Hyperuniformity of Point Patterns and Two-Phase Composite Media

Abstract

This dissertation describes the theoretical investigation of hyperuniform many-body systems, including point patterns and two-phase composite media. Hyperuniform systems encompass all perfect crystals, perfect quasicrystals, and some exotic amorphous states of matter. Hyperuniform systems arise in a broad spectrum of contexts in the physical, mathematical, and materials sciences. Using both analytical and computational methods of statistical mechanics, we devise methods to construct disordered hyperuniform systems. We also characterize the structure and physical properties of both ordered and disordered hyperuniform systems.

The first part of this dissertation (Chapters 2-5) concerns hyperuniform point patterns. In Chapter 2, we prove that the use of aspherical windows can affect the detection of hyperuniformity and propose how to resolve such problems. In Chapter 3, we study the degree to which the introduction of various types of imperfections in perfectly hyperuniform systems degrades or destroys their original perfect hyperuniformity. In Chapter 4, we study how random, uncorrelated displacements of particles on a lattice can lead to “cloaking” of the Bragg peaks in the diffraction pattern of the original lattice. We study sufficient conditions for inverting the collective coordinates of particle distributions in Chapter 5.

In the second part of the dissertation (Chapters 6-10), we study hyperuniform two-phase systems. In Chapter 6, we rank order the degree of hyperuniformity of certain two-dimensional models of Class I hyperuniform two-phase media. In Chapter 7, we devise a tessellation-based procedure to construct perfectly hyperuniform disordered packings. Chapters 8-10 concern the prediction of effective physical properties of hyperuniform and nonhyperuniform two-phase composite media. The strong-contrast expansion formalism is extended to derive exact expansions for the effective dynamic properties of a two-phase medium that account for complete microstructural information (infinite set of n-point correlation functions) and hence multiple scattering to all orders. Using these expansions, we derive accurate approximations for the effective dynamic dielectric and elastic properties that apply beyond the long-wavelength regime in Chapters 8 and 9, respectively. In Chapter 10, we establish cross-property relations linking these two effective wave properties of a given composite by utilizing the results in Chapters 8 and 9.