Nanoscale Material Interfaces:
Experiment, Theory and Simulation
(24 Nov 2004  23 Jan 2005)
~ Abstracts ~
Constitutive modeling of
microstructural fluids
Qi Wang, Florida State University
In this presentation, I will give an overview on the state
of the arts in the constitutive modeling of microstructral
fluids such as polymeric liquids, MR (magnetorheological
fluids), ER (electrorheological) fluids, liquid crystalline
polymers and polymerparticle nanocomposites. I will emphasize
on the methodology in establishing the models. I will detail
on the continuum, kinetic, GENERIC and Poisson Bracket
formulation for specific fluids. Some model predictions will
be discussed as well.
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Driven cavity flow: the slip
boundary condition
Tiezheng Qian, Hong Kong University of Science and
Technology
Molecular dynamics (MD) simulations have been carried out
to investigate the slip of fluid in the lid driven cavity flow
where the noslip boundary condition causes unphysical stress
divergence. The MD results not only show the existence of
fluid slip but also verify the validity of the Navier slip
boundary condition. To better understand the fluid slip in
this problem, a continuum hydrodynamic model has been
formulated based upon the MD verification of the Navier
boundary condition and the Newtonian stress. Our model has no
adjustable parameter because all the material parameters
(density, viscosity, and slip length) are directly determined
from MD simulations. Steadystate velocity fields from
continuum calculations are in quantitative agreement with
those from MD simulations, from the molecularscale structure
to the global flow. The main discovery is as follows. In the
immediate vicinity of the corners where moving and fixed solid
surfaces intersect, there is a core partialslip region where
the slippage is large at the moving solid surface and decays
away from the intersection quickly. In particular, the
structure of this core region is nearly independent of the
system size. On the other hand, for sufficiently large system,
an additional partialslip region appears where the slippage
varies as 1/r with r denoting the distance from the corner
along the moving solid surface. The existence of this wide
powerlaw region is in accordance with the asymptotic 1/r
variation of stress and the Navier boundary condition.
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Recent developments on phase field
method for moving interface problems
Xiaobing Feng, The University of Tennessee
Moving interface problems arise from many scientific
disciplines such as materials science, biology, fluid
dynamics, image processing, and differential geometry, just
name a few. Broadly speaking, mathematical and numerical
methods for moving interface problems can be grouped into two
categories: (i) moving interfaces are determined directly, for
that certain interface (or jump) conditions must be imposed on
the interfaces; (ii) moving interfaces are determined
indirectly as level sets of one of unknowns of a system of
evolution equations, in which the solutions are smooth but
experience large gradients. The phase field method for moving
interface problems belongs to the second category, in which a
phase function is used to identify different ``phases" (a
terminology commonly used in materials science) and their
interfaces. The main advantage of the phase field approach is
that it can easily handle singularities of moving interfaces
and is much more convenient for numerical approximations
compared to the first approach.
In this talk, I shall discuss some recent developments on
the phase field method. New models for some intriguing
applications from phase transition, image processing, and
fluid mechanics will be reviewed, recent advances on their
mathematical and numerical analysis will also be reported.
This is a joint work with Andreas Prohl of ETH Zurich
(Switzerland) and Haijun Wu of Nanjing University (China).
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Atomicscale reconstructions on metal and semiconductor
surfaces
Andrew T. S. Wee, National University of Singapore
Download/View PDF
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Steady state morphology and nanopattern
on Si(001) formed during Epitaxy
XueSen Wang, National University of Singapore
Surface morphology during material growth and removal has
been an important topic in physics and material science in
recent decades. The nearequilibrium state at high temperature
and the increasingly roughening growth at low T have been well
characterized, but the intermediate state between these two
extremes is still largely unknown. Here, using experimental
results of Si on Si(001) MBE at intermediate T and flux, we
show that a steady state in which the surface roughness
reaches a saturate value after a transition period can be
achieved. Within the parameter range where the steady state
exists, the dominant surface morphology varies from
doublelayer steps to nearly rectangular nanometerscale
multilayer vacancy islands (pits) isolated by connected
smooth area. The morphological characteristics in different
growth regimes and the transition between them are discussed.
The nanopatterns obtained in steadystate epitaxy can be used
as templates for fabrication of other nanostructures.
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Models for the thermal diode and the
thermal transistor: pave the way for heat control
Baowen Li, National University of Singapore
The invention of semiconductor transistor and its relevant
devices that control the charge flow has revolutionized our
daily life in every aspect. However, over half century has
been past, similar devices for controlling heat flow are still
lacking.
In this talk, I will give a detailed discussion about our
recent invention of thermal diode and thermal transistor
models, two fundamental devices for controlling heat flow.
Emphasis will be given on the physical principle/mechanism of
these two devices.
The thermal diode is a one way road for heat flow [1]. It
allows the heat flow from one direction, while it prohibit
heat flow for the another one. Like the electronic
counterpart, the thermal transistor [2] is a threeterminal
device with the important feature that the current through the
two terminals can be controlled by small changes we make in
the current or temperature at the third terminal. This control
feature allows us to amplify the small current or to switch
the device from an “off” (insulating) state to an “on”
(conducting) state.
The work is supported by FRG of NUS and the Temasek Young
Investigator Award of DSTA Singapore and NUS.
Refs:
[1] B Li, L Wang and G Casati, “Thermal diode:
Rectification of heat flux”, Phys. Rev. Lett 93,
xxx(2004) (in press), condmat/0407093.
[2] B Li, L Wang and G Casati, “The thermal transistor: A
switch and a modulator/amplifier for heat current”.
Submitted for publication.
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Various aspect of the Hartee Fock
approximation
Claude Bardos, Université de Paris VII
In this talk I intend to describe the state of the art for
the mathematical analysis of the Hartree Fock approximation. I
start with one component time dependent method, compare with
older results concerning the approximation of the ground state
and finally describe the multicomponents time dependent
method.
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Off lattice kineticMonteCarlo
Tim P. Schulze, University of Tennessee
Simulations of nanoscale materials and interfaces are done
by a variety of techniques that make various compromises
between accuracy and computational speed. In the classical
regime, Molecular Dynamics (MD) is both the most fundamental
and most costly in terms of computational speed. Lattice based
kinetic MonteCarlo (KMC) methods can be thought of as being
derived from MD and can be generalized into offlattice
methods appropriate for addressing elastic effects. The effect
this has on computational speed depends strongly on the amount
of strain in the system. As a prototype problem, we consider
the diffusion of an impurity in a strained FCC nanowire.
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Sharpinterface theory for nematicisotropic
phase transitions
Eliot Fried, Washington University
We derive a supplemental evolution equation for an
interface between the nematic and isotropic phases of a liquid
crystal. Our approach is based on the notion of
configurational force. As an application, we study the
behavior a spherical isotropic drop surrounded by a radiallyoriented
nematic phase: our supplemental evolution equation then
reduces to a simple ordinary differential equation admitting a
closed form solution. In addition to describing many features
of isotropictonematic phase transitions, this simplified
model yields insight concerning the occurrence and stability
of isotropic cores for hedgehog defects in liquid crystals.
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A levelset method for Epitaxial growth and
selforganization of quantum dots
Christian Ratsch, University of California, Los Angeles
We have developed an island dynamics model that employs the
levelset technique to describe epitaxial growth. The motion
of the island boundaries is described by the evolution of a
continuous levelset function. Islands are nucleated on the
surface and their boundaries are moved at rates that are
determined by the adatom density, which is obtained from
solving the diffusion equation. Thus, the individual islands
on the surface are resolved, while the adatoms are treated in
a meanfield picture.
This has significant numerical advantages: The numerical
timestep can be chosen (much) larger than in an atomistic
simulation. Thus, in our method microscopic processes with
vastly different rates can be described without an increase in
computational cost. For example, frequent detachment from and
reattachment to island boundaries is accounted for by a net
velocity of the island boundary. The important aspect is that
the simulation timestep does not have to be decreased. This
allows us to study efficiently problems with high
reversibility. Results for the scaled island size distribution
during submonolayer epitaxy will be shown.
