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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

An entropy based theory of the grain boundary character distribution

Pages: 427 - 454, Volume 30, Issue 2, June 2011      doi:10.3934/dcds.2011.30.427

 
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Katayun Barmak - Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, United States (email)
Eva Eggeling - Fraunhofer Austria Research GmbH, Visual Computing, A-8010 Graz, Austria (email)
Maria Emelianenko - Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, United States (email)
Yekaterina Epshteyn - Department of Mathematics, The University of Utah, Salt Lake City, UT 84112, United States (email)
David Kinderlehrer - Center for Nonlinear Analysis and Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213-3890, United States (email)
Richard Sharp - Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, United States (email)
Shlomo Ta'asan - Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, United States (email)

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.

Keywords:  Coarsening, Texture Development, Large Metastable Networks, Large scale simulation, Critical Event Model, Entropy Based Theory, Free Energy, Fokker-Planck Equation, Kantorovich-Rubinstein-Wasserstein Metric.
Mathematics Subject Classification:  Primary: 37M05, 35Q80, 93E03, 60J60, 35K15, 35A15.

Received: October 2010;      Revised: November 2010;      Available Online: February 2011.

 References