An empirically validated analytical model of droplet dynamics in electrowetting on dielectric devices

Explicit analytical models that describe the capillary force on confined droplets actuated in electrowetting on dielectric devices and the reduction in that force by contact angle hysteresis as a function of the three-dimensional shape of the droplet interface are presented. These models are used to develop an analytical model for the transient position and velocity of the droplet. An order of magnitude analysis showed that droplet motion could be modeled using the driving capillary force opposed by contact angle hysteresis, wall shear, and contact line friction. Droplet dynamics were found to be a function of gap height, droplet radius, surface tension, fluid density, the initial and deformed contact angles, contact angle hysteresis, and friction coefficients pertaining to viscous wall friction and contact line friction. The first four parameters describe the device geometry and fluid properties; the remaining parameters were determined experimentally. Images of the droplet during motion were used to determine the evolution of the shape, position, and velocity of the droplet with time. Comparisons between the measured and predicted results show that the proposed model provides good accuracy over a range of practical voltages and droplet aspect ratios.     

液滴在超疏水形状记忆微阵列表面上定向/非定向滚动的可逆调控

利用3D打印、 模板赋形和表面修饰的方法, 制备出具有竖直结构的超疏水形状记忆微阵列. 该微阵列的竖直状态和倾斜状态能够基于形状记忆效应进行可逆调控. 当微阵列的结构发生改变, 液滴在表面的滚动状态也随之发生变化. 液滴在竖直的微阵列上表现出非定向滚动特征, 即液滴向微阵列两侧运动的滚动角是一致的. 在倾斜的微阵列上, 液滴则表现出定向运动的功能, 即液滴沿着微阵列倾斜方向上的滚动角要小于逆着阵列倾斜方向的滚动角. 因此, 本文通过调控微阵列形态实现了对液滴定向/非定向滚动的可逆调控.     

Interfacial oil droplets

Very small, discrete oil droplets can form at the solid-liquid interface. We demonstrate this effect through formation of decane droplets at the interface between an aqueous ethanol solution and silicon wafers that have been made hydrophobic through self-assembly of octadecyltrichlorosilane (OTS). The droplets have a lens-like shape; the shape is approximately a spherical cap with a contact angle < 25 degrees. The heights of the droplets are about 2-50 nm, and diameters at the three-phase boundary are about 100-600 nm in 25% ethanol solution. The size and contact angle can be varied by changing the ethanol concentration. The contact angle of the very small droplets (height < 20 nm) is similar to the contact angle of macroscopic droplets (height approximately equal to 1 mm), so the line tension is very small. The droplets are only stable for a few hours: they gradually lose mass, presumably through Ostwald ripening. The drop perimeter is not pinned during ripening but retreats across the solid. We form the droplets by direct adsorption from an emulsion; evidence for adsorption is obtained by comparing the drop volumes in bulk to the volumes at the interface. The droplet sizes are obtained by dynamic light scattering and atomic force microscopy.     

Simulation of Sessile Water-Droplet Evaporation on Superhydrophobic Polymer Surfaces

Evaporation of sessile water-droplets on superhydrophobic polymer surfaces has been simulated in recent research. Models based on the ellipsoidal cap geometry and spherical cap geometry, which were originally put forward to describe the profile of a droplet during its evaporation process on a solid surface with a contact angle <90±, are developed to reveal the issue with an initial contact angles larger than 150±. To verify the validity of the model, experiments on superhydrophobic polycarbonate, and °uorinated polyurethane and poly (methyl methacrylate) blend surfaces were carried out. It was observed that the change trends of contact angle and height of the droplet against evaporation time on the superhydrophobic surfaces experimentally are consistent with the simulated results by ellipsoidal and spherical cap models. The ellipsoidal cap model shows the better fits due to the shape distortions of droplets.     

