材料科學基礎下英文chapt3 11b

材料科學基礎（II）Chapter3SolidificationandCrystallization-IISomecontentsareadoptedfromPPTby:ProfessorsM.E.GlicksmanandAfinaLupulescu材料科學基礎（II）DrivingforcefornucleationofonephasefromasolutionwillbeconsideredinChapter5.Inadditiontothefactorsthatlimitorpromotenucleationinpuremetals,chemicalcompositionfluctuationanddiffusionisanotherfactortoaffectthenucleationrate.§3.GrowthofsinglephasefromliquidalloysDiffusion-ControlledGrowth?Growthlimitedbytherateofdiffusion–Howfastcantheneededatomsbedelivered–Howfastcantheheatbecarriedaway–Howfastcantheatomsbereceivedbycrystalphase材料科學基礎（II）solidliquid§3.GrowthofsinglephasefromliquidalloysxSLTieline“jump”solidusliquidusCBTEquilibriumatS/LInterfacesLocalequilibriumassumption:EquilibriumcompositionalwaysmaintainsintheS/Lboundary.Itmaynotbesointhewholesystem,especiallynotinthesolidphase.From:M.E.GlicksmanandAfinaLupulescu2004右側示意圖中合金凝固界面推移的驅動力有多少？材料科學基礎（II）Partitioncoefficient:k0=CS/CL

(分配系數)Assumingphasefieldboundarytobeastraightline假想相圖，k0=Cs/CL為常數CSoluteLiquidusC0Cmaxk0C0C0/k0C

T

T3T2T1CSCLSolidusTAA§3.Growthofsinglephasefromliquidalloysk0<1不重要很重要：材料科學基礎（II）Specialcasesof1Dgrowth（三種特殊條件的一維模型描述）

（假設單位面積的長棒）T2溫度下，相應的完全平衡成分，液相增加和固相減少的溶質要守恒，即兩個陰影部分面積應該相等。C0T

T3T2T1CSCLC=massperunitvolume(單位體積中質量)assumingequaldensityintwophases1.Equilibriumfreezing固相中的溶質數=Ms=Cs

z1液相中的溶質數=ML=CL

（L–z）1T固相液相熱量CLC0CS(a)(b)單位體積中質量zL長度材料科學基礎（II）NodiffusioninsolidLocalequilibriumatS/Linterface

CS=K0CLDifferentialmassbalance:(CL–CS)dz=(L-z)dCLBoundarycondition：

z=0,CL=CoCC0k0C0SolidLiquidCC0k0C0SolidLiquidCLzLCSdzdCLstartDistancealongbar,z2.Perfectmixinginliquid

G.Gulliver(1921)andE.Scheil(1942)材料科學基礎（II）IntegrationleadstoScheileqn:

p501

non-equilibriumleverrule!!,

wherefs=z/L,fL=1–fsCompleteliquidmixingCL=C0fL(ko–1)zTaveragecompositionofsolidfordifferentTduetonon-equilibriumsolidificationk0<1,fL→0,CL

→∞,

Cs=k0CL

→∞CL

islimitedbyphasediagramC0CmaxC

SolidCC0k0C0finishCs=k0C0(1–fs)(ko–1)從成分可知結晶前沿的T材料科學基礎（II）Gulliver–ScheilSegregationEquationPurermaterialFrom:M.E.GlicksmanandAfinaLupulescu2004材料科學基礎（II）Gulliver–ScheilSegregationEquationFrom:M.E.GlicksmanandAfinaLupulescu2004材料科學基礎（II）3.Steady-state(liquiddiffusioncontrolled)freezing

§3.GrowthofsinglephasefromliquidalloysstartfinishxBoundarylayerTerminaltransient終止瞬態(tài)Initialtransient初始瞬態(tài)compositionTheboundarylayerstartsto

buildupasfreezebeginssteadystate穩(wěn)態(tài)From:M.E.GlicksmanandAfinaLupulescu2004材料科學基礎（II）startfinishsteadystate

穩(wěn)態(tài)bowwave船頭波CC0k0C0SolidLiquidXmaxInitialtransientX

CC0k0C0SteadystateDistance0CC0k0C0SolidLiquidFinaltransientD/vD/vC0/k03.Steady-state(liquiddiffusioncontrolled)freezing

材料科學基礎（II）AssumptionLocalequilibriumattheS/Linterface.Neglectdiffusioninsolid.Assumesteady–statesolidificationataconstantspeedInacoordinate,x,(x>0)movingwiththeS/Linterface1DModelofDiffusion–LimitedSteady–Statesolidification材料科學基礎（II）AtsteadystateBuildanequation

dx=RdtR=migrationspeedChangeofdCatxfordtduetointerfacemigrationis

CL(x,t+dt)–CL(x,t)=CL(x–dx,t)–CL(x,t) =–(

C/

x)dx=–(

C/

x)Rdtduetodiffusionis

CL(x,t+dt)–CL(x,t)=(

C/

t)dt=D(

C2/

x2)dtCL(x,t+dt)=CL(x–dx,t)SecondFick’sLawCC0k0C0SolidLiquid0xC0/k0t+dttdCLR-dxx材料科學基礎（II）Solvedfrom(9-32)

