Mathematical Theory and Numerical Methods for Computational Materials Simulation and Design - IMS

Mathematical Theory and Numerical Methods for Computational Materials Simulation and Design

(1 Jul 09 - 31 Aug 09)

... Jointly organized with Indian Statistical Institute, Kolkata,
and Department of Statistics & Applied Probability, NUS

~ Abstracts ~

(as of 16 Jul 2009)



A new version fast multipole method for evaluating the stress field of dislocation ensembles
Yang Xiang, Hong Kong University of Science and Technology, Hong Kong

Dislocations are the primary carriers of plastic deformation in crystals. Dislocation dynamics simulation is becoming an important tool for the study of plastic behaviors of crystalline materials. A crucial part of dislocation dynamics simulation is the calculation of the stress field of dislocation ensembles due to the long-range interaction of dislocations. We apply a new version of the fast multipole method (FMM) to compute the stress field of dislocation ensembles, in which exponential expansions are introduced to diagonalize the multipole-to-local translations. Numerical experiments show that for a dislocation ensemble discretized into N dislocation segments, the new version of the FMM is asymptotically O(N) with an optimized prefactor, and very efficient for prescribed accuracy requirements.

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Abinitio calculation of entropy and free energy in defected solids
Salvy Russo, RMIT Univeristy, Australia

Calculating the properties of materials using Density Functional Theory (DFT) has been increasing popular over the last 10-15 years. When predicting the thermodynamic stability of defects (including dopants) in solids using DFT, many studies have only considered the internal energy contribution to the free energy of defect formation.

In this talk we will consider how the various entropic contributions to the formation free energy of a defect can be calculated using DFT (or other levels of theory). These include the configurational, electronic and vibrational contributions to the total entropy of the defect. Various examples ranging from metallic nanoclusters to doped transition metal oxides will be discussed.

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Tailored finite point method for high frequency waves in heterogeneous medium
Zhongyi Huang, Tsinghua University, China

In this talk, we propose a tailored-finite-point method for the numerical simulation of the Helmholtz equation with high wave numbers in heterogeneous medium. Our finite point method has been tailored to some particular properties of the problem, then we can get the approximate solutions with the same behaviors as that of the exact solution very naturally. Especially, when the coefficients are piecewise constants, we can get the exact solution with only one point in each subdomain. Our finite-point
method has uniformly convergent rate with respect to wave number $k$ in $L^2$-norm.

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On the different phases and dynamical models of liquid crystal
Pingwen Zhang, Peking University, China

This talk provides a brief review of our basic understanding of liquid crystal phases from both macroscopic and microscopic points of view. From the macroscopic perspective, phases can be identified utilizing a series of thermodynamic properties based on classical thermodynamics; from the microscopic perspective, a phase needs to be defined on the basis of its structural symmetry and the types of order found in the phase. These two different perspectives can be reconciled by using statistical mechanics to bridge the length scale differences.

There are many different liquid crystalline phases that have been identified and characterized, the nematic phase possesses the lowest order with only long range molecular orientational order and short range positional and bond orientational order. The next class of liquid crystalline phases is the smectic phase, which possesses layered structures. In this talk, brief general descriptions are given for the thermodynamics and kinetics of phase transitions.

The Ericksen-Leslie, Tensor and Doi-Onsager kinetic models will be introduced to study the phase and phase transition, the kinetic-hydrodynamic liquid crystalline model will be used to classify the pattern formation of microstructures and the dynamics of defects. The relation of liquid crystalline models, and their limited regions, will be pointed out.

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Efficient and accurate numerical schemes for the phase-field models for multiphase complex fluids
Jie Shen, Purdue University, USA

I shall start with an introduction on an energetic variational phase field model for multiphase incompressible flows which leads to a set of coupled nonlinear system consisting a phase equation and the Navier-Stokes equations.

Then, I'll present efficient and accurate numerical schemes for solving this coupled nonlinear system. In particular, the following topics will be covered:

1. Splitting methods for unsteady incompressible Navier-Stokes equations
2. Fast spectral-Galerkin methods for elliptic problems
3. Full discretization for the coupled phase-field model
4. Spectral moving meshing method for the coupled phase-field model

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Ginzburg-Landau Equations---some recent developments and open problems
Fanghua Lin, Courant Institute, USA

We shall first go over briefly some background and earlier work on the Ginzburg-Landau equations. Then we shall describe some recent work concerning generalized Ginzburg-Landau equations including multiple components Ginzburg-Landau equations and their applications.

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Some numerical analysis issues in material and fluids simulations
Jian-Guo Liu, Duke University, USA

Many recently developed computation methods to overcome some challenges in multi-scale material/fluids simulations such as removing stiffness in time stepping due to fast waves, controlling the numerical dissipation with respect to the large parameters in the underline system, effectively local upscaling, etc. Numerical Analysis for these methods are difficulty. In this talk, I will present some results along this direction for a class of asymptotic preserving schemes.

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Existence of globally weak solutions to the flow of compressible liquid crystals system
Xian-Gao Liu, Fudan University, China

We study a simplified system for the compressible fluid of Nematic Liquid Crystals in a bounded domain in three Euclidean space and prove the global existence of the finite energy weak solutions.

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Mathematical models for liquid crystals
Daniel Phillips, Purdue University, USA

Liquid crystal materials can exist in a variety of phases that exhibit different levels of order. This order melts and defects appear if either their temperature is increased or if external stresses are applied. In this lecture we describe several mathematical models that are used to characterize and study these features.

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A level-set method for self-organized pattern formation during heteroepitaxial growth
Christian Ratsch, University of California, Los Angeles, USA

It is well know that strain leads to the formation and self organization of nanostructures and quantum dots during heteroepitaxial growth.
However, modeling these phenomena is a challenging task, because of the vastly different time and length scales involved, and the fact that elastic calculations are computationally expensive due to their long range. In this talk, we will discuss a model for heteroepitaxial growth that employs an island dynamics model with the level-set technique in combination with a fully self-consistent elastic model. Adatoms are described in a mean-field approach, and we solve a diffusion equation for the adatom concentration. At every timestep in the simulation, we solve the elastic equations for the entire system. At every lattice site strain then changes the local bonding, and thus the potential energy surface for adatoms and the microscopic parameters of the simulation, such as adatom diffusion, and the rate of detachment from island edges.We will then present results for several growth phenomena: In the submonolayer regime, islands become smaller and more regular upon increasing strain, and the island size distribution narrows and sharpens. We will discuss how a spatially varying potential energy surface can be exploited for pattern formation.
Such spatial variations are for example a result of defects or other features that are buried under the substrate. Another example is the growth of stacked quantum dots, where islands self-align on top of previously grown islands.

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Ab initio methods for quantum transport in macro-molecules
Stefano Sanvito, Trinity College Dublin, Ireland

The non-equilibrium Green's function (NEGF) method for electron transport combined with density functional theory (DFT) is at present the workhorse of material specific transport calculations. Despite some difficulties connected to the mean-field nature of the semi-local approximations to DFT [1], NEGF-DFT provides a powerful computational tool for research areas as diverse and magneto- and molecular electronics, biological sensing, and in general nano-electronics. Still challenges remain ahead in particular related to the ability of describing strong correlations, inelastic effects and in the make of algorithms scalable to large systems.

In recent years our group has developed the Smeagol code (, at present probably the most versatile code for quantum transport available worldwide [2,3]. In this talk I will review the main capabilities of Smeagol and in particular I will tackle the problems arising when dealing with large devices. I will then discuss two examples.

First I will consider spin-transport across a Mn12 molecule and demonstrate that orbital re-hybridization determines entirely the features of the I-V characteristics [4]. Interestingly the I-V presents a number of negative differential conductances at various voltages which are the fingerprint of the magnetic state of the molecule. This opens up the possibility of an electrical read-out of the molecular state.

Then I will move my attention to DNA. In particular I will discuss the transport properties of polymeric Poly-C/Poly-G strands connected to gold electrodes. The calculations will shed some light on the intrinsic nature of DNA transport and will tackle, with an effective theory, the effects due to the biological environment.

