Microstructure and dislocation structure evolution during creep life of Ni-based single crystal superalloys

Delft University of Technology

Microstructure and dislocation structure evolution during creep life of Ni-based single

crystal superalloys

Yu, Hao; Xu, Wei; van der Zwaag, Sybrand

DOI

10.1016/j.jmst.2019.11.028

Publication date

2020

Document Version

Final published version

Published in

Journal of Materials Science and Technology

Citation (APA)

Yu, H., Xu, W., & van der Zwaag, S. (2020). Microstructure and dislocation structure evolution during creep

life of Ni-based single crystal superalloys. Journal of Materials Science and Technology, 45, 207-214.

https://doi.org/10.1016/j.jmst.2019.11.028

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JournalofMaterialsScience&Technology45(2020)207–214

ContentslistsavailableatScienceDirect

Journal

of

Materials

Science

&

Technology

jou rn a l h o m e p a g e :w w w . j ms t . o r g

Research

Article

Microstructure

and

dislocation

structure

evolution

during

creep

life

of

Ni-based

single

crystal

superalloys

Hao

Yu

a

_,

_Wei

_Xu

a,b,∗

_,

_Sybrand

_van

_der

_Zwaag

a

a_{NovelAerospaceMaterialsGroup,FacultyofAerospaceEngineering,DelftUniversityofTechnology,2629HS,Delft,theNetherlands} b_{StateKeyLaboratoryofRollingandAutomation,NortheasternUniversity,110819,Shenyang,China}

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received27September2019

Receivedinrevisedform7November2019 Accepted7November2019

Availableonline17January2020 Keywords:

Nisuperalloys Microstructureevolution Dislocationbehaviour

a

b

s

t

r

a

c

t

ThehighperformanceofNisinglecrystalsuperalloysduringhightemperaturelowstresscreep

ser-vice,isintrinsicallydeterminedbythecombinedeffectsofmicrostructuralevolutionandthedislocation

behaviour.Inthefieldoftheevolutionofdislocationnetwork,twomainrecoverymechanismbased

ondislocationmigrationdominatetheprocess.Oneissuperdislocationsshearinginto␥’raftsthrough

atwo-superpartials-assistedapproach.Anotheristhecompactdislocationsmigratingalong␥/␥

inter-face.Thesetwomechanismsaresimilarlyclimb-rate-controlledprocess.Inthiswork,amodelforthe

minimumcreepratebasedonthermodynamicandkineticcalculationsandusinganexistingdetailed

dislocationdynamicsmodelhasbeenbuiltbytakingthedislocationmigrationbehavioursaswellasthe

raftedmicrostructureintoconsideration,whichcanwellreproducethe([100]tensile)creepproperties

ofexistingNisuperalloygrades,withouttheneedtomakethedislocationparametervaluescomposition

dependent.

©2020PublishedbyElsevierLtdonbehalfofTheeditorialofficeofJournalofMaterialsScience&

Technology.

1. Introduction

Ni-basedsinglecrystalsuperalloyshavebeenwidelyusedfor thebladesand otherloadedstructures ofaero-engines andgas turbinesduetotheirsuperiormechanicalproperties,inparticular theirexcellentcreepresistanceathightemperatures[1].Their out-standingcreepresistance,notonlyoriginsfromtheabsenceofgrain boundariesbutislargelydeterminedbytheunique microstruc-turecharacterisedbythepresenceofahigh-volumefractionofthe long-rangeorderedL12␥phase,whichappearsascubescoherently

embeddedinaface-centeredcubicsolidsolution␥matrix.In gen-eral,thesize,volumefractionandmorphologyof␥_precipitates

mainly determine the mechanical properties of Ni-superalloys [2,3]. In theas-produced conditionthecuboidal _␥-precipitates havea sizeof around0.4um sizeandtheyare separatedby ␥-channelswitharound0.1umsize.Thetypicalprecipitatevolume fractionatroomtemperatureis50%orhigher[1].

Whenexposedtotheirtypicaluseconditions,arelativelyhigh temperature(>950◦C) anda modeststress(<250MPa),there is directionalcoarsening of ␥ _{precipitates during theearly creep}

∗ Correspondingauthorat:NovelAerospaceMaterialsGroup,FacultyofAerospace Engineering,DelftUniversityofTechnology,2629HS,Delft,theNetherlands.

E-mailaddress:xuwei@ral.neu.edu.cn(W.Xu).

stage,whichisso-called“rafting”stage[4–7].Duringthistime,the initiallyadjacentcuboidal␥_{particlescoalesceandformplatelets}

thatturnintoplate-likeorrod-likestructures.Thislamellar␥/␥’ raftedmicrostructurewillremainmoreorlessunchangedduring thelongstablecreepstage,untilthe␥’graduallyinterconnectsand becomesthematrixphase surroundingisolated␥phaseislands [8,9].Thisprocessisknownasthe‘topologicalphaseinversion’, whichhasbeenconsideredasthemicrostructuralindicator mark-ingthetransitionfromquasi-stationarycreeptoacceleratedcreep. Thisinvertedmicrostructureismaintainedduringtheaccelerated creepstagebutrapidlylosesitregularity,themorphology evolu-tionofthephasesduringtheentirecreeplifeispresentedinFig.1 [10].

