Precise algorithms to compute surface correlation functions of two-phase heterogeneous media and their applications

TL;DR

This paper introduces efficient algorithms for computing surface correlation functions in heterogeneous media, enabling detailed microstructural analysis and property estimation from experimental data.

Contribution

The authors develop novel algorithms that reduce computational complexity for surface correlation functions, validated against benchmark models and applied to various microstructures.

Findings

Algorithms accurately compute surface correlation functions.

Excellent agreement with exact models like overlapping spheres.

Effective characterization of complex microstructures.

Abstract

The quantitative characterization of the microstructure of random heterogeneous media in $d$-dimensional Euclidean space $\mathbb{R}^d$ via a variety of $n$-point correlation functions is of great importance, since the respective infinite set determines the effective physical properties of the media. In particular, surface-surface $F_{ss}$ and surface-void $F_{sv}$ correlation functions (obtainable from radiation scattering experiments) contain crucial interfacial information that enables one to estimate transport properties of the media (e.g., the mean survival time and fluid permeability) and complements the information content of the conventional two-point correlation function. However, the current technical difficulty involved in sampling surface correlation functions has been a stumbling block in their widespread use. We first present a concise derivation of the small-$r$ behaviors of these functions, which are linked to the \textit{mean curvature} of the system. Then we demonstrate that one can reduce the computational complexity of the problem by extracting the necessary interfacial information from a cut of the system with an infinitely long line. Accordingly, we devise algorithms based on this idea and test them for two-phase media in continuous and discrete spaces. Specifically for the exact benchmark model of overlapping spheres, we find excellent agreement between numerical and exact results. We compute surface correlation functions and corresponding local surface-area variances for a variety of other model microstructures, including hard spheres in equilibrium, decorated "stealthy" patterns, as well as snapshots of evolving pattern formation processes (e.g., spinodal decomposition). It is demonstrated that the precise determination of surface correlation functions provides a powerful means to characterize a wide class of complex multiphase microstructures.