Elastic Effects in the Phase Separation of the System with Finite Size

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(a) Usual phase field models have an essential short coming that they cannot describe the formation of poly-crystals since they are lacking in the information of orientation. We proposed a vectorized phase field model which enables us to simulate a simultaneous crystallization of many particles with various orientations and a formation of grain boundaries.(b) Recrystallization takes place through the two basic processes, say, (I) grain boundary migration and (ii) grain rotation. Almost all the model of recrystallization handle the process (I) only. We proposed a totally new phase field model which can treat both of the processes (I) and (il) simultaneously.c In the above recrystallization model the equation for the evolution of angle variable expresses a non-local interaction by introducing a new mathematical concept "singular diffusivity" We justified this new equation and analyzed it mathematically.(d) It is well known that the mixture of granular materials can segregate in the rotating cylinder. We investigated this phenomena through experiments and modeling and found that it is caused by the property of strong segregation in the suface flow of binary mixture of granular materials.(e) A mathematical model of step dynamics on the facet of growing crystals are presented. This model is able to describe the motion of steps caused by the arbitrary number of screw dislocations.(f) Mathematical structure which produces self-replicating patterns were made clear by the approach of experimental mathematics. We demonstrated that the ordered saddle node bifurcation branches and the paths connecting them are essential for the formation of self-replicating patterns.