Rupture of equilibrium foam films due to random thermal and mechanical perturbations

A generalized formalism for the rupture of an equilibrium foam film due to imposed random thermal and mechanical perturbations, modeled as a Gaussian white noise, is presented. The evolution of amplitude of perturbation is described by a stochastic differential equation. The average film rupture time is the average time for the amplitude of perturbation to equal to one half the film thickness and is calculated by employing a first passage time analysis for different amplitudes of imposed perturbations, wavenumbers, film thickness, van der Waals and electrostatic interactions and interfacial mobilities. A critical film thickness is identified based on the sign of the disjoining pressure gradient, below which the film is unstable in that the rupture time is very small. There exists an optimum wavenumber for minimum rupture time only for an unstable film. The calculated values of rupture time as well as the optimum wavenumber in the present analysis agree well with the results of linear stability analysis for unstable film with immobile film interfaces. For stable films, the rupture time is found to increase dramatically with film thickness near the critical film thickness. As expected, the average rupture time was found to be higher for smaller amplitudes of imposed perturbations, larger surface potentials and smaller Hamaker constants. The interfacial mobility is accounted for through surface viscosity as well as Marangoni effect. The mean rupture time for partially mobile gas–liquid interface, as characterized by two dimensionless groups, dimensionless surface viscosity and dimensionless Marangoni number, increased with the two dimensionless groups and approached the limiting value for immobile interfaces. The second moment of the rupture time distribution was also evaluated and a beta distribution was fitted to the first two moments. The fitted rupture time distribution was found to be broader for larger film thickness, smaller pressure amplitude, lower Hamaker constant and smaller surface potential.