Phase- -field field simulations simulations of of grain grain - - PowerPoint PPT Presentation

Phase Phase-

• field

field simulations simulations of

• f grain

grain growth growth in in materials materials containing containing second second-

• phase

phase particles particles

Nele Nele Moelans Moelans

Department of metallurgy and materials engineering Department of metallurgy and materials engineering Katholieke Universiteit Leuven, Katholieke Universiteit Leuven, Belgium Belgium

28 28 February February 2008 2008

Part I Scientific context

3

Outline part I: Scientific context

• Grain

Grain growth growth in in polycrystalline polycrystalline materials materials

• Pinning

Pinning effect of effect of second second-

• phase

phase particles particles

• Phase

Phase-

• field

field method method for for simulating simulating microstructural microstructural evolution evolution

• Incentives

Incentives of the research

• f the research

4

Role of microstructures in materials science

Chemical Chemical composition composition + + Temperature Temperature, , pressure pressure, , cooling cooling rate rate, ,… …

Microstructure Microstructure

Shape Shape, , size size and and orientation

• rientation of the
• f the grains

grains, , mutual mutual distribution distribution of the

• f the phases

phases

Material Material properties properties

Strength, deformability, hardness, toughness,fatigue…

5

Polycrystalline microstructure with second-phase particles

• Mechanism

Mechanism for for controlling controlling the the grain grain size size of a

• f a material

material

• E.g.

E.g. microalloyed microalloyed steels steels – – Small Small grain grain size size required required for for high high strength strength – – Addition Addition of

• f small

small amounts amounts of

• f

Nb, Ti, Al, V, Nb, Ti, Al, V,… … – – Formation Formation of

• f NbC

NbC, , AlN AlN, , TiN TiN,... ,... – – Pinning Pinning of

• f grain

grain boundaries boundaries during during heat heat treatments treatments or

• r

welding welding

6

Normal grain growth

• Surface

Surface tension tension

P Pg

g =

= driving driving pressure pressure for for grain grain boundary boundary movement movement

gb g

P R ασ =

=> => P

Pg

g decreases

decreases in time in time

7

Normal grain growth

• Surface

Surface tension tension + + topological topological considerations considerations Isotropic Isotropic: : α α1

1 =

= α α2

2 =

= α α3

3 = 120

= 120° ° Smaller Smaller grains grains shrink shrink and and larger larger grains grains grow grow

8

Zener pinning

• Grain

Grain boundary boundary area area is is reduced reduced when when a a particle particle is is located located on

• n a

a grain grain boundary boundary

• Particles

Particles exert exert a back a back force force on

• n

moving moving grain grain boundaries boundaries

• Dimple

Dimple-

• shape

shape MnS MnS precipitate precipitate in in low low-

• C

C steel steel

max Z gb

F r π σ =

9

Final grain size – Zener relation

• Calculation

Calculation of the

• f the total

total pinning pinning pressure pressure of the

• f the particles

particles P PZ

Z

requires requires

• Number

Number of

• f particles

particles in contact in contact with with a a grain grain boundary boundary

• Angle

Angle at at which which grain grain boundary boundary meets meets the the particle particle

• Grain

Grain growth growth stops stops when when P Pg

g=P

=PZ

Z

lim

1

b V

R K f r =

From From P.A. P.A. Manohar Manohar (1998) (1998)

10

Pinning effect: experimental

• bservation
• Fe

Fe-

• 0.09 to 0.53 w% C

0.09 to 0.53 w% C-

• 0.02 w% P

0.02 w% P containing containing Ce Ce2

2O

O3

3 inclusions

inclusions

• PhD

PhD – – work work M.

• M. Guo

Guo

• Pinned

Pinned austenite austenite grain grain boundaries boundaries

20 μm

11

Phase-field simulations of microstructural evolution

Experiments Experiments, , atomistic atomistic simulations simulations and and thermodynamic thermodynamic models models

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

Phase Phase-

• field

field simulations simulations

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

Models Models that that predict predict macroscopic macroscopic material material properties properties

Strength, deformability, hardness, toughness, fatigue…

12

Phase field method

• van der Waals,

van der Waals, Cahn Cahn-

• Hilliard

Hilliard (1958), (1958), Ginzburg Ginzburg-

• Landau

Landau (1950) , (1950) , Hohenberg Hohenberg and and Halperin Halperin (1977) (1977)

• Microstructure

Microstructure evolution evolution ( (started started ± ± 20 20 years years ago ago) )

