Self-Similarity and the Dynamics of Coarsening in Materials

Two-phase mixtures, from metallic alloys to islands on surfaces, undergo coarsening wherein the total interfacial area of the system decreases with time. Theory predicts that during coarsening the average size-scale of a two-phase mixture increases with time as t^1/3 when the two-phase mixture is self-similar, or time independent when scaled by a time-dependent length. Here, we explain why this temporal power law is so robustly observed even when the microstructure is not self-similar. We show that there exists an upper limit to the length scales in the system that are kinetically active during coarsening, which we term the self-similar length scale. Length scales smaller than the self-similar length scale evolve, leading to the classical temporal power law for the coarsening dynamics of the system. Longer length scales are largely inactive, leading to a non-self-similar structure. This result holds for any two-phase mixture with a large distribution of morphological length scales.