The Phase Field Method -...

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The Phase Field Method To simulate microstructural evolutions in materials Nele Moelans Group meeting 8 June 2004

Transcript of The Phase Field Method -...

The Phase Field Method

To simulate microstructural evolutions in materials

Nele MoelansGroup meeting 8 June 2004

Outline

• Introduction• Phase Field equations• Phase Field simulations• Grain growth• Diffusion couples• Conclusions

Introduction Phase Field Methode

• Phase Field variables– Composition: c, x– Structure: φ, η

• Continuous variation in time and space – => no boundary

conditions at moving interfaces

– => diffuse interfaces

Thermodynamics for phase equilibria

• T,p,c constant• CALPHAD-method

– Thermodynamic function :– Minimization of Gm at T,p⇒

thermodynamic equilibrium– Gm-expressions according to

thermodynamic models– Optimization of parameters using

experimental data

xB0 1

GG

xB0 1

α

β

( , , ) ( ln( ) ln( )) ( , , )A B A A B B A A B B A B A BG x x T x G x G RT x x x x x x L x x T= + + + +

( , , )m kG T p x

Thermodynamics of heterogeneous systems

• Homogeneous versus heterogeneous systems:

Gibbs energy for heterogeneous systems

• Cahn-Hilliard (1958)

• Several contributions to Gibbs’ energy

• Interface energy flat surface

( , ) ( , ) ( , ) ( , ) ...hom chem elast magg c g c g c g cη η η η= + + +

( )2( ) /g c dc dx dxσ κ+∞

−∞ = ∆ + ∫

2( ) ( )homVG g c c dVκ = + ∇ ∫

2 2( , ) ( ) ( )homVG g c c dVη κ α η = + ∇ + ∇ ∫

Equilibrium for heterogeneous systems

• Conserved variables:

• Non-conserved variables:

0Gφ

∂ =∂

G ctec

∂ =∂

22 0homgG α φφ φ

∂∂ = − ∇ =∂ ∂

2 22 2homhom

fG c c ctec c

κ µ κ∂∂ = − ∇ = − ∇ =∂ ∂

Linear kinetic theory

• c,x conserved– Mass balance:

– Onsager:

• η, φ not conservedGL

η∂ ∂= −∂ ∂

kk

c Jt

∂ = −∇ ⋅∂

k jk jj

J M µ= − ∇∑

Phase Field equations

• Cahn-Hilliard

• Time dependent Ginzburg-Landau

• Cahn-Allen

22homgLtφ α φ

φ ∂∂ = − − ∇ ∂ ∂

( )22 ( , )homgL r ttη α η ζ

η ∂∂ = − − ∇ + ∂ ∂

( )22 ( , )homgc M c r tt c

κ ζ∂∂ = ∇ ∇ − ∇ + ∂ ∂ i

Phase Field simulations

• Phase field equations

• Determination of parameters• Numerical solution of partial differential equations

22homgLtη α η

η ∂ ∂ = − − ∇ ∂ ∂

22homgc M ct c

κ∂∂ = ∇ ⋅ ∇ − ∇ ∂ ∂

Phase Field simulations for real materials

• A lot of phenomenological parameters Phase equilibriaInfluence of inhomogenietiesKinetic properties

2( , )( , (2) , )homc ct

cgM ccc κ ηη η∂∂ = ∇ ⋅ ∇ − ∇ ∂ ∂

2(( , ) 2 , )( , )homgL ct

cc η α ηη

ηη η ∂ ∂ = − − ∇ ∂ ∂

Numerical solution

• Discretisation in time and space– Algebraic set of (non-

linear) equations– For each phase field

variable: Nx*Ny unknowns

• Discretisation techniques – finite differences– finite elements– spectral methods

Grain Growth

• Model of D. Fan

– Phase field variables: ηi

– Gibbs energy

– Minima (η1,η2,…,ηp)=(1,0,0,…,0),(0,1,0,…,0),…,(0,0,…,1),(-1,0,…,0),…

4 22 2

hom1 1 1

( ) 2*4 2

p p pi i

i ji i j

j i

g η η η η= = =

= − +∑ ∑∑

Grain Growth

• Influence second phase particles– Variable: φ

• Particle: φ=1• Outside particle: φ=1

– Minima Gibbs energy(φφφφ, η1,η2,…,ηp)=(1, 0,0,…0),

(0, 1,0,…,0),(0, 0,1,…,0),…

Grain Growth

• Evolution second phase particles– Composition variable: c

• Particle: c=cparticle• Outside particle: c=cmatrix

– Minima free energy(c, η1,η2,…,ηp)=(cparticle, 0,0,…0),

(cmatrix, 1,0,…,0),(cmatrix, 0,1,…,0),…– Precipitation, coarsening, dissolution of particles– Solute drag

Grain Growth simulation

Diffusion couples

• Adjustable complexity• Coupling with CALPHAD

and DICTRA⇒strategy to determine

parameters• Technically relevant• Theoretical insights

Phase equilibria: coupling with CALPHAD

• CALPHAD

• Phase field

( , )( ,0, )

( , )( ,1, )

m

m

m

m

G c Tg c TV

G c Tg c TV

α

β

=

=

( , , )g c Tφ

Coupling with CALPHAD

• Phenomenological phase field variables

––

• Physical order parameters– Landau expansion polynomial

( ) )()()()()()(1),( kkmkmkm cWgcGpcGpcG φφφφ βα ++−=2( ) (3 2 )p φ φ φ= −

22 )1()( φφφ −=g

...)( ++∑ijkl

lkjiijkl cE ηηηη

( , ) ( ,0) ( ) ( ) ( )k i i ij i j ijk i j ki ij ijk

g c g c B c C c D cη η ηη ηηη= + + +∑ ∑ ∑

Kinetics: coupling with DICTRA

• DICTRA– Diffusion in lattice reference frame:

– Atomic mobilities:

– Arrhenius expression for temperature dependence:

– Redlich-Kister expansion for composition dependence:

k Va kVaM x= Ω

kVa kk k Va

m

J x xV x

µΩ ∂= −∂

0

0

1 exp( / )

ln( ) ln( )

k k k

k k k

M M Q RTRT

RT RTM RT M Q

= −

= − = Φ + Θ

0, ,k i k i i j k ij

i i j ix x x

>

Φ = Φ + Φ∑ ∑∑

Kinetics: coupling with DICTRA

• Phase field– Onsagers’ law:

– Laboratory reference frame

• Diffusion experiments – Fick’s law

– Interdiffusion, tracer, intrinsic

heti

k kiJ Lx

µ∂= −∂∑

k kcJ Dx

∂= −∂

Conclusions

• Consistent and powerful theory• Main problems

– Large amount of parameters to determine– Computer resources

• Further work– More efficient implementation– Coupling between Phase Field Method and other

thermodynamic modeling techniques– Case studies