• Solidification

Solidification

• Ordering

Ordering reactions reactions

• Martensitic

Martensitic transformation transformation

• Nowadays

Nowadays

• Wide range of

Wide range of applicabilities applicabilities

• Quantitative

Quantitative aspects aspects – – Parameter Parameter determination determination – – Numerical Numerical implementation implementation

13

Representation of microstructures in the phase-field method

• Phase

Phase-

• field

field variables: variables: continuous continuous functions functions in in space space and time and time

• Local

Local composition composition

• Local

Local structure structure and and orientation

• rientation

Binary Binary alloy alloy A A-

• B

B

• Phase

Phase α α: : η η = 0 = 0

• Phase

Phase β β: : η η = 1 = 1 Antiphase Antiphase boundary boundary

( , )

B

x r t

• ( , )

r t η

14

Diffuse-interface description

• Sharp interface

Sharp interface

• Discontinuous

Discontinuous variation variation in in properties properties

• Requires

Requires tracking tracking of the interfaces

• f the interfaces
• Simplified

Simplified grain grain morphologies morphologies

• Diffuse interface

Diffuse interface

• Continuous

Continuous variation variation in in properties properties

• Interfaces

Interfaces implicitly implicitly given given by by local local variations variations in in phase phase-

• field

field variables variables

• Complex

Complex grain grain morphologies morphologies

15

Phase-field simulation technique

• Microstructural

Microstructural representation representation: :

• Thermodynamic

Thermodynamic and and kinetic kinetic equations equations

• Phase

Phase stabilities stabilities

• Interfaces

Interfaces

• Elastic

Elastic energy energy due due to volume to volume effects effects

• Orientation

Orientation dependence dependence

• Solute

Solute diffusion diffusion

• Parameter

Parameter determination determination

• Numerical

Numerical solution solution

( , )

B

x r t

• ( , )

r t η

16

Intermediate conclusions Incentives of the research

• Pinning

Pinning effect of effect of second second-

• phase

phase particles particles on

• n grain

grain boundaries boundaries and and final final grain grain size size still still not not understood understood

• Mesoscale

Mesoscale grain grain growth growth simulations simulations can can give give important important insights insights

• Phase

Phase-

• field

field method method for for simulating simulating microstructure microstructure evolution evolution

• General

General technique technique based based on

• n nonequilibrium

nonequilibrium thermodynamic thermodynamic principles principles

• Complex

Complex phenomena phenomena and and morphologies morphologies

⇒ ⇒ Phase Phase-

• field

field simulations simulations of

• f grain

grain growth growth in in materials materials containing containing second second-

• phase

phase particles particles

Part II Research work

Model Model description description and and simulation simulation results results

18

Outline part II: Results

• Phase

Phase-

• field

field model model for for grain grain growth growth and and Zener Zener pinning pinning

• Study

Study of the

• f the pinning

pinning mechanism mechanism

• Simulation

Simulation results results for for polycrystalline polycrystalline materials materials

• Ongoing

Ongoing/ /future future research research

19

Representation of a polycrystalline structure

• Extension

Extension grain grain growth growth model model

• D. Fan and L.
• D. Fan and L.-
• Q.
• Q. Chen

Chen (1997) (1997)

• Phase

Phase field variables: field variables:

• Particles

Particles: : Φ Φ=1 =1

• Grain

Grain i of i of matrix matrix-

• phase

phase: : Φ Φ=0 =0

1 2

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

i p

η η η η = ±

1 2

( , ,..., ,..., ) (0,0,...,0,...,0)

i p

η η η η =

1 2

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

i p

r t η η η η

20

Representation of the grain boundaries

Grain i Grain j

1

i

η =

j

η =

i

η = 1

j

η =

21

Representation of a particle

• Evolution

Evolution of

• f η

ηi

i across

across a a particle particle in in grain grain i i

Grain i

22

Free energy and kinetic equations

• Thermodynamic

Thermodynamic free free energy energy

• Equilibrium

Equilibrium

• Φ

Φ=0: =0:

• Φ

Φ=1: =1:

• Kinetic

Kinetic equations equations ( (Ginzburg Ginzburg-

• Landau

Landau) )

1 2

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

p

η η η = −

( )

