Simulations%20of%20solidification%20microstructures%20by%20the%20phase-field%20method

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Simulations of solidification microstructures by
the phase-field method
Mathis Plapp Laboratoire de Physique de la
Matière Condensée CNRS/Ecole Polytechnique, 91128
Palaiseau, France
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Solidification microstructures
Hexagonal cells (Sn-Pb)
Dendrites (Co-Cr)
Eutectic colonies
Peritectic composite (Fe-Ni)
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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 gt
10 m/s (!) Tip radius lt 0.1 mm
Succinonitrile dendrite IDGE experiment
(space) M. Glicksman et al.
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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
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Simplest case symmetric model
Assume
Define
capillary length
kinetic coefficient
Dendrites form for anisotropic interfaces
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Phase-field model physical background
Free energy functional
H energy/volume K energy/length
f order parameter or indicator function
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Phase-field model coupling to temperature
Dimensionless free energy functional
g tilting function
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Phase-field model equations
Phase-field parameters W, t, l Physical
parameters d0, b Matched asymptotic expansions
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Principle of matched asymptotic expansions
W
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
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Illustration steady-state growth
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Asymptotic matching
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Multi-scale algorithms
Adaptive meshing or multiple grids It works but
it is complicated !
Adaptive finite elements (Provatas et al.)
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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
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Adaptive step random walkers
Diffusion propagator
Convolution property
Successive jumps
For each jump, choose
(distance to boundary), with c ltlt 1
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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
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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
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An example
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Benchmark comparison to standard simulations
Numerical noise depends roughly exponentially on
the thickness of the buffere layer !
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Example in 3D A dendrite
Anisotropy
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Comparison with theory
Growth at low undercooling (D0.1)
Selection constant (depends on anisotropy)
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Tip shape
Tip shape (simulated)
Tip shape is independent of anisotropy strength
(!) Mean shape is the Ivantsov paraboloid
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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
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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

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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)
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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