PhasePhase--fieldfield simulationssimulations of of graingrain growthgrowth in in materialsmaterials containingcontaining secondsecond--phasephase particlesparticles

Nele MoelansNele MoelansDepartment of metallurgy and materials engineering, Department of metallurgy and materials engineering,

K.U.Leuven, K.U.Leuven, BelgiumBelgium

Group: Group: ThermodynamicsThermodynamics in in materialsmaterials engineeringengineering

PresentationPresentation RTGRTG--BerlinBerlinJuneJune 7, 20067, 2006

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Research group: Thermodynamics in materials engineering

•• ThermodynamicsThermodynamics in in materialmaterial developmentdevelopment•• PhasePhase diagramsdiagrams

–– ModelingModeling ((CALPHADCALPHAD--methodmethod) + ) + experimentsexperiments•• PhasePhase transformationstransformations and and microstructuralmicrostructural evolutionevolution

–– PhasePhase--fieldfield modelingmodeling

•• ThermodynamicsThermodynamics in in extractionextraction and and productionproduction of of metalsmetals•• Steel Steel cleanlinesscleanliness, , inclusionsinclusions in in liquidliquid steelsteel•• Metal and Metal and slacslac interactionsinteractions withwith refractoryrefractory materialmaterial at high at high

temperaturestemperatures•• ExperimentsExperiments + + modelingmodeling ((FactSageFactSage, , LattizeLattize BolzmanBolzman))•• Close Close collaborationcollaboration withwith industryindustry

•• 3 professors, 3 postdocs, 12 3 professors, 3 postdocs, 12 doctoraldoctoral studentsstudents, 3 , 3 technicianstechnicians

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Content

•• PhasePhase--fieldfield simulationssimulations of of graingrain growthgrowth in in materialsmaterials containingcontainingsecondsecond--phasephase particlesparticles

•• PhasePhase--fieldfield methodmethod forfor microstructuralmicrostructural evolutionevolution

•• GrainGrain growthgrowth + + ZenerZener pinningpinning

•• Model Model descriptiondescription

•• SimulationSimulation resultsresults

•• ConclusionsConclusions + outlook + outlook

Part I Phase-field method for simulating

microstructural evolution

5

Microstructures

50 μm

50 μm 25 μm

Fe-C:

a) Ferrite (wt% C < 0.022)

b) Ferrite + cementite (wt% C = 0.4, slowly cooled)

c) Ferrite + cementite (wt% C = 0.4, faster cooled)

d) Martensite (wt% C = 1.4, quenched)

250 μm

a)a) b)b)

d)d)c)c)

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Role of microstructures in materialsscience

Chemical Chemical compositioncomposition+ +

TemperatureTemperature, , pressurepressure, , coolingcooling raterate,,……

MicrostructureMicrostructureShapeShape, , sizesize and and orientationorientation of the of the grainsgrains, , mutualmutual distribution of the distribution of the phasesphases

MaterialMaterial propertiesproperties

Strength, deformability, hardness, toughness,fatigue…

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Phase-field simulations of microstructural evolution

ExperimentsExperiments, , atomisticatomistic simulationssimulationsand and thermodynamicthermodynamic modelsmodels

Crystal structure, phase stabilities, interfacial properties (energy, mobility, structure,anisotropy), diffusion properties

PhasePhase--fieldfield simulationssimulations

Morphological evolution of the grains at the mesoscale duringsolidification, precipitation, solid-state phase transformations, grain growth,…

Models Models thatthat predictpredict macroscopicmacroscopic materialmaterial propertiesproperties

Strength, deformability, hardness, toughness, fatigue…

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Representation of microstructures in the phase-field method

•• PhasePhase--fieldfield variables: variables: continuouscontinuous functionsfunctions in in spacespace and timeand time•• LocalLocal compositioncomposition

•• LocalLocal structurestructure and and orientationorientation

BinaryBinary alloyalloy AA--BB

••PhasePhase αα: : ηη = 0= 0

••PhasePhase ββ: : ηη = 1= 1AntiAnti--phasephase boundaryboundary

( , )Bx r t

( , )r tη

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Diffuse-interface description

•• Sharp interfaceSharp interface

•• DiscontinuousDiscontinuous variationvariation in in propertiesproperties

•• RequiresRequires trackingtracking of the interfacesof the interfaces

•• SimplifiedSimplified graingrain morphologiesmorphologies

•• Diffuse interfaceDiffuse interface

•• ContinuousContinuous variationvariation in in propertiesproperties

•• Interfaces Interfaces implicitlyimplicitly givengiven byby locallocalvariationsvariations in in phasephase--fieldfield variablesvariables

•• Complex Complex graingrain morphologiesmorphologies

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Phase-field simulation technique

