[PPT] - New directions in phase- -field modeling of field modeling of New PowerPoint Presentation

SLIDE 1

New directions in phase New directions in phase-

field modeling of

field modeling of microstructure evolution in polycrystalline and microstructure evolution in polycrystalline and multi multi-

component alloys

component alloys

Nele Nele Moelans Moelans Liesbeth Liesbeth Vanherpe Vanherpe, Jeroen , Jeroen Heulens Heulens, Bert , Bert Rodiers Rodiers

K.U. Leuven, K.U. Leuven, Belgium

Belgium

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

‘Quantitative’ phase-field models

Properties

Properties bulk bulk and interfaces are and interfaces are reproduced reproduced accurately accurately in the simulations in the simulations

Effect

Effect model description and model description and parameters parameters

Numerical

Numerical issues issues

Insights in the

Insights in the evolution evolution of

f complex

complex morphologies and grain morphologies and grain assemblies assemblies

Effect

Effect of

f individual

individual bulk bulk and interface and interface properties properties

Predictive

Predictive ? ?

Depends

Depends on

n availability

availability and and accuracy accuracy of input

f input

data data – – Requires Requires composition and orientation composition and orientation dependence dependence

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

General framework and goal

Experiments Experiments, , atomistic atomistic simulations simulations and and thermodynamic thermodynamic models models Crystal structure, phase diagram, interfacial properties (energy, mobility, anisotropy), diffusion properties, …

Phase Phase-

field

field simulations simulations Microstructure evolution at the mesoscale

Quantitative Quantitative characterization characterization Average grain size, grain size distribution, volume fractions, texture,… Basis for statistical and mean field theories

SLIDE 4

Some aspects of model formulation

2

2-

phase

phase systems systems (single phase (single phase-

field

field) )

Multi

Multi-

grain/phase

grain/phase systems systems (multiple phase (multiple phase-

field

field) )

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

2-phase systems

Field variables:

Field variables:

Free

Free energy energy

Bulk

Bulk energy energy

( , ) r t φ

( , )

k

c r t

2

2

( , ) ( ) | | 2

chem V

W F f c g dr ε φ φ φ   = + + ∇    

∫

Interfacial energy

Double well function

Phase

Phase : : = 0 = 0

Phase

Phase : : = 1 = 1

Composition

Composition: : c cB

B

( ) ( )

( , ) ( , ) 1

chem

f c T f h h f c T

β α

φ φ = + −     Interpolation function

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Decoupling bulk and interfacial energy

Interface

Interface treated treated as mixture of 2 phases as mixture of 2 phases

c

c-

field

field for for each each phase phase

Equal

Equal interdiffusion interdiffusion potential potential + + conservation conservation

Bulk

Bulk energy energy

Kim et al., PRE, 6 (1999) p 7186; Kim et al., PRE, 6 (1999) p 7186; Tiaden Tiaden et et al., al., Physica Physica D, D, 115 (1998) p73 115 (1998) p73

( ) ( ) f c f c c c

β β α α β α

µ ∂ ∂ = = ∂ ∂

( )

( )

( ) ( ) 1

chem

f c h f f c h

β β α α

φ φ ⇒ = + −    

( ) ( )

1 c h c h c

β α

φ φ = + −     , c c c

α β

→

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Decoupling bulk and interfacial kinetics

Kinetic

Kinetic equations equations ( (Linear Linear non non-

equilibrium

equilibrium thermodynamics thermodynamics) )

Allen

Allen-

Cahn

Cahn

Diffusion

Diffusion

Jump

Jump in in chemical chemical potential potential accross accross interface interface

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)

F M t

ϕ

ϕ ϕ ∂ ∂ = − ∂ ∂

1 1

[1 ( )] | | | |

C L k kl l i l

c h M t t φ φ φ µ α φ

− =

∂ ∂ ∇ = ∇⋅ − ∇ + ∇⋅ ∂ ∂ ∇

∑

i

i n

v µ µ ∆ ∝ ∆ ∝

Solute trapping effect

1

( ) [1 ( )]

