Coalescence of liquid drops by surface tension

A Review on the Coalescence of Confined Drops with a Focus on Scaling Laws for the Growth of the Liquid Bridge

The surface-tension-driven coalescence of drops has been extensively studied because of the omnipresence of the phenomenon and its significance in various natural and engineering systems. When two drops come into contact, a liquid bridge is formed between them and then grows in its lateral dimensions. As a result, the two drops merge to become a bigger drop. The growth dynamics of the bridge are governed by a balance between the driving force and the viscous and inertial resistances of involved liquids, and it is usually represented by power-law scaling relations on the temporal evolution of the bridge dimension. Such scaling laws have been well-characterized for the coalescence of unconfined or freely suspended drops. However, drops are often confined by solid or liquid surfaces and thus are a different shape from spheres, which affects their coalescence dynamics. As such, the coalescence of confined drops poses more complicated interfacial fluid dynamics challenges compared to that of unconfined drops. Although there have been several studies on the coalescence of confined drops, they have not been systematically reviewed in terms of the properties and geometry of the confining surface. Thus, we aim to review the current literature on the coalescence of confined drops in three categories: drop coalescence on a solid surface, drop coalescence on a deformable surface, and drop coalescence between two parallel surfaces with a small gap (i.e., Hele-Shaw cell), with a focus on power-law scaling relations, and to suggest challenges and outlooks for future research on the phenomena.

Dynamics of Droplet Coalescence on Hydrophobic Fibers in Oil: Morphology and Liquid Bridge Evolution

Although droplet self-jumping on hydrophobic fibers is a well-known phenomenon, the influence of viscous bulk fluids on this process is still not fully understood. In this work, two water droplets' coalescence on a single stainless-steel fiber in oil was investigated experimentally. Results showed that lowering the bulk fluid viscosity and increasing the oil-water interfacial tension promoted droplet deformation, reducing the coalescence time of each stage. While the total coalescence time was more influenced by the viscosity and under-oil contact angle than the bulk fluid density. For water droplets coalescing on hydrophobic fibers in oils, the expansion of the liquid bridge can be affected by the bulk fluid, but the expansion dynamics exhibited similar behavior. The drops begin their coalescence in an inertially limited viscous regime and transition to an inertia regime. Larger droplets did accelerate the expansion of the liquid bridge but had no obvious influence on the number of coalescence stages and coalescence time. This study can provide a more profound understanding of the mechanisms underlying the behavior of water droplet coalescence on hydrophobic surfaces in oil.

Evolution of Protein Assemblies Driven by the Switching of Interplay Mode

A protein assembly with the ability to switch interplay modes of multiple driving forces has been achieved. Although biomolecular systems driven by multiple driving forces have been exploited, work on such a protein assembly capable of switching the interplay modes at nanoscale has been rarely reported so far as a result of their great fabrication challenge. In this work, two sets of driving forces such as ligand-ligand interaction and protein-protein interaction were leveraged to antagonistically underpin the multilayered stackings and trigger the hollow evolution to afford the well-defined hollow rectangular frame of proteins. While these protein frames further collapsed into aggregates, the ligand-ligand interactions were weakened, and the interplay of two sets of driving forces thereby tended to switch into synergistic mode, converting the protein packing mode from porously loose packing to axially dense packing and thus giving rise to a morphological evolution toward a nanosized protein tube. This strategy not only provides a nanoscale understanding on the mechanism underlying the switch of interplay modes in the context of biomacromolecules but also may provide access for diverse sophisticated biomacromolecular nanostructures that are historically inaccessible for conventional self-assembly strategies.

Numerical Study of the Coalescence and Mixing of Drops of Different Polymeric Materials

In this study, we employ direct numerical simulation (DNS) to investigate the solutal hydrodynamics dictating the three-dimensional coalescence of microscopic, identical-sized sessile drops of different but miscible shear-thinning polymeric liquids (namely, PVAc or polyvinyl acetate and PMMA or polymethylmethacrylate), with the drops being in partially wetted configuration. Despite the ubiquitousness of the interaction of different dissimilar droplets in a variety of engineering problems ranging from additive manufacturing to understanding the behavior of photonic crystals, coalescence of drops composed of different polymeric and non-Newtonian materials has not been significantly explored. Interaction of such dissimilar droplets often involves simultaneous drop spreading, coalescence, and mixing. The mixing dynamics of the dissimilar drops are governed by interphase diffusion, the residual kinetic energy of the drops stemming from the fact that coalescence starts before the spreading of the drops have been completed, and the solutal Marangoni convection. We provide the three-dimensional velocity fields and velocity vectors inside the completely miscible, dissimilar coalescing droplets. Our simulations explicate the relative influence of these different effects in determining the flow field at different locations and at different time instances and the consequent mixing behavior inside the interacting drops. We also show the non-monotonic (in terms of the direction of migration) propagation of the mixing front of the miscible coalescing drops over time. We also establish that the overall mixing (on either side of the mixing front) speeds up as the Marangoni effects dictate the mixing. We anticipate that our study will provide an important foundation for studying miscible multi-material liquid systems, which will be crucial for applications such as inkjet or aerosol jet printing, lab-on-a-chip, polymer processing, etc.