[1] Self-interaction errors in density functional calculations of electronic transport,
C. Toher, A. Filippetti, S. Sanvito, and Kieron Burke, Phys. Rev. Lett. 95 146402 (2005)

[2] Towards Molecular Spintronics, Alexandre Reily Rocha, Victor Garcia-Suarez, Steve W. Bailey, Colin J. Lambert, Jaime Ferrer and Stefano Sanvito, Nature Materials 4, 335 (2005).

[3] Spin and Molecular Electronics in Atomically-Generated Orbital Landscapes, Alexandre Reily Rocha, Victor Garcia-Suarez, Steve W. Bailey, Colin J. Lambert, Jaime Ferrer and Stefano Sanvito, Phys. Rev. B. 73, 085414 (2006).

[4] Magnetic state electrical readout of Mn12 molecules, C.D. Pemmaraju, I. Rungger and S. Sanvito, arXiv:0905.0281 (2009)

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Grain growth, shape, and topology in all dimensions: beyond von Neumann-Mullins
David Srolovitz, Princeton University, USA and Yeshiva University, USA

Cellular microstructures are ubiquitous in nature. They can be found in polycrystalline microstructures, foams, plant epidermis, ferroelectrics, complex fluids, and even in ice cream. In many situations, the cell/grain/bubble walls move to reduce their surface area (a surface tension effect), with a velocity proportional to the wall's mean curvature. As a result, the cells evolve and coarsen. Using this relation, and little else, von Neumann gave an exact formula for the growth rate of a cell in a 2-d cellular structure, which is the basis of modern grain growth theory. Borrowing ideas from geometric probability, we present an exact solution for the same problem in 3-d using the "mean width." We then describe why the mean width is the natural linear measure of grain size and topology and is useful across broad swaths of the sciences. Next, we extend this 50 year-old theory into all dimensions. Finally, we discuss using these ideas to efficiently simulate grain growth.

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Theory and modeling of reactive events
Eric Vanden-Eijnden, Courant Institute, USA

The dynamics of a wide range of systems involve reactive events, aka activated processes, such as conformation changes of macromolecules, nucleation events during first-order phase transitions, chemical reactions, bistable behavior of genetic switches, or regime changes in climate. The occurrence of these events is related to the presence of dynamical bottlenecks of energetic and/or entropic origin which effectively partition the phase-space of the dynamical system into metastable basins. The system spends most of its time fluctuating within these long-lived metastable states and only rarely makes transitions between them. The reactive events often determine the long-time evolution of the system one is primarily interested in. Unfortunately, computing up to the time scale at which these events occur represent an enormous challenge and so there is an urgent need for developing new numerical tools for such computations.

In the first lecture, I will explain why we may need to go beyond the standard framework of transition state theory (TST) to describe activated processes and reactive events, and I will present another framework, termed transition path theory (TPT), that permits to do that. Unlike TST, which gives mainly an expression for the rate of the reactive event, TPT describes more fully the statistical properties of the reactive trajectories (i.e. those trajectory by which the reactive event occurs), in particular in terms of their probability density function and their probability current.

In the second lecture, I will describe how TPT can be use to design and/or improve numerical methods for computing the pathways and rate of reactive events. I will focus in particular on the string method and milestoning.

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An adaptive level set method for moving curves and its application to dislocation dynamics
Hanquan Wang, Yunnan University of Finance and Economics, China

The original level set equations for motions of curves both in two dimensions and in three dimensions are solved on a uniform mesh, which does not have optimal efficiency in terms of calculations since the region near the curves is of most interest. Later improvement on calculation of level set equations includes narrow-band based local level set techniques and tree-based adaptive level set techniques. Both of them have their advantages and disadvantages.

In this talk, by taking consideration into their advantages, I present an adaptive level set method for motions of curves both in two dimensions and in three dimensions. The method first applies a simple algorithm to generate an adaptive grid which is composed by a fine mesh near the curve and a sparse mesh away from the curve; it next solve the level set equations on the adaptive grid with higher-order numerical methods such as third-order total variation diminishing (TVD) Runge-Kutta method in time and third-order weighted essentially non-oscillatory (WENO) scheme in space. I test this efficient and higher-order numerical method in several numerical examples and apply the method to study dislocation dynamics in material sciences.

This is a joint work with Professor Yang Xiang at Department of Mathematics of the HongKong University of Science and Technology.

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Phase field simulations of two phase fluid flow
Xiao Ping Wang, Hong Kong University of Science and Technology, Hong Kong

In this talk, I will first describe a newly developed phase field model for two phase fluid flow based on Cahn Hilliard Navier Stokes equation with generalized Navier boundary condition. Then some numerical results on two phase flow on rough and patterned surfaces will be presented. Issues related to drop formation, dripping to jetting transition will also be discussed.

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Boundary treatment, exact Hodge decomposition and the LBB condition for MAC-like schemes on curvilinear domains
Wei-Cheng Wang, National Tsing-hua University, Taiwan

We propose a new finite difference scheme for Navier-Stokes equations in primitive formulation on curvilinear domains. With proper boundary treatment and interplay between covariant and contra-variant components, the spatial discretization admits exact Hodge decomposition and energy identity. As a result, the pressure can be decoupled from the momentum equation with explicit time stepping. No artificial pressure boundary condition is needed. In addition, the spatial discretization is shown to satisfy the Ladyzhenskaya-Babuska-Brezzi (LBB) condition, leading to a rigorous error estimate of optimal order for the pressure. Numerical experiment demonstrates the robustness and efficiency of our scheme.

This is a joint work with Yin-Liang Huang and Jian-Guo Liu.

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Homogenization of non-coercive Hamilton-Jacobi equations
Jack Xue Xin, University of California, Irvine, USA

Non-coercive Hamilton-Jacobi equations arise in modeling dynamical behavior of material interfaces in solids and fluids. Properties of homogenized Hamiltonians are compared for periodic and random media, nonlinearities and viscosity effect. Both Lagrangian and Eulerian approaches and examples will be discussed.

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Eigenvalue problem of magnetic Schrodinger operators
Xingbin Pan, East China Normal University, China

Eigenvalue problems of magnetic Schrodinger operator have played important roles in the mathematical theory of superconductivity, liquid crystals and Bose-Einstein condensates. In this talk we shall introduce basic estimates of the lowest eigenvale for large magnetic field, and give some applications of these estimates in the theory of nucleation of superconductivity and surface superconductivity, and in the theory of phase transition of liquid crystals from nematic phase to smectic phase.

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The mathematical understanding of tau-leaping algorithm
Tiejun Li, Peking University, China

The tau-leaping algorithm is proposed by D.T. Gillespie in 2001 for accelerating the simulation for chemical reaction systems. It is faster than the traditional stochastic simulation algorithm (SSA), which is an exact simulation algorithm. In this lecture, I will overview some recent mathematical results on tau-leaping done by our group, which include the rigorous analysis, construction of the new algorithm, and the systematic analysis of the error.

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Computational surface partial differential equations
Charles Elliott, University of Warwick, UK

Evolutionary PDEs on stationary and moving surfaces appear in many applications such as the diffusion of surfactants on fluid interfaces, surface pattern formation on growing domains, segmentation on curved surfaces and phase separation on biomembranes and dissolving alloy surfaces.

In this talk I discuss three numerical approaches based on:- (I) Surface Finite Elements and Triangulated Surfaces, (II)Level Set Method and Implicit Surface PDEs and (III) Phase Field Approaches and Diffuse Surfaces.

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Topics in surface and nanoscale science
Andrew Wee, National University of Singapore

In this talk, I will present some recent work from our NUS surface science group in the areas of graphene and molecular electronics. Graphene research has been proceeding at a relentless pace since it was first isolated in 2004. It is the thinnest known material in the universe and the strongest ever measured. Graphene can sustain huge current densities, and displays record thermal conductivity and stiffness. Electron transport in graphene is described by a Dirac-like equation, which allows the investigation of relativistic quantum phenomena in a benchtop experiment. However, many materials challenges to producing large area high quality graphene still remain. I will discuss our work using in situ STM, synchrotron photoemission (PES) and density functional theory (DFT) calculations to investigate the structure and growth of epitaxial graphene on 6H-SiC(0001).

