A 2D electrohydrodynamic model for electrorotation of fluid drops

A theoretical analysis of spontaneous electrorotation of deformable fluid drops in a DC electric field is presented with a 2D electrohydrodynamic model. The fluids in the system are assumed to be leaky dielectric and Newtonian. If the rotating flow is dominant over the cellular convection type of electrohydrodynamic flow, closed-form solutions for drops of small deformations can be obtained. Because the governing equations are in general nonlinear even when drop deformations are ignored, the general solution for even undeformed drop takes a form of infinite series and can only be evaluated by numerical means. Both closed-form solutions for special cases and numerical solutions for more general cases are obtained here to describe steady-state field variables and first-order drop deformations. In a DC electric field of strength beyond the threshold value, spontaneous electrorotation of a drop is shown to occur when charge relaxation in the surrounding fluid is faster than the fluid inside the drop. With increasing the strength of the applied electric field from the threshold for onset of electrorotation, the axis of drop contraction deviates from from that of the applied electric field in the direction of the rotating flow with an angle increasing with the field strength.     

Polymeric stabilized emulsions: steric effects and deformation in soft systems

Polymeric stabilizers are used in a broad range of processes and products, from pharmaceuticals and engine lubricants to formulated foods and shampoos. In rigid particulate systems, the stabilization mechanism is attributed to the repulsive force that arises from the compression of the polymer coating or "steric brush" on the interacting particles. This mechanism has dictated polymer design and selection for more than thirty years. Here we show, through direct measurement of the repulsive interactions between immobilized drops with adsorbed polymers layers in aqueous electrolyte solutions, that the interaction is a result of both steric stabilization and drop deformation. Drops driven together at slow collision speeds, where hydrodynamic drainage effects are negligible, show a strong dependence on drop deformation instead of brush compression. When drops are driven together at higher collision speeds where hydrodynamic drainage affects the interaction force, simple continuum modeling suggests that the film drainage is sensitive to flow through the polymer brush. These data suggest, for drop sizes where drop deformation is appreciable, that the stability of emulsion drops is less sensitive to the molecular weight or size of the adsorbed polymer layer than for rigid particulate systems.     

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.     

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.     

Spreading, evaporation, and contact line dynamics of surfactant-laden microdrops

An optical technique based on the reflectivity measurements of a thin film was used to experimentally study the spreading, evaporation, contact line motion, and thin film characteristics of drops consisting of a water-surfactant (polyalkyleneoxide-modified heptamethyltrisiloxane, called superspreader) solution on a fused silica surface. On the basis of the experimental observations, we concluded that the surfactant adsorbs primarily at the solid-liquid and liquid-vapor interfaces near the contact line region. At equilibrium, the completely wetting corner meniscus was associated with a flat adsorbed film having a thickness of approximately 31 nm. The calculated Hamaker constant, A = -4.47 x 10(-)(20) J, shows that this thin film was stable under equilibrium conditions. During a subsequent evaporation/condensation phase-change process, the thin film of the surfactant solution was unstable, and it broke into microdrops having a finite contact angle. The thickness of the adsorbed film associated with the drops was lower than that of the equilibrium meniscus. The drop profiles were experimentally measured and analyzed during the phase-change process as the contact line advanced and receded. The apparent contact angle, the maximum concave curvature near the contact line region, and the convex curvature of the drop increased as the drop grew during condensation, whereas these quantities decreased during evaporation. The position of the maximum concave curvature of the drop moved toward the center of the drop during condensation, whereas it moved away from the center during evaporation. The contact line velocity was correlated to the observed experimental results and was compared with the results of the drops of a pure alcohol. The experimentally obtained thickness profiles, contact angle profiles, and curvature profiles of the drops explain how the surfactant adsorption affects the contact line motion. We found that there was an abrupt change in the velocity of the contact line when the adsorbed film of the surfactant solution was just hydrated or desiccated during the phase-change processes. This result shows the effect of vesicles and aggregates of the surfactant on the shape evolution of the drops. For these surfactant-laden water drops, we found that the apparent contact angle increased during condensation and decreased during evaporation. However, for the drop of a pure liquid (n-butanol and 2-propanol) the apparent contact angle remained constant at a constant velocity during condensation and evaporation. The contact line was pinned during the evaporation and spreading of the surfactant-laden water drops, but it was not pinned for a drop of a pure alcohol (self-similar shape evolution).     

