Publication 11-CNA-001
An Entropy Based Theory of the Grain Boundary Character Distribution
Katayun Barmak
Department of Materials Science and Engineering
Carnegie Mellon University
Pittsburgh, PA 15213
katayun@andrew.cmu.edu
Eva Eggeling
Fraunhofer Austria Research GmbH
Visual Computing
A-8010 Graz, Austria
eva.eggeling@fraunhofer.at
Maria Emelianenko
Department of Mathematical Sciences
George Mason University
Fairfax, VA 22030
memelian@gmu.edu
Yekaterina Epshteyn
Department of Mathematics
The University of Utah
Salt Lake City, UT, 84112
epshteyn@math.utah.edu
David Kinderlehrer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
davidk@andrew.cmu.edu
Richard Sharp
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
rwsharp@cmu.edu
Shlomo Ta'asan
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
shlomo@andrew.cmu.edu
Abstract: Cellular networks are ubiquitous in nature. They exhibit behavior on many different length and time scales and are generally metastable. Most technologically useful materials are polycrystalline microstructures composed of a myriad of small monocrystalline grains separated by grain boundaries. The energetics and connectivity of the grain boundary network plays a crucial role in determining the properties of a material across a wide range of scales. A central problem in materials science is to develop technologies capable of producing an arrangement of grains--a texture--appropriate for a desired set of material properties. Here we discuss the role of energy in texture development, measured by a character distribution. We derive an entropy based theory based on mass transport and a Kantorovich-Rubinstein-Wasserstein metric to suggest that, to first approximation, this distribution behaves like the solution to a Fokker-Planck Equation.
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