On the thinning of films (I)

We investigate, from a mathematical perspective, the problem of a layer of fluid attracted to a horizontal plate when the layer is in equilibrium with a bulk reservoir. It is assumed that as the temperature varies, the bulk undergoes a continuous phase transition. On the basis of free energetics, this initially causes thinning of the layer but, at lower temperatures, the layer recovers and rebuilds. We provide a mathematical framework with which to investigate these problems. As an approximation, we model the layered system by a mean–field Ising magnet. The layered system is first studied in isolation (fixed thickness) and then as a system in contact with the bulk (variable thickness) with general results established. Finally, we investigate the limit of large thickness. Here, a well defined continuum theory emerges which provides an approximation to the discrete systems. In the context of the limiting theory, it is established that discontinuities in the layer thickness (as a function of temperature) or the derivative thereof are inevitable. By comparison with actual data from [8] and [7] the discontinuities may indeed be present but they are not quite in the form predicted by the theory. Finally – still in the context of the limiting theory – is shown that at low temperatures, the layer may be lost altogether; the nature of the critical binding force is elucidated.