材料科學基礎(chǔ)輔導與習題-上交課件 二元相圖及其合金的凝固

Ch4二元相圖及其合金的凝固4.1相圖的表示和測定方法4.2相圖熱力學基本要點4.3二元相圖分析4.4二元合金的凝固理論4.5高分子合金概述14.1相圖的表示和測定方法一、相圖的表示方法二、相圖的測定方法2一、相圖的表示方法1.成分、溫度和壓力三坐標立體圖2.成分與溫度兩坐標平面圖3.成分表示3二、相圖的測定方法1.通過熱力學的計算和分析建立相圖;

2.通過實驗建立相圖41.通過熱力學的計算和分析建立相圖利用已有的熱力學參數(shù)，可作出不同溫度、成分下各相的吉布斯自由能曲線，確定不同溫度、成分下平衡存在的相的狀態(tài)和成分，繪制出不同合金的相圖；或者通過熱力學計算，求出有關(guān)數(shù)據(jù)，直接作出相圖；計算機的廣泛使用為計算相圖提供了有利的條件，從長遠發(fā)展看，相圖的計算確定是有很大潛力的。

52.通過實驗建立相圖實際金屬材料或陶瓷材料相圖的建立主要依靠實驗的方法，包括熱分析法、熱膨脹法、磁性法、電阻法、金相分析法、X-射線分析法等等，這些方法可歸結(jié)為兩類基本方法--動態(tài)垂直截線法和靜態(tài)水平截線法。

6動態(tài)垂直截線法合金序號123456組元A(﹪)100806040200組元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靜態(tài)水平截線法此類方法主要用于測定固態(tài)下發(fā)生的轉(zhuǎn)變。194.2相圖熱力學基本要點一、合金相熱力學二、相平衡熱力學三、相圖熱力學20

一、合金相熱力學合金在平衡狀態(tài)下相的狀態(tài),包括相的數(shù)目和成分,由熱力學條件決定,所存在穩(wěn)定的相的狀態(tài)是系統(tǒng)吉布斯自由能最低的狀態(tài)

1.二組元固溶體相的吉布斯自由能

2.二組元中間相的吉布斯自由能3.混合相的吉布斯自由能211.二組元固溶體相的吉布斯自由能在等壓條件下H0的近似求法

S0的近似求法

溶體的自由能-成分曲線

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+⊿Gfusion342.二組元中間相的吉布斯自由能

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

1.相平衡條件

2.圖解法求化學位

3.相平衡的公切線法則371.相平衡條件二元系中,兩相平衡的熱力學條件是每個組元在各相中的化學位相等,即

μαA＝μβAμαB＝μβB二元系中,三相平衡的熱力學條件是每個組元在各相中的化學位相等,即

μαA＝μβA＝μγAμαB＝μβB＝μγB

多元復相平衡的普遍條件是每個組元在各相中的化學位都必須彼此相等,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個組元382.圖解法求化學位

393.相平衡的公切線法則一相分解為兩相平衡

兩相平衡

化合物相形成平衡

二元系中的三相平衡

40一相分解為兩相平衡41兩相平衡

42化合物相形成平衡

43二元系中的三相平衡

44三、相圖熱力學推測相圖步驟為:首先,求得各相在不同T和X時的G,并作G-X曲線;其次,根據(jù)公切線法則,作出G-X曲線的公切線,找出平衡相的成分和存在范圍,然后綜合畫在T-X坐標圖上.其規(guī)律為:1.當兩相的G-X成分曲線不相交時,表示在某T下只有穩(wěn)定單相(即G最低的那個相)存在,在相圖中對應的是單相區(qū);2.若兩相的G-X曲線相交,則必有一條公切線,兩切點相應的成分表示在此溫度下兩個平衡相的成分,在該成分范圍內(nèi)相圖上對應存在兩相區(qū);45

3.若兩相G-X曲線相交,但只在交點相切,則在相圖中與這個切點相對應的是一個相變點,表示同成分的兩相平衡;4.在有三相存在時,如三條G-X曲線依次相交,存在兩條公切線,有兩對平衡相,切點對應的成分分別表示其平衡相的成分.如三相的G-X曲線只存在一條公切線,表示三相平衡,三個切點對應的成分表示三個平衡相的成分,在相圖上對應有一條三相共存的水平線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