Modelling crystal growth with the phase-field method Mathis Plapp

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1 Modelling crystal growth with the phase-field method Mathis Plapp Laboratoire de Physique de la Matière Condensée CNRS/Ecole Polytechnique, Palaiseau, France

2 Solidification microstructures Hexagonal cells (Sn-Pb) Dendrites (Co-Cr) Eutectic colonies Peritectic composite (Fe-Ni)

3 Some basic facts Relevant physics : Transport of heat and components by diffusion and convection Interfacial properties: surface tension and kinetics Relevant length scales : Diffusion length l D = D/V ~ mm Capillary length d 0 ~ nm Structural details such as tip radius ρ ~ d 0 l D ~ µm Numerical problem : Complex time-dependent geometry

4 Some basic facts Relevant physics : Transport of heat and components by diffusion and convection Interfacial properties: surface tension and kinetics Relevant length scales : Diffusion length l D = D/V ~ mm Capillary length d 0 ~ nm ρ ~ d 0 l D Structural details such as tip radius ~ µm Numerical problem : Complex time-dependent geometry

5 Dendritic growth of a pure substance Benchmark experiments: Slow growth (Glicksman, Bilgram): Undercoolings ~ 1 K Growth speeds ~ 1 µm/s Tip radius ~ 10 µm Fast growth (Herlach, Flemings): Succinonitrile dendrite IDGE experiment (space) M. Glicksman et al. Undercoolings ~ 100 K Growth speeds > 10 m/s (!) Tip radius < 0.1 µm

6 Physics of solidification (pure substance) liquid solid Lv On the interface: Stefan condition (energy conservation) n = J Q = nˆ ( c D T c D T ) S S On the interface: Gibbs-Thomson condition (interface response) S L L L T int = T m γt L m κ v µ n k In the bulk: transport Here: assume diffusion only t T = D S,L 2 T

7 Simplest case: symmetric model Assume: D = D D c = c c S L = S L = Define: u = T T L / c m v u n int t = Dnˆ u = d = D ( u u ) 0 2 u S κ βv n L γt d0 = L c β = Lµ m 2 k c capillary length kinetic coefficient Dendrites: form for anisotropic interfaces: γ(nˆ ) β(nˆ )

8 Phase-field model: physical background Free energy functional: F = V f( φ) K 2 ( φ) + Hf ( φ) = φ 2 H : energy/volume K : energy/length / 2 + φ 4 / 4 γ ~ KH W ~ K / H φ: order parameter or indicator function

9 Phase-field model: coupling to temperature Dimensionless free energy functional: F = V W 2 g : tilting function 2 ( φ) / 2 + f( φ) + λug( φ)

10 Phase-field model: equations t u τ ( ) x,t δf = δφ ( x,t) 1 u x,t + 2 Phase-field parameters: W, τ, λ Physical parameters: d 0, β Matched asymptotic expansions: t φ ( ) 2 ( ) ( ) x,t = D W = a τ W 0 β = a1 a2 λ λw D d 1 t φ x,t

11 Principle of matched asymptotic expansions liquid W W << R W << D / vn solid inner region outer region inner region (scale W): calculation with constant κ and v n outer region (macroscale): simple solution because φ constant matching of the two solutions close to the interface

12 Illustration: steady-state growth

13 Asymptotic matching φ u x/w

14 Quantitative simulations 500 a) ρ/d W/d 0 SCN alloy (Georgelin and Pocheau) Cell spacing: 22.5 µm Capillary length: 13 nm Pulling speed: 32 µm/s Temperature gradient: 140 K/cm Ω b) W/d 0

15 Multi-scale algorithms Adaptive meshing or multiple grids: It works but it is complicated! Adaptive finite elements (Provatas et al.)

16 Hybrid Phase-field Mont-Carlo scheme

17 Example in 3D: A dendrite Anisotropy: W W( nˆ ) τ τ( nˆ ) nˆ = φ φ

18 Comparison with theory Growth at low undercooling ( =0.1) σ * = 2d ρ 0 2 D v Selection constant (depends on anisotropy)

19 Tip shape Tip shape (simulated) z(r, φ) 2 r = 2 Tip shape is independent of anisotropy strength (!) Mean shape is the Ivantsov paraboloid Q( φ)r 4

20 Rapid solidification of Nickel Kinetic parameters are important for rapid solidification Very difficult to measure Solution: use molecular dynamics (collaboration with M. Asta, J. Hoyt) Data points: circles: Willnecker et al. squares: Lum et al. triangles: simulations

21 Eutectic solidification Al-Cu (Trivedi et al) CBr 4 -C 2 Cl 6 (S. Akamatsu, G. Faivre)

22 Lamellar microstructures in Al-Cu Walker & Trivedi (2007)

23 Characteristic spacing Jackson and Hunt (1966) c α ce < < c β Important variables: Λ = λ/λ min Sample composition c T = 1 T 2 min λ λ min V λ 2 min const + λ min λ

