Chapter4Diffusion

WHYSTUDYDiffusion?Materialsofalltypesareoftenheat-treatedtoimprovetheirproperties.Thephenomenathatoccurduringaheattreatmentalmostalwaysinvolveatomicdiffusion.Often,anenhancementofdiffusionrateisdesired;onoccasion,measuresaretakentoreduceit.Heattreatingtemperaturesandtimesand/orcoolingratescanoftenbepredictedbyusingthemathematicsofdiffusionandappropriatediffusionconstants.LearningObjectivesAfterstudyingthischapter,youshouldbeabletodothefollowing:1.Nameanddescribethetwoatomicmechanismsofdiffusion.2.Distinguishbetweensteady-stateandnonsteadystatediffusion.3.(a)WriteFick’sfirstandsecondlawsinequationformanddefineallparameters.(b)Notethekindofdiffusionforwhicheachoftheseequationsisnormallyapplied.4.WritethesolutiontoFick’ssecondlawfordiffusionintoasemi-infinitesolidwhentheconcentrationofdiffusingspeciesatthesurfaceisheldconstant.Defineallparametersinthisequation.5.Calculatethediffusioncoefficientforamaterialataspecifiedtemperature,giventheappropriatediffusionconstants.4.1INTRODUCTIONdiffusion:thephenomenonofmaterialtransportbyatomicmotion.perfectmixingpartiallymixedtimedyewaterThisresultindicatesthatcopperatomshavemigratedordiffusedintothenickel,andthatnickelhasdiffusedintocopper.Theprocessbywhichatomsofonemetaldiffuseintoanotheristermedinterdiffusion,orimpuritydiffusion.Diffusionalsooccursforpuremetals,butallatomsexchangingpositionsareofthesametype;thisistermedself-diffusion.4.2DIFFUSIONMECHANISMSFromanatomicperspective,diffusionisjustthestepwisemigrationofatomsfromlatticesitetolatticesite.Infact,theatomsinsolidmaterialsareinconstantmotion,rapidlychangingpositions.Foranatomtomakesuchamove,twoconditionsmustbemet:

(1)theremustbeanemptyadjacentsite(2)theatommusthavesufficientenergytobreakbondswithitsneighboratomsandthencausesomelatticedistortionduringthedisplacement.Ataspecifictemperature,somesmallfractionofthetotalnumberofatomsiscapableofdiffusivemotion,byvirtueofthemagnitudesoftheirvibrationalenergies.Thisfractionincreaseswithrisingtemperature.twodominatemodelsformetallicdiffusionVacancyDiffusionInterstitialDiffusion(1)VacancyDiffusionOnemechanisminvolvestheinterchangeofanatomfromanormallatticepositiontoanadjacentvacantlatticesiteorvacancy,thisprocessnecessitatesthepresenceofvacancies,andtheextenttowhichvacancydiffusioncanoccurisafunctionofthenumberofthesedefectsthatarepresent;significantconcentrationsofvacanciesmayexistinmetalsatelevatedtemperaturesBecausediffusingatomsandvacanciesexchangepositions,thediffusionofatomsinonedirectioncorrespondstothemotionofvacanciesintheoppositedirection.(b)InterstitialDiffusiondiffusioninvolvesatomsthatmigratefromaninterstitialpositiontoaneighboringonethatisempty.Thismechanismisfoundforinterdiffusionofimpuritiessuchashydrogen,carbon,nitrogen,andoxygen,whichhaveatomsthataresmallenoughtofitintotheinterstitialpositionsInmostmetalalloys,interstitialdiffusionoccursmuchmorerapidlythandiffusionbythevacancymode,becausetheinterstitialatomsaresmallerandthusmoremobile.Furthermore,therearemoreemptyinterstitialpositionsthanvacancies;hence,theprobabilityofinterstitialatomicmovementisgreaterthanforvacancydiffusion.4.3FICK’SFIRSTLAWDiffusionisatime-dependentprocess—thatis,inamacroscopicsense,thequantityofanelementthatistransportedwithinanotherisafunctionoftime.Oftenitisnecessarytoknowhowfastdiffusionoccurs,ortherateofmasstransfer.Thisrateisfrequentlyexpressedasadiffusionflux(J),definedasthemass(or,equivalently,thenumberofatoms)Mdiffusingthroughandperpendiculartoaunitcross-sectionalareaofsolidperunitoftime.Inmathematicalform,thismayberepresentedasJ=-DdCdxTheunitsforJarekilogramsoratomspermetersquaredpersecond(kg/m2·soratoms/m2·s).dC/dxistheconcentrationgradient,TheconstantofproportionalityDiscalledthediffusioncoefficient,

