Dynamics of surface roughening in disordered media

In this paper the roughening of interfaces moving in inhomogeneous media is investigated by examining the corresponding stochastic differential equations using (i) numerical methods and (ii) dimensional analysis. We consider interface evolution equations where disorder is represented by quenched noise which can be both additive and multiplicative. Our main finding is that quenched noise leads to a new universality class as concerning the exponents δ and β describing respectively the spatial and temporal scaling of surface roughness. In particular, additive noise close to the pinning transition results in a behaviour with δ = 0.71 ± 0.08 and β = 0.61±0.06 up to a crossover time. These estimates are in very good agreement with the theoretical prediction β = 35 and δ = 34 that we derive from a dimensional analysis of the equation. Furthermore, we argue that multiplicative noise is the appropriate choice to describe experiments where the interface between two flowing phases is considered. By numerically integrating the proposed equation we have obtained (i) surfaces remarkably similar to those observed in the experiments and (ii) a scaling of the surface width as a function of time with an exponent β = 0.65 being in an excellent agreement with the experimental value. In addition to the exponents we discuss other relevant features of the surfaces, including the scaling of the average velocity of the surface νa close to pinning and the non-trivial, power law distribution of waiting times and noise along the interface in the stationary regime.