Percolation theory metal/insulator composite

The following simple experiment demonstrates the essential concept of the critical threshold for percolation. A mixture of small plastic and metal balls of equal size is poured into a beaker with a crumpled-foil electrode at the bottom, another crumpled-foil electrode is pressed onto the top, and the electrodes are connected to a battery through an ammeter. Current is measured as a function of the composition (i.e., fraction of metal balls) of the conductor/insulator mixture. There is a critical composition below which no current flows and above which the conductivity increases nearly exponentially. At this threshold the two electrodes suddenly become spatially connected along a statistical pathway originating in the random medium. Percolation theory tells us that the critical composition is 0.25 fraction metal balls, a remarkably low concentration. This is perhaps not an intuitive result. 

Effective medium theories characterize the frequency-dependent transport in systems with large-scale inhomogeneities such as metal particles dispersed in an insulating matrix [118,119]. An IMT in the effective medium model represents a percolation problem where a finite a c as T 0 is not achieved until metallic grains in contact span the sample. To understand the frequency dependence of the macroscopic material, an effective medium is built up from a composite of volume fraction /of metallic grains and volume fraction 1 — / of insulator grains. The effective dielectric function semaCw) and conductivity function (Tema(w) are solved self-consistently. 

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