Equilibrium of drops on inclined hydrophilic surfaces

Directional passive transport of nanodroplets on general axisymmetric surfaces

Rapid removal of small-sized droplets passively using fixed structures is a key challenge for various applications including anti-icing, rapid cooling, and water harvesting. In this work, we investigate the directional motion of nanodroplets on axisymmetric surfaces with curvature gradient through molecular dynamics (MD) simulations. It is found that as the shape of the axisymmetric surface is changed from a dome to a trumpet, the droplet velocity is greatly enhanced, by a factor of ∼14. Such an increase is mainly caused by the increment in the driving force. The droplet velocity changes nonlinearly as the surface wettability is varied and assumes the maximum at the contact angle of ∼75°. We derive a formula for the driving force of nanodroplets on general axisymmetric surfaces by evaluating the pressure gradient inside the droplet induced by the curvature gradient. Molecular dynamics simulations are performed to directly measure the driving force and confirm that the theoretical formula works well. By illustrating the reduced initial velocity of droplets as a function of a dimensionless number, which represents the ratio of the driving force to the retentive force due to contact angle hysteresis, we show that the onset of droplet motion on axisymmetric surfaces occurs when the dimensionless number is above a critical value. The dimensionless number reveals the effects of surface geometry, surface wettability, and droplet size on the droplet motion.

Drops on a Superhydrophobic Hole Hanging On under Evaporation

Drops with larger volumes placed over a superhydrophobic (SH) surface with a hole do not fall through unless they are evaporated to a size that is small enough. This feature offers the ability to preconcentrate samples for biochemical analysis. In this work, the influence of pinning on the behavior of drops placed on a 0.1 mm thick SH substrate with a 2 mm diameter hole as they evaporated was investigated. With 16 μL of water dispensed, the sessile drop component volume was initially higher than that of the overhanging drop component and maintained this until the later stages where almost identical shapes were attained and full evaporation was achieved without falling off the hole. With 15 μL of water dispensed, the volume of the sessile drop was initially higher than that of the overhanging drop component but the liquid body was able to squeeze through the hole after 180 s due to the contact line not having sufficient pinning strength when it encountered the edge of the hole. This resulted in the liquid body either falling through the hole or remaining pinned with an oval-like shape. When it did not fall-off, the liquid body had volume and contact angle characteristics for the sessile drop and overhanging drop components that were reversed. In the later stages, however, nearly identical shapes were again attained and full evaporation was achieved without falling off the hole. The effects of pinning, despite the substrate being SH, offer another path toward achieving practical outcomes with liquid bodies without the need for chemical surface functionalization. Similarities and differences could be seen in the behavior of a sessile drop on a SH plate that was inclined at 30° to the horizontal and evaporated.

Contact angles of a drop pinned on an incline

For a drop on an incline with small tilt angle α, when the contact line is a circle of radius r, we derive the relation mgsinα=γrπ/2(cosθ^{min}-cosθ^{max}) at first order in α, where θ^{min} and θ^{max} are the contact angles at the back and at the front, m is the mass of the drop and γ the surface tension of the liquid. We revisit in this way the Furmidge model for a large range of contact angles. We also derive the same relation at first order in the Bond number B=ρgR^{2}/γ, where R is the radius of the spherical cap at zero gravity. The drop profile is computed exactly in the same approximation. Results are compared with surface evolver simulations, showing a surprisingly large range of contact angles for applicability of first-order approximations.

Minimum energy shapes of one-side-pinned static drops on inclined surfaces

The shape that a liquid drop will assume when resting statically on a solid surface inclined to the horizontal is studied here in two dimensions. Earlier experimental and numerical studies yield multiple solutions primarily because of inherent differences in surface characteristics. On a solid surface capable of sustaining any amount of hysteresis, we obtain the global, and hence unique, minimum energy shape as a function of equilibrium contact angle, drop volume, and plate inclination. It is shown, in the energy minimization procedure, how the potential energy of this system is dependent on the basis chosen to measure it from, and two realistic bases, front-pinned and back-pinned, are chosen for consideration. This is at variance with previous numerical investigations where both ends of the contact line are pinned. It is found that the free end always assumes Young's equilibrium angle. Using this, simple equations that describe the angles and the maximum volume are then derived. The range of parameters where static drops are possible is presented. We introduce a detailed force balance for this problem and study the role of the wall in supporting the drop. We show that a portion of the wall reaction can oppose gravity while the other portion aids it. This determines the maximum drop volume that can be supported at a given plate inclination. This maximum volume is the least for a vertical wall, and is higher for all other wall inclinations. This study can be extended to three-dimensional drops in a straightforward manner and, even without this, lends itself to experimental verification of several of its predictions.

