材料科学基础辅导与习题-上交课件 二元相图及其合金的凝固

Ch4二元相图及其合金的凝固4.1相图的表示和测定方法4.2相图热力学基本要点4.3二元相图分析4.4二元合金的凝固理论4.5高分子合金概述14.1相图的表示和测定方法一、相图的表示方法二、相图的测定方法2一、相图的表示方法1.成分、温度和压力三坐标立体图2.成分与温度两坐标平面图3.成分表示3二、相图的测定方法1.通过热力学的计算和分析建立相图;

2.通过实验建立相图41.通过热力学的计算和分析建立相图利用已有的热力学参数，可作出不同温度、成分下各相的吉布斯自由能曲线，确定不同温度、成分下平衡存在的相的状态和成分，绘制出不同合金的相图；或者通过热力学计算，求出有关数据，直接作出相图；计算机的广泛使用为计算相图提供了有利的条件，从长远发展看，相图的计算确定是有很大潜力的。

52.通过实验建立相图实际金属材料或陶瓷材料相图的建立主要依靠实验的方法，包括热分析法、热膨胀法、磁性法、电阻法、金相分析法、X-射线分析法等等，这些方法可归结为两类基本方法--动态垂直截线法和静态水平截线法。

6动态垂直截线法合金序号123456组元A(﹪)100806040200组元B(﹪)0204060801007InterpretationofcoolingcurvesThemeltingtemperatureofanypurematerial(aone-componentsystem)atconstantpressureisasingleuniquetemperature.Theliquidandsolidphasesexisttogetherinequilibriumonlyatthistemperature.Whencooled,thetemperatureofthemoltenmaterialwillsteadilydecreaseuntilthemeltingpointisreached.Atthispointthematerialwillstarttocrystallise,leadingtotheevolutionoflatentheat

atthesolidliquidinterface,maintainingaconstanttemperatureacrossthematerial.Oncesolidificationiscomplete,steadycoolingresumes.Thearrestincoolingduringsolidificationallowsthemeltingpointofthematerialtobeidentifiedon

atime-temperaturecurve.8Mostsystemsconsistingoftwoormorecomponentsexhibitatemperaturerangeoverwhichthesolidandliquidphasesareinequilibrium.Insteadofasinglemeltingtemperature,thesystemnowhastwodifferenttemperatures,theliquidustemperatureandthesolidustemperaturewhichareneededtodescribethechangefromliquidtosolid.Theliquidustemperatureisthetemperatureabovewhichthesystemisentirelyliquid,andthesolidusisthetemperaturebelowwhichthesystemiscompletelysolid.Betweenthesetwopointstheliquidandsolidphasesareinequilibrium.Whentheliquidustemperatureisreached,solidificationbeginsandthereisareductionincoolingratecausedbylatentheatevolutionandaconsequentreductioninthegradientofthecoolingcurve.9Uponthecompletionofsolidificationthecoolingratealtersagainallowingthetemperatureofthesolidustobedetermined.Ascanbeseenonthediagrambelow,thesechangesingradientallowtheliquidustemperatureTL,andthesolidustemperatureTStobeidentified.10Whencoolingamaterialofeutecticcomposition,solidificationofthewholesampletakesplaceatasingletemperature.Thisresultsinacoolingcurvesimilarinshapetothatofasingle-componentsystemwiththesystemsolidifyingatitseutectictemperature.11Whensolidifyinghypoeutecticorhypereutecticalloys,thefirstsolidtoformisasinglephasewhichhasacompositiondifferenttothatoftheliquid.Thiscausestheliquidcompositiontoapproachthatoftheeutecticascoolingoccurs.Oncetheliquidreachestheeutectictemperatureitwillhavetheeutecticcompositionandwillfreezeatthattemperaturetoformasolideutecticmixtureoftwophases.Formationoftheeutecticcausesthesystemtoceasecoolinguntilsolidificationiscomplete.Theresultingcoolingcurveshowsthetwostagesofsolidificationwithasectionofreducedgradientwhereasinglephaseissolidifyingandaplateauwhereeutecticissolidifying.12Bytakingaseriesofcoolingcurvesforthesamesystemoverarangeofcompositionstheliquidusandsolidustemperaturesforeachcompositioncanbedeterminedallowingthesolidusandliquidustobemappedtodeterminethephasediagram.Belowarecoolingcurvesforthesamesystemrecordedfordifferentcompositionsandthendisplacedalongthetimeaxis.Theredregionsindicatewherethematerialisliquid,theblueregionsindicatewherethematerialissolidandthegreenregionsindicatewherethesolidandliquidphasesareinequilibrium.1314Byremovingthetimeaxisfromthecurvesandreplacingitwithcomposition,thecoolingcurvesindicatethetemperaturesofthesolidusandliquidusforagivencomposition.15Thisallowsthesolidusandliquidustobeplottedtoproducethephasediagram:161718静态水平截线法此类方法主要用于测定固态下发生的转变。194.2相图热力学基本要点一、合金相热力学二、相平衡热力学三、相图热力学20