Our method is ideally suited to study the formation and
selforganization of quantum dots, which is a strain driven
phenomena. The large simulation timestep makes it feasible to
solve the elastic equations at every timestep, and couple the
solution of the elastic equations to the microscopic
parameters in our model. Diffusion and attachment and
detachment are spatially varying. Our results indicate that in
a system with spatially varying diffusion rates one obtains
regions of high and low island densities.
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Mound formation and coarsening
in surface growth
Chandan Dasgupta, Indian Institute of Science
In this talk, I will present detailed numerical and
analytic results for a class of onedimensional, nonlinear,
conserved growth equations and related atomistic growth
models. Numerical integration of spatially discretized forms
of these growth equations and stochastic simulations of the
atomistic growth models show that these systems exhibit mound
formation and powerlaw coarsening with slope selection for a
range of values of the model parameters. In contrast to
previously proposed models of mound formation, the EhrlichSchwoebel
stepedge barrier, usually modeled as a linear instability in
growth equations, is absent in our models. Mound formation in
our models occurs due to a nonlinear instability. When this
instability is controlled by the introduction of an infinite
number of higherorder gradient nonlinearities, these models
exhibit a firstorder dynamical phase transition from a
kinetically rough selfaffine phase to a mounded one as the
value of a parameter that measures the effectiveness of
control is decreased. In the mounded phase, the models exhibit
powerlaw coarsening of the mounds in which a selected slope
is retained at all times. Results obtained for a model in
which both the nonlinear instability and a linear instability
representing the effect of a stepedge barrier are present
will also be presented.
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Modelling the selforganized
growth of quantum dots
YongWei Zhang, National University of Singapore
When a thin film heteroepitaxially grows on a substrate,
driven by the mismatch strain between the film and the
substrate, the film surface is intrinsically unstable again
small roughness perturbations. As a consequence, a compact
island array is formed. This selfassembled process can be
potentially used to fabricate quantum dot (QD) arrays, which
may have many applications in microelectronic and
optoelectronic devices. Although enormous efforts have been
made to achieve uniform and regular QD arrays through
selfassembly, the current uniformity and regularity of QDs is
insufficient for majority of QD device applications. Therefore
it is of interest to know what is the distribution of QDs and
how it evolves during growth.
In this talk, we will present our results of the island
formation and coarsening kinetics in the StranskiKranstanov
growth through largescale computer simulations. Our attention
is focused on island roughening kinetics through stressdriven
surface diffusion. Our simulations reveal many interesting
features of island formation and coarsening kinetics. In
particular, we have found that a bellshaped island size
distribution at the early stage of growth can give rise to an
unusual coarsening kinetics, that is, the mean island volume
increases superlinearly with time and the areal density of
islands decreases at a fasterthanlinearrate. The standard
mean field theory is used to reproduce the observed behaviors.
We have also investigated the evolution of island
distributions during the growth of quantum dot superlattices.
Our simulations show that with a proper choice of the spacer
layer thickness, the interruption time and the growth rate, a
perfectly ordered quantum dot superlattice can be achieved.
The quantum dots adopt a perfectly ordered array with an ABAB
stacking sequence. Surprisingly, the ordered selforganized
quantum dot superlattices are not controlled by the ordering
of strain energy density minima on the spacer layer surface,
but by the ordering of the strain energy density maxima.
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Dynamic of disclinations and
microstructures in liquid crystal polymer flow
Pingwen Zhang, Peking University
We study the microstructure formation and disclination
dynamics that arise when liquid crystal polymers undergo shear
flow. We use a coupled kinetichydrodynamic approach which
keeps track of the dynamics of the orientationposition
distribution functions of the rodlike molecules. The Doi
kinetic theory for homogeneous flow of rodlike liquid
crystalline polymers (LCPs) is extended to inhomogeneous
kinetic theory of rodlike LCPs through a nonlocal
intermolecular potential. An extra elastic body force
exclusively associated with the integral form of
intermolecular potential. With a few molecular parameters, we
believe, the complete system of equations is capable of
describing the evolution of the texture, the dynamics of
disclination and polydomain in flowing nematic and smectic
polymers, the phase transition and separation. In the limit of
small Debroah number, the inhomogeneous theory properly
reduces to the EricksenLeslie theory. The Leslie viscosities
are derived in terms of molecular parameters, the Ericksen
stress are derived by the body force.
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Morphological evolution of thin
films during epitaxy; largescale numerical investigations of
coarsening phenomena and scaling laws
Alex Voigt, Caesar Research Center
We consider numerical simulations of geometric evolution
laws to describe the evolution of thin films during epitaxy.
The formation of facets and corners during the evolution is
considered by higher order terms in the evolution laws for
mean curvature flow and surface diffusion, resulting from free
energy densities depending not only on orientation but also
it's gradient. A semiimplicit finite element method is used
in the parametric setting. We computationally investigate the
coarsening in the morphology of thin films, show their
selfsimilarity and predict scaling laws.
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Islands in the stream: complex shape
evolution driven by surface electromigration
Joachim Krug, Universität zu Köln
The shape evolution of twodimensional islands through
periphery diffusion biased by an electromigration force is
studied using a continuum approach, with particular emphasis
on the role of crystal anisotropy. In the absence of
capillarity effects, stationary island shapes can be computed
analytically and criteria for the existence of smooth shapes
can be formulated. The full problem including an isotropic
step stiffness and anisotropic edge atom mobility is
investigated by timedependent numerical calculations. We find
a rich variety of migration modes, which include oscillatory
and irregular behavior. A phase diagram in the plane of
anisotropy strength and island size is constructed. The
oscillatory motion can be understood in terms of stable facets
which develop on one side of the island and which the island
then slides past. The facet orientations are determined
analytically.
The talk is based on joint work with P. Kuhn, F. Hausser
and A. Voigt.
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Mathematical models and numerical
simulations of superconductivity
Qiang Du, Pennsylvania State University
Superconductivity is one of the grand challenges identified
as being crucial to future economic prosperity and scientific
leadership. In recent years, the analysis and simulations of
various mathematical models in superconductivity have
attracted the interests of many mathematicians all over the
world. Their works have helped us to understand the intriguing
and complex phenomena in superconductivity.
With the recent award of the Nobel Prize in Physics, a
renewed attention has been focused on theoretical foundations
of superconductivity, for example, the popular GinzburgLandau
theory was proclaimed as "being of great importance in physics
...". There are new and unresolved mathematical challenges be
explored further. In this tutorial, we will briefly review the
physical background of some interesting problems related to
superconductivity, in particular, the problem of quantized
vortices. Various mathematical models ranging from microscopic
BCS theory to the macroscopic critical state models will then
be described with the mesoscale GinzburgLandau model being
our emphasis. Some recent analytical and numerical results
will be surveyed. Connections to other relevant problems such
as the vortices in BoseEinstein condensation will also be
discussed.
The tutorial will be given in three continuing lectures:
Lecture 1. Superconductivity  a brief introduction to history and
phenomenon
Lecture 2. Mathematical models and numerical simulations of the vortex
state in superconductivity
Lecture 3. Quantized vortices and superfluidity.
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Coarsening dynamics and continuum
modeling for epitaxial growth of thin films
JianGuo Liu, University of Maryland
Two nonlinear diffusion equations for thin film epitaxy,
with or without slope selection, will be discussed in this
talk. We will show wellposeness of these differential
equations and some numerical simulations of coarsening
dynamics with these equations. For the case of without slop
selection, we will show rigorously that the interface width is
bounded above by $O(t^{1/2})$ and the averaged gradient is
bounded above by $O(t^{1/4})$. These bounds were predicted
previously by many experiments and numerical simulations.