C k kl kl l l

c h M h M t

β α

φ φ µ

−

∂   = ∇⋅ + − ∇   ∂

∑

Non-variational anti-trapping

current

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Multi-grain and multi-phase structures

Single phase

Single phase-

field

field models models -

> Multiple

> Multiple phase phase-

field

field models models

Model extension

Model extension – – Different Different types of interfaces types of interfaces – – Triple and Triple and higher higher order

rder junctions

junctions

Numerically

Numerically – – Same Same accuracy accuracy for all interfaces for all interfaces and phases and phases – – All interfaces All interfaces within within range of range of validity validity of the

f the thin

thin interface interface asymptotics asymptotics

{ }

1 2 3

, , ,...,

p

η η η η η →

2 2 1 2 3 1 2

( , , ,...,| | ,| | ,...) F η η η η η ∇ ∇

num

cte → =

1

2

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

i p

η η η η =

Grain i Grain j

1

i

η =

j

η =

i

η = 1

j

η =

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Multi-grain and multi-phase models: major difficulties

Third

Third-

phase contributions

phase contributions

12

12 =

= 13

13 = 7/10

= 7/10 12

12

Careful

Careful choice choice of multi

f multi-
well

well function function and gradient contribution and gradient contribution

Interpolation

Interpolation function function

Zero

Zero-

slope

slope at at equilibrium equilibrium values of the values of the phase phase fields fields

Thermodynamic

Thermodynamic consistency consistency

3 3 1 1 2 2

3

3

1 2 1 2 1 1

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

p p i chem i i i i

f h f c T h η η η η

= =

= ⇒ =

∑ ∑

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Anisotropic grain growth model

Phase

Phase fields fields

With

With grain grain i i

Free

Free energy energy

For

For each each grain grain boundary boundary

Inclination

Inclination dependence dependence

2 2 ,

( )

i j i j

η η κ η κ ≠ ⇒ =

2 2 2 1 1 , 2

( )

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

κ η η η κ η η

= < = <

= ∑∑

∑∑

( ) ( ) ( )

, , , , , , ,

, , , | |

i j i j i j i j i j i j i j i j i j

L η η γ ψ κ ψ ψ ψ η η ∇ − ∇ = ∇ − ∇

4 2 2 2 2 1 , 1 1

1 ( ) ( ) 4 2 4 2

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

F dV m κ η γ η η η η η

= = < =

    = − + + + ∇        

∑ ∑∑ ∑ ∫

1 2

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

i p

η η η η =

1 2

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

i p

r t η η η η

L.-Q. Chen and W. Yang, PRB, 50 (1994) p15752
A. Kazaryan et al., PRB, 61 (2000) p14275

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Non-variational approach – equal interface width

Ginzburg

Ginzburg-

Landau

Landau type type equations equations

Non

Non-

variational

variational with with respect to respect to

dependence

dependence of

f
Similar

Similar to to Monte Monte Carlo Carlo Potts Potts approach approach

Definition

Definition ‘ ‘grain grain boundary boundary width width’ ’

( )

2 , 3 2

( , ) ( ) 2

i i i i j i j i i j

r t t L m η η η η η η η γ κ η

≠

    ∂ = − − + − ∇     ∂      

∑

max

max

1 1 | | | |

num i j

l d d dx dx η η = =

High High controllability controllability of

f numerical

numerical accuracy accuracy ( (l lnum

num/R

/R < 5 < 5) )

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Grain boundary properties

Grain

Grain boundary boundary energy energy

Grain

Grain boundary boundary mobility mobility

Grain

Grain boundary boundary width width

Iterative

Iterative algorithm algorithm

,

, , ,

( )

i j

gb i j i j

g m

θ

γ γ κ =

,

, , , 2 ,

( ( ))

i j

i j gb i j i j

L m g

θ

κ µ γ =

g( g(γ γi,j

i,j)

