Analytical Solutions to the Problem of the Grain Groove Profile

During the last sixty years, the problem of the formation of grain boundary grooving in polycrystalline thin films, was largely studied, analyzed and commented. The thermal effect on the properties of the grain boundary grooving was first studied by Mullins in his famous paper published in 1957 and then by other authors. This paper constitutes a new contribution on the correction of Mullins problem in the case of the evaporation-condensation and proposes a more accurate solution of the partial differential equation governing the geometric profile of the grain boundary grooving. The Mullins hypothesis neglecting the first derivative (|y'|< < 1) in the main equation was defeated by our new solution. In this paper, we proved that the new proposed mathematical solution giving the solution y(x, t) is valid for all x values without any approximation on the first derivative y'