Chord-length distribution function for two-phase random media

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 ${\mathit{l}}_{\mathit{C}}$, the first moment of p(z). We also obtain corresponding formulas for the case of impenetrable (i.e., spatially correlated) polydispersed spheres.