotreproducedintheoriginalAl{ModatasetofSaunders[ 18 ],butwerefoundtobecriticalforconsistencybetweenthecalculatedphasediagramsandtheexperimentalworkofNinoetal.[ 27 ].Therefore,thethermodynamicdescriptionfortheAl{MosystemfromSaunders[ 18 ]wasre-optimizedtoreproducetheabovephaseequilibria.ThederiveddescriptionfortheAl{MosystemwasincludedastheconstituentbinarywhiletheTi{AlandTi{MobinarydescriptionswereacceptedfromSaunders[ 18 ].Theparametersforthe,,andphaseswerere-optimizedtogenerateanewthermodynamicdescriptionfortheTi{Al{Mosystem.Thecalculatedphasediagrams 206

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27 ].Toconclude,thermodynamicdescriptionsfortheTi{Al{Nb,Ti{Al{Cr,andTi{Al{Mosystemsweredevelopedinthiswork.ThesearethekeysystemsrequiredtogeneratedescriptionsforthequaternaryTi{Al{Nb{CrandTi{Al{Nb{Mosystems,and,ultimatelyforthequinaryTi{Al{Nb{Cr{Mosystem.AlthoughcalculationsusingthenewdescriptionsforTi{Al{Nb,Ti{Al{Cr,andTi{Al{Mosystemsareingoodagreementwithexperimentaldata,moreworkonotherternarysystemsisrequiredtogeneratetherespectivequaternarydescriptions. 254 ].Ofthefourternarysystems,theTi{Al{NbandTi{Al{Crsystemsarethemostimportantandwerealreadyaddressedinthepresentwork.However,thermodynamicassessmentsoftheAl{Nb{CrandTi{Nb{Crsystemshavenotyetbeenperformed,althoughcriticalevaluationsofthephaseequilibriainthesesystemsareavailablefromIvanchenko[ 255 ]andGhosh[ 256 ]respectively.TheCOST507datasetofSaunders[ 18 ]fortheAl{Nb{Crsystem,constructedbycombiningandextrapolatingtheAl{Nb,Al{Cr,andNb{Crbinarysubsystems,doesnotreproducetheextensionoftheCr2NbLavesphaseintotheternary[ 255 ]becausetheLavesphaseintheNb{Crsystemisnotmodeled.Additionally,althoughthe,,andintermetallicphasesintheNb{AlbinaryhavebeenshowntohaveasolubilityofCrbetween5and15at.%,onlythesolubilityofCrinismodeled.ThethermodynamicdescriptionfortheTi{Nb{CrsystemisalsogeneratedbycombinationandextrapolationoftheTi{Nb,Nb{Cr,andTi{Crbinariestotheternary.However,sincetheCr2NbLaves 207

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18 ].Ofthefourquaternarysystems,descriptionsfortheTi{Al{NbandTi{Al{Mosystemsweregeneratedinthepresentwork.AlthoughnocriticalphasediagramevaluationhasbeenperformedfortheTi{Mo{Nbsystem,thissystemisexpectedtodisplaycontinuoussolidsolubilityofthephaseandaTi{richsolidsolution.ThephaseequilibriaintheAl{Nb{Mosystem,however,wascriticallyevaluatedbyGuzei[ 257 ].ThereiscontinuoussolubilityofthephasesfromNb3AltoMo3Albuttheexistenceofternaryphasesisnotwellestablished.SincetheAl{Nb{ModescriptionofSaunders[ 18 ]doesnotmodelthecontinuoussolubilityofthephase,anewdatasetwouldhavetobecreatedusingtheCALPHADmethod.Onceallconstituentternarydescriptionsareoptimized,theycanbecombinedandextrapolatedtogeneratethequaternaryTi{Al{Nb{CrandTi{Al{Nb{MoandquinaryTi{Al{Nb{Cr{Mosystemdescriptions.Thesedatasetswillbeusefultoolsformaterialsdesignthroughcalculationofphasediagramsunderdierentconditions.TheymayalsobeusedtopredictmicrostructuraldevelopmentinalloysinthephaseeldmethodandtosimulatediusioninmaterialswhenusedwithsoftwareprogramssuchasDICTRA[ 29 ]. 208

