Conserved dynamics and interface roughening in spontaneous imbibition:: A phase field model

The propagation and roughening of a fluid-gas interface through a disordered medium in the case of capillary driven spontaneous imbibition is considered. The system is described by a conserved (model B) phase-field model, with the structure of the disordered medium appearing as a quenched random field alpha(x). The flow of liquid into the medium is obtained by imposing a non-equilibrium boundary condition on the chemical potential, which reproduces Washburn's equation H similar to t(1/2) for the slowing down motion of the average interface position H. The interface is found to be superrough, with global roughness exponent chi approximate to 1.25, indicating anomalous scaling. The spatial extent of the roughness is determined by a length scale xi(x) similar to H-1/2 arising from the conservation law. The interface advances by avalanche motion, which causes temporal multiscaling and qualitatively reproduces the experimental results of Horvath and Stanley (Phys. Rev. E 52, 5166 (1995)) on the temporal scaling of the interface.