4 2 2 2 2 1 1 2 1 1

( ) 4 2 2

p p p p p i i i i i j i V i i j i i

F m dV η η κ η η η ε η

= = ≠ = =

+ Φ ⎡ ⎤ ⎛ ⎞ = − + + ∇ ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦

∑ ∑∑ ∑ ∫ ∑

1 2

( , ,..., ) (0,0,...,0)

p

η η η =

2 1 2

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

i i i i

r t f F L L r t t r t r t η η η κ η η η ⎛ ⎞ ∂ ∂ ∂ = − = − − ∇ ⎜ ⎟ ⎜ ⎟ ∂ ∂ ∂ ⎝ ⎠

23

Model parameters

• Grain

Grain boundary boundary energy energy: :

• Grain

Grain boundary boundary velocity velocity: :

• Interfacial

Interfacial energy energy particles particles: :

• Parameter

Parameter choice choice: :

( ) f m ε κ

4 2 2 2 2 1 1 1

( ) 4 2

p p p p i i i j i i i j i i

m f η η η η ε η

= = ≠ =

⎛ ⎞ = − + + Φ ⎜ ⎟ ⎝ ⎠

∑ ∑∑ ∑

2 1 2

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

i i i

r t f r t t r t L η η η η η κ ⎛ ⎞ ∂ ∂ = − − ∇ ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠

• 0.58

m κ

1 2

( ) V L κ λ λ = +

0.5, 1, 1, 1 m L κ ε = = = =

24

Local interaction: spherical grain

• Grain

Grain boundary boundary passing a passing a particle particle

• 3

3-

• D

D simulation simulation

• Geometry

Geometry

• Dimple

Dimple-

• shape

shape

• Break

Break-

• free

free at at β β > > π π/4 /4

25

Temporal evolution spherical grain

8 90 76

c p

r R d − = = =

26

Large-scale 2-D simulations

• Isotropic

Isotropic grain grain boundary boundary properties properties

• Random

Random dispersion dispersion of

• f round

round particles particles

• Area

Area fraction fraction: :

• Initial

Initial microstructure microstructure: :

• Grain

Grain nucleation nucleation in in presence presence of

• f particles

particles (R (R0

0=0)

=0)

• Grain

Grain nucleation nucleation and and initial initial grain grain growth growth without without particles particles (R (R0

0>0)

>0)

• 2.5

3 r r = =

0.004 0.16

a

f = −

lim

1

b a

R K r f =

27

Large-scale 2D-simulations: initial microstructure

• Grain

Grain nucleation nucleation in the in the presence presence of

• f particles

particles

• Most

Most particles particles on

• n grain

grain boundaries boundaries

R = 3, 0.02

a

r f = =

28

Large-scale 2D-simulations: initial microstructure

• Grain

Grain nucleation nucleation and and initial initial growth growth without without particles particles

29

Large-scale 2D-simulations: initial microstructure

• Grain

Grain nucleation nucleation and and initial initial growth growth without without particles particles

• Addition

Addition of

• f particle

particle when when

• Many

Many particles particles within within grains grains

R > 3, 0.02, 13.6

a

r f R = = =

30

Large-scale 2-D simulations

• R

R0

0 = 0:

= 0:

• R

R0

0 > 0

> 0

lim

3, 0.04, 0, 22.2

a

r f R R = = = =

lim

3, 0.04, 13.6, 26.2

a

r f R R = = = =

31

Role initial grain size

• R

R0

0 important

important for for high high f fa

a

Fraction Fraction of

• f particles

particles on

• n grain

grain boundaries boundaries

32

3-D simulations for thin films

• Experimental

Experimental study study

• H.P.

H.P. Longworth Longworth and C.V. and C.V. Thompson Thompson

• Al films

Al films with with CuAl CuAl2

2 precipitates

precipitates

• Annealing

Annealing at 500 at 500° °C: C: – – Grain Grain growth growth – –> > pinning pinning – –> > abnormal abnormal grain grain growth growth

• Huge

Huge grain grain size size required required for for micro micro-

• electronic

electronic applications applications

• Grain

Grain growth growth in in thin thin films films usually usually treated treated in 2 in 2-

• D

D

Columnar Columnar grain grain structure structure Film Film preparation preparation

