材料科学基础课后习题答案8.docx

材料科学基础课后习题答案8.docx

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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