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Recent developments in modeling,
analysis and numerics of Ferromagnetism
Andreas Prohl, ETH Zurich
Micromagnetics is a continuum variational theory describing
magnetization pattern in ferromagnetic media. Its multiscale
nature due to different inherent spatiotemporal physical and
geometric scales, together with nonlocal phenomena and a
nonconvex sideconstraint leads to a rich behavior and pattern
formation. This variety of effects is also the reason for
severe problems in analysis, model validation and reductions,
and numerics, which are only accessed recently.  This is
joint work with M. Kruzik (U Prague).
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Distributions of terrace widths on
misoriented surfaces:
multipronged theory approaches to studying fluctuations in conjunction with
quantitative experiments
Theodore L. Einstein, University of Maryland,
College Park
In collaboration with Alberto Pimpinelli, Hailu Gebremariam,
Howard L. Richards, Olivier PierreLouis, Saul D. Cohen,
Robert D. Schroll, and experimentalists Ellen D. Williams and
J.E. ReuttRobey at UM, M. Giesen and H. Ibach at FZJülich,
and J.J. Métois at Marseilles
We discuss the terracewidth distribution (TWD) of
meandering steps on a vicinal surface as the prototype of a
significant materials problem, amenable to quantitative
measurements, that can be approached from multiple theoretical
perspectives and computational techniques. Stepped surfaces at
relevant temperatures are usually well described by the
terracestepkink model. The TWD of this model is analogous to
the distribution of spacings between repelling fermions in 1
dimension, leading to a meanfield solution (for nonweak
repulsions) using an equivalence to simple problems in
elementary quantum mechanics. Analogies with the distribution
of energy spacings invite application of results from
randommatrix theory [1] that describe universal properties of
fluctuations. In particular, we generalize the gammalike "Wigner
surmise," developed for a few special cases, to treat
arbitrary repulsion strength as a superior alternative to
various Gaussian approaches [26]. We use Monte Carlo
simulations as well as transfermatrix calculations, both with
finitesize scaling, to test this picture [35]. The
stepcontinuum viewpoint underlying the fermion models
diverges from the discrete character of the simulations (and
of the physical systems) only for unphysically strong
repulsions. The TWD actually corresponds to a multiparticle
correlation function. There is an exact solution for the
corresponding pair correlation function, but it is too
unwieldy to be of use in dealing with experimental or
simulated data. The Wigner distribution arises in econophysics
to describe the distribution of variances of fluctuating stock
prices in the Heston model [7]. We translate the FokkerPlanck
equation from that development to step language and thereby
make predictions about how TWDs evolve toward equilibrium [8].
Work supported primarily by the MRSEC at UM under NSF grant
DMR 0080008, and partially by NSF Grant EEC0085604. My
papers are downloadable from
http://www2.physics.umd.edu/~einstein/
1. T. Guhr, A. MüllerGroeling, and H.A. Weidenmüller,
Phys. Rep. 299 (1998) 189.
2. T.L. Einstein and O. PierreLouis, Surface Sci. 424 (1999)
L299.
3. H.L. Richards, S.D. Cohen, T.L. Einstein, and M. Giesen,
Surface Sci. 453 (2000) 59.
4. T.L. Einstein, H.L. Richards, S.D. Cohen, and O.
PierreLouis, Surface Sci. 493 (2001) 460.
5. Hailu Gebremariam, S. D. Cohen, H. L. Richards and T. L.
Einstein, Phys. Rev. B 69 (2004) 125404.
6. T.L. Einstein, Ann. Henri Poincaré 4, Suppl. 2 (2003) S811
[condmat/0306347].
7. A. A Drǎgulescu
and V. M Yakovenko, Quantitative Finance 2 (2002) 443.
8. A. Pimpinelli and T.L. Einstein, draft preprint.
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Droplet formation of two phase
flow systems inside microfluidic devices
Amy Shen, Washington University in St. Louis
Microfluidic devices offer a unique method of creating and
controlling droplets on small length scales. A microfluidic
device is used to study the effects of surface properties on
droplet formation of 2phase flow system. Four phase diagrams
are generated to compare the dynamics of the 2 immiscible
fluid system (silicone oil and water) inside microchannels
with different surface properties. Results show that the
channel surface plays an important role in determining the
flow patterns and the droplet formation of the 2phase fluid
system.
Another example of a two phase flow system shows that
Liquid crystal drops dispersed in a continuous phase of
silicon oil can be generated with a narrow distribution in
droplet size in microfluidic devices both above and below the
nematic to isotropic transition temperature. For these two
cases, we observe not only the different LC droplet generation
and coalescence dynamics, but also distinct droplet
morphology. Our experiments show that the nematic liquid
crystalline order is important for the LC droplet formation
and anchoring dynamics.
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Numerical simulations of
dislocation dynamics using level set method
Yang Xiang, Hong Kong University of Science and Technology
Dislocations are line defects in crystals. We present a
threedimensional level set simulation method for dislocation
dynamics. Since the level set method does not directly track
the individual dislocation line segments, it easily handles
topological changes of dislocations in the microstructure.
Further, the method naturally accounts for the complicated
three dimensional motion of dislocations. This method is
applied to the dislocation dynamics in the presence of a
particle dispersion. The simulations show a wide range of
dislocationparticle bypass mechanisms. Some of these
mechanisms are classic and others have never been reported
previously. We also apply this simulation method to the
formation of dislocation networks and junctions. The
simulation results agree with the experimental observations
and the results obtained using other methods.
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Variational approaches in complex
fluids
Chun Liu, Pennsylvania State University
The most common origin of different phenomena in complex
fluids are different "elastic" effects. They can be the
elasticity of deformable cells, elasticity of the molecule
alignment in liquid crystals, polarized colloids or
multicomponent phases, elasticity due to microstructures, or
bulk elasticity endowed by polymer molecules in viscoelastic
complex fluids. The physical properties are purely determined
by the interplay of entropic and structural intermolecular
elastic forces and interfacial interactions. These elastic
effects can be represented in terms of certain internal
variables, for example, the orientational order parameter in
liquid crystals (related to their microstructures), the
distribution density function in the dumbbell model for
polymeric materials, the magnetic field in
magnetohydrodynamic fluids, the volume fraction in mixture of
different materials etc. The different rheological and
hydrodynamic properties can be attributed to the special
coupling between the transport of the internal variable and
the induced elastic stress. From the point of the view of the
energetic variational formulation, this represents a
competition between the kinetic energy and the elastic energy.
In these lectures, I will study several different but related
types of problems to illustrate this unified energetic
variational approach. All the systems are related and have
common structures. However, each one posses its own distinct
features (difficulties). I will present some modeling and
analytical results, as well as those problems that remain to
be solved.
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A unified treatment of
currentinduced instabilities on Si surfaces
John Weeks, University of Maryland at College Park
Crystal surfaces with atomic steps can exhibit a number of
different morphological instabilities that may be important in
crystal growth and nanoscale device fabrication. Particularly
interesting step bunching and step wandering instabilities are
seen when Si surfaces are heated by a direct electric current.
These patterns have a strong dependence on both the current
direction and the temperature. We argue on physical grounds
that the diffusion rate in a small region around each step on
a vicinal surface can differ from that found elsewhere on the
terraces due to differences in local bonding or surface
reconstruction. We study in particular a discrete 1D hopping
model that takes into account possible differences in the
hopping rates in the region around a step and on the terraces
as well as the finite probability of incorporation into the
solid at the step site. By expanding the continuous
concentration field in a Taylor series evaluated at discrete
sites near the step, we relate the kinetic coefficients and
permeability rate in general sharp step models to the
physically suggestive parameters of the hopping models. We
find that both the kinetic coefficients and permeability rate
can be negative when diffusion is faster near the step than on
terraces. A linear stability analysis of the resulting
sharpstep model provides a unified and simple interpretation
of many experimental results for currentinduced step bunching
and wandering instabilities on both Si(111) and Si(001)
surfaces in terms of negative kinetic coefficients. We also
use a geometric representation in terms of arc length and
curvature to derive a nonlinear evolution equation for a step
in the presence of an electric field oriented at an angle to
the average step direction.
