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 $\alpha({\bf x})$. 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 $H \sim t^{1/2}$ for the slowing down motion of the average interface position $H$. The interface is found to be superrough, with global roughness exponent $\chi \approx 1.25$, indicating anomalous scaling. The spatial extent of the roughness is determined by a length scale $\xi_{\times} \sim H^{1/2}$ arising from the conservation law. The interface advances by avalanche motion, which causes temporal multiscaling and qualitatively reproduces the experimental results of Horv\a'ath and Stanley [Phys. Rev. E {\bf 52} 5166 (1995)] on the temporal scaling of the interface. 
M. Dube, M. Rost, K. R. Elder, M. Alava, S. Majaniemi, T. Ala-Nissila