Slow coarsening in unstable liquid films under gravity on a disordered substrate

We study the evolution of unstable liquid films via numerical solutions of the thin-film equation. The film is placed on a coated substrate with disorder. This is modeled by a random spatial variation of the relative value of the Hamaker constants for the substrate and coating. The free energy consists of (a) the van der Waals term for the substrate/coating interactions with the film and (b) a term due to gravity. This free energy admits a Maxwell double-tangent construction with two coexisting phases, i.e., "thin" and "thick" phases. In the absence of disorder, the film dewets by true morphological phase separation (MPS), i.e., the elimination of domain walls between the coexisting phases. The introduction of disorder may result in the trapping of these domain walls, with a drastic slowdown in growth kinetics. We present detailed numerical results in D=2 and D=3 to understand this slow coarsening, where D is the dimensionality of the liquid-film system.