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TableA-1.ThermodynamicDescriptionfortheTi{Al{NbSystem ParameterFunctionReference 10 ]GNb:Al0:75GHSERNb+0:25GHSERAl19136:3[ 17 ]GTi:Al0:75GBCCTi+0:25GHSERAl29700+9T[ 10 ]GAl:Nb0:75GHSERAl+0:25GHSERNb+5000[ 17 ]GNb:NbGHSERNb+4894:9[ 17 ]GTi:Nb0:75GBCCTi+0:25GHSERNb+10000[ 10 ]GAl:Ti0.75GHSERAl+0.25GBCCTi+297009T[ 10 ]GNb:Ti0:75GHSERNb+0:25GBCCTi[ 10 ]GTi:TiGBCCTi+5000[ 10 ]0LAl:Al,Ti4000[ 10 ]0LNb,Ti:Al13405:7098+10:6548825Tthiswork0LNb:Al,Nb13780:3+5:736T[ 17 ]0LTi:Al,Ti4000[ 10 ]:(Al,Nb,Ti)0:533(Al,Nb,Ti)0:333(Nb)0:134GAl:Al:Nb0:866GHSERAl+0:134GHSERNb[ 17 ]GNb:Al:Nb0:667GHSERNb+0:333GHSERAl27837:5+2:679T[ 17 ]GTi:Al:Nb0:533GHSERTi+0:333GHSERAl+0:134GHSERNb31453:2133+5:72T[ 10 ]GAl:Nb:Nb0:533GHSERAl+0:467GHSERNb+30837:52:679T[ 17 ]GNb:Nb:NbGHSERNb+3000[ 17 ]GTi:Nb:Nb0:533GHSERTi+0:467GHSERAl+7500[ 17 ]GAl:Ti:Nb0:533GHSERAl+0:333GHSERTi+0:134GHSERNb[ 17 ]GNb:Ti:Nb0:333GHSERTi+0:667GHSERNb+7500[ 17 ]GTi:Ti:Nb0:866GHSERTi+0:134GHSERNb+5000[ 10 ]0LAl,Nb:Al:Nb10562:2[ 17 ]0LAl,Nb:Nb:Nb10562:2[ 17 ]0LNb,Ti:Al:Nb38045:2259+24:3166036Tthiswork1LAl,Nb,Ti:*:Nb87199:5960113:299214Tthiswork:(Al,Nb,Ti)0:5(Al,Nb,Ti)0:5GAl:AlGHSERAl[ 10 ]GNb:Al0:5GHSERAl+0:5GHSERNb24133:746+2:31015467T[ 10 ]GTi:Al0:5GHSERTi+0:5GHSERAl39822+9:6T[ 18 ]GAl:Nb0:5GHSERAl+0:5GHSERNb24133:746+2:31015467T[ 10 ]GNb:NbGFCCNb[ 17 ]GTi:Nb0:5GHSERNb+0:5GHSERTi+4000thiswork 209

