M.Sc Student | Zigelman Anna |
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Subject | Verification of the Influence of Surface Energies on the Effective Mobility |

Department | Department of Applied Mathematics |

Supervisor | Professor Amy Novick-Cohen |

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 *m*^{2/3}, 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 *A*_{eff} and *A*, where *A*
is the mobility coefficient appearing in the equation for motion by mean
curvature, and *A*_{eff} 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.