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

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

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

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

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

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

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

Phase-field model: equations Phase-field parameters: W, , Physical parameters: d 0,  Matched asymptotic expansions:

Principle of matched asymptotic expansions solid liquid W inner region outer region inner region (scale W): calculation with constant  and v n outer region (macroscale): simple solution because  constant matching of the two solutions close to the interface

Illustration: steady-state growth

Asymptotic matching

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

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 3D: A dendrite Anisotropy:

Comparison with theory Growth at low undercooling (  =0.1) Selection constant (depends on anisotropy)

Tip shape 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 V p, sample composition Sequence of morphological transitions with increasing V p : 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