Structure and flow of droplets on solid surfaces

Drops on Polymer Brushes: Advances in Thin-Film Modeling of Adaptive Substrates

We briefly review recent advances in the hydrodynamic modeling of the dynamics of droplets on adaptive substrates, in particular, solids that are covered by polymer brushes. Thereby, the focus is on long-wave and full-curvature variants of mesoscopic hydrodynamic models in gradient dynamics form. After introducing the approach for films/drops of nonvolatile simple liquids on a rigid smooth solid substrate, it is first expanded to an arbitrary number of coupled degrees of freedom before considering the specific case of drops of volatile liquids on brush-covered solids. After presenting the model, its usage is illustrated by briefly considering the natural and forced spreading of drops of nonvolatile liquids on a horizontal brush-covered substrate, stick-slip motion of advancing contact lines as well as drops sliding down a brush-covered incline. Finally, volatile liquids are also considered.

Probing Surface Morphology using X-ray Grating Interferometry

X-ray reflectometry (XRR), a surface-sensitive technique widely used for characterizing surfaces, buried interfaces, thin films, and multilayers, enables determination of the electron density distribution perpendicular to a well-defined surface specularly reflecting X-rays. However, the electron density distribution parallel to the surface cannot be determined from an X-ray reflectivity curve. The electron density correlation in the lateral direction is usually probed by measuring the grazing-incidence small-angle X-ray scattering (GISAXS). GISAXS measurement, however, typically requires using a collimated X-ray point beam to distinguish the GISAXS from the specularly reflected X-rays, and so the sample must be scanned in the lateral direction with the point beam to investigate variations in the surface and interface morphology for a region larger than the size of the beam. In this paper, we report a new approach based on X-ray grating interferometry: an X-ray sheet beam is used instead of an X-ray point beam. A method using this approach can simultaneously provide one-dimensional real-space images of X-ray reflectivity, surface curvature, and ‘dark-field’ contrast with a field-of-view of more than a few millimetres. As a demonstration, a sample having a 400 nm line and space SiO₂ pattern with a depth of 10 nm on its surface was used, and the dark-field contrast due to the unresolved line and space structure, creating GISAXS in the lateral direction, was successfully observed. Quantitative analysis of these contrasts provided the real-space distribution of the structural parameters for a simple model of the grating structure. Our study paves the way to a new approach to structure analysis, providing a quantitative way to investigate real-space variations in surface and interface morphology through wavefront analysis.

Probing droplets on superhydrophobic surfaces by synchrotron radiation scattering techniques

Droplets on artificially structured superhydrophobic surfaces represent quasi contact-free sample environments which can be probed by X-ray microbeams and nanobeams in the absence of obstructing walls. This review will discuss basic surface wettability concepts and introduce the technology of structuring surfaces. Quasi contact-free droplets are compared with contact-free droplets; processes related to deposition and evaporation on solid surfaces are discussed. Droplet coalescence based on the electrowetting effect allows the probing of short-time mixing and reaction processes. The review will show for several materials of biological interest that structural processes related to conformational changes, nucleation and assembly during droplet evaporation can be spatially and temporally resolved by raster-scan diffraction techniques. Orientational ordering of anisotropic materials deposited during solidification at pinning sites facilitates the interpretation of structural data.

Use of intermediate focus for grazing incidence small and wide angle x-ray scattering experiments at the beamline P03 of PETRA III, DESY

We describe the new experimental possibilities of the micro- and nanofocus X-ray scattering beamline P03 of the synchrotron source PETRA III at DESY, Hamburg (Germany), which arise from experiments with smaller beam sizes in the micrometer range. This beamline has been upgraded recently to perform new kinds of experiments. The use of an intermediate focus allows for reducing the beam size of microfocused hard X-rays while preserving a large working distance between the focusing elements and the focus position. For the first time, this well-known methodology has been employed to grazing incidence small- and wide-angle X-ray scattering (GISAXS/GIWAXS). As examples, we highlight the applications to in situ studies using microfluidic devices in GISAXS geometry as well as the investigation of the crystallinity of thin films in GIWAXS geometry.