Ontheotherhand,thecreepresponsenotonlydependsonthe microstructureevolutionbutalsoonthechangesinthe dynam-icsandtopologyofthedislocationsanddislocationnetworks[11]. Atthebeginningofcreeploading,thedeformationisgovernedby thedislocationglideanddislocationmultiplicationinthe␥ chan-nels.Soonthereafterthemobiledislocationsstarttoaccumulate andbecomerearrangedatthe␥/␥’interface,whiletheformationof lamellarraftstakesplace,leadingtothewidelyobservedformation ofdislocationnetworksonthe␥/␥’interface[11–15].Analogousto thelamellarmicrostructure,thedislocationnetworkwillremain stableuntiltheendofthestablecreepstage(stageIIcreep)when thenetworkbeginstodegradebyhugeamountsofdislocations cut-https://doi.org/10.1016/j.jmst.2019.11.028

208 H.Yuetal./JournalofMaterialsScience&Technology45(2020)207–214

Fig.1. SchematicillustrationofthemicrostructureevolutionofNisuperalloysingle crystalsduringhightemperaturelowstresscreeploading.Theblackphaseisthe’ phase,whilethewhitephasemarksthe␥phase.Thestressisappliedinthedirection paralleltothey-axis[10].

tingintotherafted␥_{throughtheinterface.Ultimatelythischaotic}

dislocationmultiplicationprocessleadstorupture.

Summingup,thecreeppropertiesofNisinglecrystal super-alloys are strongly dependent on the combined effect of the microstructureevolutionandthedislocationbehaviour.Therefore, agoodunderstandingandsomequantificationofmicrostructure anddislocationevolutionduringcreepiscrucialtobetter compre-hendthemechanicalpropertiesofNisuperalloys.Inthiswork,the currentworkonthedependenceofmechanicalpropertiesinNi singlecrystalsonthemicrostructuralanddislocationbehaviour during isothermal creep tests is reviewed. The experimental observationsand thecorrespondingmodelsandsimulationsare combinedtoshowtheirmutualinteraction.Asimplemodelhas beenbuiltinwhichthegoverningdislocationdynamicsequation isthermodynamicallycoupledtothechemicalcompositionofsome commercialsuperalloystopredicttheirminimumcreeprateatthe loadingconditions.

2. Microstructureevolution

2.1. Initialmicrostructure

The␥ _{precipitatesinNisuperalloysundergoasuccessionof}

morphologychangesfromspherestocubesduringtheheat treat-mentsprecedingtheactualusephase.Attheverybeginning,the ␥’precipitatesnucleateasspherestominimizethesurfacearea [16].Astheparticles grow,themisfit strainenergy inducedby thelatticeandmodulusmismatchbetweenthe␥andthe␥’phase increases,andtheprecipitatesbecomecuboidsasthereductionin strainenergymorethancompensatesfortheincreaseinsurface energy.Ithasbeenproventhatcubemorphologybestminimizes thetotalenergyoftheprecipitateasitprovidesthebestbalance betweentheanisotropicstrainenergyandtheisotropicinterfacial energy[17].

Nathal[18]hasquantitativelyinvestigatedtheoptimalsizeof initial(i.e.asproduced)particleandhisworkshowsthatalloyswith aninitial␥’particlesizebetween0.35and0.5␮mcanpronouncedly outperformtheircounterpartswiththesamelevelofparticle vol-umefractionbutadifferentinitialparticlesize.Hencetheoptimal initialmicrostructureconsistsofalignedcuboidal␥’particleswith asizearound0.4_␮m.Thisvalueofsizehasbeenwidelyadopted asthetypicaldimensionofinitial␥’particlesformostofthe com-mercialNisuperalloys.Toexplorethepossible“optimal”volume

fractionof␥’phase,Murakumoetal.startedtheirstudybasedon TMS-75singlecrystalsuperalloy[2].Resultsshowthatthe depen-denceofthecreeprupturelifeontheamountof␥’wasmoreevident insinglecrystalsthaninpolycrystals,whilethe“optimal”amount of␥’phasewhichleadstothelongestcreeplifetimeisnota con-stantandvarieswithdifferentservicetemperature.Insomedesign modelsofNisuperalloys[19,20],theoptimalvolumefractionof ␥’phaseatdifferenttemperatureswasgenerallysetaround50%, whichwasmainlyduetothedependenceoftopologicalinversion onthe␥/␥’volume.

2.2. Raftingstage

Raftingisoneofthemostpronouncedcharacteristicsofhigh temperaturecreepdeformationinnickel-basedsuperalloysingle crystals[5,7,21,22].Duringcreeptestsatahightemperatureanda lowunidirectionalappliedstress,themicrostructureinNi super-alloysgraduallydegradesbyadirectionalcoarseningprocessof␥’ precipitates.Theinitiallyadjacentcuboidal␥’particlescoalesceand formplateletsthatturnintoplate-likeorrod-likestructures.The orientationofdirectionalcoarseningiscloselyrelatedtothe driv-ingforceofrafting,whichisproportionaltotheproductofapplied stressandthelatticemisfit[7,23].For negativemisfittingalloys (wherethelatticeparameterof␥_{precipitatesissmallerthanthat}

of␥matrix),N-typecoalescenceisobserved(i.e.raftsform nor-maltothedirectionoftheappliedstress)duringtensilecreeptest, whereasP-typecoalescenceisobtainedincompressiveloading(i.e. raftsformparalleltothedirectionoftheappliedload).Conversely, foralloyswithpositivelatticemisfit,thetensileloadingleadsto P-typecoalescenceandviceversa.Describingthemorphological changeinmicrostructuredimensionsduringraftingisadirectway tounderstandtheevolutionofmechanicalproperties.

