# Percolation

## Contents

## Introduction

**Percolation** describes a process by which sites or links on a (potentially random) graph or lattice can be connected. In the context of materials physics, the theory of percolation has three obvious applications:

- understanding the flow of fluid through a disordered porous medium - this could be oil flowing through the ground, or soot particles being filtered by a gas mask, for example;
- understanding the conductivity of random networks, such as granular metal films or carbon nanotube films, for example;
- understanding the formation and physical properties of gels, formed when a minimum
*percolation threshold*<math>p_c</math> fraction of bonds has been formed between the clusters making up the gel.

## Conductivity and the Elasticity of Gels

De Gennes (1976) first suggested a very elegant analogy between the elasticity of a gel and the electric conductivity of a conducting random percolating network, since both involve an abrupt transition from a 'weak' fluid-like/non-conducting to a 'strong' solid-like/conducting state. By considering the scalar case in which the spring constants associated with transverse and longitudinal displacements of the bonds holding the components of the network together are equal, de Gennes found that the elasticity and the conductivity of the network scale similarly as <math>(p-p_c)^{t}</math>, with <math>t\sim 2</math>. However, this model was later improved on several fronts: for example, Kantor and Webman (1984) later considered the more physically relevant case involving different bond-bending and bond-stretching moduli. Importantly, this difference can lead to different scaling exponents <math>t</math> of the elasticity and the conductivity of the network - for example, the elasticity exponent is typically increased, taking on a value of ~3.5-4 for singly connected networks (see, for example, Deptuck *et al.*, 1985). This shows that these two physical properties belong to different universality classes.

## Percolation of Fluids Through Disordered Media

Insert some discussion here.

## Mathematics of Percolation

Insert some discussion here.

## References

- P. G. de Gennes, "On a relation between percolation theory and the elasticity of gels",
*Journal de Physique Lettres***37,**1 (1976). - Y. Kantor and I. Webman, "Elastic Properties of Random Percolating Systems",
*Phys. Rev. Lett.***21,**1891 (1984). - W. H. Shih, W. Y. Shih, S. I. Kim, J. Liu and I. A. Aksay, "Scaling behavior of the elastic properties of colloidal gels",
*Phys. Rev. A***42,**4772 (1990). - D. Deptuck, J. P. Harrison and P. Zawadski, "Measurement of elasticity and conductivity of a three-dimensional percolation system",
*Phys. Rev. Lett.***54,**913 (1985).