Droplet motion in one-component fluids on solid substrates with wettability gradients

Droplet motion on solid substrates has been widely studied not only because of its importance in fundamental research but also because of its promising potentials in droplet-based devices developed for various applications in chemistry, biology, and industry. In this paper, we investigate the motion of an evaporating droplet in one-component fluids on a solid substrate with a wettability gradient. As is well known, there are two major difficulties in the continuum description of fluid flows and heat fluxes near the contact line of droplets on solid substrates, namely, the hydrodynamic (stress) singularity and thermal singularity. To model the droplet motion, we use the dynamic van der Waals theory [Phys. Rev. E 75, 036304 (2007)] for the hydrodynamic equations in the bulk region, supplemented with the boundary conditions at the fluid-solid interface. In this continuum hydrodynamic model, various physical processes involved in the droplet motion can be taken into account simultaneously, e.g., phase transitions (evaporation or condensation), capillary flows, fluid velocity slip, and substrate cooling or heating. Due to the use of the phase field method (diffuse interface method), the hydrodynamic and thermal singularities are resolved automatically. Furthermore, in the dynamic van der Waals theory, the evaporation or condensation rate at the liquid-gas interface is an outcome of the calculation rather than a prerequisite as in most of the other models proposed for evaporating droplets. Numerical results show that the droplet migrates in the direction of increasing wettability on the solid substrates. The migration velocity of the droplet is found to be proportional to the wettability gradients as predicted by Brochard [Langmuir 5, 432 (1989)]. The proportionality coefficient is found to be linearly dependent on the ratio of slip length to initial droplet radius. These results indicate that the steady migration of the droplets results from the balance between the (conservative) driving force due to the wettability gradient and the (dissipative) viscous drag force. In addition, we study the motion of droplets on cooled or heated solid substrates with wettability gradients. The fast temperature variations from the solid to the fluid can be accurately described in the present approach. It is observed that accompanying the droplet migration, the contact lines move through phase transition and boundary velocity slip with their relative contributions mostly determined by the slip length. The results presented in this paper may lead to a more complete understanding of the droplet motion driven by wettability gradients with a detailed picture of the fluid flows and phase transitions in the vicinity of the moving contact line.

Stability and dynamics of droplets on patterned substrates: insights from experiments and lattice Boltzmann simulations

The stability and dynamics of droplets on solid substrates are studied both theoretically and via experiments. Focusing on our recent achievements within the DFG-priority program 1164 (Nano- and Microfluidics), we first consider the case of (large) droplets on the so-called gradient substrates. Here the term gradient refers to both a change of wettability (chemical gradient) or topography (roughness gradient). While the motion of a droplet on a perfectly flat substrate upon the action of a chemical gradient appears to be a natural consequence of the considered situation, we show that the behavior of a droplet on a gradient of topography is less obvious. Nevertheless, if care is taken in the choice of the topographic patterns (in order to reduce hysteresis effects), a motion may be observed. Interestingly, in this case, simple scaling arguments adequately account for the dependence of the droplet velocity on the roughness gradient (Moradi et al 2010 Europhys. Lett. 89 26006). Another issue addressed in this paper is the behavior of droplets on hydrophobic substrates with a periodic arrangement of square shaped pillars. Here, it is possible to propose an analytically solvable model for the case where the droplet size becomes comparable to the roughness scale (Gross et al 2009 Europhys. Lett. 88 26002). Two important predictions of the model are highlighted here. (i) There exists a state with a finite penetration depth, distinct from the full wetting (Wenzel) and suspended (Cassie-Baxter, CB) states. (ii) Upon quasi-static evaporation, a droplet initially on the top of the pillars (CB state) undergoes a transition to this new state with a finite penetration depth but then (upon further evaporation) climbs up the pillars and goes back to the CB state again. These predictions are confirmed via independent numerical simulations. Moreover, we also address the fundamental issue of the internal droplet dynamics and the terminal center of mass velocity on a flat substrate.