Toquantifythekineticsofraftingprocess,thekey microstruc-turalparameters presentingthemorphologychangeneedtobe extractedandcharacterised.The␥channelwideningisan impor-tantprocessduringthecreepseeninmicrostructuresevolution, since␥channelsaretheconserveddomainswhere dislocations propagateandglideduringprimarycreep.Kamarajetal. inves-tigated the kinetics of the widening of ␥ channels in the ␥/␥’ microstructure,whoseresultsuggeststhat multi-atomdiffusion throughthe␥channelscontrolsthewideningprocess[24].Later workbySerinetal.furtherexploredtheeffectoftheleveland stateofappliedstressonthekineticsof␥channelwidening[25]. Resultsshownthat<100>tensionand(011)<011>shearcreep deformationcanequivalentlyleadtothesamemicrostructure evo-lution,while␥channelwideningratesincrease withincreasing stresslevel.

Thestructuralperiodicity␭inNisuperalloysisanother impor-tantmicrostructuralparameterandthisisdefinedasthesumofthe widthoftheN-channelsandtheextentofthe␥’phaseinthe direc-tionparalleltoappliedstress[26].Thisparametercharacterizes theglobalcoarseningof␥/␥’microstructuresince␥/␥’composite morphologyisalmostperiodicalbothfortheinitialmicrostructure andforthefullyraftedmicrostructure[27].Thisworkindicates thatthemicrostructureperioddoesnotchangeremarkably dur-ingrafting,whichmeansthatraftinginessenceisananisotropic coarseningprocess.Thegrowthof␥’cuboids,characterizedbythe firstincreasein␭,precedesthecoalescenceof␥’cuboidsintorafts takingplacewithoutagreatchangein.

Raftingisgenerallyconsideredasacreephardeningprocess, sincethemorphologicalchangein␥/␥’infactretardstheevolution ofcreepstrain[28–30].Reedetal.attributedthishardeningeffect totheclosureofvertical_␥channelsduringthemorphologychange [31];themovementofdislocationscontributingtothedeformation creepishinderedbythelamellarstructurewiththeorientation

H.Yuetal./JournalofMaterialsScience&Technology45(2020)207–214 209

normaltotheappliedstress,leadingtodislocationaccumulation andrearrangementonthe␥/␥’interface[12,13].

2.3. Stablepost-raftingstageandtopologicalinversion

Generally,therafting process is terminatedat orbeforethe stablecreepstage.Thefullyrafted lamellar␥/␥’microstructure can hereafter maintain their morphology for a relatively long time, withonlyminor coarseningbehaviourof _␥’plates taking place[27,32].Thisstablepost-raftingcontrolsthemicrostructural dynamicrecoveryprocess,leadingtoaveryslowaccumulationof creepdeformation.

Attheendofcreepstablestagewithasufficienttimeforthe rearrangementofthelamellarmorphology,the␥’phasegradually interconnectsandisnolongerconfinedbythe␥channels.This evo-lutionleadstoaneffectcalled“topologicalinversion”:the␥’phase nowsurroundsthe␥phaseandtopologicallybecomesthematrix [8,33–36].Themomentoftopologicalinversionhasbeenprovento bestronglydependentontheinitialvolumefractionof␥’phases [37].Thereasonfortheformationofinversedmicrostructurehas beenstudiedindetail bymanyresearchers[8,9,37].Itis gener-allyacceptedthataftertheraftingstage,themisfitstresseson_␥/␥’ interfacearereleasedbythedislocationnetwork,andthenthe␥’ precipitatesizeandmorphologywillevolvesuchastominimize thetotalinterfacialenergy.Thisevolutionisobtainedbya reduc-tionin␥/␥’interfaceareadrivenbydiffusion-controlledcoarsening ofthe␥particles.ForthemodernNisinglecrystalsuperalloyswith atypical␥’fractiongreaterthan50%atservicetemperature,the topologicalinversionofthe␥/␥’microstructureisduetothe min-imizationofthe total ␥/␥’interfaceareawiththeminor phase embeddedinthemajorone.Theoccurrenceoftopological inver-sionisgenerallyseenastheindicatorofonsetofacceleratedcreep [8,36].

2.4. Break-upstage

Aftertheinversionofthe␥/␥’microstructure,thecreep resis-tance degrades, rapidly which finally leads to the failure of superalloys.Mughrabiascribesthisdeteriorationtoarapidincrease in deformation induced by internal stresses, which cannot be releasedanylongerbytheinterfacedislocationsoncethe␥phase isbeingsplitintodiscreteislands[38]. Otherstudies[33,39,40] haveindicatedthatintheinversedmicrostructuretheshapeof␥/␥’ interfaceschangesfromsmoothintozigzag,leadingtothe forma-tionofnewdislocationglideplanesin␥’,whichcorrespondingly promotetheformationofdislocationpile-upsand consequently cuttingofthe␥’phase.

2.5. Modellingofmicrostructureevolution

To interpret and further predict the directional coarsening behaviourofNisuperalloys,alargenumberofmodelshavebeen proposed[32,41–56].Emphasis hasbeenputonexplainingthe orientationofraftingbasedondrivingforceofmorphology varia-tion.Andreexplainsthebehaviourinanelasticframework,where theelasticenergyiscalculatedasafunctionoftheparticleshape, theapplied stress and theratio between theYoung’s modulus oftheprecipitatesandthatofthematrix [41]. Theinfluenceof plastic strainswasadded byconsideringthe effectofunevenly distributedinterfacialdislocations.Meanwhiletheanisotropyin releasingthecoherencystresseshavebeenalsosimulated[42–44]. Theoriginallocalstressesinducedbythelatticeandmodules mis-matchdistributedevenlyin␥matrix.Whentheexternalstresses were appliedonthe materials,the local stateof stress in _␥/␥’ microstructurehasbeenmodified,whichleadtoanisotropic coars-eningbehaviours. Theorientation dependenceof elasticenergy

fordifferentmorphologieswasthenquantifiedforthe combina-tionsoftransformationstrainandmatrixplasticstrain[45].These abovementionedmodelshavepointedoutthatdrivingforceson raftingareinessencethelatticemisfit,theexternalloading,and thedifferenceinelasticconstantsbetweenthetwophases.