24 Eutectic phase-field model Based on Access approach, but with different interpolations Each phase is represented by a phase field p i with i=α,β,l Constraint : p α + p β + p L = 1

25 «Ideal» interfaces p 3 = 0 p 1 =1-p 2 One variable

26 Generic situation p 3 varies (third-phase adsorption) Two variables

27 Visualization in the Gibbs simplex One independent variable Two independent variables

28 Construction of the interpolation functions Original Access model: f = p TW i j 2 Present model: ( ) 2 f TW = i p 2 i i p 2 j 1 2 Tilting function: g = p 151 ( p ) 1+ p ( p p ) i i p i { [ ] ( )} p 9p 5 / 4 i i j k works! i i

29 Benchmark: equal volume fractions W : thickness of the diffuse interface λ : lamellar spacing ( = width of a lamella pair)

30 Large system: eutectic composition

31 From lamellae to rods and back The composition is slowly varied: c = 0.3 c = Liu,Lee,Enlow,Trivedi

32 Other applications of phase-field models Solid-solid transformation (precipitation, martensites): includes elasticity Epitaxial growth Fracture Grain growth Nucleation and branch formation: includes fluctuations Solidification with convection: includes hydrodynamics Fluid-fluid interfaces, multiphase flows, wetting Membranes, biological structures Electrodeposition: includes electric field Electromigration Long-term goal: connect length scales to obtain predictive capabilities (computational materials science)

33 Scaling transients in dendritic growth Steady-state growth : understood Ivantsov solution (1947) = Pexp P e s s ( P) ds = P = ρv 2D T m T L / c Undercooling Peclet number Solvability theory ( ) : Surface tension is singular perturbation σ * = 2Dd 2 ρ v The constant σ* depends on anisotropy 0 Question : can we understand the transient regime?

34 Simpler problem : anisotropic Laplacian growth liquid solid v = n u L 2 u nˆ = uint 0 0 [ 1 εcos( θ) ]κ = d 4 u ln r r

35 Large-scale field by conformal mapping z = x + ψ = dψ dz u = + iy iw L 4 = z 1 2 z cosh 4 z / L 1 ( ) 2 Arm shape : Y(x, t) t [ L( t )] u = dt = y t L 4 x x 4 dt

36 Scaling assumption Suppose that : ( ) α L t = At Then Y(x, t) = 1 α t αa ( 1 α) / α 1 x L x / L s ( 1 α) / α ds 1 s 4 Scaling form of the arm shape : Y(x, t) ( α x / ) β = t t Y ~ with α + β = 1 (because of constant flux)

37 Tip condition Close to the tip, the shape is a parabola Tip radius : ρ = 1 x / Y 2β α ~ t 2 2 Use that for dendrites ρ 2 v = const. : ρ 2 v ~ t 4β 2α t α 1 Therefore : 4 β α 1 = 0 α + β = 1 α = 3 / 5 β = 2 / 5

38 Unscaled shapes 600 y (lattice units) x (lattice units)

39 Scaled shapes y/t 2/ x/t 3/5

40 Back to the diffusion problem

41 Scale separation Similarity solution for diffusion : growing sphere (Zener) R 1/ 2 R ~ t = Dt 2 P s In 2D : P s / ln 0 In 3D : P s 2 0 For small undercooling, the dendrite grows in the central region of the spherical solution (almost Laplace) until the arms catch up with the spherical envelope

42 For small undercooling: scaling applies! α = d ln L d ln t (a) ν α =0.02 =0.05 =0.1 =0.25 =0.5 Laplacian ν = d ln V d ln t t/t 0

43 In contrast : noise-reduced DLA ρ ~ const. 2 β α = 0 α + β = 1 α = 2 / 3 β = 1/ 3

44 Application: 3D dendrite shape y x Idea (E. Brener) : far behind the tip, slices are like 2D evolution

45 Shape of the dendrite fins 0 5 z simulation data polynomial fit, n=10 power law fit, n= x

46 Open question : Three dimensions? Ad hoc generalization of the scaling : axisymmetric branches R(z, t) ( α z / ) β = t t R ~ Spherical diffusion field in 3D : flux ~t 3/2 α + 2 β = 3 / 2 Tip condition ρ 2 v = const. : 4 β α 1 = 0 α = 2 / 3 β = 5 / 12

47 Simulation results in 3D 2.0 (b) ν α =0.01 =0.05 =0.1 α=2/3, ν=3/ t/t 0 Prediction from scaling theory : α=2/3, ν=3/2

48 Acknowledgments Collaborators Roger Folch, Andrea Parisi, Jesper Mellenthin (Laboratoire PMC, CNRS/Ecole Polytechnique) Alain Karma, Jean Bragard, Youngyih Lee, Blas Echebarria (Physics Department, Northeastern University, Boston) Gabriel Faivre, Silvère Akamatsu, Sabine Bottin-Rousseau (INSP, CNRS/Université Paris VI) Support Centre National de la Rescherche Scientifique (CNRS) Ecole Polytechnique Centre National des Etudes Spatiales (CNES) NASA