Coalescence-Induced Droplet Jumping on Honeycomb Bionic Superhydrophobic Surfaces

Condensation-induced jumping of droplets on superhydrophobic surfaces has received extensive attention because of its great potential for applications in areas such as condensation enhancement and self-cleaning. However, the jumping efficiency of droplets on flat superhydrophobic surfaces is very low, and there is no reliable means of achieving efficient droplet jumping on large scales, which greatly limits its application. To this end, we developed a class of honeycomb bionic superhydrophobic surfaces (HBSS) that enable reliable and efficient droplet jumping on a large scale for the first time and performed experimental and simulation studies on droplet condensation and jumping on this kind of surface. Condensation experiments show that condensate droplets on HBSS can be effectively positioned under the influence of gravity and the uniformity of the droplet diameter is ensured, laying the foundation for achieving efficient jumping. The shape and geometric parameters of HBSS have a significant impact on the droplet jumping efficiency, and the maximum dimensionless jumping velocity of droplet jumping was experimentally measured to be 0.747, corresponding to an efficiency of about 45.25%. Combining with the results of simulation calculations, we found that the surface structure of HBSS can promote more of the excess surface energy to net upward kinetic energy along an extremely efficient and simple pathway (direct conversion), thus achieving an energy conversion efficiency of over 45%.

When mechanisms of coalescence and sintering at the nanoscale fundamentally differ: Molecular dynamics study

Employing classical isothermal molecular dynamics, we simulated coalescence of mesoscopic Au nanodroplets, containing from several thousands to several hundred thousands of atoms, and sintering of mesoscopic solid Au nanoparticles. For our atomistic simulations, we used the embedded atom method. The employed open access program large-scale atomic/molecular massively parallel simulator makes it possible to realize parallel graphical processing unit calculations. We have made a conclusion that the regularities and mechanisms of the nanodroplet coalescence (temperature is higher than the nanoparticle melting temperature) and of the solid nanoparticle sintering differ from each other. We have also concluded that the nanodroplet coalescence may be interpreted as a hydrodynamic phenomenon at the nanoscale whereas sintering of solid nanoparticles is a much more complex phenomenon related to different mechanisms, including collective rearrangements of atoms, the surface diffusion, and other types of diffusion. At the same time, collective rearrangements of atoms relate not only to the solid nanoparticle sintering but also to the nanodroplet coalescence. In general, our molecular dynamics results on sintering of Au nanoparticles consisting of 10 000–30 000 atoms agree with the Ferrando–Minnai kinetic trapping concept that was earlier confirmed in molecular dynamics experiments on Au nanoclusters consisting of about 100 atoms.

Spreading, pinching, and coalescence: the Ohnesorge units

Understanding the kinematics and dynamics of spreading, pinching, and coalescence of drops is critically important for a diverse range of applications involving spraying, printing, coating, dispensing, emulsification, and atomization. Hence experimental studies visualize and characterize the increase in size over time for drops spreading over substrates, or liquid bridges between coalescing drops, or the decrease in the radius of pinching necks during drop formation. Even for Newtonian fluids, the interplay of inertial, viscous, and capillary stresses can lead to a number of scaling laws, with three limiting self-similar cases: visco-inertial (VI), visco-capillary (VC) and inertio-capillary (IC). Though experiments are presented as examples of the methods of dimensional analysis, the lack of precise values or estimates for pre-factors, transitions, and scaling exponents presents difficulties for quantitative analysis and material characterization. In this tutorial review, we reanalyze and summarize an elaborate set of landmark published experimental studies on a wide range of Newtonian fluids. We show that moving beyond VI, VC, and IC units in favor of intrinsic timescale and lengthscale determined by all three material properties (viscosity, surface tension and density), creates a complementary system that we call the Ohnesorge units. We find that in spite of large differences in topological features, timescales, and material properties, the analysis of spreading, pinching and coalescing drops in the Ohnesorge units results in a remarkable collapse of the experimental datasets, highlighting the shared and universal features displayed in such flows.

