A stochastic model of grain boundary dynamics: A Fokker-Planck perspective

TL;DR

This paper develops a Fokker-Planck stochastic model to describe the evolution of grain boundary networks in polycrystalline materials, incorporating anisotropic energies and junction mobility, with analytical results on long-term behavior and equilibrium conditions.

Contribution

It introduces a novel Fokker-Planck framework for grain boundary dynamics that accounts for anisotropic energies and independent misorientation dynamics, extending previous models.

Findings

Derived long-time asymptotics of the probability density functions.

Established explicit relations at equilibrium, including a generalized Herring Condition.

Connected grain boundary energy density with network geometry.

Abstract

Many technologically useful materials are polycrystals composed of small monocrystalline grains that are separated by grain boundaries of crystallites with different lattice orientations. The energetics and connectivities of the grain boundaries play an essential role in defining the effective properties of materials across multiple scales. In this paper we derive a Fokker-Planck model for the evolution of the planar grain boundary network. The proposed model considers anisotropic grain boundary energy which depends on lattice misorientation and takes into account mobility of the triple junctions, as well as independent dynamics of the misorientations. We establish long time asymptotics of the Fokker-Planck solution, namely the joint probability density function of misorientations and triple junctions, and closely related the marginal probability density of misorientations. Moreover, for an equilibrium configuration of a boundary network, we derive explicit local algebraic relations, a generalized Herring Condition formula, as well as formula that connects grain boundary energy density with the geometry of the grain boundaries that share a triple junction. Although the stochastic model neglects the explicit interactions and correlations among triple junctions, the considered specific form of the noise, under the fluctuation-dissipation assumption, provides partial information about evolution of a grain boundary network, and is consistent with presented results of extensive grain growth simulations.