introduction to phase-field modeling of microstructure
TRANSCRIPT
Phase field simulations for grain growth in systems containing
second-phase particlesIntroduction to phaseIntroduction to
phase--field modeling of field modeling of microstructure
evolutionmicrostructure evolution
NeleNele MoelansMoelans Department of Metallurgy and Materials Engineering (MTM)Department of Metallurgy and Materials Engineering (MTM)
K.U.LeuvenK.U.Leuven, , BelgiumBelgium
2School on Computational Modeling of Materials, December 2-3, 2010
Outline
•• IntroductionIntroduction
•• Quantitative Quantitative ‘‘thinthin--interfaceinterface’’ modelsmodels
•• SummarySummary
Microstructures
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)
Role of microstructures in materials science
Chemical Chemical compositioncomposition + +
MaterialMaterial propertiesproperties
•• Diffuse interface concept Diffuse interface concept •• Phase Phase fieldfield variables variables •• ThermodynamicThermodynamic free free energyenergy functionalfunctional •• Evolution Evolution equationsequations ---- OnsagerOnsager
N. Moelan, B. Blanpain, P. Wollants, Comp. Coupling of Phase Diagrams and Thermochemistry, CALPHAD, 32, p268 (2008)
6School on Computational Modeling of Materials, December 2-3, 2010
Diffuse-interface description
•• ResultsResults independentindependent of of
•• ContinuousContinuous variationvariation in in propertiesproperties •• AvoidsAvoids numericalnumerical interface interface
trackingtracking •• Complex Complex morphologiesmorphologies feasiblefeasible
7School on Computational Modeling of Materials, December 2-3, 2010
Representation of microstructures
BinaryBinary alloyalloy AA--BB
••PhasePhase ββ: : ηη = 1= 1AntiphaseAntiphase boundaryboundary
( , ), ( , )B Bx r t c r t
( , )r tη ( , )r tφ
Polycrystalline structures
1 2
1 2
•• Single Single phasephase
•• 22--phasephase •• GrainsGrains
1 2, ,..., ( , ),...,i pr tη η η η
Grain i Grain j
9School on Computational Modeling of Materials, December 2-3, 2010
Thermodynamics for heterogeneous systems
Thermodynamics
= + = + ∇ + ∇∑ ∑∫
Homogeneous free energy
•• BinaryBinary twotwo--phasephase systemsystem
( ) ( )0 1 (( ) ( , ) ),f h hf c T f c gTβ αφ φ ω φ= + − +
f0β f0α
Homogeneous free energy
•• BinaryBinary twotwo--phasephase systemsystem
( ) ( )0 ( , ) 1 ( , ) ( )h hf f c T f c T gβ α ωφ φφ= + − +
f0β f0α
Interpolation function
Homogeneous free energy
•• BinaryBinary twotwo--phasephase systemsystem
( ) ( )0 ( ( ), ) 1 ( , )f h f c T h f gc Tβ αφ φ φω= + − +
f0β f0α
Solid-solid phase transitions
chem int elastF F F F= + + 1 ( , ) ( , ) ( , ) 2
el el elast ijkl ij kl
V
F C x x x dVη ε η ε η= ∫
15School on Computational Modeling of Materials, December 2-3, 2010
Kinetics
((→→ Interface Interface movementmovement))
k
F x xr t L r t t r t η
η ηη ξ η
x m i
F x xx r t M r t V t x r t
η η ξ
k
∂ = − − ∇ ∂
Numerical implementation
••DiscretizationDiscretization
(L. (L. VanherpeVanherpe et al., K.U.Leuven)et al., K.U.Leuven)
••DiscretizationDiscretization ((FiniteFinite differencesdifferences, , finitefinite elementselements, Fourier, Fourier--spectral spectral methodmethod, , ……))
••Adaptive Adaptive meshingmeshing ((M. M. DorrDorr et al. et al. AMPE, LLNL)AMPE, LLNL)
MICRESS, commercial software for MICRESS, commercial software for phasephase--fieldfield coupledcoupled withwith CALPHAD CALPHAD
