Vibrations of microscopic mercury droplets on glass

Personnel at the Oak Ridge Y-12 Plant, Oak Ridge, Tennessee, and at the University of Tennessee, Knoxville, have measured two types of oscillations for individual, microscopic mercury droplets within a sealed glass cell under partial vacuum. Droplet dimensions were determined by measuring the displacement of a microscope stage as the edges of either the base or the maximum diameter of a pendant droplet were sighted at high magnification. Resonance frequencies were measured by varying the input frequency until droplet oscillation was observed under a microscope. One type of vibration was identified as a special type of polygonal vibration in which the droplet is bound to a solid surface. Another type is an axisymmetric vibration of a bound droplet for which an eigenvalue notation has previously been derived. For 12 microscopic mercury droplets, each less than 9 nl in volume, the agreement between measured and calculated frequencies was within ±4.5%, but ±1% was common. This excellent agreement is consistent with assumptions that the surface tension corresponded to that of pure mercury, that these microscopic droplets were spherical sections, and that these droplets had fixed contact lines. Alternative analytical applications of resonance-frequency measurements on bound droplets are also described.     

Low-frequency dilational elasticity of the nematic 4'-pentyl-4-biphenylcarbonitrile (5CB)/water interface

Axisymmetric oscillating pendant drop shape analysis has been used to study the interfacial rheology of the liquid crystal 4'-pentyl-4-biphenylcarbonitrile (5CB) in water with homeotropic anchoring. Nearly spherical 5CB droplets were subjected to low frequency (1-5 mHz) volume oscillations, and the increase in tension with surface dilation was used to calculate the complex modulus. The droplet interface response is completely elastic, with no relaxations occurring on the experimental time scale. This surprising result is attributed to droplet storage of elastic energy in the form of distorted orientational distributions within the bulk (Frank elasticity) and on the surface (anchoring elasticity).     

Line tension effects for liquid droplets on circular surface domains

We study the morphologies of single liquid droplets wetting a substrate in the presence of the line tension of the three-phase contact line. On a homogeneous substrate, the line tension leads to a discontinuous unbinding of the droplet if its volume is decreased below a critical value. For a droplet wetting a structured surface with a circular domain, a line tension contrast gives rise to discontinuous depinning transitions of the contact line from the domain boundary as the droplet volume is varied. We calculate the corresponding free energy bifurcation diagram analytically for axisymmetric droplet shapes. Numerical minimization of the droplet free energy shows that line tension contrasts can stabilize nonaxisymmetric droplet shapes, thus modifying the bifurcation diagram. These latter shapes should be accessible to experiments and can be used to reveal the presence of a line tension contrast.     

Applying Tikhonov regularization to process pendant droplet tensiometry data

The problem of obtaining the first and second derivatives of the profile of a pendant droplet is formulated as an integral equation of the first kind. This equation is solved by Tikhonov regularization in which the method of general cross validation is used to guide the selection of the regularization parameter. These derivatives are converted into mean curvature as a function of droplet height. Surface tension is then obtained by regression computation between the mean curvature and two possible algebraic expressions suggested by the Laplace-Young equation. This way of obtaining surface tension is demonstrated by applying it to a number of published droplet profiles. Some of the problems encountered are discussed and solutions suggested.     

Surface tension of different sized single-component droplets,according to macroscopic data obtained using the lattice gas model and the critical droplet size during phase formation

Size dependences of the surface tension of spherical single-component droplets are calculated using equations of the lattice gas model for 19 compounds. Parameters of the model are found from experimental data on the surface tension of these compounds for a macroscopic planar surface. The chosen low-molecular compounds satisfy the law of corresponding states. To improve agreement with the experimental data, Lennard-Jones potential parameters are varied within 10% deviations. The surface tensions of different sized equilibrium droplets are calculated at elevated and lowered temperatures. It is found that the surface tension of droplets grows monotonically as the droplet size increases from zero to its bulk value. The droplet size R ₀ corresponding to zero surface tension corresponds to the critical size of the emergence of a new phase. The critical droplet sizes in the new phase of the considered compounds are estimated for the first time.     

