Assessment of morphological similarities for the conservative Allen–Cahn and Cahn–Hilliard equations

Abstract

Phase separations occur in various natural environments, such as materials sciences and biochemistry, and have been scientifically modeled in various fields. We often need to distinguish between two different phase separation phenomena induced by the conservative Allen–Cahn (CAC) and Cahn–Hilliard (CH) equations, which are used to describe natural phenomena. Both phase-field equations share the same properties, including phase-coarsening, mass conservation, and energy dissipation. These similarities make it difficult to determine which equation is responsible for the observed dynamics. However, there are morphological differences between the evolutionary behaviors of the two dynamics, despite their similarities.This work investigates how the dynamics can be characterized by the CAC and CH equations and proposes evaluation criteria to distinguish their features. We examine phase-coarsening in light of the morphological patterns of the phases. Specifically, we train a Convolutional Neural Network (CNN) with morphological patterns generated by the CAC and CH equations. We use Alexnet as one of the CNNs to classify the patterns. In addition, we introduce other geometric properties to characterize the phase-separation dynamics. For a comprehensive understanding of these dynamics, we propose geometrical evaluations such as a distance measure by inner product, area, arclength and anatomical complexity, as measures of similarity metrics. Furthermore, we test and investigate geometrical similarity metrics to discriminate between the CAC and CH equations through numerical experiments. The numerical evaluations of the similarity metrics show that the morphological difference between these two CAC and CH dynamics appears to be large in the middle stages of evolution, whereas the distinction is vague in the early and late stages of evolution. In addition, the geometric aspects of morphological difference and similarity are also confirmed numerically.