SCALING LAW FOR CONDUCTION IN PARTIALLY CONNECTED SYSTEMS

The electrical transport properties of systems of conducting particles embedded in an insulator are considered. For low volume fractions of the particles, the conducting matrix may only be ''partially'' connect ed, as particles may only touch at corners or edges. As a model where these connectedness questions can be precisely formulated, we consider a random checkerboard in dimensions d = 2 and 3, where the squares in d = 2 or cubes in d = 3 are randomly assigned the conductivities 1 wi th probability p or 0 < delta much less than 1 with probability 1 - p. To analyze connectedness, we introduce a new parameter, d(m), called the minimal dimension, which measures connectedness of the conducting matrix via the dimension of the dominant contacts between particles. B ased on analysis of the checker-boards, we propose a general scaling l aw for the effective conductivity sigma as delta --> 0, namely sigma* approximately delta(q), where q = 1/2(d - d(m)) for 0 less-than-or-eq ual-to d - d(m) less-than-or-equal-to 2 and q = 1 for d - d(m) greater -than-or-equal-to 2. The applicability of this law to SitUations where d(m) is non-integral, such as the checkerboards at criticality, is di scussed in detail.