Phase field simulations of the pinning effect of second-phase...

Phase field simulations of the pinning effect of Phase field simulations of the pinning effect of secondsecond--phase particlesphase particles

Effect of particle shape, stability and interfacial propertiesEffect of particle shape, stability and interfacial properties

Nele MoelansNele Moelans(1)(1), Liesbeth Vanherpe, Liesbeth Vanherpe(2)(2), Bart Blanpain, Bart Blanpain(1)(1), Patrick Wollants, Patrick Wollants(1)(1)

(1)(1)Department of metallurgy and materials engineering, Department of metallurgy and materials engineering, (2)(2)Department of computer Department of computer science, science, K.U.LeuvenK.U.Leuven, , BelgiumBelgium

2Nele MoelansCALPHAD-XXXVI, May 6 to 11, 2007State College, Pennsylvania State, USA

Outline

•• Introduction on Zener pinningIntroduction on Zener pinning

•• Phase field model for grain growthPhase field model for grain growth

•• Three modeling approaches for Zener pinningThree modeling approaches for Zener pinning

•• Simulation resultsSimulation results

•• Conclusions and further researchConclusions and further research


Zener pinning: phenomenon

•• Grain growth in polycrystalline materialsGrain growth in polycrystalline materials•• Driving pressure for grain boundary movement:Driving pressure for grain boundary movement:

•• SecondSecond--phase particles exert back forcephase particles exert back force•• DimpleDimple--shapeshape

•• Mechanism for controlling grain sizeMechanism for controlling grain size•• NbC, AlN, TiN,... in HSLANbC, AlN, TiN,... in HSLA--steels for small grain sizesteels for small grain size•• Induce abnormal grain growth in thin filmsInduce abnormal grain growth in thin films

1 2

gbgP

ασρ ρ

=+

MnSMnS precipitateprecipitate in in lowlow--CC steelsteel


Zener pinning: analytical theories

•• Zener relation for limiting grain sizeZener relation for limiting grain size•• Spherical incoherent particlesSpherical incoherent particles•• Position of particles and particles are Position of particles and particles are

not correlatednot correlated

•• Modifications forModifications for•• Particle shapeParticle shape•• Interacial propertiesInteracial properties•• Interaction between grain boundaries Interaction between grain boundaries

and particlesand particles

•• No consensus on parameters No consensus on parameters KK and and bb•• Exact description for local interactionExact description for local interaction•• Approximations required for the number Approximations required for the number

of particles that contribute of particles that contribute

lim 1b

V

R Kfr

=

Comparison with experimental Comparison with experimental data (Manohar, ISIJ Int.,1998)data (Manohar, ISIJ Int.,1998)


Evolution and stability of the particles

•• Particle coarsening and dissolution for Particle coarsening and dissolution for T>TT>Tgcgc•• E.g TiN particles in austenitic lowE.g TiN particles in austenitic low--alloyed steelsalloyed steels

Solubility Ti in austenitic matrixSolubility Ti in austenitic matrix Diffusivity Ti in austenitic matrixDiffusivity Ti in austenitic matrix


Experimental observation

•• 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 M. M. Guo (1999)Guo (1999)

•• PinnedPinned austeniteaustenite graingrainboundariesboundaries

•• Specialized and timeSpecialized and time--consuming workconsuming work

20 μm


Simulations

•• No assumptions on the number of grain boundaryNo assumptions on the number of grain boundary--particle particle interactions are required (+)interactions are required (+)

•• Exact material properties and conditions are known (+) Exact material properties and conditions are known (+) –– Simplifies interpretation of results Simplifies interpretation of results

•• Relatively easy to adapt material parameters and conditions (+)Relatively easy to adapt material parameters and conditions (+)

•• Computationally intensive (Computationally intensive (--))–– Particles can be more than 100 times smaller than grainsParticles can be more than 100 times smaller than grains–– Large grain assemblies for reliable statisticsLarge grain assemblies for reliable statistics