D

C2/

x2=–R

C/

x,atboundaryconditions:x=0,CL=C0/k0

andx=∞,CL=C0Onegets

CL(x)=C0[1+((1–k0)/k0)exp(-Rx/D)]p502Eq(9-33)D/R

=characteristicwidthcharacteristicgradientdCL/dxx=0=(C0/k0–C0)/(D/R)SolutionforCL(x)CC0k0C0SolidLiquidD/R0xC0/k0RCL(x)材料科學基礎（II）EffectsofConvectionandStirringDuringcrystalgrowthandplane–frontsolidificationstirringoccurs,whichaffectsthesegregationprocess.ThiswasanalyzedbyBurton,PrimandSlichter(BPSTheory).Theysuggestedthreeimportantzones:Solidzone(nodiffusion)A“static”layer,d,(diffusiontransport)Liquidzone(completemixing)From:M.E.GlicksmanandAfinaLupulescu2004材料科學基礎（II）4. Freezingwithpartialmixinginliquidp502,Fig.9-28

(convectioneffect,boundarylayerapproach)固體液體k0C0C0無聚集有聚集聚集(CS)i(CL)i(L)Bt1t2t3t4t5(CL)i初始狀態(tài)(a)液體中溶質的聚集對凝固圓棒的成分影響(b)初始瞬態(tài)內溶質聚集的建立接近真實結晶狀態(tài)的模型，目前研究仍然采用材料科學基礎（II）4. FreezingwithpartialmixinginliquidSet: (Cs)i=k0(CL)i,(Cs)i=ke(CL)b, (Cs)i=kL(CLst)x

ke=effectivepartitioncoefficientInanalogytosteady-stateCLst

(x)=C0[1+(1–k0)/k0)exp(-Rx/D)]kL=(Cs)i/(CLst)x=(C0)/(CLst)x=1/[1+((1–k0)/k0)exp(-Rx/D)]kL=k0/[k0+(1–k0)exp(-Rx/D)]ke=(Cs)i/(CL)b=k0/[k0+(1–k0)exp(-R

/D)]K0C0CLCsC0C(Cs)iLocusof(CL)it=0tot(finish)

Instant(CL)atti液體有部分混合時的溶質分布CLs(CL)b材料科學基礎（II）InanalogytoScheileqnforcompleteliquidmixing,Letkereplacek0

CL=C0fL(ke–1) Cs=keC0(1–fs)(ke–1)CLinarealsystemisaffectedbyvariousfactorsdvaluedependsonstirring

keCsType0k0k0C0(1–fs)(k0–1)Completeliquidmixing0<

<∞k0<ke<1keC0(1–fs)(ke–1)Partialliquidmixing∞1C0DiffusiononlyFasterstirring§3.Growthofsinglephasefromliquidalloys材料科學基礎（II）固體液體C0k0C0(CL)i(CL)B初始瞬態(tài)ke=1濃度C0k0C0(CL)i(CL)Bke=k0濃度無聚集無瞬態(tài)C0k0C0(CL)i(CL)B初始瞬態(tài)K0<ke<1濃度凝固試樣中溶質的濃度分布Comparison

有效分配系數ke值不同時，溶質的分布情況,p503C0K0C00濃度已凝固體積的分數abcd=0,ke=ko=∞,ke=1Howtoobtainpurersolids?材料科學基礎（II）zonerefinement

區(qū)域熔煉示意圖MichaelF.AshbyandDavidR.H.Jones1998材料科學基礎（II）區(qū)域熔煉1次到1000次后的溶質濃度示意圖多組加熱器可提高區(qū)域熔煉效率MichaelF.AshbyandDavidR.H.Jones,1998材料科學基礎（II）§4.Constitutionalsupercooling成分(組分)過冷(amilestonecontribution)Assumeliquidusasastraightline: dTL/dCL=-m TL=TA–mCL

T1=TA–mC0 T3=TA–mCo/k0 T1–T3=m(C0/k0–C0) C0kC0C0/k0C

T

T3TLCSCLTAT1材料科學基礎（II）ConceptofconstitutionalsupercoolingAssumeequilibriumintheinterface,atsteadystate:ConstitutionallysupercooledstableCL(x)TL(x)T

