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Abstracts and Talk Materials
NSF PIRE Summer School for Graduate Students: New frontiers in multiscale analysis and computing for materials
June 21 - 29, 2012

Dislocations are defects in solid crystalline structures that are characterized by their Burgers vectors, which describe the lattice mismatch. The interest in their study lies in the influence that their presence has on other properties of the material itself. We describe the energy and the dynamics (the model for which is due to Cermelli and Gurtin) for a system of screw dislocations subject to anti-plane shear. A variational setup is constructed to find minimizers for the energy functional associated with a system of screw dislocations in an elastic medium via a finite-core regularization of the elastic-energy functional. We give an asymptotic expansion of the minimum energy as the core radius tends to zero, allowing us to eliminate this internal length scale of the problem. The renormalized energy is the regular part of the asymptotic expansion. The motion of the dislocations is governed by a system of ordinary differential equations, which is obtained by taking the gradient of the renormalized energy with respect to the position of the dislocations. The ODE system is discontinuous because only certain directions of motion are allowed. Thus, we show existence and uniqueness in the sense of Filippov. This is joint work with Irene Fonseca, Giovanni Leoni, and Marco Morandotti.

In the context of renewable energy, there has been a significant amount of interest in organic photovoltaic (OPV) devices - solar cells made of organic materials (typically polymers) instead of traditional inorganic semiconductors. Because of the specific properties of the organic materials, the devices are more complicated than their inorganic counterparts. Most importantly, it becomes necessary to use two different materials, one for transporting electrons and one for transporting positive charge carriers analogous to the holes of semiconductor theory. The interface between these two materials becomes a vital part of the system, and appropriate modeling and simulation is of the utmost importance.

In particular, the concentrations of holes are generally significantly smaller than the concentrations of electrons in the electron-transport material (and vice-versa in the hole-transport material). Although this seems to motivate a single-carrier model (with each material containing only the dominant carrier), such a limit is difficult to justify mathematically. We present a reaction-diffusion model for electrons, holes, and excitonic states for which the interface is a thin region around the interface over which the coefficients of the equation change smoothly. We then derive 1-D asymptotic models for the charge carriers and the current in a simple bilayer device and demonstrate the role of the minority carrier. We show Hybrid Discontinuous Galerkin finite-element numerical simulations for more complicated interface geometries and discuss the relation to our simulations.

(Joint work with Klemens Fellner, Peter Markowich, and Marie-Therese Wolfram)

We derived a user friendly version of Cofactor Conditions to investigate compatibility conditions for interfaces between branching martensite variants and austenite. In twinned martensite, if either twinning plane normal or twinning shear vector is perpendicular to the eigenvector associated with the middle eigenvalue of transformation stretch tensor, there exist compatible interfaces between martensitic lamellas and austenite. The microstructures for three types of twin system are predicted based on Cofactor Conditions.

This talk is concerned with the stability and asymptotic stability at large time of solutions to a system of equations, which includes the Lifschitz-Slyozov-Wagner (LSW) system in the case when the initial data has compact support. The main result is a proof of weak global asymptotic stability for LSW like systems. Comparison to a quadratic model plays an important part in the proof of the main theorem when the initial data is critical. The quadratic model extends the linear model of Carr and Penrose. This is joint work with Barbara Niethammer.

Poster - Universal patterns in self-assembling systems
December 31, 1969

Self-assembly as a manufacturing technique buys ease of assembly with complexity in interaction design. However, it turns out some structures are universal: they appear in wide ranges of systems with some generic features, meaning they are much easier to achieve than might otherwise be believed. In our work we have used variations of the spherical spin model to predict and explain some classes of universal patterns in spin and particle systems. We are especially interested in applications towards building blocks for self-assembly.

Many materials, including most metals and ceramics, are composed of crystallites (often called grains), which are differentiated by their crystallographic orientation. Classical models describing annealing-related phenomena for these materials involve multiphase curvature-driven motion. The distance function-based diffusion-generated motion (DFDGM) algorithm has previously been demonstrated to be an accurate and efficient means for simulating the evolution described by the isotropic version of the Mullins model for grain growth. A recent extension of DFDGM to allow for unequal surface energies depending on the misorientation between adjacent grains while correctly maintaining the standard Herring boundary conditions is described here. Preliminary large-scale simulation results in two and three dimensions are presented.