Effects of particle size on the electrocoalescence dynamics and arrested morphology of liquid marbles

HYPOTHESIS

The coalescence of bare droplets when surface tension dominates always results in one larger spherical droplet. In contrast, droplets coated with particles may be stabilized into non-spherical structures after arrested coalescence, which can be achieved by different approaches, such as changing the particle surface coverage. The size of particles coating the initial liquid marbles can be used to control the coalescence dynamics and the resulting morphology of arrested droplets.

EXPERIMENT

We characterized the electrocoalescence of liquid marbles coated with particles ranging from hundred nanometers to hundred micrometers. The electrocoalescence was recorded using high-speed imaging.

FINDINGS

When the electrocoalescence initiates, particles jam and halt the relaxation of the marbles at different stages, resulting in four possible final morphologies that are characterized using the Gaussian curvature at the neck region. The four regimes are total coalescence, arrested puddle coalescence, arrested saddle coalescence, and non-coalescence. The coalescence is initiated at the center of the contact zone, independent of the particle size. Small particles show little resistance to the coalescence, while marbles coated by large particles demonstrate a viscous-like behavior, indicated by the growth of the liquid bridge and the damping. The present study provides guidelines for applications that involve the formulation of liquid marbles with complex morphologies.

When Elasticity Affects Drop Coalescence

The breakup and coalescence of drops are elementary topological transitions in interfacial flows. The breakup of a drop changes dramatically when polymers are added to the fluid. With the strong elongation of the polymers during the process, long threads connecting the two droplets appear prior to their eventual pinch-off. Here, we demonstrate how elasticity affects drop coalescence, the complement of the much studied drop pinch-off. We reveal the emergence of an elastic singularity, characterized by a diverging interface curvature at the point of coalescence. Intriguingly, while the polymers dictate the spatial features of coalescence, they hardly affect the temporal evolution of the bridge. These results are explained using a novel viscoelastic similarity analysis and are relevant for drops created in biofluids, coating sprays, and inkjet printing.

Coalescence of Microscopic Polymeric Drops: Effect of Drop Impact Velocities

In this paper, we employ the direct numerical simulation (DNS) method for probing three-dimensional, axisymmetric coalescence of microscale, power-law-obeying, and shear-thinning polymeric liquid drops of identical sizes impacting a solid, solvophilic substrate with a finite velocity. Unlike the cases of drop coalescence of Newtonian liquid drops, coalescence of non-Newtonian polymeric drops has received very little attention. Our study bridges this gap by providing (1) the time-dependent, three-dimensional (3D) velocity field and 3D velocity vectors inside two coalescing polymeric drops in the presence of a solid substrate and (2) the effect of the drop impact velocity (on the solid substrate), quantified by the Weber number (We), on the coalescence dynamics. Our simulations reveal that the drop coalescence is qualitatively similar for different We values, although the velocity magnitudes involved, the time required to attain different stages of coalescence, and the time needed to attain equilibrium vary drastically for finitely large We values. Finally, we provide detailed simulation-based, as well as physics-based, scaling laws describing the growth of the height and the width of the bridge (formed due to coalescence) dictating the 3D coalescence event. Our analyses reveal distinct scaling laws for the growth of bridge height and width for early and late stages of coalescence as a function of We. We also provide simulation-based coalescence results for the case of two unequal sized drops impacting on a substrate (nonaxisymmetric coalescence) as well as results for axisymmetric coalescence for drops of different rheology. We anticipate that our findings will be critical in better understanding events such as inkjet or aerosol jet polymer printing, dynamics of polymer blends, and many more.

Dynamics of water condensation on a switchable surface originated from molecular orientations

In heat transfer systems, how water condenses on the surface is critical to the energy efficiency of the system. With fixed surface wettability, hydrophilic surfaces enhance the nucleation rate but result in filmwise condensation due to pinning effect, which impedes the heat transfer between water vapor and surface during droplet growth. A hydrophilic surface with high drop mobility is realized with static tailored wettability surfaces, while tunable surfaces have potential in more comprehensive manipulation in condensation with different scale in time and scale. However, the mechanism has rarely been investigated and elucidated. In this paper, we investigate water condensation on a tunable surface originated from surface tension distribution control. The surface tension distribution under applied electric field is modeled and tested. We demonstrate that the surface tension manipulated by liquid crystal orientation alters the nucleation site density. Also, the periodic surface tension distribution aligns condensed water drops and decelerates the radius growth of droplets. The mechanism of active water condensation manipulation can be further applied to other tunable surfaces for various applications such as atmospheric water generator, heat transfer systems, and desalination systems.