) calculated calculated numerically numerically

, 2 ,

4 3 ( ( ))

i j i j

l m g κ γ =

N. Moelans, B. Blanpain, P.

Wollants, PRL, 101, 0025502 (2008); PRB, 78, 024113 (2008)

, , , , ,

,[ ],[ ] ,[ ],[ ],[ ]

gb gb gb i j i j i j

m L

θ θ

γ µ κ γ →

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Numerical validation

Shrinking

Shrinking grain: grain:

Triple

Triple junction junction angles: angles:

Observations

Observations

Accuracy

Accuracy controlled controlled by by

Diffuse interface

Diffuse interface effects effects for for

Angles

Angles outside

utside [100

[100° °

140

140° ° ] ] require require larger larger for for same same accuracy accuracy

2 dA dt

α αβ αβ

πµ σ = −

dA dt

α αγ αβ

µ σ = −

,

αγ βγ αγ βγ

σ σ µ µ = =

/

num

x ∆

/

5

num

R >

/

num

x ∆

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Variational approach – interface width varies with interface energy

Anisotropy

Anisotropy only

nly through

through γi,j, , κ=cte

Energy

Energy

Kinetic

Kinetic equations equations

4 2 2 2 2 1 1 1 ,

1 ( ) 4 2 4 2

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

dV m F κ η η η η γ η

= = < =

      = − + + + ∇              

∑ ∑∑ ∑ ∫

( )

, 3 2 2

( , ) 2

i j i i i i j i j i

t t L r m η γ η η η η η κ η

≠

    ∂ = − − + − ∇     ∂      

∑

wetting

wetting

13 23 12

1 5 σ σ σ = =

13 23 12

5 σ σ σ = =

1

η 168.5 θ = °

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Rotation invariance of the model

Mathematically

Mathematically, the model , the model equations equations are invariant to rotation, but are invariant to rotation, but … …

the

the order

rder parameters

parameters represent represent orientations

rientations in a

in a fixed fixed reference reference frame. frame.

The

The precision precision of

f

depends depends on the

n the numerical

numerical setup, setup,

For the model to

For the model to be be rotational rotational invariant in practice, invariant in practice, lower lower limit limit of

f

amount amount of

f order
rder parameters

parameters p p: :

L h ∆ ≈ ∆ α α cos 1 h n L p ∆ > π 2

grid spacing grid spacing h h physical width of domain physical width of domain L L rotational symmetry rotational symmetry n n

J. Heulens and N. Moelans, Scripta Mat. (2010)

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Extension to multi-component multi- phase alloys

Phase

Phase field field variables: variables:

Grains

Grains

Composition

Composition

Bulk

Bulk energy energy: :

with

with

( , ) ( )

bulk k i k

f c f c

ρ ρ ρ ρ ρ

η φ = ∑

1

, ( , ),...,

A B C

c c r t c −

1

2 1 2

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

i p

r t

α α α β β

η η η η η η

k

k

x xρ

ρ ρ

φ = ∑

2 2 , ,... i i i i ρ ρ π π α β

η φ η

=

∑ ∑ ∑

and and

( ) ( ) ...

k k

k k k

f c f c c c

α

α α β β β

µ ∂ ∂ = = = ∂ ∂

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Extension to multi-component multi- phase alloys

Bulk and interface

Bulk and interface diffusion diffusion: :

With With and and

Interface

Interface movement movement: :

Between

Between phase phase and and

(

, )

i i k i

F x L t

ρ ρ ρ

η δ η δη ∂ = − ∂

2

2 k k m k

D M f x

ρ ρ ρ

= ∂ ∂

( ) ( )

