Program on Nanoscale Material Interfaces: Experiment, Theory and Simulation - IMS

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 (magneto-rheological fluids), ER (electro-rheological) fluids, liquid crystalline polymers and polymer-particle 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.

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 no-slip 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. Steady-state velocity fields from continuum calculations are in quantitative agreement with those from MD simulations, from the molecular-scale 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 partial-slip 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 partial-slip 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 power-law region is in accordance with the asymptotic 1/r variation of stress and the Navier boundary condition.

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 Hai-jun Wu of Nanjing University (China).

Atomic-scale reconstructions on metal and semiconductor surfaces
Andrew T. S. Wee, National University of Singapore

Download/View PDF

Steady state morphology and nano-pattern on Si(001) formed during Epitaxy
Xue-Sen 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 near-equilibrium 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 double-layer steps to nearly rectangular nanometer-scale multi-layer vacancy islands (pits) isolated by connected smooth area. The morphological characteristics in different growth regimes and the transition between them are discussed. The nano-patterns obtained in steady-state epitaxy can be used as templates for fabrication of other nano-structures.

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 three-terminal 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), cond-mat/0407093.

[2] B Li, L Wang and G Casati, “The thermal transistor: A switch and a modulator/amplifier for heat current”. Submitted for publication.

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.

Off lattice kinetic-Monte-Carlo
Tim P. Schulze, University of Tennessee

Simulations of nano-scale 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 Monte-Carlo (KMC) methods can be thought of as being derived from MD and can be generalized into off-lattice 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.

Sharp-interface theory for nematic-isotropic 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 radially-oriented 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 isotropic-to-nematic phase transitions, this simplified model yields insight concerning the occurrence and stability of isotropic cores for hedgehog defects in liquid crystals.

A level-set method for Epitaxial growth and self-organization of quantum dots
Christian Ratsch, University of California, Los Angeles

We have developed an island dynamics model that employs the level-set technique to describe epitaxial growth. The motion of the island boundaries is described by the evolution of a continuous level-set 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 mean-field 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 re-attachment 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 self-organization 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.

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 one-dimensional, 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 power-law coarsening with slope selection for a range of values of the model parameters. In contrast to previously proposed models of mound formation, the Ehrlich-Schwoebel step-edge 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 higher-order gradient nonlinearities, these models exhibit a first-order dynamical phase transition from a kinetically rough self-affine 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 power-law 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 step-edge barrier are present will also be presented.

Modelling the self-organized growth of quantum dots
Yong-Wei 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 self-assembled 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 self-assembly, 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 Stranski-Kranstanov growth through large-scale computer simulations. Our attention is focused on island roughening kinetics through stress-driven surface diffusion. Our simulations reveal many interesting features of island formation and coarsening kinetics. In particular, we have found that a bell-shaped 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 faster-than-linear-rate. 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 self-organized 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.

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 kinetic-hydrodynamic approach which keeps track of the dynamics of the orientation-position distribution functions of the rod-like 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 Ericksen-Leslie theory. The Leslie viscosities are derived in terms of molecular parameters, the Ericksen stress are derived by the body force.

Morphological evolution of thin films during epitaxy; large-scale 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 semi-implicit finite element method is used in the parametric setting. We computationally investigate the coarsening in the morphology of thin films, show their self-similarity and predict scaling laws.

Islands in the stream: complex shape evolution driven by surface electromigration
Joachim Krug, Universität zu Köln

The shape evolution of two-dimensional 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 time-dependent 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.

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 Ginzburg-Landau 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 meso-scale Ginzburg-Landau model being our emphasis. Some recent analytical and numerical results will be surveyed. Connections to other relevant problems such as the vortices in Bose-Einstein 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.

Coarsening dynamics and continuum modeling for epitaxial growth of thin films
Jian-Guo 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 well-poseness 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.

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 spatio-temporal physical and geometric scales, together with nonlocal phenomena and a nonconvex side-constraint 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).

