Critical events, entropy, and the grain boundary character distribution

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 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 non-random distributions, corresponding to independent trials with respect to the energy.In this paper we derive 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. 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 to curvature driven growth, and determines a dissipation relation. A simplified coarsening model is introduced which 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 springmass-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 2D simulations and is consistent with experiment.

Simulations of anisotropic grain growth: Efficient algorithms and misorientation distributions

“…It builds on the distance function-based diffusion-generated motion (DFDGM) approach [7], which was used in earlier work [4,5] to simulate long time isotropic (equal surface energy) grain growth with very large numbers of grains. The initial simulations presented demonstrate this method recovers previous results using KMC simulations [9,12,13] and front tracking [15,2]. Subsequent simulations demonstrate the importance of the surface energy-to-misorientation map in predicting the time evolution of the MDF and suggests a framework capable of explaining the different MDFs obtained in the fiber texture and random texture cases.…”

“…Barmak et al [2] went on to further suggest that the MDF converges to a Boltzmann distribution, namely…”

A Theory and Challenges for Coarsening in Microstructure

“…Our main idea in [9], [11], [10], [17] is that the GBCD statistic for the simplified model resembles the solution of a Fokker-Planck Equation via the mass transport implicit scheme, [39]. In [9], [11], [10], [17] the simplified model is formulated as a gradient flow which results in a dissipation inequality analogous to the one found for the coarsening grain network.…”

“…In [9], [11], [10], [17] the simplified model is formulated as a gradient flow which results in a dissipation inequality analogous to the one found for the coarsening grain network. Because of this simplicity, it will be possible to 'upscale' the network level system description to a higher level GBCD description that accomodates irreversibility.…”

“…Here we describe our recent work developing an entropy based theory that suggests that the evolving GBCD satisfies a Fokker-Planck Equation, [10], [17], cf. also [11], [9], to which we refer for a more complete exposition.…”