Scaling properties of driven interfaces in disordered media

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 \ensuremath{\lambda} of the nonlinear term diverges at the depinning transition, while for the QEW universality class, either \ensuremath{\lambda}=0 or \ensuremath{\lambda}\ensuremath{\rightarrow}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 ${\mathrm{\ensuremath{\alpha}}}_{\mathit{P}}$=0.63\ifmmode\pm\else\textpm\fi{}0.03 for the pinned phase and ${\mathrm{\ensuremath{\alpha}}}_{\mathit{M}}$=0.75\ifmmode\pm\else\textpm\fi{}0.05 for the moving phase.For the growth exponent, we find ${\mathrm{\ensuremath{\beta}}}_{\mathit{P}}$=0.67\ifmmode\pm\else\textpm\fi{}0.05 for the pinned phase and ${\mathrm{\ensuremath{\beta}}}_{\mathit{M}}$=0.74\ifmmode\pm\else\textpm\fi{}0.06 for the moving phase. Furthermore, we find an anomalous scaling of the prefactor of the width on the driving force. A new exponent ${\mathit{cphi}}_{\mathit{M}}$=-0.12\ifmmode\pm\else\textpm\fi{}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 ${\mathrm{\ensuremath{\alpha}}}_{\mathit{P}}$\ensuremath{\approxeq}${\mathrm{\ensuremath{\alpha}}}_{\mathit{M}}$=0.92\ifmmode\pm\else\textpm\fi{}0.04 and ${\mathrm{\ensuremath{\beta}}}_{\mathit{P}}$\ensuremath{\approxeq}${\mathrm{\ensuremath{\beta}}}_{\mathit{M}}$=0.86\ifmmode\pm\else\textpm\fi{}0.03 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 ${\mathit{cphi}}_{\mathit{M}}$=0.44\ifmmode\pm\else\textpm\fi{}0.05 is found to relate the different values of the local ${\mathrm{\ensuremath{\alpha}}}_{\mathit{P}}$ and global roughness exponent ${\mathrm{\ensuremath{\alpha}}}_{\mathit{G}}$\ensuremath{\approxeq}1.23\ifmmode\pm\else\textpm\fi{}0.04 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.