Lineal-path function for random heterogeneous materials. II. Effect of polydispersivity

The lineal-path function L(z) for two-phase heterogeneous media gives the probability of finding a line segment of length z wholly in one of the phases, say phase 1, when randomly thrown into the sample. The function L(z) is equivalent to the area fraction of phase 1 measured from the projected image of a slab of the material of thickness z onto a plane. The lineal-path function is of interest in stereology and is an important morphological descriptor in determining the transport properties of heterogeneous media. We develop a means to represent and compute L(z) for distributions of D-dimensional spheres with a polydispersivity in size, thereby extending an earlier analysis by us for monodispersed-sphere systems. Exact analytical expressions for L(z) in the case of fully penetrable polydispersed spheres for arbitrary dimensionality are obtained. In the instance of totally impenetrable polydispersed spheres, we develop accurate approximations for the lineal-path function that apply over a wide range of volume fractions. The lineal-path function was found to be quite sensitive to polydispersivity for D≥2. We demonstrate how the measurement of the lineal-path function can yield the particle-size distribution of the particulate system, thus establishing a method to obtain the latter quantity.