Self-Similarity and the Dynamics of Coarsening in Materials

Abstract

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 t1/3 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.

Two-dimensional Cahn–Hilliard simulations for coarsening kinetics of spinodal decomposition in binary mixtures

“…For compositions up to ϕ 0 = 0.25, corresponding to morphologies made of spherical, isolated droplets, the time evolution of the energy therefore nicely matches the LSW-prediction. To the best of our knowledge, the question of growth exponents for off-critical compositions has not been resolved, even in recent studies 14,15,20 . These papers claim α = 1/3 for off-critical compositions below ϕ 0 = 0.3 and they evaluate α using the decay of the interfacial energy.…”

“…Coarsening of the morphology is therefore expected to have a significant impact on the material properties. Hence, coarsening is a long lasting subject of interest, mostly investigated together with SD in binary systems with experimental methods [7][8][9][10] , numerical simulations [11][12][13][14][15][16] , analytical models 1,17,18 and still an active area of research [19][20][21][22][23][24] . Being able to predict the actual time-dependent average domain size or characteristic length scale L(t) is crucial for understanding the morphology-property relationship.…”

A phase-field study of elastic stress effects on phase separation in ternary alloys

“…In order to determine the dynamics of coarsening of multiple domains in equiatomic alloys P 1 , P 2 and P 3 , we need to ascertain the existence of a self-similar regime where the three-phase microstructure can be dynamically scaled using a characteristic time-dependent length for each phase [49].…”

Temperature dependent extensions of the Cahn-Hilliard equation

“…During the last decades, the mathematical theory of phase separation has gained a fundamental role in understanding the behaviour of binary alloys, mixtures of materials and biological phenomena. Phase separations are processes that arise in many scientific, engineering and industrial applications, such as phase-field crystals [3], coarsening in materials [34] and segregation-like phenomena [7].…”