Solidification and Crystallization.docx
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Solidification and Crystallization.docx
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SolidificationandCrystallization
SolidificationandCrystallization
Abstract
Solidificationprocessesarecomplexinnature,involvingmultiplephaseandseverallengthscales.Thepropertiesofsolidifiedproductsaredictatedbythemicrostructure,themacrostructure,andvariousdefectspresentinthecasting.Inordertocontrolandimprovethequalityofcastproducts,itisimportanttohaveathoroughunderstandingofvariousphysicalandphysicochemicalphenomenaoccurringatvariouslengthscales,preferablythroughpredictivemodelsandcontrolledexperiments[1].Rapidadvancesinatomisticandphase-fieldmodelingtechniquesaswellasnewexperimentshaveledtomajorprogressinsolidificationscienceandcrystallizationscienceduringthepastdecadesyears.Lookingattherapidlydevelopingcommunicationsindustry,itisobviousthatalargepartofcurrentresearchforsolidificationisdevotedtotheunderstandingof[2]:
(i)solid-statetransformationsasfarastheyrelatetosolidificationmodelsandtechniques;(ii)regularandirregulareutecticandperitecticmicrostructures;(iii)keyanisotropicpropertiesofthesolid-liquidinterfacethatgovernsolidificationpatternevolution,includingthesolid-liquidinterfacefreeenergyandthethermodynamicandkineticcoefficient.What’smore,dendritesarethebeststudiedstructuresformedduringsolidificationofundercooledmelt.Additionallyothercomplexstructureshavebeenfoundinbinaryalloys,ternaryalloysandmulti-componentalloys.Inlightofthisprogresscriticalissuesthatpointtodirectionsforfutureresearchinbothsolidificationandcrystallizationtransformationsareidentified.
Keywords:
Solidificationmicrostructures,Binaryalloys,Ternaryalloys,Simulations,Undercooling,Crystallization,Segregation
1.Introduction
Materialssciencehasassumedakeypositionfornewtechnologicaldevelopments,andisthereforestronglysupportedbyindustryandgovernments.Nowadaysitoccupiesabridgingpositionphysics,chemistryandengineeringandextendsfrombasicscienceinphysicsandchemistryontheatomicscaletolarge-scaleapplicationsinindustry.Rapidadvancesinatomisticandphase-fieldmodelingtechniquesaswellasnewexperimentshaveledtomajorprogressinsolidificationscienceandcrystallizationscienceduringthepastdecadesyears.Thebasicproblemofunderstandingandcontrollingmicrostructureduringsolidificationandcrustallizationcontinuestopresentnumerousscientificandtechnologicalchallenges.Solidificationandmeltingarebothphasetransformationsinvolvingatleastoneliquidandonesolidphaserespectively.Ascomparedtomelting,researchworkonsolidificationdocumentedintheliteratureisrelativelyheavy.Thisiscertainlyduetothefactthatsolidificationisofhigherinterestfortechnicalapplications,asthepropertiesofacastmaterialaredirectlyconnectedtothemicrostructureevolutionduringsolidificationandcrystallization[3].Atlowtemperaturecrystalsurfacesareknowntoassumetheshapeofaplanefacet.Withincreasingtemperature,fluctuationsgraduallycontributeanonzerothicknesstotheinitiallyflatfacet[4].Crystallizationintomultiplegrainstructurestakesplacethroughstagesofnucleation,growth,grainimpingementandgraincoarsening.Thepresentpaperreportsonanattempttoincorporateimplicitlythesefourstagesintoasinglemodel,inwhichtheentiremicrostructureisdescribedbyonestructural,andoneorientationalorderparameter.Themodelisthusaimedtocombinethemeritsofthestandardphase-fieldandtheso-calledPottsmethods[5].Basedonthismodel,variousscenariosofcrystallizationcanbesimulatedfordifferentcombinationsofnucleationandthermalconditions.Someofthesimulationresultsareexaminedqualitativelywithrespecttotherespecttothegeneralfeaturesofmultiplegraincrystallization.
Thedesiretopredictandcontrolmicrostructureformationduringsolidificationhasmotivatednumeroustheoretical,computationalandexperimentalresearchworksonvariousaspectsofsolidificationscience,rangingfromthermodynamicmodelingandinterfacedynamicstoadvancednumericalanalysesandsimulationtechniques[5].Therehasbeenparticularlymuchprogressinsimulatingsolidificationmicrostructuresbyusingphase-fieldmethod.Inthismethod,oneormorephase-fieldvariablesareusedtodescribethecorrespondingliquidandsolidphases.Aphase-fieldvariableisnormallytakentorepresentthestructuralorder,thechemicalorderorthelocal‘crystallographic’orientationofthecorrespondingphases,andisevolvedthroughminimizationofafreeenergyfunctional.Thescopeofapplicationofthephase-fieldmethodhasbeenexpandingcontinuallyovertherecentyears,andhasincludedproblemascomplexastheformationofmultiplephaseorgrainstructures,nucleationandnon-equilibriumsolidification[6].Nevertheless,thereisstillmuchroomleftforfurtherworkinsimulatingmicrostructureformationinvariouscrystallizationprocesses.Achallengingexampleisnanocrystallizationfromundercooledliquids,wherecopiousnucleation,multiplegrainformation,growthandcoarsening,andnon-equilibriumeffectshavetobetreatedsimultaneously,allwithinarobustandphysicallyjustifiablemodel.Thispaperaddressessomeoftheissuesinthisarea,andexaminesahybridmodelofcrystallizationwiththeabovecapability.
