Structure-related Effective Properties of CLs

For materials with random morphologies, the effect of microscopic composition, i.e., of size, shape, and random distribution of phase domains and of their coimectedness, on its macroscopic properties is the topic addressed by percolation theory [129, 130], Here, we briefly describe the basic concepts of this theory and outline their application to determine the morphology and effective properties of random composite materials. We refer the interested reader to Stauffer and Aharony [46] and Sahimi [129] for detailed discussion of percolation theory and its applications. 

Percolation phenomena deal with the effect of clustering and coimectivity of microscopic elements in a disordered medium [129], Percolation theory represents a random composite material as a network or lattice structure of two or more distinct types of microscopic elements or phase domains, the so-called percolation sites. These elements represent mutually exclusive physical properties, e.g., electrically conducting vs. isolating phase domains, pore space vs. solid matrix, atoms with spin up vs. spin down states. Here, we will refer to black and white elements for definiteness. The network onto which black and white elements of the composite medium are distributed could be continuous (continuum percolation) or discrete (discrete or lattice percolation) it could be a disordered or regular network. With a probability p a randomly chosen percolation site will be 

Percolation theory deals with the size and distribution of connected black and white domains and the effects on macroscopic observable properties, e.g., eleetrie conductivity of a random composite or diffusion coefficient of a porous roek. A percolation cluster is defined by a set of connected sites of one color (e.g., white ) surrounded by sites of the complementary color (i.e., black ). If p is sufficiently small, the size of any connected cluster is likely to be small compared to the size of the sample. There will be no continuously connected path between the opposite faces of the sample. On the other hand, the network should be entirely eonnected if is close to 1. Therefore at some well-defined intermediate value, 

In the metal/isolator composite, p is the threshold above which the composite will be conducting. In the vicinity of the percolation threshold, for p p, the effective conductivity of the composite is determined by a critical law 

Percolation thresholds for several lattice types are listed in Table 8.1. In ID it is trivially, = 1. In 2D, values for p are known exactly for specific lattice types. In 3D values of p can only be found with the help of computer simulations. 

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