# American Institute of Mathematical Sciences

May  2011, 30(2): 427-454. doi: 10.3934/dcds.2011.30.427

## An entropy based theory of the grain boundary character distribution

 1 Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, United States 2 Fraunhofer Austria Research GmbH, Visual Computing, A-8010 Graz, Austria 3 Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, United States 4 Department of Mathematics, The University of Utah, Salt Lake City, UT 84112, United States 5 Center for Nonlinear Analysis and Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890 6 Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, United States, United States

Received  October 2010 Revised  November 2010 Published  February 2011

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.
Citation: Katayun Barmak, Eva Eggeling, Maria Emelianenko, Yekaterina Epshteyn, David Kinderlehrer, Richard Sharp, Shlomo Ta'asan. An entropy based theory of the grain boundary character distribution. Discrete & Continuous Dynamical Systems, 2011, 30 (2) : 427-454. doi: 10.3934/dcds.2011.30.427
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