Evolution of the Annealing Twin Density during

14 Pages·2017·4.97 MB·English

Evolution of the Annealing Twin Density during
δ-Supersolvus Grain Growth in the Nickel-Based
Superalloy Inconel™ 718
Yuan Jin, Marc Bernacki, Andrea Agnoli, Brian Lin, Gregory S. Rohrer,
Anthony D. Rollett, Nathalie Bozzolo
To cite this version:
Yuan Jin, Marc Bernacki, Andrea Agnoli, Brian Lin, Gregory S. Rohrer, et al.. Evolution of the
Annealing Twin Density during δ-Supersolvus Grain Growth in the Nickel-Based Superalloy Inconel™
718. Metals, 2016, 1, ￿10.3390/met6010005￿. ￿hal-01248299￿
HAL Id: hal-01248299
https://hal-mines-paristech.archives-ouvertes.fr/hal-01248299
Submitted on 24 Dec 2015
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Article
Evolution of the Annealing Twin Density during
δ-Supersolvus Grain Growth in the Nickel-Based
Superalloy Inconel™ 718
YuanJin1,*,MarcBernacki1,AndreaAgnoli1,BrianLin2,GregoryS.Rohrer2,
AnthonyD.Rollett2andNathalieBozzolo1,*
Received:1November2015;Accepted:17December2015;Published:24December2015
AcademicEditor:JohanMoverare
1 MINESParisTech,PSL—ResearchUniversity,CEMEF—Centredemiseenformedesmatériaux,
CNRSUMR7635,CS10207rueClaudeDaunesse,SophiaAntipolisCedex06904,France;
[email protected](M.B.);[email protected](A.A.)
2 DepartmentofMaterialsScienceandEngineering,CarnegieMellonUniversity,5000ForbesAvenue,
Pittsburgh,PA15213,USA;[email protected](B.L.);[email protected](G.S.R.);
[email protected](A.D.R.)
* Correspondence:[email protected](Y.J.);[email protected](N.B.);
Tel.:[phone](Y.J.);[phone](N.B.);Fax:[phone](Y.J.);[phone](N.B.)
Abstract: Grain growth experiments were performed on Inconel™ 718 to investigate the possible
correlation of the annealing twin density with grain size and with annealing temperature. Those
experimentswereconductedatdifferenttemperaturesintheδsupersolvusdomainandundersuch
conditions that only capillarity forces were involved in the grain boundary migration process. In
the investigated range, there is a strong inverse correlation of the twin density with the average
grain size. On the other hand, the twin density at a given average grain size is not sensitive to
annealingtemperature. Consistentwithpreviousresultsforpurenickel,thetwindensityevolution
inInconel™718islikelytobemainlycontrolledbythepropagationofthepre-existingtwinsofthe
growinggrains; i.e., thelargestonesoftheinitialmicrostructure. Almostnonewtwinboundaries
are created during the grain growth process itself. Therefore, the twin density at a given average
grainsizeismainlydependentonthetwindensityinthelargestgrainsoftheinitialmicrostructure
andindependentofthetemperatureatwhichgrainsgrow. Basedontheobservations,ameanfield
modelisproposedtopredictannealingtwindensityasafunctionofgrainsizeduringgraingrowth.
Keywords: annealingtwin;graingrowth;EBSD;meanfieldmodel
1. Introduction
Nickel-based superalloys are used for aeronautical component manufacturing because of their
performance at high temperature. Grain boundary engineering (GBE) is a possible route to
improve the properties, especially those related to intergranular damage [1]. Because of their low
energy[2],annealingtwinboundaries,observedinalmostalldeformedandsubsequentlyannealed
face-centered-cubic (FCC) metals with low to medium stacking fault energy, are fundamental for
GBE[3–5].Eventhoughthesecrystallinedefectshavebeenknownforalongtime[6],themechanisms
bywhichtheyappeararestillnotfullyunderstood. Beingabletopredictthetwindensityobtained
afteragiventhermomechanicalpath,whichhasbeenmadeonanempiricalbasisfornow,wouldbe
veryvaluablefordevelopingGBEroutes.
Thegrowthaccidentmodel, whichassertsthatacoherenttwinboundaryformsatamigrating
grain boundary due to a stacking error, is most commonly used to explain annealing twin
Metals2016,6,5;doi:10.3390/met6010005 www.mdpi.com/journal/metals
Metals2016,6,5 2of13
formation [7–9]. In the growth accident model, the amplitude of the driving force acting on grain
boundary migration and the resulting migration velocity are promoting factors for the generation
of annealing twins. However, few studies in the literature address the direct effect of the grain
boundarymigrationrate.Theaimofthepresentworkistocontributetofillingthisgap.Graingrowth
experiments are performed on a nickel-based superalloy (Inconel™ 718) at different temperatures,
so that grain boundaries migrate at different rates. The twin content evolution is discussed
quantitatively in relation to temperature, grain size increase and grain boundary migration rate.
A mean field model is proposed based on the observations to describe the average twin density
evolutionasafunctionofgrainsizeduringgraingrowth.
2. ExperimentalDetailsforQuantitativeAnalysisofTwinContent
Grain growth experiments were performed on semi-cylindrical samples machined from an
Inconel™ 718 billet at three different temperatures above the solvus for the delta phase and for
differenttimes(Table1).
