Lineal-path function for random heterogeneous materials

Random-walk approach to the d-dimensional disordered Lorentz gas

A correlated random walk approach to diffusion is applied to the disordered nonoverlapping Lorentz gas. By invoking the Lu-Torquato theory for chord-length distributions in random media [J. Chem. Phys. 98, 6472 (1993)], an analytic expression for the diffusion constant in arbitrary number of dimensions d is obtained. The result corresponds to an Enskog-like correction to the Boltzmann prediction, being exact in the dilute limit, and better or nearly exact in comparison to renormalized kinetic theory predictions for all allowed densities in d=2,3 . Extensive numerical simulations were also performed to elucidate the role of the approximations involved.

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 S2(r) and formulate a conjecture. In particular, we argue that given a complete two-point correlation function space, S2(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 S2(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.

Properties of chord length distributions across ordered and disordered packing of hard disks

Chord length distributions across packings--random or not--of equal hard disks have a universal divergence which is proportional to n(c)/square root of l for small chords, where n(c) is the mean number of contacts. A similar behavior is derived for a population of polydisperse disks. Monte Carlo simulations across various kinds of regular and random packings are in full agreement with the theoretical predictions.

Grain reconstruction of porous media: application to a low-porosity Fontainebleau sandstone

The fundamental issue of reconstructing a porous medium is examined anew in this paper, thanks to a sample of low-porosity Fontainebleau sandstone that has been analyzed by computed microtomography. Various geometric properties are determined on the experimental sample. A statistical property, namely, the probability density of the covering radius, is determined. This is used in order to reconstruct a porous medium by means of a Poissonian generation of polydisperse spheres. In a second part, the properties of the real experimental sample and of the reconstructed one are compared. The most important success of the present reconstruction technique is the fact that the numerical sample percolates despite its low porosity. Moreover, other geometrical features and conductivity are found to be in good agreement.

Stochastic reconstruction of sandstones

A simulated annealing algorithm is employed to generate a stochastic model for a Berea sandstone and a Fontainebleau sandstone, with each a prescribed two-point probability function, lineal-path function, and "pore size" distribution function, respectively. We find that the temperature decrease of the annealing has to be rather quick to yield isotropic and percolating configurations. A comparison of simple morphological quantities indicates good agreement between the reconstructions and the original sandstones. Also, the mean survival time of a random walker in the pore space is reproduced with good accuracy. However, a more detailed investigation by means of local porosity theory shows that there may be significant differences of the geometrical connectivity between the reconstructed and the experimental samples.