Ratcheting motion of liquid drops on gradient surfaces

The motions of liquid drops of various surface tensions and viscosities were investigated on a solid substrate possessing a gradient of wettability. A drop of any size moves spontaneously on such a surface when the contact angle hysteresis is negligible; but it has to be larger than a critical size in order to move on a hysteretic surface. The hysteresis can, however, be reduced or eliminated with vibration that allows the drop to sample various metastable states, thereby setting it to the path of global energy minima. Significant amplification of velocity is observed with the frequency of forcing vibration matching the natural harmonics of drop oscillation. It is suggested that the main cause for velocity amplification is related to resonant shape fluctuation, which can be illustrated by periodically deforming and relaxing the drop at low frequencies.     

Motion of drops on a surface induced by thermal gradient and vibration

It is well known that a liquid drop with a low contact angle (approximately 45 degrees ) and low wetting hysteresis moves toward the colder region of a temperature gradient substrate as a result of the thermal Marangoni force. A moderately sized water drop, however, usually does not move on such a surface because of the overwhelming effect of hysteresis. The water drop can, however, be forced to move when it is vibrated on a temperature gradient surface with its velocity exhibiting maxima at the respective Rayleigh frequencies. A simple model is presented that captures the dependence of drop velocity on hysteresis, vibration amplitude, and the forcing and resonance frequencies of vibration.     

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.     

Concentrating particles on drop surfaces using external electric fields

We propose to use an externally applied uniform electric field to alter the distribution of particles on the surface of a drop immersed in another immiscible liquid. Specifically, we seek to generate well-defined concentrated regions at the drop surface while leaving the rest of the surface particle free. Experiments show that when the dielectric constant of the drop is greater than that of the ambient liquid the particles for which the Clausius-Mossotti factor is positive move along the drop surface to the two poles of the drop. Particles with a negative Clausius-Mossotti factor, on the other hand, move along the drop surface to form a ring near the drop equator. The opposite takes place when the dielectric constant of the drop is smaller than that of the ambient liquid, namely particles for which the Clausius-Mossotti factor is positive form a ring near the equator while those for which such a factor is negative move to the poles. This motion is due to the dielectrophoretic force that acts upon particles because the electric field on the surface of the drop is nonuniform, despite the uniformity of the applied electric field. Experiments also show that when small particles collect at the poles of a deformed drop the electric field needed to break the drop is smaller than without particles. These phenomena could be useful to concentrate particles at a drop surface within well-defined regions (poles and equator), separate two types of particles at the surface of a drop or increase the drop deformation to accelerate drop breakup.     

Vibration-actuated drop motion on surfaces for batch microfluidic processes

When a liquid drop is subjected to an asymmetric lateral vibration on a nonwettable surface, a net inertial force acting on the drop causes it to move. The direction and velocity of the drop motion are related to the shape, frequency, and amplitude of vibration, as well as the natural harmonics of the drop oscillation. Aqueous drops can be propelled through fluidic networks connecting various unit operations in order to carry out batch processing at the miniature scale. We illustrate the integration of several unit operations on a chip: drop transport, mixing, and thermal cycling, which are precursor steps to carrying out advanced biological processes at microscale, including cell sorting, polymerase chain reaction, and DNA hybridization.     

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.     

The Effects of Resistance to Shift of the Equilibrium State of a Liquid Droplet in Contact with a Solid

The effect of blocking the shift of the contact surface between a liquid drop and a solid body is discussed. The model proposed in (S. D. Iliev, 1997, J. Colloid Interface Sci., 194, 287) is discussed. This equilibrium model considers the resistance to shift by adding an energy to the classical capillary equilibrium model. It is shown that the set of equilibrium shapes of static droplets is effectively modeled. Studying the set of equilibrium axisymmetric drops, located on a horizontal surface, the analysis proves that the contact angle hysteresis is described without introducing a dependence between the resistance-to-shift coefficients and the drop volume and Bond's number. A possibility of realizing a stick-slip motion and division of the equilibrium drops is studied, too. It is shown that the equilibrium model describes also the set of equilibrium nonaxisymmetric static drops. The everyday experience to obtain the various nonaxisymmetric drop shapes by the deforming of contact line with a thin rod is numerically modeled. Copyright 1999 Academic Press.     

