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

TL;DR

This paper introduces hierarchical $n$-point polytope functions to create reduced-dimension metrics that effectively quantify microstructural evolution in materials, capturing symmetry changes and pattern dynamics.

Contribution

The paper develops novel $ $-point polytope-based metrics for measuring microstructure differences and evolution pathways, enabling detailed analysis of pattern formation and symmetry changes.

Findings

Successfully applied to spinodal decomposition to analyze phase separation dynamics.

Used to study pattern evolution in vapor-deposited alloy films with complex behaviors.

Metrics reveal mechanisms and symmetry changes during microstructural evolution.

Abstract

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.