1. Introduction

Percolation is the formation of at least one continuous domain-spanning cluster of a random phase in a material containing at least two phases (and one phase can be void).

Percolation phenomena are critical in understanding the full range of materials design and response. The determination of minimum amounts of phases required for percolation is a key first step, for example, in designing materials for mechanical, filtration, and conductive properties. Percolation concepts have also been used to model disease transmission and to design of sensor arrays. Mathematically, general percolation processes and phenomena have been studied since the early part of the last century, via development of analytical solutions for percolation of particles in a finite or infinite field (Coniglio et al., 1977), and through Monte Carlo simulations (Pike and Seager, 1974).

2. Percolation of Ellipses, Ellipsoids and fibers

While there have been numerous contributions to theoretical modeling of percolation in particulate systems (circles and spheres), no analytical approximations for the problems of generalized ellipses and ellipsoids have been reported until this recent work (collaborated with Dr. A. M. Sastry at the University of Michigan) in which I (1) derived, and verified through simulation, an analytical percolation approach capable of identifying the percolation point in materials containing ellipses and ellipsoids of uniform shape and size; (2) explored the dependence of percolation on particle aspect ratio.

For fibrous networks, it has been found that waviness significantly affects the geometric percolation threshold. Increases in curl ratio lead to increase in fiber density required for percolation. This observation logically follows the findings that increases in curl ratio reduce probability of intersection within arbitrary pairs of fibers. On the other hand, percolation threshold does not differ much for various shapes of fibers having identical curl ratio. For example, sinusoidal fiber systems studied had percolation threshold very close to those of the other two shapes studied, with variations being less than 10% throughout all values of curl ratio.

My interests in percolation phenomena span a number of challenge areas, which will continue to be important for basic science advances in biology, and design of new energetic and environmentally sound materials.

Reference

Yi, Y. B. and Sastry, A. M., 2004, Analytical approximation of percolation threshold for overlapping ellipsoids, Proc. Roy. Soc. London , A460, 2353-2380.

Yi, Y. B., Berhan, L. M., and Sastry, A. M., 2004, Statistical Geometry of Random Fibrous Networks, Revisited: Waviness, Dimensionality and Percolation, J. Applied Physics, 96, 1318-1327.

Coniglio, A., Deangelis, U., Forlani, A. & Lauro, G. 1977 Distribution of physical clusters. J. Phys. A: Math. Gen. 10, 219-228.

Pike, G. E. & Seager, C. H. 1974 Percolation and conductivity: a computer study. Phys. Rev. B 10, 1421-1434.