Pillar-induced droplet merging in microfluidic circuits

A novel method is presented for controllably merging aqueous microdroplets within segmented flow microfluidic devices. Our approach involves exploiting the difference in hydrodynamic resistance of the continuous phase and the surface tension of the discrete phase through the use of passive structures contained within a microfluidic channel. Rows of pillars separated by distances smaller than the representative droplet dimension are installed within the fluidic network and define passive merging elements or chambers. Initial experiments demonstrate that such a merging element can controllably adjust the distance between adjacent droplets. In a typical scenario, a droplet will enter the chamber, slow down and stop. It will wait and then merge with the succeeding droplets until the surface tension is overwhelmed by the hydraulic pressure. We show that such a merging process is independent of the inter-droplet separation but rather dependent on the droplet size. Moreover, the number of droplets that can be merged at any time is also dependent on the mass flow rate and volume ratio between the droplets and the merging chamber. Finally, we note that the merging of droplet interfaces occurs within both compressing and the decompressing regimes.     

Chemistry of Interfaces,M. J. Jaycock and G. D. Parfitt. Halstead Press,John Wiley and Sons,New York, 1981. 279 pp. $90. Paul Becher,Wilmington

A unique physical model is proposed for relating the dimensions and properties of droplets in aqueous diesel fuel invert mlcroemulsions to the measured water vapor pressures over such systems. The model assumes discrete droplets containing surfactant-sheathed liquid cores. A dynamic equilibrium condition is visualized wherein a closed mass transfer cycle e3tists, involving the movement of water molecules from the droplet interior, through the surfactant sheath into the continuous medium and vapor space above the pool. The flat-surface fugacity of the liquid water in the aqueous core would be reduced relative to that of normal water because of Increased intermolecular association stemming from high pressure in the aqueous core caused by surface tension forces. The possible presence of dissolved surfactant constituents would reduce this fugacity even further. The mass transfer cycle is assumed to be completed by the absorption of water vapor into transitory, flat surfaces of reduced fugacity, droplet core water exposed by collapsing droplets at the pool surface. These are assumed to be continually reforming into submerged microemulsion droplets as additional droplets collapse at the pool surface.

Analytical relationships based upon the described model allowed calculation of droplet core and sheath dimensions and droplet external interfacial tension. The efficacy of the proposed model is supported by the congruity of the thus derived values.     

Relating chromatographic retention and electrophoretic mobility to the ion distribution within electrosprayed droplets

Ions that are observed in a mass spectrum obtained with electrospray mass spectrometry can be assumed to originate preferentially from ions that have a high distribution to the surface of the charged droplets. In this study, a relation between chromatographic retention and electrophoretic mobility to the ion distribution (derived from measured signal intensities in mass spectra and electrospray current) within electrosprayed droplets for a series of tetraalkylammonium ions, ranging from tetramethyl to tetrapentyl, is presented. Chromatographic retention in a reversed-phase system was taken as a measure of the analyte’s surface activity, which was found to have a large influence on the ion distribution within electrosprayed droplets. In addition, different transport mechanisms such as electrophoretic migration and diffusion can influence the surface partitioning coefficient. The viscosity of the solvent system is affected by the methanol content and will influence both diffusion and ion mobility. However, as diffusion and ion mobility are proportional to each other, we have, in this study, chosen to focus on the ion mobility parameter. It was found that the influence of ion mobility relative to surface activity on the droplet surface partitioning of analyte ions decreases with increasing methanol content. This effect is most probably coupled to the decrease in droplet size caused by the decreased surface tension at increasing methanol content. The same observation was made upon increasing the ionic strength of the solvent system, which is also known to give rise to a decreased initial droplet size. The observed effect of ionic strength on the droplet surface partitioning of analyte ions could also be explained by the fact that at higher ionic strength, a larger number of ions are initially closer to the droplet surface and, thus, the contribution of ionic transport from the bulk liquid to the liquid/air surface interface (jet and droplet surface), attributable to migration or diffusion will decrease.     

