An-introduction-to-phase-field-modeling-of-microst

ThisarticleappearedinajournalpublishedbyElsevier.Theattachedcopyisfurnishedtotheauthorforinternalnon-commercialresearchandeducationuse,includingforinstructionattheauthorsinstitutionandsharingwithcolleagues.Otheruses,includingreproductionanddistribution,orsellingorlicensingcopies,orpostingtopersonal,institutionalorthirdpartywebsitesareprohibited.Inmostcasesauthorsarepermittedtoposttheirversionofthearticle(e.g.inWordorTexform)totheirpersonalwebsiteorinstitutionalrepository.AuthorsrequiringfurtherinformationregardingElsevier’sarchivingandmanuscriptpoliciesareencouragedtovisit:(2008)268–294fieldmodelingofmicrostructureevolutionNeleMoelans∗,BartBlanpain,PatrickWollantsDepartmentofMetallurgyandMaterialsEngineering,FacultyofEngineering,KatholiekeUniversiteitLeuven,KasteelparkArenberg44,B-3001Leuven,BelgiumReceived1June2007;receivedinrevisedform10October2007;accepted1November2007Availableonline21December2007AbstractThephase-fieldmethodhasbecomeanimportantandextremelyversatiletechniqueforsimulatingmicrostructureevolutionatthemesoscale.Thankstothediffuse-interfaceapproach,itallowsustostudytheevolutionofarbitrarycomplexgrainmorphologieswithoutanypresumptionontheirshapeormutualdistribution.Itisalsostraightforwardtoaccountfordifferentthermodynamicdrivingforcesformicrostructureevolution,suchasbulkandinterfacialenergy,elasticenergyandelectricormagneticenergy,andtheeffectofdifferenttransportprocesses,suchasmassdiffusion,heatconductionandconvection.Thepurposeofthepaperistogiveanintroductiontothephase-fieldmodelingtechnique.Theconceptofdiffuseinterfaces,thephase-fieldvariables,thethermodynamicdrivingforceformicrostructureevolutionandthekineticphase-fieldequationsareintroduced.Furthermore,commontechniquesforparameterdeterminationandnumericalsolutionoftheequationsarediscussed.Toshowthevarietyinphase-fieldmodels,differentmodelformulationsareexploited,dependingonwhichismostcommonormostillustrative.c2007ElsevierLtd.Allrightsreserved.Keywords:Phase-fieldmodeling;Microstructure;Nonequilibriumthermodynamics;Kinetics;Simulation1.IntroductionMostmaterialsareheterogeneousonthemesoscale.Theirmicrostructureconsistsofgrainsordomains,whichdifferinstructure,orientationandchemicalcomposition.Thephysicalandmechanicalpropertiesonthemacroscopicscalehighlydependontheshape,sizeandmutualdistributionofthegrainsordomains.Itis,therefore,extremelyimportanttogaininsightinthemechanismsofmicrostructureformationandevolution.Howeverextensivetheoreticalandexperimentalresearchareheretorequired,asmicrostructureevolutioninvolvesalargediversityofoftencomplicatedprocesses.Moreover,amicrostructureisinherentlyathermodynamicunstablestructurethatevolvesintime.Withinthisdomain,thephase-fieldmethodhasbecomeapowerfultoolforsimulatingthemicrostructuralevolutioninawidevarietyofmaterialprocesses,suchassolidification,solid-statephasetransformations,precipitategrowthandcoarsening,martensitictransformationsandgraingrowth.Themicrostructuresconsideredinphase-fieldsimulationstypicallyconsistofanumberofgrains.Theshapeandmutual∗Correspondingauthor.Tel.:[phone];fax:[phone].E-mailaddress:[e-mail](N.Moelans).URL:(N.Moelans).distributionofthegrainsisrepresentedbyfunctionsthatarecontinuousinspaceandtime,thephase-fieldvariables.Withinthegrains,thephase-fieldvariableshavenearlyconstantvalues,whicharerelatedtothestructure,orientationandcompositionofthegrains.Theinterfacebetweentwograinsisdefinedasanarrowregionwherethephase-fieldvariablesgraduallyvarybetweentheirvaluesintheneighboringgrains.Thismodelingapproachiscalledadiffuse-interfacedescription.Theevolutionoftheshapeofthegrains,orinotherwordsthepositionoftheinterfaces,asafunctionoftime,isimplicitlygivenbytheevolutionofthephase-fieldvariables.Animportantadvantageofthephase-fieldmethodisthat,thankstothediffuse-interfacedescription,thereisnoneedtotracktheinterfaces(tofollowexplicitlythepositionoftheinterfacesbymeansofmathematicalequations)duringmicrostructuralevolution.Therefore,theevolutionofcomplexgrainmorphologies,typicallyobservedintechnicalalloys,canbepredictedwithoutmakinganyaprioriassumptionontheshapeofthegrains.Thetemporalevolutionofthephase-fieldvariablesisdescribedbyasetofpartialdifferentialequations,whicharesolvednumerically.Differentdrivingforcesformicrostructuralevolution,suchasareductioninbulkenergy,interfacialenergyandelasticenergy,canbeconsidered.Thephase-fieldmethodhasaphenomenological0364-5916/$-seefrontmatterc2007ElsevierLtd.Allrightsreserved.doi:10.1016/j.calphad.2007.11.003Author'spersonalcopyN.Moelansetal./ComputerCouplingofPhaseDiagramsandThermochemistry32(2008)268–294269character:theequationsfortheevolutionofthephase-fieldvariablesarederivedbasedongeneralthermodynamicandkineticprinciples;however,theydonotexplicitlydealwiththebehavioroftheindividualatoms.Asaconsequence,materialspecificpropertiesmustbeintroducedintothemodelthro