Materials Process Design and Control Laboratory

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ON THE CONTROL OF CRYSTAL GROWTH PROCESSES USING
MAGNETIC FIELDS AND MAGNETIC FIELD GRADIENTS
Nicholas Zabaras and Baskar Ganapathysubramanian
Materials Process Design and Control
Laboratory Sibley School of Mechanical and
Aerospace Engineering188 Frank H. T. Rhodes
Hall Cornell University Ithaca, NY
14853-3801 Email zabaras_at_cornell.edu URL
http//www.mae.cornell.edu/zabaras/
Materials Process Design and Control Laboratory
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SPONSORS
Crystals of biological macromolecules
Genome sequence
New drugs, medicine
Free adiabatic surface
Find the time history of the boundary heat fluxes
and of the imposed magnetic field/gradient, such
that diffusion-based growth is achieved in the
presence of thermocapillary and buoyant forces
T 492oC
20 mm
20 mm
insulated
liquid
solid
Materials Process Design and Control Laboratory
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OBJECTIVES OF SOLIDIFICATION PROCESS DESIGN

• Requirements for a
• better crystal
• Flat growth interface with controlled growth
  velocity (V) and thermal gradient (G)
• Homogeneous distribution of solute
• Reduction in temperature and concentration
  striations during growth
• Minimize defects and dislocations
• Minimize residual stresses in the crystal

G ,V
qos
MELT
g
SOLID
qol
B

• DEVELOP INVERSE METHODS FOR
• Controlling the growth velocity V and the
  temperature gradient G
• Improving macroscopic and microscopic homogeneity
  of the final crystal
• Eliminating or reducing the effects of convection
  on the solidification morphology
• Delaying or eliminating morphological instability

• Controllable factors
• Interface motion
• Melt flow
• Thermal conditions
• Furnace design

Materials Process Design and Control Laboratory
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SOLIDIFICATION DRIVEN BY BUOYANCY, SURFACE
TENSION ELECTROMAGNETIC FORCES

• Governing physics
• SOLID
• Heat conduction
• MELT
• Heat and solute transport
• Incompressibility
• Navier-Stokes equations with Lorentz,
• Kelvin buoyancy force terms
• Traction force on free surface due to
• surface tension variation (Marangoni
• convection)
• SOLID-LIQUID INTERFACE
• Interfacial heat and solute balance
• Thermodynamic equilibrium
• conditions

Materials Process Design and Control Laboratory
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SOLIDIFICATION DRIVEN BY BUOYANCY, SURFACE
TENSION AND ELECTROMAGNETIC FORCES
Dimensionless groups For clarity and from a
computational standpoint, the governing equations
are non-dimensionalized. Prandtl number Lewis
number Rayleigh number - Thermal - Solutal
Hartmann number Marangoni number
Materials Process Design and Control Laboratory
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PHYSICAL MECHANISMS TO BE CONTROLLED DURING
SOLIDIFICATION
Diffusion
Interfacial Thermodynamics

• MEANS FOR DESIGN
• Control the boundary heat flux
• Multiple-zone controllable furnace design
• Rotation of the furnace
• Micro-gravity growth
• Electromagnetic fields

CRYSTAL
Morphological Instability
Capillarity
INTERFACE
Buoyancy Effects
Marangoni Convection
MELT
Electromagnetic Effects
Turbulence Effects
Rotational Effects
Volume Change Induced Flow
Microgravity Effects

• Furnace design
• Time history and number of heating zones.
• Achieve growth for given V
• Furnace requirements impractical

• Heat flux design
• Sampath Zabaras (2000, ..)
• Stable growth for given V
• Design for given V G
• Required heat flux uneconomical

• Electromagnetic fields
• Constant magnetic fields- damp convection, but
  large fields required
• Rotating magnetic fields, striations
• Combination of different magnetic fields?

