  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
$\sigma^2_{_V}(R)$ associated with a spherical window of radius $R$ as an order
metric. We determine $\sigma^2_{_V}(R)$ 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.