In the field of organic and molecular electronics, the key issue is the understanding of molecule-metal and organic-organic interfaces. We used STM and synchrotron techniques to study self-assembled molecular arrays and organic heterojunctions. Some ongoing work and future perspectives will also be discussed.

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Heat/Phonon transport in nanostructures and Phononics
Baowen Li, National University of Singapore

In the first part, I will focus on several fundamental issues such as whether heat conduction in nanostructures obeys the Fourier law, how heat pulse spreads in low dimensional nanostuctures, how to reduce thermal conductivity of nanowires and nanotubes. Both computational simulation and experimental results will be presented.

In the second part, I will discuss how to build different thermal devices such as thermal rectifier/diode, thermal transistor, thermal logic gate and thermal memory.

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Molecular self-assembly on surfaces and surface nanotemplates
Wei Chen, National University of Singapore

Creation of well-ordered functional molecular arrays at the nanometer scale is one of the key issues in the development for future molecular- or nano-electronic devices, solid-state quantum computation, single-electron devices, and biosensors. Molecular self-assembly on surfaces or surface nanotemplates via selective and directional covalent or non-covalent interactions offers a promising bottom-up approach to fabricating molecular nanostructure arrays with desired functionalities over macroscopic areas.

Here, we demonstrate the formation of various well ordered C60 superstructures with tunable periodicity and symmetry using different molecular surface nanotemplates. It is found that that the formation of the tunable C60 molecular arrays arises from the delicate balance between the homointermolecular (van-der-Waals forces), hetero-intermolecular (charge transfer) and molecule-substrate interfacial interactions under different experimental conditions, which can be simply adjusted by choosing appropriate C60 and 6T, 6P or pentacene coverage and post annealing temperature. we also demonstrated a novel bottom-up approach to fabricate self-assembled 2D binary molecular networks on graphite, whose structural stability is sustained through the formation of multiple intermolecular C-F ... H-C hydrogen bonds between the electronegative periphery F atoms of F16CuPc and the electropositive periphery H atoms of 6P, pentacene or DIP. The supramolecular packing structures can be controlled by careful selection of molecular building blocks with appropriate geometry (size and shape) and molecular ratios. Our results suggest that self-assembling of molecular superstructures on surface nanotemplates represents a simple and effective method for the construction of highly ordered functional molecular nanostructure arrays, and offers a versatile route towards the fabrication of novel molecular interconnects and devices.

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Massively parallel algorithms for the simulation of complex flows
Ulrich Ruede, University Erlangen-Nuremberg, Germany

Large scale numerical simulation requires that we exploit parallelism on all levels. High performance computers will continue to be built as parallel systems with physically distributed memory, but each node will be have processors, each of which contains many cores. Additionally each core may rely on vectorization. Such systems can provide enormous compute power, but they may require a special programming and a new design of the algorithms.

The multigrid finite element solvers HHG developed at Erlangen have been run on up to tenthousand processor cores, solving systems with three hundred billion unknowns. As a second example, the talk will report on the Walberla software framework that is being developed for parallel computational fluid dynamics using on the Lattice Boltzmann method and which is being used to study multiphase flows in complex geometries and particulate flows with fully resolved fluid-structure-interaction.

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Dynamic transition theory and equilibrium phase transitions
Shouhong Wang, Indiana University, USA

In this talk, I shall present a brief overview of the dynamic transition theory developed recently by Ma and myself, and its applications to equilibrium phase transitions. The main philosophy of the theory is to search for the full set of transition states, giving a complete characterization on stability and transition. The set of transition states --physical "reality"-- is represented by a local attractor. Following this philosophy, the dynamic transition theory is developed to identify the transition states and to classify them both dynamically and physically.

The theory has a wide range of applications in both equilibrium and nonequilibrium phase transitions. The application of the theory to equilibrium phase transitions involves a combination of modeling, mathematical analysis and physical predictions. We adopt the general idea of the Ginzburg-Landau phenomenological approach, and introduce a unified time-dependent Ginzburg-Landau model, based on the le Ch\^atlier principle. Applications to non-equilibrium transitions include typical problems in classical and geophysical fluid dynamics, in climate dynamics, and in chemical reactions. The modeling and the analysis of these problems, on the one hand, provide verifications of existing experimental and theoretical studies, and, on the other hand, lead to various new physical predictions.

To demonstrate the wide range of applications of the theory, I shall present two examples: one is the phase separation of binary systems modeled by the Cahn-Hilliard equation, and the other is superfluidity of liquid helium-3. In both cases, the study leads to some specific physical predictions, which are otherwise unknown from both the physical and mathematical points of view. For example, as a physical prediction, we derive the existence of a new superfluid phase C for liquid helium-3.

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POD and CVT based reduced-order modeling for partial differential equations
Hyung Chun Lee, Ajou University, Korea

A discussion of reduced-order modeling for partial differential equations is given to provide a context for the construction and application of reduced-order bases. Review of the POD (proper orthogonal decomposition ) and CVT (centroidal Voronoi tessellation) approaches to reduced-order modeling are provided, including descriptions of POD and CVT reduced-order bases, their construction from snapshot sets, and their applications to the low-cost simulations of partial differential equations.

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The first-principles electronic structure calculations based on finite element discretizations
Aihui Zhou, Chinese Academy of Sciences, China

In this presentation, we will give a brief introduction to first-principles elec-
tronic structure calculations and talk about some recent fnite element compu-
tations in quantum chemistry and materials science. This presentation is based
on the joint works with X. Dai, X. Gong, L. Shen, Z. Yang, and D. Zhang.

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Quantum thermal transport from classical molecular dynamics
Jian-Sheng Wang, National University of Singapore

Traditional molecular dynamics, since it is based on classical mechanics, produces only classical results. It is possible to simulate quantum thermal and electronic transport using a form of molecular dynamics subject to Langevin heat baths with correlated noises. The noise spectra and the memory kernel are derived by analyzing the effect of the leads of a junction system quantum-mechanically. It is shown that quantum ballistic transport can be reproduced for linear systems, and nonlinear effect can be understood as a quasi-classical approximation to the quantum problems. Results of thermal conductance for one-dimensional lattice chain model, graphene nano-ribbons, and nanotubes will be reported.

Refs. J.-S. Wang, Phys. Rev. Lett. 99, 160601 (2007); J. T. Lü and J.-S. Wang, J. Phys.: Condens. Matter, 21, 025503 (2009).

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Introduction to molecular models
Xiantao Li, Pennsylvania State University, USA

I will provide the following lectures
1. Introduction to molecular models
2. Statistical mechanics
3. Simulation technique

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Multi-paradigm simulations at the nanoscale: methodology and applications to functional nano materials
Haibin Su, Nanyang Technological University

Multiparadigm methods to span the scales from quantum mechanics to practical issues of functional nanoassembly and nanofabrication are enabling first principles predictions to guide and complement the experimental developments by designing and optimizing computationally the materials compositions and structures to assemble nanoscale systems with the requisite properties. In this talk, we employ multi-paradigm approaches to investigate functional nano materials with versatile character, including fullerene, carbon nanotube (CNT), graphene, and related hybrid structures, which have already created an enormous impact on next generation nano devices. The topics will cover the reaction dynamics of C60 dimerization and the more challenging complex tubular fullerene formation process in the peapod structures; the computational design of a new generation of peapod nano-oscillators, the predicted magnetic state in NanoBuds; and opto-electronic properties of graphene nanoribbons.

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Predictive atomistic and coarse-grained modeling of epitaxial thin film growth
Jim Evans, Iowa State University, USA

Atomistic modeling of homoepitaxial film growth on single-element single-crystal surfaces (A on A), when combined with kinetic Monte Carlo (KMC) simulation, has achieved remarkable predictive accuracy for some systems [1]. This applies not just for the description of initial submonolayer 2D island formation, but also for subsequent multilayer growth and kinetic roughening (including formation of 3D mounds, i.e., multilayer stacks of 2D islands). However, coarse-grained modeling alternatives are appealing from the perspective of algorithmic efficiency, and also to provide deeper insight into fundamental issues such as development 2D island distributions or 3D mound coarsening dynamics. We review the current state of this field [1]. Our recent efforts have begun to explore more complex deposition processes on binary alloy surfaces (e.g., A on BC, B+C on BC) [2] and on ternary quasicrystal surfaces (A on BCD) [3].
For the former, predictive atomistic modeling has been achieved, and for the latter coarse-grained step-dynamics modeling has provided significant insight.
Of particular interest here is height-selection of multilayer islands due to quantum size effects.