• Micro-gravity growth
• Skylab experiments
• Suppression of convection
• Large, good quality crystals
• Very expensive

• Furnace rotation
• Cz growth, floating zone method
• Forced convection via Accelerated crucible
  rotation technique , etc.
• Material specific

Materials Process Design and Control Laboratory
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THERMAL DESIGN SOLIDIFICATION PROBLEM
DESIGN OBJECTIVES Find the optimal solid side
flux qos and the liquid qol side flux such that,
in the presence of coupled thermocapillary,
buoyancy, and electromagnetic convection in the
melt, a flat solid- liquid interface with desired
flux G and velocity V is achieved that is ensured
to be morphologically stable
100
Equaxied dendritic
casting
1
oriented
SOLUTION METHODOLOGY Implementation of a
mathematically rigorous inverse design method
using the adjoint method. The analysis is carried
out in functional spaces due to the infinite
dimensional nature of the problem.
dendrite
V (mm/s)
0.01
oriented cellular
VGD/?To
0.0001
NUMERICAL IMPLEMENTATION Use a whole-time
domain design approach Adjust process design
variables early to account for undesirable
effects of melt flow on solidification at later
times Object-oriented programming (OOP)
techniques for efficient implementation of
various schemes.
0.1
10
1000
G (k/mm)
Thermal gradient (G) and growth velocity (V) are
the main parameters that set the form and scale
of cast microstructures
Materials Process Design and Control Laboratory
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COMPUTATIONAL SOLIDIFICATION DESIGN EXAMPLE
FINITE ELEMENT MESH MATERIAL
SYSTEM Solid 960 bi-linear elements
Sb 8.6 Ge (near-eutectic Melt
1500 bi-linear elements

DIMENSIONLESS
GROUPS
Prandtl number 0.017
Solutal Rayleigh number
6.275e04 Lewis number 319.0
Thermal Maragoni number
-3982.02 Stefan number 0.1348
Thermal Rayleigh number 1.64405
REFERENCE BINARY ALLOY SOLIDIFICATION PROBLEM
UNDER NORMAL GROWTH CONDITIONS (NORMAL GRAVITY,
NO MAGNETIC FIELD)
1023 nodes
composition)
1581 nodes
POST PROCESS
Examination of constitutional stability
assumption on the solid-liquid interface Dimension
less contours of
? lt 0 corresponds to stable growth
STABLE REGION
UNSTABLE REGION
630.75oC
THE WELL-POSED DIRECT SIMULATION RESULTS DO NOT
SATISFY A-PRIORI ASSUMPTION OF STABILITY
Materials Process Design and Control Laboratory
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COMPUTATIONAL SOLIDIFICATION DESIGN EXAMPLE
INVERSE SOLIDIFICATION PROBLEM TO ACHIEVE
DIFFUSION-BASED STABLE GROWTH
Find the solid side optimal flux as well as the
liquid side optimal flux such that in the
presence of coupled thermo-capillary, buoyancy
and electromagnetic convection in the melt a
stable interface growth with V corresponding to a
diffusion based problem and G/V corresponding to
marginal stability is achieved
COUNTERACTS THE STRONG EFFECTS OF THERMOCAPILLARY
FLOW AT EARLIER TIMES
LIQUID SIDE OPTIMAL FLUX
BASED ON THE RESULTS OF PARAMETRIC STUDIES, A
STRONG MAGNETIC FIELD WAS SELECTED IN ORDER TO
FACILITATE THE THERMAL DESIGN PROCESS THROUGH
PARTIAL DAMPING OF MELT FLOW
Materials Process Design and Control Laboratory
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MAGNETO-HYDRO-DYNAMICS IN SOLIDIFICATION A
BRIEF LOOK
Maxwells equations
Equations of Motion Energy
Non-relativistic flow
Quasi-magneto static
Materials Process Design and Control Laboratory
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MAGNETO-HYDRO-DYNAMICS IN SOLIDIFICATION A
BRIEF LOOK
Starting from the electromagnetic stress tensor
(L. Dragos)
Taking the divergance
Maxwells relations
The total electromagnetic force acting
Magnetizing force
Lorentz force
Electromagnetic pressure
Materials Process Design and Control Laboratory
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MAGNETO-HYDRO-DYNAMICS IN SOLIDIFICATION A
BRIEF LOOK
JxB Lorenz force term
Kelvin force term Depends on the gradient of the
magnetic field and the susceptibility of the
material
Kelvin force
Joule heating Thermo-magnetic cross-effects
Behavior of a material in a magnetic
field Paramagnetic Weakly attracted towards the
field. Examples include Nitrogen
gas. Diamagnetic Weakly repelled by the field.
Examples include Water, Germanium,
Bismuth Ferromagnetic Strongly attracted towards
the field. Examples include Iron, Nickel and
Cobalt. Nature of the material corresponds to
its magnetic susceptibility, ? Diamagnetic ?
-1e-6, Paramagnetic ? 1e-7, Ferromagnetic ?
10 Interestingly, ? depends on temperature. Curie
s law for paramagnetic materials ? 1/T. That is
in the presence of a thermal gradient, a
paramagnetic substance experiences a body force
in a magnetic field Diamagnetic materials ??m
1/T (?m constant) Provides a means of control
Materials Process Design and Control Laboratory
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MAGNETO-HYDRO-DYNAMICS IN SOLIDIFICATION A
BRIEF LOOK
The application of a magnetic gradient provides a
means of control

