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

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Abstract
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 σe of n-phase heterogeneous media having phase conductivities σ1, σ2, ..., σn and volume fractions φ1, φ2, ..., φn. 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 σe 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 σe accurately with a comparatively fast execution time.

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