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momentum in crystal growth processes: Fundamentals and computational modeling
Jeffrey J. DerbyDepartment of Chemical Engineering &
15th International Summer School on Crystal Growth (ISSCG- 15)
Gdansk University of Technology
• Why do we grow crystals?• Why do we model crystal growth?• Thermodynamics and kinetics• Modeling of crystal growth• Analysis at the continuum scale
— Field equations— Modeling phase change
• Mathematical solution— Scaling analysis— Numerical modeling
• Classical models for crystal growth— Axial segregation
exhibit very uniform properties that are essential for many applications
Optical: KTP, Crystal Technology, Inc.
Mechanical: Rolls-Royce, single-crystal, nickel -
Electronic: Processed silicon wafer, ICWave, LLC
purification. Crystals will tend to incorporate li ke species; unlike species are segregated
X
since the dawn of civilization and is used today in many chemical separations
Productionof crystallinealum, Agricola (1556)
Process scaleO(m)
Crystal scaleO(µm-cm)
Surface scaleO(nm-µm)
Modern industrial crystallizer
• Why do we grow crystals?• Why do we model crystal growth?• Thermodynamics and kinetics• Modeling of crystal growth• Analysis at the continuum scale
— Field equations— Modeling phase change
• Mathematical solution— Scaling analysis— Numerical modeling
• Classical models for crystal growth— Axial segregation
phenomenon?
“ ...a little theory and calculation would have saved him ninety per cent of his labor.” --- Nikola Tesla
Genius is one percent inspiration and 99 percent perspiration. ”--- Thomas Edison
... a model can serve the purpose of defining, clar ifying, and enriching a concept.” --- Rutherford Aris
“The purpose of models is not to fit the data but to sharpen the questions.” --- Samuel Karlin
We model to quantify, to design, to optimize,
proceed? “ Art is a lie that makes us realize truth. ” --- Pablo Picasso
“In the final print of the series, Picasso reduces the bull to a simple outline that is so carefully considered through the progressive development of each image, that it captures the absolute essence of the creature in as concise an image as possible.”---www.artyfactory.com
process, modeling provides a framework to understan d and improve it
• Why do we grow crystals?• Why do we model crystal growth?• Thermodynamics and kinetics• Modeling of crystal growth• Analysis at the continuum scale
— Field equations— Modeling phase change
• Mathematical solution— Scaling analysis— Numerical modeling
• Classical models for crystal growth— Axial segregation
thermodynamic driving force, ∆∝, and controlled by non -equilibrium transport kinetics
∝ εθυιλ
∆∝
∝ σολιδ
∝ φλυιδ
∆∝∆ξ
is a gradient which drives the rate of phase change
∆ξ
Thermodynamics
KineticsGrowth is a non- equilibrium process
At equilibrium:∝ φλυιδ = ∝ σολιδ
For growth to occur, the fluid phase must be made to be unstable, so that:
∝ φλυιδ > ∝ σολιδ
or species (chemical potential) must be moved by gradients through the system
HeatFlux
Crystal
SpeciesFlux
Controlling gradients (i.e., variations of some quantity over space (and time) is a key element of a crystal growth process!
Common configurations for melt crystal growth are designed to maintain thermal gradients
keeping the solution phase supersaturated, typically by changing temperature or concentration
1
2
Over time, temperature is steadily decreased...
KDP crystals grown from aqueous solution (LLNL rapid growth system)
liquid -solid interfaces. There are two limiting behaviors of a growing solid -liquid interface
Typical for melt growth (but not always)
Typical for solution growth (but not always)
Atomically rough... Atomically smooth...
If is large...
thermodynamic basis to favor either a smooth or rough interface
Consider
where is the change in Gibbs free energy after the addition of an atom from the fluid phase to the solid phase
Favors addition of atom
If is small...
new surface is nearly complete and “smooth. ”
The outcome of growth...
In solution growth, roughening transition can be realized by a change in temperature
P. Bennema (1994)
Paraffin crystals grown from hexane solution
What are the kinetics associated with a liquid -to-crystal phase change?
If the interface is rough...
typically, the kinetics of the phase change are extremely fast compared to mass or heat transfer time scales.
If the interface is smooth...
growth proceeds layer by layer. How?
determined by surface structure, notably the existence of “steps ”
Steps are formed by...
2D nucleation of an “island” onan existing layer
Adapted from Peter Rudolph, Lecture on “Thermodynamics and
Slow! Fast!
A step “source,” such as the intersection ofa line dislocation with the crystal surface
Burton, Cabrera, and Frank (1951)
growth system move as a function of bulk and surface phenomena
• Why do we grow crystals?• Why do we model crystal growth?• Thermodynamics and kinetics• Modeling of crystal growth• Analysis at the continuum scale
— Field equations— Modeling phase change
• Mathematical solution— Scaling analysis— Numerical modeling
• Classical models for crystal growth— Axial segregation
The complete modeling of crystal growth from first principles is impossible!
