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  • 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: ,

    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

    ( , , ) ( 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 c = + + +

    ( )2( ) /g c dc dx dx + = +

    2( ) ( )homVG g c c dV = + 2 2( , ) ( ) ( )homVG g c c dV = + +

  • Equilibrium for heterogeneous systems

    Conserved variables:

    Non-conserved variables:

    0G

    =

    G ctec

    =

    22 0homgG

    = =

    2 22 2hom homfG c c cte

    c c = = =

  • Linear kinetic theory

    c,x conserved Mass balance:

    Onsager:

    , not conservedGL

    t

    =

    kk

    c Jt

    =

    k jk jj

    J M =

  • Phase Field equations

    Cahn-Hilliard

    Time dependent Ginzburg-Landau

    Cahn-Allen

    22homgLt

    =

    ( )22 ( , )homgL r tt

    = +

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

    i

  • Phase Field simulations

    Phase field equations

    Determination of parameters Numerical solution of partial differential equations

    22homgLt

    =

    22homgc M ct c

    =

  • Phase Field simulations for real materials

    A lot of phenomenological parameters Phase equilibriaInfluence of inhomogenietiesKinetic properties

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

    cgM ccc =

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

    cc

    =

  • 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: i

    Gibbs energy

    Minima (1,2,,p)=(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 = = =

    = +

  • Grain Growth

    Influence second phase particles Variable:

    Particle: =1 Outside particle: =1

    Minima Gibbs energy(, 1,2,,p)=(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, 1,2,,p)=(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 DICTRAstrategy 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

    =

    =

    ( , , )g c T

  • Coupling with CALPHAD

    Phenomenological phase field variables

    Physical order parameters Landau expansion polynomial

    ( ) )()()()()()(1),( kkmkmkm cWgcGpcGpcG ++=2( ) (3 2 )p =

    22 )1()( =g

    ...)( ++ijkl

    lkjiijkl cE

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

    g c g c B c C c D c = + + +

  • 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=

    kVa kk k Va

    m

    J x xV x

    =

    0

    0

    1 exp( / )

    ln( ) ln( )

    k k k

    k k k

    M M Q RTRT

    RT RTM RT M Q

    =

    = = +

    0, ,k i k i i j k ij

    i i j ix x x

    >

    = +

  • Kinetics: coupling with DICTRA

    Phase field Onsagers law:

    Laboratory reference frame

    Diffusion experiments Ficks law

    Interdiffusion, tracer, intrinsic

    heti

    k kiJ L x=

    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