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A secondorder
γmodel BGK scheme for
multimaterial compressible flows
Song Jiang, Institute of Applied Physics and Computational
Mathematics, China
We present a secondorder
γmodel BGK scheme for compressible multimaterial
flows, which extends the authors' earlier work on a
firstorder scheme [Int. J. Numer. Meth. Fluids 46 (2004),
163182]. The scheme is based on the incorporation of a
conservative γmodel scheme
given in [R. Abgrall, J. Comput. Phys. 125 (1996), 150160]
into the gas kinetic BGK scheme [K.H. Prendergast and K. Xu,
J. Comput. Phys. 109 (1993), 5366; and 114 (1994),917].
Numerical examples validate the scheme in numerical
simulations of compressible multifluids.
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A mortar element method for coupling
natural BEM and FEM for unbounded domain problem
Dehao Yu, Chinese Academy of Sciences
A coupling method of boundary elements and finite elements
combines the advantages of the boundary element method and the
finite element method. It is especially important for solving
problems over unbounded domains.
The natural and direct coupling of BEM and FEM was
suggested and developed first by K. Feng, D.H. Yu and H.D. Han
in early 1980. It is also known as the exact artificial
boundary condition method. Then a very similar method,
socalled DtN method, was also devised and applied in the west
by J.B. Keller, D. Givoli and others.
There are many kinds of methods to archive the coupling,
the mortar element method is one of them. Compared with other
methods, it appears to be attractive because its meshes on
different subdomains need not conform across the interface.
This method provides the flexibility of triangulation, and the
matching of discretizations on subdomains is only enforced
weakly.
In this talk, a mortar element method is presented for
coupling natural boundary element method and finite element
method for the exterior boundary value problem. Optimal error
estimate is obtained, and some numerical results are presented
to show the performance of this method.
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Perfect and Wriggled Lamellar
patterns in the Diblock copolymer problem
Juncheng Wei, The Chinese University of Hong Kong
We consider the lamellar phases in the diblock copolymer
system which can be written as a system of elliptic equations.
Using γconvergence, the
existence and stability of Kinterface solutions in 1D
are characterized. Then these solutions extend trivially to 2D
and 3D to become perfect lamellar solutions. The stability of
these lamellar solutions is completely characterized by
obtaining the asymptotic expansions of their eigenvalues and
eigenfunctions. Consequently we find that they are stable,i.e.
are local minimizers in space, only if they have sufficiently
many interfaces. Interestingly the 1D global minimizer is
near the borderline of 3D stability. Finally using
bifurcation analysis, we find wriggled lamellar solutions of
the EulerLagrange equation of the total free energy. They
bifurcate from the perfect lamellar solutions. The stability
of the wriggled lamellar solutions is reduced to a relatively
simple finite dimensional problem, which may be solved
accurately by a numerical method. Our tests show that most of
them are stable. The existence of such stable wriggled
lamellar solutions explains why in reality the lamellar phase
is fragile and it often exists in distorted forms.
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Heat conduction in onedimensional
systems  molecular dynamics and modecoupling theory
JianSheng Wang, National University of Singapore
We begin with a brief review of the studies of
onedimensional heat conduction problems. It is known that
Fourier law of heat conduction is violated, causing the
thermal conductance diverges with the length of the system.
However, its specification form of divergence is still
controversial. We study heat conduction in a onedimensional
chain of particles with longitudinal as well as transverse
motions. The particles are connected by twodimensional
harmonic springs together with bending angle interactions. The
problem is analyzed by modecoupling theory and compared with
molecular dynamics results. We find excellent agreement for
the damping of modes between modecoupling theory and
molecular dynamics. The theories predict generically that
thermal conductance diverges as N^{1/3} as the size N
increases for systems terminated with heat baths at the ends.
The N^{2/5} is also observed in molecular dynamics which we
attributed to crossover effect.
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Solving Maxwell's equations in
inhomogeneous dispersive media
Wei Cai, University of North Carolina at Charlotte
Time dependent density functional theory (TDFT) is an
important tool for studying dynamics of many particle systems
and the calculation of their excitation energies. In this
talk, we will present a self consistent high order
discontinuous Galerkin method for the TDFT and PML boundary
conditions for the boundary treatment. Numerical results will
be presented.
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Simulation of micro flows by lattice
Boltzmann method
Chang Shu, National University of Singapore
Microelectromechanical systems (MEMS) have become a
subject of active research in a growing discipline. As all the
micro devices have to operate in a fluid media, the
understanding of flow at micro level is fundamental to the
development of MEMS. In spite of its importance, the research
of micro flows is still at a preliminary stage although the
mechanical properties of some micro devices are reasonably
well studied. The main reason behind this is at microlevel,
the continuum assumption is no longer valid since the mean
free path of gas molecules is the same order as the typical
geometric dimension of the device. As a result, the
conventional governing equation of motion (the NavierStokes
equations) and numerical tools that seek to solve this
equation are not applicable. The usual ways to study the micro
flows are molecular dynamics (MD), the direct simulation Monte
Carlo (DSMC) approach and solutions of full Boltzmann equation
(BE). However, the computational effort of the MD and the DSMC
is usually very huge with the use of most powerful
supercomputer and the schemes used for solving the full BE are
more complicated than those usually used for the NavierStokes
equations.
Recently, the lattice Boltzmann method (LBM) has received
considerable attention by fluid dynamic researchers [1].
Although the LBM is intrinsically kinetic, only a few
applications for micro flows were carried out. The reasons may
be due to the difficult determination of relaxation parameter
for collision and boundary conditions. In our previous work
[2], we assumed that the collision of two particles in the LBM
happens and relaxes toward equilibrium within a mean free path
of gas molecules in a collision interval, and we established a
relationship between the relaxation parameter in the LBM and
the local Knudsen number as . In this work, we further present
a theoretical foundation of the above assumption based on the
kinetic theory [3] and the LBM theory [1]. On the other hand,
to correctly consider effects of fluidsolid interactions on
the boundary, we present a diffusescattering boundary
condition (DSBC) for the LBM to simulate micro flows according
to the classic Boltzmann assumption [1]. This boundary
condition has considered the wall equilibrium information and
is suitable for any kind of boundary geometries. To check
theoretical validity, a numerical analysis for a simple flow
is also presented. Using the LBM with present efforts, the
twodimensional (2D) pressuredriven isothermal microchannel
flows, the 2D sheardriven isothermal micro flows and the
thinfilm gas bearing lubrication were investigated. The
numerical results obtained are found to be in good agreement
with theoretical analysis, available experimental data and
numerical simulations.
References
[1] S. Chen and G. D. Doolen, Ann. Rev. Fluid Mech., 30:329,
1998
[2] C. Y. Lim, C. Shu, X. D. Niu and Y. T. Chew, Phys. Fluids,
14(7):2299, 2002
[3] C. Cercignani, Mathematical methods in kinetic theory,
Plenum, New York, 1969
« Back...
Atomistic and continuum modeling of
homoepitaxial thin film growth
Jim Evans, Iowa State University
Homoepitaxial thin film growth reveals a rich variety of
farfromequilibrium morphologies. Our goal is to develop
models which provide fundamental insight into the processes
controlling these morphologies, and which further have
predictive capability for specific systems. Atomistic
latticegas models analyzed by KMC simulation have been most
successful, but we also discuss 2D continuum formulations (BCFbased,
levelset, stochastic geometrybased) retaining discrete
layers, and 3D continuum formulations for multilayer kinetic
roughening.
Complete characterization of island formation during
submonolayer deposition remains a basic challenge due to the
failure of traditional meanfield formulations [1]. One goal
of recent multiscale approaches is to efficiently and reliably
treat the regime of highly reversible island formation. We
discuss the special features of this regime and present
results for the island size distribution obtained from the
geometrybased simulation approach [2].
Step edge (SE) barriers inhibiting downward transport
produce unstable multilayer growth characterized by mound
formation. Realistic atomistic modeling reveals that
Ag/Ag(100) regarded as the prototype for smooth growth at 300K
(due to a low SE barrier) actually grows very rough in the
1001000 layer regime [3]. Furthermore, mound dynamics is more
complex than predicted by standard 3D continuum models. We
discuss this behavior, as well as recent analyses of
Ag/Ag(111) growth involving a large SE barrier.
