# 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 www.FerroicCooling.de.

**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

phenomena.

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***CERMICS - ENPC***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

Germany

## 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