Growing interfaces in quenched disordered media

We present the microscopic equation of growing interface with quenched noise for the Tang and Leschhorn model [Phys. Rev. A 45, R8309 (1992)]. The evolution equations for the mean heigth and the roughness are reached in a simple way. Also, an equation for the interface activity density (i.e. interface density of free sites) as function of time is obtained. The microscopic equation allows us to express these equations in two contributions: the diffusion and the substratum one. All the equation shows the strong interplay between both contributions in the dynamics. A macroscopic evolution equation for the roughness is presented for this model for the critical pressure p = 0.461. The dynamical exponent β = 0.629 is analitically obtained in a simple way. Theoretical results are in excellent agreement with the Monte Carlo simulation. The growth of interfaces under nonequilibrium conditions is an interesting natural phenomenon. The interfaces has been characterized through scaling of the interface width w with time t and lateral size L. It is known that w ∼ L for t ≫ L and w ∼ t for t ≪ L . β and α are called the dynamical and the roughness exponents, respectively. The knowledge of the effects of the nonlinearities and the disorder of the media are important to describe the growing interface motion. Some experiments such as the growth of bacterial colonies and the motion of liquids in porous media, where the disorder is quenched, i.e. disorder due to the inhomogeneity of the media where the moving phase is propagating, are well described by directed percolation depinning (DPD) model proposed simultaneously by Tang and Leschhorn (TL) [1] and Buldyrev et. al [2]. In this work we will focus on TL model. This model predicts a dynamical exponent β ≃ 0.63 for qc = 1 E-mail: [email protected] Preprint submitted to Elsevier Preprint 1 February 2008 1− pc = 0.539, where pc is the critical pressure. However, some authors [2-4] found a dynamical exponent β faraway from the criticality. Yang and Hu [4] found β as function of q, analyzing the roughness as function of t. In a previous work, Braunstein and Buceta [5] show that the scaling law holds only at the criticality. We present now the Microscopic Equation (ME) for the TL model. The interface growth takes place in a lattice of edge L with periodic boundary conditions. A random pinning force g(r) uniformly distributed in [0, 1] is assigned to every cell of the lattice. For a given pressure p, the cells are divided in two groups, active (free) cells with g(r) ≤ p and inactive (blocked) cells with g(r) > p. The interface is specified by a set of integer column heights hi (i = 1, . . . , L). During the growth, a column is selected at random with probability 1/L and compared his height with those of his neighbors. The time evolution equation for the interface in a time step δt = 1/L is hi(t+ δt) = hi(t) + δt [Wi+1 +Wi−1 + Fi(h ′ i)Wi] , (1)