[1] J.W. Evans, P.A. Thiel, and M.C. Bartelt, Surf. Sci. Rep. 61 (2006) 1-128.
[2] Han, Liu, Unal, Qin, Jing, Jenks, Thiel, Evans, Phys. Rev. Lett. 100
(2008) 116105.
[3] Unal, Fournee, Thiel, Evans, Phys. Rev. Lett. 102 (2009) 196103.

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Fluid-particle interaction models: asymptotics, theory and numerics
José A. Carrillo, University Autonoma Barcerona, Spain

We are interested in coupled microscopic/macroscopic models describing the
evolution of particles dispersed in a fluid. Fluid-particle interaction is
of primarily importance in sedimentation analysis of disperse suspensions
of particles in fluids, one of the issues being the separation of the solid
grains from the fluid by external forces: gravity settling processes or
centrifugal forces. On the other hand, aerosols and sprays can be also
modelled by fluid-particle type interactions in which bubbles of suspended
substances are seen as solid particles.

The system is modelled using a Vlasov-Fokker-Planck equation to describe
the microscopic motion of the particles coupled to the Euler equations for
a compressible fluid. We investigate dissipative quantities, equilibria
and their stability properties and the role of external forces. We also study
some asymptotic problems, their equilibria and stability and the
derivation of macroscopic two-phase models. Numerical schemes capable of
dealing with the asymptotic limit situations are proposed. Numerical
simulations will be shown.

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Diffuse interface modeling and simulation of some interface problems
Qiang Du, Pennsylvania State University, USA

The diffuse interface (or phase field) method has been one of the popular approaches for the modeling and simulation of many interface problems in nature. In this series of lectures, we first present some of the basic ideas of the diffuse interface approach. Examples are shown to illustrate its effectiveness in practice. Next, we discuss the relevant mathematical theory, consider the sharp interface limit and make comparisons with other methods. We then focus on some recent works related to the application of the diffuse interface method to interface problems in materials science and biology, in particular, those problems involving elastic energy contributions. These specific applications are used to highlight the modeling and computational challenges. We also discuss a stochastic implicit interface method for modeling the effect of fluctuations in interfacial problems. Finally, we report some recent progress in the algorithmic development aspect. The lectures will be of introductory nature, with emphasis on the basic methodology, the important issues and some open problems.

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Toward quantitative analysis of nanoelectronics --- state of the art and challenges
Hong Guo, McGill University, Canada

Nanoelectronic devices operate by principles of quantum mechanics, their properties are closely influenced by the atomic and molecular structures of the device. It has been a challenge to calculate device characteristics including relevant microscopic details, especially when one wishes to predict these characteristics without relying on phenomenological parameters.

In this lecture, I will review the present status of nanoelectronic device theory, the existing theoretical and numerical difficulties, and some important problems of nanoelectronics. I will then discuss in detail a particularly useful progress achieved toward quantitative predictions of non-equilibrium and non-linear charge/spin quantum transport in nanoelectronic devices from atomic point of view. The theoretical formalism is based on carrying out real space density functional theory (DFT) analysis within the Keldysh nonequilibrium Green's function (NEGF) framework [1,2]. The theoretical background as well as numerical implementation of the NEGF-DFT formalism will be presented [1,2,3]. Quantitative comparisons to measured data will be presented. I will give many examples of calculating quantum transport in nanostructures starting from atomic arrangements of the device, including molecular wires, carbon nanostructures, magnetic tunnel junctions, disordered systems, and various spintronic systems. I will end the lecture by outlining a view on the existing challenges of nanoelectronics, and on developing tools powerful enough for nanoelectronics design automation.

I plan to cover the following technical issues in this 6-hour lecture:

(1) Quantum transport theory within NEGF. After a brief technical discussion on how to derive transport formula using NEGF, I will discuss a symbolic derivation software we have developed recently using Mathematica, that automatically derives relevant NEGF and transport formula starting from the Hamiltonian.

(2) Computational method of NEGF-DFT [1,2,3]. I will focus on important technical issues about how this formalism works and what are the important things to take care of in order to do a correct simulation. Other issues such as electron-phonon scattering during current flow, strong correlation effects in tunnelling through oxides, will also be discussed.

(3) Computational method of NEGF-DFT-NVC [4]. I will discuss how to carry out nonequilibrium vertex correction (NVC) analysis of disorder scattering in nanoelectronic devices. Since any real system has some degree of disorder, configurational average of transport properties must be done. NVC allows one
to accomplish this very important task from first principles.

Acknowledgements. The work presented here were contributed by many students and postdocs over many years: Drs. Jian Wang, Jeremy Taylor, Jose-Luis Mozos, Hatem Mehrez, Brian Larade, Gianni Taraschi, Qingrong Zheng, Wengang Lu, Eric Zhu, Lei Liu, Chaocheng Kaun, Pawel Pomorski, Yi Liu, Dan Robutsov, Nicolai Sergueev, Derek Waldron, Devrim Guclu, Vladimir Timochevski, and Chuncheng Wan. I'd like to also thank the following group members who contributed to the topics of this lecture: Dr. Yibin Hu, Dr. Wei Ji, Dr. Ferdows Zahid, Mr. Zimin Feng, Zhanyu Ning, Tao Ji, Manuel Smeu, Jesse Masson, Etienne Marcotte.

Some References:
[1] "Ab initio modelling of quantum transport properties of molecular electronic devices", Jeremy Taylor, Hong Guo, and Jian Wang, Phys. Rev. B 63, 245407 (2001).

[2] "Ab initio modelling of open systems: charge transfer, electron conduction, and molecular switching of a C60 device", Jeremy Taylor, Hong Guo and Jian Wang, Phys. Rev. B 63, R121104 (2001).

[3] "Nonlinear spin-current and magnetoresistance of molecular tunnel junctions", Derek Waldron, Paul Haney, Brian Larade, Allan MacDonald and Hong Guo, Phys. Rev. Lett. 96, 166804 (2006).

[4] "Disorder scattering in magnetic tunnel junctions: theory of nonequilibrium vertex correction", Youqi Ke, Ke Xia and Hong Guo, Phys. Rev. Lett. 100, 166805 (2008).

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Topological vorticity and conserved geometric motion
Fanghua Lin, Courant Institute, USA

After a quick review of some distinct feature of vortex dynamics in classical fluids, we shall describe a notion of topological vorticity that plays the similar important role as vorticity in the classical fluids. We can use the topological vorticity to study vortex dynamics in ferromagnets and Bose-Einstein condensates for example. Then I shall expalin my recent joint work with Jun-Cheng Wei on Kelvin motions of traveling vorticies solutions and some genral conserved geometric motion.

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Phase-field models for multiphase complex fluids: modeling, numerical analysis and simulations
Jie Shen, Purdue University, USA

I shall present an energetic variational phase field model for multiphase incompressible flows which leads to a set of coupled nonlinear system consisting a phase equation and the Navier-Stokes equations. We shall pay particular attention to situations with large density ratios as they lead to formidable challenges in both analysis and simulation.

I shall present efficient and accurate numerical schemes for solving this coupled nonlinear system, and show ample numerical results (air bubble rising in water, Newtonian bubble rising in a polymeric fluid, defect motion in a liquid crystal flow, etc.) which not only demonstrate the effectiveness of the numerical schemes, but also validate the flexibility and robustness of the phase-field model.