Quantitative ‘thin-interface’ models
18School on Computational Modeling of Materials, December 2-3, 2010
Quantitative ‘thin-interface’ models
•• →→ realisticrealistic 3D simulations3D simulations •• →→ straightforwardstraightforward relations for the model relations for the model
parametersparameters
dilutedilute systemssystems –– Karma and Rappel (1996), Karma (2001), Karma and Rappel (1996), Karma (2001), EchebarriaEchebarria
(2004), (2004), FolchFolch--PlappPlapp (2005)(2005)
(2007)(2007)
19School on Computational Modeling of Materials, December 2-3, 2010
Decoupling bulk and interfacial energy
•• Interface Interface treatedtreated as mixture of 2 phasesas mixture of 2 phases •• cc--fieldfield for for eacheach phasephase •• EqualEqual interdiffusioninterdiffusion potentialpotential + +
conservationconservation
•• BulkBulk energyenergy
Kim et al., PRE, 6 (1999) p 7186; Kim et al., PRE, 6 (1999) p 7186; TiadenTiaden et et al., al., PhysicaPhysica D, D, 115 (1998) p73115 (1998) p73
SimilarSimilar alternaltern. : Karma, PRL (2001); . : Karma, PRL (2001); Floch,PlappFloch,Plapp, PRE (2005), , PRE (2005), EchebarriaEchebarria et al., PRE (2004)et al., PRE (2004)
( ) ( )f c f c c c
β β α α
( ) ( )1c h c h cβ αφ φ= + −
,c c cα β→
Decoupling bulk and interfacial kinetics
•• JumpJump in in chemicalchemical potentialpotential accrossaccross interfaceinterface
•• Partial solutionPartial solution •Dilute, DS=0: A.Karma, PRL, 87, 115701 (2001); B. Echebarria et al., PRE, 70, 061604 (2004) •Multi-comp, DS=0: S.G. Kim, Acta Mater. 55, p4391 (2007)
1
1
[1 ( )] | | | |
l
−
=
∂ ∂ ∇ = ∇ ⋅ − ∇ + ∇ ⋅
∂ ∂ ∇∑
Anisotropic grain growth model
•• Free Free energyenergy
•• InclinationInclination dependencedependence
2 2 ,0 ( )i j i jη η κ η κ≠ ⇒ =
2 2 2
i j j i i j i
κ η η ηκη η = < = <
= ∑∑ ∑∑
( ) ( ) ( ), , , , , , ,, , , | |
i j i j i j i j i j i j i j i j
i j
∇ − ∇ =
∇ − ∇
1 ,
p p p p i i
interf i j i i i
i j j i iV
F dVm κηγη η ηη η = = < =
= − + + + ∇
∑ ∑∑ ∑∫
1 2, ,..., ( , ),...,i pr tη η η η
L.-Q. Chen and W. Yang, PRB, 50 (1994) p15752
A. Kazaryan et al., PRB, 61 (2000) p14275
22School on Computational Modeling of Materials, December 2-3, 2010
Non-variational approach – equal interface width
•• GinzburgGinzburg--LandauLandau type type equationsequations
•• DefinitionDefinition ‘‘grain grain boundaryboundary widthwidth’’
( ) 2 ,
j i i j
∂ = − − + − ∇ ∂
∑
dx dx η η= =
23School on Computational Modeling of Materials, December 2-3, 2010
Grain boundary properties
,
i j
N. Moelans, B. Blanpain, P. Wollants, PRL, 101, 0025502
(2008); PRB, 78, 024113 (2008) , , , , ,,[ ],[ ] ,[ ],[ ],[ ]gb gb gb i j i j i jm Lθ θγ μ κ γ→
24School on Computational Modeling of Materials, December 2-3, 2010
Numerical validation
•• ShrinkingShrinking grain:grain:
2dA dt
/num xΔ / 5num R >
Phase-field modelling: summary
•• MultiMulti--componentcomponent •• Transport Transport equationsequations
•• ImplementationImplementation for for realisticrealistic lengthlength scalesscales •• ParameterParameter choicechoice
→→ thinthin--interface interface modelsmodels + + couplingcoupling withwith atomisticatomistic approachesapproaches and efficient and efficient implementationsimplementations
...chem int elast magnF F F F F= + + + +
StrengthStrength
DifficultiesDifficulties
Zener pinning
•• E.g. E.g. NbCNbC, , AlNAlN, , TiNTiN,... in ,... in HSLAHSLA-- steelssteels