Si Melt
Si Crystal
300-mm diameterCZ silicon
Characteristic:
Unit cell, 5 Å Ingot length, 2 m
Range of scales:
Thermal fluctuations, 10 fs Growth time, 200 hr
Furnace -scaleComplicated geometry
Ampoule-scale Fluid mechanics
Atomic-scale Liquid configuration
analysis on many scales
EDG furnace, (m)
Melt
Crystal
CZT melt and crystal (cm)
Structure of CdTe melt (nm)
Heaters
Ampoule
Melt
Crystal
CrysVUN/Cats2D
PARSEC
1. Construct a mathematical model containing gover ning equations for the “essential physics.”- Continuum phenomena- Atomistic phenomena- Geometrical dimensionality- Temporal behavior- Boundary conditions
What are the objectives? What is to be understood? How accurate must the model be?
growth processes follows these steps
2. Solve the model equations.- Analytical approximations- Computational approaches- Convergence and accuracy- Flexibility and robustness- Cost (computational and effort)
What are available resources?
3. Interpret and apply modeling results.- Post-processing and visualization- Parametric sensitivity- Gedanken experiments- Optimization and design
• Why do we grow crystals?• Why do we model crystal growth?• Thermodynamics and kinetics• Modeling of crystal growth• Analysis at the continuum scale
— Field equations— Modeling phase change
• Mathematical solution— Scaling analysis— Numerical modeling
• Classical models for crystal growth— Axial segregation
Continuum transport equations are derived from balances over a control volume
Control volume
Diffusion
Conduction
Advection/Convection
Advection/Convection
Net generationvia reactionAccumulation
Species balance
Internal radiationAccumulation
Energy balance
Velocity,
Velocity
Boundaries can be a source of significant nonlinearities
DiffusionAdvection/Convection
Net generationvia reactionAccumulation
Species balance
ConductionAdvection/Convection
Internal radiationAccumulation
Energy balance
Solid
Chemical reactionat surface
Solid
Radiationat surface
The Navier -Stokes equations are just a manifestation of Newton ’s second law
Pressure forces Viscous forces
Buoyant forces (thermal and solutal)
Other body forces (e.g., Lorenz forces)
Fluid continuity (mass balance for incompressible fluid)
Newton ’s second law:
Mass per unit volumex acceleration
Navier -Stokes:
Typically, the interface is rough in melt growth, and...
the kinetics of the phase change are extremely fast compared to mass or heat transfer time scales.
Melting- pointisotherm
Isotherm Growth rate:
How do we model the kinetics associated with a liquid -to-crystal phase change?
If the interface is smooth...
growth proceeds layer by layer.
Mathematical representations are more difficult and depend on details...
How do we model the kinetics associated with a liquid -to-crystal phase change?
liquid interface evolution is less advanced than for rough interfaces
The challenge is to faithfully represent the surface morphology (steps and terraces) and
• Why do we grow crystals?• Why do we model crystal growth?• Thermodynamics and kinetics• Modeling of crystal growth• Analysis at the continuum scale
— Field equations— Modeling phase change
• Mathematical solution— Scaling analysis— Numerical modeling
• Classical models for crystal growth— Axial segregation
Example of scaling a governing equation and the origin of a dimensionless group
Original balance equation:
Scale variables to make O(1):
Resulting dimensionless Peclet number indicates magnitude of
estimate the importance of physical effects ---without the solution of any equations!
Grashof number = Buoyancy forces/Viscous forces
Marangoni number = Surface forces/Viscous forces
Reynolds number = Inertial forces/Viscous forces
Prandtl number = Diffusion of Momentum/Thermal
Schmidt number = Diffusion of Momentum/Solute
Peclet number = Transport by Convection/ Diffusion
dominant balances ” --- Navier -Stokes equations for buoyant flow
Pressure forces Viscous forces
Buoyant forces (thermal and solutal)
Other body forces (e.g., Lorenz forces)
Mass per unit volumex acceleration
Navier -Stokes: X X
When
Numerical solutions to differential equations are built upon principles of discretization
Dimensionless partialdifferential equation:
Discretize spatial derivatives, e.g., finite difference:
or finite element:
Coupled set of (nonlinear)differential equations:
All numerical methods are built upon principles of discretization (continued)
Coupled set of (nonlinear)differential equations:
Discretize temporalderivatives, e.g., by explicitmethod,
or implicit method,
Solve directly forsteady states,
All numerical methods which use discretization ultimately lead to linear algebra!
Steady-state analysis Transient analysis
Newton -Raphsonmethod
Linear Algebraic Equation,to be solved over and over again!
UNIVERSITY OF MINNESOTA
element methods to solve melt crystal growth problems (Cats2D)
Mathematical model 2D, Axisymmetric Quasi-steady-state (QSS) or time-
dependent Heat conduction in all materials Incompressible, Boussinesq fluid in
melt Moving boundary at melting point
for crystal-melt interface Latent heat release at interface Simple radiation/convection furnace
boundary conditions
Numerical method Galerkin FEM with isoparametric
elements Adaptive mesh (front-tracking) Implicit (trapezoid rule) time
integration Full Newton iteration method with
direct matrix solver
FEM Mesh Must do better to
Our recent efforts aim to effectively couple the strengths of different models (multi -scale)
CrysMAS
Cats2D
We think we ’ve finally figured this out! (but are still in process of implementation...)
