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PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryPhaseeldmodellingBasics: Microstructure,materialsthermodynamicsandtransformationkineticsMogadalaiPGururajan1DepartmentofMetallurgical EngineeringandMaterialsScienceIndianInstituteofTechnologyBombayMumbai 400076INDIA8December,[email protected],[email protected],IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryOutline1 Preamble2 Introduction3 Classicaldiusion4 Non-classicaldiusion5 Phaseeldmodelling6 SummaryPhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryAcknowledgementsThankstotheorganisersforinvitingmetogivethistalk!Thankstomy:Teachers: Abi,Ferdi,PeterColleagues: Shankara,Ram,Deep,Saswata,Rajdip,Chirru,Kuo-An,Ian,Vichu,Prita,andPhaniStudents: Amol,Yash,Arpit,Jayatheerdh,Kedar,Arijit,Vivek,...Fundingagencies: Volkswagen,DAAD-UGC,CSIR,ONR,IRCCatIIT-B,DRDOER&IPR,SASEPhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryAdisclaimerThisisapedagogicalpresentation;Hence,some(but,hopefully,notall:-)ofwhatIamsayingisknowntoyou;Pleaseindulgeme!PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryModelling: TimeandlengthscalesLengthAb initioMDPhase fieldNano, NanoMicron,Angstrom, FemtoMinutesFinite ElementMetres,MonthsTimeProcessFigure: TimeandlengthscalesforsomemodellingtechniquesPhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryMicronsandMinutesFigure: MicrostructureinUdimet520,Xuetal,Met. Mat. Trans. A,29,1998PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryFormationandevolutionofmicrostructuresMicrostructures: typicallyatthemicronlengthscaleEvolutionofmicrostructures: typically,inthecourseofafewsecondstoseveralhoursPhaseeldmodel: theidealtoolforformationandevolutionofmicrostructuresconsideringthetimeandlengthscalesPhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryMicrostructure: acloserlookFigure: MicrostructureinUdimet520,Xuetal,Met. Mat. Trans. A,29,1998PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryWhatdowewanttomodel?ElasticstresseectsGraingrowthGrowthandcoarseningofprecipitatesPhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryAnotherexample!Figure: Dendriticmicrostructureinanickel-basesuperalloysinglecrystalweld,Davidetal,JOM,June,2003.PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummarySolidication: phaseeldmodellingMicrostructures: typicallyatthemicronlengthscaleEvolutionofmicrostructures: typically,inthecourseofafewsecondstoseveralhoursPhaseeldmodel: theidealtoolforformationandevolutionofmicrostructuresPhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryWhatisphaseeldmodelling?Shortanswer: Partialdierentialequationsthatdescribediusion(ofatomsandheat)aswellasphasetransformationsLonganswer: Therestofthistalk!PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummarySomereadingmaterialSpinodaldecomposition,JEHilliard,Chapter12,Phasetransformations,ASM,1970.Myfavourite: Derivationsoftheimportantexpressionsaregiveninfull,onthepremisethatitiseasierforareadertoskipastepthanitisforanothertobridgethealgebraicgapbetweenitiseasilyshownthatandtheensuingequation.OriginalpapersfromCahnandco-workers: CahnandHilliardinJ.ofChem. Phys., 31,p. 688 and 28,p.258;Cahn,Trans. Met.Soc. AIME,242. p. 166;Cahn,ActaMet.,9,p795.Fewrecentreviews: KThorntonetal,ActaMat.,51,p. 5675;L-QChen,Ann. Rev. ofMat. Res.,32,p. 113;WJBoettingeretal,Ann. Rev. ofMat. Res.,32,p. 163.PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryFicksrstlawConstitutivelaw: theuxisproportionaltoconcentrationgradientJ= Dc (1)Why?PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryFicksrstlaw: atomisticpicturen n2 1JJ21If n > n ,2J > J112Figure: AnatomisticmodeltoexplainwhyFicksrstlawisreasonable!AssumerandomjumpsAssumejumpfrequencytobeindependentofcompositionHigherthenumberofatomsinaplane,highertheuxConcentrationgradient netuxPhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryFickssecondlawTheconservationofmasscoupledwiththeconstitutivelaw: theclassicaldiusionequation!ct= J= Dc (2)PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryThesurprise!Classicaldiusionlawoftheformct= Dc (3)failsincertaincases! Itneedsmodication!Whyfailure?Remember,itwasareasonableconstitutivelawplusasureconservationlaw.Whatistheneededmodication?PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryThermodynamicsofphaseseparatingsystemsRegularsolutionmodelCongurationalentropyassumingrandomarrangementEnthalpyassumingpairinteractionsandnegligiblepressureterm(i.e.,H E)G = (1 xB)GA + xBGB+ xB(1 xB) (4)+RT [xB ln xB+ (1 xB) ln (1 xB)]PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryPhaseseparatingsystems( > 0)andlowertemperaturesxBGxB BxFigure: FreeenergyofaphaseseparatingsystemPhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryMechanismsofphaseseparationxBGxBBxG 0G = 0G = 0Figure: Positivecurvature(nucleation)andnegativecurvature(spinodal)of the free energy versus composition diagram;phase separation mechanismchangesfromnucleationtospinodalatthepointofzerocurvature.PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummarySpinodal: drivingforceABCX YXYY Y XXADriving force: decrease in GB CFigure: Inregionsofnegativecurvature,A-rich(andB-rich)regionsspontaneouslybecomericherinA(andB),because,thefreeenergydecreasesinsuchaprocess.PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryClassicaldiusionxBt= D2xBx2(5)CompositionPositionCompositionPositionFigure: Evolutionofcompositionduringhomogenisationandphaseseparation. Whilehomogenisationobviouslyfollowsfromthediusionequationabove,phaseseparationimpliesD< 0.PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryChemicalpotentialOfcourseyouknew!1 22121Figure: Fluxshouldbedriventoevenoutchemicalpotentialdierences.PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryDiusionequationJ= NVM where = (B A) =GnB(6)xBt= NVM (7)NotingNV(21) =GxB,itcanbeshownxBt=_MNV__2Gx2B_2xB=_MNV_G
2xB(8)PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryArewedone?xBt=_MNV_G
2xB(9)CompositionPositionCompositionGG > 0CompositionPositionCompositionGG < 0Figure: Evolutionofcompositionduringhomogenisationandphaseseparation. Bycombiningthesignsofthecurvatureofthecompositionproleandfreeenergyversuscompositioncurve,wecanexplainboth.PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryWhywearenotyetdone!xBt=_MNV_G
2xB(10)Asolutiontotheaboveequation(whenitislinearised):xB x0B= A(, t) exp (i x) (11)where,x0Bistheaveragecomposition,A(, t)istheamplitudeofFouriercomponentofwavenumber(= 2/)attimet(beingthewavelenth)Bysubstitutingthesolutionintheequation,onecanshowA(, t) = A(, 0) exp__MNV_G
2t_ = A(, 0) exp [R()t] (12)PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryFailureofclassicaldiusionequationA(, t) = A(, 0) exp__MNV_G
2t_ = A(, 0) exp [R()t] (13)0RSolution toclassical equationSolutionthat we wantMaximally growing wavenumberCriticalWavenumberFigure: Predictedgrowthratefromtheclassicaldiusionequationandwhatwewantittobe.As 0,theA(, t) Cu-Ni-Fe, 100AandnotorderedBCCaspredictedhere!PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryWhatisthephysicsthatwehavemissed?Interfacialenergy!Attheinterfaces,therearemoreunfavourable(AB)bonds;theycostthesystemenergy. Thisenergyis,obviously,afunctionofconcentrationgradient!Inotherwords,freeenergy(G)shouldnotjustbeapolynomialincomposition(xB);itshouldinsteadbeafunctionalinxB.Ifso, =GnB,where {/nB}isthevariationalderivative.PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryThefreeenergyG=1NV_VdV_g0(xB) +|xB|2(14)g0(xB): freeenergyperatom;apolynomialinxB;hasatleasttwominimaandamaxima.: gradientenergycoecientxB: gradientPhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryThechemicalpotentialEuler-Lagrangeequation:NV =GxB_GxB_(15)NV =GxB22xB(16)PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryThenon-classicaldiusionequationxBt=_MNV__G
