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” connected, 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 with probability p or 0 < δ ⪢ 1 with probability 1 − p. To analyze connectedness, we introduce a new parameter, dm, called the minimal dimension, which measures connectedness of the conducting matrix via the dimension of the dominant contacts between particles. Based on analysis of the checkerboards, we propose a general scaling law for the effective conductivity σ∗ as δ → 0, namely σ∗∼δq, where q = 12 (d − dm for 0 ⩽ d − dm ⩽ 2 and q =1 for d − dm ⩾ 2. The applicability of this law to situations where dm is non-integral, such as the checkerboards at criticality, is discussed in detail.