The envelope of size distributions in Ostwald ripening and grain growth

In the present paper the scaling properties of microstructure coarsening by Ostwald ripening or grain growth are studied by means of the envelope of the family of evolving size distribution functions, which complements the usual scaling analysis by some new aspects and results. For a self-similar family of size distribution functions the envelope is uniquely determined by the growth exponent and an envelope parameter, where the latter is a new characteristic quantity of the coarsening system associated with the scaled size distribution function. If the family of size distributions obeys the continuity equation their envelope is the location of the maximum particle flux density in size space, while for the corresponding cumulative size distribution the envelope is the location of maximum size that a particle or grain can take during the passage through its growth path. The construction of the envelope curve therefore allows the independent determination of some coarsening parameters without recourse to the full time development of the individual growth paths. Numerical studies by means of the Monte Carlo Potts model of grain growth confirm and complement the analytical results. Especially they reveal that at the early stage of growth the envelope has a much reduced and non-integer exponent.