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ParameterFunctionReference 18 ]GNb:Ti0:5GHSERTi+0:5GHSERNb+4000thisworkGTi:TiGFCCTi[ 10 ]0LAl,Ti:Al44946+22T[ 18 ]1LAl,Ti:Al15000[ 18 ]2LAl,Ti:Al10000[ 18 ]0LAl:Al,Ti44946+22T[ 18 ]1LAl:Al,Ti15000[ 18 ]2LAl:Al,Ti10000[ 18 ]0LNb,Ti:Al90277:7229+101:070062T0:0309214441T2thiswork0LNb:Al,Ti105748:8547:1429Tthiswork0LTi:Al,Ti75671:18T[ 18 ]0LAl,Ti:Nb105748:8547:1429Tthiswork0LAl:Nb,Ti90277:7229+101:070062T0:0309214441T2thiswork0LAl,Ti:Ti75671:18T[ 18 ]0LAl,Nb:Al121471:6696Tthiswork0LAl,Nb:Nb121471:6696Tthiswork0LNb:Al,Nb121471:6696Tthiswork0LAl:Al,Nb121471:6696Tthiswork:(Al,Nb,Ti)0:25(Al,Nb,Ti)0:75:(Al,Nb,Ti)GAlGBCCAl[ 10 ]GNbGHSERNb[ 10 ]GTiGBCCTi[ 10 ]0LAl,Ti128500+39T[ 18 ]1LAl,Ti6000[ 18 ]2LAl,Ti21200[ 18 ]0LAl,Nb95384:4+20:186T[ 17 ]1LAl,Nb5995[ 17 ]0LNb,Ti13045:3[ 129 ]0LAl,Nb,Ti10844901713:2379T+0:86878T20:000147731T3thiswork1LAl,Nb,Ti1999000+3067:4802T1:56748T2+0:00026654T3thiswork2LAl,Nb,Ti480477:56721:17248T+0:256T2thiswork0:(Al,Nb,Ti)0:5(Al,Nb,Ti)0:5G0Nb,Al13273+2:22TthisworkG0Al,Nb13273+2:22TthisworkG0Ti,Al5521:47:1429Tthiswork 210

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ParameterFunctionReference 10 ]GNbGHCPNb[ 10 ]GTiGHSERTi[ 10 ]0LAl,Ti133300+39T[ 18 ]1LAl,Ti750[ 18 ]2LAl,Ti17500[ 18 ]0LAl,Nb10300thiswork0LNb,Ti11742:4[ 129 ]0LAl,Nb,Ti383571:285+0:273620318Tthiswork1LAl,Nb,Ti109614:756+0:336231919Tthiswork2LAl,Nb,Ti30860:67810:00331393828Tthiswork2:(Al,Nb,Ti)0:75(Al,Nb,Ti)0:25G2Nb:Al61257:0843+48:7576938T[ 10 ]G2Al:Nb61257:0843+48:7576938T[ 10 ]G2Nb:Ti0[ 10 ]G2Ti:Nb0[ 10 ]G2Al:Ti10461[ 10 ]G2Ti:Al17841[ 10 ]0L2*:Al,Ti0[ 10 ]1L2*:Al,Ti25001:6T[ 10 ]0L2*:Al,Nb0[ 10 ]1L2*:Al,Nb3800:06823[ 10 ]0L2*:Nb,Ti0[ 10 ]1L2*:Nb,Ti0[ 10 ]GAl2Nb214048:1738+65:0102584T[ 10 ]GAl2Ti218868[ 10 ]GNb2Ti20[ 10 ]GAlNb2Ti45555:0562+32:5051292T[ 10 ]GAl2NbTi45555:0562+32:5051292T[ 10 ] 211

