Westfälische Wilhelms Universität Münster
Strongly Anisotropic
Motion Laws, Curvature
Regularization, and
Time Discretization
Martin Burger
Johannes Kepler University Linz
SFB Numerical-Symbolic-Geometric Scientific Computing
Radon Institute for Computational & Applied Mathematics 1
Collaborations

Frank Hausser, Christina Stöcker, Axel Voigt
(CAESAR Bonn)
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Introduction
Surface diffusion processes appear in various
materials science applications, in particular in the
(self-assembled) growth of nanostructures

Schematic description: particles are deposited
on a surface and become adsorbed (adatoms).
They diffuse around the surface and can be
bound to the surface. Vice versa, unbinding and
desorption happens.

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Growth Mechanisms
Various fundamental surface growth
mechanisms can determine the dynamics, most
important:

Attachment / Detachment of atoms to / from
surfaces
-
-
Diffusion of adatoms on surfaces
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Growth Mechanisms
Other effects influencing dynamics:
- Anisotropy

-
Bulk diffusion of atoms (phase separation)
-
Exchange of atoms between surface and bulk
-
Elastic Relaxation in the bulk
-
Surface Stresses
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Growth Mechanisms

-
Other effects influencing dynamics:
Deposition of atoms on surfaces
Effects induced by electromagnetic forces
(Electromigration)
-
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Isotropic Surface Diffusion
Simple model for surface diffusion in the isotropic
case:
Normal motion of the surface by minus surface
Laplacian of mean curvature

Can be derived as limit of Cahn-Hilliard model
with degenerate diffusivity (ask Harald Garcke)

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Applications: Nanostructures

SiGe/Si Quantum Dots

Bauer

et. al. 99
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Applications: Nanostructures

SiGe/Si Quantum Dots
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Applications: Nanostructures
InAs/GaAs
Quantum Dots



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Applications: Nano / Micro

Electromigration of voids in electrical circuits
Nix et. Al. 92
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Applications: Nano / Micro
Butterfly shape transition in Ni-based
superalloys

Colin et. Al. 98
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Applications: Macro

Formation of Basalt Columns:
Panska Skala
(Czech Republic)
Giant‘s Causeway
(Northern Ireland)
See: http://physics.peter-kohlert.de/grinfeld.htmld
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Energy
The energy of the system is composed of
various terms:

Total Energy =
(Anisotropic) Surface Energy +
(Anisotropic) Elastic Energy +
Compositional Energy +
.....

We start with first term only
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Surface Energy

Surface energy is given by

Standard model for surface free energy
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Chemical Potential
Chemical potential m is the change of energy
when adding / removing single atoms

In a continuum model, the chemical potential
can be represented as a surface gradient of the
energy (obtained as the variation of total energy
with respect to the surface)

For surfaces represented by a graph, the
chemical potential is the functional derivative of
the energy

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Surface Attachment Limited Kinetics
SALK
is a motion along the negative gradient
direction, velocity
For
graphs / level sets
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Surface Attachment Limited Kinetics
Surface attachment limited kinetics appears in
phase transition, grain boundary motion, …


Isotropic case: motion by mean curvature
 Additional
curvature term like Willmore flow
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Analysis and Numerics
Existing results:
- Numerical simulation without curvature
regularization, Fierro-Goglione-Paolini 1998
- Numerical simulation of Willmore flow, Dziuk
Kuwert-Schätzle 2002, Droske-Rumpf 2004
-
Numerical simulation of regularized model
-Hausser-Voigt
2004 (parametric)
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Surface Diffusion
Surface diffusion appears in many important
applications - in particular in material and nano
science


Growth of a surface G with velocity
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Surface Diffusion
F ... Deposition flux
 Ds .. Diffusion coefficient
 W ... Atomic volume
 s ... Surface density
 k ... Boltzmann constant
 T ... Temperature
 n ... Unit outer normal
 m ... Chemical potential =
energy variation

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Surface Energy
In several situations, the surface free energy
(respectively its one-homogeneous extension)
is not convex. Nonconvex energies can result
from different reasons:

-
Special materials with strong anisotropy:
Gjostein 1963, Cahn-Hoffmann1974
-
Strained Vicinal Surfaces: Shenoy-Freund 2003
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Surface Energy
Effective surface free energy of a compressively
strained vicinal surface (Shenoy 2004)

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Curvature Regularization
In order to regularize problem (and possibly
since higher order terms become important in
atomistic homogenization), curvature
regularization has beeen proposed by several
authors (DiCarlo-Gurtin-Podio-Guidugli 1993, Gurtin
Jabbour 2002, Tersoff, Spencer, Rastelli, Von Kähnel
2003)
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Anisotropic Surface energy
Cubic anisotropy, surface energy becomes nonconvex for

e > 1/3
-
Faceting of the surface
-
Microstructure possible without curvature term
-
Equilibria are local energy minimizers only
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Chemical Potential

