Imaging, September 2005June 2006
Abstracts
IMA Workshop:
June 811, 2005
Cameron F.
Abrams (Department of Chemical and Biological
Engineering, Drexel University)
Systematic coarsegraining and concurrent multiresolution
simulation of molecular liquids
Systematic coarsegraining is a class of techniques in which
judicious
approximations are invoked to coarsen a system of many degrees
of
freedom onto one with relatively fewer in a statistical
mechanically
consistent way. Collective variables are chosen and the
resulting
potentials of mean force at a given thermodynamic state are
derived
from the atomic level potential energy hypersurface. These
techniques
are briefly reviewed in their connection with the molecular
simulation
of specific synthetic polymers. Further developments
incorporating
onthefly coarsegraining into concurrent multiresolution
simulations
of simple molecular liquids are also discussed.
Silas Alben
(Division of Engineering and Applied Sciences, Harvard
University)
A mechanical model of the teleost fin ray
This work considers the mechanics of fish swimming. In
collaboration with
the Lauder lab at Harvard, we are studying the structure of
fish fin rays.
Approximately half of all fish species utilize the same basic
structurebody segments linked by a collagen networkto
transduce fin
ray shape and motion from a given input force. We present a
simple coupled
elastica model which uses only geometry and a single elastic
constant to
obtain the scalings of forces and displacements.
Marcel
Arndt (Institut für Numerische Simulation, University of
Bonn) http://wissrech.ins.unibonn.de/people/arndt.html
Higher order gradient continuum description of atomistic
models for
crystalline solids
We propose an upscaling scheme for the passage from
atomistic to
continuum mechanical models for crystalline solids. It is
based on an
expansion of the deformation function up to a given order and
leads to a
continuum mechanical model which involves higher order
gradients. The
resulting model is an approximation of the atomistic system for
a fixed
and finite number of atoms within the quasicontinuum regime.
The
higher order terms allow the description of the microscopic
material
properties to a higher extent than commonly used continuum
mechanical
models. In particular, the discreteness effects of the
underlying
atomistic model are captured.
Our upscaling technique is compared to other upscaling schemes
and
analyzed with respect to wellposedness and the asymptotic
scale
behavior. The qualitative properties of our technique are
numerically
studied for the model problem of a onedimensional atomic
chain. The
approach is then applied to the physically more relevant
threedimensional example of a silicon crystal. The resulting
approximation properties are studied.
Paul Atzberger
(Department of Mathematical Sciences,
Rensselaer Polytechnic Institute)
A stochastic immersed boundary method for biological fluid
dynamics at microscopic length scales
With advances in cell and molecular biology there
is an increasing interest in modeling microscopic
systems at a coarse level where methods such as
molecular dynamics become infeasible as a
consequence of the wide range of active length and
time scales. An alternative approach is to use a
continuum description where neglected degrees of
freedom of the system are accounted for by an
effective model either through averaging or an
appropriate stochastic model. In this poster an
extension of the immersed boundary method
[1] is presented for this purpose
which includes appropriate stochastic forcing to
model thermal fluctuations of the fluid and
immersed structures. A stochastic numerical method is
presented which deals with stiffness in the system
by carefully handling statistical contributions of
the dynamics of the fluid and immersed structures
over long time steps. A number of physical
checks are presented for the method which show for example
that immersed particles diffuse with the appropriate scaling
in the physical parameters and have the correct equilibrium
statistics. The method is also demonstrated to
reproduce wellknown hydrodynamic effects such as the 3/2
decay in the tail of the velocity autocorelation function of
a Brownian particle. In conclusion, numerical results are
presented for specific applications to polymers and
membranes.
[1] Peskin, C.S (2002), The immersed boundary method, Acta
Numerica, 11,
pp. 139.
Marian
Bocea (Department of Mathematics, University of Utah)
A nonlinear membrane model by means of Young measures
The integral representation of a relaxed functional arising
in the derivation of
a nonlinear membrane model is obtained in terms of a special
class of Young
measures generated by sequences of scaled gradients. Algebraic
and analytical
conditions on parametrized probability measures both necessary
and sufficient
to guarantee that they belong to this class are identified, in
the spirit of
Kinderlehrer and Pedregal's characterization of gradient Young
measures. Joint
work with Irene Fonseca.