The mechanism of homogeneous dislocation nucleation in a defect free crystal under cylindrical nano-indentation has been studied by performing atomistic simulations. Previous work has shown that the nucleation process is governed by vanishing of energy associated with a single normal mode that exhibits a lengthscale that scales in an anomalous way with the geometrical loading parameters (indenter radius and film thickness). Here, we show that these scalings that were previously observed for a single particular crystallographic orientation in a two dimensional Lennard-Jones system are generic with respect to the lattice orientation.

This work in harmonic analysis addresses the study of oscillatory behavior of functions in the context of bilinear operators. Bilinear operators are transformations that combine two waves into a new one. Some new almost orthogonality estimates are obtained, which provide understanding of interactions between waves oscillating at different frequencies. Using these estimates we are able to obtain new ways of quantifying properties of the resulting wave in terms of the initial waves. Estimates of this type are a bilinear version of Littlewood-Paley estimates and are used to justify useful frequency decompositions. Among other applications, our Littlewood-Paley estimates give a complete characterization of the continuity of certain operators called bilinear Calderón-Zygmund operators which are of great relevance in harmonic analysis.

Poster - Regulatory Role of Fibrin Network in Thrombus Growth
December 31, 1969

To restrict the loss of blood following rupture of blood vessels, the human body rapidly forms a clot consisting mainly of platelets and fibrin. However, to prevent formation of a pathological clot within vessels (thrombus) as a result of vessel damage or dysfunction, the response must be regulated, and clot formation must be limited.

In the present work we tested a previously unrecognized mechanism limiting growth of small asymptomatic thrombi. The mechanism suggests that the fibrin network overlaying a thrombus limits its growth by regulating the transport of proteins and reducing the local fluidic stresses on platelets inside the thrombus. The analysis integrating experiments of protein diffusion and fluid permeation with the hemodynamic thrombus model revealed that permeability of the fibrin network and protein diffusivity are the key factors determining transport of blood proteins inside the thrombus.

Poster - Multiscale Mechanics with Long-Range Electrostatic Forces
December 31, 1969

A key challenge in modeling ferroelectrics and other electromechanically coupled materials is the long-range nature of the electrostatic fields. The electrostatic fields, in addition to being long-ranged, are not confined to the sample, but instead are present even in the surrounding medium. Typical approaches to this problem involve assumptions of periodicity or other highly restrictive approximations. We develop a real-space multiscale method to accurately and efficiently deal with electric fields in complex geometries and with realistic boundary conditions. Our approach handles the short-range atomic interactions with the quasicontinuum method. The long-range electrostatic interactions are handled by extending (from literature) the thermodynamic limits of lattices of dipoles to complex lattices of charges. We apply the method to understand the deformation of a ferroelectric with complex geometries and subject to various mechanical and electrostatic loading conditions.

Keywords of the presentation: energy minimizing patterns; competing interactions; singular perturbations; soft condensed matter

I will discuss a problem of energy-driven pattern formation, in which the appearance of two distinct phases caused by short-range attractive forces is frustrated by a long-range repulsive force. I will focus on the regime of strong compositional asymmetry, in which one of the phases has very small volume fraction, thus creating small “droplets” of the minority phase in a “sea” of the majority phase. I will present a setting for the study of Gamma-convergence of the governing energy functional in the regime leading to many droplets. The Gamma-limit and the properties of almost minimizers with prescribed limit density will then be established in the important physical case when the long-range repulsive force is Coulombic in two space dimensions. This is joint work with D. Goldman and S. Serfaty.

Poster - Young measures supported on invertible matrices
December 31, 1969

In non-linear elasticity one often encounters energy minimization on subsets of appropriate Sobolev spaces, where the energy density is not quasiconvex. Hence, the existence of a Sobolev-space minimizer is not guaranteed. A standard procedure to ensure solvability is the so-called relaxation of the energy functional through (gradient) Young measures to a broader space. This procedure usually requires the energy density to have growth of order p, 1 < p < infty, which is, however, in conflict with a natural physical blow-up requirement, namely that the deformation of a volume element into a point needs infinite energy. In this work we analyzed a subset of Young measures (in the gradient case we treated p = infty) allowing for energy densities with the aforementioned blow-up feature. This is a joint work with Barbora Benesova and Martin Kruzik.