一、合金相热力学合金在平衡状态下相的状态,包括相的数目和成分,由热力学条件决定,所存在稳定的相的状态是系统吉布斯自由能最低的状态

1.二组元固溶体相的吉布斯自由能

2.二组元中间相的吉布斯自由能3.混合相的吉布斯自由能211.二组元固溶体相的吉布斯自由能在等压条件下H0的近似求法

S0的近似求法

溶体的自由能-成分曲线

22ThermodynamicsofSolutions

Consideramechanicalmixtureoftwophases,AandB.IfthisisthentransformedintoasinglesolutionphasewithAandBatomsdistributedrandomlyovertheatomicsites,thentherewillbe,AnenthalpychangeassociatedwithinteractionsbetweentheAandBatoms,⊿Hmix

Anentropychange,⊿Smix,associatedwiththerandommixingoftheatomsAfreeenergyofmixing,⊿Gmix=⊿Hmix-T⊿Smix

AssumethatthesystemconsistsofNatoms:xANofAandxBNofB,where,xA=fractionofAatomsandxB=(1-xA)=fractionofBatoms23EnthalpyofmixingIncalculating⊿Hmixitisassumedthatonlythepotentialenergytermundergoesanysignificantchangeduringmixing.Thischangearisesfromtheinteractionsbetweennearest-neighbouratoms.ConsideranalloyconsistingofatomsAandB.Iftheatomspreferlikeneighbours,AatomswilltendtoclusterandlikewiseBatoms,soagreaternumberofA-AandB-Bbondswillform.IftheatomspreferunlikeneighboursagreaternumberofA-Bbondswillform.IfthereisnopreferenceAandBatomswillberandomlydistributed.24LetwAAbetheinteractionenergybetweenA-Anearestneighbours,wBBthatforB-BnearestneighboursandwABthatforA-Bnearestneighbours.Alloftheseenergiesarenegative,asthezeroinpotentialenergyisforinfiniteseparationbetweenatoms.LeteachatomofAandBhaveco-ordinationnumberz.Therefore,thetotalnumberofnearest-neighbourpairsisNz/2.ProbabilityofA-Aneighbours=xA2ProbabilityofB-Bneighbours=xB2ProbabilityofA-Bneighbours=2xAxBForasolidsolutionthetotalinteractionenergyis,Hs-Us=Nz/2(xA2

wAA+xB2

wBB+2xAxB

wAB)ForpureA,HA=(Nz/2)wAAForpureB,HB=(Nz/2)wBB

25Hencetheenthalpyofmixingisgivenby,⊿Hmix=Hs-(xAHA+xBHB)=(Nz/2)xAxB(2wAB-wAA-wBB)WecandefineaninteractionparameterΩ=(Nz/2)(2wAB-wAA-wBB)Therefore,⊿Hmix=ΩxAxBIfA-AandB-BinteractionsareenergeticallymorefavourablethanA-BinteractionsthenΩ>0.So,⊿Hmix>0andthereisatendencyforthesolutiontoformA-richandB-richregions.IfA-BinteractionsareenergeticallymorefavourablethanA-AandB-Binteractions,Ω<0,⊿Hmix<0,andthereisatendencytoformorderedstructuresorintermediatecompounds.Finallyifthesolutionisidealandallinteractionsareenergeticallyequivalent,thenΩ=0and⊿Hmix=0.26EntropyofmixingPermoleofsites,thisis⊿Smix=kN(-xAlnxA-xBlnxB)(thederivationofthisresultmakesuseofStirling'sapproximation)whereN=Avogadro'snumber,andkN=R,thegasconstant.Hence,⊿Smix=R(-xAlnxA-xBlnxB)Agraphof⊿SmixversusxAhasadifferentformfrom⊿Hmix.ThecurvehasaninfinitegradientatxA=0andxA=1.Thefreeenergyofmixingisnowgivenby,⊿Gmix=⊿Hmix-T⊿Smix=xAxBΩ+RT(xA

lnxA+xBlnxB)ForΩ<0,⊿Gmixisnegativeatalltemperatures,andmixingisexothermic.ForΩ>0,⊿Hmixispositiveandmixingisendothermic.27⊿Sm＝－kN（xAlnxA＋xBlnxB）即S0＝－R（xAlnxA＋xBlnxB）28Stirling'sapproximationStirling'sapproximationis:ln