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 Pierre-Louis, Saul D. Cohen, Robert D. Schroll, and experimentalists Ellen D. Williams and J.E. Reutt-Robey at UM, M. Giesen and H. Ibach at FZ-Jülich, and J.-J. Métois at Marseilles

We discuss the terrace-width 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 terrace-step-kink model. The TWD of this model is analogous to the distribution of spacings between repelling fermions in 1 dimension, leading to a mean-field solution (for non-weak repulsions) using an equivalence to simple problems in elementary quantum mechanics. Analogies with the distribution of energy spacings invite application of results from random-matrix theory [1] that describe universal properties of fluctuations. In particular, we generalize the gamma-like "Wigner surmise," developed for a few special cases, to treat arbitrary repulsion strength as a superior alternative to various Gaussian approaches [2-6]. We use Monte Carlo simulations as well as transfer-matrix calculations, both with finite-size scaling, to test this picture [3-5]. The step-continuum 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 Fokker-Planck 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 00-80008, and partially by NSF Grant EEC-00-85604. My papers are downloadable from http://www2.physics.umd.edu/~einstein/

1. T. Guhr, A. Müller-Groeling, and H.A. Weidenmüller, Phys. Rep. 299 (1998) 189.
2. T.L. Einstein and O. Pierre-Louis, 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. Pierre-Louis, 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 [cond-mat/0306347].
7. A. A Drǎgulescu and V. M Yakovenko, Quantitative Finance 2 (2002) 443.
8. A. Pimpinelli and T.L. Einstein, draft preprint.

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 2-phase 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 2-phase 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.

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 three-dimensional 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 dislocation-particle 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.

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 multi-component 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 dumb-bell model for polymeric materials, the magnetic field in magneto-hydrodynamic 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.

A unified treatment of current-induced 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 nano-scale 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 sharp-step model provides a unified and simple interpretation of many experimental results for current-induced 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.

A second-order γ-model BGK scheme for multimaterial compressible flows
Song Jiang, Institute of Applied Physics and Computational Mathematics, China

We present a second-order γ-model BGK scheme for compressible multimaterial flows, which extends the authors' earlier work on a first-order scheme [Int. J. Numer. Meth. Fluids 46 (2004), 163-182]. The scheme is based on the incorporation of a conservative γ-model scheme given in [R. Abgrall, J. Comput. Phys. 125 (1996), 150-160] into the gas kinetic BGK scheme [K.H. Prendergast and K. Xu, J. Comput. Phys. 109 (1993), 53-66; and 114 (1994),9-17]. Numerical examples validate the scheme in numerical simulations of compressible multifluids.

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, so-called 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.

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 γ-convergence, the existence and stability of K-interface 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 1-D global minimizer is near the borderline of 3-D stability. Finally using bifurcation analysis, we find wriggled lamellar solutions of the Euler-Lagrange 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.

Heat conduction in one-dimensional systems -- molecular dynamics and mode-coupling theory
Jian-Sheng Wang, National University of Singapore

We begin with a brief review of the studies of one-dimensional 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 one-dimensional chain of particles with longitudinal as well as transverse motions. The particles are connected by two-dimensional harmonic springs together with bending angle interactions. The problem is analyzed by mode-coupling theory and compared with molecular dynamics results. We find excellent agreement for the damping of modes between mode-coupling 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.

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.

Simulation of micro flows by lattice Boltzmann method
Chang Shu, National University of Singapore

Micro-electromechanical 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 micro-level, 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 Navier-Stokes 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 Navier-Stokes 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 fluid-solid interactions on the boundary, we present a diffuse-scattering 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 two-dimensional (2D) pressure-driven isothermal micro-channel flows, the 2D shear-driven isothermal micro flows and the thin-film 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

Atomistic and continuum modeling of homoepitaxial thin film growth
Jim Evans, Iowa State University

Homoepitaxial thin film growth reveals a rich variety of far-from-equilibrium 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 lattice-gas models analyzed by KMC simulation have been most successful, but we also discuss 2D continuum formulations (BCF-based, level-set, stochastic geometry-based) 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 mean-field 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 geometry-based 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 100-1000 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

Macroscopic simulation of micro(nano)-structures in finite-strain elastoplasticity
Carsten Carstensen, Humboldt-Universität zu Berlin

The computer simulation of the evolution of microstructures in finite-strain elastoplasticity requires a time-space discretization. The resulting mathematical model of each time-step 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 single-slip 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 phase-transition.

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.

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 1-2D as well as 2-3D models of HF, which involve a coupled system of degenerate nonlinear integro-partial differential equations as well as a free boundary. We demonstrate, via re-scaling the 1-2D 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 2-3D 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.

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.

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 liquid-solid 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 particle-particle contact. This is possible mainly owing to (1) the amorphous nature of the core nanoparticles, which makes the urea coating non-crystalline, 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 fluid-fluid 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 non-slip 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, 727-730 (2003).
[2] Power-Law Slip Profile of the Moving Contact Line in Two-Phase Immiscible Flows, T. Qian, X. P. Wang and Ping Sheng, Phys. Rev. Lett. 93, 094501-094504 (2004).
[3] Molecular Scale Contact Line Hydrodynamics of Immiscible Flows, T. Qian, X. P. Wang and Ping Sheng, Phys. Rev. E68, 016306 (2003).

