Characterisation of the microstructure of distributions of rigid rods and discs in a matrix

The microstructure of two-phase disordered media can be characterised in terms of a set of n-point matrix probability functions S_n which give the probability of finding n points all in the matrix phase. The authors obtain, for the first time, an exact analytical expression for S₂ for a distribution of equi-sized rigid rods in a matrix at any density and for all values of its argument. They evaluate, also for the first time, S₂ for a distribution of equi-sized rigid discs in a matrix, for a wide range of densities. Using these results for S₂ and rigorous upper and lower bounds on S₃, one may obtain bounds on S₃ for distributions of rigid rods and discs. The one- and two-dimensional results obtained here are compared to the three-dimensional results of Torquato and Stell (1983) at certain particle volume fractions.