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.     

Hydrophobic and ionic interactions in nanosized water droplets

A number of situations such as protein folding in confined spaces, lubrication in tight spaces, and chemical reactions in confined spaces require an understanding of water-mediated interactions. As an illustration of the profound effects of confinement on hydrophobic and ionic interactions, we investigate the solvation of methane and methane decorated with charges in spherically confined water droplets. Free energy profiles for a single methane molecule in droplets, ranging in diameter (D) from 1 to 4 nm, show that the droplet surfaces are strongly favorable as compared to the interior. From the temperature dependence of the free energy in D = 3 nm, we show that this effect is entropically driven. The potentials of mean force (PMFs) between two methane molecules show that the solvent separated minimum in the bulk is completely absent in confined water, independent of the droplet size since the solute particles are primarily associated with the droplet surface. The tendency of methanes with charges (M(q+) and M(q-) with q(+) = |q(-)| = 0.4e, where e is the electronic charge) to be pinned at the surface depends dramatically on the size of the water droplet. When D = 4 nm, the ions prefer the interior whereas for D < 4 nm the ions are localized at the surface, but with much less tendency than for methanes. Increasing the ion charge to e makes the surface strongly unfavorable. Reflecting the charge asymmetry of the water molecule, negative ions have a stronger preference for the surface compared to positive ions of the same charge magnitude. With increasing droplet size, the PMFs between M(q+) and M(q-) show decreasing influence of the boundary owing to the reduced tendency for surface solvation. We also show that as the solute charge density decreases the surface becomes less unfavorable. The implications of our results for the folding of proteins in confined spaces are outlined.     

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.     

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.     

Induced detachment of coalescing droplets on superhydrophobic surfaces

Coalescence of a falling droplet with a stationary sessile droplet on a superhydrophobic surface is investigated by a combined experimental and numerical study. In the experiments, the droplet diameter, the impact velocity, and the distance between the impacting droplets were controlled. The evolution of surface shape during the coalescence of two droplets on the superhydrophobic surface is captured using high speed imaging and compared with numerical results. A two-phase volume of fluid (VOF) method is used to determine the dynamics of droplet coalescence, shape evaluation, and contact line movement. The spread length of two coalesced droplets along their original center is also predicted by the model and compared well with the experimental results. The effect of different parameters such as impact velocity, center to center distance, and droplet size on contact time and restitution coefficient are studied and compared to the experimental results. Finally, the wetting and the self-cleaning properties of superhydrophobic surfaces have been investigated. It has been found that impinging water drops with very small amount of kinetic impact energy were able to thoroughly clean these surfaces.     

Sub-micrometer dropwise condensation under superheated and rarefied vapor condition

Phase change accompanying conversion of a saturated or superheated vapor in the presence of subcooled surfaces is one of the most common occurring phenomena in nature. The mode of phase change that follows such a transformation is dependent upon surface properties such as contact angle and thermodynamic conditions of the system. In present studies, an experimental approach is used to study the physics behind droplet growth on a partially wet surface. Superheated vapor at low pressures of 4-5 Torr was condensed on subcooled silicon surface with a static contact angle of 60° in the absence of noncondensable gases, and the condensation process was monitored using environmental scanning electron microscopy (ESEM) with sub-microscopic spatial resolution. The condensation process was analyzed in the form of size growth of isolated droplets before a coalescence event ended the regime of single droplet growth. Droplet growth obtained as a function of time reveals that the rate of growth decreases as the droplet increases in size. This behavior is indicative of an overall droplet growth law existing over larger time scales for which the current observations in their brief time intervals could be fitted. A theoretical model based on kinetic theory further support the experimental observations indicating a mechanism where growth occurs by interfacial mass transport directly on condensing droplet surface. Evidence was also found that establishes the presence of sub-microscopic droplets nucleating and growing between microscopic droplets for the partially wetting case.     

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.     

Phase behavior and nano-emulsion formation by the phase inversion temperature method

Formation of oil-in-water nano-emulsions has been studied in the water/C12E4/isohexadecane system by the phase inversion temperature emulsification method. Emulsification started at the corresponding hydrophilic-lipophilic balance temperature, and then the samples were quickly cooled to 25 degrees C. The influence of phase behavior on nano-emulsion droplet size and stability has been studied. Droplet size was determined by dynamic light scattering, and nano-emulsion stability was assessed, measuring the variation of droplet size as a function of time. The results obtained showed that the smallest droplet sizes were produced in samples where the emulsification started in a bicontinuous microemulsion (D) phase region or in a two-phase region consisting of a microemulsion (D) and a liquid crystalline phase (L(alpha)). Although the breakdown process of nano-emulsions could be attributed to the oil transference from the smaller to the bigger droplets, the increase in instability found with the increase in surfactant concentration may be related to the higher surfactant excess, favoring the oil micellar transport between the emulsion droplets.     

