Phase Field Method
Transcript of Phase Field Method
Phase Field Method
From fundamental theories to a phenomenological simulation method
Nele Moelans24 June 2003
Outline
• Introduction• Important concepts:
Diffuse interphase-phase field variables
• Thermodynamics of heterogeneous systemsFree energy formulation (C-H) -theoretical derivation-
interpretation
• Kinetics of heterogeneous systemsLinear kinetic theory-conserved versus non-conserved
variables
Outline
• Phase field microelasticityStress-free strain-Khachaturyan’s approach-generalisation
• Calphad and phase field modeling• Determination of the other parameters
Gradient energy coefficient-kinetic parameters -mechanical parameters
• Conclusions
Introduction
• Simulation of phase transformations and other microstructural evolutions:– Dendritic solidification, martensitic transformations,
recrystallisation, graingrowth, precipitation, …
• Possible to simulate and to predict arbitrary microstructures
• Straight forward to account for various driving forces:– Chemical free energy, interfacial energy, elastic strain
energy, external fields
Important concepts
• Diffuse interfaces– Introduced by Cahn and Hilliard
(1958)
– Continous variation in properties accross interfaces of finite thickness
– Free energy for a heterogeneous system:
( )( , , ) , , , ( / ), ( / ), ( / )i i iF c F c c x x xφ η φ η φ η→ ∂ ∂ ∂ ∂ ∂ ∂
Interface
DistancePr
oper
ty
Important concepts
• Phase field variables– Composition: ci, Ni
• Both phases same structure– spinodal decomposition
– Phase field parameter: φ• Distinction between phases with different structures:
– phase α: φ=0, phase β: φ=1– solidification
– Long range order parameter: η• Ordering:
– disordered phase: η=0, ordered phase: η=η0(c)– Ni-superalloys : γ(disorderd fcc) → γ’(ordered fcc)
• Reduction in crystal symmetry– high symmetry phase: η=0, low symmetry phase: η=η0(c)– martensitic transformation: cubic→tetragonal
NB0 1
f(NB)
f(NB)
NB0 1
α
β
Thermodynamics of heterogeneous systems
• Free energy formulation (in J):
• Theoretical derivation (Cahn-Hilliard)– f (J/mol) depends on local composition and on composition
of immediate environment – Taylor expansion around f(c,0,0,…) :
with
2 20
1 ( , ) ( ) ( )V
F f c c dVη κ α η = + ∇ + ∇ Ω ∫
2 (1) 20( , ( / ), ( / ),...) ( ) ( / ) ( / )i i j i i ij i j
i ijf c c x c x x f c L c x c x xκ∂ ∂ ∂ ∂ ∂ = + ∂ ∂ + ∂ ∂ ∂∑ ∑
(2)(1/ 2) ( / )( / ) ...ij i jij
c x c xκ + ∂ ∂ ∂ ∂ ∑
0[ / ( / )] ,i iL f c x= ∂ ∂ ∂ ∂(1) 2
0[ / ( / )] ,ij i jf c x xκ = ∂ ∂ ∂ ∂ ∂(2) 2
0[ / ( / ) ( / )]ij i jf c x c xκ = ∂ ∂ ∂ ∂ ∂ ∂ ∂
Thermodynamics of heterogeneous systems