[1] PRB 54 (96) 17359; 66 (02) 235410
[2] PRB 68 (03) 121401; Surf. Sci. 546 (03) 127; SIAM MMS 3
(04)
[3] PRL 85 (00) 800; PRB 65 (02) 193407
« Back...
Macroscopic simulation of
micro(nano)structures in finitestrain elastoplasticity
Carsten Carstensen, HumboldtUniversität
zu Berlin
The computer simulation of the evolution of microstructures
in finitestrain elastoplasticity requires a timespace
discretization. The resulting mathematical model of each
timestep yields a minimization problem with a nonconvex
energy density W. Therein, the energy minimizing (better
called infimizing) sequences of deformations develop enforced
finer and finer oscillations in the deformation gradients
called microstructures. The infimal energy is not
attained and in the limit of those infimizing sequences, the
deformation gradients yield a measure to describe
statistically the oscillations. This gradient Young measure
(GYM) acts as a generalized solution and conveys several
pieces of information about the energy infimizing process such
as the macroscopic deformation (i.e. the expected value of the
GYM) or the stress field (GYM applied to derivative DW of
energy density).
The presentation gives a simple example in finite
elastoplasticity with a singleslip mechanism and then
explains the effect of nonconvexity and the relaxation theory
from modern calculus of variations in 1D, 2D, and the vector
case in a series of Examples related to Bolza, Young, Tartar
plus one benchmark and a phasetransition.
The numerical analysis of the relaxed formulation with
adaptive finite element schemes and their stabilization is
briefly discussed. In general, however, the quasiconvex hull
is not known by some closed form expression. Instead a new
computational challenge, numerical quasiconvexification, is in
order and some new attempts towards this are discussed.
The relaxation theory allows for a macroscopic simulation
and only allows limited insight in the underlying
microstructure patterns (through the GYM). More insight in the
context of finite elastoplasticity is promised by energies
extended by some surface energy. The mathematical model of
which is less obvious in finite elastoplasticity and the
presentation briefly discusses severe difficulties even with
much simpler examples which lead to curved needles and
branching structures near interfaces.
« Back...
Hydraulic fracture: multiscale
processes and moving interfaces
Anthony Peirce, University of British Columbia
We introduce the problem of Hydraulic Fracture (HF) and
provide examples of situations in which Hydraulic Fractures
are used in industrial problems. We describe the governing
equations in 12D as well as 23D models of HF, which involve
a coupled system of degenerate nonlinear integropartial
differential equations as well as a free boundary. We
demonstrate, via rescaling the 12D model, how the active
physical processes manifest themselves in the HF model. We
also show how a balance between the dominant physical
processes leads to special solutions. We discuss the
challenges for efficient and robust numerical modeling of the
23D HF problem including: the rapid construction of Green’s
functions for cracks in layered elastic media and novel
multigrid procedures to solve the coupled system of equations.
We demonstrate the efficacy of these techniques with numerical
results.
« Back...
Computational methods for
distributed parameter estimation with application to
inversion of 3D electromagnetic data
Uri Ascher, University of British Columbia
Inverse problems involving recovery of distributed
parameter functions arise in many applications. Many
practical instances require data inversion where the forward
problem can be written as a system of elliptic partial
differential equations. Realistic instances of such problems
in 3D can be very computationally intensive and require care
in the selection and development of appropriate algorithms.
The problem becomes even harder if the model to be recovered
is only piecewise continuous.
In this talk I will describe work we have been doing in
the context of inverting electromagnetic data in frequency
and time domains for geophysical mining applications with
the objective of making such computations practically
feasible. Our techniques are applicable in a wider context,
though.
A second part of the talk will describe our efforts to
accommodate discontinuities using modified TV and Huber
function regularization. These two variants are shown to be
very similar when parameters are chosen well (we show how).
The application to diffusive foward models such as Maxwell's
equations in low frequencies is perilous, though.
This is joint work with E. Haber.
« Back...
Two recent advances in soft
condensed matter physics
Ping Sheng, Hong Kong University of Science and
Technology
An important component of nano science and technology
development is soft condensed matter physics. In this talk I
give two recent advances in this regard at HKUST.
The first is the discovery of the Giant
Electrorheological (GER) effect [1], in which nanoparticles
coated with thin layers of urea, which has a large molecular
dipole moment, have been found to exhibit very fast (~
1msec), reversible liquidsolid transitions. The yield
stress of the solid state can reach 250 kPa, more than one
order of magnitude larger than the conventional
electrorheological (ER) fluids. The GER mechanism has been
found to originate from the formation of aligned molecular
dipole layers in the regions of particleparticle contact.
This is possible mainly owing to (1) the amorphous nature of
the core nanoparticles, which makes the urea coating
noncrystalline, and (2) the electrowetting effect of the
nanoparticles' coating with the silicone oil, which effects
a "ferroelectric" type of interaction for the surface
molecular layer.
The second is the discovery of the Generalized Navier
Boundary Condition (BNBC) for the moving contact line (CL)
problem [2, 3]. Here the contact line denotes the
intersection of the immiscible fluidfluid interface with
the solid wall, and moving CL arises when one immiscible
fluid displaces the other in motion. The moving CL has been
a classical problem in hydrodynamics because it is
incompatible with the nonslip boundary condition. While
molecular dynamics (MD) simulations have clearly shown
nearly total slip of the moving CL (i.e., relative motion
with respect to the solid wall), no continuum boundary
condition was found which can reproduce the MD results,
leading to some proposal that hydrodynamics breaks down in
the vicinity of the moving CL. The inability of continuum
hydrodynamics to calculate the behavior of the moving CL
means accurate nanofluidics or microfluidics simulations
would not be possible. We report the successful resolution
of this classical problem, with continuum hydrodynamics
results in quantitative agreement with those from the MD.
Some surprising implication of this discovery will also be
presented.
Work done in collaboration with Weijia Wen, Xianxiang
Huang, Shihe Yang, Tiezheng Qian, and Xiaoping Wang.
[1] The Giant Electrorheological Effect in Suspensions of
Nanoparticles, W. Wen, X. Huang, S. Yang, K. Lu and Ping
Sheng, Nature Materials 2, 727730 (2003).
[2] PowerLaw Slip Profile of the Moving Contact Line in
TwoPhase Immiscible Flows, T. Qian, X. P. Wang and Ping
Sheng, Phys. Rev. Lett. 93, 094501094504 (2004).
[3] Molecular Scale Contact Line Hydrodynamics of Immiscible
Flows, T. Qian, X. P. Wang and Ping Sheng, Phys. Rev. E68,
016306 (2003).
« Back...
Mathematical models of interface
dynamics and coarsening
Robert Pego, Carnegie Mellon University
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« Back...
Chaos on the scaling attractor in
Smoluchowski ripening and Burgers turbulence
Robert Pego, Carnegie Mellon University
We develop a basic framework for studying dynamical
scaling that has roots in dynamical systems and probability
theory. In this framework we study Smoluchowski's
coagulation equation, a fundamental meanfield model for the
agglomeration of clusters, for the `solvable' rate kernels
2, x+y, and xy. We classify all domains of attraction in
dynamic scaling, and characterize the `scaling attractor'
(limit points modulo scaling) in terms of a remarkable LevyKhintchine
representation given by Bertoin for x+y. Via other work of
Bertoin, our results yield a complete classification of
universality classes for dynamic scaling of shock size
distributions in Burgers turbulence, for initial velocity
that is random with stationary, independent increments with
no positive jumps. This is joint work with Govind Menon.
« Back...