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Some issues in the long-time simulation of complex systems
Eric Vanden-Eijnden, Courant Institute, USA

The dynamics of many systems of interest span many spatio-temporal scales, making their analysis by numerical simulations difficult. Indeed, while we are bound to simulate the dynamics at the smallest and fastest scales, we are often interested in the system's behavior in the largest and slowest scales. These are typically not accessible to brute force simulations, both because of the computational cost involved but also because these time scales are way beyond the time at which we lose pathwise accuracy with standard numerical integrators. These difficulties force us to change perspective and look at the systems from a probabilistic viewpoint. Indeed, besides being unreliable, long time numerical trajectories are also irreproducible due to sensitivity to initial conditions and/or small perturbations, and they are typically uninformative per se. By identifying the right statistical descriptors for the systems that characterize their behavior, like e.g. their invariant measure, the time-correlation functions of suitable observable, or several statistical objects describing rare reactive, we can build algorithms that are tailor-made to calculate these descriptors accurately and efficiently. In this talk this strategy will be elucidated and illustrated on examples arising from material sciences, atmospheric sciences and molecular dynamics.

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Introduction to dislocation dynamics
Yongwei Zhang, National University of Singapore and
Institute of High Performance Computing

Dislocations are an important class of defects in crystalline solids. Therefore, the subject of dislocations is essential for an understanding of many of the physical and mechanical properties of crystalline solids. We first show experimental observation of dislocations, and then discuss physical basis for dislocations and some elementary geometric properties of dislocations. We then discuss elastic properties and movement of dislocations. Under most dynamic conditions, dislocations move so slowly that the dynamic displacements are approximated quite accurately by these static solutions. However, when dislocation moving velocity is close to the speeds of sound, the dynamic effect may become important. We will discuss displacement and stress fields of the moving screw dislocations and the moving edge dislocations, dislocation acceleration and radiation, supersonic dislocations and dislocation mobility. It should be noted that many of the theories developed for specific experiments are still tentative. The theory of dislocations presented here is in quite a general approach.

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The second-order multi-scale simulation for the physics and mechanics behaviors of composites and their structures
Junzhi Cui, Chinese Academy of Sciences, China

  • Introduction - Background.

  • Second-order two-scale simulation for the physics and mechanics behaviors of the structure of composites with periodicity and quasi-periodicty.

  • The statistically second-order two-scale simulation for the physics and mechanics behaviors of the structure of composites with consistently and inconsistently random distribution.

  • Some discussion on Multi-scale simulations of composites and their structures.

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Asymptotically correct finite difference schemes for highly oscillatory ODEs
Anton Arnold, Vienna University of Technology, Austria

We are concerned with the numerical integration of ODEs of the form
$\epsilon^2 \psi_{xx} + a(x)\psi=0$ for given $a(x)\ge\alpha>0$ in the highly oscillatory regime $0<\epsilon\ll 1$ (appearing as a stationary Schr\"odinger equation, e.g.). In two steps we derive an accurate finite difference scheme that does not need to resolve each oscillation:
1) With a WKB-ansatz the dominant oscillations are "transformed out", yielding a much smoother ODE.
2) For the resulting oscillatory integrals we devise an asymptotic expansion both in $\eps$ and $h$.
In contrast to existing strategies, the presented method has (even for a large spatial step size $h$) the same weak limit (in the classical limit $\epsilon\to 0$) as the continuous solution. Moreover, it has an error bound of the order $O(\epsilon^2 h^2)$.

Ref: A. ARNOLD, N. BEN ABDALLAH and C. NEGULESCU: WKB-based schemes for the Schr\"odinger equation in the semi-classical limit, preprint 2009.

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A deterministic solver for a hybrid quantum-classical transport model in NanoMOSFETs
José A. Carrillo, University Autonoma Barcerona, Spain

We model a nanoMOSFET by a mesoscopic, time-dependent, coupled quantum-classical system based on a sub-band decomposition and a simple scattering operator. We first compute the sub-band decomposition and electrostatic force field described by a Schrödinger-Poisson coupled system solved by a Newton-Raphson iteration using the eigenvalue/eigenfunction decomposition. The transport in the classical direction for each sub-band modeled by semiclassical Boltzmann-type equations is solved by conservative semi-lagrangian characteristic-based methods. Numerical results are shown for both the thermodynamical equilibrium and time-dependent simulations in typical nowadays nanoMOSFETs. This is a work in collaboration with N. BenAbdallah, M.J. Caceres and F. Vecil.

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Coarsening of island distributions on surfaces: Ostwald vs Smoluchowski vs unconventional ripening pathways
Jim Evans, Iowa State University, USA

Post-deposition evolution of distributions of 2D islands in pristine homoepitaxial systems provides an ideal testing ground for coarsening theories. Traditionally, Ostwald ripening (OR), i.e., dissolution of small islands and growth of larger ones, is expected to provide the dominant coarsening pathway.
Indeed, OR is often observed revealing behavior consistent with well-developed LSW theory. However, sometimes coarsening is dominated by Smoluchowski ripening (SR), i.e., diffusion and coalescence of islands. Here, typically detailed behavior deviates from predictions of classic rate equation predictions. We both elucidate the features controlling the dominant pathway and origin of the deviations from classic predictions [1]. Introducing even trace-amounts of a chemical additive often greatly accelerates the coarsening process [1,2].
We provide two possible mechanisms, either additive-catalyzed coarsening for attachment-detachment limited OR, or enhanced mass transport due to facile formation of a mobile metal-additive complex.

[1] Thiel, Shen, Liu, Evans, J. Phys. Chem. C 113 (2009) 5047 (Centennial Feature) [2] Shen, Liu, Thiel, Evans, J. Chem. Phys. 130 (2009) 094701

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Dynamic density functional theory and approximations to model crystalline materials
Axel Voigt, Technische Universit, Dresden, Germany

The phase field crystal model is by now widely used in order to predict crystal nucleation and growth. We demonstrate using a finite element discretization the possibilities for various applications. For colloidal solidification with completely overdamped individual particle motion, we show that the phase field crystal dynamics can be derived from the microscopic Smoluchowski equation via dynamical density functional theory. The different underlying approximations are discussed. We test the validity of the phase field crystal model against dynamical density functional theory. In particular, the velocities of a linear crystal front from the undercooled melt are compared as a function of the undercooling for a two-dimensional colloidal suspension of parallel dipoles. As a second application we use the phase field crystal model to compute anisotropies of surface free energies of nanocrystals as a function of temperature. The approach allows to extract anisotropy functions with cusps. Furthermore we consider liquid phase epitaxy and analyse the transition from step flow growth to step bunching, facetting and formation of dislocations. As a last example we simulate the phase field crystal model on curved surfaces to model spherical crystallography. In particular we are interested in the formation of grain boundary scars on such surfaces.

Good agreement is only obtained by a drastic scaling of the free energies in the phase field crystal model in order to match the bulk freezing transition point.

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Microscopic origins of continuum balances and peridynamics
Richard B. Lehoucq, Sandia National Labs, USA

Peridynamics (Silling 2000) is a continuum theory where the internal force density in the balance of linear momentum is given by an integral operator. This operator sums internal forces separated by a finite distance in contrast to a continuum theory where only contact forces are assumed. My presentation derives mass and momentum conservation laws for peridynamics using the principles of statistical mechanics. In particular, I show that the peridynamic force density integral operator is the phase space expected value of internal force density given by a general multibody interatomic potential. The derivations generalize the seminal work of Irving-Kirkwood (1950), and build upon the elegant ideas due to Noll (1955) that generalized the former work. The approach presented avoids the standard limitation of a pairwise interatomic potential, and the peridynamic integral operator sidesteps the significant issues associated with determining a stress tensor given a multibody interatomic potential. However, a Cauchy (symmetric) stress tensor is derived so that the divergence of this stress tensor is equal to the peridynamic force density integral operator. This represents a significant generalization of previous definitions of microscopic stress to arbitrary multibody potentials. I also discuss how the integral operator can be replaced by a force density arising from the classical notion of contact forces. Hence, the path from Newton's second law to conservation of linear momentum for simple nonpolar materials traverses peridynamics. This is joint work with Mark Sears.