Grain growth model

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

•• Based on models of Chen and Yang Based on models of Chen and Yang (1994), Fan and Chen (1997) and (1994), Fan and Chen (1997) and Kazaryan et al. (2000)Kazaryan et al. (2000)

•• Polycrystalline microstructurePolycrystalline microstructure

•• GrainGrain i of i of matrixmatrix--phasephase

1 2, ,..., ( , ),...,i pr tη η η η

Graini

Grain j

1iη =

0jη = 0iη =

1jη =


Grain growth model

•• Free energyFree energy

•• Temporal evolution: GinzburgTemporal evolution: Ginzburg--Landau equationLandau equation

•• Parameter assessmentParameter assessment•• For each interface:For each interface:

•• Related to interfacial energy (Related to interfacial energy (σσi,ji,j), interfacial mobility (), interfacial mobility (μμi,ji,j) and ) and interfacial width (interfacial width (εε))

( )4

1,

22 2 2

1 1

( )4 2 2

p p p pi i

i j iVi i i

i jj i

F dm Vκη η η η ηη

γ= = < =

⎡ ⎤⎛ ⎞⎛ ⎞= − + + ∇⎢ ⎥⎜ ⎟⎜ ⎟

⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦∑ ∑∑ ∑∫

( ) 1( ,..., )( , )( , )

pi

i

Fr tt r t

Lη η

ηηη

∂∂= −

∂ ∂

, , ,, , ,i j i j i jm Lκ γ


Three approaches for modeling Zener pinning

•• Spatially dependent parameter Spatially dependent parameter ΦΦ in free energy in free energy •• Constant particle distributionConstant particle distribution

•• Coupling with a CahnCoupling with a Cahn--Hilliard equationHilliard equation•• ΦΦ is treated as a conserved field variableis treated as a conserved field variable•• Qualitative description of the evolution of the Qualitative description of the evolution of the

particlesparticles

•• MultiMulti--phase field approach + coupling with phase field approach + coupling with diffusion equationdiffusion equation

•• ΦΦ is treated as a nonis treated as a non--conserved field variableconserved field variable•• Composition field gives local compositionComposition field gives local composition•• Quantitative treatment of phase stabilities, Quantitative treatment of phase stabilities,

interfacial properties and kineticsinterfacial properties and kinetics

Complexity and Complexity and computational computational requirements requirements

Possibilities Possibilities


Spatially dependent parameter

•• Minima free energyMinima free energy•• ΦΦ=1=1•• ΦΦ=0=0

•• AdvantagesAdvantages•• Particles can be smallParticles can be small•• Efficient and easy implementationEfficient and easy implementation

–– SemiSemi--implicit Fourierimplicit Fourier--spectral methodspectral method

•• ShortcomingsShortcomings•• Properties of the particles are ignoredProperties of the particles are ignored

1 2( , ,..., ) (0,0,...,0)pη η η =

1 2( , ,..., ) (1,0,...,0),(0,1,...,0),...(0,0,...,1),

( 1,0,...,0),...pη η η =

−


Spatially dependent parameter

•• LargeLarge--scale 3D simulations for high scale 3D simulations for high ffVV•• Bounding box algorithm (PhD L. Vanherpe)Bounding box algorithm (PhD L. Vanherpe)

–– Equations for Equations for ηηii are only solved for grain are only solved for grain ii–– 1 processor (2gb RAM): system size1 processor (2gb RAM): system size 256x256x256256x256x256

r=3,fr=3,fVV=0.05=0.05


Coupling with Cahn-Hilliard equation

•• ΦΦ is treated as conserved field variableis treated as conserved field variable•• E.g. scaled composition variableE.g. scaled composition variable

•• CahnCahn--Hilliard equation for evolution of Hilliard equation for evolution of ΦΦ