Efficient and accurate numerical
schemes for phasefield equations with applications to the
mixture of two incompressible fluids
Jie Shen, Purdue University
We shall present some highly efficient and accurate
numerical schemes for solving CahnHilliard, AllenCahn and
NavierStokes equations. The spatial discretizations will be
based on the spectralGalerkin method while the temporal
discretizations will be based on a combination of splitting
and semiimplicit schemes. We shall present ample numerical
results which not only demonstrate the effectiveness of our
numerical schemes, but also validate the flexibility and
robustness of our phasefield models for numerical
simulations of microstructural evolution and of complex
fluids.
« Back...
The vortex dynamics of a GinzburgLandau
system under pinning effect
Huaiyu Jian, Tsinghua University
In this talk, we will explain how to use an ODE or a
nonhomgeneous mean curvature flow of higher codmensional
manifolds to describe the dynamical law of a parabolic
GinzburgLandau system with pinning effect.
« Back...
Step pattern formation on Si
Vicinal surfaces with two coexisting structures
Makio Uwaha, Nagoya Univeristy
Near the structural transition temperature between 1 X 1
and 7 X 7 of the Si (111) surface, the two structures
coexist across the steps on a vicinal face. On a Si (001)
vicinal face, 2 X 1 and 1 X 2 structures appear alternately.
These vicinal faces show inphase wandering and bunching of
steps during growth or direct current heating. We study
mechanism and patterns of the wandering and bunching
instabilities by means of a linear stability analysis and
numerical simulations.
In the first system, the simplest model we adopt is the
standard step model with two terrace regions of different
diffusion coefficient. The steps show wandering instability
during growth as observed in the experiment. We also found
the possibility of drift induced step bunching during
current heating as well as inphase step wandering with
stepdown current.
In the second system, the heating current produces step
pairs, and these pairs form step bunches with either
direction of current. The structure and the growth law of
bunches are determined by the balance of the drift effect
and the repulsive interaction of steps. The repulsive
interaction brings about diffusion current between steps and
causes wandering of steps as observed in the experiment.
General features and particularity of the instabilities
will be discussed.
The talk is based on the work in collaboration with M.
Sato, R. Kato and Y. Saito.
« Back...
Error estimates of finite
element approximation for problems in unbounded domains
Weizhu Bao, National University of
Singapore
Many boundary value problems of partial differential
equaitons (PDEs) involving unbounded domain occur in many
areas of applicaitons, e.g. fluid flow around obstacles,
coupling of structures with foundation, wave propagation and
radiation, quantum physics and chemistry etc. One of the
main numerical difficulties is the unboundedness of physical
domain.
In this talk, I first review different numerical
approaches for problems in unbounded domain. Then I present
highorder nonlocal/local artificial boundary conditions
(ABCs) for secondorder elliptic PDE and reduce it to a
problem defined in a bounded computational domain. New
`optimal' error estimates for the finite element
approximation of the problem is reported. Extension of the
results to Navier system for linear elastic and Stokes
equations for incompressible material is given. Furthermore,
the method is applied successfully to NavierStokes for
incompressible viscous flow aroung obstacles. Numerical
results are also reported to confirm our error estimates.
« Back...
Numerical simulation for rotating
BoseEinstein condensate
Weizhu Bao, National University of Singapore
In this talk, we present efficient and stable numerical
methods to compute ground states and dynamics of
BoseEinstein condensates (BEC) in a rotational frame. As
preparatory steps, we take the 3D GrossPitaevskii equation
(GPE) with an angular momentum rotation, scale it to obtain
a fourparameter model and show how to reduce it to 2D GPE
in certain limiting regimes. Then we study numerically and
asymptotically the ground states, excited states and
quantized vortex states as well as their energy and chemical
potential diagram in rotating BEC. Some very interesting
numerical results are observed. Finally, we study
numerically stability and interaction of quantized vortices
in rotating BEC. Some interesting interaction patterns will
be reported.
This talk is based on joint work with Qiang Du, Peter
Markowich, Hanquan Wang and Yanzhi Zhang.
« Back...
Coarsening versus non coarsening
of growing interfaces
Chaouqi Misbah, Université Joseph Fourier
In nature there is an overabundance of systems that
spontaneously build up a pattern from a structureless state
when driven away from equilibrium. Various examples are
known for growing interfaces, such as in MBE. Two broad
classes of dynamics are known (i) selection of a length
scale, (ii) perpetual coarsening, with an intermediate
scenario "interrupted coarsening". We present a general
criterion that allows one to distinguish between these two
classes. The criterion is based on the analysis of the phase
diffusion equation of the pattern. The power of the
criterion is that there is no need to solve explicitly the
full time dependent dynamics. Rather, the knowledge of the
steadystate solutions together with the phase equation is
sufficient. Our criterion is illustrated for some generic
equations. We show in addition that the phase diffusion
equation provides us with the coarsening exponents.
References:
1. P. Politi and C. Misbah, "When does coarsening occur in
the dynamics of onedimensional fronts?”, Phys.Rev.Lett.
92, 090601 (2004)
2. G. Danker, O. PierreLouis, K. Kassner and C. Misbah,
Peculiar effects of anisotropic diffusion on dynamics of
vicinal surfaces, Phys.Rev.Lett.93, 185504 (2004)
« Back...
Quantum driftdiffusion models
derived from an entropy minimization principle
Pierre Degond, Université Paul Sabatier
In this work, we give an overview of recently derived
quantum hydrodynamic and diffusion models. A quantum local
equilibrium is defined as a minimizer of the quantum entropy
subject to local moment constraints (such as given local
mass, momentum and energy densities). These equilibria
relate the thermodynamic parameters (such as the temperature
or chemical potential) to the densities in a nonlocal way.
Quantum hydrodynamic models are obtained through moment
expansions of the quantum kinetic equations closed by
quantum equilibria. We also derive collision operators for
quantum kinetic models which decrease the quantum entropy
and relax towards quantum equilibria. Then, through
diffusion limits of the quantum kinetic equation, we
establish various classes of models which are quantum
extensions of the classical energytransport and
driftdiffusion models. We shall present 1D numerical
results which show that these model are able to reproduce
the major features of quantum transport in resonant
tunelling structures.
« Back...
Numerical methods in quantum
kinetic theory
Lorenzo Pareschi, University of Ferrara
We review some recent results on the development of
numerical methods in quantum kinetic theory. Particular care
is devoted to the development of efficient numerical schemes
for the quantum Boltzmann equation for interacting bosons.
The resulting schemes preserve the main physical features of
the continuous problem, namely conservation of mass and
energy, the entropy inequality and generalized BoseEinstein
distributions as steady states. These properties are
essential in order to develop numerical methods that are
able to capture the challenging phenomenon of bosons
condensation. We also show that the resulting methods can be
evaluated with the use of fast algorithms. The order of
accuracy of the methods and the convergence rate is also
studied.
« Back...
RKDG methods with WENO type limiter
for conservation laws
Jianxian Qiu, National University of Singapore
In the presentation we will describe our recent work on a
class of new limiters, called WENO (weighted essentially
nonoscillatory) and HWENO (Hermite WENO) limiters, for
RungeKutta discontinuous Galerkin (RKDG) methods. The goal
of designing such limiters is to obtain a robust and high
order limiting procedure to simultaneously obtain uniform
high order accuracy and sharp, nonoscillatory shock
transition for the RKDG method. We adopt the following
framework: first we use TVB limiters to identify the trouble
cells, namely those cells which might need the limiting
procedure, the cell is declared as trouble cell; then we
replace the solution polynomials in those troubled cells by
new polynomials reconstructed by WENO or HWENO finite volume
method, which maintain the original cell averages for
keeping conservation, have the same orders of accuracy as
before, but are less oscillatory. In the WENO reconstruction
procedure, the cell averages of the trouble cell and its
neighboring cells are used to reconstruct the moments of the
new polynomial in the trouble cell. In the HWENO
reconstruction procedure, the cell average of the trouble
cell and both cell averages and the first moments of its
neighboring cells are used for the reconstruction. HWENO
thus uses much fewer neighboring cells to obtain a
reconstruction of the same order of accuracy than WENO. Both
WENO limiters and HWENO limiters work quite well in our
numerical tests for both one and two dimensional cases,
which will be shown in the presentation.
