Percolation phenomena, statistical

A description of the percolation phenomenon in ionic microemulsions in terms of the macroscopic DCF will be carried out based on the static lattice site percolation (SLSP) model [152]. In this model the statistical ensemble of various... 

However, if exact results are very difficult to obtain, it is possible to use numerical simulations. The main difficulty is that every simulation is feasible only for finite lattice sizes. The percolation phenomenon is a statistical phenomenon and only mean values are relevant. Thus the simulations should ideally be done over all possible lattice configurations. This is not possible for large lattices. Monte Carlo simulation techniques are generally used to overcome this difficulty. Some sample algorithms can be found in the textbook by Stauffer and Aharony [101]. 

Composites can be divided into two subgroups statistical mixtures and matrix-inclusion type composites. The effective dielectric function of the first subgroup can be calculated by equations, which are symmetrical with respect to phase indices. Statistical mixtures exhibit the so called percolation Phenomenon which is extremely important in conductor-insulator composites. Percolation threshold is a critical... 

The concentration c is equivalent to the critical point where the crossover phenomenon occurs from randomness to order. It is also equivalent to Pc (the critical probability) in the percolation theory, where the crossover phenomenon occurs from the finite cluster (such as a macromolecule containing a finite number of monomers) to the infinite cluster (such as the network of an entangled macromolecule, which extends from one end to the other). The three regions are characterized by three important quantities the number of statistical elements per chain N, the number of statistical elements per unit volume p (density), and the correlation or screen length... 

The difference between the FS model and percolation model is in the critical phenomenon. As summarized in Table 1, if the statistical values are normalized by the equivalent distance e(= 1 — a/a ) from the gel point (the critical point), there is a significant difference in critical index for flie FS model and percolation model. This difference reflects the difference in size distribution (see Fig. 1 [6]). The difference of the structure in flie model is reflected on the fractal dimension D of the fraction that has a certain degree of polymerization x. If the radius of a sphere that corresponds to the volume of the branched polymer fiaction with the degree of polymerization x is R, the relationship between x and R is fimm the fiactal dimension D... 

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