Digital Microfluidics: Magnetic Transportation and Coalescence of Sessile Droplets on Hydrophobic Surfaces

Magnetic digital microfluidics is advantageous over other existing droplet manipulation methods, which exploits magnetic forces for actuation and offers the flexibility of implementation in resource-limited point-of-care applications. This article discusses the dynamic behavior of a pair of sessile droplets on a hydrophobic surface under the presence of a permanent magnetic field. A phase field method-based solver is employed in a two-dimensional computational domain to numerically capture the dynamic evolution of the droplet interfaces, which again simultaneously solves the magnetic and flow fields. On a superhydrophobic surface (i.e., θ_c = 150°), the nonuniform magnetic field forces the pair of sessile droplets to move toward each other, which eventually leads to a jumping off phenomenon of the merged droplet from the solid surface after coalescence. Also, there exists a critical magnetic Bond number Bo_m^cr, beyond which no coalescence event between droplets is observed. Moreover, on a less hydrophobic surface (θ_c ≤ 120°), the droplets still coalesce under a magnetic field, although the merged droplet does not experience any upward flight after coalescence. Also, the merging phenomenon at lower contact angle values (i.e., θ_c = 90°) appears significantly different than at higher contact angle values (i.e., θ_c = 120°). Additionally, if the pair of sessile droplets is dispersed to a different surrounding medium, the viscosity ratio plays a significant role in the upward flight of the merged droplet, where the coalesced droplet exhibits increased vertical migration at higher viscosity ratios.

Concentration Depth Profile-Based Multilayer Sorption Surface Tension Model for Aqueous Solutions

Surface tension of chemically complex aqueous droplets is significant to atmospheric aerosol particle dynamics and fate. Isotherm-based predictive surface tension models are available which consider one layer of solute molecules sorbed at the liquid-vapor interface. However, the concentration depth profile (CDP) of solute molecules near the surface is continuous, making the single monolayer assumption inappropriate. Here, this work extends the isotherm framework by dividing the surface region into multiple layers to capture the continuity of the spatial distribution of solute molecules for binary solutions. Partition functions are established based on the displacement of water molecules by solute molecules. The number of displaced water molecules and energy of solute molecules at the surface and in the bulk are key model parameters relating surface tension and solute activity. Number densities of surface molecules from molecular dynamic (MD) simulations available in the literature are applied to determine model parameters. Finally, the model is extended to predict surface tension for mixture solutions, considering both independent and dependent adsorptions of different solute species to the liquid-vapor interface. The proposed model works well for both electrolyte and nonelectrolyte solutions and their mixtures from pure solvent to pure solute.

Effect of a Superhydrophobic Surface Structure on Droplet Jumping Velocity

The coalescence-induced droplet jumping on superhydrophobic surfaces is fundamentally significant from an academic or practical viewpoint. However, approaches to enhance droplet jumping velocity are very limited. In this work, the effect of structural parameters of the triangular prism on droplet jumping is studied systematically. The results indicate that droplet jumping velocity can be greatly increased by exploiting structure effects, which is a promising reinforcement method. When the height and apex angle of the triangular prism are fixed, the droplet jumping velocity increases with the length of the triangular prism until a plateau is reached. The ratio of translational kinetic energy to released surface energy during droplet jumping is determined by the apex angle and the height of the triangular prism, which is more effective with a smaller apex angle and a larger height. The results are supposed to provide guidelines for optimization of superhydrophobic surfaces.

Single bubble and drop techniques for characterizing foams and emulsions

The physics of foams and emulsions has traditionally been studied using bulk foam/emulsion tests and single film platforms such as the Scheludko cell. Recently there has been a renewed interest in a third class of techniques that we term as single bubble/drop tests, which employ isolated whole bubbles and drops to probe the characteristics of foams and emulsions. Single bubble and drop techniques provide a convenient framework for investigating a number of important characteristics of foams and emulsions, including the rheology, stabilization mechanisms, and rupture dynamics. In this review we provide a comprehensive discussion of the various single bubble/drop platforms and the associated experimental measurement protocols including the construction of coalescence time distributions, visualization of the thin film profiles and characterization of the interfacial rheological properties. Subsequently, we summarize the recent developments in foam and emulsion science with a focus on the results obtained through single bubble/drop techniques. We conclude the review by presenting important venues for future research.

