# “Dynamic Measure of Hyperuniformity and Nonhyperuniformity in Heterogeneous Media via the Diffusion Spreadability” is Published in Physical Review Applied as an Editor’s Suggestion

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Time-dependent interphase diffusion processes in multiphase heterogeneous media are ubiquitous phenomena in physics, chemistry and biology. Examples of heterogeneous media include composites, geological media, gels, foams, and cell aggregates. The recently developed concept of spreadability, \(\mathcal{S}(t)\), provides a direct link between time-dependent diffusive transport and the microstructure of two-phase media across length scales [Torquato, Phys. Rev. E **104**, 054102 (2021)]. To investigate the capacity of \(\mathcal{S}(t)\) to probe microstructures of real heterogeneous media, we explicitly compute \(\mathcal{S}(t)\) for well-known two-dimensional and three-dimensional idealized model structures that span across nonhyperuniform and hyperuniform classes. Among the former class, we study fully penetrable spheres and equilibrium hard spheres, and in the latter class, we examine sphere packings derived from “perfect glasses,” uniformly randomized lattices (URLs), disordered stealthy hyperuniform point processes, and Bravais lattices. Hyperuniform media are characterized by an anomalous suppression of volume fraction fluctuations at large length scales compared to that of any nonhyperuniform medium. We further confirm that the small-, intermediate-, and long-time behaviors of \(\mathcal{S}(t)\) sensitively capture the small-, intermediate-, and large-scale characteristics of the models. In instances in which the spectral density \(\tilde{\chi}_{_V}(\mathbf{k})\) has a power-law form \(B|\mathbf{k}|^α\) in the limit \(|\mathbf{k}|\to 0\), the long-time spreadability provides a simple means to extract the value of the coefficients \(\alpha\) and \(B\) that is robust against noise in \(\tilde{\chi}_{_V}(\mathbf{k})\) at small wave numbers. For typical nonhyperuniform media, the intermediate-time spreadability is slower for models with larger values of the coefficient \(B=\tilde{\chi}_{_V}(0)\). Interestingly, the excess spreadability \(\mathcal{S}(\infty)−\mathcal{S}(t)\) for URL packings has nearly exponential decay at small to intermediate \(t\), but transforms to a power-law decay at large \(t\), and the time for this transition has a logarithmic divergence in the limit of vanishing lattice perturbation. Our study of the aforementioned models enables us to devise an algorithm that efficiently and accurately extracts large-scale behaviors from diffusion data alone. Lessons learned from such analyses of our models are used to determine accurately the large-scale structural characteristics of a sample Fontainebleau sandstone, which we show is nonhyperuniform. Our study demonstrates the practical utility of the diffusion spreadability to extract crucial microstructural information from real data across length scales and provides a basis for the inverse design of materials with desirable time-dependent diffusion properties.