Interface moving through a random background

Abstract

We study the motion of a phase interface which is driven through a random background medium, with application to immiscible-fluid displacement in porous media and to random-field Ising systems. The interface motion is described by a local stochastic differential equation, with terms corresponding to an external driving force, interface elasticity, and a random background force. The same equation has been examined by Bruinsma and Aeppli, with whose conclusions we disagree in part. In mean-field theory, we find that the interface can either translate with constant velocity and average width, or be pinned by the random background. The pinning is associated with invasion percolation in the fluid-displacement application and with metastable domains in the Ising case. Perturbation theory in the random term is consistent with the mean-field behavior above three dimensions but diverges in time at lower dimensions, suggesting a transition. The perturbation series appears to be unrenormalizable. By numerical integration of the differential equation, we find that in dimensions less than or equal to 3, the interface pins at sufficiently large randomness but translates essentially as a plane otherwise, while in four dimensions the interface always translates and is never pinned.