Extremum of the percolation cluster surface

The internal and external surfaces of a percolation cluster, as well as the total surface of the entire percolation system, are investigated numerically and analytically. Numerical simulation is carried out using the Monte Carlo method for problems of percolation over lattice sites and bonds on square and simple cubic lattices. Analytic expressions derived by using the probabilistic approach describe the behavior of such surfaces to a high degree of accuracy. It is shown that both the external and total surface areas of a percolation cluster, as well as the total area of the surface of the entire percolation system, have a peak for a certain (different in the general case) fraction of occupied sites (in the site problem) or bonds (in the bond problem). Two examples of technological processes (current generation in a fuel cell and self-propagating high-temperature synthesis in heterogeneous condensed systems) in which the surface of a percolation cluster plays a significant role are discussed.