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材料科学基础课后习题答案8
SOLUTIONSFORCHAPTER8
1.FIND:
Whenaphasetransformationoccurssuchasaliquidphasetransformingtoasolidbelowitsmeltingtemperature,whatarethetwostepsinvolvedintheprocess?
Brieflydescribeeach.
SOLUTION:
Duringaphasetransformationsuchasaliquidtransformingtosolid,therearetwostepsinvolvedintheprocess.Theyare:
1.nucleationofthenewphase,and
2.growthofthephase.
Nucleationrelatestotheformationofthenewphaseandthedevelopmentoftheinterfaceseperatingthetwophases.Nucleationcaneitheroccurrandomlythroughoutthestructure-termedhomogeneousnucleationoratspecificsitessuchasinterfaces-termedheterogeneousnucleation.
Growth-Oncethephaseshasnucleated,itbeginstogrow.Thegrowthprocessiscontrolledbydiffusionandundercooling.Asinthenucleationstep,theremaybecompetingprocessesthatleadtoamaximumgrowthrateatanintermediatetemperature.
2.FIND:
WepresentedaderivationinSection8.2.3showingthatthebarrierfornucleation,∆G*,decreaseswithincreasingundercoolingfollowingtheproportionality
Bystartingwithanexpressionforthefreeenergyofadistributionofsphericalparticlesofradiusr,deriveequation8.2-9a.Explaineachstepinthederivation.Explainanyassumptionsthataremade.
SOLUTION:
Todeterminethebarriertothenucleationprocess,∆G*webeginbynotingthatthefreeenergyasafunctionofparticlesizeforhomogeneousnucleationhastwoterms,onethatincreaseswithr2,theinterfacialenergyperunitvolumeterm,andonethatdecreaseswithr3.Amaximumoccursin∆G(r)atsomer*.Thesegraphicalrelationshipsaresketchedbelow.
Thedependenceofthevariousfreeenergytermsassociatedwithnucleationasafunctionoftemperature:
(a)therelationshipbetweenclusterradiusandsurfaceenergyofagrowingsphericalsolidphaseinliquid,(b)therelationshipbetweentheclusterradiusand(c)thesumof(a)and(b).
Thechangeinfreeenergycanbewrittenas:
Inthisequationweassumethatthenucleicanbeconsideredasarandomdistributionofspheres.Tolocatethemaximumofafunctionweequatethefirstderivativeofthefunctionwithrespecttotheparameterwhichisthevariabletozero.Hereweassumethatristheonlyvariable.TheisγSLisindependentofsizeandorentation.
Usingequation8.2-4for
wehave:
Inwriting
wehaveassumedthattheheatcapacitydifferencebetweentheliquidandsolidphasesiszero.(Note:
Althoughthismaybeareasonableassumptionatsmallundercoolings,i.e.small∆T’s,atthelargeundercoolingsthataretypicalforhomogeneousnucleationthatapproximationmaynotbevalidandamorecomplextermisrequired.Butforafirstorderapproximationthisassumptionisreasonable.)
Inordertodeterminethevalueof∆G(r)atr*,weintroducetheexpression
into
Rearranging,
Ifallthetermsinparenthesesareconstantthen,
3.FIND:
Explainthesimultaneousinfluencethatundercoolinghasonthebarriertonucleationandtheatomicrearrangementsnecessarytoinitiatethetransformation.ShowhowthesecompetingeffectsleadtoclassicalC-curvebehaviorinthenucleationofdiffusionaltransformations.
SOLUTION:
Withlargerundercoolings,bothr*and∆G*decrease,suggestingthatsimplyloweringthetemperatureofthesystemallowsnucleationtooccurevermorereadily.However,therearepracticalkineticlimitstothiseffect.Forexample,withdecreasedtemperaturethereisacorrespondingreductioninatomicmobility.Therandomfluctuationsinthelocalarrangementsofatomsistheprocessthatprovidestheclusters.Sincetheformationoftheclustersdependsonatomicmobility,areductioninthetemperaturereducestherateofclustering.Thus,asshowninFigure(a)below,theoverallnucleationrateexhibitsamaximumatanintermediatetemperature.Themaximuminthenucleationrateleadstoaminimuminthetimerequiredtonucleateaphase,asshowninFigureb.Becauseofitsshape,thiscurveisknownasaC-curve.
(a)Theinfluenceoftemperatureonthemobilitytermandthenucleationbarrierterm.Theopposingprocessesresultinamaximuminthenucleationrateatanintermediatetemperature.
(b)Sincethetimefornucleationisinverselyrelatedtothenucleationrate,thetimecurveexhibitsaminimumatanintermediatetemperature.Becauseofitsshape,thiscurveisoftenreferredtoasaC-curve.
4.FIND:
Explainhowthevalueofinterfacialenergybetweentheparentphaseandthetransformingphaseaffectsthecriticalradiusandthebarriertonucleation.
SOLUTION:
Equation8.25givesthechangeinfreeenergyasafunctionofrwhenaliquidtransformstoasolid,forexample.Inthedevelopmentofthatequationitwasassumedthatthetransformingphasewassphericalandtheinterfacialenergy,γSL,wasisotropic.Thatequationconsistsoftwotermsontherighthandside,i.e.
