Three-dimensional equilibrium shapes of drops on hysteretic surfaces

In this paper, we study equilibrium three-dimensional shapes of drops on hysteretic surfaces. We develop a function coupled with the publicly available surface energy minimization code Surface Evolver to handle contact angle hysteresis. The function incorporates a model for the mobility of the triple line into Surface Evolver. The only inputs to the model are the advancing and receding contact angles of the surface. We demonstrate this model’s versatility by studying three problems in which parts of the triple line advance while other parts either recede or remain stationary. The first problem focuses on the three-dimensional shape of a static pendant drop on a vertical surface. We predict the finite drop volume when impending sliding motion is observed. In the second problem, we examine the equilibrium shapes of coalescing sessile drops on hysteretic surfaces. Finally, we study coalescing puddles in which gravity plays a leading role in determining the equilibrium puddle shape along with hysteresis.     

Contact angle hysteresis on regular pillar-like hydrophobic surfaces

A series of pillar-like patterned silicon wafers with different pillar sizes and spacing are fabricated by photolithography and further modified by a self-assembled fluorosilanated monolayer. The dynamic contact angles of water on these surfaces are carefully measured and found to be consistent with the theoretical predictions of the Cassie model and the Wenzel model. When a water drop is at the Wenzel state, its contact angle hysteresis increases along with an increase in the surface roughness. While the surface roughness is further raised beyond its transition roughness (from the Wenzel state to the Cassie state), the contact angle hysteresis (or receding contact angle) discontinuously drops (or jumps) to a lower (or higher) value. When a water drop is at the Cassie state, its contact angle hysteresis strongly depends on the solid fraction and has nothing to do with the surface roughness. Even for a superhydrophobic surface, the contact angle hysteresis may still exhibit a value as high as 41 degrees for the solid fraction of 0.563.     

Anisotropic drop morphologies on corrugated surfaces

The spreading of liquid drops on surfaces corrugated with micrometer-scale parallel grooves is studied both experimentally and numerically. Because of the surface patterning, the typical final drop shape is no longer spherical. The elongation direction can be either parallel or perpendicular to the direction of the grooves, depending on the initial drop conditions. We interpret this result as a consequence of both the anisotropy of the contact line movement over the surface and the difference in the motion of the advancing and receding contact lines. Parallel to the grooves, we find little hysteresis due to the surface patterning and that the average contact angle approximately conforms to Wenzel's law as long as the drop radius is much larger than the typical length scale of the grooves. Perpendicular to the grooves, the contact line can be pinned at the edges of the ridges, leading to large contact angle hysteresis.     

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.     

Liquid Drops on an Inclined Plane: The Relation between Contact Angles, Drop Shape, and Retentive Force

Contact angle hysteresis, drop shape, and drop retention were studied with a tiltable plane. Contact liquids were water and ethylene glycol. Four polymers and silicon wafers were used as substrates. When the plane was inclined, the shape of drops distorted, exhibiting advancing and receding contact angles. Drops remained stationary until a critical angle of tilt was exceeded, and then they began to move. The difference in the advancing and receding contact angles, or contact angle hysteresis, ranged from 9° to 66°, depending on the liquid and the substrate. Roughness did not seem to influence the hysteresis as much as the chemical nature of the surfaces. Elongation and back-to-front asymmetry were greater on surfaces with high hysteresis. We found a linear correlation between the aspect ratio of drops and their contact angle hysteresis. Also, the retentive force increased with elongation of the drops.     

Comparison of the relaxation of sessile drops driven by harmonic and stochastic mechanical excitations

Currently, there is no conclusive evidence regarding the global equilibrium condition of vibrated drops. However, it is well-known that vibration of sessile drops effectively reduces the contact angle hysteresis. In this work, applying a recent methodology for evaluating the most-stable contact angle, we examined the impact of the type of excitation signal (random signal versus periodical signal) on the values of the most-stable contact angle for polymer surfaces. Using harmonic signals, the oscillation frequency affected the postvibration contact angle. Instead, the white noise signal enabled sessile drops to relax regardless of their initial configuration. In spite of that, the values of most-stable contact angle obtained with different signals mostly agreed. We concluded that not only the amount of relaxation can be important for relaxing a sessile drop but also the rate of relaxation. Together with receding contact angle, most-stable contact angle, measured with the proposed methodology, was able to capture the thermodynamic changes of "wetted" polymer surfaces.     

