相场理论基础

相场理论基础Long-QingChen,YunzhiWang,TheContinuumFieldApproachtoModelingMicrostructuralEvolution,JOM,48(12),13-18,1996相场模拟示例2μm1μmY.Jinet.al.Thephasefieldmethodhandleswellarbitrarymicrostructuresconsistingofdiffusionallyandelasticallyinteractingparticlesanddefectsofhighvolumefractionandaccountsself-consistentlyfortopologicalchangessuchasparticlecoalescence.Thephasefieldmethodhandleswellarbitrarymicrostructuresconsistingofdiffusionallyandelasticallyinteractingparticlesanddefectsofhighvolumefractionandaccountsself-consistentlyfortopologicalchangessuchasparticlecoalescence.0.5μm高温合金微观结构黑白:实验彩色:模拟ADEFGHBCTop:Polycrystalstructureoftheparentphasewith8randomlyorientedgrainsincomputationalvolume.Right:Thehysteresisloopandthe3Dmicrostructuresobtainedatdifferentstressesforpolycrystalwithζ=5.ParentphaseMartensiticvariant1,withthetrigonalaxisalongMartensiticvariant2,withthetrigonalaxisalongMartensiticvariant3,withthetrigonalaxisalongMartensiticvariant4,withthetrigonalaxisalongGrainboundary2[111]β2[111]β2[111]β2[111]βCubictrigonalinAuCdalloysY.M.Jin,A.Artemev,A.G.Khachaturyan,ActaMater.49(2001)2309.MartensiticTransformationFormationofγ/γ′BimodalMicrostructureinNi-AlContinuouscooling+heterogeneousnucleation0.5μm•Landaupolynomial/sublatticemodelforfreeenergy•Diffusivity,interfacialenergy,elasticconstantsandlatticeparameterfromArdell’sΔG*hetΔG*hom=0.75ApplicationsofPhaseFieldMethodSolidificationPureliquid,withfluidflowMulti-componentalloysSolid-StatePhaseTransformationsSpinodaldecompositionPrecipitationOrderingEutecticandeutectoidreactionsCoherenttransformationsMartensitictransformationsFerroelectrictransformationsFerromagnetictransformationsCoarseningandGraingrowthCoarseningandstrain-inducedcoarseningSecondphaseparticleandanti-phasedomainsGraingrowthinsingleandmulti-phasesolidsAnisotropicgraingrowthEffectofsegregation,transitionandsolute-dragDislocationnetworkcoarseningOtherApplicationsMulti-componentinterdiffusionincoatingsPhasetransformationsinthinfilmsSurface-stressinducedpatternformationSpiralgrowthCrystalgrowthunderstressSolute-dislocationinteractionsCrackpropagationElectromigrationElectrochemicalprocessOxidation,Sintering,Soldering…...RadioShack!!!DislocationsDislocationsubstructuresDislocationcorestructuresDislocation-precipitateinteractionsReferences:Y.Wang,andL.Q.Chen,SimulationofMicrostructuralEvolutionUsingtheFieldMethod,inMethodsinMaterialResearch.2000,JohnWiley&Sons.Inc.;L.Q.Chen,Annu.Rev.Mater.Res.,32(2002),p.113;A.Karma,PhaseFieldMethods,inEncyclopediaofmaterials:scienceandtechnology,2001,ElsevierOxford.p.6873-86;HeikeEmmerich,TheDiffuseInterfaceApproachinMaterialsScience,Springer,2003.微观结构的表示•一组连续场变量表示微观结构：成分、序参量、相分布、变形量、磁畴等。•场变量变化剧烈区位渐变界面所在位置。ABABc(x)xη(x)cdiscordη=0η=1微观结构演化驱动力•体化学势能(Bulkfreeenergy)的降低（相变)•界面能的降低（晶粒粗化，graingrowth,Ostwaldripening）•弹性能（晶格尺寸及位向的差异所导致）的降低•外场影响：外加应力、温度场、电场、磁场体化学自由能•局部体化学能只是当前位置处的浓度继续参数的函数。•包括平衡态和非平衡态。界面自由能[]dVkfFVchem∫∇+=2)()(φφ[]dVkffFV∫∇+−=20int)()()(φφφ两相平衡自由能密度斜率化学能梯度项界面能21()[()()]2~VVVVBAFNfXdVFNfXXdVdfdXαμμ=→=+∇−∫∫FXδδ=梯度热力学均匀体系非均匀体系~BAdfXdXμμα−−Δ弹性能•弹性能的来源：相变或有序化产生的晶格常数改变（重点），外加应立场微观结构的演化方程•转化为场变量（浓度、序参量）的演化方程∂cr,t()∂t=∇D∇δFδcr,t()⎛⎝⎜⎞⎠⎟+ξcr,t()∂ηpr,t()∂t=−LδFδηpr,t()+ξpr,t()M守恒方程（conservedequation):Cahn-Hilliardequation)非守恒方程（nonconservedequation):timedependentGinzburg-Landauequation)热噪音项变分导数微分导数系统向能量降低的方向演化物质沿化学能梯度方向迁移且物质总量守恒F=Fchem+Eel+...Fchem=dVfc,η()+κc2∇c2+κη2∇η2⎡⎣⎢⎤⎦⎥V∫守恒方程(Cahn-Hilliardequation)的物理实质:扩散方程•菲克定律:•广义菲克定律:•Cahn-Hilliard方程:22CJDxCCJDtx∂=−∂∂∂=−∇=∂∂i;:;:(,)(,)()(,)JMffdrivingforceFFfPPchemicalpotentialCrtCrtCFJMtCrtδδδδδδ==∇=−∇=−∂=−∇=∇∇∂ii;::()JcvcMffdrivingforcefPchemicalpotentialgradientCJcMPt===∇∂=−∇=∇∇∂ii计算机模拟一般步骤•确定待模拟体系的合适的场变量•建立化学能的局域能量密度及其他能量的表达式•由实验数据确定演化动力学参数•确定初始条件及边界条件，并变成进行数值模拟。