Besides,otherstudiesfocusedonquantitativelydescribingthe kinetics of the morphology changes during rafting. Svodoba’s model takesinto accounttheinteractions betweendislocations and channelsof_␥matrix [46].Through modellingthe coarsen-ingbehaviourof␥’particles,theintricatecreepbehaviourcanbe wellcaptured.Fedelichandco-workersmadesystematicstudies toquantify themicrostructuralparameters of commercialalloy CMSX-4duringraftingprocess[32,47,48].Thedirectionwidening behaviourof␥channelwidthisperfectlyfittedasafunctionof time,stressandtemperature,andthemorphologydescriptionwas furtherappliedtobuildtheconstitutivemodelbyconnectingwith creepproperties[49].

Recently,thephasefieldmethodhasemergedasapowerfulway tocapturethemorphologychangeinNisuperalloys.Earlyphase fieldmodelswerebasedonaelasticityframeworkonly[50,51], while thecontributionsfromplastic strainswereintroducedin latermodels[52–54].Theprincipalcriteriainallabovementioned modelsremainthelatticemisfit,theelasticinhomogeneityandthe appliedloadasthedrivingforceforthedirectionalcoarsening.The effectsofdiffusiononthekineticsofraftinghavebeenstudiedby couplingthephase fieldcalculations witha CALPHADapproach [55,56].ResultselucidatethatReadditionretardsthekineticsof raftingbyslowingdownthemobilityof␥/␥’interface.

3. Evolutionofdislocationstructures

3.1. Dislocationsininitialmicrostructure

ForthecommercialNisinglecrystalsuperalloyswiththe typi-calmicrostructureofalignedcubic␥’embeddedin␥matrix,the well-organized ␥/␥’ coherent initial microstructure is obtained by a multi-step solution and aging treatment followed by a slow air-cooling process. The density of dislocations in the initial microstructure will be at a sufficiently low level after thehigh-temperatureheattreatmentand theas-processed␥/␥

microstructurecanbeapproximatelyconsideredasa dislocation-depletedstate.

Whenthestressisappliedduringtheearlystagesofcreep,the dislocationsareprimarilyemittedfromthelow-angleboundaries andpercolateintothenarrow␥channels.Ascreepdeformation proceeds,themobileindividualdislocationsbecomesequentially active and start togenerate short avalanches of creep disloca-tionsinboundary-freeregions.Upontheiractivation,thedensity ofcreepdislocationsinboundary-freeregionsrisesbytwoorders ofmagnitude.Experimentalobservationatthisstageshowsthat the␥/␥_{interfacesparalleltotheappliedstressarenearlydevoid}

ofdislocation,whiledislocationsarefrequentlyobservedonthe␥ channelsnormaltothestressaxis[13].Theanisotropicdistribution ofdislocationsindifferent␥channelsisascribedtotheeffectof uni-directionalappliedstress,whichreleasesthemisfitstraininparallel channelswhileintroducinghigherstrainsinthetransverse chan-nels [14,57].The deformationmechanismduring primarycreep hasbeengenerallyinterpretedastherapidgrowthindislocation densityandthefillingof␥channelswithdislocations[33,58,59]. 3.2. Formationofinterfacialdislocationnetwork

As the microstructure turns into rafts, the dislocations will accumulateatthe␥/␥_{interfacesandformplanardislocation}

210 H.Yuetal./JournalofMaterialsScience&Technology45(2020)207–214

Fig.2. 2Dsketchforthesuperdislocationscuttinginto␥’rafts[70].

oftwo dislocations withsame Burgers vectortorearrange and generatenewdislocations[14,60–62],whileEggerlerandDlouhy alsoobservedtheformationofnewdislocationson␥/␥’interface contributedbytheinteractionoftwodislocationswithdifferent Burgersvector[63].Theformationofinterfacialdislocationsis usu-allybelievedtobedrivenbythereleaseoflatticemismatchstrain betweenthe␥’precipitateand␥matrix[12,13].Themagnitude oflatticemisfit is thereforeof vitalimportance in thesenseof determiningtheformationofthedislocationnetwork.Zhangetal.’s experimentalresults[64]providedirectevidenceforthe disloca-tionnetworksevolutionduetolatticemisfit.Inalloyswithalarge negativelatticemisfit,thedislocationsrearrangeinthe␥channels andformmorecompleteandcondenseddislocationnetworksand atahigherspeedthaninalloyswithasmallmisfitHaradaand co-workers[65]haveisolatedaquantitativecorrelationbetweenthe dislocationdensityoftheequilibriumnetworkandtheircreep per-formanceinsinglecrystal,withthedislocationspacingbeingfound tobeproportionaltothelogarithmoftheminimumcreeprate.The positivecorrelationbetweencreepstrengthandinterfacial disloca-tiondensitywasalsoprovenbyZhangetal.’swork[66].Basically, theformationofinterfacialdislocationnetwork helpstorelease the␥/␥’misfitstrain,meanwhilecontributestothestrengthening mechanismof␥/␥’lamellarstructurebyhinderingthecuttingthe mobiledislocationsinthematrix.