•• NanoNano--graingrain structuresstructures
•• InfluenceInfluence ofof •• ShapeShape of the of the particleparticle •• InterfacialInterfacial propertiesproperties of of particlesparticles •• InitialInitial distributiondistribution •• EvolutionEvolution particlesparticles
lim 1 b
20 μm
Fe-0.09 to 0.53 w% C-0.02 w% P with Ce2O3 inclusions (PhD. M. Guo)
28School on Computational Modeling of Materials, December 2-3, 2010
Representation of a polycrystalline structure
•• ExtensionExtension graingrain growthgrowth model model D. Fan and L.D. Fan and L.--Q. Q. ChenChen
•• PhasePhase field variables: field variables:
•• ParticlesParticles: : ΦΦ=1 =1
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
29School on Computational Modeling of Materials, December 2-3, 2010
Simulation results: Al thin films
•• ThinThin films films withwith CuAlCuAl22 -- precipitatesprecipitates Film preparation
33, 0.12, 21ar f l= = =
lim 0.5
11.28 a
R fr
30School on Computational Modeling of Materials, December 2-3, 2010
Simulation results: effect of particle shape and coarsening
•• EllipsoidEllipsoid particlesparticles •• EvolvingEvolving particlesparticles
b aK a
= + ffVV=0.12, L=10M=0.12, L=10M
3, 0.05Vfε = =
Modified Zener relation
Comparison with Other Studies
••In collaboration In collaboration withwith F. F. SpaepenSpaepen, , Harvard Harvard UniversityUniversity
33School on Computational Modeling of Materials, December 2-3, 2010
Grain growth in columnar films with fiber texture
•• Grain Grain boundaryboundary energyenergy:: •• FourfoldFourfold symmetrysymmetry •• Extra Extra cuspcusp atat θθ = 37.5= 37.5°° •• ReadRead--shockleyshockley
•• DiscreteDiscrete orientationsorientations
and grain and grain boundaryboundary type type distributionsdistributions
White: θ = 1.5 Gray: θ = 3 Red: θ = 37,5 Black: θ > 3, θ ≠ 37.5
1 2 60, ,..., ( , ),..., 1.5i r tη η η η θ⇒ Δ = °
2D simulation2D simulation
<0 0 1>
Simulations: 1 high-angle energy cusp
•• HighHigh--angle grain angle grain boundariesboundaries formform independentindependent networknetwork
•• LowLow--angle grain angle grain boundariesboundaries followfollow movementmovement of of highhigh--angle angle grain grain boundariesboundaries →→ elongateelongate
•• No stable quadruple No stable quadruple junctionsjunctions
White: θ = 1.5 Gray: θ = 3 Black: θ > 3, θ ≠ 37.5 Red: θ = 37,5
35School on Computational Modeling of Materials, December 2-3, 2010
Misorientation distribution function (MDF)
•• Area Area weigthedweigthed MDFMDF
36School on Computational Modeling of Materials, December 2-3, 2010
Grain growth kinetics
•• Grain Grain growthgrowth exponentexponent •• PFM: PFM: steadysteady--state state
growthgrowth withwith
•• PreviousPrevious findingsfindings::
= → ∞ +
→
+ =
N. Moelans, F. N. Moelans, F. SpaepenSpaepen, P. , P. WollantsWollants, , Phil. Mag., 90 p 501Phil. Mag., 90 p 501--523 (2010)523 (2010)
37School on Computational Modeling of Materials, December 2-3, 2010
3D simulations for wires with fiber texture
6ϑ < °
Summary
variablesvariables •• EvolutionEvolution equationsequations derivedderived followingfollowing thermodynamicthermodynamic principlesprinciples •• QuantitativeQuantitative aspectsaspects: : ‘‘thinthin interfaceinterface’’ approachapproach, parameter , parameter choicechoice, ,
Thank you for your attention ! Questions ?