∆G(r)=(4πr2)γSL+4/3πr3(∆Gv)
SinceγSL>0and∆GV<0for∆T>0,thefirsttermincreaseswithradius,andtheseconddecreases.Figure8.2-3illustratesthatthereisamaximumthatoccursatsomerwedesignatedasr*andacorresponding∆G,wedesignatedas∆G*where
r*=(-2γSL)/∆Gv
and
Boththecriticalradius,r*,andthebarriertothenucleationprocesscontainγSLinthenumerator.Thinkingintermsofthebarriertonucleation,∆G*,thereisacubicdependenceoninterfacialenergy.ThelargertheS-Linterfacialenergy,thelargerthebarriertonucleationandhencethemoredifficult.Theuseofnucleatingagentsisbasedontheprincipleofintroducingparticleswithlowerinterfacialenergiestostimulatenucleation.
5.FIND:
Comparehomogeneousnucleationandheterogeneousnucleation.
SOLUTION:
Theprocessofhomogeneousnucleationoccursatrandomlocationsintheparentphase.Thedistributionofthetransformingphaseoccurswithoutregardtospecificsites,suchasmoldwallsinsolidification.Heterogeneousnucleationoccursatspecificsites.Inthecaseofsolidificationtheycanbeatmoldwalls,unintentionaladditionssuchasceramicinclusionsfromcruciblesoritmayoccuratnucleatingagentswhichareintentionallyaddedtocontrolthesolidificationmicrostructure.
6.FIND:
Whatisthedifferencebetweenthefollowinginterfaces?
a.coherent
b.partiallycoherent,and
c.incoherent
SOLUTION:
Acoherentinterfaceisoneinwhichthereisaone-to-onecorrespondenceofatomicplanesacrosstheinterface.Thistypeofinterfaceoccurswhenthelatticeparametersinthetwophasesarethesameorveryclose.Whenthelatticeparametersaredifferentinthetwophases,theincreaseinstrainenergythatwilloccurastheparticlegrowsnecessitatestheperiodicinsertionofdislocationsattheinterfacetoaccommodatethemisfit.Thistypeofinterfaceisreferredtoaspartiallycoherent.Anincoherentinterfaceoccursbetweentwophasesofdifferentcrystalstructuresandsufficientlydifferentatomicspacingsthatcannotbeaccommodatedbydislocations.
7.FIND:
Howdoesinterfacialenergyvarywithcoherency?
SOLUTION:
Interfacialenergyissensitivetothenatureoftheinterfaceseparatingthetwophases.Theinterfacialenergyincreasesgoingfromcoherenttopartiallycoherenttoincoherent,i.e.γincoh.>γpart.coh..γcoh..
8.FIND:
Baseduponyouranswertoquestion7,explainhowtheprobabilityforheterogeneousnucleationchangesasthetypeofinterfacechangesfromcoherenttopartiallycoherenttoincoherent.
SOLUTION:
Sincethebarriertonucleation,∆G*,isrelatedtoγα/βinthefollowingway:
∆G*∝γα/β3
increasingγα/βwouldincreasethebarriertohomogeneousnucleation.Consequently,theprobabilityforheterogeneousnucleationwouldincreaseasinterfacialenergyincreases.
9.FIND:
Figure5.3-5containsaschematicillustrationofatwinboundaryinacrystal.Fromthepointofviewofcoherency,whatisthenatureofthetypeoftwinboundaryillustratedinthefigure?
Commentontherelativeenergyofatwinboundarycomparedwitharandomgrainboundaryinapolycrystallinematerial.
GIVEN:
Figure5.3-5illustratesaschematicofatwininamatrixshowingthetwotwinboundariesseparatingthematrixfromthetwinandFigure8.2-10illustratesanincoherentinterfaceseparatingthematrixfromaprecipitate.
Schematicofatwin
Schematicofanincoherentboundary
SOLUTION:
Atomsthatareonthetwinplanearepartofthestackingsequenceinthematrixabovethetwinplaneaswellasthestackingofatomsbelowthetwinplane.Sinceacoherentinterfaceisaninterfacethatoccurswhenthereisaone-to-onecorrespondenceacrosstheinterface,thenthetwinillustratedinthisfigurewouldbeclassifiedasacoherenttwinboundary.
Theincoherentboundaryillustratedaboveoccursinasystemwhenthereisnotamatchacrosstheboundaryseparatingtwophases.Sinceageneralgrainboundaryrepresentsasituationwheretheorientationoftwograinsacrossaboundaryarenotthesame,wewouldthereforeexpectthattherewouldnotbeamatchofatomsacrosstheboundary.Thus,acoherenttwinplanewouldhavelowerinterfacialenergythantheinterfacialenergyassociatedwithgrainboundaryseparatingtworandomlyorientedgrains.
10.FIND:
Incertainnickel-basesuperalloys,asecondphasecanprecipitatecoherentlyfromthematrixduringagingbecausethelatticeparametersofthetwophasesareverycloseandbothphasesarecubic.Foracoherentprecipitateinthissystem,whatisthemostlikelyrelationshipbetweenthecrystallographicaxesinthematrixphaseandthatoftheprecipitate?
Explain,usingsketches.
GIVEN:
Thematrixandprecipitatearebothcubicwithsimilarlatticeparameters.
SOLUTION:
Thebestmatch