A Mathematical Model for Grain Growth in Thin Metallic Films

We use geometrical arguments based on grain boundary symmetries to introduce crystalline interfacial energies for interfaces in polycrystalline thin lms with a cubic lattice. These crystalline energies are incorporated into a multi{phase eld model. Our aim is to apply the multi{phase eld method to describe the evolution of faceted grain boundary triple junctions in epitaxially growing microstructures. In particular, we are interested in symmetry properties of triple junctions in tricrystalline thin lms. Symmetries of triple junctions in tricrystalline lms have been studied in experiments by Dahmen and Thangaraj 6;25. In accordance with their experiments, we nd in numerical simulations that any two neighbouring triple junctions belong to diierent symmetry classes. We introduce a local equilibrium condition at triple junctions which can be interpreted as a crystalline version of Young's law. The local equilibrium condition at triple junctions is purely determined by the grain boundary energies. In particular no triple junction energies are necessary to explain which triple junctions are possible. All triple junctions observed in the experiments as well as in the simulations fulll the crystalline version of Young's law. Our approach is also capable to describe grain boundary motion in general polycrystalline thin lms.