B08
Curvature Effects in Molecular and Spin Systems

Understanding Crystallization

Many basic phenomena in solid mechanics like dislocations or plastic and elastic deformation are in fact discrete operations: small breakdowns of perfect crystalline order. The goal of this project is thereofore to address the phenomenon of crystallization, and its breakdowns, from the point of view of energy minimization.

Scientific Details+

How do the well known phenomenological continuum theories of solid mechanics (linear and nonlinear elasticity theory, plasticity theory, fracture mechanics) emerge from discrete, atomistic models? This fundamental question lies at the heart of a great deal of current research in materials science and materials engineering, yet remains very poorly understood on a mathematical level. A key bottleneck is that we don't understand crystallization, that is to say the fact that under many conditions, atoms self-assemble into crystalline order and special geometric shapes. This is a main bottleneck because all the basic phenomena in solid mechanics (dislocations, grains, fracture, plastic and elastic deformation) are small or localized breakdowns of perfect crystalline order. The goal of the project is to address the phenomenon of crystallization, and its breakdowns, from the point of view of energy minimization. In particular, we aim to extend available results on crystalline order and shape from purely combinatorial energies to soft potentials which allow for elastic modes, and develop methods for the rigorous passage from these discrete models to continuum surface energy functionals and elastic energy functionals. Our mathematical approach will rely on combining methods from three areas: (i) atomistic mechanics and its recently developed generalized convexity notions, (ii) Gamma convergence techniques from the calculus of variations, and - crucially and as far as we know for the first time in our context - (iii) discrete differential geometry, which is a central theme in other projects of the SFB-Transregio.

Publications+

Papers
From statistical polymer physics to nonlinear elasticity

Authors: Cicalese, Marco and Gloria, Antoine and Ruf, Matthias
Journal: preprint
Date: Sep 2018
Download: arXiv

Continuum limit and stochastic homogenization of discrete ferromagnetic thin films

Authors: Braides, A. and Cicalese, M. and Ruf, M.
Journal: Analysis & PDE, (2018), vol. 11, no.2, 499-553.
Date: Mar 2018
Download: external

Gamma-convergence analysis of a generalized XY model: fractional vortices and string defects

Authors: Badal, R. and Cicalese, M. and Luca, L. De and Ponsiglione, M.
Journal: Commun. Math. Phys. 358 (2018), no. 2, 705–739
Date: Mar 2018
Download: external

Hemihelical local minimizers in prestrained elastic bi-strips

Authors: Cicalese, Marco and Ruf, Matthias and Solombrino, Francesco
Journal: Zeitschrift für angewandte Mathematik und Physik, (2017),68:122
Date: Dec 2017
Download: external

Crystallization in two dimensions and a discrete Gauss-Bonnet theorem

Authors: Luca, Lucia De and Friesecke, Gero
Journal: J Nonlinear Sci 28, 69-90, 2017
Date: Jun 2017
Download: external

Classification of Particle Numbers with Unique Heitmann-Radin Minimizer

Authors: Luca, Lucia De and Friesecke, Gero
Journal: J. Stat. Phys. 167, Issue 6, 1586–1592, 2017
Date: Apr 2017
Download: external

Interfaces, modulated phases and textures in lattice systems

Authors: Braides, A. and Cicalese, M.
Journal: Arch. Rat. Mech. Anal., 223, (2017), 977-1017
Date: Feb 2017
Download: external

Crystalline Motion of Interfaces Between Patterns

Authors: A. Braides, M. Cicalese and Yip, N.K.
Journal: Journal of Statistical Physics October 2016, Volume 165, Issue 2, pp 274–319
Date: Oct 2016
Download: external

Domain formation in magnetic polymer composites: an approach via stochastic homogenization

Authors: Alicandro, R. and Cicalese, M. and Ruf, M.
Journal: Arch. Rat. Mech. Anal., 218(2):945--984
Date: 2015
Download: arXiv

Frustrated ferromagnetic spin chains: a variational approach to chirality transitions

Authors: Cicalese, M. and Solombrino, F.
Journal: Journal of Nonlinear Science, 25(291-313)
Date: 2015

Twisted x-rays: incoming waveforms yielding discrete diffraction patterns for helical structures

Authors: G. Friesecke, R. D. James, and Jüstel, D.
Date: 2015
Download: arXiv

Metastability and dynamics of discrete topological singularities in two dimensions: a Gamma-convergence approach

Authors: R. Alicandro, L. De Luca, A. Garroni and Ponsiglione, M.
Journal: Archive for Rational Mechanics and Analysis, 214(1):269--330
Date: 2014


PhD thesis
Crystalline Order, Surface Energy Densities and Wulff Shapes: Emergence from Atomistic Models

Author: Au Yeung, Yuen
Advisor: Gero Friesecke
Date: 2013
Download: external


Team+

Prof. Dr. Marco Cicalese   +

Projects: B08
University: TU München
E-Mail: cicalese[at]ma.tum.de
Website: http://www-m7.ma.tum.de/bin/view/Analysis/WebHome


Prof. Dr. Gero Friesecke   +

Projects: B08
University: TU München
E-Mail: gf[at]ma.tum.de
Website: http://www-m7.ma.tum.de/bin/view/Analysis/WebHome


Annika Bach   +

Projects: B08
University: TU München
E-Mail: annika.bach[at]ma.tum.de


Rufat Badal   +

Projects: B08
University: TU München
E-Mail: badal[at]ma.tum.de


Dr. Gianluca Orlando   +

Projects: B08
University: TU München
E-Mail: orlando[at]ma.tum.de


Arseniy Tsipenyuk   +

Projects: B08
University: TU München
E-Mail: tsipenyu[at]ma.tum.de