Drop Interface Research Articles

A triple emulsion droplet in an extensional flow

In this theoretical report we analytically explore the deformation and breakup of an initially spherical triple emulsion drop undergoing an axisymmetric extensional creeping flow. The triple emulsion, suspended in an external fluid (fluid 1), is originally made from an inner spherical drop (fluid 4), engulfed by a spherical shell (fluid 3), which is subsequently engulfed by another spherical shell (fluid 2). The problem is described by eight dimensionless parameters: the external capillary number (Ca), three viscosity ratios (λ21, λ32, λ43), two radii ratios (K, k), and two surface tensions ratios (M, m). When the triple emulsion is subjected to an external uniaxial extensional flow (Ca > 0), the outer interface (21) and the inner drop interface (43) deform into prolate spheroids, while the interface between them (32) deforms into an oblate spheroid. The reverse occurs for an external biaxial flow (Ca < 0), the outer interface (21) and the inner drop interface (43) deform into oblate spheroids and the middle interface (32) becomes a prolate spheroid. Three types of breakup mechanisms are to be expected: when the outer and the middle interfaces are in contact, when the middle and the inner interfaces are in contact, and when the inner drop breaks while the two shell phases remain continuous. Finally, an analytical expression for the effective viscosity of a dilute emulsion containing spherical triple emulsion droplets is suggested. For a triple emulsion which behaves like a solid: a very viscous or a very thin outer shell (λ21→∞ or K→ 1), the effective viscosity reduces to Einstein’s formula, and when the triple emulsion behaves like a single drop: a very thick outer shell such that the inner shell and the inner drop disappear (K→ 0), Taylor’s expression is recovered.

Marangoni instability in oblate droplets suspended on a circular frame

We study theoretically internal flows in a small oblate droplet suspended on the circular frame. Marangoni convection arises due to a vertical temperature gradient across the drop and is driven by the surface tension variations at the free drop interface. Using the analytical basis for the solutions of Stokes equation in coordinates of oblate spheroid, we have derived the linearly independent stationary solutions for Marangoni convection in terms of Stokes stream functions. The numerical simulations of the thermocapillary motion in the drops are used to study the onset of the stationary regime. Both analytical and numerical calculations predict the axially symmetric circulatory convection motion in the drop, the dynamics of which is determined by the magnitude of the temperature gradient across the drop. The analytical solutions for the critical temperature distribution and velocity fields are obtained for the large temperature gradients across the oblate drop. These solutions reveal the lateral separation of the critical and stationary motions within the drops. The critical vortices are localized near the central part of a drop, while the intensive stationary flow is located closer to its butt end. A crossover to the limit of the plane film is studied within the formalism of the stream functions by reducing the droplet ellipticity ratio to zero value. The initial stationary regime for the strongly oblate drops becomes unstable relative to the many-vortex perturbations in analogy with the plane fluid films with free boundaries.

Mass transfer and size control of partially miscible fluid drops in a flow focusing microfluidic device

We conducted laboratory experiments and numerical simulations to investigate the formation and evolution of drops formed by partially miscible two-phase fluid, n-butanol (the continuous phase) and water (the dispersed phase), in a flow focusing microfluidic system. We carefully calibrated the numerical model to obtain good agreement with experimental data in drop velocity and mass transfer, demonstrating the model's capability to capture realistic drop dynamics. Our detailed investigation of the numerical results allowed us to determine the mechanism of drop formation and obtain a relevant criterion in terms of the disperse-to-continuous flow ratio beyond which the tubing patterns would occur. Additionally, we found that the mass transfer between the two phases, specifically at the drop interface, strongly depends on the local distribution of dissolved concentration of the dispersed phase. To enhance mass transfer, we conducted numerical simulations on alternating curved channels, which allows for the lateral advection of the dispersed phase concentration in the continuous phase at the curved section. We found that this lateral movement enhances mass transfer at the drop interface. Through detailed investigation of numerical results, we addressed mechanisms of mass transfer enhancement in the curved channel. Overall, our findings provide insight into the mechanisms of drop formation and mass transfer in partially miscible two-phase fluids in microfluidic systems, which could be useful in designing and optimizing such systems for various applications.

Circulating Marangoni flows within droplets in smectic films.

