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

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Abstract
The microstructure of two-phase disordered media can be characterised in terms of a set of n-point matrix probability functions Sn 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.