Simulations of solidification microstructures by the phasefield method

Simulations of solidification microstructures by the phase-field method Mathis Plapp Laboratoire de Physique de la Matière Condensée CNRS/Ecole Polytechnique, 91128 Palaiseau, France

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

Dendritic growth of a pure substance Benchmark experiments: Slow growth (Glicksman, Bilgram): Undercoolings ~ 1 K Growth speeds ~ 1 mm/s Tip radius ~ 10 mm 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 mm

Physics of solidification (pure substance) On the interface: Stefan condition (energy conservation) liquid solid On the interface: Gibbs-Thomson condition (interface response) In the bulk: transport Here: assume diffusion only

Simplest case: symmetric model Assume: Define: capillary length kinetic coefficient Dendrites: form for anisotropic interfaces:

Phase-field model: physical background Free energy functional: H : energy/volume K : energy/length f: order parameter or indicator function

Phase-field model: coupling to temperature Dimensionless free energy functional: g : tilting function

Phase-field model: equations Phase-field parameters: W, t, l Physical parameters: d 0, b Matched asymptotic expansions:

Principle of matched asymptotic expansions liquid W solid inner region outer region • inner region (scale W): calculation with constant k and vn • outer region (macroscale): simple solution because f constant • matching of the two solutions close to the interface

Illustration: steady-state growth

Asymptotic matching

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

Hybrid Finite-Difference-Diffusion. Monte-Carlo algorithm • use the standard phase-field plus a Monte Carlo algorithm for the large-scale diffusion field only • connect the two parts beyond a buffer zone • diffusion: random walkers with adaptive step length

Adaptive step random walkers Diffusion propagator Convolution property Successive jumps For each jump, choose (distance to boundary), with c << 1

Handling of walkers • Use linked lists • A walker « knows » only its position • Data structure: position + pointer • After a jump, a walker is added to the list corresponding to the time of its next jump

Connect the two solutions • Use a coarse grid • Temperature in a conversion cell ~ number of walkers • Integrate the heat flux through the boundary • Create a walker when a « quantum » of heat is reached

An example

Benchmark: comparison to standard simulations Numerical noise depends roughly exponentially on the thickness of the buffere layer !

Example in 3 D: A dendrite Anisotropy:

Comparison with theory Growth at low undercooling (D=0. 1) Selection constant (depends on anisotropy)

Tip shape (simulated) • Tip shape is independent of anisotropy strength (!) • Mean shape is the Ivantsov paraboloid

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

Directional solidification • Experimental control parameters: temperature gradient G, pulling speed Vp, sample composition • Sequence of morphological transitions with increasing Vp : planar - cells - dendrites - cells - planar

Other applications of phase-field models • Solid-solid transformation (precipitation, martensites): includes elasticity • 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)

Acknowledgments Collaborators • Vincent Fleury, Marcus Dejmek, Roger Folch, Andrea Parisi (Laboratoire PMC, CNRS/Ecole Polytechnique) • Alain Karma, Jean Bragard, Youngyih Lee, Tak Shing Lo, Blas Echebarria (Physics Department, Northeastern University, Boston) • Gabriel Faivre, Silvère Akamatsu, Sabine Bottin-Rousseau (INSP, CNRS/Université Paris VI) • Wilfried Kurz, Stéphane Dobler (EPFL Lausanne) Support Centre National de la Rescherche Scientifique (CNRS) Ecole Polytechnique Centre National des Etudes Spatiales (CNES) NASA