2 int 2 2 2

2 ( , ) ( ) ( ) ( )

i j i

L g f c f c c c t

α β α α β β α β α α β

η η η η η µ η η   ∂   = − ∇ + − − −   ∂ +  

Curvature driven Bulk energy driven

2 2 , , , , k k i j k i j

x M t

ρ ρ ρ σ ρ ρ σ

φ η η µ

≠

    ∂ = ∇⋅ + ∇     ∂      

∑ ∑

2

3 /

interf gb interf m num k

D M f x δ δ    =      ∂ ∂    

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Numerical validation for multi- component multi-phase model

Processes

Processes for for which which v(t) v(t)

Conclusions for grain

Conclusions for grain growth growth model model remain remain – – Accuracy Accuracy controlled controlled by by – – Diffuse interface Diffuse interface effects effects for for – – Angles Angles outside

utside [100

[100° °

140

140° ° ] ] require require larger larger resolution resolution for for same same accuracy accuracy

/

num

x ∆

/

5

num

R >

/

num

x ∆

Growing

Growing sphere sphere

Coarsening

Coarsening

Triple

Triple junction junction

Intermediate

Intermediate phase phase

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Bounding Box Implementation

Basic elements

Basic elements

A grain is set of connected grid points r where |

A grain is set of connected grid points r where |eta_i(r eta_i(r)| > epsilon )| > epsilon

For each grain, the corresponding bounding box is the smallest

For each grain, the corresponding bounding box is the smallest cuboid cuboid containing the grain containing the grain Algorithm Solve the equations only locally, inside bounding boxes Only values inside boxes are kept in memory Boxes grow or shrink with grain Object Oriented C++ implementation

In collaboration

In collaboration with with L.

L. Vanherpe

Vanherpe and S. and S. Vandewalle Vandewalle, K.U. Leuven ( , K.U. Leuven (Vanherpe Vanherpe et al., et al., PRE, PRE, 76, n 76, n° °056702 (2007)) 056702 (2007))

SLIDE 20

Application examples

Grain

Grain growth growth in in anisotropic anisotropic systems systems with with a a fiber fiber texture texture

In collaboration

In collaboration with with F.

F. Spaepen

Spaepen, , Harvard Harvard University University

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Grain growth in columnar films with fiber texture

Grain

Grain boundary boundary energy energy: :

Fourfold

Fourfold symmetry symmetry

Extra

Extra cusp cusp at at = 37.5 = 37.5° °

Read

Read-

shockley

shockley

Discrete

Discrete orientations

rientations
Constant

Constant mobility mobility

Initially

Initially random random grain orientation grain orientation and grain and grain boundary boundary type type distributions distributions

White: θ = 1.5 Gray: θ = 3 Red: θ = 37,5 Black: θ > 3, θ 37.5

1 2 60

, ,..., ( , ),..., 1.5

i r t

η η η η θ ⇒ ∆ = °

2D simulation

2D simulation <0 0 1>

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Simulations: 1 high-angle energy cusp

White: Gray: Black: 37.5 Red: = 37,5

High

High-

angle grain

angle grain boundaries boundaries form form independent independent network network

Low

Low-

angle grain

angle grain boundaries boundaries follow follow movement movement of

f high

high-

angle

angle grain grain boundaries boundaries elongate elongate

No stable quadruple

No stable quadruple junctions junctions

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Misorientation distribution function (MDF)

Area

Area weigthed weigthed MDF MDF

Reaches

Reaches a a stead stead-

state

state

Low

Low energy energy boundaries boundaries lengthen lengthen + + their their number number increases increases Read Read-

Shockley

Shockley + + cusp cusp at at θ θ = 37.5 = 37.5° °

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Grain growth kinetics

Read Read-

Shockley

Shockley ( ( m

m=15

=15° ° ) + ) + cusp cusp at at =37.5 =37.5° °

Grain

Grain growth growth exponent exponent

PFM:

PFM: steady steady-

state

state growth growth with with

Previous

Previous findings findings: :

Mean

Mean field field analysis analysis: :