Controlled diffusion of nanoparticles by viscosity gradient for photonic crystal with dual photonic band gaps

Coalescence of droplets containing nanoparticles has been paid much attention regarding fabrication of functional photonic crystal (PC) patterns. However, most studies focus on the coalescence of droplets containing the same nanoparticles. Currently, an active challenge comes from the coalescence of droplets containing different nanoparticles due to the spontaneous mutual diffusion of different nanoparticles between coalescing miscible droplets driven by the released Gibbs free energy. Such diffusion breaks the self-assembly of nanoparticles into promising PCs with dual photonic band gaps (PBGs). In this work, a viscosity gradient was induced in coalescing droplets containing different nanoparticles to control the diffusion of nanoparticles and impede the diffusion across the coalescing interface. Nanoparticles diffused along the viscosity gradient to droplet surfaces and self-assembled into a period structure which enhanced the interaction of nanoparticles and contributed to impeding the random diffusion between droplets. At the same time, the high viscosity at the coalescing interface slowed down the horizontal movement of nanoparticles further and consequently the diffusion of nanoparticles across the interface was impeded. By use of such controlled diffusion of nanoparticles in the viscosity gradient, PCs with PBGs were achieved. These results demonstrate the controlled diffusion of nanoparticles during the coalescence of miscible droplets to facilely fabricate PCs with PBGs in the absence of an existing external field.

Electrohydrodynamics of droplets and jets in multiphase microsystems

Electrohydrodynamics is among the most promising techniques for manipulating liquids in microsystems. The electric stress actuates, generates, and coalesces droplets of small sizes; it also accelerates, focuses, and controls the motion of fine jets. In this review, the current understanding of dynamic regimes of electrically driven drops and jets in multiphase microsystems is summarized. The experimental description and underlying mechanism of force interplay and instabilities are discussed. Conditions for controlled transitions among different regimes are also provided. Emerging new phenomena either due to special interfacial properties or geometric confinement are emphasized, and simple scaling arguments proposed in the literature are introduced. The review provides useful perspectives for investigations involving electrically driven droplets and jets.

Coalescence of isotropic droplets in overheated free standing smectic films

A theoretical study of the interaction and coalescence of isotropic droplets in overheated free-standing smectic films (FSSF) is presented. Experimentally it is clear that merging of such droplets is extremely rare. On the basis of the general thermodynamic approach to the stability of FSSF, we determined the energy gains and losses involved in the coalescence process. The main contributions to the critical work of drop coalescence are due to the gain related to the decrease of the surface energy of the merging drops, which is opposed by the entropic repulsions of elementary steps at the smectic interface between them. To quantify the evolution of the merging drops, we use a simple geometrical model in which the volume of the smectic material, rearranged in the process of coalescence, is described by an asymmetrical pyramid at the intersection of two drops. In this way, the critical work for drop coalescence and the corresponding energy barrier have been calculated. The probability of the thermal activation of the coalescence process was found to be negligibly small, indicating that droplet merging can be initiated by only an external stimulus. The dynamics of drop merging was calculated by equating the capillary force driving the coalescence, and the Stokes viscous force slowing it down. For the latter, an approximation of moving oblate spheroids permitting exact calculations was used. The time evolution of the height of the neck between the coalescing drops and that of their lateral size are in good agreement with experiments.

Self-Enhancement of Coalescence-Induced Droplet Jumping on Superhydrophobic Surfaces with an Asymmetric V-Groove

Coalescence-induced droplet jumping on superhydrophobic surfaces have recently received significant attention owing to their potential in a variety of applications. Previous studies demonstrated that the self-jumping process is inherently inefficient, with an energy conversion efficiency η ≤ 6% and dimensionless jumping velocity V_j* ≤ 0.23. To realize a quick removal of droplets, increasing effort has been devoted to breaking the jumping velocity limit and inducing droplets sweeping. In this work, we used superhydrophobic surfaces with an asymmetric V-groove to experimentally achieve an enhanced coalescence-induced jumping velocity V_j* ≈ 0.61, i.e., more than 700% increase in energy conversion efficiency compared with droplets jumping on flat superhydrophobic surfaces, which is the highest efficiency reported thus far. Moreover, the enhanced jumping direction shows a deviation as high as 60° from the substrate normal. The induced in-plane motion is conducive to remove a considerable number of droplets along the sweeping path and significantly increase the speed of droplet removal. Numerical simulation indicated that the jumping enhancement is a joint effect resulting from the impact of the liquid bridge on the corner of the V-groove and the suppression of droplet expansion by the sidewall of the V-groove. The transient variation of the droplet velocity and the driving force of the coalescing droplets on a surface with and without the asymmetric V-groove were revealed and discussed. Furthermore, effects of groove angle, droplet pair positions, and size mismatches on the jumping velocity and direction have been studied. The novel mechanism of simultaneously increasing the coalescence-induced droplet jumping velocity and changing the jumping direction can be further studied to enhance the efficiency of various applications.

