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# “Local order metrics for two-phase media across length scales” is Published in Journal of Physics A: Mathematical and Theoretical

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The capacity to devise order metrics to characterize and classify microstructures of multiphase heterogeneous media across length scales is an outstanding but 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 $$\sigma_{_V}^2(R)$$ associated with a spherical window of radius $$R$$ as an order metric. We determine $$\sigma_{_V}^2(R)$$ as a function of $$R$$ for 22 different models across the first three space dimensions, including both hyperuniform and non-hyperuniform 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. Such order metrics could be employed to accelerate the discovery of novel heterogeneous materials by tailoring their degree of order/disorder.