Sliding Drops: Ensemble Statistics from Single Drop Bifurcations.

Ensembles of interacting drops that slide down an inclined plate show a dramatically different coarsening behavior as compared to drops on a horizontal plate: As drops of different size slide at different velocities, frequent collisions result in fast coalescence. However, above a certain size individual sliding drops are unstable and break up into smaller drops. Therefore, the long-time dynamics of a large drop ensemble is governed by a balance of merging and splitting. We employ a long-wave film height evolution equation and determine the dynamics of the drop size distribution towards a stationary state from direct numerical simulations on large domains. The main features of the distribution are then related to the bifurcation diagram of individual drops obtained by numerical path continuation. The gained knowledge allows us to develop a Smoluchowski-type statistical model for the ensemble dynamics that well compares to full direct simulations.

Dip-coating with prestructured substrates: transfer of simple liquids and Langmuir-Blodgett monolayers.

When a plate is withdrawn from a liquid bath, either a static meniscus forms in the transition region between the bath and the substrate or a liquid film of finite thickness (a Landau-Levich film) is transferred onto the moving substrate. If the substrate is inhomogeneous, e.g. has a prestructure consisting of stripes of different wettabilities, the meniscus can be deformed or show a complex dynamic behavior. Here we study the free surface shape and dynamics of a dragged meniscus occurring for striped prestructures with two orientations, parallel and perpendicular to the transfer direction. A thin film model is employed that accounts for capillarity through a Laplace pressure and for the spatially varying wettability through a Derjaguin (or disjoining) pressure. Numerical continuation is used to obtain steady free surface profiles and corresponding bifurcation diagrams in the case of substrates with different homogeneous wettabilities. Direct numerical simulations are employed in the case of the various striped prestructures. The final part illustrates the importance of our findings for particular applications that involve complex liquids by modeling a Langmuir-Blodgett transfer experiment. There, one transfers a monolayer of an insoluble surfactant that covers the surface of the bath onto the moving substrate. The resulting pattern formation phenomena can be crucially influenced by the hydrodynamics of the liquid meniscus that itself depends on the prestructure on the substrate. In particular, we show how prestructure stripes parallel to the transfer direction lead to the formation of bent stripes in the surfactant coverage after transfer and present similar experimental results.

Locking of periodic patterns in Cahn-Hilliard models for Langmuir-Blodgett transfer.

The influence of a periodic spatial forcing on the pattern formation in a generalized Cahn-Hilliard model describing Langmuir-Blodgett transfer is studied. The occurring locking effects enable a control mechanism for the pattern formation process. In the one-dimensional case the parameter range in which patterns are created is increased and the patterns' properties can be adjusted in a broader range. In two dimensions, one-dimensional stripe patterns can be destabilized, giving rise to a multitude of complex two-dimensional structures, including oblique and lattice patterns.