Dynamics of droplet motion under electrowetting actuation

The static shape of droplets under electrowetting actuation is well understood. The steady-state shape of the droplet is obtained on the basis of the balance of surface tension and electrowetting forces, and the change in the apparent contact angle is well characterized by the Young-Lippmann equation. However, the transient droplet shape behavior when a voltage is suddenly applied across a droplet has received less attention. Additional dynamic frictional forces are at play during this transient process. We present a model to predict this transient behavior of the droplet shape under electrowetting actuation. The droplet shape is modeled using the volume of fluid method. The electrowetting and dynamic frictional forces are included as an effective dynamic contact angle through a force balance at the contact line. The model is used to predict the transient behavior of water droplets on smooth hydrophobic surfaces under electrowetting actuation. The predictions of the transient behavior of droplet shape and contact radius are in excellent agreement with our experimental measurements. The internal fluid motion is explained, and the droplet motion is shown to initiate from the contact line. An approximate mathematical model is also developed to understand the physics of the droplet motion and to describe the overall droplet motion and the contact line velocities.     

Nonlinear penetration of liquid drops into radial capillaries

The nonlinear dynamics of the impact and penetration of a liquid droplet in a radial capillary is studied numerically. The radial capillary is formed by two parallel plates at a distance of delta(g). The top plate has an orifice at its center. A droplet impacting on the orifice-plate partly spreads over the top plate, and the rest penetrates into the capillary gap between the two plates. The rate of spread of the fluid on the orifice plate, xi(out), is governed by the contact angle, beta, between the liquid and the orifice plate and the droplet initial momentum, whereas the rate of fluid spread inside the capillary gap, xi(in), is decided by the contact angles with both plates and the plate gap, delta(g).     

Using the Surface Evolver to model droplet formation processes in membrane emulsification

A model was developed to describe the droplet formation mechanism in membrane emulsification from the point of view of Gibbs free energy with the help of the Surface Evolver, which is an interactive finite element program for the study of interfaces shaped by surface tension. A program to test the model was written and run which allows the user to track the droplet shape as it grows, to identify the point of instability due to free energy, and thus predict droplet size. The inputs of the program are pore geometry, oil-aqueous phase interfacial tension, and contact angle. The model reasonably predicted droplet sizes for oblong-shaped pores under quiescent conditions where the force balance approach is not applicable. The model was validated against experimental conditions from the literature where the average error of the predictions compared to the mean droplet sizes was 8%.     

Static and dynamic contact angles of water droplet on a solid surface using molecular dynamics simulation

The present study investigates the variation of static contact angle of a water droplet in equilibrium with a solid surface in the absence of a body force and the dynamic contact angles of water droplet moving on a solid surface for different characteristic energies using the molecular dynamics simulation. With increasing characteristic energy, the static contact angle in equilibrium with a solid surface in the absence of a body force decreases because the hydrophobic surface changes its characteristics to the hydrophilic surface. In order to consider the effect of moving water droplet on the dynamic contact angles, we apply the constant acceleration to an individual oxygen and hydrogen atom. In the presence of a body force, the water droplet changes its shape with larger advancing contact angle than the receding angle. The dynamic contact angles are compared with the static contact angle in order to see the effect of the presence of a body force.     

Surface-guided templating of particle assemblies inside drying sessile droplets

The particles suspended inside evaporating sessile droplets can be assembled into microscopic objects with long-ranged ordered structure. The air-water droplet interface guides the assembly and determines the shape of the resulting micropatches. We report the results of a systematic study of the mechanism of interface-templated assembly on substrates of controlled contact angle. The kinetics of drying were examined by measurements of droplet profiles, and it was found that the rate matched diffusion-limited evaporation well. The shape of the droplets and of the resulting assemblies was correlated to the dynamics of the receding contact line. The effects of major parameters controlling the process, including contact angle, particle concentration, and electrolyte, were investigated in detail. A variety of micropatch shapes were observed and categorized within the parameter space. The in-depth characterization of the process allowed the optimization of the assembly and the formulation of protocols for the deposition of nanostructured patches of different diameter, thickness, and shape.     