Molecular dynamics study of structural and thermodynamic characteristics of nanodroplets of simple fluid

Equilibrium configurations of Lennard-Jones nanodroplets composed of 10–15000 spherically symmetric molecules placed in the center of a spherical container are studied at constant temperature by the molecular dynamics method. The distribution of local density is found and size dependences of density in the center of droplet, first coordination number, and energy surface tension coinciding for equimolecular dividing surface with specific excess free energy of droplet are studied. Radial distribution function is also determined. It is established that the passage of structural characteristics to their macroscopic values is observed for droplets containing as little as about 300 molecules, while, for energy surface tension, analogous passage for energy surface tension occurs for droplets containing 700–6000 molecules.     

Systematic investigation of droplet generation at T-junctions

Droplet microfluidics has attracted much attention in recent years. For many droplet-based applications, researchers want to predict the size of the droplets in a certain experimental condition. To meet this need, van Steijn and colleagues proposed an elegant theoretical model that predicts the volume of droplets generated in a common channel configuration for forming a steady-state, continuous stream of droplets, the T-junction geometry. To determine the accuracy of this model in predicting droplet volume, we performed a systematic experimental study over two orders of magnitude in capillary number. We found that this model, albeit elegant, has a limited range of interfacial tension over which it can predict accurately the droplet volume. Our experimental results, together with fluid dynamic simulations, allowed us to highlight the importance of physical fluid properties when employing theoretical models.     

Modeling oil droplets in plateau borders

It is widely known that oil droplets can decrease the stability of aqueous films and foams. While less widely recognized, it has also been observed that oil droplets can, under certain circumstances, increase the stability of foams, especially if they are caught in the Plateau borders. In this paper, how the oil droplet deforms and is, in turn, deformed by the Plateau border is modeled using Surface Evolver. The two dimensionless parameters that affect these shapes are the size of the oil droplet relative to the Plateau border and the ratio of the oil-water interfacial tension to the air-water interfacial tension. The calculated pressures in all the phases were used to obtain the pressure exerted on the oil-water-air pseudoemulsion film, which allows the factors that influence the stability of these droplets in the Plateau border to be investigated. The final section of the paper demonstrates that the presence of an oil droplet in a Plateau border can have a major influence on the drainage of the aqueous phase along the Plateau border. This retardation of the flow would result in the oil droplets in the Plateau borders increasing the stability of foams in which they are found.     

Influence of drop volume on surface tension evaluated using the pendant drop method

The pendant drop technique is one of the most accurate methods to measure surface tension of liquids. Recently, it has been found that the value of the surface/interfacial tension found using the pendant drop method might be drop-volume dependent. In this work, the surface tension of glycerol at a 25 °C and the surface tension of polypropylene at 240 °C were measured using the pendant drop method for different drop volumes. It was shown that the values of the surface tension depend on the drop size if no calibration to take into account the anisotropy in the optical enlargement is performed. However, when a calibration procedure for optical anisotropy correction is performed, the values of the surface tension obtained do not depend on the volume drop size and they corroborate the values in the literature.     

Mass transfer from an oscillating microsphere

The enhancement of mass transfer from single oscillating aerocolloidal droplets having initial diameters approximately 40 microm has been measured using electrodynamic levitation to trap and oscillate a droplet evaporating in nitrogen gas. The frequency and amplitude of the oscillation were controlled by means of ac and dc fields applied to the ring electrodes of the electrodynamic balance (EDB). Elastic light scattering was used to size the droplet. It is shown that the mass transfer process for a colloidal or aerocolloidal particle oscillating in the Stokes flow regime is governed by a Peclet number for oscillation and a dimensionless oscillation parameter that represents the ratio of the diffusion time scale to the oscillation time scale. Evaporation rates are reported for stably oscillating droplets that are as much as five times the rate for evaporation in a stagnant gas. The enhancement is substantially larger than that predicted by quasi-steady-flow mass transfer.     

Deformation of water droplets on solid surface in electric field

The purpose of this paper is to analyze the deformation of water droplets on a solid surface under electric stress. A mathematical model making it possible to simulate the axisymmetric as well as non-axisymmetric deformations of droplets is developed. According to this model, the droplet deformation depends on several parameters such as the volume and the number of droplets, the conductivity and the permittivity of droplets, their proximity to one another, the surface of the solid material, and the location of each droplet on the dielectric surface. The results of the simulation show the disturbance of the background field through the presence of a single or multiple droplets. An experimental study is also achieved by considering one to three droplets aligned simultaneously on a dielectric smooth surface between two electrodes subjected to AC voltages. The influence of the background field and the droplet location regarding the electrodes on the deformation of water droplets are evidenced.     