• Why do we grow crystals?• Why do we model crystal growth?• Thermodynamics and kinetics• Modeling of crystal growth• Analysis at the continuum scale
— Field equations— Modeling phase change
• Mathematical solution— Scaling analysis— Numerical modeling
• Classical models for crystal growth— Axial segregation
Segregation of a dilute species occurs at a solid -liquid interface during directional solidification
AssumptionsConstant interface velocityConstant segregation and diffusion coefficientsNo solid diffusion
Analytical solution diffusion only (Smith et al., 1955)
Governing equations
Rc(x)
x
Analytical solution complete melt mixing (Scheil, 1 942)
by diffusion -limited and complete mixing
Under realistic conditionsof melt flow and heat transfer, equations must be solvednumerically
An approximation to describe mixing can be applied, the BPS mode
Rc(x)
x
accounts for melt mixing with a simple (unrealistic) parameter,
Diffusion- controlled growth, no axial segregation
Complete melt mixing, Axial segregation governed by Scheil equation
diffusion layer and results in transverse (or radia l) segregation
Growth
Solid Melt
Solute diffusion layerSolute distributionin solid
Melt flow drivesconvective transport
Solute distribution is nonuniform
unstable: Constitutional supercooling in melt growth
Unstable growth resulting in cellular interface, computed by phase- field method, Bi and Sekerka (2002)
Criterion for instability:
Tem
pera
ture Melt temperature
Liquidus temperature
Undercooled melt
supercooling
Melt flows will act tothin the concentrationboundary layer...
and eliminate undercooled region
Flow
Tem
pera
ture
Melt temperature
Liquidus temperature
Final topics
• Comments on turbulence• ACRT Bridgman growth
“Turbulence is composed of eddies: patches of zigza gging, often s wirling fluid, moving randomly around and about the overall direction of motion. T echnically, the chaotic state of fluid motion arises when the speed of the fluid exceeds a specif ic threshold, below which viscous forces damp out the chaotic behavior.”
What is turbulence?
Big whorls have little whorls, Which feed on their velocity, And little whorls have lesser whorls, And so on to viscosity.
Lewis F. Richardson
wikimedia.org
Our understanding of turbulence is “ ugly ”
I am an old man now, and when I die and go to heave n there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent mot ion of fluids . And about the former I am rather optimistic. ”
Horace Lamb, in an address to the British Associati on for the Advancement of Science, 1932
For a phenomenon that is literally ubiquitous, remarkably little of a quantitative nature is known about it. ”
P. Moin and J. Kim, Scientific American , 276, No. 1, 1997.
Turbulence is the most important unsolved problem o f classical physics. ”
fluid dynamical instabilities at discrete values of the Reynolds number, Re
Laminar, low Re
Transitional, moderate Re
Weak or soft turbulence, large Re
Strong or hard turbulence, larger Re
Kim, Witt, and Gatos (1972)
crystal growth system...
Melt
GasHotter
CoolerCrystal
Vertical Bridgman furnace
Time-dependent moving boundary problem Axisymmetric
Galerkin-finite element method
Mixed Lagrangian-Eulerian formulation
Sharp interface front-tracking
Rigorous conservation of mass and energy
Incompressible flow in melt Laminar
Three velocity components (2.5 D)
Buoyancy – Boussinesq approximationHeat transfer Heat transfer modeled throughout crucible
Simplified model of furnaceMass transfer Partitioning of chemical species at interface
crystal
UNIVERSITY OF MINNESOTA
ACRT high -pressure vertical Bridgman growth of CdZnTe (4-inch diameter)
• Acceleration generates secondary flow dominated by Ekman pumping through boundary layer at bottom and top of melt
• Cycle time based on Ekman time scale = 1.2 minutes
τ = −1/ 2 Ω−1 = 2 Ων( )1/ 2
A
B
CD
Rotation cycle
Spin -down
Spin -up
0 100 200 300
UNIVERSITY OF MINNESOTA
Görtler vortices during spin down (B) and Ekman flow reversal at the interface during spin up (C)
Crystal
Ampoule
Melt
Ekman flow
Taylor-Görtlerinstability
Ekman flow
Final comments
Theoretical models can be gainfully applied to obta in understand ing of crystal growth systems
The successful practitioner must:Be aware of model limitationsBe aware of computational issuesVerify and validate modelsCreatively apply models toward objectives
Future challenges: physicsTurbulenceExternal fieldsKinetics of crystallization (surface physics and ch emistry)Multicomponent effects (Non-Kossel crystals, liquid -phase chemistry)Features of the solid state (point defects, extende d defects, di slocations, grain boundaries, precipitates, inclusions)
Future challenges: computationBetter efficiency (parallelism, accuracy)Better reliability (robustness, convergence)Ease of use (mesh generation, visualization)Optimization, design, and control