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Analysis and computation for the semiclassical limits of the nonlinear Schrodinger equations
Weizhu Bao, National University of Singapore

In this talk, I will review recent results on analysis and efficient computation for the semiclassical limits of linear and nonlinear Schrodinger (NLS) equations. First, I will show our recent asymptotic and numerical results on the semiclassical limits of the ground and excited states of time-independent NLS with a few typical external trapping potentials. Then I will review the formal semiclassical limit of the NLS by using different approaches including WKB method, Winger transform, Grenier's
generalized WKB analysis, etc. A time-splitting spectral (TSSP) method was introduced to efficiently compute the dynamics of the NLS in the semiclassical regimes. The numerical method is explicit, unconditionally stable, time reversible and time transverse invariant. Moreover, it conserves the position density in the discretized level and has the best spatial/temporal resolution for the NLS in the semiclassical regimes. Comparison between the solutions of the NLS and its quantum hydrodynamical limit are presented, especially when the quantum hydrodynamical equations have shocks and/or vacuum. Finally, the analysis and
computation results are extended for the NLS with an angular momentum rotation term and coupled nonlinear Schrodinger equations.

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Metallic glasses: It's all in the mix
Yi Li, National University of Singapore

Materials such as glass or polystyrene are often called amorphous solids - literally ?without shape? - because their atoms are arranged randomly without the long-range repeating patterns seen in crystals. Scientists have also managed to make amorphous metal alloys known as metallic glasses, which promise great strength, elasticity and magnetic properties. But unlike the well-defined long-range order in the crystalline metals, the atomic arrangements in amorphous alloys remain mysterious at present and the details of how the atoms are packed in amorphous metals are generally far less understood than for the network-forming glasses. In this talk, I shall summarize the recent attempts made at the understanding of the atomic structure in metallic glasses and how one of our recent works gives new insight into this.

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Dynamical self-trapping of Bose-Einstein condensates in shallow optical lattices
Fong Yin Lim, Institute of High Performance Computing

We study the transport properties of a strongly repulsively interacting BEC through a shallow optical lattice of finite width. The study is carried out via numerically solving the Gross-Pitaevskii equation. Self-trapped states are observed for some sets of parameters, indicated by stopping in the expansion of BEC at certain lattice site.
Such self-trapped states can be explained analytically in terms of nonlinear Bloch waves, approximated by a truncated Bloch function. The results could help us in the design of atom-transporting wire in an atomic circuit, whose properties can be adjusted in a controllable way.

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Gate controlled non-volatile graphene ferroelectric memory and spin injection into graphene via MgO barriers
Barbaros Oezyilmaz, National University of Singapore

The discovery of graphene in 2004 has triggered enormous experimental and theoretical efforts. As a gapless semiconductor, charge carriers in graphene can be tuned continuously from electrons to holes crossing the charge neutral Dirac point using an external electric field. Unlike conventional semiconductors, the doping process does not influence the mobility of charge carriers in graphene, which can exceed 10+5 cm+2/Vs-1 at low temperature. Such doping-independent mobility leads to the field-dependent conductance in graphene. Based on these two properties, many novel graphene-based device applications have been predicted. However, a paradigm shift in the microelectronics industry from Si to graphene also requires graphene-based memory applications. Despite graphene having intrinsically a high resistance state at the Dirac point and a low resistance state when heavily doped, reports on graphene for non-volatile information storage is rarely seen. This is due to the difficulty in maintaining the resistance states in graphene without an external electric field.

I will show non-volatile switching in graphene by using ferroelectric gating without having to break the lattice symmetry. We demonstrate basic writing and reading processes of a novel graphene-ferroelectric memory device structure combining the field dependent conductance of graphene with the remnant electric field of ferroelectric thin films. A bistable state of high and low resistance value is realized by controlling the electrical doping level in graphene hysteretically, which is caused by a hysteretic switching of the polarization in the ferroelectric thin-film. Our approach of ferroelectric gating also open up the possibility of studying graphene phenomena in the limit of high carrier concentration, which is not possible with dielectric gating.

I will also report on our resent results on spin transport in graphene observed in the non-local device geometry. Here we have used MBE grown MgO as a tunneling barrier.

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Numerical simulation of coupled surface and grain boundary motion
Zhenguo Pan, National University of Singapore

A coupled surface and grain boundary motion which arises in materials science is considered in a bicrystal in the context of the quarter loop geometry. Two types of normal curve velocities are involved: motion by mean curvature and motion by surface diffusion. Three curves meet at a single point with junction conditions applied. A formulation that describes the coupled normal motion of the curves and preserves arc length parametrization up to scaling is proposed. Numerical convergence and the numerical stability of the existing travelling waves will be shown.

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Event-by-event Monte Carlo simulations: Spatial resolution limits of MeV protons & keV electrons
Chammika N B Udalagama, National University of Singapore

Progress in fabrication, experimentation and manipulation of structures at the micro/nano-meter regimes constantly push the demands made of microscopy and lithography techniques. This has promoted the use of electrons and ions, with their smaller wavelengths, to improve on the diffraction limited visible photons and not so easily focused x-ray photons. But, to what dimensions can the use of electrons and ions be pursued? Which of Nature's characteristics limit their usefulness and spatial resolution? We attempt to gain insight into these fundamental issues via computational studies in the form of Monte Carlo simulations. Our software, DEEP (Deposition of Energy due to Electrons & Protons), unlike most other Monte Carlo particle penetration simulations software adopts an event-by-event simulation formalism as opposed to that of continuous energy loss. This allows us to appreciate the richness of the energy loss process, particularly that of secondary electron or δ-ray production. DEEP incorporates the EPDL97 cross-sections together with the Hansen-Kocbach-Stolterfoht (HKS) δ-ray model for the relevant physics of ion-matter interactions. We will present recent results of the applications of DEEP to polymethylmethacrylate (PMMA) as the target along with results from recent experiments at CIBA, into high spatial resolution imaging of biological cells using MeV protons.

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Brillouin light scattering from nanostructures
Zhikui Wang, National University of Singapore

Brillouin light scattering (BLS) is a non-destructive and non-contacting technique for investigating spin and acoustic dynamics, by detecting the inelastic scattered light by the spin waves (magnons) and elastic waves (phonons) in nanostructures. The observation of well-resolved Brillouin peaks arisen from eigenvibrations of nanospheres verified Lamb?s century old theory on eigenvibrations of elastic spheres. Quantization of bulk spin waves in nanowires and bandgap of spin waves in magnonic crystals (magnetic counterpart of semiconductors, photonic crystals or phononic crystals) were first observed using BLS. Experimental data obtained provide information such as quantization of spin and acoustic waves in nanostructures, as well as their magnetic and elastic properties. We found that the magnon frequency bandgaps could be tuned by applying an external magnetic field or by varying their structure dimensions. These findings are expected to stimulate further development of the theory and applications of magnonics.

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Design of materials for sustainable manufacturing by computer modeling
Ping Wu, Institute of High Performance Computing

Although Nobel Laureate Steven Chu (currently the 12th United States Secretary of Energy) already developed methods to cool and trap atoms with laser light in late last century, it is still a dream to use them to manufacture very small electronic components. Therefore, it is widely believed that we will hit the Moore's law limit in the years 2015-2020. Furthermore, we may soon face a manufacturing cost explosion based on Moore's second law, which is doubling in cost of chip fabrication plants every four years. New paradigms of sustainable technology are needed to get us around these limits. Computer modeling is capable of exploring these sustainable manufacturing solutions. In my presentation I shall use real-life case studies to outline the design of materials based on affordable computer modeling techniques.

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Binding and patterning of organic molecules on silicon surfaces
Guo Qin Xu, National University of Singapore

Attaching functional organic layers to silicon surfaces is emerging as one of the promising approaches in the development of new semiconductor-based microelectronic devices and biosensors. It provides opportunities for incorporating molecular recognition, chirality, chemical/biological sensing, light emission/detection and lubrication for various technological needs. Recent systematic investigations on chemical reactions of organic molecules on silicon surfaces clearly demonstrated that both Si(100) and Si(111)-7x7 can act as reagent-like substrates with a high reactivity for covalent binding of different classes of organic functionalities. Reaction mechanisms including [2+2]-cycloaddition, [4+2]-cycloaddition, dative-bonding, dissociation and ene-like reactions have been revealed and will be discussed, providing a molecule-level understanding on reaction mechanisms, chemical and surface-site selectivities at organic/silicon hybrid interfaces. In addition, new approaches for fabricating organic nanopatterns using self-assembled templates on silicon surfaces will be introduced, which can be useful in growing organic nanomaterials for developing nano- or molecular-scale devices.