A modified Cassie–Baxter relationship to explain contact angle hysteresis and anisotropy on non-wetting textured surfaces

The Cassie–Baxter model is widely used to predict the apparent contact angles obtained on composite (solid–liquid–air) superhydrophobic interfaces. However, the validity of this model has been repeatedly challenged by various research groups because of its inherent inability to predict contact angle hysteresis. In our recent work, we have developed robust omniphobic surfaces which repel a wide range of liquids. An interesting corollary of constructing such surfaces is that it becomes possible to directly image the solid–liquid–air triple-phase contact line on a composite interface, using an electron microscope with non-volatile organic liquids or curable polymers. Here, we fabricate a range of model superoleophobic surfaces with controlled surface topography in order to correlate the details of the local texture with the experimentally observed apparent contact angles. Based on these experiments, in conjunction with numerical simulations, we modify the classical Cassie–Baxter relation to include a local differential texture parameter which enables us to quantitatively predict the apparent advancing and receding contact angles, as well as contact angle hysteresis. This quantitative prediction also allows us to provide an a priori estimation of roll-off angles for a given textured substrate. Using this understanding we design model substrates that display extremely small or extremely large roll-off angles, as well as surfaces that demonstrate direction-dependent wettability, through a systematic control of surface topography and connectivity.     

Size-selective sliding of sessile drops on a slightly inclined plane using low-frequency AC electrowetting

When placed on an inclined solid plane, drops often stick to the solid surface due to pinning forces caused by contact angle hysteresis. When the drop size or the plane's incline angle is small, the drop is difficult to slide due to a decrease in gravitational force. Here we demonstrate that small drops (0.4-9 μL) on a slightly inclined plane (~12°, Teflon and parylene-C surface) can be mobilized through patterned electrodes by applying low-frequency ac electrowetting under 400 Hz (110-180 V(rms)), which has a mechanism different from that of the high-frequency ac method that induces sliding by reducing contact angle hysteresis. We attribute the sliding motion of our method to a combination of contact angle hysteresis and interfacial oscillation driven by ac electrowetting instead of the minimization of contact angle hysteresis at a high frequency. We investigated the effects of ac frequency on the sliding motion and terminal sliding of drops; the terminal sliding velocity is greatest at resonance frequency. Varying the electrowetting number (0.21-0.56) at a fixed frequency (40 Hz) for 5 μL drops, we found an empirical relationship between the electrowetting number and the terminal sliding velocity. Using the relationship between the drop size and ac frequency, we can selectively slide drops of a specific size or merge two drops along an inclined plane. This simple method will help with constructing microfluidic platforms with sorting, merging, transporting, and mixing of drops without a programmable control of electrical signals. Also, this method has a potential in heat transfer applications because heat removal capacity can be enhanced significantly through drop oscillation.     

Contact angle hysteresis

In thermodynamic equilibrium, the contact angle is related by Young's equation to the interfacial energies. Unfortunately, it is practically impossible to measure the equilibrium contact angle. When for example placing a drop on a surface its contact angle can assume any value between the advancing Θ_a and receding Θ_r contact angles, depending on how the drop is placed. Θ_a − Θ_r is called contact angle hysteresis. Contact angle hysteresis is essential for our daily life because it provides friction to drops. Many applications, such as coating, painting, flotation, would not be possible without contact angle hysteresis. Contact angle hysteresis is caused by the nanoscopic structure of the surfaces. Here, we review our current understanding of contact angle hysteresis with a focus on water as the liquid. We describe appropriate methods to measure it, discuss the causes of contact angle hysteresis, and describe the preparation of surfaces with low contact angle hysteresis.     

Modeling Cassie–Baxter State on Superhydrophobic Surfaces

The goal of this work is to study the Cassie–Baxter state on the microstructure hydrophobic surfaces. The dependence of the energy barrier on the drop size, contact angle, the gap between the pillars, and pillar width is investigated. We consider the drop in three dimensions using a numerical approach to minimize the free energy of the drop in any situations. Numerical results are obtained for the wetting of a hydrophobic surface textured with a square lattice of pillars. According to the results, we found that the curvature increases allowing the liquid to touch the surface below the posts as the drop loses volume or decreased the contact angle with fixed drop volume. In the situations we considered, the pillar diagonal length is very critical and sensitive to the details of the surface patterning when the drop volume is larger.     

