|M.Sc Thesis||Department of Applied Mathematics|
|Supervisor:||Prof. Novick-Cohen Amy|
The grain boundary mobility is a crucial parameter in controlling the microstructure of solids. Classical mobility measurements rely on the motion of a U-shaped grain boundary in the "half-loop" geometry and neglect the effect of the exterior surface. Employing asymptotic analysis, we have taken the effect of the exterior surface into account and have found the leading order correction to the grain boundary mobility which gets measured experimentally.
In the "half-loop" geometry a U-shaped grain extends in an otherwise single crystal. The interface between the two grains, known as the grain boundary, contacts the exterior surface along a "groove root" where various balance laws hold. This geometry entails two types of motion: motion by mean curvature of the grain boundary, and motion by surface diffusion of the exterior surfaces. These motions can be modeled by a system of non-linear PDEs.
There is a parameter m, defined as the ratio of the free energy of the grain boundary to the free energy of the exterior surface, which is influentional in the determining boundary conditions, and m>0 causes the appearance of "thermal grooving" along the intersection of the grain boundary with the exterior surface. Typically 0<m<1/3, therefore, one may seek asymptotic solutions to this problem, based on some power of m. When m=0, the "thermal groove" vanishes and the grain boundary velocity is proportional to the grain boundary curvature.
In this thesis we have investigated the first order perturbations to the solution to the problem with m=0. Employing an arc-length parameterization of the projection of "thermal groove" on the xy-plane, and asymptotic expansions in m2/3, we have obtained a system of linear PDEs and a set of boundary conditions.
We have demonstrated that the grain boundary profile is parabolic. Additionally, we have obtained an expression for the correction term to the horizontal velocity of the U-shaped grain.
Finally, we have used these results to find the ratio between Aeff and A, where A is the mobility coefficient appearing in the equation for motion by mean curvature, and Aeff is the laboratory measurement of the mobility coefficient which neglects the influence of the exterior surfaces. The ratio that we find, takes into account the shape of the exterior surfaces and the "groove root," as well as the surface free energies of the exterior surfaces and the grain boundary. Thus, our results should improve interpretation of experiment.