Depinning transitions in interface growth models

Abstract

Pinning-depinning transitions are roughening transitions separating a growing phase and pinned (or blocked) one, and are frequently connected to transitions into absorbing states. In this review, we discuss lattice growth models exhibiting this type of dynamic transition. Driven growth in media with impurities, the competition between deposition and desorption and deposition of poisoning species are some of the physical mechanisms responsible for the transitions, leading to different types of stochastic growth rules. The growth models are classified according to the those mechanisms and possible applications are shown, which include suggestions of experimental realizations of directed percolation transitions. I IntroductionThe study of surface and interface growth processes attracts much interest from the technological point of view mainly because it helps to understand the growth mechanisms of nanostructures and, consequently, their physical properties [1]. From the theoretical point of view, they motivated significant advances in non-equilibrium Statistical Mechanics and related fields [2,3]. Theoretical modeling is usually based on stochastic differential equations or on discrete atomistic models. In various models, changing one parameter leads to a roughening transition between a rough and a smooth phase. Two well known examples are the transition in the Kardar-Parisi-Zhang equation in dimensions d ≥ 3, separating regimes with linear (smooth) and nonlinear growth, and temperature-induced roughening transitions in equilibrium conditions [2]. On the other hand, in the so-called depinning transitions, the interface propagates only in the rough phase, being pinned in the smooth phase. Several mechanisms may be responsible for this type of transition, such as the presence of impurities in the growth media, competition between deposition and evaporation or formation of poisoning species at the growing surface. One of the interesting features of depinning transitions is that they frequently can be mapped onto transitions into absorbing states, whose most prominent example is directed percolation (DP). Different connections of depinning transitions to DP are found in discrete and continuous growth models, as well as connections with other classes of transitions to absorbing states [4] and, eventually, with statistical equilibrium transitions. Besides the fundamental interest of these systems, the large variety of interface growth phenomena observed in nature suggests them as strong candidates to experimental realizations of models of dynamic transitions, such as DP (see e. g. Ref.[5]).The aim of the present work is to review the basic features of lattice growth models exhibiting depinning transitions. Before introducing these systems, we will briefly summarize the theory and phenomenology concerning interface growth and phase transitions into absorbing states (Sec. II). Then we will discuss the problem of growing interfaces in disordered media with quenched disorder, in which transitions in the DP class or in the cl...

Pinning and de-pinning phenomena in front propagation in heterogeneous media

“…In the specific case of (1) with random spatial forcing f (x, u), the model is known in the physics literature as the quenched EdwardsWilkinson model which is studied extensively (see for example [4]). …”

Multichain models of conserved lattice gas

“…In fact the weights, as writen in Eq. (5), enesures that the steady state of the CLG on a ladder has a matrix product form where each rung is represented by a matrix,…”

“…In the study of absorbing state phase transition (APT) [1], directed percolation (DP) [2] has been considered to be the most robust universality class. Critical behavior encountered in many diverse problems, like synchronization [3], damage spreading [4], depinning transition [5], catalytic reactions [6], forest fire [7], extinction of species [8] etc. belong to the DP universality class [9].…”

Multicritical absorbing phase transition in a class of exactly solvable models

“…On one hand the non-equilibrium dynamics generically makes analytical treatment of these systems highly nontrivial, giving rise to varied class of distributions as well as rich variety of novel correlations, and on the other hand the non-fluctuating disordered phase being unique to APT leads to a unconventional critical behaviour. The most robust universality class of APT is directed percolation (DP) [2], which is observed in context of synchronization [3], damage spreading [4], depinning transition [5], catalytic reactions [6], forest fire [7], extinction of species [8] etc. Recently DP critical behaviour has been observed experimentally [9] in liquid crystals.…”

“…(2). The matrix formulation is very useful here, as one can simply set E m = 0 for m > n (5) to ensure that probability of all non-recurring configurations are 0. Further, let us assume that matrix D = |α β|, where |β , α| are yet to be determined.…”