Publication 10-CNA-014
Critical Events, Entropy, and the Grain Boundary Character Distribution
Katayun Barmak
Department of Materials Science and Engineering
Carnegie Mellon University
Pittsburgh, PA 15213
katayun@andrew.cmu.edu
Eva Eggeling
Fraunhofer Austria Research GmbH
Visual Computing
A-8010 Graz, Austria
eva.eggeling@fraunhofer.at
Maria Emelianenko
Department of Mathematical Sciences
George Mason University
Fairfax, VA 22030
memelian@gmu.edu
Yekaterina Epshteyn
Department of Mathematics
The University of Utah
Salt Lake City, UT, 84112
epshteyn@math.utah.edu
David Kinderlehrer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
davidk@andrew.cmu.edu
Richard Sharp
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
rwsharp@cmu.edu
Shlomo Ta'asan
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
shlomo@andrew.cmu.edu
Abstract: Mesoscale experiment and simulation permit harvesting information about both geometric features and texture in polycrystals. The 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 growth process, an initially random distribution reaches a steady state that is strongly correlated to the interfacial energy density. In simulation, it is found that 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 a simple distribution arise from such a complex system? We derive an entropy based theory which suggests that the evolution of the GBCD satises a Fokker-Planck Equation. Cellular structures coarsen according to a local evolution law, curvature driven growth, and are limited by space lling constraints. The interaction between the evolution law and the constraints is governed primarily by the force balance at triple junctions, the natural boundary condition associated to curvature driven growth, and determines a dissipation relation. A simplied coarsening model is introduced which is driven by the boundary conditions and re ects the network level dissipation relation of the grain growth system. It resembles an ensemble of inertia-free spring-mass-dashpots. Critical application is made of the recent characterization of Fokker-Planck kinetics as a gradient ow for a free energy in deriving the theory. The theory predicts the results of large scale 2D simulations and is consistent with experiment.
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