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Publication 12-CNA-013

A Theory and Challenges for Coarsening in Microstructure

Katayun Barmak
Department of Applied Physics and Applied Mathematics
Columbia University
New York, NY 10027

Eva Eggeling
Fraunhofer Austria Research GmbH
Visual Computing
A-8010 Graz, Austria

Maria Emelianenko
Department of Mathematical Sciences
George Mason University
Fairfax, VA 22030

Yekaterina Epshteyn
Department of Mathematics
The University of Utah
Salt Lake City, UT, 84112

David Kinderlehrer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

Richard Sharp
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

Shlomo Ta'asan
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

Abstract: Cellular networks are ubiquitous in nature. Most engineered materials are polycrystalline microstructures composed of a myriad of small grains separated by grain boundaries, thus comprising cellular networks. The grain boundary character distribution (GBCD) is an empirical distribution of the relative length (in 2D) or area (in 3D) of interface with a given lattice misorientation and normal. During the coarsening, or growth, process, an initially random grain boundary arrangement reaches a steady state that is strongly correlated to the interfacial energy density. In simulation, if the given energy density depends only on lattice misorientation, then the steady state GBCD and the energy are related by a Boltzmann distribution. This is among the simplest non-random distributions, corresponding to independent trials with respect to the energy. Why does such simplicity emerge from such complexity? Here we an describe an entropy based theory which suggests that the evolution of the GBCD satisfi es a Fokker-Planck Equation, an equation whose stationary state is a Boltzmann distribution. The properties of the evolving network that characterize the GBCD must be identifi ed and appropriately upscaled or `coarse-grained'. This entails identifying the evolution of the statistic in terms of the recently discovered Monge-Kantorovich-Wasserstein implicit scheme. The undetermined diffusion coefficient or temperature parameter is found by means of a convex optimization problem reminiscent of large deviation theory.

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