A diagram technique for perturbation theory calculations of the effective conductivity of two-dimensional systems

The perturbation theory for calculating the effective conductivity of the p lane consisting of pieces of different conductivities is constructed, and a convenient diagram technique is elaborated for this perturbation theory. I t is shown that for the chessboard, perturbative calculations give results that are in agreement with the well-known formula sigma (eff) = root sigma (1)sigma (2). The components of the effective conductivity tensor for the a nisotropic three-color chessboard are calculated. It is shown that the isot ropic (symmetric) part of the effective conductivity calculated up to the s ixth order of perturbation theory satisfies the Bruggeman effective medium equation for symmetric three-color structures with equally partitioned comp onents. We also consider an isotropic three-color chessboard with nonequal weights of colors. In this case, the perturbation theory in the fourth orde r contradicts the results following from the Bruggeman equation for nonequa l weights. (C) 2000 MAIK "Nauka/Interperiodica".