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ParameterFunctionReference 10 ]1L2Al,Nb:Al0:5G2Nb:Al+1:5G2Al:Nb1:5GAl2Nb2+31L2Al:Al,Nb[ 10 ]0L2Al,Nb,Ti:Al1:5GAlNbTi21:5GAlNb2Ti+G2Nb:Al+G2Ti:Al+6GAl2NbTi1:5GAl2Nb21:5GAl2Ti21:5G2Al:Nb1:5G2Al:Ti[ 10 ]1L2Al,Nb,Ti:Al1:5GAlNbTi21:5GAlNb2Ti+G2Nb:Al+G2Ti:Al+6GAl2NbTi1:5GAl2Nb21:5GAl2Ti21:5G2Al:Nb1:5G2Al:Ti[ 10 ]2L2Al,Nb,Ti:Al1:5GAlNbTi21:5GAlNb2Ti+G2Nb:Al+G2Ti:Al+6GAl2NbTi1:5GAl2Nb21:5GAl2Ti21:5G2Al:Nb1:5G2Al:Ti[ 10 ]0L2Al,Ti:Al1:5G2Ti:Al+1:5GAl2Ti2+1:5G2Al:Ti+30L2Al:Al,Ti[ 10 ]1L2Al,Ti:Al0:5G2Ti:Al1:5GAl2Ti2+1:5G2Al:Ti+31L2Al:Al,Ti[ 10 ]0L2Nb,Ti:Al1:5GAlNbTi2+1:5GAlNb2Ti1:5G2Nb:Al1:5G2Ti:Al+30L2Al:Nb,Ti[ 10 ]1L2Nb,Ti:Al1:5GAlNbTi2+1:5GAlNb2Ti0:5G2Nb:Al+0:5G2Ti:Al+31L2Al:Nb,Ti[ 10 ]0L2Al,Nb:Nb1:5G2Nb:Al+1:5GAl2Nb21:5G2Al:Nb+30L2Nb:Al,Nb[ 10 ]1L2Al,Nb:Nb1:5G2Nb:Al+1:5GAl2Nb20:5G2Al:Nb+31L2Nb:Al,Nb[ 10 ]0L2Al,Nb,Ti:Al1:5GAlNbTi2+6GAlNb2Ti1:5G2Nb:Al1:5GAl2NbTi1:5GAl2Nb2+G2Al:Nb+G2Ti:Nb1:5GNb2Ti21:5G2Nb:Ti[ 10 ]1L2Al,Nb,Ti:Al1:5GAlNbTi2+6GAlNb2Ti1:5G2Nb:Al1:5GAl2NbTi1:5GAl2Nb2+G2Al:Nb+G2Ti:Nb1:5GNb2Ti21:5G2Nb:Ti[ 10 ]2L2Al,Nb,Ti:Al1:5GAlNbTi2+6GAlNb2Ti1:5G2Nb:Al1:5GAl2NbTi1:5GAl2Nb2+G2Al:Nb+G2Ti:Nb1:5GNb2Ti21:5G2Nb:Ti[ 10 ]0L2Al,Ti:Nb1:5GAlNbTi2+1:5GAl2NbTi1:5G2Al:Nb1:5G2Ti:Nb+30L2Nb:Al,Ti[ 10 ]1L2Al,Ti:Nb1:5GAlNbTi2+1:5GAl2NbTi0:5G2Al:Nb+0:5G2Ti:Nb+31L2Nb:Al,Ti[ 10 ]0L2Nb,Ti:Nb1:5G2Ti:Nb+1:5GNb2Ti2+1:5G2Nb:Ti+30L2Nb:Nb,Ti[ 10 ]1L2Nb,Ti:Nb0:5G2Ti:Nb1:5GNb2Ti2+1:5G2Nb:Ti+30L2Nb:Nb,Ti[ 10 ]0L2Al,Ti:Ti1:5G2Ti:Al+1:5GAl2Ti21:5G2Al:Ti+30L2Ti:Al,Ti[ 10 ]1L2Al,Ti:Ti1:5G2Ti:Al+1:5GAl2Ti20:5G2Al:Ti+31L2Ti:Al,Ti[ 10 ]0L2Al,Nb:Ti1:5GAlNb2Ti+1:5GAl2NbTi1:5G2Al:Ti1:5G2Nb:Ti+30L2Ti:Al,Nb[ 10 ]1L2Al,Nb:Ti1:5GAlNb2Ti+1:5GAl2NbTi0:5G2Al:Ti+0:5G2Nb:Ti+31L2Ti:Al,Nb[ 10 ]0L2Al,Nb,Ti:Ti6GAlNbTi21:5GAlNb2Ti1:5G2Ti:Al1:5GAl2NbTi1:5GAl2Ti2+G2Al:Ti1:5G2Ti:Nb1:5GNb2Ti2+G2Nb:Ti[ 10 ]1L2Al,Nb,Ti:Ti6GAlNbTi21:5GAlNb2Ti1:5G2Ti:Al1:5GAl2NbTi1:5GAl2Ti2+G2Al:Ti1:5G2Ti:Nb1:5GNb2Ti2+G2Nb:Ti[ 10 ]2L2Al,Nb,Ti:Ti6GAlNbTi21:5GAlNb2Ti1:5G2Ti:Al1:5GAl2NbTi1:5GAl2Ti2+G2Al:Ti1:5G2Ti:Nb1:5GNb2Ti2+G2Nb:Ti[ 10 ]0L2Nb,Ti:Ti1:5G2Ti:Nb+1:5GNb2Ti21:5G2Nb:Ti+30L2Ti:Nb,Ti[ 10 ] 212