Geometry and kinetics determine the microstructure in arrested coalescence of Pickering emulsion droplets

Arrested coalescence occurs in Pickering emulsions where colloidal particles adsorbed on the surface of the droplets become crowded and inhibit both relaxation of the droplet shape and further coalescence. The resulting droplets have a nonuniform distribution of curvature and, depending on the initial coverage, may incorporate a region with negative Gaussian curvature around the neck that bridges the two droplets. Here, we resolve the relative influence of the curvature and the kinetic process of arrest on the microstructure of the final state. In the quasistatic case, defects are induced and distributed to screen the Gaussian curvature. Conversely, if the rate of area change per particle exceeds the diffusion constant of the particles, the evolving surface induces local solidification reminiscent of jamming fronts observed in other colloidal systems. In this regime, the final structure is shown to be strongly affected by the compressive history just prior to arrest, which can be predicted from the extrinsic geometry of the sequence of surfaces in contrast to the intrinsic geometry that governs the static regime.

Numerical study on electrohydrodynamic multiple droplet interactions

We present a numerical study of inviscid multiple droplet coalescence and break-up under the action of electric forces. Using an embedded potential flow model for the droplet hydrodynamics, coupled with an unbounded exterior electrostatic problem, we are able to perform computations through various singular events and analyze the effects of the electrical field intensity on droplet interactions. Laboratory experiments on the electrodynamics of droplet pairs show a much richer, and sometimes unexpected, behavior than that of isolated droplets. For example, it has been found that opposite charged droplets tend to repel each other when the electric field intensity is above a certain critical value. Although the mathematical model employed in this work incorporates very simple flow and electric assumptions, many of the droplet coalescence patterns seen in laboratory experiments can be reproduced. In this model, the interaction pattern of two droplets of radii R_{0} separated a distance D_{0}, depends on the ratio X_{0}=D_{0}/R_{0} and the applied uniform electric field intensity, E_{∞}. By performing a vast number of numerical simulations we are able to characterize the coalescence modes before and after drop merging as a function of these two parameters. The simulations predict that droplet repulsion occurs within a narrow interval of E_{∞} values, different for each X_{0}. Surprisingly, in this E_{∞} interval, a sharp transition between two power-law precoalescence flow regimes is seen. The evolution of several flow characteristics before and after coalescence, and the shape of the deformed droplets at coalescing time and the double cone angle, are also addressed and analyzed in detail. Cone angles below 35^{∘} lead to droplet coalescence for any X_{0} value, which is in accordance with previously reported studies. Finally, it is shown that the model and algorithm can handle multiple droplet interactions. The simulations qualitatively match results from water in oil experiments in microchannels, despite the fact that the exterior fluid is not considered in the mathematical model.

Capture and re-entrainment of microdroplets on fibers

The capture of liquid microdroplets on fibers, webs, and surfaces is important in a range of natural and industrial processes. One such application is the fibrous filtration of aerosols. Contact angle and wetting dynamics have a significant influence on capture and re-entrainment, yet there is no comprehensive model that accounts for these properties and their influence on capture efficiency. In this study, a series of computational simulations using liquid droplets and air are carried out to investigate the influence of equilibrium and dynamic contact angles on the capture and re-entrainment of mist droplets. A range of operating conditions for droplet-fiber diameter ratios, flow velocities, and contact angles, encapsulating both super-oleophilic and super-oleophobic media, are considered. All simulations are carried out using the volume of fluid (VOF) interface capturing approach in the finite volume solver interFoam within OpenFOAM. The physics of microdroplet impacting on a fiber is discussed and three distinct regimes for the spreading of the droplet around the fiber-inertia, capillary, and stagnation pressure controlled-are identified. It was found that the classification of filtration media for any fluid system, rather broadly as philic or phobic, based on the equilibrium contact angle alone may be insufficient for two reasons: (i) the characteristics of droplet-fiber interaction, including capture or re-entrainment, differs significantly over the range of contact angles for both philic and phobic media; and more importantly (ii) equilibrium contact angle plays little role in the initial stages of the droplet-fiber interaction that predominantly dictates the fate of the droplet. On the contrary, it is the contact angle dynamics that influences the initial stages of droplet impact on fibers, while commercial filters are seldom characterized based on this property. The isolated influence of equilibrium, advancing and receding contact angles on the potential mechanisms that can result in full or partial capture or re-entrainment are highlighted. The influence of equilibrium and advancing and receding hystereses are summarized in the form of a capture-regime map that shows four distinct regimes: (i) likely capture, (ii) likely re-entrainment with minimal or no capture, (iii) receding contact angle assisted partial or full capture, and (iv) advancing contact angle inhibited partial or full re-entrainment.