33

3-D simulations for thin films

• 2

2-

• D

D columnar columnar grain grain structure structure

• 3

3-

• D

D interaction interaction particle particle-

• grain

grain boundary boundary

• ⇒

⇒ curvature curvature out of the

• ut of the plane

plane

• Film

Film thickness thickness

• Particles

Particles in the in the middle middle of the

• f the

film are more film are more effective effective

3

3, 0.12, 21

a

r f l = = =

34

Comparison with experimental data

lim 0.5

1 1.28

a

R f r =

35

Comparison with experimental data

• Other

Other effects effects

• Surface

Surface grooving grooving

• Semi

Semi-

• coherent

coherent particle particle-

• matrix

matrix interface interface

• Evolution

Evolution particles particles 3 3-

• D

D simulation simulation 2 2-

• D

D simulation simulation Al film: Al film: hot hot-

• stage

stage TEM TEM micrograph micrograph

(H.P. (H.P. Longworth Longworth and C.V. and C.V. Thompson Thompson, 1991) , 1991)

0.086

a

f = 0.08

a

f = 0.08

a

f =

0.3 μm

36

Ongoing /future research

• 3

3-

• D simulations for

D simulations for bulk bulk materials materials

• Bounding

Bounding box box algorithm algorithm

• Evolution,

Evolution, interfacial interfacial properties properties and and shape shape of the

• f the particles

particles

• Different

Different grain grain boundary boundary energies energies

• Misorientation

Misorientation and inclination and inclination dependence dependence

• Thermal

Thermal grooving grooving

37

Group picture

38

Arenberg Castle, Leuven, Belgium

39

End

• Thank

Thank you you for for your your attention attention ! !

• Acknowledgment

Acknowledgment: :

• Nele

Nele Moelans is Moelans is Postdoctoral Postdoctoral Fellow Fellow of the Research Foundation

• f the Research Foundation -
• Flanders

Flanders ( (FWO FWO-

• Vlaanderen

Vlaanderen) )

• Simulations

Simulations were were performed performed on

• n the

the HP HP-

• computing

computing infrastructure infrastructure

• f the
• f the K.U.Leuven

K.U.Leuven ( (operational

• perational since

since May 2005) May 2005)

• More

More information information on

• n http//

http//nele.studentenweb.org nele.studentenweb.org

40

Ongoing/future research

• 3D simulations for

3D simulations for bulk bulk materials materials

• Bounding

Bounding box box algorithm algorithm – – Equations Equations for for η ηi

i are

are only

• nly solved

solved locally locally for for grain grain i i

lim

1

m V

R K f r =

Limiting grain size

41

Ongoing /future research

• Evolving

Evolving particles particles

• Coupling

Coupling with with Cahn Cahn-

• Hilliard

Hilliard equations equations for for Φ Φ Grain Grain structure structure: : f fV

V=0.12, L=10M

=0.12, L=10M Distribution Distribution Φ Φ

42

Ongoing/future research

• Anisotropic

Anisotropic grain grain boundary boundary properties properties

Misorientation dependence of grain boundary energy

43

Ongoing/future research

• Thermal

Thermal grooving grooving

44

Interaction energy

• Energetic

Energetic consideration consideration

• Geometry

Geometry

• Theoretical

Theoretical interaction interaction energy energy

• Diffuse

Diffuse grain grain boundaries boundaries ⇒ ⇒

• Interaction

Interaction energy energy slightly slightly too too negative negative

• Lower

Lower limit limit on

• n particle

particle size size

2

2 (2 ) (3 )

gb gb

r D r D σ π σ

45

Interaction energy

46

Comparison with theory

• High

High scatter scatter for for low f low fa

a

• Fitting:

Fitting:

• Theory

Theory: :

• Phase

Phase field (R field (R0

0=0):

=0):

• Monte

Monte Carlo: Carlo:

• Front

Front-

• tracking

tracking: :

lim V

R b f r

β

=

0.48, 1.32 b β = = 0.5 β = 0.46 0.5 β β = = 0.5, 1.7 0.54, 1.2 b b β β = = = =

47

Computational considerations

• 2D:

2D: R Rlim

lim/r

/r could could be be reproduced reproduced

• High f

High fa

a:

: – – system system size size 256, 20000 time steps 256, 20000 time steps – – => 10 => 10 hours hours

• Low f

Low fa

a:

: – – system system size size 512, >60000 time steps 512, >60000 time steps – – => 10 => 10 days days

• 3D:

3D:

• R

Rlim

lim/r: x10 => system

/r: x10 => system size size: x10 : x10

• Third

Third power of system power of system size size

• Computer

Computer requirements requirements: x10 : x105

5

48

Pinning mechanism

• Zener

Zener

• Coherent/incoherent

Coherent/incoherent particles particles-

• matrix

matrix interface interface

• Rios

Rios