• Possibility of control using ferrofluids
• Addition of certain coated particles which are
  highly ferromagnetic to the fluid system. The
  application of a magnetic field causes a
  substantial force to act on the particles
  dispersed in the fluid. The particles force the
  bulk fluid also to behave predictably.
• Amount and distribution of particles
• Properties of the particles
• Problem of distribution of the particles

Commonly used fields Constant magnetic field
no gradient hence no Kelvin force Rotating
magnetic fields Kelvin force through induced
electric fields but electric field very
small Travelling magnetic fields Kelvin force
through change in magnetic field as it moves
but field highly localized Fringe effects exist
for all fields
In the present work, consider only the inherent
magnetic properties of the materials and attempt
to control certain processes by understanding the
effects of the properties (magnetic, thermal and
solute) on each other
Materials Process Design and Control Laboratory
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MAGNETIC GRADIENTS IN FLOW CONTROL

• Behavior of gas flow under gradient magnetic
  fields N.I.Wakayama (1990)
• Applied to magnetic quenching of flames
• Material processing in high static magnetic
  fields A review of an experimental study on
  levitation, phase separation, convection and
  texturation D.Braithwaith et.al. (1993)
• Developed the magnetic Rayleigh number
• Magnetizing force modeled and numerically solved
  for natural convection of air in a cubic
  enclosure-effect of the direction of the magnetic
  field T.Tagawa et.al. (2003)
• Derived a Boussinesq approximation for the Kelvin
  force
• Suppression/reversal of natural convection by
  exploiting the temperature/composition dependence
  of magnetic susceptibility J.W.Evans et.al., J.
  Appl. Phys (2000)
• Applied to reversal of flow of aqueous salt
  solution.

Here, ?Ra is the magnetic Rayleigh number
Possibility of earth based microgravity
environment through the use of magnetic
gradients Commercial gradient coils are readily
available. Coils producing specific magnetic
gradients are used in MRI Required fields can be
produced by the superposition-ing of magnetic
gradient coils and uniform field coils By
changing the power/current to the coils, the
magnitude and direction of the total magnetic
field can be varied
Materials Process Design and Control Laboratory
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MAGNETIC GRADIENTS IN FLOW CONTROL

• Suppression/reversal of natural convection by
  exploiting the temperature/composition dependence
  of magnetic susceptibility J.W.Evans et.al., J.
  Appl. Phys (2000)
• Applied to reversal of flow of aqueous salt
  solution

System specifications Natural convection in a
square cavity Cavity filled with water Left wall
at 30 C. Right wall at 10 C No magnetic gradient
applied
Materials Process Design and Control Laboratory
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MAGNETIC GRADIENTS IN FLOW CONTROL

• Suppression/reversal of natural convection by
  exploiting the temperature/composition dependence
  of magnetic susceptibility J.W.Evans et.al., J.
  Appl. Phys (2000)
• Applied to reversal of flow of aqueous salt
  solution

System specifications Natural convection in a
square cavity Cavity filled with water Left wall
at 30 C. Right wall at 10 C Magnetic gradient
corresponding to ? 1
Materials Process Design and Control Laboratory
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MAGNETIC GRADIENTS IN FLOW CONTROL

• Suppression/reversal of natural convection by
  exploiting the temperature/composition dependence
  of magnetic susceptibility J.W.Evans et.al., J.
  Appl. Phys (2000)
• Applied to reversal of flow of aqueous salt
  solution

System specifications Natural convection in a
square cavity Cavity filled with water Left wall
at 30 C. Right wall at 10 C Magnetic gradient
corresponding to ? 2
Materials Process Design and Control Laboratory
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DETRIMENTAL EFFECTS OF CONVECTION ON THE
SOLIDIFICATION SYSTEM

