Effective Theories for Materials and Macromolecules

Cameron F. Abrams (Department of Chemical and Biological Engineering, Drexel University)

Systematic coarse-graining and concurrent multiresolution simulation of molecular liquids

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.

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 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.

Marcel Arndt (Institut für Numerische Simulation, University of Bonn) http://wissrech.ins.uni-bonn.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 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.

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 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.

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 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.

Antonio DiCarlo (Department of Studies on Structures, Università degli Studi "Roma Tre")

Microtwists & nanodefects
Poster:  pdf   Paper:  pdf
Note: This paper is a (slightly corrected) reprint of the paper: A. Di Carlo, M. Monteferrante, P. Podio-Guidugli, V. Sansalone, L. Teresi: How (and why) twisting cycles make individual MWCNTs stiffer. In: Electronic Properties of Synthetic Nanostructures, Proceedings of the XVIII International Winterschool/Euroconference on Electronic Properties of Novel Materials held in Kirchberg, Tirol, Austria (March 6-13, 2004), pp. 355-358 (H. Kuzmany et al., eds.), AIP Conf. Proc. 723, American Institute of Physics, Melville (2004).

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 structure-property relationships for macromolecules

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.

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 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.

Hans Christian Öttinger (ETH Zürich, Department of Materials, Institute of Polymers,)

Thermodynamic framework for systematic coarse-graining of atomistic models for fluids

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.

Harald Pleiner (Max Planck Institute for Polymer Research)

General nonlinear hydrodynamic description of non-Newtonian fluids

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.

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 two-phase flows

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.

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 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.

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

Vivek Shenoy (Division of Engineering, Brown University)

Self-assembly 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 lattice-mismatched substrates. These shapes, which allow relaxation of mismatch strain, include nanowires that are elongated along certain crystallographic directions with widths in the single-digit 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, substrate-mediated 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 non-lithographically 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 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).

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 Peach-Koehler 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 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.

Erik van der Giessen (Department of Applied Physics, Micromechanics of Materials Group, University of Groningen)

Non-affine deformations of networks of semi-flexible 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 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.

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 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.