(x)vvIfTofliquidisbelowitsliquidus

(constitutionallysupercooled)thebumporprotuberanceextendingintotheliquidgrows.IfTofliquidisaboveitsliquidus,anybumpmeltsback.T1T3Co/k0Co溶液的液相線溫度實際溫度梯度G材料科學基礎（II）DistancexT3GCriticalgradientSolidLiquidTConstitutionalsupercoolingconditionD/RvT1TC0CC0/k0CLTheoriginofcomputationalsupercoolingaheadofaplanarsolidificationfront(a)Compositionprofileduringsteady-statesolidification.(b)ThetemperatureoftheliquidaheadofthesolidificationfrontfollowslineTL.Criticalgradient:characteristicwidthdCL/dxx=0=(C0/k0–C0)/(D/R)TLToremembertheconditionneeds:characteristicwidthD/Randdefinitionofk0andm可控參數材料參數產生成分過冷的判據：G<dTL/dxx=0材料科學基礎（II）§5.Eutecticsolidification§6.Controlandsolidificationofingot§3.Growthofsinglephasefromliquidalloys1.Equilibriumfreezing2.Perfectmixinginliquid(Scheilmodel)3.Steady-state(liquiddiffusioncontrolled)freezing4.Freezingwithpartialmixinginliquid

溫度對以上模式有什么的影響?§4.Constitutionalsupercooling合金凝固時產生枝晶原因必須是液相中有負溫度梯度嗎？

Whydendritesareoftenpresentinsolidifiedalloys!May12,實際溶液中成分過冷上述模型更容易或困難？材料科學基礎（II）

t1t2t3t4t5(cL)i初始狀態(tài)ConstitutionalsupercoolingconditionformoregeneralcasesT (Cs)i=k0(CL)i,(Cs)i=ke(CL)b,(CL)i=ke(CL)b/k0=(CL)b/[k0+(1–k0)exp(-R

/D)] ke=k0/[k0+(1–k0)exp(-R

/D)]ReplaceC0/k0with(CL)i=f((CL)b)Whend

∞,

twoexpressionsareidenticalD/RvC0CC0/k0CLcharacteristicwidth實際溶液中成分過冷更容易或困難？材料科學基礎（II）CellularanddendriticsolidificationduetoconstitutionsupercoolingandrelevantsegregationSolidLiquidTLl(x)xLiquidT

(x)T(x)TLl(x)SolidTs(x)TL(x)>T(x),themeltisthermodynamicallystable,andthesolidadvanceswithplanarinterface.TL(x)<T(x),themeltesconstitutionallysupercooled,andthesolidadvanceswithcellularinterface.From:M.E.GlicksmanandAfinaLupulescu2004材料科學基礎（II）CellularmicrostructuresPb-Sn正視側視四溴化碳Sn-0.1wt%SbAdecantedinterfaceofacellularlysolidifiedPb-SnalloyLongitudinalviewofcellsincarbontetrabromatedFrom:M.E.GlicksmanandAfinaLupulescu2004From:PorterandEasterling1992Hexagonalcells(Sn-Pb)材料科學基礎（II）Cells&DendritesAsabumpgrows,itcandevelopasequenceofdisturbancesontheS/Linterface.Thesequicklyeshallowcells,thendeepcells,andeventuallydendrites.From:M.E.GlicksmanandAfinaLupulescu2004材料科學基礎（II）SimulatedgrowthofPb-Sndendritesshowing(a)compositionfield;(b)morphologyofPb-1wt%Snalloy(c)compositionfieldand(d)morphologyofPb-5wt%Snalloy.Botharegrowingina10K/mmtemperaturegradientwitha100mm/svelocityRichinsolutesanddefects,harmfultoopticalpropertyofsinglecrystalS.Raghavanetal2003Cellulardendritesincarbontetrabromated四溴化碳（透明有機物）胞狀枝晶材料科學基礎（II）Timeevolutionofthesolid/liquidinterfacemorphologywhenacceleratingthegrowthratefrom0to3.4m/satatemperaturegradientof6.7K/mm(a)50s(b)55s(c)65s(d)80s(e)135s(f)740sKostorzedited,2001材料科學基礎（II）Oftenitisdifficulttoavoidconstitutionalsupercoolinginpracticebecausevelocitytoavoiditisverylow.Directionalsolidificationwithaplanarfrontispossibleonlyatlowgrowthrates(likesinglecrystalSi).Inmostcases,theinterfaceisunstable.G/√RInterfacemorphologydendriticcellularplanarC0影響晶體生長方式的主要因素empiricalcurvep505,Fig.9-33§4.Constitutionalsupercooling材料科學基礎（II）0.41Cu-NiWARRENandW.J.BOETTINGER,ActaMater.1995ThemicrostructureofaCu-30%Nialloyshowingacoredstructure材料科學基礎（II）Dendritesinanaluminumalloy(x50).DonaldR.Askeland,P.Phule2003Dendritesinasteel(x25).材料科學基礎（II）SolidificationmicrostructuresDendrites(Co-Cr)Ascanning-electronmicrographshowingthedevelopmentofdendritesinanickel-basedsuperalloysingle-crystalweld..A.David,S.S.Babu,andJ.M.Vitek,JOM,2003材料科學基礎（II）Ahypoeutecticstructurefromthecopper-silverphasediagramcontainingapproximate24%copper.Lightoverregionsareproeutecticdendrites,whilethegra