Annealed percolation: Determination of exponents in a correlated-percolation problem

Computer simulation of electrical conductivity of colloidal dispersions during aggregation

The computation approach to the simulation of electrical conductivity of colloidal dispersions during aggregation is considered. We use the two-dimensional diffusion-limited aggregation model with multiple-seed growth. The particles execute a random walk, but lose their mobility after contact with the growing clusters or seeds. The two parameters that control the aggregation are the initial concentration of free particles in the system p and the concentration of seeds psi. The case of psi=1, when all the particles are the immobile seeds, corresponds with the usual random percolation problem. The other limiting case of psi=0, when all the particles walk randomly, corresponds to the dynamical percolation problem. The calculation of electrical conductivity and cluster analysis were done with the help of the algorithms of Frank-Lobb and Hoshen-Kopelman. It is shown that the percolation concentration phi c decreases from 0.5927 at psi=1 to 0 at psi --> 0. Scaling analysis was applied to study exponents of correlation length v and of conductivity t. For all psi>0 this model shows universal behavior of classical 2d random percolation with v approximately t approximately 4/3. The electrical conductivity sigma of the system increases during aggregation reaching up to a maximum at the final stage. The concentration dependence of conductivity sigma(phi) obeys the general effective medium equation with apparent exponent ta(psi) that exceeds t. The kinetics of electrical conductivity changes during the aggregation is discussed. In the range of concentration Pc(phi)<p<0.9527 the time of percolation cluster formation pi c decreases with increasing phi.

Finite-size scaling analysis of percolation in three-dimensional correlated binary Markov chain random fields

Percolation and finite-size scaling properties in three-dimensional binary correlated Markov-chain random fields on a cubic lattice are computed by extensive Monte Carlo simulation. At short correlation scales, the percolation threshold in correlated random fields decreases as the correlation scale increases. The rate of decrease rapidly diminishes for correlation lengths larger than 2-3 lattice sites. At correlation scales of 4-6 lattice sites, the percolation threshold is found to be 0.126 +/- 0.001 for the Markov chain random fields, similar to that for sequential Gaussian and indicator random fields, which are evaluated for comparison. The average percolation threshold in finite-size lattices is a function of both, the correlation length and the finite lattice size. The universal scaling constants for mean cluster size and backbone fraction are found to be consistent with results on uncorrelated lattices. But prefactors of scaling relationships vary with correlation length. The squared radius of gyration of non-percolating clusters is found to scale with gamma/nu and its scaling prefactors are independent of the correlation scale. Prefactors are similar between the three random field generators evaluated. The percolation properties derived here are useful to account for finite-size effects on percolation in natural or man-made correlated systems.