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

Slide electrification of drops at low velocities

Slide electrification of drops is mostly investigated on tilted plate setups. Hence, the drop charging at low sliding velocity remains unclear. We overcome the limitations by developing an electro drop friction force instrument (eDoFFI). Using eDoFFI, we investigate slide electrification at the onset of drop sliding and at low sliding velocities ≤ 1 cm s-1. The novelty of eDoFFI is the simultaneous measurements of the drop discharging current and the friction force acting on the drop. The eDoFFI tool facilitates control on drop length and width using differently shaped rings. Hereby, slide electrification experiments with the defined drop length-to-width ratios >1 and <1 are realized. We find that width of the drop is the main geometrical parameter which determines drop discharging current and charge separation. We combine Kawasaki-Furmidge friction force equation with our finding on drop discharging current. This combination facilitates the direct measurement of surface charge density (σ) deposited behind the drop. We calculate σ ≈ 45 μC m-2 on Trichloro(1H,1H,2H,2H-perfluorooctyl)silane (PFOTS) and ≈20 μC m-2 on Trichloro(octyl)silane (OTS) coated glass surfaces. We find that the charge separation by moving drops is independent of sliding velocity ≤ 1 cm s-1. The reverse sliding of drop along the same scanline facilitates calculation of the surface neutralization time constant. The eDoFFI links two scientific communities: one which focuses on the friction forces and one which focuses on the slide electrification of drops.

Sliding of drops on mesoporous thin films

There is great interest in developing surfaces with enhanced properties for the sliding of liquid droplets. Here we show that both water and oil droplets placed on mesoporous thin film surfaces slide at relatively small tilt angles with respect to non-porous surfaces of the same material. The effect arises from a particular soft pinning at the contact line, which is a consequence of the fact that sessile droplets are partially "floating" onto a locally self-imbibed mesoporous film. Therefore, droplets present a reduced sliding angle and an enhanced sliding velocity in comparison to droplets on non-porous surfaces of the same material. The formed droplet-substrate interface is different to those observed on superhydrophobic or oil-infused surfaces, and involves a particular sliding dynamic. These findings would help to improve technical developments that require the precise handling of droplet mobility, whose interest span from chemical and biological assays in open microfluidic platforms to applications in energy and the environment.

Contact-angle-hysteresis effects on a drop sitting on an incline plane

We study the contact-angle hysteresis and morphology changes of a liquid drop sitting on a solid substrate inclined with respect to the horizontal at an angle α. This one is always smaller than the critical angle, α_{crit}, above which the drop would start to slide down. The hysteresis cycle is performed for positive and negative α's (|α|<α_{crit}), and a complete study of the changes in contact angles, free surface, and footprint shape is carried out. The drop shape is analyzed in terms of a solution of the equilibrium pressure equation within the long-wave model (lubrication approximation). We obtain a truncated analytical solution describing the static drop shapes that is successfully compared with experimental data. This solution is of practical interest since it allows for a complete description of all the drop features, such as its footprint shape or contact angle distribution around the drop periphery, starting from a very small set of relatively easy to measure drop parameters.

Numerical Study for a Large-Volume Droplet on the Dual-Rough Surface: Apparent Contact Angle, Contact Angle Hysteresis, and Transition Barrier

The profile, apparent contact angle (ACA), contact angle hysteresis (CAH), and wetting state transmission energy barrier (WSTEB) are important static and dynamic properties of a large-volume droplet on the hierarchical surface. Understanding them can provide us with important insights into functional surfaces and promote the application in corresponding areas. In this paper, we establish three theoretical models (models 1-3) and the corresponding numerical methods, which were obtained by the free energy minimization and the nonlinear optimization algorithm, to predict the profile, ACA, CAH, and WSTEB of a large-volume droplet on the horizontal regular dual-rough surface. In consideration of the gravity, the energy barrier on the contact circle, the dual heterogeneous structures and their roughness on the surface, the models are more universal and accurate than the previous models. It showed that the predictions of the models were in good agreement with the results from the experiment or literature. The models are promising to become novel design approaches of functional surfaces, which are frequently applied in microfluidic chips, water self-catchment system, and dropwise condensation heat transfer system.