We present a theoretical study and numerical simulation of Marangoni convection within ellipsoidal isotropic droplets embedded in free-standing smectic films (FSSFs). The thermocapillary flows are analyzed for both isotropic droplets spontaneously formed in FSSF overheated above the bulk smectic-isotropic transition and oil lenses deposited on the surface of the smectic film. The realistic model for which the upper drop interface is free from the smectic layers, while at the lower drop surface the smectic layering persists is considered in detail. For isotropic droplets and oil lenses this leads effectively to a sticking of fluid motion at the border with a smectic shell. The above mentioned asymmetric configuration is realized experimentally when the temperature of the upper side of the film is higher than at the lower one. The full set of stationary solutions for Stokes stream functions describing the Marangoni convection flows within the ellipsoidal drops are derived analytically. The temperature distribution in the ellipsoidal drop and the surrounding air is determined in the frame of the perturbation theory. As a result, the analytical solutions for the stationary thermocapillary convection are obtained for different droplet ellipticity ratios and the heat conductivity of the liquid crystal and air. In parallel, the numerical hydrodynamic calculations of the thermocapillary motion in drops are made. Both analytical and numerical simulations predict the axially symmetric circulatory convection motion determined by the Marangoni effect at the droplet-free surface. Due to a curvature of the drop interface a temperature gradient along its free surface always exists. Thus, the thermocapillary convection within the ellipsoidal droplets in overheated FSSF is possible for the arbitrarily small Marangoni numbers. Possible experimental observations enabling the checking of our predictions are proposed.

Interfacial Dilational Rheology of Molecular Films in DC Electric Fields.

We present a novel technique to measure the interfacial dilational rheology of a molecular film adsorbed at a water-oil interface in a direct-current (dc) electric field. A film of a highly polar subfraction of asphaltenes was allowed to adsorb at a water drop interface, surrounded by an organic phase and subjected to a dc electric field. The measurements involved calculations of the dynamic interfacial tension (IFT), while the drop was sinusoidally oscillated, using our in-house axisymmetric drop shape analysis (ADSA) algorithm adapted for electric fields. The amplitude of the IFT waveform over equilibrium IFT and the phase difference from the applied area oscillations were used in the estimation of surface moduli. The asphaltene films were found to become more elastic on increasing bulk concentrations and electric field strengths. However, the effect was not monotonous and observed to be governed by combinations of these parameters. The Lucassen-van den Tempel (LVDT) model was used to further elucidate the experimentally obtained interfacial dilational moduli.

Influence of Surfactant Concentration on Spontaneous Emulsification Kinetics

The kinetics of spontaneous emulsification is investigated on aqueous pendant drops in paraffin oil. Optical microscopy in transmission mode is used for high-spatial-resolution image recording. The influence of a lipophilic surfactant (Span 80) and two water-soluble surfactants (CTAB and SDS) is investigated. As time runs, the drop interface turns opaque due to the formation of microstructures associated with spontaneous emulsification. The time evolution of this phenomenon is shown to depend upon temperature and surfactant concentration, which leads to an overall shrinkage due to gradual water uptake and transport into paraffin oil. Spontaneous emulsification kinetics depends upon the chemical composition. Higher concentrations of Span 80 and CTAB (resp. SDS) are shown to promote (resp. hinder) water transport. This work provides new insights into the understanding of spontaneous emulsification when combining the properties of non-ionic and ionic surfactants.

A model for the electric field-driven flow and deformation of a drop or vesicle in strong electrolyte solutions

A model is constructed to describe the flow field and arbitrary deformation of a drop or vesicle that contains and is embedded in an electrolyte solution, where the flow and deformation are caused by an applied electric field. The applied field produces an electrokinetic flow, which is set up on the charge-up time scale $\tau _{*c}=\lambda _{*} a_{*}/D_{*}$ , where $\lambda _{*}$ is the Debye screening length, $a_{*}$ is the inclusion length scale and $D_{*}$ is an ion diffusion constant. The model is based on the Poisson–Nernst–Planck and Stokes equations. These are reduced or simplified by forming the limit of strong electrolytes, for which dissolved salts are completely ionised in solution, together with the limit of thin Debye layers. Debye layers of opposite polarity form on either side of the drop interface or vesicle membrane, together forming an electrical double layer. Two formulations of the model are given. One utilises an integral equation for the velocity field on the interface or membrane surface together with a pair of integral equations for the electrostatic potential on the outer faces of the double layer. The other utilises a form of the stress-balance boundary condition that incorporates the double layer structure into relations between the dependent variables on the layers’ outer faces. This constitutes an interfacial boundary condition that drives an otherwise unforced Stokes flow outside the double layer. For both formulations relations derived from the transport of ions in each Debye layer give additional boundary conditions for the potential and ion concentrations outside the double layer.

Transport Behavior of Commercial Anticancer Drug Protein-Bound Paclitaxel (Paclicad) in a Micron-Sized Channel.