High-speed, clinical-scale microfluidic generation of stable phase-change droplets for gas embolotherapy

In this study we report on a microfluidic device and droplet formation regime capable of generating clinical-scale quantities of droplet emulsions suitable in size and functionality for in vivo therapeutics. By increasing the capillary number-based on the flow rate of the continuous outer phase-in our flow-focusing device, we examine three modes of droplet breakup: geometry-controlled, dripping, and jetting. Operation of our device in the dripping regime results in the generation of highly monodisperse liquid perfluoropentane droplets in the appropriate 3-6 μm range at rates exceeding 10(5) droplets per second. Based on experimental results relating droplet diameter and the ratio of the continuous and dispersed phase flow rates, we derive a power series equation, valid in the dripping regime, to predict droplet size, D(d) approximately equal 27(Q(C)/Q(D))(-5/12). The volatile droplets in this study are stable for weeks at room temperature yet undergo rapid liquid-to-gas phase transition, and volume expansion, above a uniform thermal activation threshold. The opportunity exists to potentiate locoregional cancer therapies such as thermal ablation and percutaneous ethanol injection using thermal or acoustic vaporization of these monodisperse phase-change droplets to intentionally occlude the vessels of a cancer.     

Droplet morphologies on particles with macroscopic surface roughness

The equilibrium configuration of liquid droplets on the surface of macroscopically rough solid particles was determined by numerical simulations using the volume-of-fluid (VOF) method. The fractional surface coverage of the particle as a function of the droplet size, equilibrium contact angle, and the particle surface roughness amplitude and correlation length has been systematically investigated. Droplet size and contact angle were found to generally have a stronger effect on surface coverage than particle surface roughness. Because of droplet coalescence, a relatively large variation in surface coverage was observed for any given total liquid volume, particularly for larger values of the equilibrium contact angle.     

Use of helium nanodroplets for assembly, transport, and surface deposition of large molecular and atomic clusters

The utility of continuous beam of helium droplets for assembly, transport, and surface deposition of metal and molecular clusters is studied. Clusters of propyne having from about 10 to 10(4) molecules were obtained via sequential pickup of molecules by He droplets with average sizes in the range of 10(4)-10(7) atoms. The maximum attainable flux of the propyne molecules carried by He droplets was found to be in the range of (5-15)x10(15) molecules sr(-1) s(-1), being larger in larger droplets. The size of the clusters and the flux of the transported species are ultimately limited by the evaporative extinction of the entire helium droplet upon capture of particles. It is shown that the attenuation of the He droplet beam in the process of the cluster growth can be used in order to obtain the average size and the binding energy of the clusters. Furthermore, we used He droplets for assembling and surface deposition of gold and silver clusters having about 500 atoms. Typical deposition rate of metal atoms of about 3 x 10(15) atoms sr(-1) s(-1) is comparable to or larger than obtained with other beam deposition techniques. We propose that doping of He droplets by Au and Ag atoms in two separate pickup chambers leads to formation of the bimetal clusters having core-shell structure.     

Monte carlo simulation of macromolecular ionization by nanoelectrospray

Electrospray ionization (ESI) is commonly used in macromolecular mass spectrometry, yet the dynamics of macromolecules in ESI droplets are not well understood. In this study, a Monte Carlo based model was developed, which can predict the efficiency of electrospray ionization for macromolecules, i.e., the number of macromolecular ions produced per macromolecules electrosprayed. The model takes into account ESI droplet evaporation, macromolecular diffusion within the droplet, droplet fissions, and the statistical nature of the ESI process. Two idealized representations of macromolecular analytes were developed, describing cluster prone, droplet surface inactive macromolecules and droplet surface active macromolecules, respectively. It was found that surface active macromolecules are preferentially ionized over surface inactive cluster prone macromolecules when the initial droplet size is large and the analyte concentration in solution is high. Simulations showed that ESI efficiency decreases with increasing initial droplet size and analyte molecular weight, and is influenced by analyte surface activity, the properties of the solvent, and the variance of the droplet size distribution. Model predictions are qualitatively supported by experimental measurements of macromolecular electrospray ionization made previously. Overall, this study demonstrates the potential capabilities of Monte Carlo based ESI models. Future developments in such models will allow for more accurate predictions of macromolecular ESI intensity.     

