The work on Marangoni-driven films investigated the dynamics of a film rising from a reservoir onto a tilted substrate, in particular, near the meniscus that connects the film to the reservoir. Numerical simulations for the time dependent meniscus shape equation revealed the nature of a 'pinch-off' transition observed in a classical experiment by Ludviksson and Lightfoot, showing that this phenomenon gives rise to a wave combination with a new reverse undercompressive front. These findings prompted a new experimental survey of the pinch-off transition by B. Behringer's research group (Duke University, N.C., USA) that confirmed our predictions. This work was published in Physical Review Letters in the first year of the funding period [M03].
P. L. Evans and A. Münch discussed the equilibrium solutions for the meniscus at different inclination angles of the substrate using phase space methods [ME05]. We classified the rich solution structure, including a non-monotonic equilibrium solution and multiple non-monotonic solutions with the same far-field film thickness. A second paper [EM06] analyzed the interaction of these solutions with the classical and non-classical wave structures in the thin film. This work also generated some insight into the classical Landau-Levich drag-out problem [JAM05].
The experience gained from this work currently benefits an adjunct project. Here, we (K. Afanasiev, A. Münch and B. Wagner) investigate the flow of a thin polymer film on a vertical rotating disc that drags the liquid out of a reservoir. This problem is part of an ongoing collaboration with process engineers, the group of Th. Rieckmann at the FH Köln, who are doing experiments using a model reactor, which are linked to an industrial partner. We have set up a lubrication model for the flow of viscous liquid including the meniscus region. K. Afanasiev has developed a finite-element-code that will be able to solve the model on the necessary complicated geometry. This code, in turn, will be beneficial for other future and ongoing thin film projects.
We have simultaneously explored new application areas in key technologies, in particular, we studied thin polymer films on a hydrophobized silicon wafer. These films tend to recoil from the substrate in a process that is initiated by the formation of holes. The film thickness is reduced from its initial thickness of typically a few tens to hundreds of nanometers to a residual film of one nm or less. Intermolecular forces dominate at these scales and drive the increase of the dewetted area. The process of dewetting and the eventual patterns of droplets depend on the properties of substrate and the liquid and have attracted considerable experimental and theoretical research in the physics (and surface-chemistry) community. Nevertheless, many of the experimental observations are not yet well understood.
It is perceived that further progress is needed on the mathematical side, by systematically investigating and setting up new lubrication models that describe the dewetting dynamics.
Our work on the rim propagation and instability revealed that slippage at the liquid/solid interface may play a key role for this instability. Our numerical simulations and stability analysis recover the characteristic asymmetric growth of the finger instability if the liquid film is allowed to slip along the substrate, which is absent if the usual no-slip condition is used [MW04], [M04].
The lubrication model (or generalized lubrication model) that we used in this situation is actually one out of a whole family of such models that are derived from the Navier-Stokes free boundary problem with a Navier slip condition at the interface for different regimes of the slip length [MWW05]. These models cover the range of weak or no-slip, the intermediate regime used in [MW04], to a strong-slip regime which has the free slip regime observed for e.g. soap films as a limit. The investigation of (intermediate and strong) slip has hence opened up a new era in lubrication models with applications to nanoscopic thin films, and has already had an impact on understanding nanofluidic flow in open confined systems. For the case of strong slippage we found that for large enough slip lengths, the spatially oscillating structure of the dewetting rim is replaced by monotonic decay. Our results are in very good agreement with physical experiments carried out by our collaborators R. Konrad and K. Jacobs at the Universität Saarland, that could not be explained before. This is demonstrated in a recent joint paper (in PRL) [FJMWW05].
The numerical simulations in particular for the three-dimensional flow become very hard due to the wide separation of scales (a large ridge accompanied by the thinner original film and the even much thinner residual film in the hole) and also because in many cases the interesting behaviour appears only if the evolution is followed for long times. For this reason we have carried out a further model reduction to a sharp contact-line model via matched asymptotics [MWW05], [KMW05].