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Recent Developments in Material Microstructure: a Theory of Coarsening

Published online by Cambridge University Press:  11 June 2015

K. Barmak,
E. Eggeling,
M. Emelianenko,
Y. Epshteyn,
D. Kinderlehrer,
R. Sharp and
S. Ta’asan
K. Barmak
Affiliation:
Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027 E-mail address: kb2612@columbia.edu
E. Eggeling
Affiliation:
Fraunhofer Austria Research GmbH, Visual Computing, A-8010 Graz, Austria E-mail address: eva.eggeling@fraunhofer.at
M. Emelianenko
Affiliation:
Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030 E-mail address: memelian@gmu.edu
Y. Epshteyn
Affiliation:
Department of Mathematics, The University of Utah, Salt Lake City, UT, 84112 E-mail address: epshteyn@math.utah.edu
D. Kinderlehrer
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213 E-mail address: davidk@cmu.edu
R. Sharp
Affiliation:
Globys Corporation, Seattle, WA, 98104 E-mail address: rsharp@gmail.com
S. Ta’asan
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213 E-mail address: shlomo@andrew.cmu.edu
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Abstract

Cellular networks are ubiquitous in nature. Most engineered materials are polycrystalline microstructures composed of a myriad of small grains separated by grain boundaries, thus comprising cellular networks. The recently discovered grain boundary character distribution (GBCD) is an empirical distribution of the relative length (in 2D) or area (in 3D) of interface with a given lattice misorientation and normal. During the coarsening, or growth, process, an initially random grain boundary arrangement reaches a steady state that is strongly correlated to the interfacial energy density. In simulation, if the given energy density depends only on lattice misorientation, then the steady state GBCD and the energy are related by a Boltzmann distribution. This is among the simplest non-random distributions, corresponding to independent trials with respect to the energy. Why does such simplicity emerge from such complexity? Here we describe an entropy based theory which suggests that the evolution of the GBCD satisfies a Fokker-Planck Equation, an equation whose stationary state is a Boltzmann distribution.

Keywords

microstructuregrain boundariestexture

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