Phase Field Modelling

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1、MaterialsScience&MetallurgyMasterofPhilosophy,MaterialsModelling,CourseMP6,KineticsandMicrostructureModelling,H.K.D.H.BhadeshiaLecture15:PhaseFieldModellingIntroductionImaginethegrowthofaprecipitatewhichisisolatedfromthema-trixbyaninterface.Therearethreedistinctquant

2、itiestoconsider:theprecipitate,matrixandinterface.Theinterfacecanbedescribedasanevolvingsurfacewhosemotioniscontrolledaccordingtothebound-aryconditionsconsistentwiththemechanismoftransformation.Theinterfaceinthismathematicaldescriptionissimplyatwo–dimensionalsurfacew

3、ithnowidthorstructure;itissaidtobeasharpinterface.Inthephase–fieldmethod,thestateoftheentiremicrostructureisrepresentedcontinuouslybyasinglevariableknownastheorderpa-rameterφ.Forexample,φ=1,φ=0and0<φ<1representtheprecipitate,matrixandinterfacerespectively.Thelatterist

4、hereforelocatedbytheregionoverwhichφchangesfromitsprecipitate–valuetoitsmatrix–value(Fig.1).Therangeoverwhichitchangesisthewidthoftheinterface.Thesetofvaluesoftheorderparameteroverthewholemicrostructureisthephasefield.Theevolutionofthemicrostructurewithtimeisassumedto

5、beproportionaltothevariationofthefreeenergyfunctionalwithrespecttotheorderparameter:∂φ∂g=M∂t∂φFig.1:(a)Sharpinterface.(b)Diffuseinterface.whereMisamobility.Thetermgdescribeshowthefreeenergyvariesasafunctionoftheorderparameter;atconstantTandP,thistakesthetypicalform(Ap

6、pendix1)†:Z·¸2g=g{φ,T}+²(∇φ)dV(1)0VwhereVandTrepresentthevolumeandtemperaturerespectively.Thesecondterminthisequationdependsonlyonthegradientofφandhenceisnon–zeroonlyintheinterfacialregion;itisadescriptionthereforeoftheinterfacialenergy.Thefirsttermisthesumofthefreeen

7、ergiesoftheprecipitateandmatrix,andmayalsocontainatermdescribingtheactivationbarrieracrosstheinterface.Forthecaseofsolidification,SLg=hg+(1−h)g+Qf0wheregSandgLrefertothefreeenergiesofthesolidandliquidphases†Ifthetemperaturevariesthenthefunctionalisexpressedintermsofen

8、tropyratherthanfreeenergy.respectively,Qistheheightoftheactivationbarrierattheinterface,222h=φ(3−2φ)andf=φ(1−φ)NoticethatthetermhgS