# Effective conductivity of a random suspension of highly conducting spherical particles

## TL;DR

This paper refines the analytical understanding of the effective conductivity in random suspensions of highly conducting spheres, correcting previous formulas and developing a new computational model to account for inclusion interactions.

## Contribution

It corrects Jeffrey's formula for effective conductivity by properly analyzing the conditionally convergent sum and introduces a new symbolic computation model for higher-order terms.

## Key findings

- Corrected Jeffrey's formula up to O(f^3)
- Developed an algorithm for effective conductivity tensor up to O(f^{10/3})
- Showed that previous universal formulas are limited to dilute or non-interacting composites.

## Abstract

Randomly distributed non-overlapping perfectly conducting spheres are embedded in a conducting matrix with the concentration of inclusions $f$. Jeffrey (1973) suggested an analytical formula valid up to $O(f^3)$ for macroscopically isotropic random composites. A conditionally convergent sum arose in the spatial averaging. In the present paper, we apply a method of functional equations to random composites and correct Jeffrey's formula. The main revision concerns the proper investigation of the conditionally convergent sum and correction the $f^2$-term. A new model of symbolic computations is developed in order to compute the effective conductivity tensor. The corresponding algorithm is realized up to $O(f^{\frac{10}3})$. The obtained formulae explicitly demonstrate the dependence of the effective conductivity tensor on the deterministic and probabilistic distributions of inclusions in the $f^2$-term, and in the $f^3$-term. This leads to the conclusion that some previous formulae presented as universal, i.e., valid for all random composites, may be actually applied only to dilute or to special composites when interaction between inclusions do not matter.