Table 1. Annealing conditions (annealing times for each temperature) applied on the initial
microstructureshowninFigure1toobtainthoseshowninFigure2.
Temperature AppliedAnnealingTime
1025˝C 10min 25min
1065˝C 3min 6min 9min
1100˝C 1min 2min 3min
All of the analyzed samples (longitudinal section) were metallographically prepared with a
prolongedfinalmechanicalpolishusinga0.5-µmcolloidalSiO suspension. Theelectronbackscatter
2
diffraction(EBSD)mapswereobtainedwithaZEISSSUPRA40FEGSEM(Jena,Germany)equipped
with a Bruker CrystAlign EBSD system (Berlin, Germany). The step size for EBSD map acquisition
wassetto0.46µmfortheinitialmicrostructure. However,astheaveragegrainsizeintheannealed
samplesbecamelarger,thestepsizewasaccordinglyincreasedto1.44µm,whichappearedasagood
compromise for measuring a sufficient number of grains, but still completing each map within a
reasonableacquisitiontime. Themappedareaoftheinitialmicrostructurecontainsmorethan5000
grains.TheEBSDmapsrecordedaftergraingrowthinclude150to800grains(twinsbeingexcludedin
graincounting).TheOIM™software(EDAX,Mahwah,NJ,USA)wasusedtoanalyzetheEBSDdata.
Grainboundariesweredefinedasboundarieswithamisorientationangleabove5˝. Annealing
twinboundariesaredefinedbyamisorientationof60˝aboutthe<111>axiswithatoleranceof8.66˝,
according to Brandon’s criterion [10], regardless of their coherent versus incoherent character. The
Brandon’scriterionwasappliedbecauseanF.C.C.twincanalsobedepictedasanΣ3coincidencesite
lattice. Two different quantities were used to quantify annealing twins: the annealing twin density
(N )andthenumberofannealingtwinboundariespergrain(N ),respectivelydefinedas:
L G
L 2
N “ tb ˆ (1)
L S π
N ´N
N “ 2 1 (2)
G
N
1
whereL isthetwinboundarylengthdetectedinagivensamplesectionareaS,N isthenumberof
tb 1
grainsignoringtwinboundariesinthegraindetectionprocedureandN isthenumberofgrainsby
2
consideringtwinboundariesasgrainboundaries. Theformulationtocalculatethetwindensity(N )
L
isderivedfrom[11].
The recrystallized grains were identified in the EBSD maps using a criterion that the grain
orientationspread(GOS)waslessthan1˝ [12]. Twinboundarieswereignoredinthegraindetection
Metals2016,6,5 3of13
Metals 2016, 6, 5 3 of 13
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atdlaercgreearsgesr aaitn lasrigzeers .grTaihne sitzwesin. Tdheen tswitiyn idnenthsietyg irna itnhes gorfaainsg iovfe an ggivraenin gsraizine scilzaes scliasssc aisl ccuallcauteladtewd ith
thewittwh itnheb towuinnd baoryunldeanrgyt hleningthth iens ethgesrea ignrsaiannsd antdh ethseu srufarcfaeceo coccucuppieidedb byy tthheessee ggrraaiinnss. .ThTish iisniitniailt ial
microstructure results from the prior recrystallization process. It has been established that the twin
microstructure results from the prior recrystallization process. It has been established that the twin
density after recrystallization is mainly controlled by the stored energy level [13,14] and that the
densityafterrecrystallizationismainlycontrolledbythestoredenergylevel[13,14]andthatthetwin
twin formation event occurs more often when the grains are growing faster [9,15]. The migration
formation event occursmore often when the grainsare growing faster [9,15]. The migration rate of
rate of the recrystallization front decreases during the recrystallization process because the
therecrystallizationfrontdecreasesduringtherecrystallizationprocessbecausetheremainingstored
remaining stored energy level in the deformed matrix is decreasing. Therefore, the growth rate of a
energylevelinthedeformedmatrixisdecreasing. Therefore,thegrowthrateofarecrystallizedgrain
recrystallized grain decreases, and fewer twins are formed. Consequently, the twin density
decreases, and fewer twins are formed. Consequently, the twin density decreases with increasing
decreases with increasing the size of the recrystallized grains. The reason why small grains also
the size of the recrystallized grains. The reason why small grains also have a lower twin density is
have a lower twin density is different. The small grains observed in a 2D section after complete
different. The small grains observed in a 2D section after complete recrystallization are either truly
recrystallization are either truly small (in 3D) and have a nucleated late in the recrystallization
small (in 3D) and have a nucleated late in the recrystallization process or they are sections near the
process or they are sections near the edge of a large grain. In both cases, they correspond to a
edgeofalargegrain. Inbothcases,theycorrespondtoavolumethathasbeensweptlate,andthus,
volume that has been swept late, and thus, relatively slowly, by the recrystallization front because
reloaft itvheel yloswlo lwevleyl, obfy rtehmeariencirnygs statollriezda teionnerfgryo.n tbecauseofthelowlevelofremainingstoredenergy.