1 n ≈

0.6...1 n =

eff

n eff

A A k t − =

, (0) 1

eff h

n kt t A kt = → ∞ + →

1 ’ /

f f ef ef

dA d k k t N N = + =

N. Moelans, F.
N. Moelans, F. Spaepen

Spaepen, P. , P. Wollants Wollants, ,

Phil. Mag., 90 p 501
Phil. Mag., 90 p 501-
523 (2010)

523 (2010)

SLIDE 25

Application examples

Coarsening

Coarsening of Al

f Al6

6Mn

Mn precipitates precipitates located located on a

n a

recrystallization recrystallization front in Al front in Al-

Mn

Mn alloys alloys

In collaboration In collaboration with with A.

A. Miroux

Miroux, E. , E. Anselmino Anselmino, S. van der , S. van der Zwaag Zwaag, T. U. Delft , T. U. Delft

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Jerky motion during recrystallization in Al-Mn alloy

In

In-

situ EBSD observation of

situ EBSD observation of recrystallization recrystallization in AA3103 in AA3103 at at 400 400 ° ° C C

CamScan

CamScan X500 Crystal Probe X500 Crystal Probe FEGSEM FEGSEM

Jerky

Jerky grain grain boundary boundary motion motion

Stopping

Stopping time: 15 time: 15-

25 s

25 s

Pinning

Pinning by second by second-

phase

phase precipitates precipitates – – Al Al6

6(

(Fe,Mn Fe,Mn), ),

Al

Al12

12(

(Fe,Mn Fe,Mn) )3

3Si

Si

Added

Added to phase to phase field field model model

Grain

Grain boundary boundary diffusion diffusion

Driving

Driving force for force for recrystallization recrystallization

20m

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Phase field model

Multiple order parameter

Multiple order parameter representation representation: :

Mn

Mn composition composition field: field:

Homogeneous

Homogeneous driving driving pressure pressure for for recrystallization recrystallization: : md

Bulk

Bulk diffusion diffusion + + Surface Surface diffusion diffusion

,1 ,2 ,

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

m m p i

r t r t r t η η η

( , )

Mn

x r t

,1

,2 ,

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

m m p i

η η η =

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Material properties at 723K

D D0,p

0,p = D

= D0,

0,bulk bulk,

, Q Qp

p = 0.65Q

= 0.65Qbulk

bulk

D Dp

p =

= 1.2195 1.2195· ·10 10-

12

12 m

m2

2/s

/s Pipe diffusion Pipe diffusion high high angle angle boundaries boundaries, , precipitate precipitate/ /matrix matrix interface interface D D0,

0,bulk bulk = 10

= 10-

2

2 m

m2

2/s,

/s, Q Qbulk

bulk = 211 kJ/mol

= 211 kJ/mol D Dbulk

bulk = 5.5973

= 5.5973· ·10 10-

18

18 m

m2

2/s

/s Mn diffusion in Mn diffusion in fcc fcc Al Al γ γpr

pr = 0.3 J/m

= 0.3 J/m2

2

Interfacial Interfacial energy energy Al Al6

6Mn

Mn precipitates precipitates A Am

m = 6

= 6· ·10 1011

11; x

; xm

m 0 = 0.000258

= 0.000258 A Ap

p = 6

= 6· ·10 1012

12; x

; xp

p 0 = 0.1429

= 0.1429 Bulk Bulk energy energy density density: : f

fρ

ρ = A

= Aρ

ρ(x

(x-

x

xρ

ρ 0)

)2

2

c cMn

Mn = 0.3 w% (0.1474

= 0.3 w% (0.1474 at at%) %) ( (PhD PhD thesis thesis Lok Lok 2005) 2005) Actual Actual composition of composition of matrix matrix ( (supersaturated supersaturated) ) c cMn,eq