Pinning of a drop by a junction on an incline

The shape of a drop pinned on an inclined substrate is a long-standing problem where the complexity of real surfaces, with heterogeneities and hysteresis, makes it complicated to understand the mechanisms behind the phenomena. Here we consider the simple case of a drop pinned on an incline at the junction between a hydrophilic half plane (the top half) and a hydrophobic one (the bottom half). Relying on the equilibrium equations deriving from the balance of forces, we exhibit three scenarios depending on the way the contact line of the drop on the substrate either simply leans against the junction or overfills (partly or fully) into the hydrophobic side. We draw some conclusions on the geometry of the overlap and the stability of these tentative equilibrium states. In the corresponding retention force factor, we find that a major role is played by the wetted length of the junction line, in the spirit of Furmidge's observations. The predictions of the theory are compared with extensive molecular dynamics simulations.

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.

Evaporation of inclined water droplets

When a drop is placed on a flat substrate tilted at an inclined angle, it can be deformed by gravity and its initial contact angle divides into front and rear contact angles by inclination. Here we study on evaporation dynamics of a pure water droplet on a flat solid substrate by controlling substrate inclination and measuring mass and volume changes of an evaporating droplet with time. We find that complete evaporation time of an inclined droplet becomes longer as gravitational influence by inclination becomes stronger. The gravity itself does not change the evaporation dynamics directly, whereas the gravity-induced droplet deformation increases the difference between front and rear angles, which quickens the onset of depinning and consequently reduces the contact radius. This result makes the evaporation rate of an inclined droplet to be slow. This finding would be important to improve understanding on evaporation dynamics of inclined droplets.

Droplets on inclined plates: local and global hysteresis of pinned capillary surfaces

Local contact line pinning prevents droplets from rearranging to minimal global energy, and models for droplets without pinning cannot predict their shape. We show that experiments are much better described by a theory, developed herein, that does account for the constrained contact line motion, using as an example droplets on tilted plates. We map out their shapes in suitable phase spaces. For 2D droplets, the critical point of maximum tilt depends on the hysteresis range and Bond number. In 3D, it also depends on the initial width, highlighting the importance of the deposition history.

Solute concentration-dependent contact angle hysteresis and evaporation stains

The presence of nonvolatile solutes in a liquid drop on a solid surface can affect the wetting properties. Depending on the surface-activity of the solutes, the extent of contact angle hysteresis (CAH) can vary with their concentration and the pattern of the evaporation stain is altered accordingly. In this work, four types of concentration-dependent CAH and evaporation stains are identified for a water drop containing polymeric additives on polycarbonate. For polymers without surface-activity such as dextran, advancing and receding contact angles (θa and θr) are independent of solute concentrations, and a concentrated stain is observed in the vicinity of the drop center after complete evaporation. For polymers with weak surface-activity such as poly(ethylene glycol) (PEG), both θa and θr are decreased by solute addition, and the stain pattern varies with increasing PEG concentration, including a concentrated stain and a mountain-like island. For polymers with intermediate surface-activity such as sodium polystyrenesulfonate (NaPSS), θa descends slightly, but θr decreases significantly after the addition of a substantial amount of NaPSS, and a ring-like stain pattern is observed. Moreover, the size of the ring stain can be controlled by NaPSS concentration. For polymers with strong surface-activity such as poly(vinylpyrrolidone) (PVP), θa remains essentially a constant, but θr is significantly lowered after the addition of a small amount of PVP, and the typical ring-like stain is seen.

On the onset of motion of sliding drops

In this article we numerically investigate the onset of motion of liquid drops in contact with a plane and homogeneous substrate with contact angle hysteresis. The drops are driven by a body force F = ρgV, where ρ is the density of the liquid, g is the acceleration of gravity, and V is the volume of the drop. We compare two protocols to vary the bond number Bo = λ(v)/λ(c) by changes of either the drop size λ(v) = V(1/3) or the capillary length λ(c) = (γ/ρg)(1/2) where γ is the interfacial tension, revealing that the transition between pinned and steady moving states can be either continuous or discontinuous. In a certain range both pinned and moving states can be found for a given bond number Bo, depending on the history of the control parameters g and V. Our calculations are extended to arbitrary combinations of static advancing and receding contact angles and provide a comprehensive picture of the depinning transition induced by a quasi-static variation of the control parameters. Finally, we demonstrate that the particular form of the contact line mobility in our model has an impact on the interfacial shape of steady moving drops.

Bioinspired ultrahigh water pinning nanostructures

Rose petal mimetic surfaces with ultrahigh water pinning forces have been fabricated via nanoimprinting process onto three different polymer films. Water pinning forces ranging from 104 to 690 μN are obtained on free-standing polycarbonate films with imprinted nanostructures. Through a systematic variation of the surface structures, this study provides experimental evidence that an ultrahigh water pinning force can be achieved by combining two surface topographical designs: (1) conical- or parabolic-shaped nanoprotrusions and (2) isotropic and continuous nanoprotrusions. These design criteria ensure that a continuous solid-liquid contact line is achieved and provide a rule-of-thumb to engineer surfaces with tunable water pinning forces. The ultrahigh water pinning film is further demonstrated to mitigate the "coffee ring" effect, a phenomenon associated with nonuniform deposition from a drying solute-laden liquid droplet.

