Disordered hyperuniform heterogeneous materials

Abstract

Disordered hyperuniform many-body systems are distinguishable states of matter that lie between a crystal and liquid: they are like perfect crystals in the way they suppress large-scale density fluctuations and yet are like liquids or glasses in that they are statistically isotropic with no Bragg peaks. These systems play a vital role in a number of fundamental and applied problems: glass formation, jamming, rigidity, photonic and electronic band structure, localization of waves and excitations, self-organization, fluid dynamics, quantum systems, and pure mathematics. Much of what we know theoretically about disordered hyperuniform states of matter involves many-particle systems. In this paper, we derive new rigorous criteria that disordered hyperuniform two-phase heterogeneous materials must obey and explore their consequences. Two-phase heterogeneous media are ubiquitous; examples include composites and porous media, biological media, foams, polymer blends, granular media, cellular solids, and colloids. We begin by obtaining some results that apply to hyperuniform two-phase media in which one phase is a sphere packing in d-dimensional Euclidean space [Formula: see text]. Among other results, we rigorously establish the requirements for packings of spheres of different sizes to be 'multihyperuniform'. We then consider hyperuniformity for general two-phase media in [Formula: see text]. Here we apply realizability conditions for an autocovariance function and its associated spectral density of a two-phase medium, and then incorporate hyperuniformity as a constraint in order to derive new conditions. We show that some functional forms can immediately be eliminated from consideration and identify other forms that are allowable. Specific examples and counterexamples are described. Contact is made with well-known microstructural models (e.g. overlapping spheres and checkerboards) as well as irregular phase-separation and Turing-type patterns. We also ascertain a family of autocovariance functions (or spectral densities) that are realizable by disordered hyperuniform two-phase media in any space dimension, and present select explicit constructions of realizations. These studies provide insight into the nature of disordered hyperuniformity in the context of heterogeneous materials and have implications for the design of such novel amorphous materials.