物理冶金第12章-替代型固溶体在扩散VIP

Chapter12.DiffusioninSubstitutionalSolidSolutions,Diffusioninanidealsolution:Consideranidealbinarysystem(AandBatoms),thejumpsareentirelyrandom.Ajumpismadewhenavacancyisnextthetheatom(vacancymechanismformostcrystalstructureslikeFCC,BCC,andHCP)!Truerandomjump=jumprateisconcentrationindependent!Assumeasimplecubicstructurewithdirectionalongthejumpaxis;furtherassumetheconcentrationgradientisdnA/dx,gradientonlyexistinonedirection.=thegradientacrossplaneXandplaneYisadnA/dx.a:interatomicdistanceorlatticespacing;assumethemeantimeofstayofanatominalatticesitebe,=theattemptfrequencyfortheatomsjumpis1/.Forasimplecubicstructure,anatomcouldjumpinsixdirections.ForatomsinplaneY,onlyjumpsin-xdirectioncouldchangetheconcentration,whileforatomsinplaneX,onlyjumpsinxdirectioncouldchangetheconcentration.=theaveragefrequencyforusefuljumpsis1/(6).ThefluxofatomsfromplaneXtoplaneYisJXY:nA:concentrationofAatominplaneX(crosssection:A),a,PlaneX,PlaneY,highnA,lownA,x,TheconcentrationofAatomsinplaneYis,=thefluxofatomsfromplaneYtoplaneXisJYX:,=thedifferenceinfluxofatomsbetweenplaneXandplaneYis:,-signrepresentsthefluxisagaintheconcentrationgradient,i.e.,atomsmovefromhighconcentrationtolowconcentration(fortypicalcase,however,thereisspecialcasecallup-hilldiffusion!),Ficksfirstlaw,D:diffusivityordiffusioncoefficient;typicallyDisaconstantatafixedtemperatureforgassystem;whileDisseldomaconstantatafixedtemperatureforliquidorsolidsystems.See,Table12.1andFig.12.3.,TheKirkendallEffects:Diffusioncouple:heatedthesampletohightemperaturesforalongtimeandthencooleddown.Sectionedthesampleandmeasuredthecomposition=compositionasafunctionofsamplepositionatafixedtemperatureforafixedtime!,B,A,Wire:marker,IfAdiffusefasterthanB=markermovetowardAandviceversa.xissmallbutmeasurable;xt1/2.Kirkendalleffectconfirmsthatthevacancymechanismexistedfordiffusionofsubstitutionalsolidsolution.Why!Seebelow.*possiblemechanismsincludediffusionsinvolveasingleatom(vacancyandinterstitialsmechanism)andthoseinvolvecooperativemovementoftwoormoreatoms.,100%A,100%B,Originalinterface,0%A,0%B,CA,CB,x,DirectexchangediffusionmechanismWith2andsimultaneousrotationof3atoms(ringmechanism),Zenerringmechanisminvolvefouratoms,*Theenergyrequiredformechanismsinvolvedirectexchangeof2atomsaremuchlargerthanthatforthesingleatom.=notanimportantmechanisminthediffusionofmetals.*Theringmechanismshavereasonableenergiesinvolved=feasiblemechanism,especiallyforBCCmetalthestructureismoremoreopen!Butnoexperimentaldatatodirectproveityet!,Vacancymechanism=differentdiffusionrateforAatomsandBatoms(why!)=markermove!,Whileringmechanism=samediffusionrateforAatomsandBatoms=markerwontmove!But,evenindiffusioncouplesofBCCmetals,markermove=evenringmechanismispossible,vacancymechanismstillinvolvesinthediffusionofBCCmetal!Experimentaldatashownthattheelementwiththelowermeltingpointdiffusesfaster!(weakerbonding)=unbalanceflowofmass(volume)=regionofcompressionandregionoftension(accompaniedbypores),A,B,FasterB,*Onthesideofthefastermovingspecies=masslosingmorethanarriving=lessmaterialsinthesamevolume=tension*Onthesideoftheslowermovingspecies=massarrivingmorethanlosing=morematerialsinthesamevolume=compression,Theformationofporescomesfromthefactthatthefluxofvacancyisoppositetothatoftheatom=morevacanciesonthesideofslowerspecies(super-saturationisonly1%abovetheequilibriumvacancyconcentration)=segregation=voidsorpores(believedtobeheterogeneouslynucleatedwiththeassistanceoftensilestress)!Thedifferentsourcesforvacancycreationandannihilationaresurfaces,grainboundaries,anddislocationclimb(mostimportantone);=vacancycreationbydislocationclimbandannihilationbyvoidformation.,DimensionalconstraininArea,notinthedirectionofflux,DarkensEquations:Diffusioncouplecomposedoftwometalsmoveatdifferentrates=onefluxforAatomandtheotherfluxforBatom,;DAandDBarecompositiondependent!,Assumptions:(1)allofthevolumeexpansionandcontractionoccursinthedirectiontotheweldinterface;(2)nA+nB=constant(inunitvolume)(thisassumptionisnottruebutcouldbecompensatedbycomputation);(3)porositydoesnotoccurinthespecimenduringthediffusionprocess.