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ParameterFunctionReference 10 ](Al):(Al,Nb,Ti)[ 10 ]Liq:(Al,Nb,Ti)0:5GLiqAlGLIQAl[ 10 ]GLiqNbGLIQNb[ 10 ]GLiqTiGLIQTi[ 10 ]0LLiqAl,Ti108250+38T[ 18 ]1LLiqAl,Ti6000+5T[ 18 ]2LLiqAl,Ti15000[ 18 ]0LLiqAl,Nb106943+28:263T[ 17 ]1LLiqAl,Nb8552[ 17 ]0LLiqNb,Ti7406:1[ 129 ]0LLiqAl,Nb,Ti49466:3231:237688777Tthiswork1LLiqAl,Nb,Ti508740:877242:168466Tthiswork2LLiqAl,Nb,Ti28182:97333:30348699TthisworkAl11Ti5:(Al)17(Ti)8[ 18 ]Al2Ti:(Al)2(Ti)[ 18 ] 213

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TableB-1.ThermodynamicDescriptionfortheTi{Al{CrSystem ParameterFunctionReference 18 ]GCrGHSERCr[ 18 ]GTiGBCCTi[ 18 ]0LAl,Cr54900+10T[ 18 ]0LAl,Ti128500+39T[ 18 ]1LAl,Ti6000[ 18 ]2LAl,Ti21200[ 18 ]0LCr,Ti19100[ 18 ]1LCr,Ti5500[ 18 ]2LCr,Ti1750[ 18 ]0LAl,Cr,Ti512:38194thiswork1LAl,Cr,Ti25860:671thiswork2LAl,Cr,Ti10974:107thiswork0:(Al,Cr,Ti)0:5(Al,Cr,Ti)0:5G0Cr,Al10808:85[ 18 ]G0Al,Cr10808:85[ 18 ]G0Ti,Al175002:5T[ 18 ]G0Al,Ti175002:5T[ 18 ]G0Ti,Cr0[ 18 ]G0Cr,Ti0[ 18 ]0L0Cr,Ti:Al0[ 18 ]0L0Al:Cr,Ti0[ 18 ]0L0Al,Ti:Cr0[ 18 ]0L0Cr:Al,Ti0[ 18 ]0L0Al,Cr:Ti67292:0+45:0Tthiswork0L0Ti:Al,Cr67292:0+45:0Tthiswork{TiCr2:(Al,Cr,Ti)2(Al,Cr,Ti)G{TiCr2Al:Al3GHSERAl+15000thisworkG{TiCr2Cr:Cr3GHSERCr+15000thisworkG{TiCr2Ti:Ti3GHSERTi+15000thisworkG{TiCr2Cr:Ti2GHSERCr+GHSERTi2803:853575:13838702TthisworkG{TiCr2Ti:Cr2GHSERTi+GHSERCr+32803:85357+5:13838702Tthiswork 214

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ParameterFunctionReference 215

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ParameterFunctionReference 216

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TableC-1.ThermodynamicDescriptionfortheTi{Al{MoSystem ParameterFunctionReference 18 ]GMoGHSERMo[ 18 ]GTiGBCCTi[ 18 ]0LAl,Mo73113:2883+22:7704371Tthiswork1LAl,Mo16584:5446thiswork2LAl,Mo18877:1914thiswork0LAl,Ti128500+39T[ 18 ]1LAl,Ti6000[ 18 ]2LAl,Ti21200[ 18 ]0LMo,Ti2000[ 18 ]1LMo,Ti2000[ 18 ]0LAl,Mo,Ti34665:339662:7530956Tthiswork1LAl,Mo,Ti436018:268+199:763999Tthiswork2LAl,Mo,Ti701116:243457:111853Tthiswork:(Al,Nb,Ti)3(Al,Nb,Ti)GAl:Al4GHSERAl+80000thisworkGMo:Al3GFCCMo+GHSERAl+80000thisworkGTi:Al3GHSERTi+GHSERAl+30459237:024T[ 18 ]GAl:Mo3GHSERAl+GHSERMo138710:769+40:9417405TthisworkGMo:Mo4GFCCMo+80000[ 18 ]GTi:Mo3GFCCTi+GFCCMo[ 18 ]GAl:Ti3GHSERAl+GHSERTi144592+37:024T[ 18 ]GMo:Ti3GFCCMo+GFCCTi[ 18 ]GTi:Ti4GHSERTi+80000[ 18 ]0LAl,Mo:Mo80000[ 18 ]0LAl,Mo:Ti80000thiswork0LAl,Ti:Ti60000[ 18 ]0LAl,Ti:Mo60000thiswork0LAl:Mo,Ti238049:751+120:0Tthiswork:(Al,Mo,Ti)0:75(Al,Mo,Ti)0:25GAl:AlGHSERAl+10000thiswork 217