• INFLUENCE OF COUPLED THEMOCAPILLARY AND BUOYANCY
  DRIVEN CONVECTION ON THE SOLIDIFICATION SYSTEM
• Coupled capillary and buoyancy forces increase
  the strength of the melt convection leading
  to higher distortion of the freezing front
• Transition to unsteady behavior is hard to
  control due to the presence of capillary
  convection
• Buoyancy and thermo-capillary forces can couple
  in a complicated manner leading to hydrodynamic
  instabilities
• Complex melt flow structures lead to segregation
  patterns that impact the solidified product
  quality
• COMPUTATIONAL DESIGN APPROACH
• Implementation of a mathematically rigorous
  inverse design method for achieving
  diffusion-like stable growth conditions in the
  presence of melt convection through optimal
  boundary heat fluxes and magnetic fields
• Use a whole-time domain design approach
  Adjust process design variables early to account
  for undesirable effects of melt flow on
  solidification at later times

Materials Process Design and Control Laboratory
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MATERIAL SOLIDIFICATION PROCESS DESIGN UNDER
DIFFUSION CONDITIONS

• Growth under diffusion dominated conditions
  ensures
• Flat solid-liquid interface. This is crucial in
  crystal growth
• Uniform temperature gradients along the
  interface. This results in reduced stress in the
  cooling crystal. Found to be directly related to
  the life time of the component
• Uniform solute distribution. This leads to a
  homogeneous crystal. Further, this also results
  in reduced dislocations
• Suppression in temperature and solute
  fluctuations leading to reduced defects in the
  crystal

DESIGN OBJECTIVES Find the optimal magnetic field
such that, in the presence of coupled
thermocapillary, buoyancy, and electromagnetic
convection in the melt, a flat solid- liquid
interface with diffusion dominated growth is
achieved
Micro-gravity based growth is purely diffusion
based Objective is to achieve some sort of
reduced gravity growth
Materials Process Design and Control Laboratory
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INVERSE-DESIGN PROBLEM
INVERSE PROBLEM STATEMENT Find the magnetic field
b(t) in 0, tmax such that melt convection is
suppressed
B(t)

• With a guessed magnetic field, solve the
  following direct problem for
• Melt region
• Temperature field T(x, t b)
• Concentration field c(x, t b)
• Velocity field v(x, t b)
• Electric potential ?(x, t b)
• Solid region
• Temperature field Ts(x, t b)

Measure of deviation from diffusion based growth
Materials Process Design and Control Laboratory
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NONLINEAR OPTIMIZATION APPROACH TO THE INVERSE
SOLIDIFICATION PROBLEM
Define the inverse solidification problem as a
unconstrained spatio- temporal optimization
problem
Find a quasi- solution bo ? L2 (0, tmax) such
that S (bo) ? S (b ) ? bo ? L2 ( 0, tmax)
Solve the above unconstrained minimization
problem using the nonlinear Conjugate Gradient
Method (CGM)
Needs design gradient information
Needs descent step size
Continuum adjoint problem
Continuum sensitivity problem
Materials Process Design and Control Laboratory
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ACHIEVING MAGNETIC GRADIENTS
Rapidly varying, significantly large, magnetic
gradients superimposed on a uniform magnetic
field are required for Magnetic Resonance
Imaging. Rapid advances in MRI have resulted in
easily controllable, economical gradient coils
Schematic diagram showing a longitudinal section
of system composed of resistive magnets and
copper shields to generate magnetic gradients (S.
De Luzio et.al.)
High magnetic field facilities Research on
Ultra large magnetic gradients Levitation
Materials Process Design and Control Laboratory
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DIRECT CONTINUUM PROBLEM AND THE CONTINUUM
SENSITIVITY PROBLEM
DIRECT EQUATIONS
SENSITIVITY EQUATIONS
Materials Process Design and Control Laboratory
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DIRECT CONTINUUM PROBLEM AND THE CONTINUUM
SENSITIVITY PROBLEM
SENSITIVITY EQUATIONS
DIRECT EQUATIONS
Materials Process Design and Control Laboratory
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THE CONTINUUM ADJOINT PROBLEM
Find an analytical expression for the gradient of
the cost functional
Find operators (similar to the corresponding
sensitivity operators) that satisfy the following
relationships
Using integration by parts Greens theorem
Reynolds transport theorem and some vector algebra
Materials Process Design and Control Laboratory
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THE CONTINUUM ADJOINT PROBLEM
Using integration by parts Greens theorem
Reynolds transport theorem and some vector algebra
1 2 3 4
Adjoint equations
Gradient of the cost functional given in terms of
the direct and the adjoint fields
Materials Process Design and Control Laboratory
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NUMERICAL IMPLEMENTATION