3.3. Sustenanceofthestabledislocationnetwork

Experimentalobservationshaveshownthatthenetworkscan maintaintheirmorphologyanddensityduringthesecondarycreep stage[62,67,68].Asmentionedabove,theinterfacialdislocation networkcanactasdislocationsinkstoabsorb/accommodatethe matrixdislocations,therebyeffectivelypreventingmatrix disloca-tionsfrompilingupattheinterface[69].Moreover,thedislocation networksalsoprovidedislocationpinstopreventthematrix dis-locationsfromcuttingintothe␥_{precipitate[11].Thestabilityof}

dislocationnetworkhelpstocorrespondinglystabilizethe lamel-lar␥/␥’structureduringthestablecreepstage.Soclearlythereis mutualstabilisation.

The stable stage for ␥/␥’ interfacial dislocation network corresponds to the creep stable stage, which presents the

microstructural dynamic recovery process with a very slow accumulation ofcreep deformation.Srinivasan investigatedthe recoverybehaviourofmobiledislocations,focusingonthe mech-anismof ␥’ rafts cutting[70]. Experimental observations show thattwodislocationswithdifferentBurgers’vectorsin␥channel jointlyshearthe_␥’phasebyforminganon-compacted superdis-locationduring[001]tensile creep.Thetwosuperpartialsmove into the ␥’ phase by a combined process of glide and climb, which isequivalenttothemigration ofsinglesuperdislocation. The2Dschematicillustration isshownin Fig.2.Similarresults wereprovedbyexperimentsfrommanyotherresearches[71–74], whilethesuperdislocation-cutting-␥’mechanismhasbeenwidely acceptedasthedominaterecoverymechanismduringcreepstable stage.Accordingtothisclimb-plus-glidecuttingprocess,the rate-controllingstepincreepisessentiallytheclimbrateofdislocations in␥’.

Inaddition,EpishinandLinkproposedanothermechanismfor dislocationmovementduringstablecreep[33].Heredislocations inthe_␥channelssegregateon_␥/␥’interfaceduringprimarycreep, thenmovetransverselytotheappliedstressbyacombinationof glideandclimb,whiletheglidingdislocationsfirstgetpinnedby␥’ rafts.Butwiththehelpofosmoticforcesproducedbytemperature andstress,trappeddislocationscanclimbsawayfromitsoriginal slipplane.Thentheclimbingdislocationcanglideinanewslip planeagainonceitsclimbdistanceislargeenoughtoaffordanew glidestepbeforeitisblockedagainby␥’rafts.The2Dsketchof dis-locationmovementisshowninFig.3.Asaresult,thedislocationsin thematrixchannelscanmovealongthe␥/␥’interfaceinazig–zag mannerasshowninthefigure.Unlikethecollectiveshearingof␥’ rafts,dislocationsinthismechanismarecompact,andmigrating onlyin␥matrix.Asimilardescriptionofthedislocationmigration on_␥/␥’interfacescanalsobefoundinRefs.[75,76].

3.4. Break-upofinterfacialdislocationstructure

The␥/␥’microstructurebecomesinversedwhencreepenters the tertiary stage. A large part of the dislocation network is still present onthe _␥/␥’ interfaces, but some are locally dam-agedinalater periodofcreep.Thedamaged networkslosethe co-ordinating role of maintaining the dynamic equilibrium, so

H.Yuetal./JournalofMaterialsScience&Technology45(2020)207–214 211

Fig.3.2Dsketchforthemotionofadislocationon␥/␥’interface[75].

thatalargenumberofdislocationsarepiledupatsomeregions wherethenetworkshavebeendamaged,givingrisetothelocal stressconcentration[14].Experimentalresultshaveshownthat the{111}-typedistortedinterfaceformsasaconsequenceofthe inversed microstructure [73,74,77]. The zigzag interface forms throughdislocationswhichcutintothe␥’raftvia{111}planesof interface[40].Theacceleratedcreepratecanthenbeattributedas theincreaseddislocationactivitywiththebreak-downof disloca-tionnetworksandtheformationofnewglidingplanes.

3.5. Modellingthedislocationevolution

Based on a constitutive model for precipitate-strengthened alloysfromDyson[78],Zhuetal.hasdevelopedamodelby simulat-ingtheinteractionbetweenthedislocationandthe␥’particleswith acuboidmorphology[76]. Therate-controllingstepis assumed tobeclimbofdislocationsatthematrix/particleinterface,hereby thecreepratecanbemodelledasa functionofmicrostructural parameteraswellasintrinsicdislocationdiffusivity.

