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