PAGE 218

ParameterFunctionReference 18 ]GAl:Mo0:75GHSERAl+0:25GHSERMo+10000thisworkGMo:MoGHSERMo+10000thisworkGTi:Mo0:75GHSERTi+0:25GHSERMo+10000[ 18 ]GAl:Ti0:75GHSERAl+0:25GHSERTi+10000thisworkGMo:Ti0:75GHSERMo+0:25GHSERTi+10000thisworkGTi:TiGHSERTi+10000thiswork0LAl,Mo:Al0:455738495Tthiswork0LMo:Al,Mo2:63250337Tthiswork0LMo,Ti:Al22021:7991Tthiswork(Al):(Al,Nb,Ti)G(Al)AlGHSERAl[ 18 ]G(Al)MoGFCCMo[ 18 ]G(Al)TiGHSERTi+60000:1T[ 18 ]0L(Al)Al,Mo146174:503+75:6992933Tthiswork0L(Al)Al,Ti128970+39T[ 18 ]1L(Al)Al,Ti5000[ 18 ]2L(Al)Al,Ti20000[ 18 ]0L(Al)Mo,Ti16500[ 18 ]Liq:(Al,Nb,Ti)GLiqAlGLIQAl[ 18 ]GLiqMoGLIQMo[ 18 ]GLiqTiGLIQTi[ 18 ]0LLiqAl,Mo148756:738+60:614292Tthiswork1LLiqAl,Mo84284:591126:0603279Tthiswork2LLiqAl,Mo96206:0414+26:5383283Tthiswork3LLiqAl,Mo55040:960023:0458461Tthiswork0LLiqAl,Ti108250+38T[ 18 ]1LLiqAl,Ti6000+5T[ 18 ]2LLiqAl,Ti15000[ 18 ]0LLiqMo,Ti9000+2T[ 18 ]0LLiqAl,Mo,Ti100000[ 18 ]1LLiqAl,Mo,Ti100000[ 18 ]2LLiqAl,Mo,Ti100000[ 18 ]Al63Mo37:(Al)63(Mo)37

PAGE 219

ParameterFunctionReference 219

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DamianM.CupidwasborninCascade,St.Ann's,TrinidadandTobagoin1980.InAugust1998,hestartedhisundergraduateworkatMorehouseCollegeinAtlanta,Georiga.HewasawardedaBachelorofScienceinphysicsinMay2002.DamianstartedgraduateworkatClarkAtlantaUniversity,Atlanta,GeorgiainAugust2002andwasawardedaMasterofScienceinphysicsinAugust2004.ThetopicofhisresearchwastheinvestigationoftheshapememoryeectofNiTialloysusingdensityfunctionaltheory.DamianwasacceptedintothePh.D.programinmaterialsscienceandengineeringattheUniversityofFloridainAugust2004.HisgraduateadvisorwasProf.Dr.Hans-JurgenSeifert.InOctober2006,Prof.Dr.FereshtehEbrahimijoinedhiscommitteeasco-chair.InDecember2006,DamianmovedtoFreiberg,GermanytocontinuehisPh.D.researchattheFreibergUniversityofMiningandTechnologywherehewascloselysupervisedbyDr.Dr.OlgaFabrichnaya.DamianreceivedhisPh.D.inDecember2009. 233