Comparison of contact angle hysteresis of different probe liquids on the same solid surface

Advancing and receding contact angles of water, formamide and diiodomethane were measured on 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) layers deposited on three different solid supports—glass, mica and poly(methyl methacrylate). Up to five statistical monolayers were deposited on the surfaces by spreading DPPC solution. It was found that even on five statistical DPPC monolayers, the hysteresis of a given liquid depends on the kind of solid support. Also on the same solid support the contact angle hysteresis is different for each probe liquid used. The AFM images show that the heights of roughness of the DPPC films cannot be the primary cause of the observed hysteresis because the heights are too small to cause the observed hystereses. It is believed that the hysteresis is due to the liquid film present right behind the three-phase solid surface/liquid drop/gas (vapour) contact line and the presence of Derjaguin pressure. The value of contact angle hysteresis depends on both the solid surface and liquid properties as well as on intermolecular interactions between them.     

Gibbs free energy of liquid drops on conical fibers

Small drops can move spontaneously on conical fibers. As a drop moves along the cone, it must change shape to maintain a constant volume, and thus, it must change its surface energy. Simultaneously, the exposed surface area of the underlying cone must also change. The associated surface energies should balance each other, and the drop should stop moving when it reaches a location where the free energy is a minimum. In this paper, a minimum Gibbs free energy analysis has been performed to predict where a drop will stop on a conical fiber. To obtain the Gibbs free energies of a drop at different locations of a conical fiber, the theoretical expressions for the shape of a droplet on a conical fiber are derived by extending Carroll's equations for a drop on a cylindrical fiber. The predicted Gibbs free energy exhibits a minimum along the length of the cone. For a constant cone angle, as the contact angle between the liquid and the cone increases, the drop will move toward the apex of the cone. Likewise, for a constant contact angle, as the cone angle increases, the drop moves toward the apex. Experiments in which water and dodecane were placed on glass cones verify these dependencies. Thus, the final location of a drop on a conical fiber can be predicted on the basis of the geometry and surface energy of the cone, the surface tension and volume of the liquid, and the original location where the drop was deposited.     

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.     

Experiments on the motion of drops on a horizontal solid surface due to a wettability gradient

Results from experiments performed on the motion of drops of tetraethylene glycol in a wettability gradient present on a silicon surface are reported and compared with predictions from a recently developed theoretical model. The gradient in wettability was formed by exposing strips cut from a silicon wafer to dodecyltrichlorosilane vapors. Video images of the drops captured during the experiments were subsequently analyzed for drop size and velocity as functions of position along the gradient. In separate experiments on the same strips, the static contact angle formed by small drops was measured and used to obtain the local wettability gradient to which a drop is subjected. The velocity of the drops was found to be a strong function of position along the gradient. A quasi-steady theoretical model that balances the local hydrodynamic resistance with the local driving force generally describes the observations; possible reasons for the remaining discrepancies are discussed. It is shown that a model in which the driving force is reduced to accommodate the hysteresis effect inferred from the data is able to remove most of the discrepancy between the observed and predicted velocities.     

Dynamic self-assembly of polymer colloids to form linear patterns

A dynamic self-assembling process is reported which involves drying a droplet of positively charged colloidal suspension on a flat negatively charged hydrophilic surface. This extremely simple method affords lines of colloidal particles with regular 1.5-4.5 microm line spacing and smaller than 2 microm line width over a broad surface area. The ordered region diffracts light to display an iridescent appearance and generates first-order diffraction spots when illuminated by a He-Ne laser. A periodic stick-slip motion of the drying liquid front is observed during the drying process using optical microscopy. The periodic motion must be related to the periodic particle deposition. We propose that the simultaneous deposition of the particles at a fixed distance (i.e., the line spacing) behind the previous line of particles where the contact line is pinned is in turn responsible for the periodic stick-slip motion. The key distinguishing feature of the present system is the attractive interaction between the particles and the surface, which instigates the periodicity of the particle deposition.     