•• MicrostructuralMicrostructural representationrepresentation::

•• ThermodynamicThermodynamic and and kinetickinetic equationsequations•• PhasePhase stabilitiesstabilities•• InterfacesInterfaces•• ElasticElastic energyenergy duedue toto volume volume effectseffects•• OrientationOrientation dependencedependence•• SoluteSolute diffusiondiffusion

•• Parameter Parameter determinationdetermination

•• NumericalNumerical solutionsolution of a set of coupled of a set of coupled partialpartial differentialdifferentialequationsequations

( , )Bx r t( , )r tη

Part IIGrain growth and Zener pinning

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Polycrystalline microstructure withsecond-phase particles

•• MechanismMechanism forfor controllingcontrolling the the graingrain sizesize of a of a materialmaterial

•• E.g. E.g. microalloyedmicroalloyed steelssteels–– SmallSmall graingrain sizesize requiredrequired forfor

high high strengthstrength–– AdditionAddition of of smallsmall amountsamounts of of

Nb, Ti, Al, V,Nb, Ti, Al, V,……–– FormationFormation of of NbCNbC, , AlNAlN, , TiNTiN,...,...–– PinningPinning of of graingrain boundariesboundaries

duringduring heat heat treatmentstreatments ororweldingwelding

•• E.g. E.g. thinthin films in films in micromicro--electronicelectronicapplicationsapplications

–– LargeLarge graingrain sizesize requiredrequired totoreducereduce failurefailure

–– ToTo promotepromote abnormalabnormal graingraingrowthgrowth

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Normal grain growth

•• SurfaceSurface tensiontension

PPgg = = drivingdriving pressurepressure forfor graingrain boundaryboundarymovementmovement

gbgP

Rασ

=

=> => PPgg decreasesdecreases in timein time

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Normal grain growth

•• SurfaceSurface tensiontension + + topologicaltopological considerationsconsiderations

IsotropicIsotropic: : αα11 = = αα22 = = αα33 = 120= 120°°

Smaller Smaller grainsgrains shrinkshrinkandand

largerlarger grainsgrains growgrow

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Zener pinning

•• GrainGrain boundaryboundary areaarea is is reducedreducedwhenwhen a a particleparticle is is locatedlocated onon a a graingrain boundaryboundary

•• ParticlesParticles exertexert a back a back forceforce ononmovingmoving graingrain boundariesboundaries

•• DimpleDimple--shapeshape

MnSMnS precipitateprecipitate in in lowlow--CC steelsteel

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Final grain size – Zener relation

•• GrainGrain growthgrowth stops stops whenwhen PPgg=P=PZZ

•• CalculationCalculation of the of the totaltotal pinningpinningpressurepressure of the of the particlesparticles PPZZrequiresrequires

•• NumberNumber of of particlesparticles in contact in contact withwith a a graingrain boundaryboundary

•• AngleAngle at at whichwhich graingrain boundaryboundarymeetsmeets the the particleparticle

•• ImportanceImportance of computer of computer simulationssimulations

lim 1b

V

R Kfr

=

FromFrom P.A. P.A. ManoharManohar (1998)(1998)

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Pinning effect: experimentalobservation

•• FeFe--0.09 0.09 toto 0.53 w% C0.53 w% C--0.02 w% P 0.02 w% P containingcontaining CeCe22OO33 inclusionsinclusions

•• PhDPhD –– workwork of M. of M. GuoGuo

•• PinnedPinned austeniteaustenite graingrainboundariesboundaries

20 μm

Part IIIA phase-field model for grain growth in the

presence of second-phase particles

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Representation of a polycrystallinestructure

•• ExtensionExtension graingrain growthgrowth model model D. Fan and L.D. Fan and L.--Q. Q. ChenChen

•• PhasePhase--fieldfield variables: variables:

•• ParticlesParticles: : ΦΦ=1 =1

•• GrainGrain i of i of matrixmatrix--phasephase: : ΦΦ=0=0

1 2( , ,..., ,..., ) (0,0,..., 1,...,0)i pη η η η = ±

1 2( , ,..., ,..., ) (0,0,...,0,...,0)i pη η η η =

1 2, ,..., ( , ),...,i pr tη η η η

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Representation of the grain boundaries

Grain i Grain j

1iη =

0jη = 0iη =

1jη =

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Representation of a particle

•• EvolutionEvolution of of ηηii acrossacross a a particleparticle in in graingrain ii

Grain i

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Free energy and kinetic equations

•• ThermodynamicThermodynamic freefree energyenergy

•• EquilibriumEquilibrium•• ΦΦ=0: =0: •• ΦΦ=1:=1:

•• KineticKinetic equationsequations ((GinzburgGinzburg--LandauLandau))