Analysis of the relationship between liquid droplet size and contact angle

The purpose of this paper is to present a consistent theoretical concept that can explain the various physical phenomena associated with the effect of droplet size on contact angle for droplets on solid surfaces, and with the geometry of the liquid/gas/solid contact line in general. Two droplet geometries have been considered: uniformly elongated droplets and axisymmetric droplets. It has been shown that the contact angle for elongated droplets is size-independent and, thus, satisfies the Young equation for constant material and interfacial properties. On the other hand, whereas the contact angle for axisymmetric droplets is size-dependent and does not satisfy the original Young equation, it is shown that this contact angle can still be predicted for any combination of droplet and substrate materials, and a given mass of the droplet. The theoretical work has been combined with the development of numerical schemes of solving the Laplace-Young equation for various droplet geometries. The proposed approach has been applied to different material/substrate combinations and validated against several sets of experimental data. As a result, a method has been developed for predicting the contact angle of both long and axisymmetric sessile droplets of arbitrary sizes for given liquid/solid/gas properties.     

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.     

Droplet shape analysis and permeability studies in droplet lipid bilayers

We apply optical manipulation to prepare lipid bilayers between pairs of water droplets immersed in an oil matrix. These droplet pairs have a well-defined geometry allowing the use of droplet shape analysis to perform quantitative studies of the dynamics during bilayer formation and to determine time-dependent values for the droplet volumes, bilayer radius, bilayer contact angle, and droplet center-line approach velocity. During bilayer formation, the contact angle rises steadily to an equilibrium value determined by the bilayer adhesion energy. When there is a salt concentration imbalance between droplets, there is a measurable change in the droplet volume. We present an analytical expression for this volume change and use this expression to calculate the bilayer permeability to water.     

Analysis of the equilibrium droplet shape based on an ellipsoidal droplet model

The extent of a droplet's spreading over a flat, smooth solid substrate and its equilibrium height in the presence of gravity are determined approximately, without a numerical solution of the governing nonlinear differential equation, by assuming that the droplet takes on the shape of an oblate spheroidal cap and by minimizing the corresponding free energy. The comparison with the full numerical evaluations confirms that the introduced approximation and the obtained results are accurate for contact angles below about 120° and for droplet sizes on the order of the capillary length of the liquid. The flattening effect of gravity is to increase the contact radius and decrease the height of the droplet, with these being more pronounced for higher values of the Bond number.     

Dynamic contact angles and hysteresis under electrowetting-on-dielectric

By designing and implementing a new experimental method, we have measured the dynamic advancing and receding contact angles and the resulting hysteresis of droplets under electrowetting-on-dielectric (EWOD). Measurements were obtained over wide ranges of applied EWOD voltages, or electrowetting numbers (0 ≤ Ew ≤ 0.9), and droplet sliding speeds, or capillary numbers (1.4 × 10(-5) ≤ Ca ≤ 6.9 × 10(-3)). If Ew or Ca is low, dynamic contact angle hysteresis is not affected much by the EWOD voltage or the sliding speed; that is, the hysteresis increases by less than 50% with a 2 order-of-magnitude increase in sliding speed when Ca < 10(-3). If both Ew and Ca are high, however, the hysteresis increases with either the EWOD voltage or the sliding speed. Stick-slip oscillations were observed at Ew > 0.4. Data are interpreted with simplified hydrodynamic (Cox-Voinov) and molecular-kinetic theory (MKT) models; the Cox-Voinov model captures the trend of the data, but it yields unreasonable fitting parameters. MKT fitting parameters associated with the advancing contact line are reasonable, but a lack of symmetry indicates that a more intricate model is required.     

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.     