(a) (b)
0.06 70
0.05 60
n 0.04 -1)m50
iftrercaaoA 00..0023 it(seydnNmL234000
0.01 inw10
T
0 0
2.6 5.0 7.4 9.8 12.2 14.6 17.0 19.4 21.8 24.2 26.6 29.1 31.5 33.9 36.3 38.7 41.9 2.6 5.0 7.4 9.8 12.2 14.6 17.0 19.4 21.8 24.2 26.6 29.1 31.5 33.9 36.3 38.7 41.9
Grain size (µm) Grain size (µm)
(c) (d)
Figure 1. The initial microstructure: (a) orientation color-coded EBSD map (vertical direction of the
Figure1. Theinitialmicrostructure: (a)orientationcolor-codedEBSDmap(verticaldirectionofthe
map projected in the standard triangle); (b) grain orientation spread (GOS) map; grain boundaries in
mapprojectedinthestandardtriangle);(b)grainorientationspread(GOS)map;grainboundariesin
black and twin boundaries in white; (c) grain size distribution histogram (average grain size = 13 μm);
blackandtwinboundariesinwhite;(c)grainsizedistributionhistogram(averagegrainsize=13µm);
(d) twin density as a function of grain size; the dashed line is a tendency guideline.
(d)twindensityasafunctionofgrainsize;thedashedlineisatendencyguideline.
Metals 2016, 6, 5 4 of 13
The microstructure evolution at the three different annealing temperatures is shown in
Figure 2. Grain growth kinetics are described in Figure 3. As expected, grain growth is faster when
increasing the annealing temperature. The annealing times were adapted accordingly to result in
similar grain sizes (in each column of Figure 2).
Metals2016,6,5 4of13
Metals 2016, 6, 5 4 of 13
a: 102TT5hh-1ee0 mmicircorostsrtuructcutureree veovluoltuiotnioantb :ta h1t0e 2tt5hh-2er5e ethdrieffee rdeniftfearnennet alainngnetaelminpge rtaetmurpeesriastsuhroesw nisi nshFoigwunre i2n.
GFirgauinreg 2ro. wGtrhaikni ngerotiwcsthar keidneesticcrsi baerde dinesFcirgiubreed3 i.nA Fsigeuxrpee c3t.e Ad,s gerxapinecgterodw, gthraiisnf gasrotewrtwh hies nfaisntcerre washinegn
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ssiizmesila(irn geraacinh sciozleusm (inn oefaFchig cuorleu2m).n of Figure 2). (cid:9)
y x
y
a: 1025-10 b: 1025-25
D = 33 µm, N = 31.1 mm-1 D = 50 µm, N = 21.1 mm-1
L L
c: 1065-3 d: 1065-6 e: 1065-9
(cid:9)
y x
y
D = 33 µm, N = 31.1 mm-1 D = 50 µm, N = 21.1 mm-1
L L
c: 1065-3 d: 1065-6 e: 1065-9
D = 35 µm, N = 29.5 mm-1 D = 49 µm, N = 21.4 mm-1 D = 59 µm, N = 16.6 mm-1
L L L
f: 1100-1 g: 1100-2 h: 1100-3
D = 35 µm, N = 29.5 mm-1 D = 49 µm, N = 21.4 mm-1 D = 59 µm, N = 16.6 mm-1
L L L
f: 1100-1 g: 1100-2 h: 1100-3
D = 42 µm, N = 24.1 mm-1 D = 48 µm, N = 21.2 mm-1 D = 64 µm, N = 15.5 mm-1
L L L
Figure 2. Microstructure evolution after annealing (same color code as in Figure 1a). (a) at 1025 °C
D = 42 µm, N = 24.1 mm-1 D = 48 µm, N = 21.2 mm-1 D = 64 µm, N = 15.5 mm-1
L L L
for 10 min, (b) at 1025 °C for 25 min, (c) at 1065 °C for 3 min, (d) at 1065 °C for 6 min, (e) at 1065 °C
Figure2.Microstructureevolutionafterannealing(samecolorcodeasinFigure1a).(a)at1025˝Cfor
for 9 mFiignu,r(ef) 2 .a Mt ic1r1o0s0tr u°cCtu rfeo erv o1l umtioinn ,a f(tger) aantn e1a1li0n0g (°sCam feo cro l2o r mcoidne, a(sh i)n aFtig u1r1e0 10a )°. C(a ) faotr 1032 5m °Cin . The
10min,(b)at1025˝Cfor25min,(c)at1065˝Cfor3min,(d)at1065˝Cfor6min,(e)at1065˝Cfor
correspfoorn 1d0i nmgi na, v(be)r aatg 1e0 2g5r a°Cin f osri z2e5 manind, (acn) ante 1a0l6in5 g°C t wfoirn 3 dmeinn,s (idty) aat r1e0 6s5p °eCc iffoier d6 munind, e(er) eata c1h06 5m °aCp . The
9min,(f)at1100˝Cfor1min,(g)at1100˝Cfor2min,(h)at1100˝Cfor3min. Thecorresponding
for 9 min,(f) at 1100 °C for 1 min, (g) at 1100 °C for 2 min, (h) at 1100 °C for 3 min. The
white lines denote twin boundaries; the black lines are grain boundaries with a disorientation
averagegrainsizeandannealingtwindensityarespecifiedundereachmap. Thewhitelinesdenote
corresponding average grain size and annealing twin density are specified under each map. The
greatert wthinanbo 5u°n. daries;theblacklinesaregrainboundarieswithadisorientationgreaterthan5˝.
white lines denote twin boundaries; the black lines are grain boundaries with a disorientation
greater than 5°.
