Scaling Properties of Driven Interfaces in Disordered Media

Recently the growth of rough interfaces has witnessed an explosion of theoretical, numerical, and experimental studies, fueled by the broad interdisciplinary aspects of the subject. Applications can be so diverse as imbibition in porous media, fluid--fluid displacement, fire front motion, and the motion of flux lines in superconductors.

In general, a d-dimensional self-affine interface, described by a single-valued function h(x,t), evolves in a (d+1)-dimensional medium. Usually, some form of disorder affects the motion of the interface leading to its roughening. Two main classes of disorder have been discussed in the literature. The first, called thermal or ``annealed,'' depends only on time. The second, referred to as ``quenched,'' is frozen in the medium. Early studies focused on time-dependent disorder as being responsible for the roughening. Here, we consider in detail the effect of quenched disorder on the growth [H. A. Makse and L. A. N. Amaral, Scaling Behavior of Driven Interfaces Above the Depinning Transition, Europhys. Lett. 31, 379-384 (1995); and L. A. N. Amaral, A.-L. Barabasi, H. A. Makse, and H. E. Stanley, Scaling Properties of Driven Interfaces in Disordered Media, Phys. Rev. E 52, 4087-4104 (1995); L. A. N. Amaral and H. A. Makse, Comment on: Kinetic Roughening in Slow Combustion of Paper, Phys. Rev. Lett. 80, 5706 (1998)].

We perform a systematic study of several models that have been proposed for the purpose of understanding the motion of driven interfaces in disordered media. We identify two distinct universality classes: (i) One of these, referred to as directed percolation depinning (DPD), can be described by a Langevin equation similar to the Kardar-Parisi-Zhang equation, but with quenched disorder. (ii) The other, referred to as quenched Edwards-Wilkinson (QEW), can be described by a Langevin equation similar to the Edwards-Wilkinson equation but with quenched disorder. We find that for the DPD universality class the coefficient of the nonlinear term diverges at the depinning transition, while for the QEW universality class the coefficient goes to 0 as the depinning transition is approached. The identification of the two universality classes allows us to better understand many of the results previously obtained experimentally and numerically. However, we find that some results cannot be understood in terms of the exponents obtained for the two universality classes at the depinning transition. In order to understand these remaining disagreements, we investigate the scaling properties of models in each of the two universality classes above the depinning transition. For the DPD universality class, we find for the roughness exponent alpha_P = 0.63 for the pinned phase, and alpha_M = 0.75 for the moving phase. For the growth exponent, we find beta_P = 0.67 for the pinned phase, and beta_M = 0.74 for the moving phase. Furthermore, we find an anomalous scaling of the prefactor of the width on the driving force. A new exponent varphi_M = -0.12 0.06, characterizing the scaling of this prefactor, is shown to relate the values of the roughness exponents on both sides of the depinning transition. For the QEW universality class, we find that alpha_P ~ alpha_M = 0.92 and beta_P ~ beta_M = 0.86 are roughly the same for both the pinned and moving phases. Moreover, we again find a dependence of the prefactor of the width on the driving force. For this universality class, the exponent varphi_M = 0.44 is found to relate the different values of the local alpha_P and global roughness exponent alpha_G ~ 1.23 at the depinning transition. These results provide us with a more consistent understanding of the scaling properties of the two universality classes, both at and above the depinning transition. We compare our results with all the relevant experiments.

Singularities and Avalanches in Interface Growth with Quenched Disorder

I proposed cellular automaton model for an interface moving in a disordered medium [H. A. Makse, Singularities and Avalanches in Interface Growth with Quenched Disorder, Phys. Rev. E 52, 4080-4086 (1995)]. The model exhibits a transition between the two universality classes of interface growth in the presence of quenched disorder. Using this model, it is shown that the application of constraints to the local slopes of the interface produces avalanches of growth, that become relevant in the vicinity of the depinning transition. The study of these avalanches reveals a singular behavior at the depinning transition that explains a recently observed divergency in the equation of motion of the interface. The anisotropy in the medium is also studied as a possible source of the divergency in the equation of motion.

(a) (b) (c) Three different interfaces at the depinning transition of the models considered in this paper. (a) A typical DPD interface at the depinning transition. The holes left behind by the interface correspond to the application of the generalized SOS condition after a longitudinal motion of a site at the interface occurs. The holes are situated in regions of strong disorder strength, so that the interface overcomes these pinning regions by longitudinal motions plus the application of the generalized SOS condition. (b) A typical QEW interface at the depinning transition. Since no longitudinal motions occur, the holes have disappeared. Notice the large slopes developed due to the absence of constraints to the local slopes of the interface. (c) A typical anisotropic flux line ``interface'' characterized by anisotropic random forces of strength Delta_x=2.0, and Delta_y=3. The trajectories of the even beads are plotted in black, and the trajectories of the odd beads are plotted in gray. The flux line takes advantage of the allowed longitudinal motions to surround the strong pinning configurations, as can be observed in the trajectories of the beads. Also, since Delta_x