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  1. 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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    Free, publicly-accessible full text available December 1, 2024
  2. 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. 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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