In this dissertation, we present a combination of computational and theoretical results concerning the characterization of the microstructure of heterogeneous materials and hard-particle packings. An overview of the dissertation is provided in Chapter 1. In Part I of this dissertation, we focus on the characterization of multi-phase heterogeneous materials. In Chapter 2, we present a detailed discussion of the correlation functions that statistically characterize the microstructure of a heterogeneous material. Examples of such materials include composites, colloids, foams and biological media. In Chapter 3, we introduce a microstructure reconstruction/constructiion procedure developed by Yeong and Torquato and devise a powerful universal sampling scheme, called the lattice-point scheme, that enables one to incorporate the widest class of lower-order correlation functions known to date into the Yeong-Torquato procedure, which opens the door to many fruitful applications. In Chapter 4, we present two major applications of our lattice-point scheme including modelling heterogeneous materials via two-point correlation functions and identifying superior microstructure descriptors of random media. These developments suggest novel approach for material design and more accurate rigorous structure-property relations;​ they also have ramifications in atomic and molecular systems.

In Part II of this dissertation, we focus on quantitatively describing the structure of hard-particle packings, which have been employed to model a wide spectrum of condensed matters such as simple liquid, disordered/crystalline solids and granular media as well as biological systems. In Chapter 5, we present two major numerical packing protocols, namely the Donev-Torquato-Stillinger (DTS) event-driven molecular dynamics (MD) algorithm for smooth convex particles and the adaptive-shrinking-cell (ASC) scheme for hard polyhedral particles. In Chapter 6, the DTS event-driven MD algorithm is employed to find the optimal packings of superball and the ASC scheme is employed to obtain dense packings of polyhedral particles. We also derive a simple upper bound on the optimal packing density for an arbitrary nonspherical particle and apply it to the superballs and polyhedra. Our simulation results and theoretical arguments lead to important organizing principles for nonspherical particles in terms of conjectures, which are very useful for designing nano-structured materials with desired structure and properties. In Chapter 7, we generate packings of binary superdisks, circular disks and ellipses in two-dimensions and monodisperse superballs in three dimensions that represent the maximally random jammed (MRJ) state of these particles and investigate their characteristics. We show that MRJ packings of nonspherical particles can be hypostatic and hyperuniformity is a universal feature of MRJ hard-particle packingss. In Chapter 8, we make concluding remarks and point out several directions for future research.