Liquid marble coalescence via vertical collision

The coalescence process of liquid marbles is vital to their promising roles as reactors or mixers in digital microfluidics. However, the underlying mechanisms and critical conditions of liquid marble coalescence are not well understood. This paper studies the coalescence process of two equally-sized liquid marbles via vertical collision aided by dielectrophoretic handling. A liquid marble was picked up using the dielectrophoretic force and then dropped vertically onto another liquid marble resting on a hydrophobic powder bed. The whole collision process was recorded by a high-speed camera and the recorded images were then analysed to derive the generalised conditions of liquid marble coalescence. By varying the marble volume, impact velocity and offset ratio in the experiments, we concluded that liquid marble coalescence may occur through the coating pore opening mechanism. We quantitatively measured the radius change versus time of the liquid neck formed between two coalescing marbles and estimated the maximum deformation of impacting marbles before rupture in rebound cases. We also qualitatively described the redistribution of coating particles at the impact area during coalescence as well as the consequent ejection of particles. Finally, we summarised the critical conditions for liquid marble coalescence, providing a frame for future applications involving liquid marbles as micromixers and microreactors.

Electrohydrodynamic coalescence of droplets using an embedded potential flow model

The coalescence, and subsequent satellite formation, of two inviscid droplets is studied numerically. The initial drops are taken to be of equal and different sizes, and simulations have been carried out with and without the presence of an electrical field. The main computational challenge is the tracking of a free surface that changes topology. Coupling level set and boundary integral methods with an embedded potential flow model, we seamlessly compute through these singular events. As a consequence, the various coalescence modes that appear depending upon the relative ratio of the parent droplets can be studied. Computations of first stage pinch-off, second stage pinch-off, and complete engulfment are analyzed and compared to recent numerical studies and laboratory experiments. Specifically, we study the evolution of bridge radii and the related scaling laws, the minimum drop radii evolution from coalescence to satellite pinch-off, satellite sizes, and the upward stretching of the near cylindrical protrusion at the droplet top. Clear evidence of partial coalescence self-similarity is presented for parent droplet ratios between 1.66 and 4. This has been possible due to the fact that computational initial conditions only depend upon the mother droplet size, in contrast with laboratory experiments where the difficulty in establishing the same initial physical configuration is well known. The presence of electric forces changes the coalescence patterns, and it is possible to control the satellite droplet size by tuning the electrical field intensity. All of the numerical results are in very good agreement with recent laboratory experiments for water droplet coalescence.

Coalescence Processes of Droplets and Liquid Marbles

The coalescence process of droplets and, more recently, of liquid marbles, has become one of the most essential manipulation schemes in digital microfluidics. This process is indispensable for realising microfluidic functions such as mixing and reactions at microscale. This paper reviews previous studies on droplet coalescence, paying particular attention to the coalescence of liquid marbles. Four coalescence systems have been reviewed, namely, the coalescence of two droplets freely suspended in a fluid; the coalescence of two sessile droplets on a solid substrate; the coalescence of a falling droplet and a sessile droplet on a solid substrate; and liquid marble coalescence. The review is presented according to the dynamic behaviors, physical mechanisms and experimental parameters of the coalescence process. It also provides a systematic overview of how the coalescence process of droplets and liquid marbles could be induced and manipulated using external energy. In addition, the practical applications of liquid marble coalescence as a novel microreactor are highlighted. Finally, future perspectives on the investigation of the coalescence process of liquid marbles are proposed. This review aims to facilitate better understanding of the coalescence of droplets and of liquid marbles as well as to shed new insight on future studies.