Atthemesoscopicscale,themaincomputationaltooltostudy thedynamiccollectiveevolutionofdislocationsunderthe exter-nalloadingisdiscretedislocationdynamics(DDD)simulation.The applicationofDDDinNisuperalloyswasprimarilyconcernedwith superdislocations in the␥’ phase, focusing onthe role of cube sizeandshape,phasevolumefraction,temperatureandanti-phase boundary energy [79–81]. Attention hasbeen paid tothe raft-ingdomaintotackledislocationplasticityin␥channelsof␥/␥’ microstructures,whichmakesdislocationclimbandvacancy dif-fusionbecomeunavoidablekeyissues.InHafezHaghighatetal.’s work,themovementofdislocationsalongthe␥/␥_{interfaceclose}

tothe␥’cubewasinterpretedasacombinationofdislocationglide andclimb,whilethenetdislocationmotionisclimbcontrolled[82]. WorkdonebyGaoetal.explicitlytakesintoaccountthe vacancy-diffusion-coupleddislocationclimb,whichshowsthatdislocation climbcanrearrangethedislocationconfigurationtorelaxthe hard-eningduetodislocationfilling␥channels[83].Yangetal.’swork alsoputsemphasis onsimulatingthediffusion-induced disloca-tionclimbtostudytheprimarycreepandearlyplasticbehaviour [75]. The influencesof microscale vacancysupersaturation and mesoscalephasemorphologyweredescribedindetail.UsingaDDD model,theeffectofinterfacialdiffusiononthecreepbehaviourhas beeninvestigatedbyShishvanetal.[84].Resultsshowthat interfa-cialdiffusionhelpstorearrangethedislocationdistribution,aswell astorelaxtheback-stressesinducedbydislocationspile-up.The DDDmodelhasalsobeencombinedwithfiniteelementmethodby Songtoinvestigatetheinfluenceofinterfacialdislocationnetworks andlatticemismatchonthedynamicsofdislocationevolutionin thematrixchannels[85].

Fig.4.TheminimumcreeprateofcommercialNisinglecrystalsuperalloysasa functionofinterfacialdislocationspacingduring1100◦_{C,137MPacreep[11,57,65].}

4. Discussion

4.1. Thedependenceofminimumcreeprateoninterfacial dislocationdensity

As shown in Fig.4, theminimum creeprate of commercial Nisinglecrystalgradesincreaseslinearlywiththespacingofthe interfacialdislocations[11,57,65].Thisphenomenological relation-shipcangiveanintuitivefirst-orderconnectionbetweenthecreep propertiesandthedislocationproperties.However,theparameter dislocationspacingishighlydependentontheaccuracyof experi-mentalobservation.Moreover,theapplicationofthisrelationship toquantifythecreepbehaviourofexistingsuperalloysorfurther predictthepropertiesofnovelsuperalloys,islargelyinfeasible.To buildaconstitutiverelationshipwithphysicalmeaningsinstead offittingequations,moredetaileddescriptioninthedependence ofdislocationbehaviourneedstobemade,whilethe dislocation-behaviour-baseddeformationmechanismneedstobetakeninto consideration.

4.2. Simulationofdislocationbehaviour

Formal statementshave shown that the layeredfully-rafted structure coupled with␥/␥ _{interfacial dislocationnetwork can}

remainstableandkeepalmostunchangedforarelativelylongtime untilrupture.Therefore,theinvestigationofcreeppropertiesas thedependenceofdislocationbehaviourshouldbefocusedonthe post-raftingstageasthemostlong-lastingstage.Asstatedbefore, Srinivasanetal.’sresearchhasshownthatthe␥’raftscanbecut bythesuperdislocationthroughacombinedprocessofglideand climb,wheretheclimbingvelocityofsuperpartialsarethe control-lingfactorsofcreeprate[70].Thefollowingdiscussionaboutcreep isbydefaultreferredtothe[001]tensilecreep.Basedonthisresults, theclassicdislocationclimbmodelfromAndersonetal.[86]was employedasafirstorderapproximationtoestimatethedislocation mobility,wheretheclimbingrateofdislocationsin␥’phase

v

chas

beenpresentedasfollows:

v

c=

2Ds’(e xxVa’

kT −₁₎

bln(R/b) (1)

whereDs isthediffusivityof␥,xx istheappliedstress,Va is

theatomicvolumeof␥_{,bistheBurger’svector,R}_istheaverage

dislocationspacingin␥_andR’≈



1

_⁄

_

’ and␥isthedislocation densityin␥_phase.

212 H.Yuetal./JournalofMaterialsScience&Technology45(2020)207–214

Forthedislocationzig-zagmovementalongthe␥/␥_interface

proposedbyEpishinandLink[33],thecreepratecontrolling fac-torofthismechanismissimilarlythedislocationclimbrate.Here thecompactdislocationsmovealong␥/␥_{interfacewithout}

super-partials,andthe␥matrixisthedomainofdislocationmigration. Andersonetal.’stheorywasagainemployed[86],andtheclimbing rateofdislocationon␥/␥_{interfacewasshownasfollows:} vc= 2Ds(e

xxVa

kT −₁₎

bln(R/b) (2)

whereDsisthediffusivityof␥,Vaistheatomicvolumeofg,Rstands

fortheaveragedistanceofdislocationsonthe␥/␥’interface.The interfacialdislocationdensityofNisuperalloyshasbeenreported bymanyresearchersasafunctionoflatticemisfit[68],wherethe spacingofdislocationnetworkR=



b



_⁄



_ı



_{.Hereıisthelatticemisfit} presentingbyı= a’−a

a .a␥anda␥arethelatticeparameterof␥’

and␥phase,respectively.Sotheequationwasrepresentedinthe followingway:

v

c= 2Ds(e xxVa kT −₁₎ −bln



ı



(3)

In this work,the lattice misfit is obtained, from the lattice parametersof␥/␥_{phasescalculatedthroughtheirmolarvolumes}

a= 3



4Vm

NA .Forthecalculationofdiffusioncoefficientinthe

multi-elementalloysystem,aharmonicmeanofcalculationisselected basedonZhuetal.’swork[76],i.e.,Ds= 1

⁄



_mx_Dmm

0

.