Shaping of gelling biopolymer drops in an elongation flow

Shaping, defined as deformation in combination with gel formation of gelatine and kappa-carrageenan drops in an elongation flow, was studied. The focus was to investigate the possibility of shaping and fixating small drops in the diameter range 20 to 229 mum. In the shaping progress and the influence of experimental properties, the viscosity, temperature, and flow of the deforming fluid were examined on the final drop shape. In the experiments a hot emulsion of an aqueous biopolymer solution in silicone oil was injected into cold silicone oil where a deforming elongation flow field existed. After injection, a temperature decrease in the drops resulted in a gel formation of the biopolymer and a fixation of the deformed drop in the flow. The shape was measured and the effect on the drop aspect ratio was determined by image analysis. Over the total drop diameter range, kappa-carrageenan was more ellipsoid-shaped than gelatine, with a maximum aspect ratio of 6 compared to 4 for gelatine. For small drops, around 22 mum, it is possible to shape kappa-carrageenan, but for gelatine small drops tend to be unaffected. An increase in viscosity, temperature, and flow resulted in an increase in the final fixated shape of the drops. The differences in drop deformation between the biopolymers were explained by drop-viscosity/oil differences and differences in the kinetics of gel formation. The different gel formation kinetics resulted in a short, well-defined, shaping process for kappa-carrageenan, while for gelatine the process was more complex, with both deformation and relaxation present at different stages.     

The dynamics of evaporating sessile droplets

Experiments on sessile drops evaporating in a normal atmosphere without an applied thermal gradient are reported and compared with an available theoretical model. The liquids used are alkanes; water; and, more recently, polydimethylsiloxane oligomers. The substrates are silicon wafers, completely wetted by the liquid. Experiments with hanging drops allow us first to discard any influence of convection in the gas phase on the drop dynamics. The model assumes the process to be controlled by the stationary diffusion of the evaporating molecules in the gas phase. For alkanes and water, and in a limited range of drop sizes where gravity can be ignored, the model accounts very well for the dynamics of the drop radius, and rather well for the contact angle. This is no longer the case with the polydimethylsiloxane oligomers, where the very small contact angles require a more elaborated analysis of the drop edge. The text was submitted by the authors in English.     

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.     

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.     

Emulsification in turbulent flow 2. Breakage rate constants

Systematic experimental study of the effects of several factors on the breakage rate constant, k(BR), during emulsification in turbulent flow is performed. These factors are the drop size, interfacial tension, viscosity of the oil phase, and rate of energy dissipation in the flow. As starting oil-water premixes we use emulsions containing monodisperse oil drops, which have been generated by the method of membrane emulsification. By passing these premixes through a narrow-gap homogenizer, working in turbulent regime of emulsification, we study the evolution of the number concentration of the drops with given diameter, as a function of the emulsification time. The experimental data are analyzed by a kinetic scheme, which takes into account the generation of drops of a given size (as a result of breakage of larger drops) and their disappearance (as a result of their own breakage process). The experimental results for k(BR) are compared with theoretical expressions from the literature and their modifications. The results for all systems could be described reasonably well by an explicit expression, which is a product of: (a) the frequency of collisions between drops and turbulent eddies of similar size, and (b) the efficiency of drop breakage, which depends on the energy required for drop deformation. The drop deformation energy contains two contributions, originating from the drop surface extension and from the viscous dissipation inside the breaking drop. In the related subsequent paper, the size distribution of the daughter drops formed in the process of drop breakage is analyzed for the same experimental systems.     

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°.     

Groovy drops: effect of groove curvature on spontaneous capillary flow

Spontaneous capillary flow (SCF) of a drop in a groove with an ideally sharp corner is possible when the Concus-Fin (CF) condition is fulfilled. However, since ideally sharp corners do not exist in reality, it is important to understand the effect of finite corner curvature on SCF. This effect is analytically studied for long drops in a V-shaped groove with a curved corner, leading to a generalization of the CF condition for such drops. The generalized condition implies that SCF depends on the geometry of the corner as well as on the dimensionless length of the drop, in addition to its dependence on the opening angle and contact angle that is covered by the CF condition. Specific calculations are presented for rounded corners. In addition, this effect is numerically calculated for short drops in V-shaped grooves with rounded corners, using the Surface Evolver software. The results of both types of calculations show that even a relatively small corner radius strongly affects the possibility of SCF: when the corner is not ideally sharp, SCF requires conditions that are more difficult to achieve than predicted by the CF condition; also, the spreading of the drop stops at a finite length and does not proceed indefinitely.     

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.     

