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

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.

Point process microstructural model of metallic thin films with implications for coarsening

“…Third, once we have additional experimental microstructural data at different annealing times, our aim is to track triple-junction motion to quantify directly triple-junction dynamics. It is expected that nonequilibrium correlation functions of the triple-junction density at spatial positions r and r 0 , such as hρ r; t ð Þρ r 0 ; t ð Þi, will embody the dynamics at time t. Using this information, we expect to obtain a more complete, physical picture of triple-junction motion that will inform the development of an ongoing, related modeling effort 26 . Finally, we wish to compare the results of our proposed DDFT grain growth model with spin (e.g., Q-state Potts) [5][6][7][8] and phasefield models [9][10][11] .…”

“…Our aim here is to propose a dynamical equation for the triple junction density that is informed by the correlation functions and related quantities obtained above. For this purpose, we build on recent work 26 that describes grain growth in terms of triplejunction locations and associated disorientations. In its current form, this model lacks information regarding the interactions among triple junctions and the role of disorientations in such interactions, and we will incorporate such interactions via F β; ρ ½ ð Þ below.…”

Local well-posedness of a nonlinear Fokker–Planck model

“…In our previous work we derived Fokker-Planck type systems as a part of grain growth models of polycrystalline materials, e.g. [1,2,4,18].…”

“…Each grain boundary has a lattice misorientation which is the difference between lattice (lined grids on the figure) orientations α ( j) , j = 1, 2, 3 of the grains that share the grain boundary. In [18], a grain boundary network was considered as a system of such triple junctions and the grain boundaries misorientations, and was modeled by the Fokker-Planck equation for the joint distribution function of the position of the triple junctions and the misorientations.…”

“…Such systems arise in many physical and engineering applications, e.g. [1,2,4,11,12,18,34]. One example of the application of Fokker-Planck systems is the modeling of grain growth in polycrystalline materials.…”