Insoluble surfactant spreading on a thin viscous film: shock evolution and film rupture

Lubrication theory and similarity methods are used to determine the spreading rate of a localized monolayer of insoluble surfactant on the surface of a thin viscous film, in the limit of weak capillarity and weak surface diffusion. If the total mass of surfactant increases as t(alpha), then at early times, when spreading is driven predominantly by Marangoni forces, a planar (axisymmetric) region of surfactant is shown to spread as t(1 + alpha)/3 (t(1 + alpha)/4) . A shock exists at the leading edge of the monolayer; asymptotic methods are used to show that a wavetrain due to capillary forces exists ahead of the shock at small times, but that after a finite time it is swamped by diffusive effects. For alpha < 1/2 (alpha < 1), the diffusive lengthscale at the shock grows faster than the length of the monolayer, ultimately destroying the shock; subsequently, spreading is driven by diffusion, and proceeds as t1/2. The asymptotic results are shown to be good approximations of numerical solutions of the governing partial differential equations in the appropriate limits. Additional forces are also considered: weak vertical gravity can also destroy the shock in finite time, while effects usually neglected from lubrication theory are important only early in spreading. Experiments have shown that the severe thinning of the film behind the shock can cause it to rupture: the dryout process is modelled by introducing van der Waals forces.