• FINITE ELEMENT SOLUTION METHODS
• SUPG/PSPG fluid flow formulation with
  additional terms from buoyancy, surface tension
  and electromagnetic forces
• Consistent SUPG formulation for transport
  equations
• Moving/deforming FEM to explicitly track
  the advancing solid-liquid interface
• Predictor-corrector scheme for time
  integration of heat/mass transport eqs
• T1 formulation for time integration of
  flow equations
• Object-oriented programming (OOP)
  techniques for efficient implementation of
  various schemes

Preconditioned GMRES based fast parallel solver
Element-by-element parallel solver
Optimization algorithm Conjugate Gradient
method- Fletcher-Reeves formula
Navier-Stokes
Convection-Diffusion
Materials Process Design and Control Laboratory
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AN OBJECT-ORIENTED APPROACH TO MULTIDISCIPLINARY
DESIGN OPTIMIZATION
DirectFlow
Direct
DirectHeat
LinearSolver
DirectConc
StabNavierStokes
MenuUDC
Store4Plotting
AdjointFlow
Material Process Design Simulator
Adjoint
AdjointHeat
AdjointConc
FEM
ConvectionDiffusion
MenuUDC
SensitivityFlow
Sensitivity
Store4Plotting
SensitivityHeat
SensitivityConc
Materials Process Design and Control Laboratory
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VALIDATION OF THE CONTINUUM SENSITIVITY PROBLEM
Run direct problem with field b
Find difference in all properties
Run direct problem with field b?b
Compare the properties
Run sensitivity problem with b ?b
Material characteristics Prandtl number
0.007 Thermal Rayleigh number 82931 Solutal
Rayleigh number 0.0 Lewis number 330 Marangoni
number -8000 Stefan number 0.034
Setup Specifications Solidification in a
rectangular cavity Dimensions 2cm x 2cm Fluid
initially at 40 C superheat Left wall kept at 40
C below melting Magnetic fields corresponding to
? 0.5 and ? ? 0.2
Materials Process Design and Control Laboratory
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VALIDATION OF THE CONTINUUM SENSITIVITY PROBLEM

Isotherms
Validation of the continuum sensitivity problem
by comparison with a finite difference
formulation. Run the direct problem for two
different fields. Evaluate the differences in
various properties to find the sensitivity of
that property to the change in the control field.
Compare it with the continuum sensitivity
properties. ? 0.5, ??0.2
Isochors
Isobars
Streamline contours
Materials Process Design and Control Laboratory
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INVERSE DESIGN PROBLEM 1 - NON CONDUCTING MATERIAL
Design definition Find the time history of the
imposed magnetic field/gradient, such that
diffusion- based growth is achieved in the
presence of thermocapillary and buoyant
forces Material characteristics Binary
alloy/pure material, Non-conducting
Material characteristics Prandtl number
0.007 Thermal Rayleigh number 82931 Solutal
Rayleigh number 0.0 Lewis number 330 Marangoni
number -8000 Stefan number 0.034 Setup
specifications Solidification in a rectangular
cavity Dimensions 2cm x 2cm Fluid initially at 40
C superheat Left wall kept at 40 C below
melting Top surface free.

Driven purely by thermal or solutal buoyancy
(with surface tension effects)
Materials Process Design and Control Laboratory
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INVERSE DESIGN PROBLEM 1 - NON CONDUCTING
MATERIAL- RESULTS
Reference case
Optimal magnetic field
Significant improvement in the shape of the solid
liquid interface. In the growth with no field the
interface becomes skewed due to the
thermo-capillary effects.
Interface positions at different times
Concentration profile at two positions on the
interface
Large reduction in the fluctuations in
concentration at the interface Directly
responsible for improvement in quality of crystals
Materials Process Design and Control Laboratory
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INVERSE DESIGN PROBLEM 1 - NON CONDUCTING
MATERIAL- RESULTS
Comparison of stream-function and temperature
contours
Reference case
Optimal magnetic field
Reduction in vorticity by 2 orders of magnitude.
??max 6.7 ?max 0.07
Streamline contours
The reduced vorticity results in negligible
convection. This leads to diffusion based heat
transfer in the melt as seen by the planar
isotherms
Isotherms
Materials Process Design and Control Laboratory
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INVERSE DESIGN PROBLEM 2 - NON CONDUCTING MATERIAL
Design definition Find the time history of the
imposed magnetic field/gradient, such that
diffusion- based growth is achieved in the
presence of thermocapillary and buoyant
forces Material characteristics Binary
alloy/pure material, Non-conducting