whichisexpressedinsquaremeterspersecond.Fick’sfirstlaw.Thenegativesigninthisexpressionindicatesthatthedirectionofdiffusionisdowntheconcentrationgradient,fromahightoalowconcentration.(4.1)Fick’sfirstlawmaybeappliedtothediffusionofatomsofagasthroughathinmetalplateforwhichtheconcentrations(orpressures)ofthediffusingspeciesonbothsurfacesoftheplateareheldconstant.Thisdiffusionprocesseventuallyreachesastatewhereinthediffusionfluxdoesnotchangewithtime—thatis,themassofdiffusingspeciesenteringtheplateonthehigh-pressuresideisequaltothemassexitingfromthelow-pressuresurface—suchthatthereisnonetaccumulationofdiffusingspeciesintheplate.Thisisanexampleofwhatistermedsteady-statediffusion.WhenconcentrationCisplottedversusposition(ordistance)withinthesolidx,theresultingcurveistermedtheconcentrationprofile;furthermore,concentrationgradientistheslopeataparticularpointonthiscurve.Onepracticalexampleofsteady-statediffusionisfoundinthepurificationofhydrogengas.Onesideofathinsheetofpalladiummetalisexposedtotheimpuregascomposedofhydrogenandothergaseousspeciessuchasnitrogen,oxygen,andwatervapor.Thehydrogenselectivelydiffusesthroughthesheettotheoppositeside,whichismaintainedataconstantandlowerhydrogenpressure.H2+othergaseousc1palladiumH2EXAMPLEPROBLEM4.1Aplateofironisexposedtoacarburizing(carbon-rich)atmosphereononesideandadecarburizing(carbon-deficient)atmosphereontheothersideat700℃.Ifaconditionofsteadystateisachieved,calculatethediffusionfluxofcarbonthroughtheplateiftheconcentrationsofcarbonatpositionsof5and10mm(5×10-3

and10-2

m)beneaththecarburizingsurfaceare1.2and0.8kg/m3,respectively.Assumeadiffusioncoefficientof3×10-11m2/satthistemperature.SolutionFick’sfirstlaw,isusedtodeterminethediffusionflux.4.4FICK’SSECONDLAW—NONSTEADY-STATEDIFFUSIONMostpracticaldiffusionsituationsarenonsteady-stateones—thatis,thediffusionfluxandtheconcentrationgradientatsomeparticularpointinasolidvarywithtime,withanetaccumulationordepletionofthediffusingspeciesresulting.Concentrationprofilesfornonsteady-statediffusiontakenatthreedifferenttimes,t1,t2,andt3.Underconditionsofnonsteadystate,thepartialdifferentialequationknownasFick’ssecondlaw,isusedIfthediffusioncoefficientisindependentofcomposition,EquationsimplifiestoThislawstatesthattherateofcompositionalchangeisequaltothediffusivitytimestherateofchangeoftheconcentrationgradient.(4.2)(4.3)SolutionsoftheFick’ssecondlaw

Solutionstothisexpression(concentrationintermsofbothpositionandtime)arepossiblewhenphysicallymeaningfulboundaryconditionsarespecified.Onepracticallyimportantsolutionisforasemi-infinitesolid

inwhichthesurfaceconcentrationisheldconstant.Frequently,thesourceofthediffusingspeciesisagasphase,thepartialpressureofwhichismaintainedataconstantvalue.Furthermore,thefollowingassumptionsaremade:1.Beforediffusion,anyofthediffusingsoluteatomsinthesolidareuniformlydistributedwithconcentrationofC0.2.Thevalueofxatthesurfaceiszeroandincreaseswithdistanceintothesolid.3.Thetimeistakentobezerotheinstantbeforethediffusionprocessbegins.Initialcondition:Fort=0,C=C0at0≤x≤∞Boundaryconditions:Fort>0,C=Cs(theconstantsurfaceconcentration)atx=0Fort>0,C=C0

atx=∞Theseconditionsaresimplystatedasfollows:ApplicationoftheseconditionstoEquation,

yieldsthesolutionwhereCxrepresentstheconcentrationatdepthxaftertimet.