HerefirstlyIwouldintroduceaphase-fieldmodelforcrystallizationintomultiplegrainstructures;andthenIwouldintroducethesolidificationandthecrystallizationofthebinaryalloysandtheternaryalloyset.al;inthelastsectionofthistitle,Iwouldtalkaboutthesegregationwhenthealloysgeneratesolidificationandcrystallization.
2.Theoreticalbackgroundandmodeldevelopment
Formationofnanocrystallinemicrostructuresfromhighlyundercooledliquidsisassociatedwithextremelyhighnucleationrates,oftenordersofmagnitudeslargerthanthosepredictedfromtheclassicalnucleationtheory.Diffuseinterfacemodelsofnucleationprovideanexplanationforthisnucleationbehavior,intermsofthevariationofthesolid/liquidinterfacialenergywithtemperature.Figure1showsanexampleofsuchvariations[7].
Fig.1Variationsofthenormalizedinterfacialenergy(cubed)withundercooling,calculatedbasedonadiffuse-interfacemodelofnucleation.
Basedonthesemodels,thephenomenonofcopiousnucleation,andtheresultingnanocrystallinemicrostructure,canbeattributedtothevanishingoftheinterfacialenergyandhencethebarriertonucleationatacriticalundercooling,ΔTc.Thedecreaseintheinterfacialenergywithincreasingundercoolingisacharacteristicfeatureofthediffuseinterfacemodels.Thus,phase-fieldsimulationofcrystallizationwouldbeexpectedtoillustrateappropriatelycopiousnucleationandnanocrystallization[5].Nevertheless,formationofananocrystallinemicrostructureisnotonlyinfluencedbynucleationkinetics,butalsobysubsequentgraingrowthorcoarsening.Inordertosimulatenanocrystallization,therefore,amodelshouldalsobeabletotreatappropriatelytheproblemofgraingrowthorcoarsening.
Simulationofgraingrowthwassubsequentlytackledbyusingrariousphase-fieldmodels.Examplesincludemodelingbymultipleorderparameterphasefield.Inthismethod,afinitenumberoforientationsarerepresentedbyuniquecombiaationsofthesamenumberoforderparameters[1].MultipleorderparametermodelsarerobustandareinperfectagreementwithPottsmethod.Neverthess,intheseapproaches,thefreeenergydensitydependsontheorientationofthegrainsanmeasuredinafixedframe.Incontrasttomultipleorderparametermodels,thelocalenergydensityinthelattermodelisnotafunctionoforientation.Theframe-invariantapproachisthusbasedonamoresolidphysicaljustification.However,earlierversionsofthismodelranintotheproblemofbeingunstablewithrespecttoacompletelydiffusegrainboundary[5].Thisproblemwassolvedinthelaterversionsofframe-invariantapproach,ensuringgrainboundarieswithafinitethickness(Fig.2).
Fig.2Profilesofthestructuralandorientationalorderparameters,øandθ,acrossagrainboundary.Solidcurvescorrespondtoatypicalequilibriumsolutionfortheframe-invariantmodelofKobayashietal.Thethincurveshowsahypotheticalorientationprofilewithafinitegradientattheboundary,correspondingtoanarbitrarynon-equilibriumcondition[8].
AsshowninFig.3,thesolutionforθisastepfunctionatthemiddleofgrainboundary,whereøisminimal.Thissolutionhasclearphysicaljustifications.Nevertheless,itcanbeshownthat,atleastforacertaincombinationofparameters,themodelmaydivergefromtheabovesolutionandresultinanon-physicalsolution.Thiscanbeexaminedbycalculatingthefreeenergyofthesystemasafunctionofthenon-equilibriumthicknessofthe‘orientationalboundary’,asrepresentedbytheinverseoforientationgradient.Figure3showstheresultofsuchananalysis[8].Asshowninthefigure,theenergyincreaseswithincreasingthethicknessuptoamaximumpoint.Forsuchacase,thephase-fieldequationswouldconvergetothephysicalequilibriumsolution,ifandonlyiftheinitialorientationalboundarythicknessisbelowthismaximumpoint,ifitlieswithinthehatchedregioninFig.3.
Fig.3Variationofthetotalgrainboundaryenergywiththeinverseoforientationgradientattheboundary,calculatedforafixedstructuralorderparameterprofile,suchasthatshowninFig.2.Thehatchindicatestheregionwheretheenergyisminimizedthroughanincreaseoftheorientationgradientuptoitsequilibriumvalue.
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