From Grain Boundaries to Stochastic Homogenization: PIRE Workshop

Abstracts for the talks

Benny Davidovitch
University of Massachusetts Amherst
Imposing curvature on crystalline and non-crystalline sheets: shape deformations, grain boundaries, and asymptotic isometry
Imposing a shape with Gaussian curvature on a solid sheet, generates
in it elastic stress. This coupling between geometry (curvature) and
mechanics (stress) is a basic demonstration of Gauss’ theorema
Egregium, and underlies the morphological richness observed in solid
sheets, and their nontrivial response to exerted forces.
In this talk I will attempt to provide a unifying framework for
morphological transitions in elastic sheets with imposed curvature, by
using asymptotic analysis around ``tension field theory”. This singular
limit assumes a sheet with zero bending modulus under finite tensile
load. Considering crystalline sheets with small, finite bending modulus,
we predict a transition between patterns of wrinkles (shape
deformation) and grain boundaries. Considering a vanishing tensile
load, we predict a transition between two types of “asymptotic
isometry”: a developable type (which repels Gaussian curvature) and a
non-developable on (which accommodates the imposed Gaussian
curvature). The predicted transitions will be demonstrated through
examples from a few realistic systems.

Kaushik Dayal
Carnegie Mellon University
Applications of Objective Structures to Study Nanostructures
I will describe the Objective Structures framework introduced by James, and further developed by in our collaborations with him. The framework has enabled a systematic approach to a broad variety of problems relevant to the design and characterization of materials at the atomic level. I will describe some of these activities from the research groups at University of Minnesota and Carnegie Mellon. These applications include efficient new computational methods for the study of electron structure and transport in low dimensional materials, theoretical insights and computational methods for non-equilibrium molecular dynamics and Boltzmann equation, and the self-assembly of complex nanostructures.

Nicolas Dirr
Cardiff University
Entropic and gradient flow formulations of nonlinear diffusion
We consider nonlinear diffusion equations that arise as scaling limit of certain interacting particle systems,
the zero range process. We show that they are the gradient flow of the thermodynamic entropy of the process with respect to a weighted Wasserstein metric, where the weight is related to the diffusion of a tagged particle.
(Joint work with Mark Peletier, Marios Stamatakis and Johannes Zimmer)

Selim Esedoglu
University of Michigan Ann Arbor
Algorithms for anisotropic mean curvature flow of networks
Threshold dynamics is a very efficient algorithm for moving an interface (e.g. a surface in 3D) by mean curvature motion. It was proposed by Merriman, Bence, and Osher in 1989, and also extended to networks of surfaces in the same paper. This dynamics arises as gradient flow for the sum of the areas of the surfaces in the network, and plays a prominent role in materials science applications where it describes the motion of grain boundaries in polycrystals (such as most metals) under heat treatment.

Further extension of the algorithm to weighted mean curvature flow of networks, where the surface tension of each interface in the network may be distinct (unequal) and may depend on the direction of the normal, is of great interest for applications, but has remained elusive. We describe how to extend threshold dynamics to unequal and anisotropic (normal dependent) surface tensions. Joint work with Matt Elsey and Felix Otto.

Gero Friesecke
Technical University of Munich
Identifying atomic structure via diffraction
For the past decades, the method of choice for the determination of atomic structure in structural biology and nanoscience has been X-ray diffraction. This exploits the spectacular Bragg/Von Laue phenomenon that plane waves scattered at crystals yield discrete diffraction patterns. The catch is that a native assembly of proteins has to be broken and the protein needs to be crystallized, which is difficult and may lead to non-native forms. In my talk I discuss alternatives such as fiber diffraction which avoid crystallization at the expense of lower resolution in angular direction (Cochran, Crick, Vand 1952), and a recent theoretical advance which proposes incoming waveforms which exhibit fully discrete diffraction patterns when scattered at helices (Friesecke, James, Juestel, arXiv 1506.04240, 2015).