We obtain
Energy variation corresponds to fourth-order
term (due to curvature variation)

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Curvature Term

Derivative

with matrix
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Analysis and Numerics
Existing results:
- Studies of equilibrium structures, Gurtin 1993,

Spencer 2003, Cecil-Osher 2004
Numerical simulation of asymptotic model
(obtained from long-wave expansion), Golovin-
Davies-Nepomnyaschy 2002 / 2003
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Discretization: Gradient Flows
SD and SALK can be obtained as the limit of
minimizing movement formulation (De Giorgi)

with different metrics d between surfaces, but
same surface energies
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Discretization: Gradient Flows
Natural first order time discretization.
Additional spatial discretization by constraining
manifold and possibly approximating metric and
energy

Discrete manifold determined by
representation (parametric, graph, level set, ..)
+ discretization (FEM, DG, FV, ..)

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Gradient Flow Structure

Expansion of the shape metric (SALK / SD)
where
denotes the surface
obtained from a motion of all points in normal
direction with (given) normal velocity Vn

Shape metric translates to norm (scalar
product) for normal velocities !
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Gradient Flow Structure

Expansion of the energy (Hadamard-Zolesio
structure theorem)
where
denotes the surface
obtained from a motion of all points in normal
direction with (given) normal velocity Vn
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MCF – Graph Form

Rewrite energy functional in terms of u

Local expansion of metric

Spatial discretization: finite elements for u
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MCF – Graph Form

Time discretization in terms of u

Implicit Euler: minimize
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MCF – Graph Form
Time discretization yields same order in time
if we approximate
to first order in t
 Variety of schemes by different
approximations of shape and metric


Implicit Euler 2: minimize
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MCF – Graph Form

Explicit Euler: minimize

Time step restriction: minimizer exists only if
quadratic term (metric) dominates linear term
This yields standard parabolic condition
by interpolation inequalities
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MCF – Graph Form

Semi-implicit scheme: minimize
with quadratic functional B
 Consistency and correct energy dissipation
if B is chosen such that B(0)=0 and
quadratic expansion lies above E
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MCF – Graph Form

Semi-implicit scheme: with appropriate choice
of B we obtain minimization of

Equivalent to linear equation
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MCF – Graph Form
Semi-implicit scheme is unconditionally
stable, only requires solution of linear system
in each time step
 Well-known scheme (different derivation)

Deckelnick-Dziuk 01, 02
 Analogous
for level set representation
 Approach
can be extended automatically to
more complicated energies and metrics !
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Minimizing Movement: SD
SD can be obtained as the limit (t →0) of
minimization


subject to
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Minimizing Movement: SD

Level set / graph version:
subject to
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Numerical Solution
Basic idea: Semi-implicit time discretization +
Splitting into two / three second-order equations +
Finite element discretization in space

Natural variables for splitting:
Height u, Mean Curvature k, Chemical potential m

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Spatial Discretization
Discretization of the variational problem in
space by piecewise linear finite elements

and P(u) are piecewise constant on the
triangularization, all integrals needed for
stiffness matrix and right-hand side can be
computed exactly

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SALK e = 3.5, a = 0.02, 10t = 5 10-4
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SD e = 3.5, a = 0.02, 10t = 5 10-5
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SALK e = 3.5, a = 0.02, 10t = 2.8 10-3
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SD e = 3.5, a = 0.02, 10t = 2.8 10-5
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SALK e = 1.5, a = 0.02, 10t = 6.66 10-3
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SALK e = 1.5, a = 0.02, 10t = 6.66 10-3
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SALK e = 1.5, a = 0.02, 10t = 6.66 10-3
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SD e = 1.5, a = 0.02, 10t = 3.33 10-3
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SD e = 1.5, a = 0.02, 10t = 1.66 10-3
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Faceting
Graph Simulation: mb JCP 04,
 Level Set Simulation: mb-Hausser-Stöcker-Voigt 06


Adaptive FE grid around zero level set
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Faceting
 Anisotropic
mean curvature flow
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Faceting of Thin Films
Anisotropic
Mean Curvature
Anisotropic
Surface Diffusion
mb 04,
mb-HausserStöcker-Voigt-05
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Faceting of Crystals
 Anisotropic
surface diffusion
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Obstacle Problems
Numerical schemes obtained again by
approximation of the energy and metric for
time discretization, finite element spatial
discretization
 Local optimization problem with bound
constraint (general inequality constraints for
other obstacles)
 Explicit scheme: additional projection step
 Semi-implicit scheme: quadratic problem with
bound constraint, solved with modified CG

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MCM with Obstacles

Obstacle
Evolution
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MCM with Obstacles

Obstacle
Evolution
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MCM with Obstacles

Obstacle
Evolution
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Download and Contact

Papers and Talks:
www.indmath.uni-linz.ac.at/people/burger
from October:
wwwmath1.uni-muenster.de/num

e-mail: [email protected]
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