The depth of a steep evaporating grain boundary groove: Application of comparison theorems

The nonlinear Mullins diffusion equation for the development of a surface groove by evaporation-condensation is reconsidered. Comparison theorems for differential equations are employed to obtain upper and lower bounds on the profile of the groove itself and on the growth rate of the groove at the grain boundary. It is shown that as the dihedral angle at the grain boundary approaches zero, the groove growth rate diverges logarithmically. This phenomenon is in disparity with the finite growth rate found in the development of grooves by surface diffusion.