80
)m 80
(reµ 60 )µm
taem t(reem 60
id a
n id
ia 40 n
l traeng l itraeang 40 1100°1C1 00°C
v v
iu 20iu 20 1025°1C0 25°C
q q
E E
1065°1C0 65°C
0
0 0 5 10 15 20 25 30
0 5 An10n ealing tim1e5 ( min) 20 25 30
Annealing time (min)
Figure 3. Grain growth kinetics at different annealing temperatures.
FiguFreig 3u.r eG3r.aGinra ginrogwrotwht hkiknineetitcicss aatt ddiiffffeerreenntta nannenaelianlgintegm tpemeraptuerreast.ures.
The twin density is plotted as a function of the average grain size in the whole microstructure
Thine Ftwiguinre d 4e. nTshiet yto itsa l ptwloitnt ebdou ansd aar yfu lenncgtitohn a nodf tthhee s uarvfearcae goef tghrea oivne rsaizlle m iinc rtohsetr uwcthuorele a rme iucsreods ttoru cture
in Figu re 4. The total twin boundary length and the surface of the overall microstructure are used to
Metals2016,6,5 5of13
The twin density is plotted as a function of the average grain size in the whole microstructure
Metals 2016, 6, 5 5 of 13
inFigure4. Thetotaltwinboundarylengthandthesurfaceoftheoverallmicrostructureareusedto
ccaallccuullaattee tthhee oovveerraallll ttwwiinn ddeennssiittyy vviiaa EEqquuaattiioonn ((11)).. TThheerree iiss aa ssttrroonngg iinnvveerrssee ccoorrrreellaattiioonn ooff tthhee ttwwiinn
ddeennssiittyy wwiitthh tthhee aavveerraaggee ggrraaiinn ssiizzee;; ssiimmiillaarr ttrreennddss hhaavvee aallrreeaaddyy bbeeeenn rreeppoorrtteedd iinn tthhee lliitteerraattuurree ffoorr
ddiiffffeerreenntt FFCCCC mmeettaallssa anndda allllooyyss[ [77,1,155––1188]]..
60
Initial state
)
1 1025-10
-m45
1065-3
m
( 1100-1
L
N 1025-25
30
y 1065-6
t
is 1100-2
n
e 1065-9
d15
n 1100-3
iw Pande's model
T
0
0 30 60 90
Equivalent grain diameter (µm)
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gb
According to Pande [18], the annealing twin evolution depends on the grain boundary
AccordingtoPande[18],theannealingtwinevolutiondependsonthegrainboundarymigration
migration driving force and on the grain boundary migration distance (and therefore, on grain size).
driving forceand on thegrain boundary migrationdistance (and therefore, on grain size). In order
In order to test the influence of grain size on annealing twin development, microstructures with an
totesttheinfluenceofgrainsizeonannealingtwindevelopment, microstructureswithanidentical
identical average grain size of about 50 μm, but obtained at different annealing times and
averagegrainsizeofabout50µm,butobtainedatdifferentannealingtimesandtemperatures,were
temperatures, were compared (highlighted by the red rectangle in Figures 3 and 4). Despite the
compared(highlightedbytheredrectangleinFigures3and4).Despitethedifferenceingraingrowth
difference in grain growth kinetics at the three temperatures, the twin densities for the same
kinetics at the three temperatures, the twin densities for the same average grain size (50 µm) are
average grain size (50 μm) are identical. Therefore, the twin density evolution appears to be
identical. Therefore, thetwindensityevolutionappearstobeindependentofgraingrowthkinetics
independent of grain growth kinetics within the considered range. This observation is consistent
within the considered range. This observation is consistent with experimental data reported in the
with experimental data reported in the early literature [19].
earlyliterature[19].
IInn aaddddititioionnt otot htehetw twinind ednesnitsyi,tyth, ethneu mnubmerboefr aonfn aenanlienaglitnwgi ntwboinu nbdouarniedsaprieers gpreari ng(rNain) (wNaGs) awlsaos
G
uaslseod ufosreda nfonre aalninngeatlwining qtwuainn tqifiucaanttiiofinc.atNion.i sNcGa licsu claatlceudlautseidn gutshinegt othtael tgortaailn gnrauimn bneurmsNberws Nith2 wanitdh
G 2
Nandw iNth1 owutitchoonusti dceorninsgidtewriinngb otwunind abroieusnadsagrriaeisn abso ugnradianr ibeosuinntdhaerioevse irna llthmei corovsetrraullc tmuriec.roInstorurdceturrtoe.
