Liquid drops on vertical and inclined surfaces; II. A method for approximating drop shapes

In this article, a new method is proposed to approximate the shapes of liquid drops on vertical and inclined surfaces. Based on observations from Part I, the profile of a drop at a given azimuthal angle is approximated by two circles sharing a common tangent at the maximum height. The drop volume is obtained by integrating all profiles over the circumference of the base. The volume is thus described as a function of the contact angles and the three-phase contact line. The new method accurately predicts the volumes of drops tested in Part I and independent measurements from the literature. Simplifying the drop shape to a spherical cap can lead to a 75% error in drop-volume prediction. The proposed method is used to study the effect of drop parameters on volume prediction. The two-circle geometry can also be used to measure contact angles from profile images.     

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°.     

A method for correcting the contact angle from the θ/2 method

The θ/2 method, a widely used technique on measuring the contact angle of a sessile drop, assumes that the drop profile is part of a sphere. However, the shape profile of a sessile drop is governed by the Young–Laplace equation and is different from a sphere, especially for drops with a large bound number (e.g. large volume or small surface tension). The spherical assumption, therefore, causes errors on evaluating the contact angles. The deviation of contact angle from the θ/2 method is evaluated from a theoretical calculation in this work. A simple means is given for correcting the measurement error. The corrected angle results from the drop volume, surface tension, liquid density and the contact angle from θ/2 method. An algorithm for finding the correct contact angle without knowing the density and surface tension is also given. At the end, two examples of pendant drops are given for the illustration.     

Simultaneous measurement of contact angle and surface tension using axisymmetric drop-shape analysis-no apex (ADSA-NA)

Axisymmetric drop-shape analysis-no apex (ADSA-NA) is a recent drop-shape method that allows the simultaneous measurement of contact angles and surface tensions of drop configurations without an apex (i.e., a sessile drop with a capillary protruding into the drop). Although ADSA-NA significantly enhanced the accuracy of contact angle and surface tension measurements compared to that of original ADSA using a drop with an apex, it is still not as accurate as a surface tension measurement using a pendant drop suspended from a holder. In this article, the computational and experimental aspects of ADSA-NA were scrutinized to improve the accuracy of the simultaneous measurement of surface tensions and contact angles. It was found that the results are relatively insensitive to different optimization methods and edge detectors. The precision of contact angle measurement was enhanced by improving the location of the contact points of the liquid meniscus with the solid substrate to subpixel resolution. To optimize the experimental design, the capillary was replaced with an inverted sharp-edged pedestal, or holder, to control the drop height and to ensure the axisymmetry of the drops. It was shown that the drop height is the most important experimental parameter affecting the accuracy of the surface tension measurement, and larger drop heights yield lower surface tension errors. It is suggested that a minimum nondimensional drop height (drop height divided by capillary length) of 1.7 is required to reach an error of less than 0.2 mJ/m(2) for the measured surface tension. As an example, the surface tension of water was measured to be 72.46 ± 0.04 at 24 °C by ADSA-NA, compared to 72.39 ± 0.01 mJ/m(2) obtained with pendant drop experiments.     

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.     

Dip-coating crystallization on a superhydrophobic surface: a million mounted crystals in a 1 cm2 array

Silicon wafers (silicon dioxide surfaces) were patterned by photolithograpy to contain 3 μm (width) × 6 μm (length) × 40 μm (height) staggered rhombus posts in a square array (20 μm center-to-center spacing). These surfaces were hydrophobized using a vapor phase reaction with tridecafluorooctyldimethylchlorosilane and exhibit "superhydrophobicity" (water contact angles of θ(A)/θ(R) = 169°/156°). When a section of a wafer is submerged in and withdrawn from water, the superhydrophobic surface emerges, apparently completely dry. If the same procedure is performed using aqueous sodium chloride as the liquid bath, individual crystals of the salt can be observed on the top of each of the posts. "Dip-coating crystallization" using an aqueous sodium chloride solution of 4.3 M produces crystals with ～1 μm dimensions. A less concentrated solution, 1 M NaCl, renders crystals with ～500 nm dimensions. These experiments suggest that superhydrophobic surfaces that emerge from water and are "apparently completely dry" are, in fact, decorated with micrometer-size (several femtoliters) sessile water drops that rapidly evaporate. This simple technique is useful for preparation of very small liquid drops or puddles (of controlled composition) and for preparation of arrays of controlled size, crystalline substances (dip-coating crystallization).     

