GEOMETRICAL PERCOLATION-THRESHOLD OF OVERLAPPING ELLIPSOIDS

A recurrent problem in materials science is the prediction of the perc olation threshold of suspensions and composites containing complex-sha ped constituents. We consider an idealized material built up from free ly overlapping objects randomly placed in a matrix, and numerically co mpute the geometrical percolation threshold p(c) where the objects fir st form a continuous phase. Ellipsoids of revolution, ranging from the extreme oblate limit of platelike particles to the extreme prolate li mit of needlelike particles, are used to study the influence of object shape on the value of p(c). The reciprocal threshold 1/p(c) (p(c) equ als the critical volume fraction occupied by the overlapping ellipsoid s) is found to scale linearly with the ratio of the larger ellipsoid d imension to the smaller dimension in both the needle and plate limits. Ratios of the estimates of p(c) are taken with other important functi onals of object shape (surface area, mean radius of curvature, radius of gyration, electrostatic capacity, excluded volume, and intrinsic co nductivity) in an attempt to obtain a universal description of p(c). U nfortunately, none of the possibilities considered proves to be invari ant over the entire shape range, so that p(c) appears to be a rather u nique functional of object shape. It is conjectured, based on the nume rical evidence, that 1/p(c) is minimal for a sphere of all objects hav ing a finite volume.