1 2( , ,..., ) (1,0,...,0), (0,1,...,0),...(0,0,...,1), ( 1,0,...,0),...pη η η = −

( )4 2

22 2

1 1

2

1 1( )

4 2 2

p p p pp

ii

i ii j iV

i i j i iF m dVη η κη η ηε η

= = ≠ ==

+ Φ⎡ ⎤⎛ ⎞

= − + + ∇⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

∑ ∑∑ ∑∫ ∑

1 2( , ,..., ) (0,0,...,0)pη η η =

20 1 2( , ) ( , ,...) ( , )( , ) ( , )

ii

i i

r t fFL L r tt r t r t

η η η κ ηη η

⎛ ⎞∂ ∂∂= − = − − ∇⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠

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Model parameters

•• GrainGrain boundaryboundary energyenergy::

•• GrainGrain boundaryboundary velocityvelocity::

•• InterfacialInterfacial energyenergy particlesparticles::

•• InterfacialInterfacial thicknessthickness: :

•• Parameter Parameter choicechoice: :

( )f mε κ

4 22 2 2

1 1 1( )

4 2

p p p pi i

i j ii i j i i

mf η η η η ε η= = ≠ =

⎛ ⎞= − + + Φ⎜ ⎟

⎝ ⎠∑ ∑∑ ∑21 2( , ) ( , ,...) ( , )

( , )i

ii

r t f r tt r t

Lη η η ηη

κ⎛ ⎞∂ ∂

= − − ∇⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

0.58 mκ

1 2( )V Lκ λ λ= +

0.5, 1, 1, 1m Lκ ε= = = =

mκ

∝

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Numerical solution: semi-implicitFourier-spectral method

•• ProcedureProcedure•• PhasePhase--fieldfield equationequation in 1Din 1D

•• FourierFourier transformtransform

•• SemiSemi--implicitimplicit time time steppingstepping

•• Inverse Inverse FourierFourier transformtransform

2

2

( , ) ( , )x t f x tLt x

η ηκη

⎛ ⎞∂ ∂ ∂= − −⎜ ⎟∂ ∂ ∂⎝ ⎠

2( , ) ( , )k t fL k k tt

η κ ηη

⎛ ⎞⎛ ⎞∂ ∂= − +⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠

112

nn nnfL L k

tη η κ η

η

++⎛ ⎞− ∂

= − −⎜ ⎟Δ ∂⎝ ⎠

1

21

nn

n

fL t

L tk

ηη

ηκ

+

⎛ ⎞∂− Δ ⎜ ⎟∂⎝ ⎠=+ Δ

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Numerical solution: semi-implicitFourier-spectral method

•• AdvantagesAdvantages•• ImplicitImplicit treatmenttreatment of the of the laplacianlaplacian ⇒⇒ largerlarger time step time step •• No No needneed toto solvesolve anan algebraicalgebraic system of system of equationsequations•• EfficientEfficient fftfft –– algorithmsalgorithms availableavailable in most in most programmingprogramming

languageslanguages•• HigherHigher spatialspatial accuracyaccuracy ⇒⇒ largerlarger gridgrid spacingspacing

•• LimitationsLimitations•• PeriodicPeriodic boundaryboundary conditionsconditions•• Constant model parametersConstant model parameters

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Localinteraction: spherical grain

50 450 1200

2200 2900 3100

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Temporal evolution spherical grain

0

890

76c p

rRd −

===

Part IVDiscussion of the simulation results

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Large-scale 2-D simulations

•• IsotropicIsotropic graingrain boundaryboundary propertiesproperties

•• RandomRandom dispersiondispersion of of roundround particlesparticles

•• AreaArea fractionfraction: :

•• InitialInitial microstructuremicrostructure::•• GrainGrain nucleationnucleation in in presencepresence of of particlesparticles (R(R00=0)=0)•• GrainGrain nucleationnucleation and and initialinitial graingrain growthgrowth without without particlesparticles (R(R00>0)>0)

••

2.53

rr==

0.004 0.16af = −

lim 1b

a

R Kr f

=

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Initial microstructure

•• GrainGrain nucleationnucleation in the in the presencepresence of of particlesparticles

•• Most Most particlesparticles onon graingrainboundariesboundaries

0 0R =

3, 0.02ar f= =

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Initial microstructure

•• GrainGrain nucleationnucleation and and initialinitialgrowthgrowth without without particlesparticles

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Initial microstructure

•• GrainGrain nucleationnucleation and and initialinitialgrowthgrowth without without particlesparticles

•• AdditionAddition of of particlesparticles whenwhen

•• ManyMany particlesparticles withinwithin grainsgrains

0 0R >

03, 0.02, 13.6ar f R= = =

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Large-scale 2-D simulations

•• RR00 = 0:= 0: •• RR00 > 0> 0

0 lim3, 0.04, 0, 22.2ar f R R= = = = 0 lim3, 0.04, 13.6, 26.2ar f R R= = = =

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Role initial grain size

•• RR00 important important forforhigh high ffaa

•• May May explainexplainlargelargeexperimentalexperimentalscatterscatter at high at high ffVV

FractionFraction of of particlesparticles onon graingrain boundariesboundaries

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3-D simulations for thin films

•• ExperimentalExperimental studystudy•• H.P. H.P. LongworthLongworth and C.V. and C.V.