Smoluchowski's coagulation equation is related in remarkable ways to models of random shock clustering and the merging of ancestral trees. I plan to discuss a) work with Iyer and Leger on criteria for the asymptotic self-similarity of clan size distributions in branching processes, and b) how kinetic coagulation models are tied to Menon and Srinivasan's discoveries that indicate complete integrability for Burgers turbulence.

Poster - Finite Element Methods for Geometric Problems
December 31, 1969

We focus on the approximation of nonlinear partial differential equations occurring from geometric problems. Main interest is a compactness result for discrete saddle points of conformally invariant nonlinear elliptic functionals. Two applications have been analyzed and numerically realized so far, harmonic maps into submanifolds and surfaces of prescribed variable mean curvature.

In a second project we work on a mathematical model for biological cells and their behavior in the presence of chemical and physical forces. We introduce a novel elasticity functional that describes the coupling of a director field on a membrane and its curvature.

We present a unified KMC model of compound semiconductor growth. The crucial feature of this model is that it explicitly takes the different species into account and is not bound by the typical solid-on-solid constraint. Although this work is generally applicable to many systems, here we have focussed GaAs. The model was primarily calibrated with our experiments on homoepitaxial growth. Nevertheless it was able capture a wide range of physical phenomena known to occur in such systems. In particular, we can successfully predict the surface termination as function growth conditions. The simulations were then applied to droplet epitaxy and crystallization by As over-pressure, over a broad range of growth conditions. Statistics derived from the simulations such as droplet size and density compare well with experimental data. The model faithfully also captured several experimentally observed phenomena including formation quantum dot formation and nanorings. Other structures include polycrystalline shells and liquid Ga cores were also observed. An analytical model is developed that explains the relationship between such phenomena and growth conditions, yielding key insight into the mechanisms behind their appearance which would be impossible to infer from experiments alone.

Poster - Gamma-convergence approach to quasi-continuum methods
December 31, 1969

In this poster we present some results about the verification of quasi-continuum methods in the context of fracture mechanics. To this end we start from an one-dimensional system and consider a chain of atoms with nearest and next-to-nearest neighbour interactions of Lennard-Jones type. To mimic QC methods we approximate the second neighbour interactions by certain nearest neighbour interactions in some regions and pick some atoms (repatoms) and let the deformation of the other atoms depend just on the deformation of this repatoms. We derive a development by Gamma-convergence of this QC approximation and com- pare the limiting functional and its minimizers with those obtained for a fully atomistic system, which was derived by Scardia, Schlömerkemper and Zanini.

(Joint work with Anja Schlömerkemper)

Keywords of the presentation: microstructures; infinite-rank laminates; T3 configurations; Monoclinic-I martensite

The most common shape memory alloys are monoclinic-I martensite. We study their zero energy states and have two surprising results:

First, there is a five-dimensional continuum in which the energy minimising microstructures are T3s, i.e. in nite-rank laminates. To our knowledge, this is the fi rst real material in which T3s occur. We discuss some of the consequences of this discovery.

Second, there are in fact two types of monoclinic-I martensite, which di ffer by their convex poytope structure but not by their symmetry properties. It happens that all known materials belong to one of the two types. We explore whether materials belonging to the other type would have superior properties since they have di erent zero-energy states.

Our analysis uses algebraic methods, in particular the theory of convex polytopes.

Poster - Passage from discrete to continous systems
December 31, 1969

I present some recent work on a discrete to continuum modelling in the context of fracture mechanics. In joint work with L. Scardia and C. Zanini we start from a one-dimensional chain of atoms with nearest and next-to-nearest neighbour interactions of Lennard-Jones type. The work is based on Gamma-convergence methods. In particular we derive the Gamma-limit and the Gamma-limit of first order, which gives formulas for the surface energy contributions. A so-called uniformly Gamma-equivalent theory then yields a rigorous derivation of Griffith' model of fracture mechanics.

Rayleigh--Benard convection is the buoyancy-driven and conduction-limited flow of a Newtonian fluid that is heated from below and cooled from above. The strength of the temperature forcing is encoded in one dimensionless parameter: the system height $H$. We are interested in the regime where the temperature forcing is strong, $Hgg1$. In this case, the flow pattern shows a clear separation of the relevant heat transfer mechanisms: thin laminar boundary layers, in which heat is essentially conducted, and a large bulk, in which convection is dominant. While the temperature field shows a linear profile in the boundary layers, the bulk dynamics are rather chaotic.