Rayleigh and depinning instabilities of forced liquid ridges on heterogeneous substrates

Depinning of two-dimensional liquid ridges and three-dimensional drops on an inclined substrate is studied within the lubrication approximation. The structures are pinned to wetting heterogeneities arising from variations of the strength of the short-range contribution to the disjoining pressure. The case of a periodic array of hydrophobic stripes transverse to the slope is studied in detail using a combination of direct numerical simulation and branch-following techniques. Under appropriate conditions the ridges may either depin and slide downslope as the slope is increased, or first break up into drops via a transverse instability, prior to depinning. The different transition scenarios are examined together with the stability properties of the different possible states of the system.

Simulation of drop movement over an inclined surface using smoothed particle hydrodynamics

Smoothed particle hydrodynamics (SPH) is used to numerically simulate the movement of drops down an inclined plane. Diffuse interfaces have been assumed for tracking the motion of the contact line. The asymmetric shape of the three-dimensional drop and the variation of contact angle along its periphery can be calculated using the simulation. During the motion of a liquid drop down an inclined plane, an internal circulation of liquid particles is observed due to gravitational pull which causes periodic change in the drop shape. The critical angle of inclination required for the inception of drop motion is also evaluated for different fluids as a function of drop volume. The numerical predictions exhibit a good agreement with the published experimental results.

Mimicking the stenocara beetle--dewetting of drops from a patterned superhydrophobic surface

This paper describes the preparation of superhydrophobic surfaces that have been selectively patterned with circular hydrophilic domains. These materials mimicked the back of the stenocara beetle and collected drops of water if exposed to mist or fog. Under the effect of gravity, the drops dewetted from the hydrophilic regions once a critical volume had been reached. The surface energy in the hydrophilic regions was carefully controlled and assumed various values, allowing us to study the behavior of drops as a function of the superhydrophobic/hydrophilic contrast. We have investigated the development of drops and quantitatively analyzed the critical volumes as a function of several parameters.

Effect of transient pinning on stability of drops sitting on an inclined plane

We report on new instabilities of the quasistatic equilibrium of water drops pinned by a hydrophobic inclined substrate. The contact line of a statically pinned drop exhibits three transitions of partial depinning: depinning of the advancing and receding parts of the contact line and depinning of the entire contact line leading to the drop's translational motion. We find a region of parameters where the classical Macdougall-Ockrent-Frenkel approach fails to estimate the critical volume of the statically pinned inclined drop.

Driven drops on heterogeneous substrates: onset of sliding motion

Pinning and depinning of driven drops on heterogeneous substrates is studied as a function of the driving and heterogeneity amplitude. Two types of heterogeneity are considered: a "hydrophobic" defect that blocks the drop in front and a "hydrophilic" one that holds it at the back. Two different types of depinning leading to sliding motion are identified, and the resulting stick-slip motion is studied numerically.

Directing droplets using microstructured surfaces

Systematic variation of microscale structures has been employed to create a rough superhydrophobic surface with a contact angle gradient. Droplets are propelled down these gradients, overcoming contact angle hysteresis using energy supplied by mechanical vibration. The rough hydrophobic surfaces have been designed to maintain air traps beneath the droplet by stabilizing its Fakir state. Dimensions and spacing of the microfabricated pillars in silicon control the solid-liquid contact area and are varied to create a gradient in the apparent contact angle. This work introduces the solid-liquid contact area fraction as a new control variable in any scheme of manipulating droplets, presenting theory, fabricated structures, and experimental results that validate the approach.

Drops down the hill: theoretical study of limiting contact angles and the hysteresis range on a tilted plate

The limiting inclination angle (slip angle), for which a two-dimensional water drop may be at equilibrium on a chemically heterogeneous surface, is exactly calculated for a variety of cases. The main conclusion is that, in the cases studied, the contact angles at the upper and lower contact line do not always simultaneously equal the receding and advancing contact angles, respectively. On a hydrophobic surface, the lowest contact angle (at the upper contact line) tends to be approximately equal to the receding contact angle, while the highest contact angle (at the lower contact line) may be much lower than the advancing contact angle. For hydrophilic surfaces, the opposite is true. These conclusions imply that the hysteresis range cannot in general be measured by analyzing the shape of a drop on an inclined plane. Also, the limiting inclination angle cannot in general be calculated from the classical equation based only on the advancing and receding contact angles.