An Entropy Based Theory of the Grain Boundary Character Distribution

K. Barmak Department of Materials Science and Engineering Carnegie Mellon University Pittsburgh, PA 15213 katayun@andrew.cmu.edu

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

M. Emelianenko Department of Mathematical Sciences George Mason University Fairfax, VA 22030 memelian@gmu.edu

Y. Epshteyn Department of Mathematics The University of Utah Salt Lake City, UT, 84112 epshteyn@math.utah.edu

D. Kinderlehrer Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213 davidk@andrew.cmu.edu

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

S. 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 ditechnologically 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 rst approximation, this distribution behaves like the solution to a Fokker-Planck Equation.