Grain Growth and the Effect of Different Time Scales

Barmak, K.; Dunca, Anastasia; Epshteyn, Yekaterina; Li, Chun; Mizuno, Masashi

doi:10.1007/978-3-031-04496-0_2

Cited by 3 publications

(2 citation statements)

References 34 publications

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“…As a part of our future research, we will further extend such Fokker-Planck systems to the modeling of the evolution of the grain boundary network that undergoes disappearance/critical events, e.g. [3,18].…”

Section: Discussionmentioning

confidence: 99%

“…Grain growth is a very complex multiscale and multiphysics process influenced by the dynamics of grain boundaries, triple junctions and the dynamics of lattice misorientations (difference in the lattice orientations between two neighboring grains that share the grain boundary, figure 1), e.g. [3,38,39]. In case of the grain growth modeling [18], in the Fokker-Planck system, f may describe the joint distribution function of the lattice misorientation of the grain boundaries and of the position of the triple junctions, ϕ may describe the grain boundary energy density, and D is related to the absolute temperature of the entire system [32] (it can be viewed as a function of the fluctuation parameters of the lattice misorientations and of the position of the triple junctions due to fluctuation-dissipation principle [18]).…”

Section: Introductionmentioning

confidence: 99%

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Local well-posedness of a nonlinear Fokker–Planck model

Epshteyn

Liu²,

Li³

et al. 2023

Nonlinearity

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Noise or fluctuations play an important role in the modeling and understanding of the behavior of various complex systems in nature. Fokker–Planck equations are powerful mathematical tools to study behavior of such systems subjected to fluctuations. In this paper we establish local well-posedness result of a new nonlinear Fokker–Planck equation. Such equations appear in the modeling of the grain boundary dynamics during microstructure evolution in the polycrystalline materials and obey special energy laws.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Local well-posedness of a nonlinear Fokker–Planck model

Epshteyn

Liu²,

Li³

et al. 2023

Nonlinearity

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

Epshteyn¹,

Li²,

Mizuno³

2022

Math. Models Methods Appl. Sci.

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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.

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Nonlinear inhomogeneous Fokker–Planck models: Energetic-variational structures and long-time behavior

Epshteyn¹,

Liu²,

Li³

et al. 2022

Anal. Appl.

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Inspired by the modeling of grain growth in polycrystalline materials, we consider a nonlinear Fokker–Plank model, with inhomogeneous diffusion and with variable mobility parameters. We develop large time asymptotic analysis of such nonstandard models by reformulating and extending the classical entropy method, under the assumption of periodic boundary condition. In addition, illustrative numerical tests are presented to highlight the essential points of the current analytical results and to motivate future analysis.

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Grain Growth and the Effect of Different Time Scales

Cited by 3 publications

References 34 publications

Local well-posedness of a nonlinear Fokker–Planck model

Local well-posedness of a nonlinear Fokker–Planck model

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

Nonlinear inhomogeneous Fokker–Planck models: Energetic-variational structures and long-time behavior

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