Coarsening kinetics and the envelope theorem

Abstract The scaling properties of microstructural coarsening are studied by means of the envelope theorem, which connects structural features of the projections of an evolving size distribution function in size space and time space. This is made possible by extending the envelope treatment of [P. Streitenberger, D. Zollner, Acta. Mater. 88 (2015) 334–345] to the time domain, thus establishing a new method of the scaling analysis. An important new finding is that there exists a duality between the pictures of the family of distribution functions in size and time space, according to which the envelope in size space equals to the location of the maxima in time space, and vice versa. For self-similar coarsening the scaling properties are completely reflected by the associated envelopes and maxima in the two complementary representations. The analysis of cumulative size distribution functions for the determination of the growth path of individual particles or grains is extended to the time space. The construction of the envelope curves both, in size space as well as in time space, allows a new and efficient numerical determination of the coarsening kinetics from large four-dimensional datasets of experiments and simulations. This is demonstrated by numerical studies of the results of Monte Carlo Potts model simulations of grain growth, which confirm and complement the analytical results.