Modeling heterogeneous materials via two-point correlation functions: Basic principles

Heterogeneous materials abound in nature and man-made situations. Examples include porous media, biological materials, and composite materials. Diverse and interesting properties exhibited by these materials result from their complex microstructures, which also make it difficult to model the materials. Yeong and Torquato [Phys. Rev. E 57, 495 (1998)] introduced a stochastic optimization technique that enables one to generate realizations of heterogeneous materials from a prescribed set of correlation functions. In this first part of a series of two papers, we collect the known necessary conditions on the standard two-point correlation function ${S}_{2}(\mathbf{r})$ and formulate a conjecture. In particular, we argue that given a complete two-point correlation function space, ${S}_{2}(\mathbf{r})$ of any statistically homogeneous material can be expressed through a map on a selected set of bases of the function space. We provide examples of realizable two-point correlation functions and suggest a set of analytical basis functions. We also discuss an exact mathematical formulation of the (re)construction problem and prove that ${S}_{2}(\mathbf{r})$ cannot completely specify a two-phase heterogeneous material alone. Moreover, we devise an efficient and isotropy-preserving construction algorithm, namely, the lattice-point algorithm to generate realizations of materials from their two-point correlation functions based on the Yeong-Torquato technique. Subsequent analysis can be performed on the generated images to obtain desired macroscopic properties. These developments are integrated here into a general scheme that enables one to model and categorize heterogeneous materials via two-point correlation functions. We will mainly focus on basic principles in this paper. The algorithmic details and applications of the general scheme are given in the second part of this series of two papers.