N!=NlnN-N,forlargeNTheentropy,S=k

lnw

wherewisthenumberofpossibleconfigurationsforasystem.Foramechanicalmixturew=1astheonlyarrangementisAatomsonAsitesandBatomsonBsites.ForasolidsolutionofAandBcontainingxANAatomsandxBNBatomsthevalueofwiscalculatedasfollows

N!

{xAN}!{(1-

xA)N}!AssumingthatthethermalentropyofthesystemremainsunchangedwhenAandBgointosolution2930FreeenergycurvesofSolutionsSolutionscontainmorethanonecomponentandinthesesituationsthefreeenergyofthesolutionwillbecomedependentonitscompositionaswellasthetemperature.Itisshownabovethatthefreeenergyofmixingis:⊿Gmix=⊿Hmix-T⊿Smix=xAxBΩ+RT(xAlnxA+xBlnxB)Theshapeofthe⊿Gmixcurveisdependentontemperature.Forthecurveshownbelowthevalueof⊿Hmixispositive,leadingtoamaximumonthecurveatlowtemperatures.⊿Gmixisalwaysnegativeforlowsoluteconcentrationsasthegradientof⊿SmixisinfiniteatxA=0andxA=1.31Athightemperaturesthereisacompletesolutionandthecurvehasasingleminimum.Atlowtemperaturesthecurvehasamaximumandtwominima.Inthecompositionrangebetweenthetwominima(denotedbythedashedlines)amixtureoftwophasesismorestablethanasingle-phasesolution.Thefreeenergyofaregularsolidsolution,⊿Gsol,isthesumofthefreeenergyofmixing⊿Gmixandthefreeenergyoffusion⊿Gfus.32FreeenergyoffusionWhenaliquidsolidifiesthereisachangeinthefreeenergyoffreezing,astheatomsmoveclosertogetherandformacrystallinesolid.Forapurecomponent,thiscanbeempiricallycalculatedusingRichard'sRule:⊿Gfusion=-9.5(Tm-T)Tm=meltingtemperature

T=currenttemperature⊿Gfusion=0atthemeltingtemperatureofthecomponent.

⊿Gfusion<0belowthemeltingtemperatureofthecomponent.

⊿Gfusion

>0abovethemeltingtemperatureofthecomponent.Inanalloy,ifboththeliquidandsolidsolutionsareidealthen⊿Gfusionforthealloycanbeinterpolatedbetweenthevaluesforthetwocomponents.33Nowwecanplotthefreeenergyofaregularsolidsolutionfromtheequation,⊿Gsol=⊿Gmix+⊿Gfusion342.二组元中间相的吉布斯自由能

353.混合相的吉布斯自由能x=(n1x1+n2x2)/(n1+n2)36二、相平衡热力学

1.相平衡条件

2.图解法求化学位

3.相平衡的公切线法则371.相平衡条件二元系中,两相平衡的热力学条件是每个组元在各相中的化学位相等,即

μαA＝μβAμαB＝μβB二元系中,三相平衡的热力学条件是每个组元在各相中的化学位相等,即

μαA＝μβA＝μγAμαB＝μβB＝μγB

多元复相平衡的普遍条件是每个组元在各相中的化学位都必须彼此相等,即μαi＝μβi＝μγi＝…＝μPi

其中,α、β、γ…P表示合金中存在的相,i代表合金中的第i个组元382.图解法求化学位

393.相平衡的公切线法则一相分解为两相平衡

两相平衡

化合物相形成平衡

二元系中的三相平衡

40一相分解为两相平衡41两相平衡

42化合物相形成平衡

43二元系中的三相平衡

44三、相图热力学推测相图步骤为:首先,求得各相在不同T和X时的G,并作G-X曲线;其次,根据公切线法则,作出G-X曲线的公切线,找出平衡相的成分和存在范围,然后综合画在T-X坐标图上.其规律为:1.当两相的G-X成分曲线不相交时,表示在某T下只有稳定单相(即G最低的那个相)存在,在相图中对应的是单相区;2.若两相的G-X曲线相交,则必有一条公切线,两切点相应的成分表示在此温度下两个平衡相的成分,在该成分范围内相图上对应存在两相区;45

