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1. Grain Growth and the Effect of Different Time Scales

   https://doi.org/10.1007/978-3-031-04496-0_2  

   Barmak, Katayun; Dunca, Anastasia; Epshteyn, Yekaterina; Liu, Chun; Mizuno, Masashi (April 2022, Association for Women in Mathematics series)

   Español, M; Lewicka, M; Scardia, L; Schlömerkemper, A (Ed.)

   Many technologically useful materials are polycrystals composed of a myriad of small monocrystalline grains separated by grain boundaries. Dynamics of grain boundaries play a crucial role in determining the grain structure and defining the materials properties across multiple scales. In this work, we consider two models for the motion of grain boundaries with the dynamic lattice misorientations and the triple junctions drag, and we conduct extensive numerical study of the models, as well as present relevant experimental results of grain growth in thin films. 

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2. Local well-posedness of a nonlinear Fokker–Planck model

   https://doi.org/10.1088/1361-6544/acb7c2  

   Epshteyn, Yekaterina; Liu, Chang; Liu, Chun; Mizuno, Masashi (February 2023, Nonlinearity)

   Abstract 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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3. Point process microstructural model of metallic thin films with implications for coarsening

   https://doi.org/10.1038/s41524-023-00986-w  

   Rickman, J. M.; Barmak, K.; Epshteyn, Y.; Liu, C. (December 2023, npj Computational Materials)

   Abstract We develop a thin-film microstructural model that represents structural markers (i.e., triple junctions in the two-dimensional projections of the structure of films with columnar grains) in terms of a stochastic, marked point process and the microstructure itself in terms of a grain-boundary network. The advantage of this representation is that it is conveniently applicable to the characterization of microstructures obtained from crystal orientation mapping, leading to a picture of an ensemble of interacting triple junctions, while providing results that inform grain-growth models with experimental data. More specifically, calculated quantities such as pair, partial pair and mark correlation functions, along with the microstructural mutual information (entropy), highlight effective triple junction interactions that dictate microstructural evolution. To validate this approach, we characterize microstructures from Al thin films via crystal orientation mapping and formulate an approach, akin to classical density functional theory, to describe grain growth that embodies triple-junction interactions. 

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4. An experimentally informed continuum grain boundary model

   https://doi.org/10.1177/10812865231223921  

   Ansari, S Syed; Acharya, Amit; Alankar, Alankar (August 2024, Mathematics and Mechanics of Solids)

   A continuum grain boundary model is developed, which uses experimentally measured grain boundary energy data as a function of misorientation to simulate idealized grain boundary evolution in a one-dimensional (1D) grain array. The model uses a continuum representation of the misorientation in terms of spatial gradients of the orientation as a fundamental field. The grain boundary energy density employed is non-convex in this orientation gradient, based on physical grounds. Simple gradient descent dynamics of the energy are utilized for idealized microstructure evolution, which requires higher-order regularization of the energy density for the model to be well set; the regularization is physically justified. Microstructure evolution is presented using two plausible energy density functions, both defined from the same experimental data: a “smooth” and a “cusp” energy density. Results of grain boundary equilibria and microstructure evolution representing grain reorientation in 1D are presented. The different shapes of the energy density functions representing a common data set are shown to result in different overall microstructural evolution of the system. Mathematically, the constructed energy functional formally is of the Aviles–Giga/Cross–Newell type but with unequal well depths, resulting in a difference in the structural feature of solutions that can be identified with grain boundaries, as well as in the approach to equilibria from identical initial conditions. This study also investigates the metastability of grain boundaries. It supports the general thermodynamics belief that they persist for extended periods before eventually vanishing due to the lowest energy configuration favored by fluctuations over infinite time. 

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5. A deep learning method for solving Fokker-Planck equations

   Jiayu Zhai, Matthew Dobson (January 2022, Proceedings of Machine Learning Research)

   Joan Bruna, Jan S (Ed.)

   The time evolution of the probability distribution of a stochastic differential equation follows the Fokker-Planck equation, which usually has an unbounded, high-dimensional domain. Inspired by Li (2019), we propose a mesh-free Fokker-Planck solver, in which the solution to the Fokker-Planck equation is now represented by a neural network. The presence of the differential operator in the loss function improves the accuracy of the neural network representation and reduces the demand of data in the training process. Several high dimensional numerical examples are demonstrated. 

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