|M.Sc Thesis||Department of Applied Mathematics|
|Supervisors:||Prof. Novick-Cohen Amy|
This thesis focuses on numerical simulation and analysis of grain boundary migration in an idealized system of 5 grains embedded in a thin film. Since even the system of 5 grains in the special geometry is very difficult to study in full generality, we make symmetry assumptions which reduce our problem to a simpler 3 grain system. We consider the motion of the grain boundaries and the external surfaces which couple along thermal grooves. The grain boundaries and external surfaces are assumed to be governed by motion by mean curvature and by motion by surface diffusion respectively. Along the groove line, we assume the boundary conditions are given by Young's law, continuity of the surface chemical potential, and balance of mass flux. At the quadruple junction and along the exterior boundary of the domain, boundary conditions are formulated in accordance with the symmetry assumptions and the physical principles indicated above.
We define a physical parameter, m, to be the ratio of the free energy of the grain boundary to the free energy of the exterior surface. Theoretically the parameter m may vary between 0 and 2. When m > 0 "thermal grooves" appear along the intersection of the grain boundary with the exterior surface. Our mathematical formulation is valid within the range m Î [0, 31/2] Ì [0, 2]. For many materials, in particular for most metals, m Î [0, 1/3], hence this does not constitute a serious constraint.
The assumptions described above yield a nonlinear system of partial differential algebraic equations. To solve this nonlinear system numerically, we discretize the system using a finite difference scheme, and then employ various numerical algorithms (the backward Euler method, Newton's method). Since computer memory and CPU time is somewhat problematic, we parallelize parts of our computational algorithm.
The aim of the numerical simulations is to get insight into the coupled motion, and to understand certain phenomena such as grain grooving, break-up of thin films, pitting at quadruple junctions, jerky motion, etc., which influence the robustness and stability of polycrystalline materials. While many phenomena remain to be studied in greater depth, our results so far indicate that we have developed a framework in which the phenomena listed above may be investigated. We have been able, for example, to follow pitting depths as a function of time and to verify the predictions of the von Neumann-Mullins law in the limit m ↓ 0.