Andrea
Braides (Dipartimento di Matematica, II Universita degli
studi di Rome (Tor Vergata)) http://mat.uniroma2.it/~braides
Simple lattice systems with complex
macroscopic description
I will present a homogenized description of the simplest
lattice systems,
where the lattice energy depends on a variable that possesses
only two
states (without loss of generality we may take these two values
as
+1 and 1). The overall behavior of the system, as the number
of nodes
increases can be described, upon scaling, by a continuum
expansion.
in terms of Gammaconvergence. In the limit energies we may
recognize
bulk terms, interfacial energies, antiphase boundaries and
microscopical
oscillations, depending on the lattice parameters and shape.
Some comments on the random case will also be given.
Antonio
DiCarlo
(Department of Studies on Structures, Università degli
Studi "Roma
Tre")
Microtwists &
nanodefects Poster: pdf Paper: pdf
Joint work with Luciano
Teresi.
Prototype nanoelectromechanical devices
incorporating individual
multiwall carbon nanotubes as torsion bars or rotary bearings
have
been fabricated and tested by various groups. Typical length
scales
are: 1 micron for the overall span, 25 nm for the diameter, 0.5
nm
for the interwall gap. The experimental evidence collected so
far is
puzzling, pointing out a need for a better understanding of the
interwall mechanical coupling mechanisms. We speculate that the
basic
mechanism behind progressive interwall coupling is the
formation of
bridging defects, that is, covalent links between adjacent
walls,
triggered by inward migration of chromium atoms (which are
evaporated
onto the outer wall of the nanotube when fabricating the
device).
Antoine
Gloria (CERMICS  ENPC)
A direct approach to numerical homogenization in finite
elasticity
We present a direct approach to tackle the numerical
simulation of a homogenized
problem in nonlinear elasticity at finite strain. We provide an
approximation
result for this problem and derive an error estimate in the
particular case of
convex energy densities. We have implemented this approach in a
nonlinear
elasticity solver and performed several numerical tests on
idealized rubber
foams.
Claude Le
Bris
(CERMICS,
Ecole Nationale des Ponts et Chaussees (ENPC))
http://cermics.enpc.fr/~lebris/home.html
Inserting computational chemistry in materials science: a
guided tour
We will introduce the audience to the basic modelling in
computational quantum chemistry. We will overview the
mathematical and
numerical aspects, pointing out some recent works. A special
emphasis will be
laid upon the approaches used for large size systems, and,
beyond, the
different methods coupling quantum chemistry models and models
of materials
science.
Xiantao Li
(Institute for Mathematics and its Applications, University of
Minnesota)
Boundary conditions for molecular dynamics
At the atomic scale, crystalline solids can be modelled by
Molecular
dynamics (MD), which provides a very useful tool to study
crystal
structure and defect dynamics. MD simulations can be conducted
either in
isolation, with some experimental loading condition applied to
its
boundary, or they can be coupled with a continuum model
replacing all the
atoms outside of the atomistic region. In both cases, a key
issue is to
eliminate the reflection of phonons at the boundary or the
continuum/atomistic interface.
In this talk, I will present a variational formulation for
constructing
boundary conditions that suppress phonon reflections. Local
boundary
conditions, which are practical for computational purpose, are
obtained
from this formulation. A few examples, including 1D chain, 2D
triangular
lattice, 3D BCC lattice and Graphene (complex lattice) will be
given.
Finally we apply these boundary conditions to fracture
simulations.
This is joint work with Weinan E (Princeton).
Dionisios
Margetis (Department of Mathematics, Massachusetts Institute of Technology)
Continuum approach to crystal surface
morphology evolution
The design of small devices with novel properties relies on the
synthesis
and stability of nanoscale surface structures. At temperatures
below
roughening crystal surfaces have flat, macroscopic regions
known as
"facets'' and evolve via the motion of interacting atomic
steps. This
talk describes macroscopic evolution laws on the basis
of the microscopic step motion. First, continuum evolution
equations
in (2+1) dimensions are derived from kinetic considerations;
the surface height profile outside facets satisfies a nonlinear
PDE
that accounts for mass fluxes parallel and transverse to steps
via an
appropriate tensor mobility. Second, the PDE is tested through
comparisons
of analytical predictions with experimental results and
numerical simulations
that follow the motion of individual steps. The challenging
problem of
suitable boundary conditions at the boundaries of facets is
discussed.
Kevin W.