•• AcknowledgementsAcknowledgements •• Research Foundation Research Foundation -- FlandersFlanders ((FWOFWO--VlaanderenVlaanderen)) •• FlemishFlemish SupercomputingSupercomputing Center (VSC)Center (VSC)
•• More More informationinformation onon http//http//nele.studentenweb.orgnele.studentenweb.org
Workshop on Workshop on MultiscaleMultiscale simulation simulation atat K.U.LeuvenK.U.Leuven
http://www.cs.kuleuven.behttp://www.cs.kuleuven.be /conference/multiscale11//conference/multiscale11/
Outline
Microstructures
Principles of phase field modeling
Diffuse-interface description
Anisotropic grain growth model
Grain boundary properties
Simulation results: effect of particle shape and coarsening
Comparison with Other Studies
Simulations: 1 high-angle energy cusp
Misorientation distribution function (MDF)
Summary
NeleNele MoelansMoelans Department of Metallurgy and Materials Engineering (MTM)Department of Metallurgy and Materials Engineering (MTM)
K.U.LeuvenK.U.Leuven, , BelgiumBelgium
2School on Computational Modeling of Materials, December 2-3, 2010
Outline
•• IntroductionIntroduction
•• Quantitative Quantitative ‘‘thinthin--interfaceinterface’’ modelsmodels
•• SummarySummary
Microstructures
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)
Role of microstructures in materials science
Chemical Chemical compositioncomposition + +
MaterialMaterial propertiesproperties
•• Diffuse interface concept Diffuse interface concept •• Phase Phase fieldfield variables variables •• ThermodynamicThermodynamic free free energyenergy functionalfunctional •• Evolution Evolution equationsequations ---- OnsagerOnsager
N. Moelan, B. Blanpain, P. Wollants, Comp. Coupling of Phase Diagrams and Thermochemistry, CALPHAD, 32, p268 (2008)
6School on Computational Modeling of Materials, December 2-3, 2010
Diffuse-interface description
•• ResultsResults independentindependent of of
•• ContinuousContinuous variationvariation in in propertiesproperties •• AvoidsAvoids numericalnumerical interface interface
trackingtracking •• Complex Complex morphologiesmorphologies feasiblefeasible
7School on Computational Modeling of Materials, December 2-3, 2010
Representation of microstructures
BinaryBinary alloyalloy AA--BB
••PhasePhase ββ: : ηη = 1= 1AntiphaseAntiphase boundaryboundary
( , ), ( , )B Bx r t c r t
( , )r tη ( , )r tφ
Polycrystalline structures
1 2
1 2
•• Single Single phasephase
•• 22--phasephase •• GrainsGrains
1 2, ,..., ( , ),...,i pr tη η η η
Grain i Grain j
9School on Computational Modeling of Materials, December 2-3, 2010
Thermodynamics for heterogeneous systems
Thermodynamics
= + = + ∇ + ∇∑ ∑∫
Homogeneous free energy
•• BinaryBinary twotwo--phasephase systemsystem
( ) ( )0 1 (( ) ( , ) ),f h hf c T f c gTβ αφ φ ω φ= + − +
f0β f0α
Homogeneous free energy
•• BinaryBinary twotwo--phasephase systemsystem
( ) ( )0 ( , ) 1 ( , ) ( )h hf f c T f c T gβ α ωφ φφ= + − +
f0β f0α
Interpolation function
Homogeneous free energy
•• BinaryBinary twotwo--phasephase systemsystem
( ) ( )0 ( ( ), ) 1 ( , )f h f c T h f gc Tβ αφ φ φω= + − +
f0β f0α
Solid-solid phase transitions
chem int elastF F F F= + + 1 ( , ) ( , ) ( , ) 2
el el elast ijkl ij kl
V
F C x x x dVη ε η ε η= ∫
15School on Computational Modeling of Materials, December 2-3, 2010
Kinetics
((→→ Interface Interface movementmovement))
k
F x xr t L r t t r t η
η ηη ξ η
x m i
F x xx r t M r t V t x r t
η η ξ
k
∂ = − − ∇ ∂
Numerical implementation
••DiscretizationDiscretization
(L. (L. VanherpeVanherpe et al., K.U.Leuven)et al., K.U.Leuven)
••DiscretizationDiscretization ((FiniteFinite differencesdifferences, , finitefinite elementselements, Fourier, Fourier--spectral spectral methodmethod, , ……))
••Adaptive Adaptive meshingmeshing ((M. M. DorrDorr et al. et al. AMPE, LLNL)AMPE, LLNL)
