New solution of the partial differential equation of the grain groove profile problem in the case of evaporation/condensation

Abstract

This paper constitutes a new contribution on the resolution of Mullins problem in the case of the evaporation-condensation and gives an exact and explicit solution of the second partial differential equation relative to the geometric profile of the grain boundary grooving. New analytical expressions of the solution, the groove profile, the derivative and the groove deep were obtained:y(x,t)=−πCtsinθ[erfc(x2Ct)+∑n=1∞(2n)!(n!)222n3nsin2nθ(erfc(x3n2ct))]y′(x,t)=+sinθex2/(2ct)−sin2θandε0(θ)=πctsinθ[1+∑n=1∞(2n)!(n!)222n3nsin2nθ]It was proved that the found solution gave more accurate results relative to those obtained by Mullins that neglected the first derivative (|y′| ≪ 1) relative to 1. The results obtained by this new solution can be advantageously used to give more precise solution of the general problem when combining the two phenomena relative to the evaporation-condensation and the surface diffusion in thin polycrystalline films.