ThompsonThompson

•• Al films Al films withwith CuAlCuAl22 precipitatesprecipitates

•• AnnealingAnnealing at 500at 500°°C:C:–– GrainGrain growthgrowth ––> > pinningpinning

––> > abnormalabnormal graingrain growthgrowth

•• GrainGrain growthgrowth in in thinthin films films usuallyusually treatedtreated in 2in 2--DD

ColumnarColumnar graingrain structurestructure

Film Film preparationpreparation

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3-D simulations for thin films

•• 22--D D columnarcolumnar graingrain structurestructure

•• 33--D D interactioninteraction particleparticle--graingrainboundaryboundary

•• ⇒⇒ curvaturecurvature out of the out of the planeplane

•• Film Film thicknessthickness

•• ParticlesParticles in the in the middlemiddle of the of the film are more film are more effectiveeffective

33, 0.12, 21ar f l= = =

37

Comparison with experimental data

lim0.5

11.28a

Rfr

=

38

Comparison with experimental data

•• OtherOther effectseffects•• SurfaceSurface groovinggrooving•• SemiSemi--coherent coherent particleparticle--matrixmatrix

interfaceinterface

33--D D simulationsimulation 22--D D simulationsimulationAl film: Al film: hothot--stagestage TEM TEM micrographmicrograph(H.P. (H.P. LongworthLongworth and C.V. and C.V. ThompsonThompson, 1991), 1991)

0.086af =

0.08af = 0.08af =

0.3 μm

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Computational considerations

•• 22--D: D: realisticrealistic RRlimlim//rr--ratioratio couldcould bebe reproducedreproduced•• High fHigh faa: :

–– system system sizesize 256, 20000 time steps256, 20000 time steps–– ⇒⇒ few few hourshours

•• LowLow ffaa: : –– system system sizesize 512, >60000 time steps 512, >60000 time steps –– ⇒⇒ upup toto 10 10 daysdays

•• 33--D:D:•• RRlimlim//rr--ratioratio: x10 : x10 ⇒⇒ system system sizesize: x10: x10•• ThirdThird power of system power of system sizesize•• Computer Computer requirementsrequirements: x10: x1055

•• Parallel code Parallel code requiredrequired

Part VConclusions and suggestions for further

research

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Conclusions

•• A A phasephase--fieldfield model model forfor graingrain growthgrowth in the in the presencepresence of a of a fixedfixeddispersiondispersion of of particlesparticles was was developeddeveloped and and implementedimplemented

•• SimulationsSimulations werewere performedperformed forfor thinthin films films withwith realisticrealisticdimensionsdimensions and and forfor realisticrealistic volume volume fractionsfractions and and RRlimlim//rr--ratiosratios

•• The The initialinitial graingrain sizesize is important is important forfor high volume high volume fractionsfractions

•• The The interactioninteraction betweenbetween particlesparticles and and columnarcolumnar graingrainboundariesboundaries in in thinthin films is 3films is 3--DD•• The The pinningpinning effect of effect of particlesparticles in in thinthin films is films is lowerlower thanthan in 2in 2--D D

structuresstructures

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Further research

•• OutlookOutlook•• 33--D D simulationssimulations forfor high volume high volume fractionsfractions•• CouplingCoupling withwith concentrationconcentration fieldfield

–– EvolutionEvolution of the of the particlesparticles–– ParticleParticle--matrixmatrix interface interface

•• SurfaceSurface energyenergy forfor thinthin filmsfilms

•• SuggestionsSuggestions forfor furtherfurther researchresearch•• AnisotropyAnisotropy•• FurtherFurther experimentalexperimental studystudy of of pinningpinning mechanismmechanism and and particleparticle--

matrixmatrix interfaceinterface

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End

•• Thank you for your attention !Thank you for your attention !

•• More information: http://More information: http://nele.studentenweb.orgnele.studentenweb.org//

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Properties of diffuse grain boundaries

•• Interfacial widthInterfacial width

•• Numerical accuracyNumerical accuracy

mκ

∝

45

Pinning mechanism

•• ZenerZener

•• Coherent/incoherent Coherent/incoherent particlesparticles--matrixmatrix interfaceinterface

•• RiosRios