1
In order to determine the density of twins formed during a given process, i.e., to describe the
determinethedensityoftwinsformedduringagivenprocess,i.e.,todescribetheoccurrenceoftwin
occurrence of twin formation events, the number of twin boundaries per developing grain provides
formation events, the number of twin boundaries per developing grain provides a better indicator
a better indicator than any of the other quantification parameters. During mean curvature-driven
than any of the other quantification parameters. During mean curvature-driven grain growth, the
grain growth, the bigger grains grow at the expense of the smaller ones. Therefore, twin formation
bigger grains grow at the expense of the smaller ones. Therefore, twin formation in the largest
in the largest grains (about 100 grains) in each analyzed sample was investigated. In the sample
grains (about 100 grains) in each analyzed sample was investigated. In the sample with the largest
with the largest average grain size (1100-3), the 105 largest grains occupied more than 95% of the
average grain size (1100-3), the 105 largest grains occupied more than 95% of the EBSD mapped
EBSD mapped area (Figure 5). The number of twin boundaries per grain in the largest grains does
area (Figure 5). The number of twin boundaries per grain in the largest grains does not increase
not increase during grain growth (Figure 6); if anything, it decreases, although the counts are not
during grain growth (Figure 6); if anything, it decreases, although the counts are not large enough
large enough to have high confidence in this conclusion. This observation is consistent with the
to have high confidence in this conclusion. This observation is consistent with the results of in situ
results of in situ annealing on pure nickel reported in [15]. Indeed, in this previous work, most
annealingonpurenickelreportedin[15]. Indeed,inthispreviouswork,mostannealingtwinswere
annealing twins were formed during recrystallization, and very few new twins were observed
formed during recrystallization, and very few new twins were observed during grain growth. In
during grain growth. In the grain growth regime, small grains containing several twin boundaries
the grain growth regime, small grains containing several twin boundaries are consumed by large
are consumed by large grains, which grow, but only very rarely produce new twins. A new
grains, which grow, but only very rarely produce new twins. A new mesoscopic model, which
mesoscopic model, which relates annealing twin formation to the topology of the moving grain
relates annealing twin formation to the topology of the moving grain boundaries, was proposed
boundaries, was proposed in [13,15,16] to explain this phenomenon. According to this model,
in[13,15,16]toexplainthisphenomenon. Accordingtothismodel,coherenttwinscannucleateonly
coherent twins can nucleate only along grain boundary segments that migrate opposite to their
alonggrainboundarysegmentsthatmigrateoppositetotheircurvature. Duringgraingrowth, this
curvature. During grain growth, this may happen only at triple junctions: in other words, only
mayhappenonlyattriplejunctions: inotherwords,onlytriplejunctionscanbepotentialnucleation
triple junctions can be potential nucleation sites for annealing twins. This idea was confirmed by a
sites for annealing twins. This idea was confirmed by a recent 3D in situ study [20]. In this study,
recent 3D in situ study [20]. In this study, the near-field high-energy X-ray diffraction microscopy
(nf-HEDM) technique was used to follow the microstructure evolution during grain growth in a
pure nickel sample. In this 3D experiment, only very few twin formation events were observed and
all of them were at triple junctions.
Metals2016,6,5 6of13
thenear-fieldhigh-energyX-raydiffractionmicroscopy(nf-HEDM)techniquewasusedtofollowthe
microstructure evolution during grain growth in a pure nickel sample. In this 3D experiment, only
Metals 2016, 6, 5 6 of 13
veryfewtwinformationeventswereobservedandallofthemwereattriplejunctions.
Metals 2016, 6, 5 6 of 13
ThTehefa cfatctt htahtatt htehel alragrgee ggrraaininss ddeevveelloopp aallmmoosstt wwiitthhoouutt ccrreeaatitningg nnewew twtwinisn, sa,llaielldie dwiwthi tthhet he
concosunsmupmTtpihoteni ofnoa fcott fh tthehaett w ttwhineinn lneaderdgse sm mgaralalliln ggsr raadiinenvsse,, lcocoopnn tatrrliimbbuousttete ssw ttoiot htthohuee ti nicnvrveeaertsriesne gc ocnorrerewrlea lttaiwotinion nos,f o atfhllteihe tdew twiwni tidhne dnthseeint ysi ty
wiwthictthohn etshuienm cinprectriaoesnain soignf gath vaeev rtewargainegnege grdar aisnminsa silzilz egeri aninint thsh,e ec wownootrrrikkb uootnne sII nntocc ootnhneee lli™™nv 7e71r1s88e s schhoorowrwenlna itniion nFF iogigfu utrhere e4 t.4 wF.iuFnru tdhretehnrsemirtmoyr oe,r e,
