The Phase Field Method

Transcription

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

2 Outline Introduction Phase Field equations Phase Field simulations Grain growth Diffusion couples Conclusions

3 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

4 Thermodynamics for phase equilibria T,p,c constant CALPHAD-method Thermodynamic function : Gm( T, p, xk) Minimization of G m at T,p thermodynamic equilibrium G m -expressions according to thermodynamic models Optimization of parameters using experimental data G G 0 x B 1 α β 0 x B 1 G( x, x, T) = x G + x G + RT( x ln( x ) + x ln( x )) + x x L( x, x, T) A B A A B B A A B B A B A B

5 Thermodynamics of heterogeneous systems Homogeneous versus heterogeneous systems:

6 Gibbs energy for heterogeneous systems Cahn-Hilliard (1958) Several contributions to Gibbs energy g (, c η) = g (, c η) + g (, c η) + g (, c η) +... hom chem elast mag Interface energy flat surface + ( ) 2 σ = g() c + κ dc/ dx dx = + 2 G () ( ) V ghom c κ c dv 2 2 G = g (, ) ( ) ( ) V hom c η + κ c + α η dv

7 Equilibrium for heterogeneous systems Conserved variables: Non-conserved variables: G g = hom = φ φ 2 2α φ 0 G = cte c G fhom 2 2 = 2κ c= µ hom 2κ c= cte c c G = 0 φ

8 Linear kinetic theory c,x conserved Mass balance: Onsager: η, φ not conserved c t J = M µ k η G = L t η = J k jk j j k

9 Phase Field equations Cahn-Hilliard c g M hom = i κ c + r t t c Time dependent Ginzburg-Landau η g L hom = α η + ζ t η Cahn-Allen φ g hom 2 = L 2α φ t φ ( ζ ) 2 2 (, ) ( r t ) 2 2 (, )

10 Phase Field simulations Phase field equations c g hom 2 = M 2κ c t c η g hom 2 = L 2α η t η Determination of parameters Numerical solution of partial differential equations

11 Phase Field simulations for real materials A lot of phenomenological parameters Phase equilibria Influence of inhomogenieties Kinetic properties c ghom(, c η) 2 = M(, c η) 2κ( c, η) c t c η g 2 (, ) hom (, c η) = L c η 2 α( c, η) η t η

12 Numerical solution Discretisation in time and space Algebraic set of (nonlinear) equations For each phase field variable: Nx*Ny unknowns Discretisation techniques finite differences finite elements spectral methods

13 Grain Growth Model of D. Fan Phase field variables: η i g hom Gibbs energy 4 2 ηi ηi = ( ) + 2* 4 2 Minima p p p i= 1 i= 1 j= 1 j i η η 2 2 i j (η 1,η 2,,η p )=(1,0,0,,0),(0,1,0,,0),,(0,0,,1),(-1,0,,0),

14 Grain Growth Influence second phase particles Variable: φ Particle: φ=1 Outside particle: φ=1 Minima Gibbs energy (φ, η 1,η 2,,η p )=(1, 0,0, 0), (0, 1,0,,0),(0, 0,1,,0),

15 Grain Growth Evolution second phase particles Composition variable: c Particle: c=c particle Outside particle: c=c matrix Minima free energy (c, η 1,η 2,,η p )=(c particle, 0,0, 0), (c matrix, 1,0,,0),(c matrix, 0,1,,0), Precipitation, coarsening, dissolution of particles Solute drag

16 Grain Growth simulation

17 Diffusion couples Adjustable complexity Coupling with CALPHAD and DICTRA strategy to determine parameters Technically relevant Theoretical insights

18 Phase equilibria: coupling with CALPHAD CALPHAD G gc (,0, T) = G gc (,1, T) = Phase field g(, c φ, T) α m β m (, c T) V m (, c T) V m

19 Coupling with CALPHAD G Phenomenological phase field variables α β, φ) = ( 1 p( φ) ) G ( c ) + p( φ) G ( c ) + g( φ) W ( c ) m( ck m k m k k p φ φ φ 2 ( ) = (3 2 ) g 2 2 ( φ) = φ (1 φ) Physical order parameters Landau expansion polynomial gc (, η ) = gc (,0) + Bc () η + C() cηη + D () cηηη k i i ij i j ijk i j k i ij ijk + Eijkl( c) ηηη i j kηl +... ijkl

20 DICTRA Kinetics: coupling with DICTRA Diffusion in lattice reference frame: Atomic mobilities: M = x Ω k Va kva J = x x k k Va ΩkVa µ k V x m Arrhenius expression for temperature dependence: 1 0 Mk = Mk exp( Qk / RT) RT 0 RT ln( RTM ) = RT ln( M ) Q =Φ+Θ k k k Redlich-Kister expansion for composition dependence: Φ = xφ + xx Φ k i 0 ki, i j kij, i i j> i

21 Kinetics: coupling with DICTRA Phase field Onsagers law: J k = L ki µ x het i Laboratory reference frame Diffusion experiments Fick s law c Jk = Dk x Interdiffusion, tracer, intrinsic

22 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