Quantifying Microstructural Evolution via Time-Dependent Reduced-Dimension Metrics Based on Hierarchical $n$-Point Polytope Functions

We devise reduced-dimension metrics for effectively measuring the distance between two points (i.e., microstructures) in the microstructure space and quantifying the pathway associated with microstructural evolution, based on a recently introduced set of hierarchical $n$-point polytope functions $P_n$. The $P_n$ functions provide the probability of finding particular $n$-point configurations associated with regular $n$-polytopes in the material system, and a special sub-set of the standard $n$-point correlation functions $S_n$ that effectively decomposes the structural features in the systems into regular polyhedral basis with different symmetry. The $n$-th order metric $\Omega_n$ is defined as the $\mathbb{L}_1$ norm associated with the $P_n$ functions of two distinct microstructures. By choosing a reference initial state (i.e., a microstructure associated with $t_0 = 0$), the $\Omega_n(t)$ set quantifies the evolution of distinct polyhedral symmetries and can in principle capture emerging polyhedral symmetries that are not apparent in the initial state. To demonstrate their utility, we apply the $\Omega_n$ metrics to a 2D binary system undergoing spinodal decomposition to extract the phase separation dynamics via the temporal scaling behavior of the corresponding $\Omega_n(t)$, which reveals mechanisms governing the evolution. Moreover, we employ $\Omega_n(t)$ to analyze pattern evolution during vapor-deposition of phase-separating alloy films with different surface contact angles, which exhibit rich evolution dynamics including both unstable and oscillating patterns. The $\Omega_n$ metrics have potential applications in establishing quantitative processing-structure-property relationships, as well as real-time processing control and optimization of complex heterogeneous material systems.