Droplet size scaling of water-in-oil emulsions under turbulent flow

The size of droplets in emulsions is important in many industrial, biological, and environmental systems, as it determines the stability, rheology, and area available in the emulsion for physical or chemical processes that occur at the interface. While the balance of fluid inertia and surface tension in determining droplet size under turbulent mixing in the inertial subrange has been well established, the classical scaling prediction by Shinnar half a century ago of the dependence of droplet size on the viscosity of the continuous phase in the viscous subrange has not been clearly validated in experiment. By employing extremely stable suspensions of highly viscous oils as the continuous phase and using a particle video microscope (PVM) probe and a focused beam reflectance method (FBRM) probe, we report measurements spanning 2 orders of magnitude in the continuous phase viscosity for the size of droplets in water-in-oil emulsions. The wide range in measurements allowed identification of a scaling regime of droplet size proportional to the inverse square root of the viscosity, consistent with the viscous subrange theory of Shinnar. A single curve for droplet size based on the Reynolds and Weber numbers is shown to accurately predict droplet size for a range of shear rates, mixing geometries, interfacial tensions, and viscosities. Viscous subrange control of droplet size is shown to be important for high viscous shear stresses, i.e., very high shear rates, as is desirable or found in many industrial or natural processes, or very high viscosities, as is the case in the present study.     

Stochastic diffusion interactions and coarsening in a system of droplets growing from a supersaturated gas mixture

In this work we study diffusion interactions among liquid droplets growing in stochastic population by condensation from supersaturated binary gas mixture. During the postnucleation transient regime collective growth of liquid droplets competing for the available water vapor decreases local supersaturation leading to the increase of critical radius and the onset of coarsening process. In coarsening regime the growth of larger droplets is prevailing noticeably broadening the droplet size-distribution function when the condensation process becomes more intensive than the supersaturation yield. Modifications in the kinetic equation are discussed and formulated for a stochastic population of liquid droplets when diffusional interactions among droplets become noteworthy. The kinetic equation for the droplet size-distribution function is solved together with field equations for the mass fraction of disperse liquid phase, mass fraction of water vapor component of moist air, and temperature during diffusion-dominated regime of droplet coarsening. The droplet size and mass distributions are found as functions of the liquid volume fraction, showing considerable broadening of droplet spectra. It is demonstrated that the effect of latent heat of condensation considerably changes coarsening process. The coarsening rate constant, the droplet density (number of droplets per unit volume), the screening length, the mean droplet size, and mass are determined as functions of the temperature, pressure, and liquid volume fraction.     

Droplet size effects on film drainage between droplet and substrate

When a droplet approaches a solid surface, the thin liquid film between the droplet and the surface drains until an instability forms and then ruptures. In this study, we utilize microfluidics to investigate the effects of film thickness on the time to film rupture for water droplets in a flowing continuous phase of silicone oil deposited on solid poly(dimethylsiloxane) (PDMS) surfaces. The water droplets ranged in size from millimeters to micrometers, resulting in estimated values of the film thickness at rupture ranging from 600 nm down to 6 nm. The Stefan-Reynolds equation is used to model film drainage beneath both millimeter- and micrometer-scale droplets. For millimeter-scale droplets, the experimental and analytical film rupture times agree well, whereas large differences are observed for micrometer-scale droplets. We speculate that the differences in the micrometer-scale data result from the increases in the local thin film viscosity due to confinement-induced molecular structure changes in the silicone oil. A modified Stefan-Reynolds equation is used to account for the increased thin film viscosity of the micrometer-scale droplet drainage case.     

