The Phase Field Method -...
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
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
tη
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
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
∂= −∂