材料热力学与动力学-2

Chapter2.DiffusionProf.Dr.X.B.ZhaoDepartmentofMaterialsScienceandEngineeringZhejiangUniversityChapter2:Diffusion2-2DiffusionProcess12ABGAB@T01G12G2x0G3G4mA1mA2mB1mB2Chapter2:Diffusion2-3Down-HillDiffusion12AB@T0G1G2G3G4mA1mA2mB1mB22AB1x0GCBdistanceInitialcompositiondistributiondown-hilldiffusionFinaldistributionChapter2:Diffusion2-4DiffusionandFreeEnergy12ABG@T0AB12G1G2G3G412Up-HillDiffusionmA1mA2mB1mB2Chapter2:Diffusion2-5Up-HillDiffusion12ABG@T0G1G2G3G4mA1mA2mB1mB2AB1122CBdistanceinitialcompositiondistributionup-hillinterfacefinaldistributionChapter2:Diffusion2-62.1AtomicMechanismsofDiffusion2.1.1VacancyMechanismGyaDGmChapter2:Diffusion2-7GyaDGm2.1.2InterstitialMechanismChapter2:Diffusion2-8ABCBy12ayCaBCB1CB2=CB1+yCaB2.1.3RandomJumpofAtomsChapter2:Diffusion2-9CB1twoadjacentatomicplaneswithunitareayCaBCB1+a12NumberofBatomsinplane:aCB1Jumpintoplanepersecond:aCB1·nB/6NumberofBatomsinplanejumpintoplanepersecond:6BB2B1n+yCaaCyCajnB2BB61NetfluxofBatomsfromtopersecond:Substituting2BB61aDnyields:Fick’sfirstlawofdiffusionyCDBBBjChapter2:Diffusion2-101-Dmodel:alargenumberof(sayN)Batomsinthelocalofy=0att=0,andtheywilljumponceperDtwiththedistanceofdforwardorbackwardrandomly.yCt=0t=t1t=t2(t2t1)2.1.4StatisticsoftheRandomJumpProcessChapter2:Diffusion2-11Sinceatanytime,wecalculate,wehave:0)(ty)(2ty2222)(2)()()(yyttyttyyttytyDDDDDD+++Dy:distanceperjumpofanatom,Dy=±d=d20)()()(BACKWARDFORWARDttyttyNdyttyDDDD=022)(dtty+DThismeanswillincreased2iftheatomsjumponetime.Ifthejumpfrequencyoftheatomsisn,anatomwilljumpnttimesfromt=0tot=t,andthenwehave:)(2tytdtyn22)(Einsteinhasdemonstratedthatd2nequalsto2Dforonedimensionalandto4Dand6Dfortwo-andthree-dimensionalcaserespectively.Dty22Averagediffusiondistance(equivalently)Chapter2:Diffusion2-12RTQDDexp0Arrhenius’law2.1.5ThermalActivationChapter2:Diffusion2-13DtSincehasthesamedimensionofdistance,a=L2/(Dt)isadimensionlessnumber,whereLisacharacteristiclengthinadiffusionsystem.TheavaluesmustbesameforanysimilardiffusionsystemsaccordingtothePrincipleofSimilarity.[Example]:Afterbeingheat-treatedat900ºCfor10minutes,theZn-contentofaCu-Znsampleat1mmdepthunderthesurfacehasdecreasedtosomeextend,sincetheboilingpointofZnisrelativelow(907ºC).Howlongtimeitwillbe,afterwhichtheZn-contentat0.5depthwilldecreasetothesameextendifthesampleisheatedat800ºC.Sincea900=a800,or10732117325.0101DtDandRTQDDexp0Wehave:RQRQt11731073exp5.2(min.)IfQ=20000R,t12(min.)2.1.6DimensionalAnalysisChapter2:Diffusion2-142.2.1DiffusionthroughthecylinderwallC2C1H2Lengthofthevessel:LForsteady-statediffusionthefluxthroughacylinderwallwithradiusofr(r1rr2)isaconstant.Const.2drdCrLDdtdmor21212CCrrdCLDrdrdtdm1212ln2rrCCLDdtdmFick’sfirstlawinCylinder-SystemrCC1C2Notlineardistributed!If(r2-r1)r1(thinwalled),ln(r2/r1)=ln(1+(r2-r1)/r1)=(r2-r1)/r1121212rrCCDLrdtdm2.2Steady-StateDiffusionChapter2:Diffusion2-15TT2CC2aqAfterbeinghomogenizedatT2,alloyC2iscooledtoT1.qparticlesareprecipitated.T1CqC1r1yinterfaceCompositionsofqparticlesandamatrixaredeterminedbythephasediagram.qatq/aaata/qafarawayThereisacompositiongradientinanearq,whichleadstothediffusionofsoluteatomsfromfarawayplacestothea/qinterface.Whereinathecompositionlowerthanfarawayiscalled“solute-poorzone”(radiusr2).r2Thediffusioninthesolute-poorzoneissimilartothatthroughathickwallsphericalshell:1122112211221121221444rCCDrrrrrCCDrrrCCDrrdtdm1sincer2r1pseudodiffusionareapseudodiffusiondistance2.2.2DiffusionduringprecipitationChapter2:Diffusion2-16200160014001200100080060040001020304050607080901000102030405060708090100gaLd1538ºC1514ºC1394ºC1440ºC1455ºC912ºC770ºC517ºC347ºC354.3ºC66(FeNi3)47496372TCaTCgatomicpercentnickelweightpercentnickeltemperature,ºCNiFe2.2.3DiffusioninmultiphasesystemsChapter2:Diffusion2-170102030405060708090100ga(FeNi3)300ºCxNiyFeNit=0(FeNi3)t0Chapter2:Diffusion2-182.3.1Fick’sSecondLaw22yCDtCtwo-orderpartialdifferentialequationinfinitesolutions,ifany,butthesimplestoneshouldbeusedboundaryconditionsandinitialconditionsareneedlinearcombinationofsomesolutionsisalsoasolutionFick’sFirstLaw&MatterConservationLawmustbesatisfied2.3Fick’ssecondLawanditsApplicationsChapter2:Diffusion2-19infinitelylongsampleGaussFunction(Mathematics)Formulam222exp21xymMathematicalExpectationVarianceRootintegration1+ydxSoluteElementLocationofSoluteatt=00CharacteristicLengthDt2AmountofSoluteSDtyDtSC42exp42.3.2theGaussFunctionSolutionChapter2:Diffusion2-20A.Gauss-FormIInfinitelyLongSampleAmountSiny=0att=0DtyDtSC42exp4yCt=0t=t1t=t2(t2t1)Chapter2:Diffusion2-21B.Gauss-FormIIHalfInfinitelyLongSampleAmountSiny=0att=0(1)UndeterminedCoefficientDtyDtBC4exp42LetwehaveB=2S,sinceSCdy0DtyDtSC42exp42(2)FictitiousSampleRealSFictitiousS+2SRealFictitious(3)MirrorImageThe“fictitioussample”isnotnecessary.IfaMirrorisputaty=0,wehavean“ImageSample”identicaltotherealsampleiny0.Thenweconsidertheimageandtherealsampleasawh