Self-pinning protein-laden drops

Proteins dissolved in a drop induce and enhance the pinning of the drop contact line. This effect dramatically increases the volume of drops that are vertically pinned on a flat siliconized substrate. It was found that this drop pinning behavior exhibits two regimes: for low protein content in a drop the pinning increases as the contact angle hysteresis increases, and for high protein content the pinning decreases as the surface tension of the protein solution decreases.     

Directional self-cleaning superoleophobic surface

In this work, we report the creation of a grooved surface comprising 3 μm grooves (height ～4 μm) separated by 3 μm from each other on a silicon wafer by photolithography. The grooved surface was then modified chemically with a fluorosilane layer (FOTS). The surface property was studied by both static and dynamic contact angle measurements using water, hexadecane, and a polyethylene wax ink as the probing liquids. Results show that the grooved surface is both superhydrophobic and superoleophobic. Its observed contact angles agree well with the calculated Cassie-Baxter angles. More importantly, we are able to make a replica of the composite wax ink-air interface and study it by SEM. Microscopy results not only show that the droplet of the wax ink "sits" on air in the composite interface but also further reveal that the ink drop actually pins underneath the re-entrant structure in the side wall of the grooved structure. Contact angle measurement results indicate that wetting on the grooved surface is anisotropic. Although liquid drops are found to have lower static and advancing contact angles in the parallel direction, the drops are found to be more mobile, showing smaller hysteresis and lower sliding angles (as compared to the FOTS wafer surface and a comparable 3-μm-diameter pillar array FOTS surface). The enhanced mobility is attributable to the lowering of the resistance against an advancing liquid because 50% of the advancing area is made of a solid strip where the liquid likes to wet. This also implies that the contact line for advancing is no longer smooth but rather is ragged, having the solid strip area leading the wetting and the air strip area trailing behind. This interpretation is supported by imaging the geometry of the contact lines using molten ink drops recovered from the sliding angle experiments in both the parallel and orthogonal directions. Because the grooved surface is mechanically stronger against mechanical abrasion, the self-cleaning effect exhibited in the parallel direction suggests that groove texturing is a viable approach to create mechanically robust, self-cleaning, superoleophobic surfaces.     

Dynamics of a disturbed sessile drop measured by atomic force microscopy (AFM)

A new method for studying the dynamics of a sessile drop by atomic force microscopy (AFM) is demonstrated. A hydrophobic microsphere (radius, r ～ 20-30 μm) is brought into contact with a small sessile water drop resting on a polytetrafluoroethylene (PTFE) surface. When the microsphere touches the liquid surface, the meniscus rises onto it because of capillary forces. Although the microsphere volume is 6 orders of magnitude smaller than the drop, it excites the normal resonance modes of the liquid interface. The sphere is pinned at the interface, whose small (<100 nm) oscillations are readily measured with AFM. Resonance oscillation frequencies were measured for drop volumes between 5 and 200 μL. The results for the two lowest normal modes are quantitatively consistent with continuum calculations for the natural frequency of hemispherical drops with no adjustable parameters. The method may enable sensitive measurements of volume, surface tension, and viscosity of small drops.     

Spreading Dynamics of Polydimethylsiloxane Drops: Crossover from Laplace to Van der Waals Spreading

The spreading dynamics of small polydimethylsiloxane (PDMS) drops was studied on substrates with varying surface energies. For experimental parameters near the wetting transition, we observed small PDMS drops of different drop volumes as a function of time using interference video microscopy. While for large drops the contact angle θ decreases with the well-established power-law relation θ approximately t(-0.3) (Tanner's law), the effect of dispersive van der Waals (VW) interactions must be taken into account when interpreting the evolution of small drops. Two signatures of the VW forces are observed. For a positive Hamaker constant, the disjoining pressure acts as an additional driving force, leading to an acceleration of droplet spreading as soon as the drop height becomes comparable to the range of the VW interactions. In addition, a precursor film forms ahead of the contact line, leading to an apparent volume loss, particularly noticeable for very small drops. Contact line pinning may be a problem and we describe its effect on our experimental results. We present a theory that discusses the interplay of surface tension and VW forces in the case of a spreading drop. This model predicts a new spreading regime for very thin drops, in agreement with our experimental results. Copyright 2001 Academic Press.     

