Hyperuniformity and Diffusion Spreadability of Quasiperiodic Tilings

Abstract

Hyperuniform materials are characterized by an anomalous suppression of large-scale density compared to typical disordered materials, and embody all crystals and quasicrystals as well as exotic disordered varieties. Such correlated materials are endowed with novel photonic, phononic, electronic, thermal, mechanical, and diffusion properties. This work explores the hyperuniformity and antihyperuniformity in one- and two-dimensional quasiperiodic tilings and systems, and methods to extract their values of α, which are the exponents that characterize the scaling behavior of the structure factor and spectral density at small wave numbers. In addition to using the number variance σ 2 N (R) to numerically validate previous theoretical predictions of the hyperuniform scaling properties of one-dimensional tilings, we explore for the first time, two-phase quasiperiodic packings. Using the volume-fraction variance σ 2 V (R), we find that all two-phase systems considered belong to the same hyperuniform class as their underlying point patterns, and that the class II, III and antihyperuniform systems possess the same values of α as their point patterns. We find no advantage in using the volume-fraction variance over the number variance to extract α, since both quantities involve integrals on the spectral density and structure factor respectively, that smooth the spectral functions in similar ways. We use the long-time behavior of the spreadability - a newly introduced concept which serves as a powerful tool to characterize heterogeneous media across length scales - to extract values of α for the different two-phase systems from their timedependent properties. We find that that the Gaussian kernel of the spreadability provides smoothing of the spectral density that proves to be more robust against “microscopic-scale” oscillations than all other quantities used. The spreadability is found to be a more accurate tool for extracting values of α than either the number and volume-fraction variances or the spectral intensity function Z(k). The latter function is numerically similar to the spreadability and can yield values of α of class I systems, while the variances, by definition, cannot. Still, since the spreadability is much less variable at large time than Z(k), which contains significant variation at small k, it is found to be more accurate when compared to theory