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 = ∂ ∂ ∂ ∂ ∂ ∂ ∂


– 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 + ∂ ∂ ∂ ∂ ∑


– 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


• 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


• 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


– 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

Phase Field Method

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