A numerical study of the ζ₂ parameter for random suspensions of disks

The effective conductivity of two-component composites can be tightly bounded through the knowledge of structural parameters. While the first- and second-order parameters are known analytically for isotropic materials, the third and higher order parameters are generally not. Their evaluation has, therefore, become the subject of much research. In particular, the third-order structural parameter ζ₂ has been computed many times. Interface methods, beginning with Rayleigh, have proven successful for periodic composites with simple unit cells. Statistical methods, involving three-point correlation functions, work well for dilute random suspensions. Composites consisting of complicated, dense suspensions have been much more difficult to treat. In this article, we illustrate how one can greatly accelerate the computation of structural parameters with interface methods, so that these methods can be applied to dense suspensions with tens of thousands of randomly placed inclusions per unit cell. We implement a numerical scheme, based on the fast multipole method, for which the amount of work grows linearly with the number of inclusions per unit cell and quadratically with the logarithm of the desired precision. By incorporating a Monte Carlo sampling technique, we have computed values of ζ₂ for the random suspension of disks at 20 volume fractions between 0.50 and 0.69. These tabulated values are accurate to at least three digits and improve on the best previous estimates by a factor of between 30 and 100.