Mathematical models of interface dynamics and coarsening
Robert Pego, Carnegie Mellon University

Download/View PDF file

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 mean-field 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 Levy-Khintchine 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.

Efficient and accurate numerical schemes for phase-field 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 Cahn-Hilliard, Allen-Cahn and Navier-Stokes equations. The spatial discretizations will be based on the spectral-Galerkin method while the temporal discretizations will be based on a combination of splitting and semi-implicit 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 phase-field models for numerical simulations of microstructural evolution and of complex fluids.

The vortex dynamics of a Ginzburg-Landau 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 Ginzburg-Landau system with pinning effect.

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 in-phase 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 in-phase step wandering with step-down 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.

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 high-order nonlocal/local artificial boundary conditions (ABCs) for second-order 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 Navier-Stokes for incompressible viscous flow aroung obstacles. Numerical results are also reported to confirm our error estimates.

Numerical simulation for rotating Bose-Einstein condensate
Weizhu Bao, National University of Singapore

In this talk, we present efficient and stable numerical methods to compute ground states and dynamics of Bose-Einstein condensates (BEC) in a rotational frame. As preparatory steps, we take the 3D Gross-Pitaevskii equation (GPE) with an angular momentum rotation, scale it to obtain a four-parameter 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.

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 steady-state 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 one-dimensional fronts?”, Phys.Rev.Lett. 92, 090601 (2004)
2. G. Danker, O. Pierre-Louis, K. Kassner and C. Misbah, Peculiar effects of anisotropic diffusion on dynamics of vicinal surfaces, Phys.Rev.Lett.93, 185504 (2004)

Quantum drift-diffusion 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 non-local 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 energy-transport and drift-diffusion 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.

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 Bose-Einstein 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.

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 non-oscillatory) and HWENO (Hermite WENO) limiters, for Runge-Kutta 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, non-oscillatory 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.

On transport in micro-macro 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.

Topics in the mathematical analysis of nematic elastomers
Georg Dolzmann, University of Maryland

We analyze mathematical models for a special class of polymers, so-called 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).

Vortex nucleation in 3-dimensional 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 well-understood. In this talk we examine the system for a bulk superconductor occupying a bounded 3-dimensional domain. Our results suggest that, vortices nucleate at the boundary of the sample where the applied field is tangential to the surface.

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.

One-dimensional interfaces in two-dimensional materials structures
Ellen D. Williams, University of Maryland

Steps, island edges and domain boundaries are one-dimensional 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 one-dimensional 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 DOE-BES, NSF-NIRT and NSF-MRSEC

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_xu=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_xλ + λ² = <l, D²F r>. We shall outline three prototype cases.
We begin with the n-dimensional Restricted Euler equations, obtaining [n/2]+1 global invariants which precisely characterize the local topology at breakdown time. Next we introduce the corresponding n-dimensional Restricted Euler-Poisson (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 two-dimensional initial REP configurations with global smooth solutions. And finally, we show how rotation prevents finite-time breakdown. Our study reveals the dependence of the CT phenomenon on the initial spectral gap, λ₂(0)-λ₁(0).

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 velocity-surface 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 so-called 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.

Dynamics of the Becker-Doering equation
Barbara Niethammer, Humboldt-University of Berlin

The Becker-Doering 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 Becker-Doering 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.

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.

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 self-assembly mechanism. Some applications of the functional materials in tissues engineering, drug delivery, etc. will be given.

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 Ehrlich-Schwoebel 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 non-equilibrium 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 large-system-size 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.

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.

Projection-free Jacobi-Davidson method for Maxwell's equation
Wei-Cheng Wang, National Tsing Hua University

The band structure of a photonic-crystal 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 non-zero 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 non-zero 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 de-Rham 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.

Explore 2D Wigner crystal stable near room temperature
Xue-sen Wang, National University of Singapore

Periodic structures on surfaces, such as surface reconstructions, have been used as self-assembled nano-scale 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).

A quasi-continuum 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 so-called 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.

First-principles study of high-k oxide – Si and metal silicide – strained Si interfaces
Yuanping Feng, National University of Singapore

First-principles total energy calculations were used to study interfaces of high-k 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 up-to-date strained-Si device fabrication.

Numerical analysis of coarse-grained 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 coarse-grained 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 coarse-grained 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 path-wise agreement to MC simulations at the microscopic level. The presented results are joint work with M. Katsoulakis (UMASS), A. Sopasakis (UMASS).