Interfacial thin films rupture and self-similarity

Abstract

Two superposed thin layers of fluids are prone to interfacial instabilities due to London-van der Waals forces. Evolution equations for the film thicknesses are derived using lubrication theory. Using the intrinsic scales, for a single layer, results in a system with parametric dependence of four ratios of the two layers: surface tension, Hamaker constant, viscosity, and film thickness. In contrast to the single layer case, the bilayer system has two unstable eigenmodes: squeezing and bending. For some particular parameter regimes, the system exhibits the avoided crossing behavior, where the two eigenmodes are interchanged. Based on numerical analysis, the system evolves into four different rupture states: basal layer rupture, upper layer rupture, double layer rupture, and mixed layer rupture. The ratio of Hamaker constants and the relative film thickness of the two layers control the system dynamics. Remarkably, the line of avoided crossing demarks the transition region of mode mixing and energy transfer, affecting the scaling of the dynamical regime map consequentially. Asymptotic and numerical analyses are used to examine the self-similar ruptures and to extract the power law scalings for both the basal layer rupture and the upper layer rupture. The scaling laws for the basal layer rupture are the same as those of the single layer on top of a substrate. The scaling laws for the upper layer rupture are different: the lateral length scale decreases according to (tr-t)1/3 and the film thickness decreases according to (tr-t)1/6.