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Critical events, entropy, and the grain boundary character distribution

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


Physical Review. B 83 (2011), No.13, Art. 134117, 12 pp.
ISSN: 0163-1829
ISSN: 1098-0121
ISSN: 0556-2805
Journal Article
Fraunhofer IGD ()
Fraunhofer Austria ()
coarsening; large scale simulation; texture

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 two dimensions (2D)] or area (in 3D) of an interface with a given lattice misorientation and normal. During the growth process, an initially random distribution of boundary types 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 nonrandom distributions, corresponding to independent trials with respect to the energy. In this paper, we derive an entropy-based theory that suggests that the evolution of the GBCD satisfies a Fokker-Planck equation, an equation whose stationary state is a Boltzmann distributi on. Cellular structures coarsen according to a local evolution law, curvature-driven growth, and are limited by space-filling 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 with curvature-driven growth, and determines a dissipation relation. A simplified coarsening model is introduced that is driven by the boundary conditions and reflects the network level dissipation relation of the grain growth system. It resembles an ensemble of inertia-free spring-mass dashpots. Application is made of the recent characterization of Fokker-Planck kinetics as a gradient flow for a free energy in deriving the theory. The theory predicts the results of large-scale two-dimensional simulations and is consistent with experiment.