An analytical nonlinear theory of thin film rupture and its application to wetting films

Abstract

A formalism is developed to investigate the instability and the time of rupture of thin films which are subjected to finite amplitude perturbations. The hydrodynamic and the interaction force nonlinearities cannot be ignored in such an event. The key idea of the approach is that, for a finite amplitude perturbation, one needs to study the instability of the corresponding finite amplitude, spatially nonhomogeneous stationary solution of the governing equations. It is in the event of infinitesimal perturbations that the nonhomogeneous stationary state reduces to the planar interface, the stability issue of which is addressed by the linear stability analysis. Thus the essential idea is to account for a part of the finite disturbance of the interface in the base state itself, thereby facilitating the solution for a finite amplitude disturbance. This procedure brings the base state (or the zeroth order solution) closer to the initial perturbation, in contrast to the choice of the base state as an undisturbed, planar interface. The formalism gives the growth rate of the initial perturbations as a function of various system parameters, the wave number, and the amplitude of the initial perturbation. As opposed to the linear theory then, both the neutrally stable state of the thin film and the dominant growth rate of the perturbations are functions of the amplitude of the perturbations. The time of rupture is derived for a thin film in contact with a solid and bounded by a thick bulk phase, with soluble surfactants present in both the film and the bounding fluid. The analytical expressions for a thin film devoid of surfactants are in good agreement with the numerical calculations of Williams and Davis for the same case. The nonlinear effects of the surfactant concentration driven Marangoni-motion (surface elasticity) and the surface shear viscosity are then considered. It is shown that for large initial perturbations, both the Marangoni-flow and the surface viscosity may prolong the time of rupture of a thin film by as much as a factor of 10, whereas the linear theory predicts this factor to be 4. Finally, the time of rupture of the mucus layer covering the corneal epithelium is derived and the effects of alterations in various characteristics of the external eye on the time of tear film breakup are studied. It is shown that large quantitative discrepancies exist between the time of tear film breakup computed by the linear and the nonlinear theories, if the amplitude of perturbations at the mucousaqueous interface is relatively large.