Mn,eq = 0.0524 w% (0.02456

= 0.0524 w% (0.02456 at at%) %) ( (PhD PhD thesis thesis Lok Lok 2005) 2005) Equilibrium Equilibrium composition of composition of matrix matrix M Mh

h = 2.94

= 2.94· ·10 10-

11

11 m

m2

2s/kg

s/kg

( (Miroux Miroux et et al.,Mater al.,Mater. . Sci

Sci. Forum,467

. Forum,467-

470,393(2004))

470,393(2004))

Mobility Mobility high high angle grain angle grain boundary boundary At At solute solute content 0.3w% Mn content 0.3w% Mn γ γh

h = 0.324 J/m

= 0.324 J/m2

2

Grain Grain boundary boundary energy energy high high angle angle

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Precipitate coarsening and unpinning

P

PD

D < P

< PZS

ZS (

(P PD

D

P PZS

ZS)

)

Pinning

Pinning: P : PZS

ZS=3.6

=3.6 MPa MPa

Rex: P

Rex: PD

D=3.1

=3.1 MPa MPa

Unpinning

Unpinning mainly mainly through through surface surface diffusion diffusion around around precipitates precipitates

0.9µm x 0.375µm

2 2 x y

J J J = +

6 s

7·10-10

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Precipitate coarsening and unpinning

P

PD

D <<< P

<<< PZS

ZS

Pinning

Pinning: P : PZS

ZS = 3.6

= 3.6 MPa MPa

Rex: P

Rex: PD

D = 1.1

= 1.1 MPa MPa

0.9µm x 0.375µm

Unpinning

Unpinning through through grain grain boundary boundary diffusion diffusion

2 2 x y

J J J = +

8 s

1.7·10-10

SLIDE 31

Application examples

Coarsening

Coarsening and diffusion and diffusion controlled controlled growth growth in in lead lead-

free solder joints

free solder joints

Within Within the the framework framework of COST MP

f COST MP-
0602

0602 ( (Advanced Solder Materials for High Advanced Solder Materials for High Temperature Application Temperature Application), Chairs A. ), Chairs A. Kroupa Kroupa and A. Watson and A. Watson

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Coarsening in Sn(-Ag)-Cu solder joints

IMC formation and

IMC formation and growth growth – – precipitate precipitate growth growth – – Kirkendal Kirkendal voids voids – – stresses stresses – – grain grain boundary boundary diffusion diffusion

– – CALPHAD description CALPHAD description – – Diffusion coefficients, Diffusion coefficients, growth growth coefficient for IMC coefficient for IMC-

layers

layers

SEM-image of Sn – 3.8Ag–0.7 Cu alloy after annealing for 200h at 150° C (Peng 2007)

Annealing temperature: 180 ° C Eutectic Composition: Sn-2at%Cu Cu Sn

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Phase field model

( ),1 ( ),2 ,

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

Cu Cu i ρ

η η η =

3 3 6 5 6 5

( ),1 ( ),2 ( ), ,1 ,2 ,1 ,2 ( ),1 ( ),2 ( )

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

Cu Cu Cu i Cu Sn Cu Sn Cu Sn Cu Sn Sn Sn Sn i

r t η η η η η η η η η η

Multiple order parameter model:

Multiple order parameter model:

Grains

Grains and and phases phases – – with with

Composition

Composition field: field: ( , )

Sn

x r t

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

CALPHAD Gibbs energies

COST 531

COST 531-

v3

v3-

0 database database (+ (+ parabolic parabolic extensions) extensions)

Cu Cu-

Sn

Sn T=180 T=180 ° ° C C

A. Dinsdale, et al. COST 531-Lead

Free Solders: Atlas of Phase Diagrams for Lead-Free Soldering,

vols. 1,2 (2008) ESC-Cost office

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

IMC-layer growth (1D)