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Modelling the heteroepitaxial growth of novel surface nanostructures
Yongwei Zhang, National University of Singapore and
Institute of High Performance Computing

Surface roughening of a mismatched thin film leads to the formation of three-dimensional islands, or quantum dots. This roughening is via surface diffusion driven by strain energy relaxation. This spontaneous roughening process was proposed to grow uniform and regular quantum dot arrays since uniform and regular quantum dot arrays with precisely controlled positions and sizes are desired for making the template for the next generation of nanoelectronic devices. However, numerical experimental results have shown that unguided self-assembled growth of quantum dots usually fails to realize perfectly ordered dot arrays. Hence how to reliably and reproducibly achieve ordered quantum dot arrays through surface pre-patterning is an interesting topic.

In the present work, three-dimensional finite element method was developed to investigate the self-assembly of heteroepitaxial quantum structures during Stranski-Krastonov growth. In the model, various factors such as strain energy, surface energy, wetting effect, surface energy anisotropy and elastic anisotropy, and substrate surface pre-patterning were taken into account, and the SiGe/Si material system was used as a model system. We consider quantum structures growing on both flat substrate surfaces without patterns or with patterns. Our computer simulations showed that various quantum dot surface patterns can be obtained by the guided self-assembled growth, including some novel surface structures such as quantum-dot automata arrays, fortress-enclosed quantum-dot automata arrays, and ordered quantum dot arrays. Parametric studies have been performed to obtain the phase diagrams for obtaining various surface patterns. The present simulation work demonstrates that the coupling of surface pre-patterns, surface energy anisotropy and elastic anisotropy strongly influences the surface roughening morphology, self-assembly and shape transition of epitaxial quantum dots, resulting in diverse evolution pathways.

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The Q-tensor theory of liquid crystals
John Ball, University of Oxford, UK

The lectures will survey what is known about the mathematics of the de Gennes Q-tensor theory for describing nematic liquid crystals. This theory, despite its popularity with physicists, has been little studied by mathematicians and poses many interesting questions. In particular the lectures will describe the relation of the theory to other theories of liquid crystals, specifically those of Oseen-Frank and Onsager/Maier-Saupe.

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Quantum drift-diffusion models in semiconductor simulation
Li Chen, Tsinghua University, China

As the dimensions of semiconductor device decrease, quantum effects have to be included in the modeling of the transport phenomena. A brief outline of the quantum fluid dynamic model in semiconductor simulation will be shown in the first part of this talk. Then a list of the mathematical results on one of these models- quantum drift-diffusion model will be introduced, including the existence of global weak solution with different boundary conditions, long time behavior and semiclassical limit. The ideas ?entropy based methods- used in getting those results will be listed in the end.

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Sub-linear scaling algorithms for the study of the electronic structure of material
Carlos Garcia-Cervera, University of California, Santa Barbara, USA

I will discuss asymptotic-based algorithms for the study of the electronic structure of materials, in the context of density functional theory. I will illustrate the ideas using both the Kohn-Sham and orbital-free formulations.

This is joint work with Weinan E (Princeton University), and Jianfeng Lu (Princeton University).

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A nonlocal vector calculus with application to nonlocal boundary value problems
Max Gunzburger, Florida State University, USA

We develop a calculus for nonlocal operators that mimics Gauss' theorem and the Green's identities of classical vector calculus. The operators we treat do not involve derivatives. We then apply the nonlocal calculus to define variational formulations of nonlocal "boundary" value problems that mimic the Dirichlet and Neumann problems for second-order scalar elliptic partial differential equations. For the nonlocal variational problems, we show how one can derive existence and uniqueness results and also how, under appropriate limits, they reduce to their classical analogs. Although we do not report on this in this talk, the results are easily extended to vector elliptic equations, and in particular, to the peridynamics model for materials.

This is joint work with Richard Lehoucq.

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Blow-up criteria for the Navier-Stokes equations of multidimensional compressible fluids
Song Jiang, Institute of Applied Physics and Computational Mathematics, China

We study an initial boundary value problem and the Cauchy problem for the multidimensional Navier-Stokes equations of compressible isentropic and non-isentropic flows. We will report some recent blow-up criteria for the local strong solutions, including a criterion similar to the Beale-Kato-Majda criterion for ideal incompressible flows.

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Some recent theoretical and numerical contributions to stochastic homogenization and related problem
Claude Le Bris, École Nationale Des Ponts Et Chaussées, France

The talk will overview some recent contributions on several theoretical aspects and numerical approaches in stochastic homogenization. In particular, some variants of the theory of classical stochastic homogenization will be introduced. The relation between such homogenization problems and other multiscale problems in materials science will be emphasized. On the numerical front, some approaches will be presented, for acceleration of convergence in stochastic homogenization (representative volume element, variance reduction issues, etc) as well as for approximation of the stochastic problem when the random character is only a perturbation of a deterministic model. The talk is based upon a series of joint works with X. Blanc (CEA, Paris), PL. Lions (College de France, Paris), and F. Legoll, A. Anantharaman, R. Costaouec (ENPC, Paris).

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Quantum simulation of materials at micron scales and beyond
Gang Lu, California State University Northridge, USA

We present a multiscale modeling approach that can simulate multimillion atoms effectively via density functional theory. The method is based on the framework of the quasicontinuum (QC) approach with the density-functional theory (DFT) as its sole energetics formulation - there is no empirical input. The local QC part is formulated by the Cauchy-Born hypothesis with DFT calculations for strain energy and stress. The nonlocal QC part is treated by a DFT-based embedding approach, which couples DFT nonlocal atoms to local region atoms. The method-QCDFT-is applied to a nanoindentation study of an Al thin film, and the results are compared to a conventional QC approach. The results suggest that QCDFT represents a new direction for the quantum simulation of materials at length scales that are relevant to experiments. If time permits, an application of QCDFT to a novel magnetism-induced dislocation motion in NiAl alloys will be also presented.

1. Q. Peng, X. Zhang, L. Hung, E.A. Carter and G. Lu, "Quantum simulation of materials at micron scales and beyond", Phys. Rev. B, 78, 054118 (2008).
2. X. Zhang, C.Y. Wang and G. Lu, "Electronic structure analysis of self-consistent embedding theory for quantum mechanics/molecular mechanics simulations", Phys. Rev. B, 78, 235119 (2008).
3. X. Zhang and G. Lu, "Quantum mechanics/molecular mechanics methodology for metals based on orbital-free density functional theory", Phys. Rev. B, 76, 245111 (2007).
4. G. Lu, E. Tadmor and E. Kaxiras, ?From electrons to finite-elements: a concurrent multiscale approach for metals?, Phys. Rev. B 73, 024108 (2006).

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Long-time asymptotic behaviour of a multiscale rod-like model of polymeric fluids
Hui Zhang, Beijing Normal University, China

We investigate the long-time asymptotic behaviour of some multiscale rod-like models for polymeric fluids in various settings which include no flow, shear flow, general bounded domain with homogeneous and non-homogeneous Dirichlet boundary conditions on the velocity. We use entropy approach to obtain the decay rates of dynamic states converging to the equilibrium in some norms by virtue of the Csiszar-Kullback inequality.

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Surface diffusions on elastically stressed axisymmetric rods
Xiao Fan Li, Illinois Institute of Technology, USA

Many applications in materials involve surface diffusion of elastically stressed solids. Study of singularity formation and long-time behavior of such solid surfaces requires accurate simulations in both space and time.
Here we present a high-order boundary integral method for an elastically stressed solid with axi-symmetry due to surface diffusions. Using the high-order boundary integral method, we demonstrate that in absence of elasticity the cylinder surface pinches in a finite time at the axis of the symmetry and the universal cone angle of the pinching is found to be consistent with the previous studies based on a self-similar assumption. In the presence of elastic stress, we show that a finite-time, geometrical singularity occurs well before the cylindrical solid collapses onto the axis of symmetry, and the angle of the corner singularity on the cylinder surface is also estimated.

