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Local Order Metrics for Two-Phase Media Across Length Scales

The capacity to devise order metrics for microstructures of multiphase heterogeneous media is a highly challenging task, given the richness of the possible geometries and topologies of the phases that can arise. This investigation initiates a program to formulate order metrics to characterize the degree of order/disorder of the microstructures of two-phase media in $d$-dimensional Euclidean space $\mathbb{R}^d$ across length scales. In particular, we propose the use of the local volume-fraction variance $σ^2_{_V}(R)$ associated with a spherical window of radius $R$ as an order metric. We determine $σ^2_{_V}(R)$ as a function of $R$ for 22 different models across the first three space dimensions, including both hyperuniform and nonhyperuniform systems with varying degrees of short- and long-range order. We find that the local volume-fraction variance as well as asymptotic coefficients and integral measures derived from it provide reasonably robust and sensitive order metrics to categorize disordered and ordered two-phase media across all length scales.