*Velocityofthemarker:,x,x,t0,t0+t,*,;,TotalfluxofAatoms:(1)diffusionfluxand(2)Theinterfaceismovingwithavelocityv.,x,x,dx,Diffusionflux,*Thefluxthroughxperunitarea:,Thefluxthroughx+dxperunitarea:,=Therateatwhichthenumberofatomsinsidethevolumedxunitareachangesis:,Fickssecondlaw:Basicequationforstudiesofisothermaldiffusion,*,Ifisconstant(indilutesolutionordiffusioninsystemswithsmallconcentrationgradient.,Grubemethod:i.e.constantD.,Initialcondition:,DiffusionofSemi-infinitemedium:,Multiplyingbothsidesby,Laplacetransform:theLaplacetransformoff(t)isdefinedas,p:anumberlargeenoughtomaketheintegralconverge,Someexamples:,Laplacetransform,Initialcondition:,Assume,Findinmathtable,SolutionoftheequationforthecaseofadiffusioncoupleconsistingoriginallyoftwoalloysofelementAandB,onehavingthecompositionNA1andtheotherthecompositionNA2atthestartofthediffusionprocess:,for-,*Diffusionequation=,A1,A2,x=0,NA,x=0,NA,C1,NA1,*Procedures:TodeterminethediffusivityatconcentrationNA:(i)definetheMantanointerface(A1=A2=thepositionofx=0);(ii)findtheslope(dNA/dx)atNAand(ii)findtheareainthecross-hatchedregion=computeusingequation!Seetextbookforanexample!SeeFig.12.15forconcentrationdependenceofof.,*Problem:Uncertaintyatnearthelimitingvalueisverylarge!Therearewaystoimproveit.See,J.Crank,“TheMathematicsofDiffusion”,2ndEd.,OxfordUniversityPress,Ch.10,1975.,DeterminationoftheIntrinsicDiffusivities:Toobtaintheindividualdiffusivityinthediffusioncouple:coupletwoequations,oneisrelatedtothemakervelocityandtheotherrelatedtothediffusivity.,*Markervelocity:,*Diffusivity:determinedfromtheMatanomethod,=DAandDB.,SelfDiffusioninPureMetals:Studiesofasoluteconsistingofaradioactiveisotopeinasolventthatisanonradioactiveisotopeofthesameelement.Theisotopesarechemicallyidentical=thesolutionistrulyideal,isotopeshavedifferentmasses=differentjumprate(theresonancefrequencyofaspring(C/m)1/2,seechapter3).Thedifferenceisreallysmall!,Toagoodapproximation,theintrinsicdiffusivityareequal=interdiffusioncoefficient=intrinsicdiffusivity,PreviouslyobtainedDinsimplecubicstructure(randomwalks),Ingeneral,*assumethevacancymechanismisthemajorprocessfordiffusion!,*thevacancyjumprateis,qf:enthalpyofformation;qm:enthalpyofmigration;,AcouldbereplacedbyZwhere:attemptfrequency;Z:coordinationnumber=Z:totalnumberofattempt!,*includeentropyterm=,=1/Z,TemperaturedependenceoftheD:,lnD,1/T,0,Slope=-Q/k,ChemicalDiffusionatLow-soluteconcentration:system:verylittleBinAelement=NB0,NA1,=GetinformationaboutdiffusionofBelementDB,QB.,SeeTable12.5.,Sofar,allcasesareconsideredtobethecaseofdiffusioninidealsolidsolutions.,FicksFirstLawintermsofaMobilityandanEffectiveForce:Ficksfirstlawforoneofthecomponents(A),WhereD*AisthediffusivityofAinanidealsolution(equivalenttothatmeasuredwithradioactivetracersinasolutionofconstantchemicalcomposition),NA:molefraction,B:mobility,Fornon-idealsolution,TheStudyofChemicalDiffusionusingRadioactiveTracers:Thediffusioninannon-idealsolidsolution.,Ax1By1alloy,Ax2By2alloy,SomeBisradioactive,DAandDBaretheintrinsicdiffusivity;D*AandD*Barethetracerdiffusivityfromtheselfdiffusionexperiment!,*SeeFig.12.18:thetracerdiffusioncoefficientofNiandAu;Fig.12.19:thethermodynamicfactor=combiningdatafromFig.12.18andFig.12.19=Fig.12.20.,DiffusionalongGrainBoundaryandFreeSurface:Ds:surfacediffusivity;Db:grainboundarydiffusivity,Typically,diffusioninsurfaceisfastestandthenfollowedbythegrainboundarydiffusion(lowerqsandqb);howeverthesitedensityislimited(lowDs0andDb0).Thesurfacediffusionismoreimportantinthinnerfilms!Diffusioninpolycrystallinesampleisacombinationoflatticediffusionandgrainboundarydiffusion.SeeFig.12.21.Thetotaldiffusivityis,qb:activationenergyforgrainboundarydiffusion;qs:activationenergyforsurfacediffusion;,DL0Db0=DLdominatesathighTandDbdominatesatlowT!,1/T,lnD,DL,Db,Apparentdiffusivity,DiffusioninNon-isomorphicAlloySystems:Assumeasystem(AandBcomponents)withasingleintermediatephase()andtwotermina