Kinetic roughening of interfaces in driven systems

Abstract

We study the dynamics of an interface driven far from equilibrium in three dimensions. First we derive the Kardar-Parisi-Zhang equation from the Langevin equation for a system with a nonconserved scalar order parameter, for the cases where an external field is present, and where an asymmetric coupling to a conserved variable exists. The relationship of the phenomena to self-organized critical phenomena is discussed. Numerical results are then obtained for three models that simulate the growth of an interface: the Kardar-Parisi-Zhang equation, a discrete version of that model, and a solid-on-solid model with asymmetric rates of evaporation and condensation. We first make a study of crossover effects. In particular, we propose a crossover scaling ansatz and verify it numerically. We then estimate the dynamical scaling exponents. Within the precision of our study, the Kardar-Parisi-Zhang equation and the solid-on-solid model have the same asymptotic behavior, indicating that the models share a dynamical universality class. Furthermore, the discrete models exhibit a kinetic roughening transition. We study this by monitoring the surface step energy, which shows a dramatic jump at a finite temperature for a given driving force. At the same temperature, a finite-size-scaling analysis of the bond-energy fluctuation shows a diverging peak

Stochastic growth equations and reparametrization invariance

“…Among the notable properties of the KPZ equation are its relations with a large number of other problems, including the statistics of directed polymers in random media (Kardar and Zhang, 1987, Fisher and Huse, 1991, Halpin-Healy and Zhang, 1995 and the Burgers equation for fluid dynamics (Forster et al, 1977). The Galilean invariance in the latter problem has profound consequences on the properties of the KPZ equation, and leads to the exponent identity z + α = 2 (Forster et al, 1977;Meakin et al, 1986).…”

“…A derivation of the KPZ equation from the functional derivative of a free energy with a volume and a surface term, was also obtained by Grossmann et al (1991) in a more complex way. An alternative derivation was also given in Keblinski et al, (1996).…”

“…The effects of a random substrate on wetting phenomena have been considered among others by Pfeifer et al, (1989) and Giugliarelli andStella (1991, 1994). Recent work by Halpin-Healy and Zhang (1995) concentrates on the Kardar-Parisi-Zhang (1986) equation and reviews the efforts of the last decade devoted to elucidating its scaling behavior. From this central theme, it extends to cover the field in a pedagogical manner, with special emphasis on directed polymers in random media.…”

“…There has been a great deal of recent work on the formation, growth and geometry of interfaces Barabási and Stanley, 1995;Halpin-Healy and Zhang, 1995). These studies are relevant to a variety of experimental situations including biological growth (Eden, 1958), the propagation of flame fronts (Sivashinsky, 1977(Sivashinsky, , 1979, fluid flow in porous media Martys et al, 1991) and atomic deposition processes (Tu and Harris,1991;Messier and Yehoda, 1985;Bales et al 1990;Tang et al 1990) such as the technologically important molecular beam epitaxy (MBE).…”

“…The first (see, e.g., ) is based on computer simulations of discrete models and often it provides useful links between analytic theory and experiments. The second approach (Krug and Spohn, 1990;Halpin-Healy and Zhang, 1995) is to describe the dynamical process by stochastic differential equations. This procedure neglects the short length-scale details but provides a coarse grained description of the interface which is suitable for characterizing the asymptotic scaling behavior.…”