We study fully three-dimensional droplets that slide down an incline by employing a thin-film equation that accounts for capillarity, wettability, and a lateral driving force in small-gradient (or long-wave) approximation. In particular, we focus on qualitative changes in the morphology and behavior of stationary sliding drops. We employ the inclination angle of the substrate as control parameter and use continuation techniques to analyze for several fixed droplet sizes the bifurcation diagram of stationary droplets, their linear stability, and relevant eigenmodes. The obtained predictions on existence ranges and instabilities are tested via direct numerical simulations that are also used to investigate a branch of time-periodic behavior (corresponding to repeated breakup-coalescence cycles, where the breakup is also denoted as pearling) which emerges at a global instability, the related hysteresis in behavior, and a period-doubling cascade. The non trivial oscillatory behavior close to a Hopf bifurcation of drops with a finite-length tail is also studied. Finally, it is shown that the main features of the bifurcation diagram follow scaling laws over several decades of the droplet size.

We briefly review selected mathematical models that describe the dynamics of pattern formation phenomena in dip-coating and Langmuir-Blodgett transfer experiments, where solutions or suspensions are transferred onto a substrate producing patterned deposit layers with structure length from hundreds of nanometres to tens of micrometres. The models are presented with a focus on their gradient dynamics formulations that clearly shows how the dynamics is governed by particular free energy functionals and facilitates the comparison of the models. In particular, we include a discussion of models based on long-wave hydrodynamics as well as of more phenomenological models that focus on the pattern formation processes in such systems. The models and their relations are elucidated and examples of resulting patterns are discussed before we conclude with a discussion of implications of the gradient dynamics formulation and of some related open issues.

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 synchronization 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 novel complex two-dimensional structures, including oblique and lattice patterns.

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.

When a solid substrate is withdrawn from a bath of simple, partially wetting, nonvolatile liquid, one typically distinguishes two regimes, namely, after withdrawal the substrate is macroscopically dry or homogeneously coated by a liquid film. In the latter case, the coating is called a Landau-Levich film. Its thickness depends on the angle and velocity of substrate withdrawal. We predict by means of a numerical and analytical investigation of a hydrodynamic thin-film model the existence of a third regime. It consists of the deposition of a regular pattern of liquid ridges oriented parallel to the meniscus. We establish that the mechanism of the underlying meniscus instability originates from competing film dewetting and Landau-Levich film deposition. Our analysis combines a marginal stability analysis, numerical time simulations and a numerical bifurcation study via path-continuation.

typical phase behavior of a Langmuir monolayer at the airwater interface. During the transfer of DPPC with a low surface density from the air/water interface onto a solid substrate, regular stripe patterns can be formed. The resulting stripe patterns are composed of DPPC molecules in the low density liquid expanded (LE) phase in the channels and DPPC molecules in the higher density liquid condensed (LC) phase in the stripes, respectively, as illuminated in the sketch of Figure 1 e. The formation of stripes horizontal with respect to the threephase contact line was attributed to the substrate-mediated phase transition while vertical stripes were formed due to a secondary transversal instability of the monolayer, similar to the Rayleigh instability of liquid ridges. [ 7 ] Vertical stripes were often obtained at slightly higher surface pressures and lower transfer velocities. Using the resulting DPPC patterns, the patterning of nanocrystals and nanoparticles as well as guiding of cell growth was demonstrated. [ 8 ] Because no masks or stamps are required as for other patterning methods like photolithography or microprinting, the LB patterning has the advantages of low cost and high throughput. However, one of the major challenges for such methods employing self-assembly and self-organization is the lacking of pattern quality, refl ected by irregular structures or defects. For self-organized LB patterns formation onto fl at surfaces, irregularity is frequently observed owing to the Rayleigh instability, structural defects on the substrates or impurities on the water surface. Specifi cally in the case of the vertical DPPC stripe structures on fl at surfaces, branched patterns with a certain irregularity of the overall pattern are often obtained.Herein, we report an effective method to suppress the branching of self-organized molecular monolayer stripes obtained with the LB technique by using prestructured surfaces. Previously, prestructures were extensively used to create patterns of functional molecules and particles. [ 9 ] However, less attention has been paid to improve the pattern quality of self-assembly and self-organization patterning. Our concept is to use prestructures to introduce a spatially periodic forcing, where certain properties of the substrate (such as the wettability or the strength of interaction between the monolayer and the substrate) vary with a well-defi ned spatial frequency, [ 10 ] subsequently infl uencing the self-organized DPPC patterns. With the spatial confi nement provided by the prestructures, branched structures in vertical LB can be well suppressed, leading to LB patterns with a perfectly regular periodicity and orientation. We further demonstrate that the spacing in between the prestructure lines has a signifi cant infl uence on the LB patterns, where the periodicity of the LB patterns can be adjusted according to the periodicity of the prestructures or Micro/nanoscopic surface patterning is of great importance owing to its applications in many fi elds such as electronics, optic...

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.

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