Poresizes in random line networks

Abstract

Many natural fibrous networks with fiber diameters much smaller than the average poresize can be described as three-dimensional (3D) random line networks. We consider here a `Mikado' model for such systems, consisting of straight line segments of equal length, distributed homogeneously and isotropically in space. First, we derive analytically the probability density distribution $p(r_{no})$ for the `nearest obstacle distance' $r_{no}$ between a randomly chosen test point within the network pores and its closest neighboring point on a line segment. Second, we show that in the limit where the line segments are much longer than the typical pore size, $p(r_{no})$ becomes a Rayleigh distribution. The single parameter $\sigma$ of this Rayleigh distribution represents the most probable nearest obstacle distance and can be expressed in terms of the total line length per unit volume. Finally, we show by numerical simulations that $\sigma$ differs only by a constant factor from the intuitive notion of average `pore size', defined by finding the maximum sphere that fits into each pore and then averaging over the radii of these spheres.