MICRESS, commercial software for MICRESS, commercial software for phasephase--fieldfield coupledcoupled withwith CALPHAD CALPHAD
Quantitative ‘thin-interface’ models
18School on Computational Modeling of Materials, December 2-3, 2010
Quantitative ‘thin-interface’ models
•• →→ realisticrealistic 3D simulations3D simulations •• →→ straightforwardstraightforward relations for the model relations for the model
parametersparameters
dilutedilute systemssystems –– Karma and Rappel (1996), Karma (2001), Karma and Rappel (1996), Karma (2001), EchebarriaEchebarria
(2004), (2004), FolchFolch--PlappPlapp (2005)(2005)
(2007)(2007)
19School on Computational Modeling of Materials, December 2-3, 2010
Decoupling bulk and interfacial energy
•• Interface Interface treatedtreated as mixture of 2 phasesas mixture of 2 phases •• cc--fieldfield for for eacheach phasephase •• EqualEqual interdiffusioninterdiffusion potentialpotential + +
conservationconservation
•• BulkBulk energyenergy
Kim et al., PRE, 6 (1999) p 7186; Kim et al., PRE, 6 (1999) p 7186; TiadenTiaden et et al., al., PhysicaPhysica D, D, 115 (1998) p73115 (1998) p73
SimilarSimilar alternaltern. : Karma, PRL (2001); . : Karma, PRL (2001); Floch,PlappFloch,Plapp, PRE (2005), , PRE (2005), EchebarriaEchebarria et al., PRE (2004)et al., PRE (2004)
( ) ( )f c f c c c
β β α α
( ) ( )1c h c h cβ αφ φ= + −
,c c cα β→
Decoupling bulk and interfacial kinetics
•• JumpJump in in chemicalchemical potentialpotential accrossaccross interfaceinterface
•• Partial solutionPartial solution •Dilute, DS=0: A.Karma, PRL, 87, 115701 (2001); B. Echebarria et al., PRE, 70, 061604 (2004) •Multi-comp, DS=0: S.G. Kim, Acta Mater. 55, p4391 (2007)
1
1
[1 ( )] | | | |
l
−
=
∂ ∂ ∇ = ∇ ⋅ − ∇ + ∇ ⋅
∂ ∂ ∇∑
Anisotropic grain growth model
•• Free Free energyenergy
•• InclinationInclination dependencedependence
2 2 ,0 ( )i j i jη η κ η κ≠ ⇒ =
2 2 2
i j j i i j i
κ η η ηκη η = < = <
= ∑∑ ∑∑
( ) ( ) ( ), , , , , , ,, , , | |
i j i j i j i j i j i j i j i j
i j
∇ − ∇ =
∇ − ∇
1 ,
p p p p i i
interf i j i i i
i j j i iV
F dVm κηγη η ηη η = = < =
= − + + + ∇
∑ ∑∑ ∑∫
1 2, ,..., ( , ),...,i pr tη η η η
L.-Q. Chen and W. Yang, PRB, 50 (1994) p15752
A. Kazaryan et al., PRB, 61 (2000) p14275
22School on Computational Modeling of Materials, December 2-3, 2010
Non-variational approach – equal interface width
•• GinzburgGinzburg--LandauLandau type type equationsequations
•• DefinitionDefinition ‘‘grain grain boundaryboundary widthwidth’’
( ) 2 ,
j i i j
∂ = − − + − ∇ ∂
∑
dx dx η η= =
23School on Computational Modeling of Materials, December 2-3, 2010
Grain boundary properties
,
i j
N. Moelans, B. Blanpain, P. Wollants, PRL, 101, 0025502
(2008); PRB, 78, 024113 (2008) , , , , ,,[ ],[ ] ,[ ],[ ],[ ]gb gb gb i j i j i jm Lθ θγ μ κ γ→
24School on Computational Modeling of Materials, December 2-3, 2010
Numerical validation
•• ShrinkingShrinking grain:grain:
2dA dt
/num xΔ / 5num R >
Phase-field modelling: summary
•• MultiMulti--componentcomponent •• Transport Transport equationsequations
•• ImplementationImplementation for for realisticrealistic lengthlength scalesscales •• ParameterParameter choicechoice
→→ thinthin--interface interface modelsmodels + + couplingcoupling withwith atomisticatomistic approachesapproaches and efficient and efficient implementationsimplementations
...chem int elast magnF F F F F= + + + +
StrengthStrength
DifficultiesDifficulties
Zener pinning
•• E.g. E.g. NbCNbC, , AlNAlN, , TiNTiN,... in ,... in HSLAHSLA-- steelssteels
•• NanoNano--graingrain structuresstructures
•• InfluenceInfluence ofof •• ShapeShape of the of the particleparticle •• InterfacialInterfacial propertiesproperties of of particlesparticles •• InitialInitial distributiondistribution •• EvolutionEvolution particlesparticles
lim 1 b
20 μm
Fe-0.09 to 0.53 w% C-0.02 w% P with Ce2O3 inclusions (PhD. M. Guo)