•• AdvantagesAdvantages•• Particles evolve in timeParticles evolve in time•• Relatively easy and efficient implementationRelatively easy and efficient implementation

–– SemiSemi--implicit Fourierimplicit Fourier--spectral method spectral method

•• ShortcomingsShortcomings•• Only for diffusion limited processesOnly for diffusion limited processes•• Grain boundary segregation exagerated Grain boundary segregation exagerated

matrix

particle matrix

c cc c

−Φ =

−


Coupling with Cahn-Hilliard equations

•• Evolution grain structureEvolution grain structure •• Distribution Distribution ΦΦ

•• Grain boundary segregation Grain boundary segregation exageratedexagerated

ffVV=0.12, L=10M=0.12, L=10M


Coupling with multi-phase field approach

•• ΦΦ nonnon--conserved field variable, composition field conserved field variable, composition field cc•• Interface consists of 2 phasesInterface consists of 2 phases

–– with for each componentwith for each component

•• Diffusion equationDiffusion equation

•• AdvantagesAdvantages•• Interfacial and bulk properties are decoupledInterfacial and bulk properties are decoupled•• Extendable to multiExtendable to multi--phase systemsphase systems

•• ShortcomingsShortcomings•• Computationally intensiveComputationally intensive•• Grain boundary segregation is neglected Grain boundary segregation is neglected •• Artificial wettingArtificial wetting

,

fc Mcα

α αα β α

φ ∂= ∇ ⋅ ∇

∂∑

Steinbach, Physica D. 127 (2006) 153Steinbach, Physica D. 127 (2006) 153--160160

α βμ μ=cc

ccββ

ccαα

c c cα α β βφ φ= +


Coupling with multi-phase field approach

•• Evolution grain structureEvolution grain structure •• Composition profile Composition profile

0 50 100

0

0.2

0.4

0.6

0.8

1

distance

conc

entr

atio

nc = 0.003

c = 0.999

CCeq,parteq,part =0.001, C=0.001, Ceq,matrixeq,matrix = 0.999= 0.999DDpartpart = 0.01, D= 0.01, Dmatrixmatrix = 0.1= 0.1σσgbgb = 0.25,= 0.25,σσintint = 0.2 = 0.2 cctotaltotal = 0.05= 0.05


Conclusions

•• The effect of secondThe effect of second--phase particles on grain growth is still phase particles on grain growth is still not understoodnot understood

•• Simulation studies are important for a better understanding Simulation studies are important for a better understanding

•• Three phase field models that account for the effect of secondThree phase field models that account for the effect of second--phase particles on grain growth have been discussedphase particles on grain growth have been discussed•• Particle distribution is constant in timeParticle distribution is constant in time•• Qualitative treatment of particle evolutionQualitative treatment of particle evolution•• Quantitative treatment of particle evolution and interfacial Quantitative treatment of particle evolution and interfacial

propertiesproperties


Future research

•• Further validation of the three modeling approachesFurther validation of the three modeling approaches

•• Orientation dependence of interfacial energy of the particlesOrientation dependence of interfacial energy of the particles

•• Systematic studies of particular phenomena for 3D structuresSystematic studies of particular phenomena for 3D structures

•• Towards quantitative simulations for real alloy systems Towards quantitative simulations for real alloy systems


End

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

•• Acknowledgment:Acknowledgment:•• Nele Moelans is Postdoctoral Fellow of the Research Foundation Nele Moelans is Postdoctoral Fellow of the Research Foundation --

Flanders (FWOFlanders (FWO--Vlaanderen)Vlaanderen)•• Simulations were performed on the HPSimulations were performed on the HP--computing infrastructure computing infrastructure

of the K.U.Leuven (operational since May 2005) of the K.U.Leuven (operational since May 2005)

•• More information on More information on http//nele.studentenweb.orghttp//nele.studentenweb.org

Phase field simulations of the pinning effect of second-phase...

Documents

Transcript of Phase field simulations of the pinning effect of second-phase...