Protein-bound paclitaxel has been developed clinically as one of the most successful chemotherapy drugs for the treatment of a wide variety of cancers. However, these medications, due to their nanoscale properties, may often induce capillary blocking while migrating through minute blood vessels. Considering the detrimental impact of this restriction, we investigated the transport of protein-bound paclitaxel, Paclicad, in a 7 μm microchannel mimicking the identical mechanical confinement of the blood capillaries. The drug was reported to migrate through a constricted microchannel without obstruction at a solution flow rate of 20-50 μL/h. The onset of an agglomeration site was observed at higher flow rates of 70-90 μL/h, while complete capillary obstruction was observed at 100 μL/h. The mobility of the particles was also calculated, and the results suggested that the presence of varying cross-sections affects the mobility of the drug particles. The trajectory of the particle migration was observed to be less tortuous at the higher flow rate, but the tortuous nature appeared to increase with the presence of agglomeration sites in the flow field. The experimental results were also compared with the computational model of the drug particle. The drug particle was modeled both as Newtonian and as an FENE-P viscoelastic drop. The drop interface tracking was done by the VOF method using the open source software Basilisk. The particle displacement was better estimated by both the FENE-P and Newtonian model at a flow rate of 30 μL/h, while deviation was observed at a flow rate of 50 μL/h. The FENE-P model was observed to show higher deformation than the Newtonian model at both flow rates. The experimental results provided better insight into the agglomeration tendency of Paclicad, migrating through a constricted microchannel at higher flow rates. The numerical model could be further employed to understand the more complex intravenous transport of drugs.

Interface topology and evolution of particle patterns on deformable drops in turbulence

The capture of neutrally buoyant, sub-Kolmogorov particles at the interface of deformable drops in turbulent flow and the subsequent evolution of particle surface distribution are investigated. Direct numerical simulation of turbulence, phase-field modelling of the drop interface dynamics and Lagrangian particle tracking are used. Particle distribution is obtained considering excluded-volume interactions, i.e. by enforcing particle collisions. Particles are initially dispersed in the carrier flow and are driven in time towards the surface of the drops by jet-like turbulent fluid motions. Once captured by the interfacial forces, particles disperse on the surface. Excluded-volume interactions bring particles into long-term trapping regions where the average surface velocity divergence sampled by the particles is zero. These regions correlate well with portions of the interface characterized by higher-than-mean curvature, indicating that modifications of the surface tension induced by the presence of very small particles will be stronger in the highly convex regions of the interface.

Lattice Boltzmann simulation of drop splitting in a fractal tree-like microchannel

Fractal tree-like microchannel is advantageous to faster emulsification and droplet production in microchannels. Although computer simulation is becoming a powerful tool for design and optimization, precise treatment of a large number of walls in fractal tree-like microchannel is troublesome. The numerical roughness caused by traditional bounce-back boundary condition may accumulate and generate large errors in flow pattern recognization. In this paper, an approach integrating Immersed Boundary Method (IBM), Phase-Field Model (PFM) and Lattice Boltzmann Method (LBM) is developed, aiming to accurately simulate droplet splitting in fractal tree-like microchannel system involving surfactants and wetting boundary walls. Then the operating conditions including capillary number, flow rate ratio and wetting boundary conditions were optimized. We found that the capillary number in 0.02 ∼ 0.05 and flow rate ratio in 1/3.6 – 1/9 can accelerate emulsification. Hydrophobic-lipophilic walls generate a slug-like water drop of bullet shape and thin oil film between the drop and walls, facilitating drop movement in microchannel. The effects of surfactant, Marangoni stress and interfacial tension force on droplet splitting were investigated. The contact walls at the corner and forks resist the surfactant migration, and therefore enrich surfactant at the upstream end of drop interface, leading to uneven distribution of interfacial tension. The breaking interfaces and the interfaces contacting with wall or fork are subjected to opposite reaction of Marangoni stress and tend to be thinner and finally deform or break.

Diffusion-limited osmotic swelling of droplets

Swelling phenomena due to permeation appear in problems, such as the swelling of hydrogels and water-in-oil-in-water (W/O/W) emulsions. In the osmotic swelling of W/O/W emulsions driven by an inner salt concentration, diffusive effects inside the drop can decrease its expansion rate considerably. Although these inner-diffusion effects can play a large role on hindering drop swelling, they have not usually been taken into account in most works concerning the swelling kinetics of W/O/W emulsions. We perform numerical simulations of the expansion-diffusion problem governing the diffusion inside an expanding spherical droplet containing salt and with a semi-permeable interface. We also present asymptotic solutions for the limiting cases of slow and fast diffusion, which we compare with our numerical results. The results indicate that diffusive resistance significantly reduces the swelling kinetics of droplets. Moreover, in the regime of large Péclet numbers, diffusive effects are localized near the drop's interface in a concentration boundary layer, as predicted by our theory. The numerical results presented in this paper are in agreement with the behavior observed in recent experiments on W/O/W emulsion swelling.

