Department of Physics, Princeton University, Princeton, New Jersey 08544, USA.
Department of Chemistry, Department of Physics, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA.
Phys Rev E. 2018 Jul;98(1-1):013307. doi: 10.1103/PhysRevE.98.013307.
The quantitative characterization of the microstructure of random heterogeneous media in d-dimensional Euclidean space 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 mean curvature of the system. Then we demonstrate that one can reduce the computational complexity of the problem, without sacrificing accuracy, by extracting the necessary interfacial information from a cut of the d-dimensional statistically homogeneous and isotropic 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.
在 d 维欧几里得空间 R^{d}中，通过各种 n 点相关函数对随机非均匀介质的微观结构进行定量描述非常重要，因为各自的无限集确定了介质的有效物理性质。特别是，表面-表面 F_{ss}和表面-空穴 F_{sv}相关函数（可从辐射散射实验中获得）包含关键的界面信息，使人们能够估计介质的输运性质（例如，平均存活时间和流体渗透率），并补充了传统两点相关函数的信息含量。然而，目前采样表面相关函数所涉及的技术困难一直是其广泛应用的绊脚石。我们首先简要推导了这些函数的小 r 行为，这些行为与系统的平均曲率有关。然后我们证明，可以通过从具有无限长线的 d 维统计均匀各向同性系统的切割中提取必要的界面信息，来降低问题的计算复杂度，而不会牺牲准确性。因此，我们根据这一思想设计了算法，并在连续和离散空间中的两相介质中对其进行了测试。具体对于重叠球体的精确基准模型，我们发现数值结果与精确结果之间有极好的一致性。我们计算了各种其他模型微结构的表面相关函数和相应的局部表面面积方差，包括平衡态的硬球、装饰的“隐形”图案以及演化图案形成过程的快照（例如，旋节分解）。结果表明，表面相关函数的精确确定提供了一种强有力的方法来描述广泛的复杂多相微结构。