Material specification 27 NaCl aqueous
solution Prandtl number 0.007 Thermal Rayleigh
number 200000 Solutal Rayleigh number
10000 Lewis number 3000 Marangoni number
0 Stefan number 0.12778 Ratio of thermal
diffusivites 1.25975 Setup specifications Solidi
fication in a rectangular cavity Dimensions 2cm x
2cm Fluid initially at 1 C Left wall kept at -10
C
Driven by thermal and solutal buoyancy
Minimize the cost functional
Materials Process Design and Control Laboratory
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INVERSE DESIGN PROBLEM 2 - NON CONDUCTING
MATERIAL - RESULTS
Stopping tolerance 5e-4. Initially quadratic
convergence, superlinear later.
Gradient of the cost functional
Cost functional
Optimal field
Optimal field Reduces initially because of the
increased solutal buoyancy due to the solute
rejection into the melt at the interface. At
later times, the concentration of the solute
along the interface becomes uniform and hence
solutal buoyancy decreases
Materials Process Design and Control Laboratory
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INVERSE DESIGN PROBLEM 2 - NON CONDUCTING
MATERIAL - RESULTS

Comparison of interface positions at different
times for the two growth cases. With the optimal
magnetic field the interface is perfectly planar
at all time instances
Interface positions at different times
Streamline contours
Huge reduction in the velocity. Comparison of the
stream function contours for the two cases shows
a reduction in vorticity of more than a factor of
200
Materials Process Design and Control Laboratory
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INVERSE DESIGN PROBLEM 2 - NON CONDUCTING
MATERIAL - RESULTS

Comparison of the evolution of the velocity and
temperature fields for the reference case (Left)
and the optimal case (Right). Velocity is damped
out to a large extent. The maximum velocity for
the optimal case is 0.76 compared to 36.0 for the
reference case. There is some amount of vorticity
near the interface due to the local gradients in
temperature and concentration Temperature
evolution is primarily conduction based as can be
seen by the motion of the isotherms.
Materials Process Design and Control Laboratory
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INVERSE DESIGN PROBLEM 3- NON CONDUCTING MATERIAL
Design definition Find the time history of the
imposed magnetic field/gradient, such that
diffusion- based growth is achieved in the
presence of thermocapillary and buoyant
forces Material characteristics Binary
alloy/pure material, Non-conducting

Material specification 27 NaCl aqueous
solution Prandtl number 0.007 Thermal Rayleigh
number 200000 Solutal Rayleigh number
500000 Lewis number 3000 Marangoni number
0 Stefan number 0.12778 Ratio of thermal
diffusivites 1.25975 Setup specifications Solidi
fication in a rectangular cavity Dimensions 2cm x
2cm Fluid initially at 1 C Left wall kept at -10
C
Driven by thermal and solutal buoyancy
Minimize the cost functional
Materials Process Design and Control Laboratory
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INVERSE DESIGN PROBLEM 3- NON CONDUCTING
MATERIAL- RESULTS
Optimal field

Comparison of the evolution of the velocity and
temperature fields for the reference case (Left)
and the optimal case (Right). Notice that the
maximum velocity reduces from 60.0 to 0.01 by the
application of the optimal magnetic field. The
temperature evolution is purely diffusion based.
Materials Process Design and Control Laboratory
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VARIATION IN SOLUTAL RAYLEIGH NUMBER

• Optimization problem for thermal Raleigh number
  of 200000 and solutal Raleigh numbers of 10000,
  20000, 30000
• Simulation run for 0.1 dimensionless time units

Comparison of optimal magnetic field required for
various solutal Rayleigh numbers. Solutal and
thermal buoyant forces act in opposing
directions. At early times the solute is rejected
into the melt at the interface. This leads to a
nonuniform distribution of solute at the
interface. The increased solutal buoyancy acts
against the thermal gradient driven flow. But due
to the application of the magnetic gradient, the
concentration profile at the interface quickly
becomes uniform and the driving mechanism becomes
purely thermal gradient driven. Interestingly,
the magnitude of the lowest requires field as
well as the time of its occurrence follow a
linear relationship with the solutal Rayleigh
number.
Materials Process Design and Control Laboratory
41
IS A LARGE MAGNETIC FIELD ALONE ENOUGH?
Is complete damping of flow through the
application of very large magnetic fields
possible?

• Compare growth due to a magnetic gradient vs.
  large magnetic field
• Sampath Zabaras Large magnetic fields damp
  convection to some extent. Formation of cell
  structured flows. Reduced gravity plus large
  magnetic fields cause significant damping.
• Incropera et.al Extremely large fields needed
  to prevent segregation
• W.J.Evans et.al Magnetic gradients can produce
  reduced-gravity effects.

Antimony-doped Germanium growth Prandtl number
0.007 Thermal Rayleigh number 82931 Solutal
Rayleigh number 0.0 Lewis number 330 Marangoni
number -8000 Stefan number 0.034
Setup specifications Solidification in a
rectangular cavity Dimensions 4cm x 2cm Fluid
initially at 40 C superheat Left wall kept at 40
C below melting Case 1 uniform magnetic field
100 Ha Case 2 optimal magnetic field
superimposed gradient
Materials Process Design and Control Laboratory
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IS A LARGE MAGNETIC FIELD ALONE ENOUGH?
magnetic field
magnetic field gradient
Formation of flow near the walls and the
interface. Near quiescent flow in the rest of the
melt. This leads to flow circulation along the
walls back to the interface. Vorticity reduction
better by a factor of 4
Comparison of the solute distribution at the
interface for the two cases show that the
application of a uniform magnetic field still
results in fluctuations of the solute
concentration
Comparison of the standard deviation of the
interface from a planar front. The effect of a
magnetic gradient is far more prominent than the
effect of a large magnetic field alone.
Materials Process Design and Control Laboratory
43
INVERSE DESIGN PROBLEM 4 - CONDUCTING MATERIAL
Design definition Find the time history of the
imposed magnetic field/gradient, such that
diffusion based growth is achieved in the
presence of thermocapillary, buoyancy and
electromagnetic forces Material
characteristics Binary alloy/pure
material, Conducting

Antimony-doped Germanium growth Prandtl number
0.007 Thermal Rayleigh number 200000 Solutal
Rayleigh number 10000 Lewis number
1000 Marangoni number 0 Stefan number
0.034 Setup Specifications Solidification in a
rectangular cavity Dimensions 2cm x 2cm Fluid
initially at 40 C superheat Left wall kept at 40
C below melting
Driven by thermal and solutal buoyancy along with
electromagnetic effects
Minimize the cost functional
Materials Process Design and Control Laboratory
44
INVERSE DESIGN PROBLEM 4 - CONDUCTING MATERIAL-
RESULTS

Optimal magnetic field (x60 Ha) to be applied
along with a magnetic gradient of 2 mT/m
Stopping tolerance 5e-4. Super linear
convergence using the Fletcher Reeves CG method
Materials Process Design and Control Laboratory
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INVERSE DESIGN PROBLEM 4 - CONDUCTING MATERIAL-
RESULTS
Comparison of isotherms and total velocity
contours
The uniform magnetic field damps out convection
to some extent. But there is still some amount of
convection resulting in skewed isotherms. Compare
with the optimal magnetic field along with the
superimposed magnetic gradient
In the case of a uniform magnetic field, the
Lorentz force inhibits flow only in the
horizontal direction, so due to the buoyant
forces and continuity, there is some motion in
the horizontal direction on the top.
Materials Process Design and Control Laboratory
46
INVERSE DESIGN PROBLEM 4 - CONDUCTING MATERIAL-
RESULTS

Comparison of the evolution of the velocity and
temperature fields for a large constant magnetic
field (Left) and the optimal magnetic field
(Right). Velocity is damped out to a large
extent. The maximum velocity for the optimal case
is 0.80 compared to 16.0 for the other case.
Notice that the large magnetic field leads to a
correspondingly large Lorentz force which
restricts motion in one direction. Temperature
evolution is primarily conduction based as can be
seen by the motion of the isotherms.
Materials Process Design and Control Laboratory
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CRYSTAL GROWTH HORIZONTAL BRIDGEMAN GROWTH

• Design definition
• Find the time history of the imposed magnetic
  field/gradient, such that
• diffusion-based growth is achieved in the
  presence of thermocapillary, electromagnetic and
  buoyant forces
• the growth velocity is maximized
• Material characteristics
• - Doped Germanium
• - Conducting

• Subject to
• Surface tension effects on the free surface
• Radiative heating
• Given rate of crystal pulling

Materials Process Design and Control Laboratory
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CRYSTAL GROWTH HORIZONTAL BRIDGEMAN GROWTH