Molecular dynamics simulations of nanoparticle-laden drop–interface electrocoalescence behaviors under direct and alternating current electric fields

Electrocoalescence is an energy-efficient and environmentally friendly process for separating water-in-oil emulsions. In this study, nanoparticle-laden drop–interface electrocoalescence behaviors under direct current (DC) and alternating current (AC) electric fields were numerically investigated using molecular dynamics methods. Good qualitative agreement was obtained between numerical results and experimental validation work. In this report, the influences of the electric field strength, nanoparticle concentration, and electric field frequency on the electrocoalescence process are systematically examined, analyzed, and discussed from the perspective of intermolecular interactions. The coupling of the hydration effect of ions and the strong interactions between silicon dioxide (SiO2) nanoparticles and water molecules was found to give rise to partial drop–interface electrocoalescence. The critical cone angle at which a transition from complete to partial electrocoalescence occurs was found to be 34.41°. A larger water intermolecular distance and a greater tendency for partial coalescence was observed in the nanoparticle-laden (NP-laden) droplets than in a pure water drop. Compared to a DC electric field, an AC electric field tended to cause complete drop–interface electrocoalescence, and the efficiency of complete coalescence was found to increase with increasing frequencies ranging from 10 to 200 GHz (GHz). The results of this work will be potentially useful for optimizing the design of compact and efficient oil–water separators.

Dynamics of a viscous drop under an oscillatory uniaxial extensional Stokes flow

We quantify the transient deformation and breakup of a neutrally-buoyant drop with viscosity λμ∗ immersed in another Newtonian fluid with viscosity μ∗ undergoing oscillatory uniaxial extension at zero Reynolds number. The interfacial tension acting between drop phase and medium phase is γ∗. The drop is initially a sphere of radius a∗. Since the external flow oscillates harmonically with a frequency ω∗, the strength of the imposed flow is characterized by an instantaneous capillary number, Ca=Ca0cos(Det), where Ca0=μ∗ε̇a∗/γ∗ and De=ω∗μ∗a∗/γ∗ is the dimensionless frequency, or Deborah number. Here, ε̇ is the rate of extension in the imposed flow. We utilize boundary-integral computations to calculate the evolution of drop interface as a function of Ca0 and De, focusing primarily on the case where the drop and surrounding fluid have equal viscosities. The computations suggest two families of behavior for the drop deformation. First, below a critical Deborah number (which we determine to be in the interval 0.375<De<1.0), the drop breaks up in a finite time at a critical capillary number that is a function of De. At sufficiently small De the critical capillary number increases linearly with De and the breakup mode is that of “center-pinching”. On increasing De the break-up mode switches to “end-pinching”, and on further increasing De it appears that the critical capillary number diverges at a critical Deborah number between 0.375 and 1.0. This divergence signals the transition to the second family of behavior where the drop attains a long-time periodic state, or alternance, regardless of Ca0. However, at large Ca0 the drop dynamics exhibits a two time-scale behavior: the drop deforms instantaneously at a fast capillary relaxation time-scale τ∗=μ∗a∗/γ∗, whereas the maximum deformation attained during a cycle of the imposed flow grows at the slow time-scale τ∗Ca0=ε̇(μ∗a∗/γ∗)2. As such, the state of alternance is approached exceedingly slowly at large Ca0. Lastly, we perform calculations for different viscosity ratios and find the drop dynamics to be in qualitative agreement with above observations.

Self-Propulsion of a Volatile Drop on the Surface of an Immiscible Liquid Bath.

We demonstrate the self-propulsion of a volatile drop on the surface of a bath of an immiscible liquid. Evaporative heat pumping is converted into directed motion through thermocapillary stresses, which arise from the coupling between surface-tension-driven flows and temperature advection. A propulsive force arises from convection-sustained temperature gradients along the drop interface, resulting in a warmer pool of liquid being advected by the hydrodynamic flow in the underlying bath toward the back of the drop. The dependence of the drop speed on the activity source, i.e., the evaporation flux, is derived with scaling arguments and captures the experimental data.