« Back...
On transport in micromacro models
for polymeric fluids
Chun Lu, National University of Singapore and Institute
of High Performance Computing, Singapore
In this talk, we will discuss the multiscale coupled
models for polymeric materials. The focus will be on the
transport of the microscopic variables and the induced
elastic stress.
« Back...
Topics in the mathematical
analysis of nematic elastomers
Georg Dolzmann, University of Maryland
We analyze mathematical models for a special class of
polymers, socalled nematic elastomers. These materials
combine the elastic properties of rubbers with the
instabilities observed in liquid crystals. In this
presentation we focus on the derivation of macroscopic
models, thin film theories, and their numerical simulation.
This is joint work with A. DeSimone (SISSA) and S. Conti
(Duisburg).
« Back...
Vortex nucleation in 3dimensional
superconductors
Xingbin Pan, East China Normal University
Type II superconductors undergo phase transition from the
Meissner state to the mixed state in an increasing applied
magnetic field. A simplified model of partial differential
system has been used to describe vortex nucleation in
superconductors. For a cylindrical sample in an applied
field parallel to the axis, this system is reduced to a
single equation and has been wellunderstood. In this talk
we examine the system for a bulk superconductor occupying a
bounded 3dimensional domain. Our results suggest that,
vortices nucleate at the boundary of the sample where the
applied field is tangential to the surface.
« Back...
Isentropic modeling of unsteady
cavitating flow
Tiegang Liu, Institute of High Performance Computing,
Singapore
In this talk, I will introduce a physically
reasonable and mathematically sound cavitation model for
unsteady cavitating flow driven by pressure drop. Such
cavitating flow is widely observed in underwater explosions
and biological flow. The model will be validated by
experimental data and applied to various problems related to
underwater explosions.
« Back...
Onedimensional interfaces in
twodimensional materials structures
Ellen D. Williams, University of Maryland
Steps, island edges and domain boundaries are
onedimensional interfaces that serve as the locus of
material transport, and as interfacial barriers for electron
transport. These interfaces fluctuate under thermal
excitation, with length and time scales that can be observed
directly using scanning probe imaging. Quantitative
characterization of these fluctuations using the tools of
statistical mechanics yields energetic and kinetic
parameters that can be used to predict evolution of
structure under external driving forces (e.g. temperature
gradient, growth or sublimation, electromigration). In
addition, as the size of the bounded structure decreases
into the nanoscale, the stochastic aspects of the
fluctuations themselves become a significant component of
the material properties.
Scanned probe measurement of fluctuations, correlation,
autocorrelation, survival and persistence, will be presented
for steps (on Ag, Pb and C60/Ag) and domain boundaries (Pb/Si,
Ag/Si and C60/Ag). The meaning of system size in designing,
evaluating and using these results will be explained. The
impact of the onedimensional structures on electron flow
will also be presented. Direct measurements of step
fluctuations in the presence of an electromigration current
density of up to 105 A/cm2 will be shown and interpreted in
terms of the limits on effective charge for mass
displacement at the line boundary. Measurements of the noise
and resistivity in electron transport will be shown and
characterized in terms of structural fluctuations in a film
near the percolation threshold.
* Different aspects of this work have been supported
respectively by the DOEBES, NSFNIRT and NSFMRSEC
« Back...
Critical thresholds in Eulerian dynamics
Eitan Tadmor, University of Maryland We study the questions of global regularity vs. finite time breakdown
in Eulerian dynamics, u_{t}+u·Ñ_{x}u=ÑF,
which shows up in different contexts dictated by different
modeling of F's.
To adders these questions, we propose the notion Critical Threshold (CT),
where a conditional finite time breakdown
depends on whether the initial configuration crosses an
intrinsic, O(1) critical threshold.
Our approach is based on a main new tool of spectral dynamics, where the
eigenvalues, λ:=λ(Ñ u), and eigenpairs (l,r), are
traced by the forced Riccati equation
λ_{t} +u·Ñ_{x}λ + λ^{2} = <l, D^{2}F
r>.
We shall outline three prototype cases.
We begin with the ndimensional Restricted Euler equations,
obtaining [n/2]+1 global invariants
which precisely characterize the local topology at breakdown time.
Next we introduce the corresponding ndimensional Restricted
EulerPoisson (REP) system, identifying a set of [n/2] global
invariants, which yield
(i) sufficient conditions for finite time breakdown, and (ii)
a remarkable characterization of twodimensional initial REP configurations with
global smooth solutions.
And finally, we show how rotation prevents
finitetime breakdown.
Our study reveals the dependence of the CT phenomenon on the initial spectral gap,
λ_{2}(0)λ_{1}(0).
« Back...
Faceted island and film growth:
a level set approach
David Srolovitz, Princeton University
In this presentation, I will discuss two approaches to
modeling the growth of faceted thin films and islands.
Faceted surfaces are routinely observed for a wide range of
materials  especially covalently and ionically bound
materials. We have used two methods to simulate this type of
growth. In the first, originally proposed by Russo and
Smereka, we choose which types of facets may occur and
constrain the system to only allow the chosen sets of
surface normals. In the second, we construct a full velocity
vs. surface normal profile, and evolve the system according
to this. The evolution of the systems is simulated within
the level set method framework. This approach was chosen to
allow for easy modeling of topology changes, as compared
with front tracking methods. In the fixed set of facets
simulations, we concentrate on the chemical vapor deposition
of polycrystalline diamond films. We predict the evolution
of surface morphology, grain microstructure, grain size,
roughness, and crystallographic texture. Comparisons with
experimental observations will be presented. In the velocity
vs. surface normal simulations, we focus on the growth of
GaN islands via the epitaxial lateral overgrowth method
(widely used experimentally). In this case, islands grow
through shaped holes in the substrate. Coltrin and Mitchell
performed a series of experiments for different hole shapes.
Based on their observations, we propose a form of the
velocitysurface normal profile and parameters to specify
it. Using this, we monitor the entire morphology evolution.
The correspondence with experiment is remarkable. From these
series of simulations, some general features of anisotropic
surface can be deduced. For example, the facets observed in
the growth of a convex surface correspond to cusp
orientations in the velocity profile and the resultant shape
is found by convexifying the plot of velocity vs. surface
normal orientation in a spherical geometry. This is
equivalent to determining the equilibrium shape of a crystal
from the surface energy vs. orientation plot – the socalled
Wulff shape. For convex growth, the fastest growing surface
dominates the surface morphology. Nearly perfectly flat
facets can be obtained in this type of growth – however, in
this case, the facet normals do not correspond to cusps in
the velocity vs. surface normal orientation. Curved surfaces
can form where different types of facets meet. These,
however, always grow out, leaving flat facets behind.
« Back...
Dynamics of the BeckerDoering
equation
Barbara Niethammer, HumboldtUniversity of Berlin
The BeckerDoering equations are a model for cluster
formation in a set of identical particles. The main
assumption in this model is that clusters gain or shed only
one particle at a time. Despite its simplicitz the BeckerDoering
equations can describe the relevant stages in a first order
phase transformation, such as nucleation, metastability and
coarsening. We give an overview on the underlying assumption
of the model, on the main results on the large time behavior
and discuss some open questions.
« Back...
Hydrodynamics of suspensions and
nematic polymers
Qi Wang, Florida State University
I will discuss the developement of hydrodynamic theories
for anisotropic particle suspensions and nematic polymers in
solutions. I will focus on the phase transition in chiral
nematic polymers and suspensions under the imposed magnetic
field. Nematodynamics of the suspension and nematic polymers
in simple flows will be discussed in the end.
« Back...