Modeling contact angle hysteresis of a liquid droplet sitting on a cosine wave-like pattern surface

A liquid droplet sitting on a hydrophobic surface with a cosine wave-like square-array pattern in the Wenzel state is simulated by using the Surface Evolver to determine the contact angle. For a fixed drop volume, multiple metastable states are obtained at two different surface roughnesses. Unusual and non-circular shape of the three-phase contact line of a liquid droplet sitting on the model surface is observed due to corrugation and distortion of the contact line by structure of the roughness. The contact angle varies along the contact line for each metastable state. The maximum and minimum contact angles among the multiple metastable states at a fixed viewing angle correspond to the advancing and the receding contact angles, respectively. It is interesting to observe that the advancing/receding contact angles (and contact angle hysteresis) are a function of viewing angle. In addition, the receding (or advancing) contact angles at different viewing angles are determined at different metastable states. The contact angle of minimum energy among the multiple metastable states is defined as the most stable (equilibrium) contact angle. The Wenzel model is not able to describe the contact angle along the three-phase contact line. The contact angle hysteresis at different drop volumes is determined. The number of the metastable states increases with increasing drop volume. Drop volume effect on the contact angles is also discussed.     

Contact angles on spherical surfaces

In this paper, we explore the influence of curved surfaces on contact angles. Small liquid drops were deposited at the apex of spheres. Liquid was added to advance the contact line (or withdrawn to cause recession). As drop volume increased, the contact line advanced outward and downward. With the addition of each increment of liquid, the contact line encountered a steeper slope and showed progressively larger apparent advancing contact angles. Observed apparent contact angles could be explained in terms of intrinsic contact angles and surface orientation. We found that if curvature and geometry were correctly accounted for, the classic Gibbs relation held. The experimental approach and analysis used here for estimating intrinsic wettability from curved surfaces could easily be integrated into automated contact angle measurement systems.     

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.     

Microscopic treatment of a barrel drop on fibers and nanofibers

The microscopic approach of Berim and Ruckenstein (J. Phys. Chem. B 108 (2004) 19330, 19339) regarding the shape and stability of a liquid drop on a planar bare solid surface is extended to a liquid barrel drop on the bare surface of a solid cylinder (fiber) of arbitrary radius. Assuming the interaction potentials of the liquid molecules between themselves and with the molecules of the solid of the London-van der Waals form, the potential energy of a liquid molecule with an infinitely long fiber was calculated analytically. A differential equation for the drop profile was derived by the variational minimization of the total potential energy of the drop by taking into account the structuring of the liquid near the fiber. This equation was solved in quadrature and the shape and stability of the barrel drop were analyzed as functions of the radius of the fiber and the microscopic contact angle theta(0) which the drop profile makes with the surface of the fiber. The latter angle is dependent on the fiber radius and on the microscopic parameters of the model (strength of the intermolecular interactions, densities of the liquid and solid phases, hard core radii, etc.). Expressions for the evaluation of the microcontact angle from experimentally measurable characteristics of the drop profile (height, length, volume, location of inflection point) are obtained. All drop characteristics, such as stability, shape, are functions of theta(0) and a certain parameter a which depends on the model parameters. In particular, the range of drop stability consists of three domains in the plane theta(0)-a, separated by two critical curves a=a(c)(theta(0)) and a=a(c1)(theta(0)) [a(c)(theta(0))h(m1) cannot exist, whereas in the third domain (between those curves) the drop can have values of h(m) either smaller than h(m1) or larger than h(m2), where h(m2)>h(m1) is a second critical height. For sufficiently large fiber radii, R(f)1 >/= microm, the critical curves almost coincide and only two domains, the first and the second, remain. The smaller the radius, the larger is the difference between the critical curves and the larger is the second domain of drop stability. The shape of the drop depends on whether the point (theta(0),a) on the theta(0)-a plane is far from the critical curve or near it. In the first case the drop profile has generally a large circular part, while in the second case the shape is either almost planar or contains a long manchon that is similar to a film on the fiber.     

