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 Garcia and Chan (1998) [1] and Ganshin et al. (2006) [2] 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–it is shown that at low temperatures, the layer may be lost altogether; the nature of the critical binding force is elucidated.