On the analytical solution for self-similar grain size distributions in two dimensions

In a recent publication an analytical solution of the Fokker–Planck continuity equation for the grain size distribution for two-dimensional grain growth in the long time limit (self-similar state) was provided. It used von Neumann–Mullins law and the results of Rios and Glicksman, but was based on a stochastic formulation first proposed by Pande. In this paper this analytical solution is compared with experimental and computer simulation distributions. It is found that grain size distribution, as obtained by simulations of two-dimensional grain growth, although in agreement with our analytical results, may in fact differ from experimentally obtained grain size distributions in thin films. It is also shown mathematically that in the two limiting cases the general solution is reduced to the Hillert or Rayleigh distributions.