Evaporative properties and pinning strength of laser-ablated, hydrophilic sites on lotus-leaf-like, nanostructured surfaces

Wetting, evaporative, and pinning strength properties of hydrophilic sites on superhydrophobic, nanostructured surfaces were examined. Understanding these properties is important for surface characterization and designing features in self-cleaning, lotus-leaf-like surfaces. Laser-ablated, hydrophilic spots between 250 mum and 2 mm in diameter were prepared on silicon nanowire (NW) superhydrophobic surfaces. For larger circumference pinning sites, initial contact angle measurements resemble the contact angle of the surface within the pinning site: 65-69 degrees . As the drop volume is increased, the contact angles approach the contact angle of the NW surface without pinning sites: 171-176 degrees . The behavior of water droplets on the pinning sites is governed by how much of the water droplet is being influenced by the superhydrophobic NW surfaces versus the hydrophilic areas. During the evaporation of sinapic acid solution, drops are pinned by the spots except for the smaller circumference sites. Pinning strengths of the hydrophilic sites are a linear function of the pinning spot circumference. Protein samples prepared and deposited on the pinning sites for analysis by matrix-assisted laser desorption ionization indicate an improvement in sensitivity from that of a standard plate analysis by a factor of 5.     

Suspension of water droplets on individual pillars

We report results of extensive experimental and numerical studies on the suspension of water drops deposited on cylindrical pillars having circular and square cross sections and different wettabilities. In the case of circular pillars, the drop contact line is pinned to the whole edge contour until the drop collapses due to the action of gravity. In contrast, on square pillars, the drops are suspended on the four corners and spilling along the vertical walls is observed. We have also studied the ability of the two geometries to sustain drops and found that if we compare pillars with the same characteristic size, the square is more efficient in pinning large volumes, while if we normalize the volumes to pillar areas, the opposite is true.     

Spreading of a suspension drop on a horizontal surface

Experimental studies were performed on the contact line motion of a suspension of PS particles on a glass surface. The base liquids were silicone oil and glycerin. The particle size was in the range of 1-6 μm and the particle loading was 0.5-5 vol %. The drop shape was determined by using a drop image and its reflection and the drop outline was traced to the subpixel level. The Tanner-Voinov-Hoffman relation was valid for suspensions as well as for pure liquids. Silicone oil suspensions showed almost no noticeable change compared with the pure fluid. Glycerin suspensions showed an increase in contact line speed at low particle loading. The difference was due to the microstructure change at the contact line region, and the microstructure change was originated from the wetting characteristics of particles. Particle alignment occurred during the spreading stage for partially wetting particles. The contact line showed a stop-and-go fashioned motion due to surface irregularities. This result can be used as the boundary condition at the contact line in the numerical simulation of suspension spreading.     

Modeling contact angle hysteresis on chemically patterned and superhydrophobic surfaces

We investigate contact angle hysteresis on chemically patterned and superhydrophobic surfaces, as the drop volume is quasistatically increased and decreased. We consider both two (cylindrical drops) and three (spherical drops) dimensions using analytical and numerical approaches to minimize the free energy of the drop. In two dimensions, we find, in agreement with other authors, a slip, jump, stick motion of the contact line. In three dimensions, this behavior persists, but the position and magnitude of the contact line jumps are sensitive to the details of the surface patterning. In two dimensions, we identify analytically the advancing and receding contact angles on the different surfaces, and we use numerical insights to argue that these provide bounds for the three-dimensional cases. We present explicit simulations to show that a simple average over the disorder is not sufficient to predict the details of the contact angle hysteresis and to support an explanation for the low contact angle hysteresis of suspended drops on superhydrophobic surfaces.     

Determination of line tension for systems near wetting

Contact angle measurements for three n-alkanes, heptane, octane, and nonane, on two different self-assembled surfaces (SAM) are reported as a function of drop size. These liquids all formed low contact angles (below 20 degrees ); the measurements were performed using an accurate method for systems with low contact angle, ADSA-D. The observed drop size dependence of the contact angles was interpreted using the modified Young equation. It was concluded that the observed drop size dependence of contact angles was due to line tension. The choice of systems also provided the opportunity to examine the behavior of the line tension for systems near wetting (i.e., low contact angles). It was determined that the line tension is positive and ranges from below 10(-7) to just below 10(-6) J/m for the systems studied; the observations suggested that the line tension decreases as the contact angle decreases and likely vanishes at complete wetting.     

Microspreading studies on rough surfaces by scanning electron microscopy

The use of scanning electron microscopy for direct observation of the effects of surface roughness on the spreading of liquids is described, making it possible to view moving liquid drops at distances less than 1 μm from the advancing contact line. Various surfaces were examined including several with simple forms of roughness which can assist in explaining the behavior of more complex surfaces. Spreading is shown to be highly dependent on the orientation and texture of the roughness; in particular, the presence of sharp edges of step height 0.05 μm are shown to influence spreading significantly. These observations reinforce our previously stated doubts of the significance of conventionally measured macroscopic contact angles.