Sebastian Fähler
Leibniz Institute for Solid State and Materials Research Dresden
Formation of a hierarchical martensitic microstructure
The martensitic microstructure is decisive for most functionalities of shape memory alloys, including pseudoelastic, pseudoplastic, magnetic shape memory and caloric effects. Here we examine the formation of a hierarchical microstructure by a combination of high resolution ex-situ and in-situ experiments and theoretical models. As model system we select epitaxial Ni-Mn-Ga films as the high fraction of surface to volume makes surface experiments representative. Moreover, the high aspect ratio of a film minimizes mechanical interactions between different regions, making films a statistically relevant ensemble of many nucleation events.
The need to form phase boundaries requires introducing nanotwin boundaries. Interaction energy results in an ordered arrangement, appearing as a modulated phase and forming a-b twin boundaries. Nucleation and growth within the volume requires mesoscopic type I and II twin boundaries. As a consequence of different nucleation sites also macroscopic twin boundaries form. To conclude, the resulting hierarchical microstructure is not result of a global minimization of energy but determined by the most easy transformation path, ending in a metastable configuration containing a well-defined arrangement of many different types of twin boundaries.
This work is supported by DFG through SPP 1599

Elizabeth Holm
Carnegie Mellon University
New mesoscale approaches to microstructural evolution and analysis
As the link between the atomic and continuum scales,
microstructural models and simulations are a
critical element of computational materials science.
However, while standardized computational tools have become
widely accepted at the electronic, atomic, and continuum
scales, mesoscale simulation methods remain diverse and
are still evolving. In part, this stems from the breadth of
phenomena included under the microstructural umbrella, but it is also due to intrinsic limitations in the prevailing methods. Discrete microstructural evolution simulations (i.e. the Monte Carlo Potts model, probabilistic cellular automata) are computationally efficient and easy to implement; they suffer, however, from various artifacts of the underlying computational lattice. This talk will introduce a lattice-­‐free, discrete kinetic Monte Carlo method. The Material Point Monte Carlo (MPMC) method uses
randomly placed material points to overcome the unphysical
effects of lattice anisotropy on interfacial and volumetric
energies and to enable the correct evolution of systems that
undergo shape distortion. MPMC simulations retain most of the computational benefits of other discrete methods and
provably reproduce the physics of interface motion by surface and bulk driving forces. One goal of mesoscale simulations is to understand rare events, such as failure
initiation, hot spot formation, and abnormal grain nucleation. Because these phenomena typically arise from the
localization of long-­‐range interactions, identifying an
incipient rare event is challenging. This talk will present a network theory approach to analyzing long-­‐range grain
neighborhoods using the random walk graph kernel. Utilizing
data from microstructural evolution simulations, a machine-­‐learning system can be trained to classify potential abnormal growth events within the grain network. By operating beyond a nearest-­‐neighbor or mean-­‐field interaction distance, this method has promise for characterizing a number of long-­‐range microstructural
Elizabeth A. Holm, Philip Goins, and Brian DeCost

Hans Knüpfer
Heidelberg University
Low density phases in a uniformly charged liquid
We consider a macroscopic limit for the Ohta-Kawasaki energy. This model has been used to
described to describe phase separation for diblock-copolymers. We first investigate existence and shape of
minimizers of the energy with prescribed volume (of the one phase) in the full space setting. We then consider situation of
periodic configurations with prescribed density of the minority phase. We show that in a certain regime, the energy
Gamma-converges to a homogenized problem. This is joint work with C. Muratov and M. Novaga.

Roman Kotecký
University of Warwick
Metastability of an interacting system of particles in continuum
One of the obstructions to truly microscopic statistical physics theory of elasticity
lies in the fact that one should consider a metastable system of particles. While metastability is reasonably
well understood for lattice systems, particles in continuum are more difficult to analyse.
Even a rigorous proof of just the existence of a phase transition has been achieved only for few simple models.
We will discuss the need for considering a metastable system as well as formulate/explain the results
concerning metastability for the particular case of the Widom-Rowlinson model.
Based on a joint work with F. den Hollander, S. Jansen, and E. Pulvirenti.

