An old-standing topic in grain growth is the evolution of the grains size distribution. Inspired by the work of Lifshitz and Slyozov on Ostwald ripening, Hillert has suggested that in a normal grain growth process the size distribution function must converge to a self-similar solution, i.e. it becomes invariant when the grain size is scaled with a characteristic length of the system that is proportional to the average grain size .
Several simulation studies, however, have shown that the size distribution deviates from the Hillerts self-similar solution. Hence a search for a new possible self-similar solution was initiated that indeed led to a new class of self-similar distribution functions. Large-scale phase field simulations have shown that there is indeed a self-similar behavior possible within the new distribution functions. It was shown that the origin of the deviation from Hillerts distribution is indeed the geometry of grains specially when they are shrinking.