Mclaughlin (Department of Chemistry, University of
Wisconsin  River Falls)
On the use of topological indices in quantitative
structureproperty relationships for macromolecules
Chemical graph theory has been extensively applied to
predicting
the physical properties of small molecules through
quantitativestructure
property relationships (QSPR). This has been accomplished by
demonstrating
strong correlations between physical properties and one or more
topological
indices. Extending the application of topological indices to
macromolecules can lead to potential problems for such models.
Using the
Hosoya index, we illustrate both the problems associated with
degeneracy of
a topological index as molecular size and complexity increases,
and what
limits should be placed on the development of such models.
Barbara
Niethammer (Institut fuer Mathematik,
Humboldt Universitaet zu Berlin)
The effect of screening and correlations in
Ostwald Ripening
The classical theory by Lifshitz, Slyozov and Wagner describes
diffusion
limited coarsening of particles in the limit of
vanishing volume fraction.
Due to several shortcomingis of the LSW theory
first order corrections in terms of the volume
fraction should be taken into account.
We
discuss a new method to effeciently identify firstorder
corrections
in a statistically homogeneous system.
The key idea is to relate the full system of particles to
systems
where a finite number of particles has been removed.
This method allows to decouple screening and correlation
effects
and allows to effiently evaluate conditional expected values of
the particle
growth rates.
Hans Christian
Öttinger
(ETH Zürich, Department of Materials,
Institute of Polymers,)
Thermodynamic framework for systematic
coarsegraining of atomistic models for fluids
For this talk, I distinguish between two fundamentally
different simulation approaches in materials science,
"bruteforce simulations"
and "thermodynamically guided simulations." Bruteforce
simulations can be thought of as computer experiments mimicking
the physical
situation of interest directly on a computer; thermodynamically
guided simulations rely on a nonequilibrium statistical
ensemble
containing the variables of some coarsegrained description of
the system of interest. The availability of an appropriate
coarsegrained
level of description is thus crucial for thermodynamically
guided simulations and should be considered as a reasonable
price to pay for
bridging widely separated time scales (and deeper
understanding).
The above remarks are elaborated in the context of molecular dynamics simulations
of Lennard Jones fluids and of polymer melts. It is shown how simulations based
on nonequilibrium ensembles can help to bridge the wide range of time scales
from monomer motions to polymer processing. The importance of coarsegrained
models for specifying an ensemble and for identifying suitable quantities of
interest is illustrated.
Harald
Pleiner (Max Planck Institute for Polymer Research)
General nonlinear hydrodynamic description of nonNewtonian
fluids
We review conventional constitutive equations for nonNewtonian
fluids
from a hydrodynamic point of view. Using general thermodynamic
and
symmetry arguments and applying valid physical principles we
describe
viscoelasticity by setting up nonlinear dynamic equations
either for a
relaxing (Eulerian) strain tensor or for a transient
orientational
order parameter tensor. This covers the usual nonNewtonian
effects,
like shear thinning, strain hardening, stress overshoot, normal
stress
differences and non exponential stress relaxation. In both
cases an
effective dynamic equation for the stress tensor can be derived
in
terms of a power series and compared with conventional
nonNewtonian
rheological models. It is more general in structure than those,
comprises most, restricts some, and discards a few of them.
In addition, we generalize this approach into a 2fluid
description for
multicomponent fluids, which is appropriate, when the relative
velocity of the different components is relaxing slowly.
Special
emphasis is laid on nonlinearities involving velocities that
are
governed by symmetry and other general invariance principles.
It is
shown that the proper velocities, with which the dynamic
quantities are
transported and convected, cannot be chosen at will, since
there are
subtle relations among them. Within allowed combinations the
convective
velocities are generally material dependent and not fixed by
general
principles. The socalled stress division problem, i.e. how the
total
stress is distributed between the different components, is
shown to
depend partially on the choice of the convected velocities, but
is
otherwise also material dependent. A set of reasonably
simplified
equations is given for viscoelastic fluids, polymeric gels, and
ferrofluids focusing on an effective concentration dynamics
that may be
used for comparison with experiments.
