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1. Grain Growth and the Effect of Different Time Scales

   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. Grain boundary energies in yttria‐stabilized zirconia

   https://doi.org/10.1111/jace.18283  

   Dillon, Shen J. ; Shen, Yu‐Feng ; Rohrer, Gregory S. ( December 2021 , Journal of the American Ceramic Society)

   The three‐dimensional microstructure of 8% yttria‐stabilized zirconia (YSZ) was measured by electron backscatter diffraction and focused ion beam serial sectioning. The relative grain boundary energies as a function of all five crystallographic grain boundary parameters were determined based on the assumption of thermodynamic equilibrium at the internal triple junctions. Grain boundaries with (100) orientations have low energies compared to boundaries of other orientations, and all [100] twist boundaries have relatively low energies. Other classes of boundaries with lower than average energies include [100] symmetric tilt boundaries with disorientations less than 40° and [111] twist boundaries with disorientations greater than 20°. At fixed misorientations, the relative areas of boundaries are inversely correlated to the relative grain boundary energy. The results suggest that texturing microstructures to increase the relative areas of [100] twist boundaries might increase the oxygen ion conductivity of YSZ ceramics.

    

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3. 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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4. 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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5. Nonadiabatic escape and stochastic resonance

   https://doi.org/10.1088/1751-8121/ab6aee  

   Moon, W. ; Balmforth, N. ; Wettlaufer, J. S. ( February 2020 , Journal of Physics A: Mathematical and Theoretical)

   We analyze the fluctuation-driven escape of particles from a metastable state under the influence of a weak periodic force. We develop an asymptotic method to solve the appropriate Fokker–Planck equation with mixed natural and absorbing boundary conditions. The approach uses two boundary layers flanking an interior region; most of the probability is concentrated within the boundary layer near the metastable point of the potential and particles transit the interior region before exiting the domain through the other boundary layer, which is near the unstable maximal point of the potential. The dominant processes in each region are given by approximate time-dependent solutions matched to construct the approximate composite solution, which gives the rate of escape with weak periodic forcing. Using reflection we extend the method to a double well potential influenced by white noise and weak periodic forcing, and thereby derive a two-state stochastic model—the simplest treatment of stochastic resonance theory—in the nonadiabatic limit.

    

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