An Experimentally Informed Grain Boundary Model in 2–D: Addressing Triple Junctions, Mobility and Invariance of Misorientation

Abstract

A novel 2-D continuum model for grain boundaries is presented, incorporating experimen- tally obtained data on grain boundary energy variation with misorientation. The model is employed to simulate the idealized evolution of grain boundaries within a 2-D grain ar- ray, following the methodology outlined in a previous study by us [1]. The approach of the model involves representing misorientation in a continuum scale through spatial gra- dients of orientation, considered a fundamental field. Based on experimental findings, the dependence of grain boundary energy density on the orientation gradient is found to be generically non-convex. The model employs gradient descent dynamics for the energy to simulate idealized microstructure evolution, necessitating the energy density to be regular- ized with a higher-order term to ensure the model’s well-posedness. From a mathematical perspective, the formulated energy functional fits the Aviles-Giga (AG)/Cross-Newell (CN) category, albeit with non-uniform well depths, leading to unique structural characteristics in solutions linked to grain boundaries in equilibria. The presented results showcase mi- crostructure evolution, and grain boundary equilibria, illustrating reorientation of grains in two dimensional space. Idealized features such as equilibrium high–angle grain boundaries (HAGBs), curvature-driven grain boundary motion, grain rotation, grain growth, and triple junctions that satisfy the Herring condition in our 2-D simulations are also demonstrated.

Keywords: Microstructure evolution, Grain rotation, Grain boundaries, Coarsening