Claude Le Bris
Local profiles for elliptic problems at different scales
This is joint work with Xavier Blanc (University Denis Diderot, Paris) and Pierre-Louis Lions (College de France, Paris).
We present a general approach to approximate at the fine scale the solution to an elliptic equation with oscillatory
coefficient when this coefficient consists of a ”nice” (in the simplest possible case say periodic) function which is, in some sense to be made precise, perturbed. The approach is based on the determination of a local profile, solution to an equation similar to the corrector equation in classical homogenization. We prove that this equation has a unique solution, in various functional settings depending upon the perturbation: local
perturbation, two different periodic structures separated by a common interface, etc.

Govind Menon
Brown University
A kinetic model for 2D grain boundary coarsening
A fundamental aspect of 2D cellular networks with isotropic line tension is the Mullins-von Neumann n 6 rule: the rate of change of the area of a (topological) n-gon is proportional to n 6. As a consequence, cells with fewer than 6 sides vanish in finite time, and the network coarsens. Numerical and physical experiments have revealed a form of statistical self-similarity in the long time dynamics.

We propose a kinetic description for the evolution of such networks. The ingredients in our model are an elementary N particle system that mimics essential features of the von Neumann rule, and a hydrodynamic limit theorem for population densities when N →∞. This model is compared with a set of models derived in the physics and materials science communities, as well as extensive numerical simulations by applied mahematicians.

This is joint work with Joe Klobusicky (Brown University and Geisinger Health Systems) and Bob Pego (Carnegie Mellon University).


Eckhard Quandt
Christian-Albrechts-Universität zu Kiel
Ultra low fatigue shape memory films
Functional shape memory alloys need to operate reversibly and repeatedly. This is especially crucial for many future applications such as artificial heart valves or elastocaloric cooling, where more than ten million transformation cycles will be required. Here we report on the discovery of an ultra-low fatigue shape memory alloy film system based on TiNiCu that allows at least ten million transformation cycles. We found these films contain Ti2Cu precipitates embedded in the base alloy that serve as sentinels to ensure complete and reproducible transformation in the course of each memory cycle. In addition we found an almost perfect fulfillment of the compatibility conditions in TiNiCuCo thin films that also show ultra low fatigue.

Lenya Ryzhik
Stanford University
The weakly random Schroedinger equation: homogenization and the kinetic limit
We consider a Schroedinger equation with a weakly random time-independent
potential. When the correlation function of the potential is, roughly speaking,
of the Schwartz class, it has been shown by Spohn (1977), and Erdos
and Yau (2001) that the kinetic limit holds -- the expectation of the
phase space energy density of the solution converges
to the solution of a kinetic equation. We "extend" this result to potentials
whose correlation functions satisfy (in some sense) "sharp" conditions,
and also prove a parallel homogenization result for slowly varying initial conditions.
I will explain the quotation marks above and make some speculations
on the genuinely sharp conditions on the random potential that separate
various regimes. This talk is a joint work with T. Chen and T. Komorowski

Angkana Rüland
University of Oxford
Surface Energies Arising in Microscopic Modeling of Martensitic Transformations
In this talk I present and analyze a two-well Hamiltonian on a 2D atomic lattice for low energy states. The two wells of the Hamiltonian consist of the SO(2) orbit of two rank-one connected matrices. Seeking to obtain an understanding of the origin of surface energies, I focus on special (1 +

Date and Location

July 20 - 23, 2015
University of Leipzig
Augustusplatz 10
04109 Leipzig

Scientific Organizers

  • Irene Fonseca, Carnegie Mellon University
  • Richard James, University of Minnesota
  • Stephan Luckhaus, Universität Leipzig, Institut für Mathematik
  • Felix Otto, MPI für Mathematik in den Naturwissenschaften
  • Peter Smereka, University of Michigan

Administrative Contact

Valeria Hünniger
Max Planck Institute for Mathematics in the Sciences

Saskia Gutzschebauch
Max-Planck-Institut für Mathematik in den Naturwissenschaften

Katja Heid
Max Planck Institute for Mathematics in the Sciences

05.04.2017, 12:53