Simulation analysis of contact angles and retention forces of liquid drops on inclined surfaces

A simulation study of liquid drops on inclined surfaces is performed in order to understand the evolution of drop shapes, contact angles, and retention forces with the tilt angle. The simulations are made by means of a method recently developed for dealing with contact angle hysteresis in the public-domain Surface Evolver software. The results of our simulations are highly dependent on the initial contact angle of the drop. For a drop with an initial contact angle equal to the advancing angle, we obtain results similar to those of experiments in which a drop is placed on a horizontal surface that is slowly tilted. For drops with an initial contact angle equal to the mean between the advancing and the receding contact angles, we recover previous results of finite element studies of drops on inclined surfaces. Comparison with experimental results for molten Sn-Ag-Cu on a tilted Cu substrate shows excellent agreement.

Drops sitting on a tilted plate: receding and advancing pinning

The wetting behavior of a liquid drop sitting on an inclined plane is investigated experimentally and theoretically. Using Surface Evolver, the numerical simulations are performed based on the liquid-induced defect model, in which only two thermodynamic parameters (solid-liquid interfacial tensions before and after wetting) are required. A drop with contact angle (CA) equal to θ is first placed on a horizontal plate, and then the plate is tilted. Two cases are studied: (i) θ is adjusted to the advancing CA (θ(a)) before tilting, and (ii) θ is adjusted to the receding CA (θ(r)) before tilting. In the first case, the uphill CA declines and the downhill CA remains unchanged upon inclination. When the tilted drop stays at rest, the pinning of the receding part of the contact line (receding pinning) and the depinning of the advancing part of the contact line (advancing depinning) are observed. The free energy analysis reveals that upon inclination, the reduction of the solid-liquid free energy dominates over the increment of the liquid-gas free energy associated with shape deformation. In the second case, the downhill CA grows and the uphill CA remains the same upon inclination. Advancing pinning and receding depinning are noted for the tilted drop at rest. The free energy analysis indicates that upon inclination, the decrease of the liquid-gas free energy compensates the increment of the solid-liquid free energy. The experimental results are in good agreement with those of simulations.

Modulating contact angle hysteresis to direct fluid droplets along a homogenous surface

The shape and motion of drops on surfaces is governed by the balance between the driving and the pinning forces. Here we demonstrate control over the motion of droplets on an inclined surface by exerting control over the contact angle hysteresis. The external modulation of contact angle hysteresis is achieved through a voltage-induced local molecular reorganization within the surface film at the solid-liquid interface. We show that tuning contact angle hysteresis alone is sufficient to direct and deform drops when subjected to a constant external driving force, here gravity, in the absence of a pre-defined surface energy gradient or pattern. We also show that the observed stretching and contraction of the drops mimic the motion of an inchworm. Such reversible manipulation of the pinning forces could be an attractive means to direct drops, especially with the dominance of surface forces at micro-/nanoscale.

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.

Anomalous contact angle hysteresis of a captive bubble: advancing contact line pinning

Contact angle hysteresis of a sessile drop on a substrate consists of continuous invasion of liquid phase with the advancing angle (θ(a)) and contact line pinning of liquid phase retreat until the receding angle (θ(r)) is reached. Receding pinning is generally attributed to localized defects that are more wettable than the rest of the surface. However, the defect model cannot explain advancing pinning of liquid phase invasion driven by a deflating bubble and continuous retreat of liquid phase driven by the inflating bubble. A simple thermodynamic model based on adhesion hysteresis is proposed to explain anomalous contact angle hysteresis of a captive bubble quantitatively. The adhesion model involves two solid–liquid interfacial tensions (γ(sl) > γ(sl)′). Young’s equation with γ(sl) gives the advancing angle θ(a) while that with γ(sl)′ due to surface rearrangement yields the receding angle θ(r). Our analytical analysis indicates that contact line pinning represents frustration in surface free energy, and the equilibrium shape corresponds to a nondifferential minimum instead of a local minimum. On the basis of our thermodynamic model, Surface Evolver simulations are performed to reproduce both advancing and receding behavior associated with a captive bubble on the acrylic glass.

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.