Colloidosomes from the controlled interaction of submicrometer triglyceride droplets and hydrophilic silica nanoparticles

The self-assembly of hydrophilic silica nanoparticles at the surface of charged submicrometer triglyceride droplets has been investigated with the aim to optimize the preparation of stable colloidosomes. The droplet charge, oil phase volume fraction, droplet/nanoparticle ratio, and salt concentration play important roles in controlling nanoparticle interactions and are reflected in the colloidosome zeta potential, size, stability, and interfacial structure (visualized by freeze-fracture SEM). Silica nanoparticle interactions with negatively charged droplets are weak, and partially covered droplets are identified. Positively charged droplets are strongly coated by silica nanoparticles and undergo charge reversal at specific droplet to nanoparticle ratios and electrolyte concentrations. Droplets at volume fractions (varphi) <10 (-4) undergo time-dependent limited coalescence until nanoparticle coverage is complete. For varphi in the range 10 (-4) to 2.5 x 10 (-4) and at certain critical droplet to nanoparticle ratios, droplets undergo neutralization or charge reversal coupled with aggregation and precipitation; this occurs in a time-independent manner. Specific conditions have been identified where stable 1-3 mum colloidosomes can be phase separated from heterocoagulates of droplets and nanoparticles.     

Spontaneous size selection in cholesteric and nematic emulsions

We study the spontaneous size selection in lyotropic cholesteric (W/O) and thermotropic nematic (O/W) liquid crystal emulsions. The droplet sizes have been characterized by dynamic light scattering, which indicates a narrow monomodal distribution of droplets achieved spontaneously even without emulsion filtration. Anchoring of the director, provided by the chosen surfactant on the interface, may generate a topological defect inside the droplet. Below the critical radius R = K/W, determined by the ratio of Frank elastic and the surface anchoring constants, the effective anchoring strength is weak and droplets are not topologically charged; this allows them to coalesce freely, depleting the size distribution in this range. Large droplets possess a topological charge of +1 and present a high elastic energy barrier for pair coalescence; the resulting size distribution is skewed, with R > R, and effectively frozen.     

Monte Carlo computer simulations and electron microscopy of colloidal cluster formation via emulsion droplet evaporation

We consider a theoretical model for a binary mixture of colloidal particles and spherical emulsion droplets. The hard sphere colloids interact via additional short-ranged attraction and long-ranged repulsion. The droplet-colloid interaction is an attractive well at the droplet surface, which induces the Pickering effect. The droplet-droplet interaction is a hard-core interaction. The droplets shrink in time, which models the evaporation of the dispersed (oil) phase, and we use Monte Carlo simulations for the dynamics. In the experiments, polystyrene particles were assembled using toluene droplets as templates. The arrangement of the particles on the surface of the droplets was analyzed with cryogenic field emission scanning electron microscopy. Before evaporation of the oil, the particle distribution on the droplet surface was found to be disordered in experiments, and the simulations reproduce this effect. After complete evaporation, ordered colloidal clusters are formed that are stable against thermal fluctuations. Both in the simulations and with field emission scanning electron microscopy, we find stable packings that range from doublets, triplets, and tetrahedra to complex polyhedra of colloids. The simulated cluster structures and size distribution agree well with the experimental results. We also simulate hierarchical assembly in a mixture of tetrahedral clusters and droplets, and find supercluster structures with morphologies that are more complex than those of clusters of single particles.     

Uniform-sized silicone oil microemulsions: preparation, investigation of stability and deposition on hair surface

Emulsions are commonly used in foods, pharmaceuticals and home-personal-care products. For emulsion based products, it is highly desirable to control the droplet size distribution to improve storage stability, appearance and in-use property. We report preparation of uniform-sized silicone oil microemulsions with different droplets diameters (1.4-40.0 μm) using SPG membrane emulsification technique. These microemulsions were then added into model shampoos and conditioners to investigate the effects of size, uniformity, and storage stability on silicone oil deposition on hair surface. We observed much improved storage stability of uniform-sized microemulsions when the droplets diameter was ≤22.7 μm. The uniform-sized microemulsion of 40.0 μm was less stable but still more stable than non-uniform sized microemulsions prepared by conventional homogenizer. The results clearly indicated that uniform-sized droplets enhanced the deposition of silicone oil on hair and deposition increased with decreasing droplet size. Hair switches washed with small uniform-sized droplets had lower values of coefficient of friction compared with those washed with larger uniform and non-uniform droplets. Moreover the addition of alginate thickener in the shampoos and conditioners further enhanced the deposition of silicone oil on hair. The good correlation between silicone oil droplets stability, deposition on hair and resultant friction of hair support that droplet size and uniformity are important factors for controlling the stability and deposition property of emulsion based products such as shampoo and conditioner.