Bynowthecreeprate-controllingbehavioursofdislocationsin ␥and␥ _{phase havebeenpresentedrespectively.Afterfiguring}

outtheunderlyingdeformationmechanismoriginatingfrom dis-locationbehaviour,thecreepbehaviourofsuperalloyscanbethen interpretedasthecombinationofthesetwomechanisms. 4.3. Simulationofminimumcreeprate

For the fully rafted ␥/␥ _{lamellar microstructure, the creep}

behaviourcanbeapproximatelyequaltothecompositesreinforced bythecontinuouslamellae,whichareorientedperpendicularto theappliedstress.Assuminganiso-stresscondition[87],thestrain rateofalloycanbepresentedbythefollowingequation:

˙ε= ˙˙εV+ ˙εV_’ (4)

whereVandVarethevolumefractionof␥and␥_phase

respec-tively.AccordingtotheOrowanlaw,thestrainrateingandg’phase canbewrittenas:

˙

εε˙=Mbvc (5)

where␥ stands for thedislocationdensityin ␥; and M isthe Schmidfactor.

However,the abovementioned equation describesthecreep propertiesofalloyswitha lamellarmicrostructurewithout tak-ingintoconsiderationthedistributionoflamellaespacing,andthe effectofinter-spacingoflamellaeneedtobefurtherinterpreted. WhitelyproposedamodelbasedontheBailey-Nortoncreep equa-tionto describeAl-CuAl2 eutectic alloywith lamellar structure

[88],where dislocation-motion related microstructural parame-ter,inter-lamellarspacingL,wasadded,asshownbythefollowing equation

˙ε∝Ln (6)

wherenisthelamellarspacingexponent.Theoreticallynshould beequalto1foraperfectlyalignedandcontinuousfiber(orplate)

Table1

Microstructuralparametersandphysicalconstantsusedinequation[70,76].

Parameter Value

Lamellarspacingexponent,n 1.5

Schmidfactor,M 1/√6

Burger’sfactor,b 2.5×10−10_m

Dislocationdensityin␥’phase,␥’ 109m−2

Dislocationdensityin␥phase,␥ 1011_m−2

Initialsizeof␥’particles,␻0 4×10−7m

reinforcedcomposite,butitwillbeashighas3iffiber(orplate) rup-tureoccurs.Theapplicationofequationcanalsobefound[89,90] withsimilarwell-alignedlamellarstructure.Inthisworkthe lamel-larinter-spacingcanbedefinedasthechannelwidthof␥_phaseto

presenttheperiodicalmicrostructure.Accordingtothegeometrical relationship,Lcanbeexpressedinthefollowingway:

L= V’ 1−V’1/3

ω0 (7)

whereω0istheinitialsizeof␥particles.

Combiningall,thecreepratecanbeexpressedbythefollowing equation: ˙ε=Ln( 2MVDs’



exxVa’kT −1



ln



R_b



− 2MVD_s(e xxVa kT −₁₎ ln



ı



)(8) ThecreeppropertiesofNisuperalloyscanbepresentedby dis-locationbehaviourthroughbuildingtheclimb-assistedequation withtheconsiderationoflamellarmicrostructure.Inequation,the parameterssuchasphasevolumefractionVandV’,diffusivity DsandDs,andlatticemisfitıarethermodynamicfactorsthatare

stronglyrelatedtochemicalcomposition.Themechanical prop-ertiesofsuperalloyscanbethenconnectedwiththecomposition whenthesethermodynamicparameterscanbeproperlyobtained asaconsequenceofcomposition.WhilethemodelasgivenbyEq. (8)isprobablyconceptuallycorrect, thelargenumberof physi-calparameterswhichvalues cannotbeobtainedindependently, impliesthatthemodelcannoteasilybeusedforthecompositional optimisationofexistingalloys,northedesignofnewsuperalloys.

Inthefield ofsuperalloys, theimplementationof CALPHAD-basedthermodynamicmodelsisnowanemergingapproach,where equilibriummicrostructuralfeaturesofcomplexmulti-component alloys,suchasphasefraction,elementpartitionanddiffusivity,can bewellcapturedbyusingGibbsfreeenergydatabases.Valuesfor theparameterssuchasVf,Vm,xm,Dswerecalculatedvia

Thermo-CalcusingTCNI9andMOB2database.Sincetheminimumcreep rateisalsoafunctionofthe(a-prioriunknown)dislocationdensity in␥and␥’phaseinthefollowingsimulationsitwaspre-setasa constant.Thisisaslightsimplificationbuthelpsustoillustratethe effectofcomposition-relatedfactors.Themicrostructural param-etersandphysicalconstantsusedinequationarelistedinTable1 [70,76].

4.4. ValidationofminimumcreeprateinexistingNicommercials Toillustratethemodel’scapability inreproducing thecreep propertiesof superalloys,existing commercialgradesofNi sin-glecrystalshavebeenemployedwiththeirchemicalcompositions showninTable2.Basedonthecomposition,thevolumefractionas wellasthediffusivityof␥and␥_{phaseinexistingalloysat1100}◦_C

arecalculatedandshowninFig.5.

InFig.5(a),thesquaredotswithblackandgreycolourindicate thecalculateddiffusioncoefficientin_␥and_␥phasesrespectively. Forallalloys,almostnodifferencecanbefoundinthediffusivity of␥_{phase,duetotherelativelysmallsolubilityofalloying}

ele-H.Yuetal./JournalofMaterialsScience&Technology45(2020)207–214 213 Table2

ChemicalcompositionsofcommercialNisinglecrystalsuperalloys(wt%)[65,77,91–94].