(4.4)

Concentrationprofilefornonsteady-statediffusion;concentrationparametersrelatetoEquationSupposethatitisdesiredtoachievesomespecificconcentrationofsolute,C1,inanalloy;theleft-handsideofEquationandsubsequentlytheright-handsideisalsoaconstantSomediffusioncomputationsarefacilitatedonthebasisofthisrelationship(4.5)EXAMPLEPROBLEM4.2Forsomeapplications,itisnecessarytohardenthesurfaceofasteel(oriron–carbonalloy)abovethatofitsinterior.Onewaythismaybeaccomplishedisbyincreasingthesurfaceconcentrationofcarboninaprocesstermedcarburizing;thesteelpieceisexposed,atanelevatedtemperature,toanatmosphererichinahydrocarbongas,suchasmethane(CH4).Consideronesuchalloythatinitiallyhasauniformcarbonconcentrationof0.25wt%andistobetreatedat950℃.Iftheconcentrationofcarbonatthesurfaceissuddenlybroughttoandmaintainedat1.20wt%,howlongwillittaketoachieveacarboncontentof0.80wt%ataposition0.5mmbelowthesurface?Thediffusioncoefficientforcarboninironatthistemperatureis1.6×10-11

m2/s;assumethatthesteelpieceissemi-infinite.SolutionBecausethisisanonsteady-statediffusionprobleminwhichthesurfacecompositionisheldconstant,Valuesforalltheparametersinthisexpressionexcepttimetarespecifiedintheproblemasfollows:Wemustnowdeterminethevalueofzforwhichtheerrorfunctionis0.4210.Aninterpolationisnecessary,asz=0.392EXAMPLEPROBLEM4.3Thediffusioncoefficientsforcopperinaluminumat500℃

and600℃

are4.8×10-14and5.3×10-13m2/s,respectively.Determinetheapproximatetimeat500℃

thatwillproducethesamediffusionresult(intermsofconcentrationofCuatsomespecificpointinAl)asa10-hheattreatmentat℃.EquationmaybewrittenasSolutionBecauseatboth500℃and600℃

thecompositionremainsthesameatsomeposition,sayx0withtheresultthator4.5EFFECTOFTEMPERATUREONDIFFUSIONINSOLIDSSinceatomdiffusioninvolvesatomicmovements,itistobeexpectedthatbyincreasingthetemperatureofadiffusionsystemwillincreasethediffusionrate.Byexperiment,ithasbeenfoundthatthetemperaturedependenceofthediffusionrateofmanydiffusionsystemscanbeexpressedbythefollowingArrhenius-typeequation:(4.6)where:D0=atemperature-independentpreexponential(m2/s)Qd

=theactivationenergyfordiffusion(J/moloreV/atom)R=thegasconstant,8.31J/mol·Kor8.62×10-5eV/atom·KT=absolutetemperature(K)Theactivationenergymaybethoughtofasthatenergyrequiredtoproducethediffusivemotionofonemoleofatoms.Alargeactivationenergyresultsinarelativelysmalldiffusioncoefficient.(4.6)TakingnaturallogarithmsofEquation4.6yields(4.7)or,intermsoflogarithmstothebase10,(4.8)BecauseD0,Qd,andRareallconstants,Equation4.8takesontheformofanequationofastraightline:whereyandxareanalogous,respectively,tothevariableslogDand1/T.Thus,iflogDisplottedversusthereciprocaloftheabsolutetemperature,astraightlineshouldresult,havingslopeandinterceptof-Qd/2.3RandlogD0,respectively.Thisis,infact,themannerinwhichthevaluesofQd

andD0aredeterminedexperimentally.

Fromsuchaplotforseveralalloysystems,itmaybenotedthatlinearrelationshipsexistforallcasesshown.EXAMPLEPROBLEM4.4UsingthedatainTable,computethediffusioncoefficientformagnesiuminaluminumat550℃.SolutionThisdiffusioncoefficientmaybedeterminedbyapplyingEquation4.6;thevaluesofD0andQd

fromTabl