Tiezheng
Qian (Department of Mathematics,
The Hong Kong University of Science and Technology)
http://www.math.ust.hk/~maqian
Slip boundary condition for the moving contact
line in immiscible
twophase flows
From extensive molecular dynamics simulations on immiscible
twophase
flows, we find the relative slipping between the fluids and the
solid wall
everywhere to follow the generalized Navier boundary condition,
in which
the amount of slipping is proportional to the sum of tangential
viscous
stress and the uncompensated Young stress. The latter arises
from the
deviation of the fluidfluid interface from its static
configuration. We
give a continuum formulation of the immiscible flow
hydrodynamics,
comprising the generalized Navier boundary condition, the
NavierStokes
equation, and the CahnHilliard interfacial free energy. Our
hydrodynamic
model yields interfacial and velocity profiles matching those
from the
molecular dynamics simulations at the molecularscale vicinity
of the
contact line.
Weiqing Ren
(Mathematics department and PACM, Princeton University)
Multiscale modeling of contact line dynamics
The moving contact line (MCL) problem is analyzed using the
multiscale methods developed recently by Ren and E.
It is wellknown that the noslip boundary condition results in
a nonintegrable singularity in the stress at the contact line.
Numerous empirical slip models have been proposed to remove
the stress singularity. For the multiscale method,
we solve the NavierStokes (NS) equations for
the macroscale flow field, and
calculate the needed boundary condition, e.g., the shear
stress,
at the contact line region based on molecular dynamics (MD).
In the talk, we will discuss the details of the multiscale
method,
the validation study, and the results on large scale contact
line problem.
James P.
Sethna (Laboratory of Atomic and Solid State Physics
(LASSP), Cornell University) http://www.lassp.cornell.edu/sethna/sethna.html
Estimating systematic errors: sloppy
models
Joint work with Søren
Fredericksen, Karsten W.
Jacobsen, and Kevin
S. Brown
Science is filled with multiparameter models that must be
fit to observations. An ecosystem has many interacting
species, a cell has interacting proteins and genes, and
a material has many atoms whose forces are governed by
quantummechanical electronic calculations. A key question
for these models is when we can trust their predictions:
usually only wisdom and experience can judge for which
problems a given model will likely be reliable. One source
of unreliability in these models is that they are sloppy:
the parameters are illdetermined by the data, with enormous
ranges giving roughly equivalent fits. These parameters
giving roughly equivalent fits, however, do not yield the
same predictions! By using an ensemble of good parameter sets,
we have been able to produce `sloppy model' estimates of the
errors in one particular system: the interatomic potential
for the element Molybdenum. Our error estimates capture most
of the systematic error in this system, for three different
forms of the interatomic potential.
Vivek Shenoy
(Division of Engineering, Brown University)
Selfassembly and shape transitions of epitaxial
nanowires and strained monolayer islands
Several interesting shape transitions have been recently
observed during the growth of
submonolayer islands on latticemismatched substrates. These
shapes, which allow
relaxation of mismatch strain, include nanowires that are
elongated along certain
crystallographic directions with widths in the singledigit
nanometer range, shapes that
show concave boundaries in equilibrium and formation of highly
ramified or branched
structures during growth. In this talk, I will present a sharp
interface model and a phasefield
model to study the shapes (kinetic and equilibrium) of
individual islands and
stability and coarsening kinetics of monolayer island arrays.
Our model includes the
kinetics of adatom diffusion on the terraces and island edges,
attachment kinetics to
islands, substratemediated elastic interactions between the
islands and anisotropies
associated with the creation of island edges. Particular
emphasis will be given to selfassembled
growth of regular arrays of epitaxial nanowires that have
potential applications
as nonlithographically fabricated interconnects.
Peter
Smereka (Department of Mathematics, University of
Michigan) http://www.math.lsa.umich.edu/~psmereka/
Computations of strained heteroepitaxy in 3 dimensions
using kinetic Monte Carlo
The growth of strained heteroepitaxial films in 3 dimensions
using a
SolidonSolid model is discussed. Elastic effects are included
by
using a ball and spring model. The system is evolved in time
using a
kinetic Monte Carlo method. Discrete models of this form
naturally
include nanoscale effects, such as nucleation, which are
difficult to
incorporate in continuum models. On the other hand, it is more
computationally intensive to use these discrete models simulate
film
growth on experimentally relevant length scales. This talk will
discuss
some of the computational challenges and approaches we have
developed
for simulation of heteroepitaxy. In addition, some preliminary
results
of film growth will be presented which shows that when the
elastic
effects are small the film grows in a layerbylayer fashion.
However,
when the elastic effects become strong we observe mound
formation
(selfassembled quantum dots).
