Geometry of coincident-site lattice boundaries and fractal aspect in grain boundary energy landscape

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New mathematical method which can describe systematically various lattice patterns showing up at a grain boundary between the two crystallines was devised. This method is based on the conformal mapping regarding the grain boundary as a Gauss-Argand plane, and is easily extended to a case of resultant lattice patterns which are made of low symmetry crystalline structures. In fact, the method was applicable to the twist boundaries as well which are constructed from (110)-plane twist of simple cubic structure. By using the two-dimensional version of Frenkel-Kontrova Hamiltoinan upon evaluation of grain boundary energy, the energy landsape, which is spanned on the parameter space (misorientation angle in the twist boundary case) specifying the precise location of the geometry of the boundary, is shown to have the exact self-similarity consisting of energy cusps. That is the the energy surface is made up of fractal aggregates of energy cusps with various depth, the largest ones being observed experimentally.Application of this grain boundary energy as a function of misorientation angle between the two grains divided by the boundary to the two-dimensional vertex model that is capable of describing grain growth has led to a dicovery of quite intriguing property that a relaxation process to the stationary growth process is 'stretched'. This is found to be ascribed to a hierarchical nature in the relaxation processes, and is supposed to have some connection with the recent experimental findings regarding the glassy properties at grain boundaries.A series of this research will be presented in a talk in Third International Conference on Grain Growth held in Pittsburgh, USA in coming June. A primary purpose, which is to make my powerful scheme be worldwide, is therefore said to be achieved.