3.若两相G-X曲线相交,但只在交点相切,则在相图中与这个切点相对应的是一个相变点,表示同成分的两相平衡;4.在有三相存在时,如三条G-X曲线依次相交,存在两条公切线,有两对平衡相,切点对应的成分分别表示其平衡相的成分.如三相的G-X曲线只存在一条公切线,表示三相平衡,三个切点对应的成分表示三个平衡相的成分,在相图上对应有一条三相共存的水平线46Phasediagrams1

Freeenergycurvescanbeusedtodeterminethemoststablestateforasystem,i.e.thephaseorphasemixturewiththelowestfreeenergyforagiventemperatureandcomposition.Belowisaschematicfree-energycurveforthesolidphaseofanalloy.47ThesolidshowncouldeitherexistasamixtureorasahomogeneoussolutionofAandB.ThefiguresbelowshowthatanalloyofcompositionCcanexistindifferentconfigurationswithdifferingfreeenergies.Inthefirstfigure(below)thefreeenergyofunmixedAandBisshownasthediagonalblackline.ThefreeenergyofthismixtureatcompositionCisshownasaredpoint.48Thesystemcanreduceitsfreeenergybyexistingasamixtureoftwophases

Thoughthesystemhasreduceditsfreeenergyfromthatofthemixture,themoststableconfigurationforthesystemisasolidsolution.Thisallowsthefreeenergyofthesystemtositonthefreeenergycurve.49Formostsystemstherewillbemorethanonephaseandassociatedfree-energycurvetoconsider.Atagiventemperaturethemoststablephaseforasystemcanvarywithcomposition.Whilethesystemcouldconsistentirelyofthephasewhichismoststableatagivencompositionandtemperature,ifthefreeenergycurvesforthetwophasescross,themoststableconfigurationmaybeamixtureoftwophaseswithcompositionsdifferingfromthatoftheoverallsystem.Thetotalfreeenergyofthesysteminanygiventwo-phaseconfigurationcanbefoundbylinkingthetwophasesinquestionwithastraightlineonafree-energyplot.5051Takingalinethatisacommontangenttothetwofree-energycurvesproducesthelowestpossiblefreeenergyforthesystemasawhole.Wherethelinemeetsthefreeenergycurvesdefinesthecompositionofeachphase.52Forpositionswhereitisnotpossibletodrawacommontangentbetweenthetwofree-energycurvesthesystemwillsitentirelyinthephasewiththelowestfreeenergy.Thebordersbetweenthesingle-andtwo-phaseregionsmarkthepositionsofthesolidusandliquidusonthephasediagram.53Whenthetemperatureisalteredthecompositionsofthesolidandliquidinequilibriumchangeandbuilduptheshapeofthesolidusandliquiduscurvesonaphasediagram.Below,abinarysystemcanbeseenalongwiththefree-energycurvesfortheliquidandsolidphasesatarangeoftemperaturesshownonthephasediagram.54

55Phasediagrams2Thefree-energycurvesandphasediagramsdiscussedinPhaseDiagrams1wereallforsystemswherethesolidexistsasasolutionatallcompositionsandtemperatures.Inmostrealsystemsthisisnotthecase.Thisisduetoapositive⊿Hmixcausedbyunfavourableinteractionsbetweenunlikeneighbouratoms.Asthetemperatureisreducedthe⊿Hmixtermbecomesmoresignificantandthecurveturnsupwardatintermediatecompositions,resultinginacurvewithtwominimaandonemaximumasdescribedearlier.Acommontangentcanthenbedrawnbetweenthetwominimashowingthatthesystemcanreduceitsfreeenergythroughexistingasamixtureoftwodistinctphases.56ThefreeenergyofasystemofcompositionCocanbeminimisedbyexistingasamixtureoftwosolidphasesofcompositionC1andC2:Thiseffectcanresultinasystemwhich,thoughsingle-phaseuponsolidification,willseparateintotwosolidphasesoncooling(e.g.Cr-W).57Anotherpossibleresultisthatthefree-energycurvefortheliquidwillintersecttheupturnedsectionofthefree-energycurveforthesolidbeforethetemperatureishighenoughtoinducetheformationofasolidsolution.Asthetemperatureisincreased,thefree-energycurvefortheliquidmovesdownwardrelativetothesolidcurveandreachesapositionwhereit