Numerical analysis of a three-dimensional radially symmetric grain attached to a free crystal surface

We study a three-dimensional bicrystal containing an axially symmetric shrinking grain which is initially a spherical segment attached along a circular groove root to the flat exterior surface of a second grain. Following Mullins [Mullins W. J Appl Phys 1956;27:900; Mullins W. J Appl Phys 1956;28:333; Mullins W. Acta Metall 1958;6:414], a time-dependent problem is formulated for the coupled motion of the grain boundary, the groove root, and the external surface. Numerical solutions calculated using an implicit finite difference scheme indicate that the grain shrinks and disappears in finite time, no non-trivial limiting motion is seen, and there is no pinning of the grain boundary that might be associated with so-called “jerky” motion. Surprisingly enough, the surface groove seems only to minimally affect the grain boundary motion, the groove depth varies non-monotonically in time, and after annihilation of the grain the exterior surface has a profile which depends on the system's history and contains certain features which can be interpreted as so-called “ghost lines.”