Motion of generated dumbbell-shaped satellite droplets during liquid dielectrophoresis

Actuation of sub-millimeter sized droplets has important implications in heat transfer, microfluidics, and self-cleaning surfaces. Here, we report the motion of satellite droplets generated around a primary droplet actuated using liquid dielectrophoresis (L-DEP). Satellite droplets spanning the electrode gap grow due to the electric field induced enhanced merging of droplets. These satellite droplets attain a dumbbell shape due to the applied field. Interestingly, satellite droplets follow the primary drop interface maintaining a constant gap between themselves and the primary drop. The motion of satellite droplets was captured using high-speed imaging, and the role of temperature gradient was verified by measuring temperature distribution using an IR camera. Through observations and simulations, we qualitatively believe that this behavior arises from (a) repulsive electrostatic and temperature gradient forces; and (b) attractive vapor concentration gradient forces. Simulations were used to estimate the direction of individual forces. Interfacial charges from the applied electric field give rise to the electrostatic repulsion. During L-DEP, the primary droplet heats due to induced current. Thermal conduction through the substrate sets up a temperate gradient, which adds to the repulsion forces between the droplets. Evaporation of the primary droplet leads to the vapor concentration gradient, responsible for the attraction force between the droplets.

Water-in-Water Emulsions, Ultralow Interfacial Tension, and Biolubrication

Water-in-Water Emulsions, Ultralow Interfacial Tension, and Biolubrication

Adsorption of nanoparticles suspended in a drop on a leaf surface of Perilla frutescens and their infiltration through stomatal pathway

Particulate matter (PM) has become a severe environmental issue, and ultrafine PM particles such as PM2.5 or PM1 can cause various complications and respiratory diseases to human beings. In particular, heavy metals contained in PM particles can contaminate edible plants; for example, plant leaves are exposed to PM particle-laden raindrops. The contaminated edible plants can injure the human health by ingestion, so a detailed understanding on the accumulation of PM particles inside edible plants is essential. In this study, we investigate the infiltration of PM particles in plant tissues with a hypothesis that ultrafine PM particles are absorbed through stomatal pathways. As an edible test plant, Perilla frutescens is selected. Drops of gold nanoparticle (AuNP) suspension are deposited on a leaf of P. frutescens to simulate the scenario where PM particle-laden raindrops fall on patulous stomata of the test plant. To examine AuNP adsorption on the P. frutescens foliar surface and diffusional AuNP absorption through stomatal apertures, we investigate three physical dynamics of AuNPs suspended in a sessile drop: sedimentation, evaporation-driven convective flow, and shrinkage of the drop interface. Quantitative information on the 3D spatial distribution of AuNPs in plant tissues was measured by X-ray imaging and two-photon excitation microscopy.

Particle capture by drops in turbulent flow

Three-phase turbulent flows are crucial in a number of practical problems involving particulate abatement, from scavenging of air pollutants by precipitation to scrubbing processes. These flows are extremely rich in physics and challenging to simulate. Through direct numerical simulations of turbulence, coupled with a phase field interface description and Lagrangian particle tracking, the capture dynamics of small solid particles by large deformable drops is examined in detail. The role of the topologically changing drop interface in connection with the local turbulence structure is highlighted, and a simple transport model for predicting capture efficiency is derived.

Taking a Cooler Breath: Moderate Hypothermia Boosts Lung Surfactant Activity

Reducing temperature to values close to lung surfactant melting point may improve the dynamic interfacial properties of the system. in this work, we have analysed the temperature-mediated enhancement of the interfacial performance of purified porcine surfactant (PS) under plasma inhibition in a Constrained Drop Surfactometer. We have also explored the temperature-related rescue of surfactant performance. To do so, the drop interface was previously covered by plasma and PS until impairing surfactant dynamic properties.

Electrohydrodynamic assembly of colloidal particles on a drop interface.

A mathematical model to simulate the dynamics of colloidal particles on a drop interface in an applied electric field is presented. The model accounts for the electric field driven flow within the drop and suspending fluid, particle-particle electrostatic interaction, and the particle motion and rotation due to the induced flow and the applied electric field. The model predicts the formation of chains in the case of conducting particles or an undulating band around the equator in the case of dielectric particles. The model results are in agreement with recent experimental work. A study is presented on the impact of particle concentration and electric field strength on the collective motions of the particles. In the case of non-conducting particles, we find that in the presence of Quincke rotation, the amplitude of the undulations of the observed equatorial particle belt increases with particle concentration but decreases with electric field strength. We also show that the wavelength of the undulations appears independent of the applied field strength.