Antimony-doped Germanium growth Prandtl number
0.007 Thermal Rayleigh number 82931 Solutal
Rayleigh number 0.0 Lewis number 330 Marangoni
number -8000 Stefan number 0.034 Heat
conductivity ratio 0.4358 Heat diffusivity
ratio 0.4358
Furnace specification Temperature gradient in the
melt 10 K\cm Temperature gradient in the solid
30 K\cm Pulling rate 10 cm/hr Ampoule
dimensions 10cm x 5 cm Size of the seed
1mm Initial temperature of the melt 40 C
superheat Cooled until temperature gradient
reaches 120 C below melting
Materials Process Design and Control Laboratory
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CRYSTAL GROWTH HORIZONTAL BRIDGEMAN GROWTH
RESULTS
Reference case
Optimal magnetic field

The large pulling rate coupled with the
thermo-capillary flow leads to a badly skewed
solidifying front. The rate of solidification is
different at different heights. This has an
adverse impact on the quality of the crystal. On
the other hand, the magnetic gradient stabilized
flow is quite planar even at these high pulling
rates and marginal furnace temperature gradients
Comparison of the standard deviation of the
interface from a planar shape. Under the
influence of the magnetic gradient, the growth is
able to keep up with the higher pulling rate.
This is due to the damping out of the bulk
convection and reduction in the thermocapillary
forces
Materials Process Design and Control Laboratory
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CRYSTAL GROWTH HORIZONTAL BRIDGEMAN GROWTH
RESULTS
Reference case
Optimal magnetic field
Comparison of streamline contours, isotherms and
isochors during the growth


Streamline contours
There is significant reduction in vorticity
(reduction by a factor of 200). Notice that in
the reference growth, the larger flow near the
walls causes a stratification of temperature as
seen in the isotherms. This is avoided in the
optimal growth. The temperature contours are more
evenly distributed. The melt is almost quiescent.
The concentration of the solute is more evenly
distributed with the application of the magnetic
gradient.
Isotherms
Isochors
Materials Process Design and Control Laboratory
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CRYSTAL GROWTH HORIZONTAL BRIDGEMAN GROWTH
RESULTS

Under normal growth conditions, are fluctuations
in the temperature and concentration fields in
the melt. This leads to striations or formation
of banded solute layers in the solid. This has an
implicit relation with the dislocation density,
stress and defects in the crystal. The two
figures show the concentration profiles at the
interface during the time of growth.

A frequency domain representation of the
concentration at the interface. (log(power) vs.
frequency). The application of the magnetic
gradient damps out fluctuations to a great
extent. This has a direct effect on the quality
of the crystal.
Materials Process Design and Control Laboratory
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APPLICATIONS AND FURTHER RESEARCH EFFORTS

• Computational design of crystal growth processes
• Optimize crystal growth with improved growing
  speeds
• Coupling of models to predict stresses in the
  cooling crystal with growth simulator
• Design for improved quality and defect control
• Computational design of binary alloy
  solidification processes
• Melt flow control
• Control of thermal, flow and segregation
  conditions within the mushy zone
• Control of segregation patterns and defects in
  the product
• A data-driven Bayesian approach to robust
  solidification process design
• Multi-length scale design
• Coupling of macro design algorithms with
  microstructure-sensitive design algorithms
• Use level set methods to model and design
  solidification microstructures

Casting of metals
Controlled solidification of metals
Transport and manufacturing Industries
Materials Process Design and Control Laboratory
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A FEW RELATED REFERENCES

• Suppression/reversal of natural convection by
  exploiting the temperature/composition dependence
  
• of magnetic susceptibility J. W. Evans, C. D.
  Seybert, F. Leslie, W. K. Jones, J. App. Phys.
  88(7)
• (2000) 4347.
• Behavior of gas flow under gradient magnetic
  fields N. I. Wakayama, J. App. Physics 69 (4)
• (1991) 4337.
• Material processing in high static magnetic
  fields A review of an experimental study of
  levitation,
• phase separation, convection and texturation
  D. Braithwaith et al., J. de Physique 1 (1993).
• Magnetizing force modeled and numerically solved
  for natural convection of air in a cubic
  enclosure
• effect of the direction of the magnetic field
  T. Tagawa, R. Shigemitsu, H. Ozoer, Int J Heat
  Mass
• Transfer 45 (2002) 267.
• Magnetically damped convection during
  solidification of a binary metal alloy, P. J.
  Prescott, F. P.
• Incropera, Transactions of the ASME 115 (1993)
  202.
• Numerical study of convection in the directional
  solidification of a binary alloy driven by the
  combined

Materials Process Design and Control Laboratory