– Most general mathematical expression
– Cubic or isotropic symmetry
– Hence for a cubic lattice (J/mol):
1 2(1) (2)( ) ( )0, ,
0( ) 0( )i ij ij
i j i jL
i j i jκ κ
κ κ= =
= = =≠ ≠
( )22 20 1 2( , , ,...) ( ) ...f c c c f c c cκ κ∇ ∇ = + ∇ + ∇ +
2 (1) 20( ,( / ),( / ),...) ( ) ( / ) ( / )i i j i i ij i j
i ijf c c x c x x f c L c x c x xκ∂ ∂ ∂ ∂ ∂ = + ∂ ∂ + ∂ ∂ ∂∑ ∑
(2)(1/ 2) ( / )( / ) ...ij i jij
c x c xκ + ∂ ∂ ∂ ∂ ∑
Thermodynamics of heterogeneous systems
– Total free energy (J):
– Divergence theorem:
– ∇c•n=0 at the boundary ⇒
( )220 1 2
1 ( ) ...V
F f c c c dVκ κ = + ∇ + ∇ + Ω ∫
( ) ( )( ) ( )221 1 1/
V V Sc dV d dc c dV c n dSκ κ κ⇒ ∇ = − ∇ + ∇∫ ∫ ∫ i
( )20
1V
F f c dVκ = + ∇ Ω ∫
( ) ( )1 1S Vc n dS c dVκ κ∇ = ∇ ∇∫ ∫i
Thermodynamics of heterogeneous systems
• Flat interfase (C-H): – 1D:
– Free energy of a flat interfase:
– Equilibrium:σ minimal
( )20
1 ( ) /B BF A f N dN dx dxκ+∞
−∞ = + Ω ∫
( ) ( )20
1 ( ) / (1 )e eB B B B B Af N dN dx N N dxσ κ µ µ
+∞
−∞ = + − + − Ω ∫
( )21 ( ) /B Bf N dN dx dxσ κ+∞
−∞ = ∆ + Ω ∫
NB0 1
f 0(N
), (J
/mol
)
∆f
NBβNB
α
µBe
µAe
( )2( ) /B Bf N dN dxκ∆ =
[ ]1/ 2
1/ 22 ( )B
B
N
BN
f N dxβ
α
σ κ δ κ= ∆ ⇒ ∝Ω ∫
Distance
Nbα
Nbβ
α
β
Thermodynamic equilibrium in heterogeneous systems
• η, φ not conserved ⇒
• c conserved ⇒
• Euler’s equation
• ⇒
0Fφ
∂ =∂
F ctec
∂ =∂
20 2 0fF α φφ φ
∂∂ = − ∇ =∂ ∂
20 2fF c ctec c
κ∂∂ = − ∇ =∂ ∂
2 20
1 ( , ) ( ) ( )V
F f c c dVη κ α η = + ∇ + ∇ Ω ∫
( ( )) ( , ( ), '( )) 0( ) '( )F L d LF y x L x y x y x dxy x y dx y x∂ ∂ ∂= ⇒ = − =
∂ ∂ ∂∫
and
Kinetics of heterogeneous systems
• ci,Ni conserved– Mass balance
– Linear kinetic theory (Onsager)
– Cahn-Hilliard equation
kk
c Jt
∂ = −∇∂
k jkj j
FJ Mc
∂= − ∇∂∑
20 2fc M ct c
κ∂∂ = ∇ ∇ − ∇ ∂ ∂ i
Kinetics of heterogeneous systems
• η, φ not conserved– Linear Kinetic theory:
• Cahn-Allen equation:
• Time dependant Ginzburg-Landau equation:
• Equivalent formulation: Langevin equations
20 2fd Ldtφ α φ
φ ∂= − − ∇ ∂
20 2fd Ldtη α η
η ∂= − − ∇ ∂
d FLdtη
η∂= −∂
20 2 ( , )fc M c r tt c
κ ζ∂∂ = ∇ ∇ − ∇ + ∂ ∂ i
Phase field microelasticity theory
• Important for phase transformations in solids
• Elastic contribution to free energy
ε0
tot chem elF F F= + ∫=V
elkl
elijijkl
el dVCF εε21
Phase field microelasticity theory
• Khachaturyan’s approach → Fel(c,η)– Stress-free strain: εij
0(ck,ηl)
– Elastic strain/total strain:– Total strain is sum of homogeneous and heterogeneous
strains
Homogeneous strain=macroscopic strain
2000 )()( ηεδεε η+∂= ijc
ij rcr)()()( 0 rrr ij
totij
elij εεε −=
∂∂
+∂
∂=∂
→∂
∂+=
∫
i
j
l
iij
ijV ij
ijijtotij
rru
rrur
dVr
rr
)()(21)(
)(
)()(
ε
εε
εεε
Phase field microelasticity theory
