Structure Formation in Thin Liquid Films

We outline some recent developments in the theoretical description of structure formation in thin liquid films. The main focus is systems involving a single layer of liquid on a solid substrate that can be described using an evolution equation for the film thickness profile. We review the history of the subject and we sketch important experimental and theoretical results and practical applications. After discussing the classification of the different cases, we introduce the common mathematical framework for studies of thin films of soft matter, namely by deriving the generic evolution equation for such films from the Navier-Stokes equations. In the main part we first introduce the different possible geometries and the transitions between them, i.e. from homogeneous to inhomogeneous substrates, or from horizontal to inclined substrates. We then present the physical questions posed by the individual systems and discuss approaches and results for:

• Dewetting on a horizontal homogeneous substrate. We investigate the solution structure and its consequences for the system behavior. For the initial film rupture we distinguish nucleation-dominated and instability-dominated behavior for linearly unstable thin films.

• Dewetting on a horizontal inhomogeneous substrate. The solution structure of the governing equation is analysed as a function of the strength of a chemical heterogeneity. We describe a pinning-coarsening transition with a large range of multistability, implying a large hysteresis and strong dependence on initial conditions and noise.

• Heated thin films on a horizontal homogeneous substrate. We discuss nucleation and drop solutions and show that it is possible to construct all drop solutions separated by dry regions. Incorporating a disjoining pressure allows to study the coarsening behaviour of the drop pattern.

• Sliding drops on an inclined homogeneous substrate. Using a model that incorporates a disjoining pressure allows to calculate the frequently used adhoc parameters of models for moving contact lines from surface chemistry. The involved transition from a Cahn-Hilliard-like to a Kuramoto-Sivashinsky-like dynamics that occurs for increasing inclination angle is analyzed for heated films.

• Transverse instabilities of a liquid ridge are discussed encompassing all the above geometries. Particular attention is given on the stabilization of such an instability due to stripe-like heterogeneities for a resting ridge on a horizontal substrate and on the drastic change in the mode type when inclining the substrate. It changes from a symmetric varicose mode (horizontal substrate) via an asymmetric varicose mode and via an asymmetric zigzag mode to decoupled front and back modes.

Finally, we shortly discuss extensions of thin film studies beyond the case of a single evolution equation. In particular, we introduce two different models based on two coupled evolution equations describing the dynamics of dewetting of a two-layer thin film and the chemically driven self-propelled movement of droplets, respectively.