Commun.Comput.Phys.doi:10.4208/cicp.2009.09.099Vol.7,No.4,pp.674-682Ap - PDF document

Commun.Comput.Phys.doi:10.4208/cicp.2009.09.099Vol.7,No.4,pp.674-682April2010SimultaneousPredictionofMorphologiesofaCriticalNucleusandanEquilibriumPrecipitateinSolidsLeiZhang1,Long-QingChen2andQiangDu1,2,1DepartmentofMathematics,PennStateUniversity,PA16802,USA.2DepartmentofMaterialsScienceandEngineering,PennStateUniversity,PA16802,USA.Received11June2009;Accepted(inrevisedversion)4September2009CommunicatedbyJieShenAvailableonline12October2009 Abstract.Weinvestigatethecriticalnucleusandequilibriummorphologiesduringprecipitationofasecond-phaseparticleinasolid.Weshowthatacombinationofdiffuse-interfacedescriptionandaconstrainedstringmethodisabletopredictboththecriticalnucleusandequilibriumprecipitatemorphologiessimultaneouslywith-outaprioriassumptions.Usingthecubictocubictransformationasanexample,itisdemonstratedthatthemaximumcompositionwithinacriticalnucleuscanbeeitherhigherorlowerthanthatofequilibriumprecipitatewhilethemorphologyofanequi-libriumprecipitatemayexhibitlowersymmetrythanthecriticalnucleusresultedfromelasticinteractions.AMSsubjectclassications:74G15,74B20Keywords:Phaseeld,diffuseinterface,nucleation,criticalnucleus,constrainedstringmethod,elasticity. 1IntroductionPrecipitationisacommon,naturalprocesswhichtakesplaceinasupersaturatedsolidorliquidsolution,e.g.,duringisothermalannealingofaquenchedhomogeneousal-loywithinatwo-phaseeldofaphasediagram.Itisthebasicprocessthatunderliesthedevelopmentofmanyadvancedmaterialssuchashigh-temperaturesuperalloysandultralightaluminumandmagnesiumalloys.Theprecipitatemicrostructure(thenum-berdensity,volumefraction,andmorphology)isthedominantfactorthatdetermines Correspondingauthor.Emailaddresses:zhang l@math.psu.edu(L.Zhang),lqc3@psu.edu(L.-Q.Chen),qdu@math.psu.edu(Q.Du)http://www.global-sci.com/674c\r2010Global-SciencePress L.Zhang,L.-Q.ChenandQ.Du/Commun.Comput.Phys.,7(2010),pp.674-682675themechanicalpropertiesofasolid.Oneofthemainchallengesinpredictingprecipi-tatemicrostructuresinsolidshasbeenthedeterminationofprecipitateparticlemorphol-ogybecauseofthepresenceofbothinterfacialenergyanisotropyandanisotropicelasticinteractions.Asthemajorityofprecipitationreactionsinsolidstakeplacethroughanucleation-and-growthmechanismfollowedbyparticlecoarsening,therearetwother-modynamicallywell-denedmorphologies:themorphologyofacriticalnucleusandtheequilibriummorphologyofaprecipitateparticle.Inclassicalnucleationmodels,acriticalnucleusisusuallyassumedtobesphericalandcriticalradiusisdeterminedbyacompetitionbetweenabulkfreeenergydecreasewhichisproportionaltovolumeandaninterfacialenergyincreasewhichisproportionaltointerfacialarea.Inadiffuse-interfacedescription,acriticalnucleusisdenedasthecompositionororderparameteructuationhavingtheminimumfreeenergyincreaseamongalluctuationswhichleadtonucleation,i.e.,thesaddlepointcongurationalongtheminimumenergypath(MEP)betweenthemetastableinitialphaserepresentedbyalocalminimuminthefreeenergylandscapeandtheequilibriumphaserepresentedbytheglobalminimum.Therefore,nucleationofnewprecipitateparticlesrequiresover-comingathermodynamicbarrier.Themagnitudeofthenucleationbarrier,andthusthenucleationrate,ortheresultedprecipitateparticledensity,isstronglydependentonthemorphologyofcriticalnuclei.Ontheotherhand,followingnucleationandgrowth,themorphologyandvolumefractionofprecipitateparticlesduringcoarseningaregenerallyclosetoequilibrium.Theparticlemorphologyandvolumefractionduringcoarseningtogetherwiththeparticledensitypredictedfromnucleationprovidealltheinformationthatisneededforpredictingthestrengthofasolidinmechanisticmodels.Therehavebeenextensivestudies,particularlynumericalsimulations,ofequilib-riumshapesofaprecipitateparticleinsolidsusingbothsharp-anddiffuse-interfaceapproaches[5,7,8,11–13].Attemptshavealsobeenmadetopredictthemorphologyofacriticalnucleusinsolidsbytakingintoaccountbothinterfacialenergyanisotropyandanisotropicelasticinteractions[9,10,14–16].Forexample,weshowedthatonecanpredictthemorphologyofacriticalnucleusinasystemgoingthroughaphasetransi-tion[14–16]usingacombinationofthediffuse-interface(phase-eld)descriptionandtheminimaxalgorithmbasedonthemountainpasstheorem.Themainobjectiveofthislet-teristoreportarstattempttopredictthemorphologyofacriticalnucleusaswellastheequilibriummorphologyofaprecipitatesimultaneouslywithinthesamephysicalmodelandmathematicalformulation.Aconcentrationeldthatconservestheaverageconcentrationisconsideredasanillustration.Weextendthestringmethod[3,4]tosys-temswithconstraintsthroughanovelaugmentedLagrangemultiplierformulation.Thisleadstoaneffectiveconstrainedstringmethodwhichmaybeusefulinthestudyofmanyconstrainedbarriercrossingproblemsinphysics,chemistryandbiology.Inthiswork,wedemonstratethatacombinationofdiffuse-interfacedescriptionandtheconstrainedstringmethodcansimultaneouslypredictthemorphologiesofacriticalnucleusandanequilibriumprecipitatewhichcanbedramaticallydifferent. 676L.Zhang,L.-Q.ChenandQ.Du/Commun.Comput.Phys.,7(2010),pp.674-6822DiffuseinterfacemodelFollowingthediffuse-interfacetheoryofCahn-Hilliard[1],weconsideraconservedeldwhichdescribestheconcentrationdistributioninabinarysolid.Thechangeofthetotalfreeenergy,Ft,arisingfromthecompositionaluctuationinaninitiallyhomogeneousstatewith0isgivenbyFt()=ZW1 2jArj2+df()dx+bEe().(2.1)WeusethedomainW=(1,1)dwithdbeingthespacedimension.Aperiodicboundaryconditionisusedforwiththeperiodsufcientlylargeincomparisonwiththesizeofthenucleusandtheequilibriumparticlesotheeffectofboundaryconditionsisnegligible.ThegradientenergycoefcientAisaconstantdiagonaltensorforisotropicinterfacialenergy,whileforanisotropicinterfacialenergy,itcanbemadetobeeitherdirection-allydependentordependentonthederivativesof.In[14],theeffectofanisotropicinterfacialenergyonthecriticalnucleimorphologyhasbeenexaminedinthecaseofanon-conservedeld.Inthiswork,wechoosetofocusonthecaseofisotropicinterfa-cialenergywithAbeingaconstantmultipleoftheidentitytensor.Thelocalfreeenergydensitychangedf(),arisingfromacompositionaluctuationaroundthehomogeneousstatewithcomposition0,isgivenbydf()=1 4k[(21)2(201)24(0)(300)],wherekisacoefcientofenergydensity.Theplotsofdf=df()aregiveninFig.1fordifferent0atk=0.03,withs=p 3/3beingthespinodalcomposition. Figure1:Freeenergychangeforc0=1,0.9andcs.Assumingthattheelasticmodulusisanisotropicbuthomogeneous,themicroscopicelasticitytheoryofKhachaturyan[6]canbeconvenientlyemployedtoefcientlycalcu-latetheelasticstrainenergyforsimplyconnectedcoherentinclusionsinasolid.Forthe L.Zhang,L.-Q.ChenandQ.Du/Commun.Comput.Phys.,7(2010),pp.674-682677caseofcubicprecipitatesinacubicmatrix,theelasticenergycontributioncanbewrittenasEe()=1 2(2p)dZˆWdkB(n)jˆ(k)ˆ0(k)j2.