A generalized Rayleigh-Ritz method combined with the Galerkin functional space approximation was elaborated by using the extended and modified Laguerre functions manifold for the weak solution of the asymmetric grain-boundary thermal grooving problem with the Dirichlet boundary. This new hybrid RRG approach, which resembles the front-tracking method, reveals the fine features of the grain boundary groove-root topography (rough or faceted regions) more accurately than the previous approach under the severe non-analyticity of the surface stiffness anisotropy, and showing almost excellent in accord with the experimental observations made by atomic force microscopy and scanning tunnelling microscopy. The large deviations from Mullins’ t1/4 scaling law combined with the self-trapping (quasifaceting) were observed especially at low values of the normalized longitudinal mobilities, where the kinetics rather than the energetic considerations were found to be the dominating factor for the whole topographic appearances. For very high longitudinal mobilities, the smooth and symmetric groove profiles (no faceting) were found to be represented by the Mullins’ function for the fourfold symmetry in the stationary state with great precision, if one modifies the rate parameter by the anisotropy constant and simultaneously utilizes the anisotropic complementary dihedral angle in the calculation of the slope parameter. A recently developed analytical theory fully supported this observation rigorously, and furnishes the quantitative determination of the threshold level of the anisotropy constant for the ridge formation, and as well as the penetration depth evaluation.

Dirichlet Extremum Problem Associated with the Asymmetric Grain-Boundary Thermal Grooving under the Dirac δ-type Anisotropic Surface Stiffness in Bicrystal Thin Solid Films. T.O.Ogurtani: Journal of Applied Physics, 2007, 102[6], 063517