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Simulations of solidification microstructures by the phase-field 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.

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Simulations of solidification microstructures by the phase-field method

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

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

  3. 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): Undercoolings ~ 100 K Growth speeds > 10 m/s (!) Tip radius < 0.1 mm Succinonitrile dendrite IDGE experiment (space) M. Glicksman et al.

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

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

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

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

  8. Phase-field model: equations Phase-field parameters: W, t, l Physical parameters: d0, b Matched asymptotic expansions:

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

  10. Illustration: steady-state growth

  11. Asymptotic matching

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

  13. 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

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

  15. 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

  16. 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

  17. An example

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

  19. Example in 3D: A dendrite Anisotropy:

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

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

  22. 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

  23. 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

  24. 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)

  25. 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

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