Determination of the effective conductivity of heterogeneous media by Brownian motion simulation

A new Brownian motion simulation technique developed by Torquato and Kim [Appl. Phys. Lett. 55, 1847 (1989)] is applied and further developed to compute "exactly" the effective conductivity σ_e of n-phase heterogeneous media having phase conductivities σ₁, σ₂,..., σ_n and volume fractions φ₁, φ₂,., φ_n. The appropriate first passage time equations are derived for the first time to treat d-dimensional media (d=1, 2, or 3) having arbitrary microgeometries. For purposes of illustration, the simulation procedure is employed to compute the transverse effective conductivity σ_e of a two-phase composite composed of a random distribution of infinitely long, oriented, hard cylinders of conductivity σ₂ in a matrix of conductivity σ₁ for virtually all volume fractions and for several values of the conductivity ratio α=σ₂/σ₁, including perfectly conducting cylinders (α=∞). The method is shown to yield σ_e accurately with a comparatively fast execution time.