Hyperuniformity of the Optimal Multiscale Hashin-Shtrikman Two-Phase Tessellations

Disordered hyperuniform two-phase systems are characterized by anomalous suppression of volume-fraction fluctuations at infinite long wavelengths. They provide fertile ground for fundamental research and have attracted considerable practical interest because they often are endowed with exotic physical properties, including possessing optimal or nearly optimal physical properties. The Hashin-Shtrikman (HS) two-phase multiscale dispersions are derived from special tessellations of space that endow these two-phase systems with the optimal effective transport and elastic properties for given phase properties and phase volume fractions. Using a new tiling formulation that ensures perfect hyperuniformity, we rigorously establish the hyperuniformity of the optimal HS structures. We analytically show that these structures are strongly hyperuniform by deriving the small-wavenumber scaling behavior of their spectral density. We verify these theoretical results numerically in two dimensions by constructing extremely dense polydisperse disk packings. Our work provides insights about the relationship between hyperuniformity in two-phase systems and property optimization.