28School on Computational Modeling of Materials, December 2-3, 2010
Representation of a polycrystalline structure
•• ExtensionExtension graingrain growthgrowth model model D. Fan and L.D. Fan and L.--Q. Q. ChenChen
•• PhasePhase field variables: field variables:
•• ParticlesParticles: : ΦΦ=1 =1
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
29School on Computational Modeling of Materials, December 2-3, 2010
Simulation results: Al thin films
•• ThinThin films films withwith CuAlCuAl22 -- precipitatesprecipitates Film preparation
33, 0.12, 21ar f l= = =
lim 0.5
11.28 a
R fr
30School on Computational Modeling of Materials, December 2-3, 2010
Simulation results: effect of particle shape and coarsening
•• EllipsoidEllipsoid particlesparticles •• EvolvingEvolving particlesparticles
b aK a
= + ffVV=0.12, L=10M=0.12, L=10M
3, 0.05Vfε = =
Modified Zener relation
Comparison with Other Studies
••In collaboration In collaboration withwith F. F. SpaepenSpaepen, , Harvard Harvard UniversityUniversity
33School on Computational Modeling of Materials, December 2-3, 2010
Grain growth in columnar films with fiber texture
•• Grain Grain boundaryboundary energyenergy:: •• FourfoldFourfold symmetrysymmetry •• Extra Extra cuspcusp atat θθ = 37.5= 37.5°° •• ReadRead--shockleyshockley
•• DiscreteDiscrete orientationsorientations
and grain and grain boundaryboundary type type distributionsdistributions
White: θ = 1.5 Gray: θ = 3 Red: θ = 37,5 Black: θ > 3, θ ≠ 37.5
1 2 60, ,..., ( , ),..., 1.5i r tη η η η θ⇒ Δ = °
2D simulation2D simulation
<0 0 1>
Simulations: 1 high-angle energy cusp
•• HighHigh--angle grain angle grain boundariesboundaries formform independentindependent networknetwork
•• LowLow--angle grain angle grain boundariesboundaries followfollow movementmovement of of highhigh--angle angle grain grain boundariesboundaries →→ elongateelongate
•• No stable quadruple No stable quadruple junctionsjunctions
White: θ = 1.5 Gray: θ = 3 Black: θ > 3, θ ≠ 37.5 Red: θ = 37,5
35School on Computational Modeling of Materials, December 2-3, 2010
Misorientation distribution function (MDF)
•• Area Area weigthedweigthed MDFMDF
36School on Computational Modeling of Materials, December 2-3, 2010
Grain growth kinetics
•• Grain Grain growthgrowth exponentexponent •• PFM: PFM: steadysteady--state state
growthgrowth withwith
•• PreviousPrevious findingsfindings::
= → ∞ +
→
+ =
N. Moelans, F. N. Moelans, F. SpaepenSpaepen, P. , P. WollantsWollants, , Phil. Mag., 90 p 501Phil. Mag., 90 p 501--523 (2010)523 (2010)
37School on Computational Modeling of Materials, December 2-3, 2010
3D simulations for wires with fiber texture
6ϑ < °
Summary
variablesvariables •• EvolutionEvolution equationsequations derivedderived followingfollowing thermodynamicthermodynamic principlesprinciples •• QuantitativeQuantitative aspectsaspects: : ‘‘thinthin interfaceinterface’’ approachapproach, parameter , parameter choicechoice, ,
Thank you for your attention ! Questions ?
•• AcknowledgementsAcknowledgements •• Research Foundation Research Foundation -- FlandersFlanders ((FWOFWO--VlaanderenVlaanderen)) •• FlemishFlemish SupercomputingSupercomputing Center (VSC)Center (VSC)
•• More More informationinformation onon http//http//nele.studentenweb.orgnele.studentenweb.org
Workshop on Workshop on MultiscaleMultiscale simulation simulation atat K.U.LeuvenK.U.Leuven
http://www.cs.kuleuven.behttp://www.cs.kuleuven.be /conference/multiscale11//conference/multiscale11/
Outline
Microstructures
Principles of phase field modeling
Diffuse-interface description
Anisotropic grain growth model
Grain boundary properties
Simulation results: effect of particle shape and coarsening
Comparison with Other Studies
Simulations: 1 high-angle energy cusp
Misorientation distribution function (MDF)
Summary