Lev
Steinberg (Department of Mathematics, University of
Puerto Rico)
Mesoelastic deformation with strain singularities
The poster will present our study on the continuous
distribution
of singularities in the strain field, which we describe in
terms of surface
densities and fluxes. We define the mass mesodensity tensor and
induce the
constitutive relationships between the strain singularity
current and the
linear mesomomentum. Based on the modification of PeachKoehler
formula we
consider the constitutive relation between the line mesostress
tensor and
the strain singularities density. These constitutive
relationships allow us
to model stresses in mesoelastic materials.
Erik van
der Giessen (Department of Applied Physics,
Micromechanics of Materials Group, University of Groningen)
Dislocation field theory in 2D: formulation
and validation
Plastic deformation of crystalline metals at the scale of
micrometers and
smaller is often size dependent, particularly in the presence
of strain
gradients. Standard local theories are not able to capture
these size
effects because they lack a material length scale. Therefore,
numerous
nonlocal models have been proposed during the last decade, the
majority of
which are phenomenological straingradient theories. An
alternative
description will be presented here, which is termed Dislocation
Field
Theory (DFT). It is based on a rigorous statistical averaging
of the
dislocation motion of edge dislocations gliding on a single
slip system.
The resulting field equations are then coupled to a crystal
plasticity
model using Orowan's relation. The resulting DFT is
subsequently
generalized to multipe slip, albeit in two dimensions. The main
characteristic that distinguishes DFT from other plastic
straingradient
theories is that the material length scale the average
dislocation
spacing is not a material constant but evolves with
deformation.
It will be shown how the few material parameters in DFT are
fitted to
discrete dislocation results for one particular problem, and
how the
subsequent predictions of the theory for other problems
compares with
discrete dislocation simulations. The problems addressed
include shearing
of a model composite; constrained shear; bending; and stress
relaxation in
a thin film.
Erik van
der Giessen (Department of Applied Physics,
Micromechanics of Materials Group, University of Groningen)
Nonaffine deformations of networks of
semiflexible polymers
There is a deep interest in the mechanical response of
biological
tissues and gels in view of the importance for biological
functions
such as cell motility and mechanotransduction. Many
networklike
biological tissues respond to deformation by exhibiting an
increasing
stiffness. The current paradigm is that stiffening is primarily
due to
the stiffening of the filaments themselves: As the filament is
stretched, the mean amplitude of transverse thermal undulations
reduces
and, as a consequence, the stiffness increases; in the limit
that the
filament is pulled straight, all subsequent axial deformation
would
have to originate from axial straining of the chain, but at an
enormous energy cost. Along with the assumption that the
initially
randomly oriented filaments deform in an affine manner, this
leads to
drastic stiffening once the slack has been pulled out.
Networks of discrete filaments, however, reveal another
explanation for
stiffening, which lies in the network rather than in its
constituents.
During deformation, the filaments rotate in the direction of
straining, thus inducing a transition from a bendingdominated
response at small strains to a large strain response that is
controlled by
stretching of aligned filaments. Thus, the network response is
not at all
affine and that the presence of thermal undulations merely
postpones the
stiffening transition.
Qi Wang
(Department of Mathematics,
Florida State University) http://www.math.fsu.edu/~wang
A paradigm of kinetic theories for suspensions
and nematic polymers
In this talk, I will present a systematic approach to the
development of
kinetic theories for suspensions and nematic polymers ranging
from rigid
bodies to deformable ones. The theories account for the
molecular
configuration of the suspensions and polymers. For example, the
kinetic
theory for biaxial liquid crystal polymers account for the
broken symmetry
at the molecular level in the transport equation for the number
density
function as well as the mesoscopic stress tensor calculation.
We will
discuss the connection with the existing kinetic theories and
give some
examples in simple flows.
Hong Zhou
(Department of Applied Mathematics,
Naval Postgraduate School)
Anchoring distortions coupled with plane Couette and
Poiseuille flows of natmic polymers
The aim of this work is to model and simulate
Processinginduced heterogeneity in rigid, rodlike nematic
polymers
In viscous solvents. We employ a mesoscopic orientation tensor
model
due to
Doi, Marrucci and Greco which extends the small molecule,
liquid crystal
theory of LeslieFrank to nematic polymers. We focus
simulations in the
regime of weak flow and strong distortional elasticity to
expose the
effects due to wall anchoring conflicts. A remarkably simple
diagnostic
emerges in this physical parameter regimes, in which salient
morphology
features are controlled by the amplitude and sign of the
difference in
plate anchoring angles of the director field at the two plates.
