A sharp interface model for solidstate phase transformations under homogenization conditions

1
A sharp interface model for solid-state phase
transformations under homogenization conditions
E. Javierre, S. van der Zwaag Fundamentals of
Advanced Materials
F.J. Vermolen, C. Vuik Delft Institute of
Applied Mathematics
Delft University of Technology
EUROMAT 2007 Nürnberg, Germany September 10-13,
2007
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Introduction and objective
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Base assumptions

• Dissolution due to Fickian diffusion,
  cross-diffusion neglected
• Precipitate remains stoichiometric
• Thermodynamic local equilibrium at the interface
• Mass conservation

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Model requirements
Should capture morphological changes during
dissolution
Efficiency The numerical algorithm should have a
moderate computational cost
(cpu-time)
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Option 1 Phase Field Model
Computational cell parameterized according to
the local phase Sharp interface replaced by a
diffusive interface where phase transformation
takes place Interface motion
Transformation kinetics determined by free energy
functional F, which shall account for the free
energy density of the pure components in each
phase F such that sharp-interface problem
recovered in the limit of zero interfacial
thickness
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Option 2 Level Set Model
Computational domain parameterized by the
distance to the interface
Interface motion
Sharp interface boundary fitting necessary
to discretize equations at the interface
Cut-cell method
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Advantages / Disadvantages
- Adaptive mesh strategy required
- Direct implementation of interface conditions
avoided
- Suitable parameters hard to derive
- Velocity extension each time step
- Direct implementation of interface conditions
unavoidable
- Interfacial concentrations part of the solution
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Comparison with self-similar solutions
Dissolution of a 1D-plate

• Alloying species sequentially added
• Early stages of dissolution not affected by the
  size of the computational cell
• Long-term dissolution reaches equilibrium
  predicted by mass conservation

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Shape-preserving transformations
Dissolution of a Mg2Si elliptical particle in an
Al-alloy
Thermal regime isothermal, 560oC Initial area
Eccentricity Initial aspect ratio
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Non-shape-preserving transformations
Dissolution of a dumbbell-shaped precipitate in a
ternary alloy

• Thermal regime isothermal
• D2 2 D1
• Morphological change due to initial geometry
• Full dissolution of only one of the
  sub-particles
• Steady-state in agreement with mass conservation
  hypothesis

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Non-shape-preserving transformations
Spheroidization in cracked lamellar structures (I)
No crack
t0.084
t0
t0.027
Linear
t0
t0.009
t0.016
Sinusoidal
t0
t0.024
t0.036
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Non-shape-preserving transformations
Spheroidization in cracked lamellar structures
(II)

• Dissolution of a cementite plate
• Diffusion of Cr disregarded
• Isothermal transformation, T800oC

• Precipitate volume fraction 8.33
• Spheroidization by induced cracks
• Transient shapes depending on the surface
  perturbation

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Conclusions
A sharp interface model for phase transformations
in multicomponent alloys based on the Level Set
Method Fixed-grid algorithm combined, if
necessary, with a straightforward interface
fitting Additional work invested in maintaining
a distance function is rewarded with a
straightforward solution of the interfacial
conditions Results for simple precipitate
morphologies in agreement with self-similar
solutions Precipitate morphology has a large
impact on dissolution kinetics and macroscopic
evolution of the interface
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References Contact

• Selected publications
• E. Javierre, C. Vuik, F.J. Vermolen, S. van der
  Zwaag, A comparison of numerical models for
  one-dimensional Stefan problems, Journal of
  Computational and Applied Mathematics, vol. 192,
  pp. 445-459 (2006)
• E. Javierre, C. Vuik, F.J. Vermolen and A.
  Segal, A level set method for three dimensional
  vector Stefan problems Dissolution of
  stoichiometric particles in multi-component
  alloys, Journal of Computational Physics, vol.
  224, pp. 222-240 (2007)
• F.J. Vermolen, E. Javierre, C. Vuik, L. Zhao and
  S. van der Zwaag, A three dimensional model for
  particle dissolution in binary alloys,
  Computational Materials Science, vol. 30, pp.
  767-774 (2007)
• L. Zhao, E. Javierre, F.J. Vermolen, J. Sietsma,
  Simulation of NbC dissolution in micro-alloyed
  low carbon steel, In Proceedings of SteelSim
  2007, submited.
• Further information
• E-mail e.javierre_at_ewi.tudelft.nl
• Web site http//ta.twi.tudelft.nl/nw/users/pe
  rez/

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