Kinetic cross-coupling between non conserved and conserved fields in phase field models

Transcription

1 Kinetic cross-coupling between non conserved and conserved fields in phase field models Guillaume Boussinot 1,2,3, Efim A. Brener 1 1 P. Gruenberg Institut, Forschungszentrum Juelich, Germany 2 Max-Planck-Institut fur Eisenforschung, Duesseldorf, Germany 3 ACCESS e.v., Aachen, Germany

2 Diffusional growth and kinetic effects Transport equation in the bulk (diffusion equation) + boundary conditions at the interfaces : - energy/mass conservation - local equilibrium + kinetic effects 1 φ(x) thin interface analysis : kinetic effects depend strongly on W 1/2 0 W x which boundary conditions the phase field model corresponds to? φ φ

3 Phase field models for diffusional growth φ : non-conserved field C : conserved field Ċ + J =0 F = V dv f(φ)+w 2 φ 2 + g(φ,c) thermodynamic coupling between phase field and diffusion field classical phase field model : δf δφ = φ δf = J D(φ) 2 velocity scales : W/, D/W

4 Macroscopic approach (sharp-interface description) of diffusional transformations in binary alloys A1-CBC Diffusion equation in the bulk : Conservation of mass at the interface : Ċ = D i 2 C D 1 ( C 1 n)+vc 1 = D 2 ( C 2 n)+vc 2 = J B κ n D 1 D 2 C 1 C 2 total number of atoms number of solute atoms Interface kinetics : Φ 2 Φ 1 = AV + BJ B + d 0 κ grand potential: Φ i = f i (C i ) C i µ i µ 2 µ 1 = BV + CJ B chemical potential: µ i = f i(c i ) 3 velocity scales describing interface kinetics

5 Non diagonal model (1) classical diagonal model : δf δφ = φ +0 δf =0+ J D(φ) δf δφ = φ + non diagonal model : 3 velocity scales : δf = (MW φ) φ (MW φ) J + J D(φ) W/, D/W, 1/M positiveness of the Onsager matrix: M 2 < max[d(φ)w 2 ( φ) 2 ] Phys. Rev. E 86, 60601(R) (2012)

6 Non diagonal model (2) determinant > 0} 1 D(φ)(MW φ)2 φ = 1 δf δφ + MWD(φ) φ δf Ċ = D(φ) δf + MWD(φ) φ φ

7 Non diagonal model (2) determinant > 0} 1 D(φ)(MW φ)2 φ = 1 δf δφ + MWD(φ) φ δf Ċ = D(φ) δf + MWD(φ) φ φ } same structure as anti-trapping current

8 Non diagonal model (2) determinant > 0} 1 D(φ)(MW φ)2 φ = 1 δf δφ + MWD(φ) φ δf Ċ = D(φ) δf + MWD(φ) φ φ } same structure as anti-trapping current one-sided model: D 1 D 2 D 1 ( C 1 n)+vc 1 = J B = D 2 ( C 2 n)+vc 2 J B = VC eq 1 no requirement concerning Onsager symmetry

9 Thin interface limit A = [ 2MWC eq (x)][φ eq(x)] 2 + C 2 eq (x) D(φ eq ) (Ceq 1 )2 2D 1 (Ceq 2 )2 2D 2 B = MW [φ eq(x)] 2 Ceq (x) D(φ eq ) Ceq 1 Ceq 2 2D 1 2D 2 C = 1 D(φ eq ) 1 1 2D 1 2D 2 Phys. Rev. E 88, (2013) Equilibrium profiles : Switching function : { x φ eq(x) = tanh 2W C eq (x) = C + C p[φ eq(x)] 2 p(φ) = 15 8 φ 2φ 3 /3+φ 5 /5 C = Ceq 1 + Ceq 2 2 C = C eq 1 Ceq 2

10 1 D(φ) = 1 D + 1 Contrast of diffusivity { 1 D = ; 2D 1 2D 2 D = 1 1 2D 1 2D 2 D q(φ) with q(φ) = q( φ) ; q(±1) = ±1 A = α W 1 β C 2 W 2 D 2 CB B = αm γ C W D C =0 α = W β = 1 α γ = φ eq(x) 2 4W 1 p 2 (φ eq ) 2W [p(φ eq)q(φ eq ) 1]

11 1 D(φ) = 1 D + 1 Contrast of diffusivity { 1 D = ; 2D 1 2D 2 D = 1 1 2D 1 2D 2 D q(φ) with q(φ) = q( φ) ; q(±1) = ±1 A = α W C =0 α = W β = 1 α 1 β C 2 W 2 B = αm γ C W D γ = φ eq(x) 2 4W D 1 p 2 (φ eq ) 2W [p(φ eq)q(φ eq ) 1] 2 CB } Almgren: µ 2 µ 1 = BV + CJ B =0 if B =0i.e. M = γ C α then Φ 2 Φ 1 = AV + d 0 κ A = α 1 β C 2 W 2 with W D Karma-Rappel: 1/ D =0 B = C =0 if M =0 W D Phys. Rev. E 89, 60402(R) (2014)

12 Conclusion - The kinetic cross coupling is necessary to fully describe interface kinetics. allows to have the same number of degrees of freedom in phase field model and sharp-interface description solves problems: - elimination of temperature jump when finite contrast of diffusivity - introduce the Ehrlich-Schwoebel effect in MBE - tuning the solute trapping effect in alloys open question: introduction of kinetic cross coupling for multiphase systems (treatment of the triple junction?) (t) MW/ =0.4 MW/ = t/ v