2xB 24xB(17)Thefourthordertermiswhatmakessurethatforlarger,thesolutionremainsnite!Cahn-HilliardequationGibbsvanderWaalsTimeDependentGinzburgLandauAllen-CahnHohenbergandHalperin: ModelsA,B,C,...PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryThenon-classicaldiusionequationxBt=_MNV__G
2xB 24xB(18)Physically,theinterfacialenergysetsthelowerlimitonthewavelengthofcompositionmodulations;phaseseparationreducesoverallfreeenergy;however,newinterfacesareformedasphasetransformationproceeds;theseinterfacescostthesystemenergy;so,unlessthegaininenergycanmorethancompensatefortheinterfacialenergyincrease,phaseseparationwillnotproceed.Howaboutcoherency?Miscibilitygap,chemicalspinodalandcoherentspinodalPhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryAmovie!AsolutiontotheCahn-Hilliardequation:xBt=_MNV_G
2xB 24xB(19)ObtainedusingFouriertransforms!SpinodaldecompositionPhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryTherecipeOrderparameter: Aeldvariablecontinuousandhasvaluesforallpositionsandtime;couldbeconservedornon-conserved;describesthetopology(Microstructure)Freeenergyfunctional: Todescribethefreeenergyforagivenmicrostructure;afunctionaloftheorderparameters(Thermodynamics)Variational derivativeequatedtothechemical potential,whichinturn,determineshowtheorderparameterschange: Microstructureevolvesinsuchawaythatthefreeenergykeepsdecreasing(Kinetics)Notethatthereisnothephaseeldmodel;anymodel,whichisformulatedaccordingtotheprinciplesaboveisaphaseeldmodel;and,forthesamephenomena,dependingonthefreeenergyfunctionalthatonewrites,therecouldbeseveralphaseeldmodels!PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryCharacteristicsofaphaseeldmodelInterfacesarenotsharp;diuseinterfacemodelNotrackingofinterface: numericalsolutionsareeasierGradientenergycoecient: interfacialenergycontributions(Gibbs-Thomson,forexample)areautomaticallyaccountedforTopologicalsingularities(splittingordisappearanceofinterfaces): naturallytakencareofElasticstress,magneticandelectriceld: canbecoupledbyaddingtherelevantfreeenergyterm!PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryAcoupleofmoremoviesSolidication: four-foldandsix-folddendritesPhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryOrderparameters: non-conserved order parameter;can be thought ofas solid fractionorascrystallanityT: temperature;anothernon-conservedorderparameterPhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryThermodynamicsFreeenergyminimization: isothermalsystemsEntropymaximization: Non-isothermalThermodynamicallyconsistentphase-eldmodels: entropyisafunctionalofandTPhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryKineticsNon-classicalFourierlawofheatconductionTt= 2T+ Lslt(20)Equationofphasetransformation t= + g
(, T) (21)PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryWhatcanphaseeldmodelsnotdo?Necessityofnon-dimensionalizationAdvantageofnon-dimensionalizationDisadvantageofnon-dimensionalizationMeasuringquantitiesthatenterphaseeldmodelPhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryIdealproblemsforphaseeldmodellingMicrostructuralevolution: qualitativeParametricspaceexploration: exampleofelasticstresseect.Giventhatelasticmodulidierenceandelasticanisotropyaectthinlminstability,whichoneaectsitmoreforagiveneigenstrainandinterfacialproperties?ComplexinteractionsbetweendierentphenomenawherecontrolledexperimentsoranalyticalexplorationsaredicultPhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryAwordaboutnumericalmethodsFFTW:whythebestWhyFiniteElementtechniquesarenotthatsuccessfulFinitevolume/ntedierencetechniquesPhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummarySummaryPhaseeldmodel: non-classicalpartialdierentialequationsIdealforthestudyofmicrostructuralevolutionIdealfordiscerningtrendsandparameterrangesthatcangiverisetospecicphenomenaHugescopeforexploration!PhaseeldmodellingMPGururajan,IIT-BPreambleIntroductionClassical diusionNon-classicaldiusionPhaseeldmodellingSummaryTHANKYOU!