Effect

Effect bulk bulk diffusion coefficient diffusion coefficient

( ) 25 2 3 13 2 6 5 13 2 ( ) 12 2

10 m /s 10 m / 10 m /s 10 m /s

Cu Sn Cu Sn Sn Cu Sn Sn Sn Sn

D D s D D

− − − −

= = = =

( ) 25 2 3 13 2 6 5 13 2 ( ) 14 2

10 m /s 10 m / 10 m /s 10 m /s

Cu Sn Cu Sn Sn Cu Sn Sn Sn Sn

D D s D D

− − − −

= = = =

6 3 6 6 5

0.0301 10 0.0833 10

Cu Sn Cu Sn

k k

− −

⇒ = ⋅ = ⋅

6 3 6 6 5

0.0306 10 0.0849 10

Cu Sn Cu Sn

k k

− −

⇒ = ⋅ = ⋅

( ) 12 2

10 m /s

Sn Sn

D

−

=

( ) t s

( ) h m

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Comparison with experimental data

0.0071 10 0.0071 10-

6

6

0.0043 10 0.0043 10-

6

6

200 200 ° ° C C 0.0038 10 0.0038 10-

6

6

0.0032 10 0.0032 10-

6

6

180 180 ° ° C C 0.00032 10 0.00032 10-

6

6

0.0010 10 0.0010 10-

6

6

150 150 ° ° C C k_Cu6Sn5 k_Cu6Sn5 k_Cu3Sn k_Cu3Sn T T

( ) 25 2 3 15 2 6 5 15 2 ( ) 12 2

10 m /s 10 m /s 10 m /s 10 m /s

Cu Sn Cu Sn Sn Cu Sn Sn Sn Sn

D D D D

− − − −

= = = = Parabolic Parabolic growth growth constant constant experiments experiments Cu3Sn, T = 180 Cu3Sn, T = 180 ° ° C C Parabolic Parabolic growth growth constant in constant in simulations simulations with with

6 3 6 6 5

0.0023 10 0.0073 10

Cu Sn Cu Sn

k k

− −

⇒ = ⋅ = ⋅

Cu6Sn5, T = 180 Cu6Sn5, T = 180 ° ° C C

J.
J. Janckzak

Janckzak, EMPA , EMPA

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Effect of grain boundary diffusion

( ) 25 25 2 3 15 13 2 6 5 15 13 2 ( ) 12 12 2

2 10 ,*2 10 m /s 2 10 ,*2 10 m /s 2 10 ,*2 10 m /s 2 10 ,*2 10 m /s

Cu Sn Cu Sn Sn Cu Sn Sn Sn Sn

D D D D

− − − − − − − −

= ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅

2

0.25 J/m

gb

γ =

Sn

J ↑

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Growth behavior Cu3Sn ?

(

) 25 2 3 15 2 6 5 15 2 ( ) 12 2 12 2

2 10 m /s 2 10 m /s 2 10 m /s 2 10 m /s D 2 10 m /s

Cu Sn Cu Sn Sn Cu Sn Sn Sn Sn surf Sn

D D D D

− − − − −

= ⋅ = ⋅ = ⋅ = ⋅ = ⋅

Grain structure Grain structure Composition: Composition: x xSn

Sn

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Conclusions

What

What do do we we need need next next to to improve improve the quantitative the quantitative accuracy accuracy of

f

phase phase-

field

field models models ? ?

Accurate

Accurate representation representation of triple

f triple-
junction

junction angles angles outside

utside [100

[100° °

140

140° ° ] ]

CALPHAD Gibbs

CALPHAD Gibbs energies energies over full composition

ver full composition domain

domain, , also also for for stoichiometric stoichiometric phases and phases and metastable metastable regions regions

Composition and orientation

Composition and orientation dependent dependent expressions for expressions for diffusion and diffusion and interfacial interfacial properties properties

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Thank you for your attention !

