The fourth-order nonlinear partial differential equation governing the formation of a groove on the surface of a polycrystal at the intersection of a grain boundary has previously been solved only for a limited number of cases. By the use of Lie symmetry methods it was hoped that further solutions to the evolution equation could be developed.
If the surface of a crystalline material is disturbed from equilibrium by the presence of another crystal surface due to, say, a crack or contact with another surface then the surfaces will reconfigure in such a way as to minimise the surface energy. This may occur through the processes of plastic or viscous flow, evaporation and condensation, volume diffusion, or surface diffusion (Herring 1951). Only surface relaxation due to surface diffusion is considered here as this is dominant for a single species of material.
At the intersection of a stationary grain boundary and the surface of a polycrystalline material a groove will develop such that the surface tension and the grain boundary tension vanish along the root of the groove. This produces an equilibrium angle along the line of intersection of the grain boundary with the surface (Bailey & Watching, 1950) The higher curvature of the groove compared with the relative flatness of the surface away from it leads to a difference in the chemical potential between the region and causes a drift in material between the two. Consequently, the equilibrium angle is continually disturbed, forcing the groove to deepen.
Mullins (1957) considered the surface groove for a material with isotropic properties using the small slope approximation. Tritscher (1996a) showed that the linearized governing equation can be formed using constitutive relations which model a theoretical liquid crystal material. For this material Broadbridge & Tritscher (1994) demonstrated that for the evolution of grooves with zero dihedral angle by surface diffusion have finite growth rates. This was not predicted by linear models such as Mullin's (1957).
Tritscher & Broadbridge (1995) modelled the development of a symmetric grain boundary groove for an isotropic material by approximating the isotropic material with Tritscher's (1996a) liquid crystal. The problem was then reduced to the solution of a system of transcendental equations. Tritscher (1996b) compared this solution with known numerical methods and found the latter to suffer from a high degree of error. Lee (1995) proved that the 2nd order finite difference schemes have complex valued solutions for groove slopes greater than 0.5.
The difficulty faced with modelling the grain boundary groove problem is the limited range of theoretical materials for which solutions have been found, namely isotropic and Tritscher's (1996a) liquid crystal. Further models are required, especially for the purpose of developing error bounds on the numerical methods. It was hoped that Lie symmetry methods could be used to develop solutions to the nonlinear groove equation that could model a larger range of materials. |