incionhcwoeihrtehenr tethntetw itnwincrineba obsuionnugdn aadvraeireriasegsm em agayryami nm isgigizrraeat tiene iitnnh eaa w ddoiirrreekcc totiinoo nnIn ttchhoaantte rlre™edd u7uc1ce8es ss thwtowwininn l eilnne gnFtgihgt uharnead n4 d.t oFttuaorltt aihnletirenmrtfeoarrcefia,a cl ial
eneenrgienyrcg,owyh, hewricehnhitci tshw aiinsn aabdno duanidtdidoainrtiiaeolsn rmaelaa sryeo mansiowgnrha wyte thihnye att hwdeii rnetwcdtiienonn ds tiehtnyastd irteeycd rdueecaecssree tsawsdienus r ldiennuggrtighnr gaa nigndr agtiornota wgl ritnohtwe(rittfhai cs(iiwat l iosr th
mewnoteirnotehnr ignmyg,e hwnethiroiecnhtih niagst athnhe iarseda ddtihtdiaoittn ioathln iarsel amasdoendc ihwtiaohnnyias tmlh emi stewpchionas nsdiiebsnmlys itmiys a dtpeeocrrsiaesailbsalenys d dmtuearmtineprgie agrlra ataiunnrd eg rdtoeewmptephne (driate tniustr )e. In
dewpeonrtdhe nmt)e. nInti oandindgit iohner, et hteh afat ctt htihsa ta dthdeit itownianl emvoelcuhtainoins mis iins dpeopsesnibdleyn tm oaft egrriaailn agnrdo wtethm kpienraettuicrse is
addition, the fact that the twin evolution is independent of grain growth kinetics is consistent with
condseipsetenndte nwti)t. hI nt haed dabitoiovne,- dthees cfaricbt etdh amt tehceh atwniisnm e vwohluetrieo nt wisi nin ddeenpseintyd eenvt oolfu gtiroanin i sg rmowainthl yk icnoentitcrso lilse d
theabove-describedmechanismwheretwindensityevolutionismainlycontrolledbytheextension
byc tohnes iestxetnetn swioitnh othf ep arbeo-evxei-sdtiensgcr itbweidn sm (etchhoasnei sfmor mwehder de utwriinng d pernisoirty r eecvroylsuttaiollniz iast imonai nalnyd c oexnitsrtoilnlegd i n
of pre-existing twins (those formed during prior recrystallization and existing in the large grains at
theb yla trhgee egxrtaeinnssi oant tohfe pbreeg-einxinsitningg o tfw thines g(rthaoins eg rfoorwmtehd r edguirmineg) . pHrioowr rmecurychst athlleiz tawtioinns aenxdte enxdis dtienpge innd s
thebeginningofthegraingrowthregime). Howmuchthetwinsextenddependsonhowmuchthe
ont hheo wla rmgeu gchra tihnse agtr athine sb gegroinwn,i nbgu to nf otht eo gnr haionw g rfoaswt tthh ereyg gimroew). .H ow much the twins extend depends
grainsgrow,butnotonhowfasttheygrow.
on how much the grains grow, but not on how fast they grow.

(a) (b)
(a) (b)

((cc))
FigFuigrFeuigr5ue. r5eT. 5Th.h eTeh( a(eab (boaoubuottu1 1t0 01000))0 l)la arlraggregessetts tgg grraaraiinninsss iininn (((aaa))) ttthhheee iiinnniiitttiiiaaall lmmmiiccircroorossttsrrtuurcucttcuutruree,r ,e ((b,b)( )bt ht)hete hs aesamsmapmplel epa lnaennaenanelnaeldeea dal eta dt at
10215021˝50C 2°5Cf o° Cfro 1rfo 01r0m 1 m0in minai nna ndadn( dc()c ()tc ht) hethes esa asmammpplpleele aa nannnnneeeaaalleleeddd aaattt 111111000000 °°˝CCC ffooforr r 33 3 mmmiinni.n. SS.aaSmmaeme c ecoolcoloorl rco ocrodcdeo ead sae sin ai nsF iiFgniugFrueigr 1eu. 1re. 1.
6
6
Initial state
Initial state
4
4 1025-10
1025-10
G
GN 1065-3
N 1065-3
2
2 1100-1
1100-1
0
0 0 30 60 90 120
0 30 Equivalent g60ra in diameter 9(0µ m) 120
Equivalent grain diameter (µm)
Figure 6. Number of twin boundaries per grain in the 100 largest grains expressed as a function of
Figure6.Numberoftwinboundariespergraininthe100largestgrainsexpressedasafunctionofthe
Figthuer ea v6e. rNaguem gbraeirn o sfi ztew. in boundaries per grain in the 100 largest grains expressed as a function of
avethrea gaevegrraagine gsrizaein. size.
Metals2016,6,5 7of13
4. LiteratureModelsforPredictingTwinDensityEvolution
Twomainmathematicalmodelstopredicttwindensityexistintheliterature, namelyGleiter’s
model [21] and Pande’s model [18], which are both consistent with the growth accident theory. In
Gleiter’s model, derived from classical nucleation theory, the twin formation probability mainly
depends on two factors: the activation enthalpy for migration and the grain boundary migration
driving force. However, it is shown in [16] that contrary to the experimental results described in
this study and in other works in the literature [19], the twin density predicted by this model is
very sensitive to temperature. Meanwhile, Pande’s model is shown to be consistent with various
experimental data [7,17,18]. Consequently, Pande’s model is most often used to compare to the
experimental data in the literature. Originally formulated to consider curvature-driven growth,
Pande’s model has also been adapted to account for the stored energy contribution to the grain
boundarymigrationdrivingforce[7].
Pande’s model [18] is derived from the assumption that the increment of annealing twin
boundary number per grain (∆N ) is proportional to the grain boundary migration driving force
G
(F)andtheincreaseingrainsize(∆D).Therefore,themodelassumesthatnewtwinsareformed(N
G
increasing)whilegrainsgrow(Dincreasing). Thisisnotconsistentwithourobservationswithinthe
graingrowthregime,presentedhereforInconel™718andinthepreviouspaperforpurenickel[15].
