Random networks, electrical properties

The special electrical properties of a-AgI inevitably led to a search for other solids exhibiting high ionic conductivity preferably at temperatures lower than 146°C. The partial replacement of Ag by Rb, forms the compound RbAgJs. This compound has an ionic conductivity at room temperature of 25 S m , with an activation energy of only 0.07 eV. The crystal structure is different from that of a-AgI, but similarly the Rb and T ions form a rigid array while the Ag ions are randomly distributed over a network of tetrahedral sites through which they can move. 

Increasing the concentration of metal particles in an insulating adhesive matrix changes the electrical properties of the composite in a discontinuous way. Assuming a random dispersion of the metal filler, as the concentration increases no significant change occurs until a critical concentration, pc, is reached. This point, where the electrical resistivity decreases dramatically, called the percolation threshold, has been attributed to the formation of a network of chains of conductive particles than span the composite. A two-dimensional cartoon of a conductive adhesive below p and just above pc is shown in Fig. 3. A typical plot showing the relationship between particle concentration and electrical resistivity is shown in Fig. 4. 

We have studied the the fracture properties of such elastic networks, under large stresses, with initial random voids or cracks of different shapes and sizes given by the percolation statistics. In particular, we have studied the cumulative failure distribution F a) of such a solid and found that it is given by the Gumbel or the Weibull form (3.18), similar to the electrical breakdown cases discussed in the previous chapter. Extensive numerical and experimental studies, as discussed in Section 3.4.2, support the theoretical expectations. Again, similar to the case of electrical breakdown, the nature of the competition between the percolation and extreme statistics (competition between the Lifshitz length scale and the percolation correlation length) is not very clear yet near the percolation threshold of disorder. 

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 rationalizes sizes and distribution of connected black and white domains and the effects of cluster formation on macroscopic properties, for example, electric conductivity of a random composite or diffusion coefficient of a porous rock. A percolation cluster is defined by a set of connected sites of one color (e.g., white ) surrounded by percolation 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 connected if p is close to 1. Therefore, at some well-defined intermediate value of p, the percolation threshold, pc, a transition occurs in the topological structure of the percolation network that transforms it from a system of disconnected white clusters to a macroscopically connected system. In an infinite lattice, the site percolation threshold is the smallest occupation probability p of sites, at which an infinite cluster of white sites emerges. 

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