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1、PhysicaD214(2006)144–156www.elsevier.com/locate/physdAthermodynamicapproachtonon-isothermalphase-fieldevolutionincontinuumphysicsMauroFabrizioa,ClaudioGiorgib,AngeloMorroc,∗aDipartimentodiMatematica,PiazzadiPortaS.Donato5,40127Bologna,ItalybDipartimentodiMatematica,ViaValotti9,25133Brescia,ItalycDI
2、BE,ViaOperaPia11a,16145Genova,ItalyReceived19May2005;receivedinrevisedform5October2005;accepted6January2006CommunicatedbyJ.LegaAbstractPhasetransitionsbetweentwophasesaremodelledasspaceregionswhereanorderparameter,orphasefield,changessmoothly.Athermodynamicapproachisdevelopedbyallowingforthenonloca
3、lcharacterofthecontinuum.Thephasefieldisregardedasaninternalvariableandthekineticorevolutionequationisviewedasaconstitutiveequation.Alongwiththeotherconstitutiveequations,theunknownevolutionequationisrequiredtosatisfythesecondlawofthermodynamics.Necessaryandsufficientrestrictionsplacedbythermodynami
4、csarederivedfortheconstitutiveequationsand,furthermore,ageneralformoftheevolutionequationfortheorderparameterisobtainedwithintheschemesofanon-conservedoraconservedphasefield.Basedonthethermodynamicrestrictions,amodelfortheice–watertransitionisestablishedwhichallowsforsuperheatingandundercooling.Amo
5、delisalsoprovidedforthetransitioninsuperconductingmaterials.c2006ElsevierB.V.Allrightsreserved.Keywords:Phasetransitions;Phase-fieldmodel;Evolutionequation;Freeenergy;Continuumthermodynamics1.Introductiontoselectaquantity,sayϕ,whichdiffersinthetwophases.SinceLandau,suchaquantityiscalledanorderparam
6、eter.TheTheaimofthispaperisfourfold.First,toestablishatermphasefieldfortheorderparameteremphasizesthesmoothgeneralthermodynamicalsettingforphasetransitionmodelsvariationofϕinthepertinentregion.inconservedandnon-conservedtwo-phasesystems.Secondly,Concerningtheliquid–solidtransition,thephasefieldwasto
7、contrastthethermodynamicschemewithothersappearingintroducedbyLanger[4],Fix[5]andCaginalp[6].Toremedyintheliterature.Thirdly,todevelopageneralmodelforthethelackofaproperthermodynamicsetting,inthe1990sice–watertran
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