June 8 - 11, 2005
Systematic coarse-graining 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 on-the-fly coarse-graining into concurrent multiresolution simulations of simple molecular liquids are also discussed.
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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
structure--body segments linked by a collagen network--to 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.
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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 quasi-continuum 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 well-posedness and the asymptotic scale
behavior. The qualitative properties of our technique are numerically
studied for the model problem of a one-dimensional atomic chain. The
approach is then applied to the physically more relevant
three-dimensional example of a silicon crystal. The resulting
approximation properties are studied.
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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 well-known 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. 1-39.
Further information can be found on my
website at
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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.
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 Gamma-convergence. In the limit energies we may recognize bulk terms, interfacial energies, anti-phase boundaries and microscopical oscillations, depending on the lattice parameters and shape. Some comments on the random case will also be given.
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Abstract:
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).
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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.
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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.
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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).
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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.
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Chemical graph theory has been extensively applied to predicting
the physical properties of small molecules through quantitative-structure
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.
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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 first-order 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.
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For this talk, I distinguish between two fundamentally different simulation approaches in materials science, "brute-force simulations" and "thermodynamically guided simulations." Brute-force 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 coarse-grained description of the system of interest. The availability of an appropriate coarse-grained 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 coarse-grained models for specifying an ensemble and for identifying suitable quantities of interest is illustrated.
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We review conventional constitutive equations for non-Newtonian 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 non-Newtonian 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 non-Newtonian
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 2-fluid description for
multi-component 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 so-called 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.
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From extensive molecular dynamics simulations on immiscible two-phase 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 fluid-fluid interface from its static configuration. We give a continuum formulation of the immiscible flow hydrodynamics, comprising the generalized Navier boundary condition, the Navier-Stokes equation, and the Cahn-Hilliard interfacial free energy. Our hydrodynamic model yields interfacial and velocity profiles matching those from the molecular dynamics simulations at the molecular-scale vicinity of the contact line.
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From extensive molecular dynamics simulations on immiscible two-phase 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 fluid-fluid interface from its static configuration. We give a continuum formulation of the immiscible flow hydrodynamics, comprising the generalized Navier boundary condition, the Navier-Stokes equation, and the Cahn-Hilliard interfacial free energy. Our hydrodynamic model yields interfacial and velocity profiles matching those from the molecular dynamics simulations at the molecular-scale vicinity of the contact line.
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The moving contact line (MCL) problem is analyzed using the
multiscale methods developed recently by Ren and E.
It is well-known that the no-slip boundary condition results in
a non-integrable 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 Navier-Stokes (NS) equations for
the macro-scale 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.
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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 quantum-mechanical 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 ill-determined 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.
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The growth of strained heteroepitaxial films in 3 dimensions using a
Solid-on-Solid 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 layer-by-layer fashion. However,
when the elastic effects become strong we observe mound formation
(self-assembled quantum dots).
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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 strain-gradient 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 strain-gradient 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.
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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 network-like
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 bending-dominated
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.
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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.
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The aim of this work is to model and simulate
Processing-induced heterogeneity in rigid, rod-like 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 Leslie-Frank 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.
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