An entropy based theory of the grain boundary character distribution

Katayun Barmak; Eva Eggeling; Maria Emelianenko; Yekaterina Epshteyn; David Kinderlehrer; Richard Sharp; Shlomo Ta'asan

doi:10.3934/dcds.2011.30.427

Article Contents

2011, Volume 30, Issue 2: 427-454. Doi: 10.3934/dcds.2011.30.427

This issue Previous Article Necessary optimality conditions for fractional difference problems of the calculus of variations Next Article Existence of minimizers for nonautonomous highly discontinuous scalar multiple integrals with pointwise constrained gradients

An entropy based theory of the grain boundary character distribution

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

Received: September 30, 2010

Revised: October 31, 2010

Published: May 2011

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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.

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References

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