Acknowledgements

Acknowledgements

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 Flemisch Flemisch Super Computer (VSC) Super Computer (VSC)

Projects

Projects – – OT/07/040 (Quantitative phase field OT/07/040 (Quantitative phase field modelling modelling of coarsening in lead

f coarsening in lead-
free solder joints)

free solder joints) – – IUAP Program DISCO ( IUAP Program DISCO (Dynamical Dynamical Systems, Control, and Systems, Control, and Optimization Optimization – – IWT IWT grant grant SB SB-

73163 (

73163 (Phase Phase-

Field

Field Modelling Modelling of the Solidification of

f the Solidification of

Oxidic Oxidic Systems Systems) )

More

More information information on

n http://

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

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Webpage Webpage: : http://www.cs.kuleuven.be http://www.cs.kuleuven.be /conference/multiscale11/ /conference/multiscale11/

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Multi-grain and multi-phase models

Multi

Multi-

phase

phase-

field

field model model

Phase

Phase fields fields

Free

Free energy energy – – Double obstacle, Double obstacle, higher higher order

rder terms

terms, gradient , gradient term term non non-

variational

variational – – Interpolation: Interpolation: zero zero-

slope

slope or

r thermodynamic

thermodynamic consistency consistency

Multi

Multi-

order
rder parameter

parameter models models

Order

Order parameters parameters

Interfacial

Interfacial energy energy

1 2 1

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

p i p i i

r t η η η η η

=

  ≠    

∑

1

2 3 1

, , ,... , 1

p p i i

ϕ ϕ ϕ ϕ ϕ

=

∑

, int , , 2 2

4 | |

i i j i j i j j i j j i

f φ φ φφ π σ η η

≠

    = ∇ ⋅∇ +      

∑

Steinbach et al. MICRESS phase-field code

H. Garcke, B.Nestler,
B. Stoth, SIAM J. Appl.
Math. 60 (1999) p 295.

,

1

i j

φ < <

( )

4 2 2 2 2 int 1 1 1 ,

1 ( ) 4 2 4 2

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

m f η η η η κ γ η η

= = < =

    = − + + + ∇        

∑ ∑∑ ∑

L.-Q. Chen and W. Yang,

PRB, 50 (1994) p15752

A. Kazaryan et al., PRB,

61 (2000) p14275

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Multi-grain and multi-phase models

Vector

Vector valued valued model model

Orientation

Orientation field field ( ) and phase ( ) and phase field field ( ) ( )

Free

Free energy energy

2

2-

phase solidification

phase solidification

Phase

Phase fields fields

Fifth

Fifth order

rder interpolation

interpolation functions functions g gi

i(

( 1

1,

, 2

2,

, 3

3)

) – – Zero Zero-

slope

slope and and thermodynamic thermodynamic consistent consistent – – Order Order g gi

i increases

increases with with number number of phase

f phase-
fields

fields

Multi

Multi-

order
rder parameter

parameter + 4th + 4th order

rder gradient

gradient terms terms

Phase

Phase field field crystal crystal and amplitude and amplitude equations equations

int

( ,| |,| |) f f φ φ θ = ∇ ∇

φ

R. Kobayashi, J.A. Warren,

W.C. Carter, Physica D, 119 (1998) p415

3 1 2 3 1

, , , 1

i i

ϕ ϕ ϕ ϕ

=

∑

θ

R. Folch and M. Plapp, PRE, 72

(2005) n° 011602 I.M. McKenna, M.P. Gururajan, P.W. Voorhees, J. Mater. Sci., 44 (2009) p2206

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Nele Moelans Third annual workshop HERO-M, Saltsjöbaden, Sweden, May 17-18, 2010

Bounding Box Algorithm

Initialization by random nucleation

Initialization by random nucleation

Generate sphere

Generate sphere-

shaped grain uniformly over microstructure

shaped grain uniformly over microstructure

Generate particles uniformly over microstructure

Generate particles uniformly over microstructure

Set

Set-

up sparse data structure

up sparse data structure

Determine bounding box for every grain

Determine bounding box for every grain

Create object for every grain