Wetting phenomena on micro-grooved aluminum surfaces and modeling of the critical droplet size

The behavior of water droplets on aluminum surfaces with parallel grooves tens of microns in width and depth is considered, and a mechanistic model is developed for predicting the critical droplet size-droplets at incipient sliding due to gravity. The critical droplet size is nearly 50% smaller on micro-grooved surfaces than on the same surface without micro-grooves. The application of existing models fails to predict this behavior, and a new model based on empiricism is developed. The new model provides reasonable predictions of the critical droplet size for a given inclination angle, advancing contact angle, and maximum contact angle. When the grooves are aligned parallel to gravity, the maximum apparent contact angle does not occur at the advancing front but rather along the side of the droplet because of contact-line pinning. Droplets on these surfaces are elongated and possess a parallel-sided base contour shape. Novel data are provided for droplets in a Wenzel state, a Cassie-Baxter state, and combined state on micro-grooved surfaces, and the ability of the empirical model to handle these variations is explored. These findings may be important to a broad range of engineering applications.     

Magnetic levitation of liquid crystals

The principle of magnetic levitation is demonstrated using a large magnetic field gradient to elevate a polycrystalline sample of dodecyloxycyanobiphenyl against gravity. Additionally, a nematic droplet of pentylcyanobiphenyl clinging to a vertically oriented wire is elevated against gravity. The contact angle and length of the droplet are extracted from the droplet shape in the context of a gravitation-free model.     

Simultaneous spreading and evaporation: Recent developments

The recent progress in theoretical and experimental studies of simultaneous spreading and evaporation of liquid droplets on solid substrates is discussed for pure liquids including nanodroplets, nanosuspensions of inorganic particles (nanofluids) and surfactant solutions. Evaporation of both complete wetting and partial wetting liquids into a nonsaturated vapour atmosphere are considered. However, the main attention is paid to the case of partial wetting when the hysteresis of static contact angle takes place. In the case of complete wetting the spreading/evaporation process proceeds in two stages. A theory was suggested for this case and a good agreement with available experimental data was achieved. In the case of partial wetting the spreading/evaporation of a sessile droplet of pure liquid goes through four subsequent stages: (i) the initial stage, spreading, is relatively short (1–2 min) and therefore evaporation can be neglected during this stage; during the initial stage the contact angle reaches the value of advancing contact angle and the radius of the droplet base reaches its maximum value, (ii) the first stage of evaporation is characterised by the constant value of the radius of the droplet base; the value of the contact angle during the first stage decreases from static advancing to static receding contact angle; (iii) during the second stage of evaporation the contact angle remains constant and equal to its receding value, while the radius of the droplet base decreases; and (iv) at the third stage of evaporation both the contact angle and the radius of the droplet base decrease until the drop completely disappears. It has been shown theoretically and confirmed experimentally that during the first and second stages of evaporation the volume of droplet to power 2/3 decreases linearly with time. The universal dependence of the contact angle during the first stage and of the radius of the droplet base during the second stage on the reduced time has been derived theoretically and confirmed experimentally. The theory developed for pure liquids is applicable also to nanofluids, where a good agreement with the available experimental data has been found. However, in the case of evaporation of surfactant solutions the process deviates from the theoretical predictions for pure liquids at concentration below critical wetting concentration and is in agreement with the theoretical predictions at concentrations above it.     

凹槽状PDMS基底上水滴的挥发及Cassie-Wenzel转变行为

通过在线跟踪水滴在凹槽状聚二甲基硅氧烷(PDMS)基底上的挥发行为, 研究了蒸馏水的挥发规律Cassie-Wenzel转变行为. 结果表明, 初始阶段, 水滴处于Cassie状态, 且在垂直于凹槽方向(V)和平行于凹槽方向(P)上存在显著的各向异性. 水滴的挥发过程依次表现出接触直径不变模式、 接触角不变模式及共同减小模式, 与平滑基底上水滴的挥发规律类似. 在挥发过程中, 发生了Cassie-Wenzel转变, 转变发生的时间与PDMS基底上突起部分的面积分数(即固相率)呈现良好的线性关系. 随着挥发的进行, 水滴的各向异性在接触角不变模式阶段消失, 即挥发导致水滴从开始的椭球缺状变为球缺状.