Assessing the accuracy of contact angle measurements for sessile drops on liquid-repellent surfaces

Gravity-induced sagging can amplify variations in goniometric measurements of the contact angles of sessile drops on super-liquid-repellent surfaces. The very large value of the effective contact angle leads to increased optical noise in the drop profile near the solid-liquid free surface and the progressive failure of simple geometric approximations. We demonstrate a systematic approach to determining the effective contact angle of drops on super-repellent surfaces. We use a perturbation solution of the Bashforth-Adams equation to estimate the contact angles of sessile drops of water, ethylene glycol, and diiodomethane on an omniphobic surface using direct measurements of the maximum drop width and height. The results and analysis can be represented in terms of a dimensionless Bond number that depends on the maximum drop width and the capillary length of the liquid to quantify the extent of gravity-induced sagging. Finally, we illustrate the inherent sensitivity of goniometric contact angle measurement techniques to drop dimensions as the apparent contact angle approaches 180°.     

Contact line and contact angle dynamics in superhydrophobic channels

The dynamics of the wetting and movement of a three-phase contact line confined between two superhydrophobic surfaces were studied using a mean-field free-energy lattice Boltzmann model. Principle features of superhydrophobic surfaces, such as trapped vapor/air between rough microstructures, high contact angles, reduced contact angle hysteresis, and low resistance to fluid flow, were all observed. Movement of the three-phase contact line over a well-patterned superhydrophobic surface displays a periodic stick-jump-slip behavior, while the dynamic contact angle changes accordingly from maximum to minimum. Two regimes were found for the flow velocity as a function of surface roughness and can be related directly to the balance between driving force and flow resistance. This work provides a better understanding of dynamic wetting and fluid flow behaviors over superhydrophobic surfaces and hence could be useful in related applications.     

Sliding behavior of liquid droplets on tilted Langmuir-Blodgett surfaces

The sliding behavior of liquid droplets on inclined Langmuir-Blodgett surfaces was investigated. The critical sliding angle defined as the tilt angle of the surface at which the drop slides down as well as the advancing and receding contact angles was measured for five different liquids on five surfaces. In addition, the contact line geometry was analyzed at critical sliding angle. The experimental relationship between the surface tension forces resulting from contact angle hysteresis and the weight of the drop was compared to theoretical predictions. Even though the shape of the drop bases was found as skewed ellipses, a model assuming parallel-sided elongated drops is shown to describe reasonably the experimental values. This result probably indicates the main influence of the capillary forces at the rear and front edges of the drop with respect to that exerted on the lateral sides.     

Evaporation of pure liquid sessile and spherical suspended drops: a review

A sessile drop is an isolated drop which has been deposited on a solid substrate where the wetted area is limited by a contact line and characterized by contact angle, contact radius and drop height. Diffusion-controlled evaporation of a sessile drop in an ambient gas is an important topic of interest because it plays a crucial role in many scientific applications such as controlling the deposition of particles on solid surfaces, in ink-jet printing, spraying of pesticides, micro/nano material fabrication, thin film coatings, biochemical assays, drop wise cooling, deposition of DNA/RNA micro-arrays, and manufacture of novel optical and electronic materials in the last decades. This paper presents a review of the published articles for a period of approximately 120 years related to the evaporation of both sessile drops and nearly spherical droplets suspended from thin fibers. After presenting a brief history of the subject, we discuss the basic theory comprising evaporation of micrometer and millimeter sized spherical drops, self cooling on the drop surface and evaporation rate of sessile drops on solids. The effects of drop cooling, resultant lateral evaporative flux and Marangoni flows on evaporation rate are also discussed. This review also has some special topics such as drop evaporation on superhydrophobic surfaces, determination of the receding contact angle from drop evaporation, substrate thermal conductivity effect on drop evaporation and the rate evaporation of water in liquid marbles.     

Partial wetting of chemically patterned surfaces: the effect of drop size

Partial wetting of chemically heterogeneous substrates is simulated. Three-dimensional sessile drops in equilibrium with smooth surfaces supporting ordered chemical patterns are considered. Significant features are observed as a result of changing the drop volume. The number of equilibrated drops is found either to remain constant or to increase with growing drop volume. The shape of larger drops appears to approach that of a spherical cap and their three-phase contact line seems, on a larger scale, more circular in shape than that of smaller drops. In addition, as the volume is increased, the average contact angle of drops whose free energy is lowest among all equilibrium-shaped drops of the same volume appears to approach the angle predicted by Cassie. Finally, contrary to results obtained with two-dimensional drops, contact angle hysteresis observed in this system is shown to exhibit a degree of volume dependence in the advancing and receding angles. Qualitative differences in the wetting behavior associated with the two different chemical patterns considered here, as well as differences between results obtained with two-dimensional and three-dimensional drops, can possibly be attributed to variations in the level of constraint imposed on the drop by the different patterns and by the dimensionality of the system.     

