Chord-length distribution function for two-phase random media

Abstract

A statistical correlation function of basic importance in the study of two-phase random media (such as suspensions, porous media, and composites) is the chord-length distribution function p(z). We show that p(z) is related to another fundamentally important morphological descriptor studied by us previously, namely, the lineal-path function L(z), which gives the probability of finding a line segment of length z wholly in one of the phases when randomly thrown into the sample. We derive exact series representations of the chord-length distribution function for media comprised of spheres with a polydispersivity in size for arbitrary space dimension D. For the special case of spatially uncorrelated spheres (i.e., fully penetrable spheres), we determine exactly p(z) and the mean chord length Ic, the first moment of p(z). We also obtain corresponding formulas for the case of impenetrable (i.e., spatially correlated) polydispersed spheres. PACS number(s): 47.55.Mh, 05.20. -y, 61.20.GyThe characterization of the microstructure of twophase random media, such as suspensions, composites, and porous media, is of great fundamental as well as practical importance [1 -11]. The goal ultimately is to ascertain what is the essential morphological information, quantify it either theoretically or experimentally, and then employ the information to estimate the desired macroscopic properties of the heterogeneous material.In this Brief Report, we concern ourselves with the socalled chord length -distribution function p (z).Specifically, p (z)dz is the probability of finding a chord of length between z and z+dz in one of the phases, say phase 1. Chords are distributions of lengths between intersections of lines with the two-phase interface (see Fig. 1). Knowledge of the chord-length distribution function is of basic importance in transport problems involving "discrete free paths" and thus has an application in FIG. 1. Schematic of chord-length measurements for a cross section of a two-phase random medium. The chords are defined by the intersection of lines with the two-phase interface.Knudsen diffusion and radiative transport in porous media [12 -15]. The function p(z) has also been measured for sedimentary rocks [16] for the purpose of studying Auid Aow through such porous media. The chordlength distribution function p(z) is also a quantity of great interest in stereology [11]. For example, the mean chord (or intercept) length lc is the first moment of p (z).We first show that p (z) is related to another important morphological descriptor of random media studied by us earlier [17,18], namely, the lineal path functio-n L (z) which gives the probability of finding a line segment of length z wholly in phase 1 when randomly thrown into the sample. For heterogeneous media composed of spheres with a polydispersivity in size, we find an exact series representation of p (z). In the special case of fully penetrable (i.e., spatially uncorrelated) spheres, we determine exact expressions for p (z) and lc. Corresponding analytical formulas are als...

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