Static lattice site percolation model

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... 

Here the dimensionless time z=t/t is normalized by the characteristic relaxation time t, the time required for a charge carrier to move the distance equal to the size of one droplet, which is associated with the size of the unit cell in the lattice of the static site-percolation model. Similarly, we introduce the dimensionless time zs = ts/t where ts is the effective correlation time of the s-cluster, and the dimensionless time z = tm/t. The maximum correlation time t, is the effective correlation time corresponding to the maximal cluster sm. In terms of the random walker problem, it is the time required for a charge carrier to visit all the droplets of the maximum cluster sm. Thus, the macroscopic DCF may be obtained by the averaging procedure... 

Figure 34 shows the temperature dependencies of the static fractal dimensions of the maximal cluster. Note that at percolation temperature the value of the static fractal dimension Ds is extremely close to the classical value 2.53 for a three-dimensional lattice in the static site percolation model [152]. Moreover, the temperature dependence of the stretch parameter v (see Fig. 34) confirms the validity of our previous result [see (62)] Ds = 3v obtained for the regular fractal model of the percolation cluster [47]. 

Note that the classic SLSP model for a square lattice ABCO would provide a percolation trajectory connecting opposite sites OA and CB (see Fig. 32). On the other hand, for visualization of dynamic percolation we shall consider an effective three-dimensional static representation of a percolation trajectory connecting ribs OA and ED spaced distance Lh (see Fig. 33). Such a consideration allows us to return to the lattice OEDA, with the initial dimension <7 = 2, which is non-square and characterized by the two dimensionless sizes Lh 11 and L/l. The new lattice size of the system can be determined from the rectangular triangle OEQ by... 

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