Thecorrelationbetweentheannealingtwindensityandtheaveragegrainsizecannevertheless
befittedquitewell(seeFigure4)bytheformuladerivedfromPande’smodel(Equation(3))[18]:
1 D
P“ Kγ ln (3)
D gb D
0
with γ « 1 J¨m´2, K « 0.3 m3¨J´1 and D « 2 µm, where the parameter values were identified
gb 0
basedonexperimentaldata(Figure4)byaninversemethod. AMATLABfunction“fminsearch”was
used to identify the parameters that minimize the cost function, which is defined as the sum of the
squared residuals. A residual is defined here as the difference between a value in the experimental
data and the corresponding value predicted by Pande’s model. Even though Pande’s model can
indeed describe the correlation between the annealing twin density and the average grain size, the
underlyingmodelingassumptionsarenotconsistentwiththeexperimentalobservationofannealing
twin evolution. Accordingly, a mean field model will be attempted in the next section to address
thisgap.
5. DescriptionoftheNewMeanFieldModel
As presented in the Section 3, there are almost no annealing twins formed in the grain growth
regime. This is the basis statement for the new model proposed here that aims at describing the
average twin density evolution in the overall microstructure as a function of the average grain size
duringgraingrowth.
Inthepresentstudyandformostoftheexperimentaldataintheliterature,twinboundariesare
quantifiedin2D.Accordingly,forthesakeofbrevity,onlythe2Dversionofthemeanfieldmodelis
presentedhere. Thepossibilityofthe3DextensionwillbediscussedinSection7.
ThemicrostructureisdiscretizedintoncategoriesofrepresentativegrainsG,iP{1,... ,n},based
i
on the same principle applied by [22], but here, representative grains are defined by two variables:
the grain diameter D and the assumed related twin boundary length Li . Representative grains G
i tb i
standfor theaverage stateof anumber N ofassumed identical(in termsofdiameter and, so, twin
i
boundarylength)circulargrainspresentinallofthemicrostructure.
In mean curvature-driven grain growth with isotropic mobility and grain boundary energy
(whichcanbeassumedsincetwinboundariesarenotconsideredasgrainboundaries,butasinternal
defectsofthegrowinggrains),thetwo-dimensionalaveragegrowthrateoftherepresentativegrain
Metals2016,6,5 8of13
G, i.e., of the N grains belonging to the i-th category, can be approximated using Hillert’s classical
i i
meanfieldmodel[23]: ˆ ˙
dD 1 1
i “2¨Mγ ´ (4)
dt gb D D
i
whereMisthegrainboundarymobilityandDistheaveragegrainsizeofthemicrostructure.
In this context, the representative grains that are larger than the average size grow, and those
smaller than the average size shrink. This deterministic growth rate thus represents the average
behavioroflargepopulationsofgrains.
As presented previously, essentially no twins formed during grain growth; thus, the twin
evolutionismainlycontrolledbytheevolutionofthepre-existingtwins. Moreconcretely,annealing
twin boundaries intersecting grain boundaries are extended or reduced in length as the grain
boundary migrates. The basic principle of the model is to describe by how much the twin length
in a grain changes when the grain size either increases or decreases. The real topology of twin
boundaries can be complex [24], especially that of incoherent twins, which obviously differ from
coherentones[25].Itwouldthusbequitecomplicated,ifnotimpossible,toderiveaproperanalytical
description of the change in twin boundary length associated with a change in grain size. Instead,
we have tested a rough, but simple assumption, which is as follows. The change in twin boundary
length Li is considered as proportional to the change in grain size D @i P {1, ... ,n} and in each
tb i
timeincrement.
Inmathematicalterms,theassumptionsofthemodelbecome:
Li pt`∆tq´Li ptq D pt`∆tq´D ptq
tb tb “k i i ,@iPt1,...,nu (5)
Ltibptq Diptq
Inthefirstapproximation,kwillbeassumedtobeconstant,i.e.,identicalforallofthecategories
andidenticalforgrowinggrainsandshrinkinggrains,whichagainarestrongassumptions.
When a grain category G is fully consumed by other categories, i.e., D pt`∆tq “ 0, its
k k
correspondingtwinboundarylength,Lk pt`∆tqisfixedtozero(whichisnotautomaticallyobtained
tb
usingEquation(5)fork ‰1). Foridealizedcirculargrains,theproportionalityfactorkseemsmainly
influencedbytwofactors,illustratedinFigure7:
‚ For a coherent twin boundary spanning to the opposite side of the representative grain, since
∆Li ě ∆D (equality occurs when Li is a diameter of the considered circular representative
tb i tb
grains)andLi ď D,wehave∆Li {Li ě∆D{D;thuskshouldinprinciplebehigherthanone.
tb i tb tb i i
However,ifthereismorethanonecoherenttwinboundaryinsidetherepresentativegrain,the
totaltwinboundarylengthisnotnecessarilysmallerthantherepresentativegraindiameter. In
thiscase,thevalueofkdependsalsoontheinitialratiobetweenthetwinboundarylengthand
therepresentativegraindiameter.
‚ Incoherent twin boundary segments may migrate inside representative grains to decrease the
total twin boundary length. This may lead to the shortening of the considered twin boundary,
eventhoughtherepresentativegrainboundarymigrationtendstolengthenit,whichisareason
forwhichkmightbesmallerthanone.
The possible material and temperature dependencies of incoherent twin boundary migration
behavior are implicitly considered in the parameter k. In the following, the constant k will be
identifiedbasedontheexperimentalresultspresentedinSection3. Thealgorithmofthemeanfield
modelcanbesummarizedasfollows.
Duringatimestep:
‚ ThediameterchangeofeachrepresentativegrainiscalculatedusingEquation(4). Thevolume
conservationisnaturallyverified.