Rails and anchors: guiding and trapping droplet microreactors in two dimensions

This paper presents a method to control the motion of nanolitre drops in a wide and thin microchannel, by etching fine patterns into the channel's top surface. Such control is possible for drops that are squeezed by the channel roof, by allowing them to reduce their surface energy as they enter into a local depression. The resulting gain in surface energy pulls a drop into the groove such that localized holes can be used as anchors for holding drops, while linear patterns can be used as rails to guide them along complex trajectories. An anchored drop can remain stationary indefinitely, as long as the driving flow rate is below a critical value which depends on the hole and drop sizes. By micro-fabricating holes into a grid pattern, drops can be arrayed and held in the observation field of a microscope against the mean carrier flow. Their contents can then be modulated by gas exchange with the flowing carrier oil. We demonstrate in particular how the pH or the oxygen levels within the drops can be controlled spatially and temporally, either by exposing rows of drops to two streams of oil at different gas concentrations or by periodically switching oil inputs to vary the gas concentration of drops as a function of time. Oxygen control is used to selectively deoxygenate droplets that encapsulate red blood cells from patients suffering from sickle cell disease, in order to study the polymerization of intracellular hemoglobin. Cycles of oxygenation and deoxygenation of anchored droplets induce depolymerization and polymerization of the hemoglobin, thus providing a method to simulate the cycling that takes place in physiological flows.     

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.     

Mass and momentum transport in extra-luminal flow (ELF) membrane devices for blood oxygenation

In this paper, we report on the characterisation of transport in membrane modules for blood oxygenation where blood is circulated outside hollow fibre membranes arranged in double layer cross-laid mats at an angle with respect to the main direction of blood flow. The effect of design and operating variables on module performance was investigated with respect to oxygen transfer into water, as gaseous oxygen and water are circulated counter-currently, respectively inside the membrane lumen and through the membrane assembly.Increasing water flow rates and membrane angles enhanced oxygen transfer across the membrane and resulted in robust operation but also in high pressure drops.Module pressure drop and oxygen transfer rate were correlated to module geometry, fibre packing density, water flow rate and membrane angle with respect to the main direction of the liquid flow in non-dimensional equations that can be used by membrane module manufacturers for the design of optimal ELF blood oxygenators. The results suggest that an optimum membrane angle exists, beyond which module operation is not convenient in terms of energy.     

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.     

A robust algorithm for the simultaneous parameter estimation of interfacial tension and contact angle from sessile drop profiles

The pendant and sessile drop profile analysis using the finite element method (PSDA-FEM) is an algorithm which allows simultaneous determination of the interfacial tension (gamma) and contact angle (theta(c)) from sessile drop profiles. The PSDA-FEM algorithm solves the nonlinear second-order spherical coordinate form of the Young-Laplace equation. Thus, the boundary conditions at the drop apex and contact position of the drop with the substrate are required to solve for the drop profile coordinates. The boundary condition at the position where the drop contacts the substrate may be specified as a fixed contact line or fixed contact angle. This paper will focus on the fixed contact angle boundary condition for sessile drops on a substrate and how this boundary condition is used in the PSDA-FEM curve-fitting algorithm. The PSDA-FEM algorithm has been tested using simulated drop shapes with and without the addition of random error to the drop profile coordinates. The random error is varied to simulate the effect of camera resolution on the estimates of gamma and theta(c) values obtained from the curve-fitting algorithm. The error in the experimental values for gamma from sessile drops of water on acrylic and Mazola corn oil on acrylic falls within the predicted range of errors obtained for gamma values from simulated sessile drop profiles with randomized errors that are comparable in magnitude to the resolution of the experimental setup.     

Liquid drops on vertical and inclined surfaces; I. An experimental study of drop geometry

Experiments have been conducted to investigate the geometric parameters necessary to describe the shapes of liquid drops on vertical and inclined plane surfaces. Two liquids and eight surfaces have been used to study contact angles, contact lines, profiles, and volumes of drops of different sizes for a range of surface conditions. The results show the contact-angle variation along the circumference of a drop to be best fit by a third-degree polynomial in the azimuthal angle. This contact-angle function is expressed in terms of the maximum and minimum contact angles of the drop, which are determined for various conditions. The maximum contact angle, thetamax, is approximately equal to the advancing contact angle, thetaA, of the liquid on the surface. As the Bond number, Bo, increases from 0 to a maximum, the minimum contact angle, thetamin, decreases almost linearly from the advancing to the receding angle. A general relation is found between thetamin/thetaA and Bo for different liquid-surface combinations. The drop contour can be described by an ellipse, with the aspect ratio increasing with Bo. These experimental results are valuable in modeling drop shape, as presented in Part II of this work.