Sample Al Co Cr Hf Mo Re Ru Ta Ti W Ni TMS-75 6 12 3 0,1 2 5 – 6 – 6 Bal. CMSX-4 5,6 9 6,5 0,1 0,6 3 – 6,5 1 6 ERBO/39 4,4 8,92 5,11 – 0,97 – – 6,7 3 9 ERBO/38 5,54 8,71 5,12 – 0,95 – – 6,54 0,79 9,05 ERBO/37 6 8,74 5,14 – 0,95 – – 6,56 – 9,09 ERBO/36 5,65 8,89 5,23 – 0,96 – – 6,67 – 6,19 ASTRA100 6,13 8,92 5,25 – 0,97 – – 6,7 – 6,19 CMSX-10K 5,7 3 2 0,03 0,4 6 – 8 0,2 5 TMS-138 5,9 5,9 2,9 0,1 2,9 4,9 2 5,6 – 5,9 TMS-162 5,8 5,8 2,9 0,1 3,9 4,9 6 5,6 – 5,8 LSC-15 4 6 7 0,1 1,5 – – 5,5 – 10 SX-0Ru 6 – 4 – 1 4 0 5 0,5 5 SX-2Ru 6 – 4 – 1 4 2 5 0,5 5 SX-4Ru 6 – 4 – 1 4 4 5 0,5 5

Fig.5.Calculateddiffusivityandvolumefractionof␥and␥’phase,aswellasthelatticemisfitofexistingNisinglecrystalgradesat1100◦C(a)andtheexperimentalminimum creepratesofalloysat1100◦C,137MPa[65,91,92](b).Thelinesconnectingtheindividualdatapointshavenophysicalmeaningandareonlyaddedtoguidetheeye.The orderofthesuperalloysalongthex-axisisbasedonthegenerationofNicommercialsinglecrystalsuperalloys.

mentsintheNi3Alintermetallic.Incomparison,thediffusivityin

TMS-138andTMS-162alloysissignificantlysmallerthanthatin theotheralloys,whichisunderstandableduetothemoreheavily alloyingbyrefractoryelements(suchasReandMo)inTMSseries. Also,thecalculatedresultsdemonstratethatdifferenceof diffusiv-ityin␥_{issmallbutthatin␥isbig,whichimpliesthediffusion}

behaviourin␥phaseisthedominatemechanismindeformation whichleadstothedifferenceincreepbehaviourforNicommercial grades.Therounddotspresentthephasevolumefractionsof list-ingalloys,whereASTRA100,ERBO/36,ERBO/37andERBO/38alloys havearelativelylowvaluesof␥_{volumefractionaround35%.The}

redtriangledotsshowthecalculatedlatticemisfitsofallalloys.The TMSalloysgenerallypossesslargenegativemisfits,whilethelattice mismatchofotheralloysfluctuatesaroundzerowithsmall differ-ence.Fig.5(b) showstheexperimentalminimumcreepratesof thesealloysat1100◦Cand137MPa.TMS-162alloypronouncedly outperformsotheralloyswhileERBO/36hastheworstperformance increep.Theinfluencefactorscorrespondingtothe composition-relatedparametershavebeenanalysed.

Fig.6showsthecomparisonbetweentheexperimental min-imum creep rates of existing Ni grades, and the calculated minimumcreepratesobtainedfromthermodynamicsimulations. Theselectedexperimentaldataareobtainedfromdifferent temper-atureandstressranges[65,77,91–94].Resultsshowthatcompared totheexperimentalresults, thecreeprates aregenerally over-estimatedbythesimulations,asindicatedbythereddashline. Butitisworthpointingoutthismodelbasedonthesimulationof dislocationmovementcanwellreproducethecreeppropertiesof existingNisuperalloysatdifferenttemperatureandstressranges. Hence,throughthermodynamicandkineticcalculations,the

chem-Fig.6.SimulatedminimumcreeprateofexistingNicommercialgradescompared withtheexperimentalresultsobtainedfromtheliterature[65,77,91–94].

icalcompositionsofNisuperalloyscanbesuccessfullycoupledto theircreepperformancebythismodelwithoutmakingthe dislo-cationspecificationsthemselvescompositiondependent.

5. Conclusions

(1)ThehighperformanceofNisinglecrystalsuperalloysduring hightemperaturelow stresscreepservice, isprimarily con-trolledbythecombinedeffectsofmicrostructuralevolution, namelytheformationofraftinglamellae,andthedislocation

214 H.Yuetal./JournalofMaterialsScience&Technology45(2020)207–214

behaviour,i.e.,thewell-arrangeddislocationnetworklocated onthe␥/␥_interface.

(2)Duringthesecondarycreepstagewhichtakeslongesttimeof creeplife,two main recoverymechanism basedon disloca-tionmigrationdominatetheprocess.Oneissuperdislocations shearing into ␥’ rafts through a two-superpartials-assisted approach.Anotheristhecompactdislocationsmigratingalong ␥/␥_{interface.Thesetwomechanismsaresimilarly}

climb-rate-controlledprocess.

(3)Amodelfortheminimumcreepratebasedonthermodynamic andkineticcalculationsandusinganexistingdetailed disloca-tiondynamicsmodelhasbeenbuiltbytakingthedislocation migrationbehavioursaswellastheraftedmicrostructureinto consideration,whichcanwellreproducethecreepproperties ofexistingNisuperalloygrades.

Acknowledgements

ThisworkwasfinanciallysupportedbytheNationalNatural Sci-enceFoundationofChina(No.51722101)andtheKeyResearchand DevelopmentProject(No.2017YFB0703001).

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