扩散控制球形沉淀的溶解

SaiptaMetallurgieaeiMat&h,Vol.32,No.5,pp.787-j91,1995copyright01995ElsevierScienceLtdprintedintheUSA.Allrightsreserved0956-716X95$9.50t.OODIFFUSION-CONTROLLEDDISSOLUTIONOFASPHERICALPRECIPITATEINANINFINITEBINARYALLOYN.NojiriandM.EnomotoDepartmentofMaterialsScience,FacultyofEngineering,IbarakiUniversity,Hitachi316,Japan(ReceivedJune10,1994)(RevisedOctober6,1994)IntroductionIntheanalysesofthekineticsofdiffusion-controlledphasetransformationtheavailabilityofanalyticalsolutionsfordissolutionislimitedcomparedtothoseforgrowth.Thedifficultyseemstoarisefromthefactthatonehastodealwithanon-zeroinitialradiusinthedissolutionofprecipitates.Toalleviatethisdifficultyseveralapproximateanalyses,i.e.invariant-field,invariant-size(stationary-interface,SI)andlinearizedgradient(LG)approximationsetc.,weredevisedandtheaccuracyofeachapproximationhasbeendiscussedatlength(l-4).TheGreen’sfunctionmethodwasshowntobeausefulmeanstosolveafreeboundaryproblemofheatequation(5)andassuch,maybeappliedtotheanalysisofkineticsofvariousdiffusion-controlledphasetransformations.OneofthepresentauthorsappliedKolodner’smethodtothegrowthofprecipitatesinafinitematrix(6).Themethodinvolvesnumericalintegrationofanintegro-differentialequationandpermitstheboundaryvelocityandadvancementtobeobtainedwithremarkableaccuracy.ThepurposeofthisnoteistoobtainanumericalsolutionfordissolutionofprecipitatesinaninfinitelylargematrixfromtheKolodner’smethodandtocomparewiththeapproximatesolutionsheretoforeutilized.Thesameequationscanbeusedforthegrowthofsphericalprecipitateshavinganon-zeroinitialradius,theresultsofwhicharealsoincluded.CalculationMethodUsingdimensionlessquantities,undertheassumptionofconstantdiffusivityofsolute,thediffusionequationforradiallysymmetricdissolution(orgrowth)inthreedimensionsiswrittenas,(1)whereR=r/r,,,T=Dt/r,’andu=(C-C,,)/(C,-C,)arethenormalizedradialcoordinate,timeandsoluteconcentrationinthematrix,respectively.r,r,,tandDare‘theradialcoordinate,initialradiusofprecipitate,realtimeandsolutediffusivity,andC,andC,arethesoluteconcentrationinthematrixatboundariesandatinfinity,respectively.Theconditionoffluxbalanceatboundariesiswrittenas,I;=&?(1aRP(2)whereS=(C,-C,)/(C,-C,),andC,istheconcentrationofsoluteintheprecipitates,andP(T)istheprecipitatesradius.787788DISSOLUTIONOFPRECIPITATEVol.32,No.5OnputtingU=uR,e+.(l)and(2)become,aua2u-=-aTaPand,au*-=!IaRP-F+l=g(T)(3)(4)respectively.UndertheassumptionthatC,isconstant(capillarityisneglected),lJ=p=f(r)(5)atR=pand,U=Q(6)atinfinity.AccordingtoKolodner(S),giventheconditiononUanditsderivativeaU/aXatanunknownmovingboundary,thesoluteconcentrationinthematrixcanbeexpressedas,U(R,T)=h’(G'f-G,’(g+fP)lds(7)where,G(R,p,T,t)=!1Tr2dm0%0TstistheGreen’sfunctionofdiffusionequationinonedimension.G,isthederivativeofGwithrespecttoP.f(T)andg(T)representther.h.s.ofeqs.(4)and(5),respectively.WCdefineU’intheoriginaldomainandU-inavirtualdomainontheoppositesideoftheunknownboundaryas,U’(R,T)=U(R,TI*;fVl.(9)OnthereductiontheoremtheboundarypositionP(T)isobtainedbysolvingtheequationU,-=O,whichreducesto(5,6),U,Q(P,‘l?=$-I.(10)Inthiscaseeqs.(7)and(9)areexplicitlywrittenas,U=-@@@(TM))+Rj-;zrl(o)do-$arp(~)q(Z(~r))~~(11)and,Vol.32,No.5DISSOLUITONOFPRECIPlTATE789P(T)P(T)=-2q_z(p(nso)q(o)do-;Tq(z(p(TLo)+J,~P(~-~r)Z(p(T),T)S(Z(P(T),c))dr(12)respectively.Hcrc:,rl(O)andZ(R,r)aredefinedas,q(a)=Lxp(-02)andZ(Rr)=R-p(r)fi2$fTT(13)respectively.Eq.(12)canbesolvednumericallybysuccessivelyincreasingtheupperlimitofintegralbyasmallamount,AT.Atveryinitialtimesthemotionofsegmentsofsphericalshellcanbeapproximatedbythatofaplanarboundary.Hence,Pisreplacedbythesolutionofdissolutionofaplanarboundaryas,p=l-210(14)fornumericalintegrationnearthelowerlimitofintegral,where1satisfiesthefollowingequation,a=J;;Aexp(J.2)erfc(-I).(15)Thedetailsofthecalculationprocedurehavebeenshownelsewhere(6).Rqs.(ll)and(12)canbeusedtoanalyzethegrowthofprecipitatesofaninitiallynon-zeroradius.Followingtheconvention,11isnowdefinedasn=(C,-C&C,-C,)forgrowth.Hence,5!variesfromzerotoinfinityindissolutionandfromzerotounityingrowth.Incontrasttothefinitedifferencecalculationtheboundaryvelocityisobtainedsuccessivelywithoutcalculatingthediffusionfieldinthismethod.Thediffusionfieldisobtainedfromeq.(ll)atanyspecifiedtimeduringthecalculationofboundaryvelocity.Stationary-interface(SI)approximation-Amongtheapproximatesolutionsthestationary-interfaceapproximationisconsideredasthebestinmostcasesofpracticalinterest(3,7).Usingthecurrentnotation,undertheSIapproximationtheprecipitateradiussatisfiesthefollowingrelation,h(p2+2pJTip+T’)=-A43wherep2=R/2zandT=2nT(3).Itcanbeshownthatthecorrespondingequationforgrowthbecomes,ln(p2-2p@p-T’)=-pinWi&Y-EP-p+Jq@wherepandTarcdcfincdinthesamemannerasabove.(17)790DISSOLUTIONOFPRECIPITATEVol.32,No.5TheLGapproximationwasnotinvokedbecausetheextentoflinearizeddiffusionfieldbecomesacomplexfunctionofprecipitateradiusandthus,thetimedependenceofprecipitateradiusmaynotbeobtainedinananalyticalform.ResultsKineticsofdissolution-Figures1and2showthecalculatedvariationwithtimeofprecipitateradiusa