Surfaces of percolation systems in lattice problems

Cluster size distribution in percolation theory and fractal Cantor dust

Results of numerical simulation of cluster size distribution in the site percolation problem are presented. These results disagree with the theoretical data obtained on the basis of the standard drop model of finite cluster structure, in particular, they give a different value of exponent zeta (lnn{s} approximately -s{zeta}). Therefore, a more precise fractal model for describing the structure of clusters in a percolation system is proposed. The consideration is based on the solution of a kinetic equation for the number of finite clusters. In the framework of the proposed approach (fractal model together with kinetic equation), a correct value of exponent zeta is obtained and an explanation is given to the dependence of this exponent on the fraction of occupied sites p, which was revealed by numerical simulations. Additionally, a relation is established between the characteristics of cluster size distribution and fractal dimension of the Cantor dust constructed on the percolation cluster.

Effect of random internal structure on combustion of binary powder mixtures

Combustion of a mixture of two solid reactants is considered. A combined geometrical and physicochemical model of mixing of these reactants is proposed. The model takes into account random distribution of reactants and voids over the system. This allows description of incomplete burning of reactants. The model is used for studying the combustion wave propagation in a binary heterogeneous mixture. The obtained results are compared with those calculated from the perfect-mixing model, which implies that the deficient (with respect to the stoichiometry) component burns completely. It is shown that micrononuniformity of mixing not only reduces the combustion wave temperature and velocity but also can lead to a shift of the maximal wave velocity away from the stoichiometric composition of the mixture. The results of this study allow us to suggest that one of the main reasons for such a shift, which was observed in a number of experimental works, is structural disorder of binary mixtures.