– Mechanical equilibrium is established much faster than chemical equilibrium
• Elasticity theory:
→
→
• Extension to elastically inhomogeneous systems:
[ ] [ ])()()()(0)( 0 rrCrCr
rr
ijtotijijkl
elijijkl
elij
j
elij εεεσ
σ−===
∂∂
)(rui
∫=V
elkl
elijijkl
el dVrrcrrcCF ))(),(())(),((21 ηεηε
αβ
αβ
αβ
βα
eqeq
eqijkl
eqeq
eqijklijkl cc
crcC
ccrcc
CrC−−
+−
−=
)()()(
Generalisation to a phenomenological theory
• Multiple components and multiple phases: ck, φl or ηl
• For condensed matter f(J/mol)≈Gm(J/mol)
⇒CALPHAD approach to obtain free energy expressions
• Interfacial energy
22 22( , ) ( ) ( )
2 2m k l k l
k lVk lm
G cG c dVV
φ κ α φ
= + ∇ + ∇
∑ ∑∫
2 22 2 2 2( , ) ( , )( , ) ( / ) ( / )
2 2pij k l rij k lm k l
p i j r i jVp ij r ijm
c cG cG c x x x x dVV
κ φ α φφ φ
= + ∂ ∂ ∂ + ∂ ∂ ∂
∑∑ ∑∑∫
CALPHAD and the phase field method
• Transformation α(φ=0)→β(φ=1)
– Calphad:
– Wheeler-Boettinger
• p(φ) smooth function with p(0)=0,p(1)=1, for example
• double well
?),,(10),(),1,(
),(),0,(
TcGTcGTcGTcGTcG
km
kmkm
kmkm
φφ
β
α
→<<=
=
( ) )()()()()()(1),( kkmkmkm cWgcGpcGpcG φφφφ βα ++−=
)23()( 2 φφφ −=p
22 )1()( φφφ −=g
φ0 1
g(φ)
CALPHAD and the phase field method
• Landau expansion polynomial
– Symmetry (for example cubic→tetragonal phase transformation)
– Calphad:
– Other parameters must be determined by fitting
...)( ++∑ijkl
lkjiijkl cE ηηηη
∑∑∑ +++=ijk
kjiijkij
jiiji
iik cDcCcBcfcf ηηηηηηη )()()()0,(),(
3242 )()()()0,(),(
+++= ∑∑∑k
kk
kk
kk cGcEcCcfcf ηηηη
( ,0) ( )cubicmf c G c≈
Determination of the other parameters
• The gradient energy coefficient • Experimental measurement of interfacial energy and thickness• From theoretical calculation (starting from regular solution model, first
principles)• Kinetic parameters
– Mobilities in the Cahn-Hilliard equation• Relation between mobilities and diffusivities:• DICTRA-software
– Relaxation parameter in TDGL equation• Very hard to determine
• Elastic parameters– Elastic moduli Cijkl
• From mechanical experiments– Stress-free strains
• Related to the difference in lattice parameters between different phases
FJ Mc
∂= − ∇∂
BB RTMD =
d FLdtη
η∂= −∂
2 20
1 ( , ) ( ) ( )V
F f c c dVη κ α η = + ∇ + ∇ Ω ∫
Conclusions
• The Phase field method for modeling microstructural evolutions has become very popular
• Achievements– A consistent and general theory (thermodynamics, kinetics) has
been worked out– Simulations for simple cases give promising results– Link with CALPHAD and DICTRA
• Remaining problems: – Systems with multiple components and phases with different
orienations– Determination of the parameters– Computational intensive method