Metals 2016, 6, 5 9 of 13
During a time step:
M etalsT2h01e6 ,d6i,a5meter change of each representative grain is calculated using Equation (4). The vo9luofm13e
conservation is naturally verified.
‚ TThhee cchhaannggee inint wtwininl elnengtghthf ofroer aecahchca cteagteogroyrNy ∆NLiiLiisc aisl ccuallactuedlatbeyd Ebqyu aEtqiounat(i5o)nu (s5in) gutshinegt wthine
i tb tb
btwouinn dbaoruynldenagryth leinngththis icna tethgiosr ycaNteigLotibr,ys inNcieLiN,i ∆siLntibce{ NNiLitibL“i ∆NLitiLbi{Ltib(Lwi heLni a (gwrhaienn caa teggroariny
G isfullyconsumedbyothercategories,i.e.,Dtb=0,itscorresptbondintgb twintbboutbndarylength,Li ,
caitegory Gi is fully consumed by other categoiries, i.e., Di = 0, its corresponding twin boundartby
isfixedtozero).
length, Li , is fixed to zero).
‚ Thetwindtbensityineachcategory(Ni)iscalculatedviathetwinboundarylengthinthiscategory
L
 (TNheL itw)ainn ddietnssaitryea i(nN eSac)ha scafotellgoowrys: (Ni ) is calculated via the twin boundary length in this
i tb i i L
category (NiLi ) and its area (NiSi) as follows:
tb NNLLii “2π2¨ πNNiNiLDˆiitbLD2tibi2˙22“NπN2iiL¨SitNibNi iLStiib (6()6 )
N  i 
i 2 
‚ Theaveragetwindensityintheoverallmicrostructureiscalculatedfromthesummationoftwin
 bTohuen advaerryagleen gtwthisn odfeenascihtyc ainte gthoery oavnedratlhl emoivcerorasltlruarcetuar(eS )isa scafolclluolwatse:d from the summation of
twin boundary lengths of each category ařnd the overall řarea (S) as follows:
NLi NLi
2řNi Lii tb 2 Ni Li i tb
NNL“2π ¨i iNtibSi 2“πi¨ i tSb (7()7 )
L  Ni S  S
i i
i
TTaakkeenn ttooggeetthheerr,, ttoo pprreeddiicctt tthhee eevvoolluuttiioonn ooff tthhee aavveerraaggee ttwwiinn ddeennssiittyy,, wwee nneeeedd aa ggrraaiinn ssiizzee
ddiissttrriibbuuttiioonn wwiitthh tthhee aarreeaa aanndd tthhee ttwwiinn bboouunnddaarryy lleennggtthh ooff eeaacchh ccaatteeggoorryy aass tthhee iinnppuutt..

FFiigguurree 77.. AAnnnneeaalliinngg ttwwiinn eevvoolluuttiioonn mmeecchhaanniissmmss dduurriinngg ggrraaiinn ggrroowwtthh..
6. Mean Field Modeling of Twin Density Evolution during Grain Growth in Inconel™ 718
6. MeanFieldModelingofTwinDensityEvolutionduringGrainGrowthinInconel™718
The grain size distribution and the twin density in each grain size class of the initial
The grain size distribution and the twin density in each grain size class of the initial
microstructure of Inconel™ 718, shown in Figure 1c,d, was used as input. The EBSD data were
microstructure of Inconel™ 718, shown in Figure 1c,d, was used as input. The EBSD data were
discretized into 49 grain size categories (Figure 1c), and the twin density was measured for each
discretized into 49 grain size categories (Figure 1c), and the twin density was measured for each
category (Figure 1d).
category(Figure1d).
With regard to the physical parameters of the model, the grain boundary mobility M and the
With regard to the physical parameters of the model, the grain boundary mobility M and
gthreaing rabionubnoduanryd aernyeergnyer gγygb γvaluveaslu weserwee rcehocsheons ewniwthiitnh inthteh erarnagneg eofo ftytyppiciacal lvvaalulueess ffoorr mmeettaallss
gb
((MMγγgb == 88..2222 ˆ× 1100´−1133 mm22//s)s.) .TThhiiss vvaalluuee ththeererefoforered odeosens ontoret ferrefteora tsop eac isfipcemciafitce rmiaal.teSriniacle. tShiinscme otdheisl
gb
model aims at predicting the twin density evolution as a function of grain size, but not as a function
aimsatpredictingthetwindensityevolutionasafunctionofgrainsize,butnotasafunctionoftime,
of time, it is not necessary to stick to the real grain growth kinetics of a particular
itisnotnecessarytosticktotherealgraingrowthkineticsofaparticularmaterial. Furthermore,fora
material. Furthermore, for a given material, the twin density evolution is not influenced by the
givenmaterial,thetwindensityevolutionisnotinfluencedbythegraingrowthkinetics(withinthe
grain growth kinetics (within the present modeling assumptions and consistent with our
presentmodelingassumptionsandconsistentwithourexperimentalobservations).
experimental observations).
Theaverageannealingtwindensityevolutionintheoverallmicrostructuremodeledbythemean
field model with the time step dt = 1 s is compared in Figure 8 to the Inconel™ 718 grain growth
experimentaldata. Thevalueof0.9forkwasdeterminedasprovidingthebestfitbetweenthemodel