From architecture of nanostructure
to engineering of functional materials
Xiang Yang Liu, National University of Singapore
Biological systems can fabricate special biomaterials,
such as tissues and other biomaterials, with properties far
superior to the original ones found in Nature. For instance,
spider silk, an extremely strong material fabricated by
organisms, outer performs almost any synthetic material in
its combination of strength and elasticity. It is also on
weight basis stronger than steel. It is found that the
superiority in the strength and elasticity of these
biomaterials is attributed to the formation of the
interconnecting structure of nano fibril networks, which is
controlled by a special type of nucleation. The above
progress has enabled us to identify roles for bio substrates
and additives in promoting the formation of self organized
microstructures, so as to provide a guideline for producing
a comparable robust synthetic system. In this contribution,
I will present the architecture of three dimensional
interconnecting self organized nanofiber networks from
separate needle like crystals, based on the above concept of
branching creation by a trace amount of additives (branching
promoters) (a few ppm). We demonstrate that this novel
technique enables us to produce previously unknown self
supporting supramolecular functional materials with tailor
made micro/nano structures, possessing significantly
modified macroscopic properties, by utilizing “useless”
materials. Our results show for the first time that the
formation of the interconnecting 3D self organized network
structure is controlled by a new mechanism, so called
crystallographic mismatch branching mechanism, as opposed to
the conventionally adopted molecular selfassembly
mechanism. Some applications of the functional materials in
tissues engineering, drug delivery, etc. will be given.
« Back...
On continuum models of thin film
epitaxy with biased diffusion: variational properties and
bounds for the dynamic scaling
Bo Li, University of California, San Diego
In epitaxial growth of thin films, often the adatom
diffusion to and from step edges is biased due to the
EhrlichSchwoebel barrier. This kinetic asymmetry can
significantly affect the coarsening rate and dynamic scaling
law of an underlying system. There have been several
continuum models of the nonequilibrium dynamics of thin
film epitaxy after the roughening transition that describe
such an atomistic effect. These models are diffusion
equations for the height profile of film surfaces, and often
possess Liapunov functionals. They have a sequence of
equilibrium solutions with increasing wavelength that are of
lowest "free energy" among surface profiles with the same or
shorter wavelength. The system then evolves in such a way
that it stays always near such equilibria but evolves from
one to another to increase its wavelength and reduce its
energy. The theory developed in this work is along these
lines of thinking and includes two parts: (1) Variational
properties of the free energies, in particular, their
largesystemsize asymptotics, showing the unboundedness of
surface slope and revealing the relation between some of the
models; (2) Rigorous bounds for scaling laws on the
roughness, the rate of increase of surface slope, the rate
of energy dissipation, and the dynamic scaling.
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The mathematics of scientific
computation
Eitan Tadmor, University of Maryland
Before emails and media players, the sole purpose of
computers was to perform scientific computations. That
purpose remains the central task of today’s high performance
computers. Indeed, scientific computation has emerged as one
of the fundamental tools of scientific investigation, and it
has revolutionized the scientific methodology through its
interplay with experiments and theory.
Numerical algorithms are at the heart of this revolution.
They simulate quantitative assembly of different small scale
dynamics and convert it into accurate predictions of large
scale phenomena. It is here that mathematics, modeling and
experiments interact through scientific computation. In this
talk, the speaker will provide a bird’s eye view on the
mathematics behind numerical algorithms. He will review
applications ranging from computational fluid dynamics and
image processing to weather prediction and computational
tomography.
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Projectionfree JacobiDavidson
method for Maxwell's equation
WeiCheng Wang, National Tsing Hua University
The band structure of a photoniccrystal material is
given by the distribution of eigenvalues for the time
harmonic Maxwell's equation. This operator is degenerate
with an enormous null space. Since we are only interested in
nonzero eigenvalues, this null space is strongly attractive
and often becomes a pitfall for the approximate
eigenfunctions of the nonzero eigenvalues. Thus plain
iteration usually converges slowly or does not converge at
all. The conventional treatment to this problem is to employ
an orthogonal project for the approximate eigenfunctions
onto the nonzero eigenspace by way of the Hodge
decomposition at each iteration. Since the Poisson equation
needs to be solved accurately, this projection step
contributes a significant portion of CPU time, especially
for large systems.
In this talk, we review the discrete analogue of the deRham
Theorem associated with certain finite volume
discretizations of Maxwell's operator. We show that there
exists a discrete vector potential for each nonzero
eigenfunction. The crucial step is to derive a correction
equation for the vector potential. By shifting the
correction equation into the vector potential, we can bypass
the projection step and significantly improve the
efficiency.
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Explore 2D Wigner crystal stable near room temperature
Xuesen Wang, National University of Singapore
Periodic structures on surfaces, such as surface
reconstructions, have been used as selfassembled nanoscale
templates to facilitate fabrication of other nanostructures.
Here, we explore the formation of 2D periodic structures by
Wigner crystallization of surface accumulated charge that
can be stable near room temperature. If the surface
segregation of impurity atoms, driven by reduction of
surface energy, is accompanied with charge transfer, the
surface density of impurity and charge will reach an optimal
value in thermal equilibrium due to the balance between
surface energy reduction and Coulomb energy increase. Within
certain charge density range, Wigner crystallization can
occur for the segregated charge on the surface, and
subsequently the segregated atoms may also form a periodic
superstructure. We try to explain the superstructures
observed on Ge/Ru(0001) and other systems with this
scenario.
References:
[1] E. Wigner, Phys. Rev. 46, 1002 (1934).
[2] C.C. Grimes, G. Adams, Phys. Rev. Lett. 42, 795
(1979).
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A quasicontinuum approximation for
material problems and its analysis
Ping Lin, National University of Singapore
In many applications materials are modeled by a large
number of particles (or atoms) where any one of particles
interacting with all others. The computational cost is very
high since the number of atoms is huge. Recently much
attention has been paid to a socalled quasicontinuum (QC)
method which is a mixed atomistic/continuum model.
The QC method solves a fully atomistic problem in regions
where the material contains defects (or larger deformation
gradients), but used continuum finite elements to
effectively integrate out the majority of the atomistic
degrees of freedom in regions where deformation gradients
are small. However, numerical analysis is still at its
infancy. In this talk we will conduct a convengence analysis
of the QC method in the case that there is no defect or that
the defect region is small. The difference of our analysis
form conventional one is that our exact solution is not a
solution of a continuous partial diffeeential equation but a
discrete lattice scale solution which is not approximately
related to any conventional partial differential equation.
We will consider both one dimensional and two dimensional
cases.
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Firstprinciples study of highk
oxide – Si and metal silicide – strained Si interfaces
Yuanping Feng, National University of Singapore
Firstprinciples total energy calculations were used to
study interfaces of highk oxide and silicon, metal silicide
and strained silicon. Various model interfaces satisfying
the general bonding rules were considered. The interface
formation energies, band offsets, and Schottky barrier
heights were evaluated. We focus in particular the strain
mode and interface structure effects on band offsets of ZrO2
– Si interfaces and strain effects on silicide – Si
interfaces. Our studies show possibility of atomic control
of interface structures by altering the chemical
environment. Band offsets of ZrO2 – Si interfaces were found
strongly dependent on the strain modes and interface
structures. These results suggest that in epitaxial growth
of ZrO2 on Si for gate dielectric applications, the chemical
environment should be well controlled to get reproducible
band offsets. It was also found that strain affects the
Schottky barrier heights of the silicide – Si interfaces,
providing important guidance for the uptodate strainedSi
device fabrication.
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Numerical analysis of
coarsegrained stochastic lattice dynamics
Petr Plechac, University of Warwick
Coupling microscopic simulations with description at
larger scales has been one of the principal tasks in many
areas of computational modelling. We discuss some general
mathematical issues arising in problems where the
microscopic Markov process is approximated by a hierarchy of
coarsegrained processes. We provide both analytical and
numerical evidence that the hierarchy of the coarse models
is built in a systematic way that allows for the error
control of quantities that may also depend on the path. We
also demonstrate that coarsegrained MC leads to significant
CPU speed up of simulations of metastable phenomena, e.g.,
estimation of switching times or nucleation of new phases.
Numerical evidence guided by analytical results suggests
that CGMC probes energy landscape in pathwise agreement to
MC simulations at the microscopic level. The presented
results are joint work with M. Katsoulakis (UMASS), A.
Sopasakis (UMASS).
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