We present two new results: 1) In joint work with F. Otto, we show that --- despite of complex pattern in the chaotic regime --- the average upward heat flux, measured in the so-called Nusselt number, is independent of the system height up to double-logarithmic corrections. This result improves earlier works of Constantin & Doering, and Doering, Otto & Reznikoff. 2) Based on new estimates on the temperature field in the boundary layer, we show that the temperature profile is indeed essentially linear in the boundary. Again, the results are uniform in $H$ up to logarithmic corrections.

Coarsening dynamics are gradient flow dynamics. This means, the evolution of an out-of-equilibrium system towards relaxation follows the steepest descent in an energy landscape. In many two-phase systems, the energy is proportional to the area of the interfacial layer between these phases. Therefore, descent in an energy landscape corresponds to reduction of the interface --- the pattern formed by the two phases "coarsens".

We consider the demixing process of a binary mixture of two viscous liquids as an example of such coarsening dynamics. Here, diffusion and convection are two possible transport mechanisms. It turns out, that each mechanism becomes relevant at a different stage of the evolution, which gives rise to two different coarsening rates. Using a method proposed by R. V. Kohn and F. Otto, we investigate the crossover between these rates in terms of (time averaged) upper bounds. Our analysis makes use of a Monge--Kantorowicz distance with linear/logarithmic cost as a proxy for the intrinsic distance, which is not known explicitly. This is joint work with Y. Brenier, F. Otto, and D. Slepcev.

Predicting the evolution of complex systems with a range of temporal and spatial scales requires a multi-scale computational approach. A multi scale framework typically consists of a macroscale and microscale description of the system. For concurrent computations, efficient scale communication procedure (reconstruction) between the macro and micro-scales is required. Existing multiscale frameworks assume equilibrated microscale and hence trivial reconstruction procedures suffice. For non-equilibrated systems an objective estimation of the microscale is required. The present work employs an Information theoretic approach based on Bregman divergence for reconstruction of the microscale. The multi-scale algorithm uses a learning strategy to adaptively incorporate the microscale information into the generator function of the Bregman divergence.

Percolation in nanorod dispersions induces extreme properties, with large variability near the percolation threshold. We investigate electrical properties across the dimensional percolation phase diagram of sheared nanorod dispersions [Zheng et al., Adv. Mater. 19, 4038 (2007)]. We quantify bulk average properties and corresponding fluctuations over Monte Carlo realizations, including finite size effects. We also identify fluctuations within realizations, with special attention on the tails of current distributions and other rare nanorod subsets which dominate the electrical response.

In this poster we present the models of von Karman plate, von Karman shell and bending plate model for the thin elastic body made of oscillatory changing material. We derive the models by means of Gamma convergence, starting from the equations of 3D elasticity. In the asymptotic analysis we deal with two small parameters, namely the thickness of the body and the oscillations of the material, and the obtained models depend on the relation between these two parameters. Here we present the case when they are on the same scale. Since we are in small strain regime one expects that the model depends only on the second derivative of the stored energy density function, which is convex in strain, and thus one can use two-scale analysis (following the oscillations of the material) to justify the models.

We propose an unconditionally stable semi-implicit time discretization of the phase field crystal evolution. It is based on splitting the underlying energy into convex and concave parts and then performing $H^{-1}$ gradient descent steps implicitly for the former and explicitly for the latter. The splitting is effected in such a way that the resulting equations are linear in each time step and allow an extremely simple implementation and efficient solution. We provide the associated stability and error analysis as well as numerical experiments to validate the method's efficiency.

(Joint work with Matt Elsey)

A new fracture model which takes into the account a presence of a curvature dependent surface tension on the crack boundary is studied on the examples of a curvilinear crack and an interface crack between two dissimilar materials. A linear elasticity model is assumed for the behavior of the material in the bulk. A non-linear boundary condition which includes a surface tension depending on the curvature of the crack is given on the crack surface. Both problems are reduced to the systems of integro-differential equations. It is shown that the presence of the surface tension eliminates the classic power singularities of the order 1/2 at the tips of the crack and, additionally, removes oscillating singularities in the case of an interface crack. Some components of the stresses and the strains may still posses weaker logarithmic singularities.

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