Anisotropic Effects on Dendrites

1
Numerical Simulation of Dendritic Solidification
2
Outline

• Problem statement
• Solution strategies
• Governing equations
• FEM formulation
• Numerical techniques
• Results
• Conclusion

3
Problem Statement
Solid
T lt TM
Liquid
TW lt TM
TW lt TM
S
Liquid
Solid
T gt TM
S
When the solidification front grows into a
superheated liquid, the interface is stable, heat
is conducted through the solid
When the solidification front grows into an
undercooled melt, the interface is unstable, heat
is conducted through the liquid
4
Solution Strategies

• Most analytical solutions are limited to
  parabolic
• cylinder or a paraboloid of
  revolution
• Experimental studies are usually limited to
  transparent
• materials with ideal thermal
  properties such as succinonitrile
• Numerical solution can change the various
  material
• characteristics and modify the
  governing equations to gain
• more insight to the actual underlying
  physics

5
Governing Equations
Heat conduction is the dominant mode of thermal
transport
The interface temperature is modified by local
capillary effects
The second boundary conditions on the interface
preserves the sensible heat transported away from
the interface and the latent heat of fusion
released at the interface
6
Finite Element Formulation
For the energy equation in each domain,
the Galerkin Weighted Residual finite element
method formulation can be expresses as
7
Finite Element Formulation - Continued
Since a moving mesh is used to track the
interface, the motion of the coordinate system
must be incorporated
The FEM formulation of the heat equation can be
rewritten as
8
The time derivative is treated in standard finite
difference form,
and the heat equation is evaluated at time
with
9
Finite Element Formulation - Continued
The interface motion balances the heat diffused
from the front and the heat liberated at the
front
Numerically
10
Numerical Simulation Domain

• As the solidification
• front advances into the
• undercooled melt,
• unstable perturbations
• emerge and develop into
• dendrite
• Infinitesimal perturbation
• is seeded in the middle of
• the interface
• Boundaries are extended far
• enough to simulate a relatively
• unconfined environment

y
Free surface
x
T 0
T DT
Frozen region
Unfrozen region
11
Simulation Results
Sequences of Dendrite Growth from a
Crystallization Seed
12
Finite Element Mesh
Liquid
Solid
13
Numerical Technique -- Interface node adjustment
Coarse Interface Resolution
Refined Interface Resolution
14
Numerical Technique --Remeshing
Remeshed Mesh
Distorted Mesh
15
Numerical Technique -- Solution Mapping
k
P
j
i
16
Simulation Results
--Thermal Field in the Vicinity of the Dendrite
Tip
17
Simulation Results
-- Isotherm Coarsening
o C
.01 -.02 -.06 -.09 -.12 -.16 -.19 -.23 -.26 -.29
-.33 -.36 -.40 -.43 -.47 -.50
I
J
K
0.1 mm
18
Side Branch Steering
Steering direction
0.02mm
19
Simulation Results
Scaling Relationship of Dendrite Size with the
Critical Instability Wavelength ---
Dimensional
DT 5.92
DT 10
20
Simulation Results
Scaling Relationship of Dendrite Size with the
Critical Instability Wavelength
--- Non-dimensional
10
DT 10
(DU 0.42)
DT 5.92
(DU 0.25)
21
Simulation Results
Remelting
Remelted Branch Tip
22
Simulation Results
-- Dendrite Tip Velocity and Tip Radius
Relationship
0
1
0
Experimental Data
2
Simulated Data
-
1
1
0
2
-
2
1
0
Dendrite Tip Velocity (cm/s)
2
-
3
1
0
2
-
4
1
0
Simulated undercooling range 1.45-10 oC
Experimental undercooling range 0.0043-1.8 oC
2
-
5
1
0
-
4
-
3
-
2
-
1
1
0
2
3
4
5
1
0
2
3
4
5
1
0
2
3
4
5
1
0
Dendrite Tip Radius (cm)
23
Conclusions

• A numerical model was developed for dendritic
  growth simulation
• during solidification based on the physics
  governing the conduction
• of heat.
• The model accounts for interfacial curvature,
  surface energy,
• and the latent heat of fusion released during
  solidification.
• The moving interface was tracked with moving
  mesh, automatic
• remeshing and refinement techniques.
• The simulated dendrite growth characteristics
  agree with experimental
• observation
• dendrite growth tip velocity and radius
  relationship
• dendrite size scaling relationship
• side branch competition and remelting,
  isothermal coarsening.