An Enthalpy Model for Dendrite Growth

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An Enthalpy Model for Dendrite Growth Vaughan
Voller University of Minnesota
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Growing Numerical Crystals
Objective Simulate the growth (solidification)
of crystals from a solid seed placed in an
under-cooled liquid melt
Some Physical Examples
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In terms of the process
Sub grid scale
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Simulation can be achieved using modest models
and computer power
Solved in ¼ Domain with A 200x200 grid
Box boundaries are insulated Since the thermal
boundary layer is thin Boundaries do not affect
growth until seed approaches edges
Growth of solid seed in a liquid melt Initial
dimensionless undercooling T -0.8 Resulting
crystal has an 8 fold symmetry
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By changing conditions can generate any number of
realistic shapes in modest times
PC CPU 5mins
Complete garbage
BUTWHY do we get these shapes ?WHAT is Physical
Bases for Model ? Solution is with a FIXED grid
-HOW does the Numerical solution work ? IS the
solution correct ?
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First recognize that there are two under-coolings
The bulk liquid is under-cooled, i.e., in a
liquid state below the equilibrium liquidus
temperature TM
The temperature of the solid-liquid interface is
undercooled
Kinetic vn normal interface velocity
Conc. of Solute, m lt 0 slope of liquidus

• Gibbs Thomson
• curvature,
• g surface tension

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Can describe process with the Sharp Interface
Modelfor pure material
With dimensionless numbers
Assumed constant properties
Insulated domain Initiated with small solid seed
The heat flows from warm solid into the
undercooeld liquid drives solidification Preferre
d growth direction and interplay
between curvature and liquid temperature gradient
determine Growth rate and shape.
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A Diffusive interface model Usually implies
Phase FieldHere we mean ENTHALPY
(see Tacke 1988, Dutta 2006)
Use smooth interpolation for Liquid fraction f
across interface
Smear out interface
f1
f0
Can be solved on a Fixed grid
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A microstructure model
Growth of Equiaxed Crystal In under-cooled melt
Sub-grid models Account for Crystal
anisotropy and smoothing of interface jumps
Phase change temperature depends on
interface curvature, speed and concentration
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Very SimpleCalculations can be done on regular PC
Numerical Solution
Sub grid constitutive
ENTHALPY
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Some Results
Physical domain 2-10 microns
Typical grid Size 200x200 ¼ geometry
seed
Initially insulated cavity contains liquid metal
with bulk undercooling T0 lt 0. Solidification
induced by placing solid seed at center.
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Tricks and Devices
Problem range of cells with 0 lt f lt 1
restricted to width of one cell Accuracy in
curvature calc?
Remedial scheme smear out f value, e.g.,
Remedial Scheme Use nine volume stencil to
calculate derivatives
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Produces nice answers BUT are they correct
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Dendrite shape with 3 grid sizes shows reasonable
independence
3.25do (black)
4do (blue)
2.5do (red)
e 0.05, T0 -0.65
Dimensionless time t 6000
a 0.25, b 0.75
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Long term tip dynamics approaches theory
BUT results begin to deteriorate if grid is made
smaller !!
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Verify solution coupling by Comparing with one-d
solidification of an under-cooled melt
T0 -.5
Compare with Analytical Similarity Solution
Carslaw and Jaeger



Front Movement
Temperature at dimensionless time t 250
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Looks Right!!
Verification 1
Enthalpy Calculation
k 0 (pure), e 0.05, T0 -0.65, Dx 3.333d0
Dimensionless time t 0 (1000) 6000
Red my calculation for these parameters With
grid size
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Low Grid Anisotropy
The Solid color is solved with a 45 deg twist on
the anisotropy and then twisted back the white
line is with the normal anisotropy
e 0.05, T0 -0.65
Dimensionless time t 6000
Note Different smear parameters are usedin 00
and 450 case
Tip position with time
Not perfect In 450 case the tip velocity at
time 6000 (slope of line) is below the
theoretical limit.
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For binary system need to consider solute
transport, discontinuous diffusivities and solutes
m
Single Domain Eq.
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Comparison with one-d Analytical Solution
Constant Ti, Ci
k 0.1, Mc 0.1, T0 -.5, Le 1.0
Concentration and Temperature at dimensionless
time t 100
Front Movement
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Effect of Lewis Number small Le ? interface
concentration close to C0
k 0.15, Mc 0.1, T0 -.65

• 0.05,
• Dx 3.333d0

All predictions at time t 6000
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Concentration
k 0.15, Mc 0.1, T0 -.55, Le 20.0
e 0.02, Dx 2.5d0
Profile along dashed line
Concentration field at time t 30,000
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FAST-CPU
This
On This
In 60 seconds !
time t 6000
e 0.05, T0 -0.65
a 0.25, b 0.75, Dx 4d0
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Conclusion Score card for Dendritic Growth
Enthalpy Method (extension of original work by
Tacke)
Ease of Coding Excellent
CPU Excellent (runs shown here took between 1 and 2 hours on a regular PC)
Convergence to known analytical sol. Excellent
Convergence to known operating state Very Good (if grid is not too fine and remedial parameters well chozen
Grid Anisotropy Good (see comment above)
Alloy Promising
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Playing Around
A Problem with Noise
Multiple Grains-multiple orientations
Grains in A Flow Field
Thses calculations were performed by Andrew Kao,
University of Greenwich, London Under supervision
of Prof Koulis Pericleous and Dr. Georgi
Djambazov.
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Extensions
Grain Growth
100 mm
seconds
Can this work be related to other physical cases ?
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NCEDs purpose

• to catalyze development of an integrated,
  predictive science of the processes shaping the
  surface of the Earth, in order to transform
  management of ecosystems, resources, and land use

The surface is the environment!
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Who we are 19 Principal Investigators at 9
institutions across the U.S.
Lead institution University of Minnesota

• Research fields
• Geomorphology
• Hydrology
• Sedimentary geology
• Ecology
• Civil engineering
• Environmental economics
• Biogeochemistry

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Moving Boundaries in Sediment Transport
Two Sedimentary Moving Boundary Problems of
Interest
Shoreline
Fans Toes
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Examples of Sediment Fans Moving Boundary
Badwater Deathvalley
1km
How does sediment- basement interface evolve
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A Sedimentary Ocean Basin
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Melting vs. Shoreline movement
An Ocean Basin
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Experimental validation of shoreline boundary
model
3m
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Base level
1-D finite difference deforming grid vs.
experiment
(n calculated from 1st principles)
Measured and Numerical results
Shoreline balance
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Is there a connection