Brownian motion of a drop with hysteresis dissipation

Small water drops placed on a low-energy substrate with a slight tilt were vibrated parallel to the support with bands of Gaussian white noise of different powers. The drops drifted downward on the inclined support accompanied with random forward and backward movements. For a hysteresis free surface, the drift velocity should only be the product of the component of the gravitational acceleration and the Langevin relaxation time, being independent of the power of noise. On the other hand, in the presence of hysteresis, as is the case here, the drift velocity depends strongly on the power of the noise. This result illustrates the role of hysteresis in the drifted motion of drops on a surface subjected to vibration, which has important bearings on various forms of work fluctuation relations.     

Impact of a superhydrophobic sphere onto water

When a water drop hits a superhydrophobic solid surface, it bounces off the substrate like an elastic ball. Here we show that when a tiny superhydrophobic solid sphere impacts with water, it can bounce off the free surface just as it impacts with an elastic membrane. The motion of a sinking sphere is analytically calculated by solving a potential flow whose free boundary is determined by the Young-Laplace equation. To find conditions under which the solid sphere should sink, bounce off, or oscillate upon impact with water, we construct simple scaling laws which are shown to agree well with experimentally found boundaries between the distinct impact behaviors in a regime map based on dimensionless parameters.     

Contact angle hysteresis of bovine serum albumin (BSA) solution/metal (Au-Cr) coated glass substrate

Bovine serum albumin (BSA) has an extraordinary property to carry biomolecules. An experimental study on the wettability of BSA is presented in this study. The variations in the surface tension and the equilibrium contact angle with the change in BSA concentration are also reported. The surface tension and the contact angle are measured with pendant and sessile drop techniques, respectively. A nonlinear decrement in the surface tension with the increment in the BSA concentration is observed. An equilibrium contact angle of a BSA solution with particular concentration is determined by studying the hysteresis in the contact angle from dynamic contact angle measurements. The needle-in-drop technique is used to study the hysteresis of the contact angle. It is observed that the obtained surface tension and the equilibrium contact angle vary with the BSA concentration. In this reported study, for the considered combination of the BSA concentration and solid surface, the liquid drop does not recede as the drop volume decreases, which represents nonreceding contact angle condition. The increment in the contact angle with the increment in the BSA concentration is observed. Finally, it is observed that the inclusion of the proteins not only changes the surface tension but also changes the contact angle.     

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.     

Modeling of the moving deformed triple contact line: influence of the fluid inertia

For partial wetting, motion of the triple liquid-gas-solid contact line is influenced by heterogeneities of the solid surface. This influence can be strong in the case of inertial (e.g., oscillation) flows where the line can be pinned or move intermittently. A model that takes into account both surface defects and fluid inertia is proposed. The viscous dissipation in the bulk of the fluid is assumed to be negligible as compared to the dissipation in the vicinity of the contact line. The equations of motion and the boundary condition at the contact line are derived from Hamilton's principle. The rapid capillary rise along a vertical inhomogeneous wall is treated as an example.     

Wetting and absorption of water drops on Nafion films

Water drops on Nafion films caused the surface to switch from being hydrophobic to being hydrophilic. Contact angle hysteresis of >70 degrees between advancing and receding values were obtained by the Wilhelmy plate technique. Sessile drop measurements were consistent with the advancing contact angle; the sessile drop contact angle was 108 degrees . Water drop adhesion, as measured by the detachment angle on an inclined plane, showed much stronger water adhesion on Nafion than Teflon. Sessile water and methanol drops caused dry Nafion films to deflect. The flexure went through a maximum with time. Flexure increased with contact area of the drop, but was insensitive to the film thickness. Methanol drops spread more on Nafion and caused larger film flexure than water. The results suggest that the Nafion surface was initially hydrophobic but water and methanol drops caused hydrophilic sulfonic acid domains to be drawn to the Nafion surface. Local swelling of the film beneath the water drop caused the film to buckle. The maximum flexure is suggested to result from motion of a water swelling front through the Nafion film.