Droplets wetting on filament rails: Surface energy and morphology transition

The morphology of liquid droplets wetting on filaments depends on the filament configuration, droplet volume, and contact angle. A stable morphology is the one that minimizes the potential energy of the droplet–filament system, while morphology transition may happen when an intermediate state exists which corresponds to a higher potential energy. This paper aims to explore such morphology transition of droplet wetting on filament rails made of two parallel identical microfilaments. Detailed numerical simulations were performed to extract the surface energy of the droplet–filament system at varying filament spacings, droplet volumes, and contact angles. Critical conditions of the morphology transition between two symmetrical wetting morphologies (i.e., liquid droplet bridge and barrel-shaped droplet) were determined. A family of characteristic curves in terms of the dimensionless droplet volume vs the filament spacing at varying contact angles was obtained, which can be used as a universal law to govern the morphology transition for such droplet–filament rail systems. The results and concepts presented in this work can be extended to broad wetting systems and utilized for the analysis and design of microfluidic devices and testers based on droplet–filament systems.     

Influence of dynamic interfacial tension on droplet formation during membrane emulsification

Membrane emulsification is a promising and relatively new technique for producing emulsions. The purpose of this study was to better understand the influence of interfacial tension on droplet formation during membrane emulsification. Droplet formation experiments were carried out with a microengineered membrane; the droplet diameter and droplet formation time were studied as a function of the surfactant concentration in the continuous phase. These experiments confirm that the interfacial tension influences the process of droplet formation; higher surfactant concentrations lead to smaller droplets and shorter droplet formation times (until 10 ms). From drop volume tensiometer experiments we can predict the interfacial tension during droplet formation. However, the strong influence of the rate of flow of the to-be-dispersed phase on the droplet size cannot be explained by the predicted values. This large influence of the oil rate of flow is clarified by the hypothesis that snap-off is rather slow in the studied regime of very fast droplet formation.     

Interfacial effects in the spreading kinetics of liquid droplets on solid substrates

Several theories deal with the spreading kinetics of liquids on solid substrate, most of which relate the rate of spreading to the surface tension and the viscosity of the liquid. Measurements of the spreading of a number of liquids exhibiting a wide range of surface tension and viscosity on dry soda-lime glass have been carried out to validate the proposed models. The measurements used a small droplet of constant volume to minimize gravitational effects. The contact radius was acquired as a function of time by an image analysis system. It was noted that power law theories describe the spreading rate for silicone oil on glass. However, significant departures were noted in the case of other liquids. Mechanistic considerations of our data suggest that equal volume droplets of similar surface tension and of diverse viscosity spread to the same area but at different rates. On the other hand, the spreading rate of glycerine, which exhibits incomplete spreading on glass, and that of silicone oil, with comparable viscosity behave similarly. These observations seemingly support the view that surface tension acts to retain the spherical shape of the droplet, whereas the difference between the solid-liquid and solid-vapor interfacial energies acts to enlarge the contact area. In the meantime, viscous dissipation acts to retard the spreading rate, past a constant rate regime.     

Delayed coalescence behavior of droplets with completely miscible liquids

Because of capillary forces, sessile droplets usually fuse instantaneously after contact. We find however a delay of the droplet fusion by many seconds if the droplets consist of different but completely miscible liquids. After the initial contact, the main bodies of the droplets remain separated, connected only through a shallow conduit with a flow from the low to the high surface tension liquid. Sporadically, this connecting film can thicken with turbulent or pulsating flows. The droplets will finally fuse when the flow has sufficiently reduced the difference in composition and surface tension. We present calculations which explain this delayed droplet fusion with the compensation of the fusion-promoting capillary pressure by a droplet-separating dynamic pressure caused by the flow between the droplets. Droplets with high contact angles fuse instantaneously. In this case, no separation-stabilizing dynamic pressure can build up because the interdroplet flow becomes turbulent.     

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.