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Interfaces, surface energy and solid phase transformations
John Ball, University of Oxford, UK

Phase transformations in solids lead to interfaces between different variants of the product phase. In some materials these are atomistically sharp, while in others the interface thickness extends over a number of atomic spacings. The talk will discuss different models for describing such interfaces. This is joint work with Elaine Crooks (Swansea) and with Carlos Mora-Corral (Bilbao).

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Modeling nanoelectronics from atomic first principles
Hong Guo, McGill University, Canada

In this talk, I will discuss an aspect of nanoelectronic device theory, namely modeling nanoelectronics from atomic first principles. Some important issues of quantum transport theory in the atomic limit will be reviewed. I will then report a recent theoretical development for treating atomistic disorder in nonlinear and non-equilibrium quantum transport modeling. The theory uses non-equilibrium vertex corrections to handle the configurational average of random disorder at the density matrix level. Using this technique, we have analyzed spin injection in magnetic tunnel junctions with interface roughness as well as with oxygen vacancies in the tunnel barrier. Disorder effect is found to significantly alter spin polarized tunneling. We will also present results concerning surface roughness scattering in Cu interconnects. Comparison with existing experimental data will be presented.

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Analysis and computation for the ground states of two-component Bose-Einstein condensates with an external driving field
Yongyong Cai, National University of Singapore

In this talk, we prove existence and uniqueness results for the ground states of the coupled Gross-Pitaevskii equations for describing two-component Bose-Einstein condensates with an external driving field and obtain the limiting behavior of the ground states with large parameters. Efficient and accurate numerical methods based on continuous normalized gradient flow and gradient flow with discrete normalization are presented for computing the ground states numerically. A modified backward Euler finite difference scheme is proposed to discretize the gradient flows. Numerical results are reported to demonstrate the efficiency and accuracy of the numerical methods and show the rich phenomena of the ground sates in the problem.

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Meissner states of 3-dimensional superconductors
Xingbin Pan, East China Normal University, China

Meissner state of bulk superconductors of type II loses its stability as the applied magnetic field increases to a critical field H_S. The Meissner states are described by the solutions of a partial differential system in 3-dimensional space, which are called Meissner solutions. In this talk we shall show the existence of the Meissner solutions, and examine the convergence of the Meissner solutions to a solution of the limiting system as the Ginzburg-Landau parameter increases to infinity. Our results suggest that in order to determine the critical field H_S one needs to measure the maximum value of the tangential component of the induced magnetic field on the surface of the superconductor.

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Material prediction based on first principles method
Yuan Ping Feng, National University of Singapore

Materials design based on computational methods is playing a very important role in today?s materials science and engineering. Among various methods, the first-principles electronic structure method based on density functional theory is ideal for designing new materials because such methods do not require experimental inputs and prior knowledge on the materials. We have been using first-principles method to study properties of materials for future advanced technologies and to design new materials. Examples related to high dielectric constant oxides for applications in complementary metal-oxide-semiconductor (CMOS) devices, magnetic semiconductors for spintronics applications, etc., will be discussed.

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Coarse-graining molecular dynamics for crystalline solids
Xiantao Li, Pennsylvania State University, USA

I will discuss a coarse-grained model of molecular dynamics in crystalline solids. Such system usually has an underlying lattice structure, and the empirical potentials are of quite simple forms. Thus it provides a nice setting in which coarse-graining (CG) methods can be considered. The goal of the CG method is to reduce the huge
number of atomic degrees of freedom, and arrive at an effective model in which only a small number of atoms are explicitly involved. I will discuss how the CG is done in both the static and dynamic settings, the connections to continuum elasticity models, and the application to fracture and dislocation dynamics.

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Smectic energies and existence results for lquid crystals
Daniel Phillips, Purdue University, USA

We examine the problem of minimizing the Chen-Lubensky liquid crystal energy. This is a Ginzburg-Landau type energy where the smectic layering is described by a complex valued order parameter. In the case that the energy has a smectic C ground state we show that anchoring conditions on the smectic layering at the boundary are needed in order for minimizers to exist. We further give examples of strong and weak anchoring conditions that suffice.

This is joint work with Patricia Bauman and Jinhae Park.

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Stable solutions to the Lawrence-Doniach equations in tilted magnetic fields
Patricia Bauman, Purdue University, USA

We consider minimizers to the Lawrence-Doniach energy for layered uperconductors with nonlinear Josephson coupling. When the exterior magnetic field is nearly parallel to the layers and the Josephson constant is sufficiently small, we show that the global minimizer has no vortices in the layers. We estimate the upper and lower critical magnetic fields in different directions, and identify the pattern of the order parameters in this case.

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Applied partial differential equations: the Cambridge PDE group
Peter Markowich, University of Cambridge, UK

We report on the work of the Cambridge based PDE group, covering a wide range of applications, from nonlinear Schrödinger equations to social science modeling

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A generalization of the MacPherson - Srolovitz formula
Duc Thinh Le, Pennsylvania State University, USA

The MacPherson-Srolovitz formula has been recently established as a generalization of the two dimensional von Neumann - Mullins relation for microstructure coarsening. In this talk we present an extension of the MacPherson-Srolovitz formula under more general junction conditions.

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Traction and outflow boundary conditions for the incompressible Navier-Stokes equations
Jie Liu, National University of Singapore

We present numerical schemes for the incompressible Navier-Stokes equations (NSE) with open and traction boundary conditions. We use pressure Poisson equation (PPE) formulation and propose new boundary conditions for the pressure on the open or traction boundaries. After replacing the divergence free constraint by this pressure Poisson equation, we obtain an unconstrained NSE. For Stokes equation with open boundary condition on a simple domain, we prove unconditional stability of a first order semi-implicit scheme where the pressure is treated explicitly and hence is decoupled from the computation of velocity. Using either boundary condition, the schemes for the full NSE that treat both convection and pressure terms explicitly work well with various spatial discretizations including spectral collocation and $C^0$ finite elements. Moreover, when Reynolds number is of $O(1)$ and when the first order semi-implicit time stepping is used, time step size of $O(1)$ is allowed in benchmark computations for the full NSE. Besides standard stability and accuracy check, various numerical results including flow over a backward facing step, flow past a cylinder and flow in a bifurcated tube are reported. Numerically we have observed that using PPE formulation enables us to use the velocity/pressure pairs that do not satisfy the standard inf-sup compatibility condition. Our results extend that of H. Johnston and J.-G. Liu (J. Comp. Phys. 199 (1) 2004, 221--259) which deals with no-slip boundary conditions only.

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Comparisons between the sine-Gordon equation and perturbed nonlinear Schrodinger equations for modeling light bullets
Xuanchun Dong, National University of Singapore

Comparisons between the solution of the sine-Gordon (SG) equation and the solutions of perturbed nonlinear Schrodinger equations are studied numerically for the propagation of two space dimensional (2D) localized pulses in nonlinear dispersive optical media. We begin with the (2+1) sine-Gordon (SG) equation obtained as an asymptotic reduction in the two level dissipationless Maxwell-Bloch system, followed by the review on the perturbed nonlinear Schrodinger (NLS) equation in 2D for the SG pulse envelopes, which is globally well-posed and has all the relevant higher order terms to regularize the collapse of the standard critical NLS. The perturbed NLS is approximated by truncating the nonlinearity into finite higher order terms undergoing focusing-defocusing cycles.

Energy conservative and efficient semi-implicit Fourier pseudospectral discretizations for the SG and perturbed NLS are proposed. Numerical comparison results between the solutions of the SG and perturbed NLS as well as critical (cubic focusing) NLS are reported for the propa- gation of light bullets. Based on our extensive numerical experiments, the solution of the perturbed NLS as well as its finite terms truncations are in qualitative and quantitative agreement with the solution of SG for the light bullets propagation even after the critical collapse of the cubic focusing NLS. In contrast, the standard critical NLS is in qualitative agreement with SG only before its collapse.

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