The effect of contact angle hysteresis on droplet coalescence and mixing

In this work, droplet coalescence and the subsequent mixing in superhydrophobic surfaces is studied over a range of impact velocities and impact angles. Sanded Teflon surfaces are used as a novel two-dimensional microfluidics platform. These superhydrophobic surfaces exhibit a constant advancing contact angle of θ(A)=150° over a broad range of contact angle hysteresis. As a result, the effect of contact angle hysteresis on droplet coalescence and mixing can be studied. Based on the observed characteristics of coalescence, three different regimes of coalescence are identified as a function of both Weber number and impact angle. These regimes include oscillation dominated, rotation dominated, and mixed dynamics. It is shown that within Weber number ranges achievable in this experiment, hysteresis greatly reduces the deformation of the droplet coalescence process and the subsequent mixing. In head-on collisions, higher hysteresis is found to decrease the frequency at which the resulting dr oscillates. In the case of glancing collisions, where the resulting droplet is found to rotate, higher hysteresis increases the rate of rotation although the overall angular momentum is found to be independent of contact angle hysteresis.     

Formation of superhydrophobic surfaces by biomimetic silicification and fluorination

The amazing water repellency of many biological surfaces, exemplified by lotus leaves, has recently received a great deal of interest. These surfaces, called superhydrophobic surfaces, exhibit water contact angles larger than 150 degrees and a low contact angle hysteresis because of both their low surface energy and heterogeneously rough structures. In this paper, we suggest a biomimetic method, "biosilicification", for generating heterogeneously rough structures and fabricating superhydrophobic surfaces. The superhydrophobic surface was prepared by a combination of the formation of heterogeneously rough, nanosphere-like silica structures through biosilicification and the formation of self-assembled monolayers of fluorosilane on the surface. The resulting surface exhibited the water contact angle of 160.1 degrees and the very low water contact angle hysteresis of only 2.3 degrees, which are definite characteristics of superhydrophobic surfaces. The superhydrophobic property of our system probably resulted from the air trapped in the rough surface. The wetting behavior on the surface was in the heterogeneous regime, which was totally supported by Cassie-Baxter equation.     

Facile preparation of superhydrophobic coatings by sol-gel processes

Different organic/inorganic compositions and deposition methods were used to prepare superhydrophobic surfaces using metal alkoxides and the sol-gel process. Both surface roughness and composition had to be adjusted in order to obtain very high contact angles and low contact angle hysteresis as a necessary requirement for superhydrophobicity. Multilayer samples with a fluorinated organic-inorganic top layer showed water contact angles of about 157 degrees with low hysteresis (2 degrees ). Water drops rolled easily off their surface at a tilt angle as low as 4 degrees .     

Creation of “Rose Petal” and “Lotus Leaf” Effects on Alumina by Surface Functionalization and Metal‐Ion Coordination

Functional differences between superhydrophobic surfaces, such as lotus leaf and rose petals, are due to the subtle architectural features created by nature. Mimicry of these surfaces with synthetic molecules continues to be fascinating as well as challenging. Herein, we demonstrate how inherently hydrophilic alumina surface can be modified to give two distinct superhydrophobic behaviors. Functionalization of alumina with an organic ligand resulted in a rose‐petal‐like surface (water pinning) with a contact angle of 145° and a high contact angle hysteresis (±69°). Subsequent interaction of the ligand with Zn²⁺ resulted in a lotus‐leaf‐like surface with water rolling behavior owing to high contact angle (165°) and low‐contact‐angle‐hysteresis (±2°). In both cases, coating of an aromatic bis‐aldehyde with alkoxy chain substituents was necessary to emulate the nanowaxy cuticular feature of natural superhydrophobic materials.     

Contact angle hysteresis: surface morphology effects

Irradiation of metallic surfaces using ultra-short pulse laser results in a dual-scale structure. While metallic surfaces are superhydrophilic immediately after laser irradiation, prolonged exposure to air renders surfaces superhydrophobic due to surface reactions and deposition of carbonaceous materials onto the surface. In this work, we have fabricated a paraboloid microstructure, which is analyzed thermodynamically through the use of the Gibbs free energy to obtain the equilibrium contact angle and contact angle hysteresis. The effects of the geometrical details on maximizing the superhydrophobicity of the nanopatterned surface are also discussed in an attempt to design surfaces with desired and/or optimum wetting characteristics.