(2.2)ˆ(k)istheFouriertransformof(x).Theintegrationin(2.2)isoverthereciprocalspaceˆWofthereciprocallatticevectork,n=k/jkj=(n1,n2,n3)isthenormalizedunitvectorandB(n)isgivenby[6]B(n)=3(11+212)e20(11+212)2e20(1+2zs(n)+3z2n21n22n23) 11+z(11+12)s(n)+z2(11+212+44)n21n22n23,(2.3)wherez=(1112244)/44istheelasticanisotropicfactorwith11,12,44beingelasticconstantsintheVoigt'snotation,e0isthelatticemismatchbetweenthenewnucleatingcubicphaseandtheparentcubicphase,ands(n)=n21n22+n21n23+n22n23.Weset,inparticularthat,n=0ifk=0.Ratherthanvaryingthemagnitudeoflatticemismatchandelasticconstants,afac-torbisintroducedin(2.1)tostudytheeffectofrelativeelasticenergycontributiontochemicaldrivingforceonthecriticalnucleusmorphologyandequilibriumparticlemor-phology.Foraconservedeldwithprole=(x),thecomputationofsaddlepointsandtheminimumenergypathfortheenergyfunctional(2.1)subjecttotheconstraintZW((x)0)dx=0.(2.4)iscarriedoutviatheconstrainedstringmethodwhichisanaturalextensionofthesimpli-edstringmethodoriginallydevelopedbyE,RenandVanden-Eijnden[3,4].Weoutlinethealgorithmicprocedureshere.Somerelatedmathematicaltheorycanbefoundin[2]whiledetailednumericalanalysiswillbegivenelsewhere.Thestringmethodsproceedbyevolvingastring,i.e.,asmoothcurvewithintrin-sicparametrization,totheMEPbetweentwometastable/stableregionsincongurationspace.Specically,letj(,t)denotetheinstantaneousposition(representingthecom-positionproleinourcase)ofthestringwithbeingasuitableparametrization.ForanenergyE=E(j),theevolutionofthestringisbasedonrsttakingagradientdecentdirectionviathedynamicequationjt=dE dj(j),thenfollowedbyaprojectionstepthatmapsjbacktoacongurationsatisfyingthespeciedparametrization[4].Here,dE djrepresentsthevariationalderivativeoftheen-ergyEwithrespecttoj.Inpractice,acommonlyusedparametrizationforastringdiscretizedbyanitenumberoflinesegmentsistoenforceanequalsegmentlengthcon-ditionthroughaninterpolationprocedure[4].Sufcientnumberofsegmentsareneeded 678L.Zhang,L.-Q.ChenandQ.Du/Commun.Comput.Phys.,7(2010),pp.674-682toensureboththeconvergenceandtheaccuracyofthealgorithm.Basedonsuchanidea,wedevelopedaconstrainedstringmethodtondtheMEPongeneralconstrainedmanifolds.Itfollowsessentiallythestringmethodwithadditionaltreatmentofthecon-straints.TheconstrainedstringmethodallowsseveralequivalentformulationssuchasthepenaltyorLagrangemultipliermethods.Yet,someformulationsaremorenaturalandrobustthanothersandrequirelessparametertuning.OneparticularlyeffectiveapproachisbasedontheaugmentedLagrangemultipliermethod.Itsapplicationtotheenergy(2.1),subjecttothesimpleconstraint(2.4),amountstoconsideramodiedtotalenergyinvolvingtwoparametersandM:El(j)=Ft(j)+ZW(j0)dx+M(ZW(j0)dx)2.ForaxedpositivepenaltyconstantM,wesolvefortheconstrainedstring,viathefol-lowingiterations:rst,givenj,weapplythestringmethod[4]tothemodiedenergyElj=Elj(j)tosolveforjj;then,withjjknown,weupdatejbyj+1via:j+1=j+2MZW(jj0)dx.Weiteratebetweenthesetwostepsuntilconvergence.Attheendofiteration,thecon-strainedMEPisfoundwiththeequation(2.4)satisedalongthestring,andthelimitofjgivesthecorrespondingLagrangemultiplier.Adoptingthisformulation,theimplemen-tationoftheconstrainedstringmethodisstraightforwardanditassuresthesatisfactionoftheconstraintwithoutrequiringMtobeexceedinglylarge,thusreducingthestiffnessofthedynamicsystem.TheconstrainedstringmethodincludingtheaugmentedLa-grangemultiplierformulationcanbederivedforverygeneralenergiesandconstrainedmanifoldsandthushavemanypotentialapplications.Forthecaseoftheenergyfunc-tionalFt,eachpointofthestringcorrespondstoacompositionprolealongtheMEP.Thecriticalnucleusisdeterminedbythecompositionprole=(x)whichisrecoveredfromthesaddlepointcorrespondingtothepointontheconvergedMEPwiththehighestenergy.3NumericalsimulationsThemodelandalgorithmdescribedaboveallowsustodetermineboththecriticalnu-cleusandequilibriumprecipitate.Asanillustration,wefocusonthetwo-dimensionalexampleofacubictocubictransformation.Wexoneendofthestringtobetheinitialstaterepresentingauniformcompositionwith(x)=0inW,whileallowingtheotherendtomovebutgenerallywithintheenergywellofthegroundstateorequilibriumsolu-tion.Weuse31points(30linesegments)todiscretizethestringandtheFourierspectralmethodwitha256256gridforcomputingeachcompositionprole,i.e.,pointonthe L.Zhang,L.-Q.ChenandQ.Du/Commun.Comput.Phys.,7(2010),pp.674-682679 Figure2:Criticalnucleus,equilibriumandMEPforc0=0.9. Figure3:Criticalnucleus,equilibriumandMEPforc0=0.88.string.Anadaptiverescalingofthecomputationaldomainalongthestringcanbeusedtofurtherimprovetheresolution.Numericaltestswereconductedtoensurethatsufcientresolutioncanbeachieved.WextheparametersA1=A2=1.56104,11=250,12=150,44=200ande0=0.02.SincebothcriticalnucleusandequilibriumsolutionarerelativelysmallincomparisontothespatialdomainW,theirplotsaremagniedbyafactorof2inordertogetabetterview.InFig.2,fork=0.7andb=0.5,weplotthecriticalnucleus(left)andequilibriumso-lution(center)andtheMEP(right)inthepresenceofthelong-rangeelasticinteractionscorrespondingtoanaveragecomposition0=0.9.Oneoftheinterestingobservationsisthatthemaximumcompositionwithinthecriticalnucleusisabout5%higherthanthatoftheequilibriumprecipitate.Forthis0,however,boththecriticalnucleusandequilib-riumprecipitatehavethesamecubicsymmetryduetotheelasticenergyinteractions.AnotherexampleisshowninFig.3when0ischangedto0.88.As0isclosetothespinodalpoint,theinterfaceofcriticalnucleusbecomesmorediffusive.Thecompositionvalueatthecenterofacriticalnucleusdecreasesandisabout5%smallerthanthecom-positionoftheequilibriumprecipitate.Moreover,thesizeoftheequilibriumprecipitateislargerfor0=0.88thanfor0=0.9asaresultofhighersupersaturation.InbothFigs.2and3,theMEPplotsrevealhowtheenergyvalueschangefromtheinitialstatetothenalequilibriumstatealongpointsonthestring(correspondingtototal31differentcompositionproles).Thevalueofcriticalenergyneededtonucleateanewparticlefor0=0.88is0.1282whichissignicantlylowerthanthevalueof0.1792for0=0.9. 680L.Zhang,L.-Q.ChenandQ.Du/Commun.Comput.Phys.,7(2010),pp.674-682 Figure4:CalculatedMEPsforb=0.5,1and1.5withinsertsshowingthecriticalnucleiandequilibria. Figure5:Criticalnucleationenergywithchangingc0forb=0.125and0(left)andMEPs(right)forc0=0.93,b=0.125andc0=0.945,b=0.Toexaminetheeffectofelasticenergycontributions,wexthechemicaldrivingforcewith0=0.85andk=1,andincreasebtocomputetheMEPs.InFig.4(left),weplottheMEPs(withthecompositionalprolesforthecriticalnucleusandequilibriumprecipitateasinserts)fordifferentvaluesofb.Atarelativelysmallelasticenergycontribution,boththecriticalnucleusandtheequilibriumprecipitatedisplayacubicsymmetry.Withhigherelasticstrainenergycontribution,whilethecriticalnucleusmaintainsthecubicsymmetry,theequilibriumprecipitateisplate-likewithonlytwo-foldsymmetry(Fig.4,center).Aswefurtherincreasetheelasticenergycontribution,forexample,b=1.5,boththecriticalnucleusandtheequilibriumprecipitateexhibitplate-shapedparticles(Fig.4,right).Wealsoobservetheincreasesinboththecriticalnucleussize(seetheinserts)andthenucleationenergybarrier(from0.1222to0.1708and0.2449)withincreasingelasticenergycontributions.Theinuenceofelasticenergycontributionsonthemorphologiesofbothcriticalnu-cleiandequilibriumprecipitatescanbeunderstoodfromthecompetitionbetweenin-terfacialenergyandelasticstrainenergy.Thetotalinterfacialenergyisproportionaltointerfacialareabetweenaparticleandthematrixwhilethetotalelasticstrainenergyisproportionaltothevolumeoftheparticle.Sincethesizeofacriticalnucleusissigni-cantlysmallerthanthatofanequilibriumprecipitate,itisexpectedthattheelasticenergy L.Zhang,L.-Q.ChenandQ.Du/Commun.Comput.Phys.,7(2010),pp.674-682681willhavealesserinuenceonthecriticalnucleus(assumingtheinterfacialcoherencybe-tweentheparticleandmatrixisalwaysmaintainedduringtheentireevolutionprocess).Minimizationofelasticstrainenergyleadstoplate-shapedparticleswhileminimizationofinterfacialenergy(assumingisotropic)leadstosphericalshapes.Therefore,astheelasticstrainenergycontributionincreases,theshapeoftheequilibriumprecipitatebi-furcatesrstfrombeingcubictoplate-shapedbeforethecriticalnucleusdoes.Tofurtherunderstandtheelasticenergycontributions,inFig.5(left),thecriticalfreeenergyofformationasafunctionofaveragecomposition0isplottedforthecasewith-outtheelasticitycontributionb=0(redsquares),andforcriticalnucleiwithb=0.125(bluecircles).Wetakek=1inbothcases.Asexpected,withtheincreaseoftheaveragecom-position,thesizeofcriticalnuclei(withcubicsymmetry)isreducedandthecriticalnu-cleationenergydecreases.Thisdependenceissimilartothatpredictedfromtheclassicalnucleationtheoryforsphericalparticles.Wealsonoticethat,forthegivenparametersandelasticenergy,thesmallest0whichallowsnucleationtohappenis0.93,wheretheenergyofequilibriumsolutionisveryclosetotheinitial-stateenergy(Fig.5,topright).If0issmallerthan0.93,theenergyofanequilibriumprecipitatebecomeshigherthantheinitialhomogeneousstate,indicatingthattheinitialuniformstatecouldbegloballystablesothattheelasticenergycontributioncanpreventthenucleationprocessfromoc-curring,i.e.coherencystrainenergycontributionshiftstheequilibriumphaseboundary.Withoutelasticity,nucleationcanstilltakeplacewithanevensmaller0=0.945(Fig.5,bottomright).4SummaryInsummary,wereportanewapproachforcomputingthemorphologiesofbothcriticalnucleiandequilibriumprecipitateswithoutapriorishapeassumptions.Ourcalculationsrevealthatthemorphologyofacriticalnucleuscanbedramaticallydifferentfromtheequilibriumoneduetotheelasticenergycontributions.Weplantoextendtheapproachtotreatsystemswithdefectssuchasdislocationsandinterfaces,i.e.,processesofhet-erogeneousnucleation.Moreover,whilethefocusofthisletterisontheprecipitatenu-cleationandtheequilibriumstate,themathematicalandcomputationalframeworkcanbepotentiallyappliedtootherconstrainedbarriercrossingproblemsinphysics,chem-istryandbiology,includingexampleslikethesaddlepointsearchforactivatedstatesinsolidstatediffusionusingdensityfunctiontheory,andthedeterminationofdomainmor-phologyofacriticalnucleusandaswitchedstateinferroelectricsolidsunderanappliedelectriceld.AcknowledgmentsThisresearchissupportedinpartbyNSF-DMS0712744,NSFDMR-0710483andNSF-IIP541674CenterforComputationalMaterialsDesign(CCMD). 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