Dynamics of simultaneously impinging drops on a dry surface: Role of impact velocity and air inertia

Three dimensional simulations are performed to investigate the interaction dynamics between two drops impinging simultaneously on a dry surface. Of particular interest in this study is to understand the effects of impact velocity and surrounding gas density on droplet interactions. To simulate the droplet dynamics and morphologies, a computational framework based on the phase-field lattice Boltzmann formulation is employed for the two-phase flow computations involving high density ratio. Two different coalescence modes are identified when the impinging droplets have different impact speeds. When one of the droplet has a tangential impact velocity component, asymmetric ridge formation is observed. Influence of droplet impact angle on the interaction dynamics of the central ridge is further investigated. Traces of different fluid particles are seeded to analyse internal flow dynamics in oblique impact scenarios. Greater overlapping between the fluid particles is observed with increase in the impact angle. Finally, the present simulations indicate that the ambient gas density has a significant influence to determine the final outcome of the droplet interactions.

Surfactant effects on the coalescence of a drop in a Hele-Shaw cell

In this work the coalescence of an aqueous drop with a flat aqueous-organic interface was investigated in a thin gap Hele-Shaw cell. Different concentrations of a nonionic surfactant (Span 80) dissolved in the organic phase were studied. We present experimental results on the velocity field inside a coalescing droplet in the presence of surfactants. The evolution of the neck between the drop and the interface was studied with high-speed imaging. It was found that the time evolution of the neck at the initial stages of coalescence follows a linear trend, which suggests that the local surfactant concentration at the neck region for this stage of coalescence can be considered quasiconstant in time. This neck expansion can be described by the linear law developed for pure systems when the surfactant concentration at the neck is assumed higher than in the bulk solution. In addition, velocity and vorticity fields were computed inside the coalescing droplet and the bulk homophase using a high-speed shadowgraphy technique. The significant wall effects in the Hele-Shaw cell in the transverse axis cause the two vertical velocity components towards the singularity rupture point, from the drop and from the bulk homophase, to be of the same order of magnitude. This movement together with the neck expansion creates two pairs of counteracting vortices in the drop and in the bulk phase. The neck velocity is the average of the advection velocities of the two counteracting vortex pairs on each side of the neck. The presence of the surfactant slows down the dynamics of the coalescence, affects the propagation direction of the pair of vortices in the bulk phase, and reduces their size faster compared to the system without surfactant.

Coalescence of sessile microdroplets subject to a wettability gradient on a solid surface

While there are intensive studies on the coalescence of sessile macroscale droplets, there is little study on the coalescence of sessile microdroplets. In this paper, the coalescence process of two sessile microdroplets is studied by using a many-body dissipative particle dynamics numerical method. A comprehensive parametric study is conducted to investigate the effects on the coalescence process from the wettability gradient, hydrophilicity of the solid surface, and symmetric or asymmetric configurations. A water bridge is formed after two microdroplets contact. The temporal evolution of the coalescence process is characterized by the water bridge's radii parallel to the solid surface (W_{m}) and perpendicular to the solid surface (H_{m}). It is found that the changes of both H_{m} and W_{m} with time follow a power law; i.e., H_{m}=β_{1}τ^{β} and W_{m}=α_{1}τ^{α}. The growth of H_{m} and W_{m} depends on the hydrophilicity of the substrate. W_{m} grows faster than H_{m} on a hydrophilic surface, and H_{m} grows faster than W_{m} on a hydrophobic surface. This is due to the strong competition between capillary forces induced by the water-bridge curvature and the solid substrate hydrophobicity. Also, flow structure analysis shows that regardless of the coalescence type once the liquid bridge is formed the liquid flow direction inside the capillary bridge is to expand the bridge radius. Finally, we do not observe oscillation of the merged droplet during the coalescence process, possibly due to the significant effects of the viscous forces.

Controlling Pickering Emulsion Destabilisation: A Route to Fabricating New Materials by Phase Inversion

The aim of this paper is to review the key findings about how particle-stabilised (or Pickering) emulsions respond to stress and break down. Over the last ten years, new insights have been gained into how particles attached to droplet (and bubble) surfaces alter the destabilisation mechanisms in emulsions. The conditions under which chemical demulsifiers displace, or detach, particles from the interface were established. Mass transfer between drops and the continuous phase was shown to disrupt the layers of particles attached to drop surfaces. The criteria for causing coalescence by applying physical stress (shear or compression) to Pickering emulsions were characterised. These findings are being used to design the structures of materials formed by breaking Pickering emulsions.

Self-enhancement of droplet jumping velocity: the interaction of liquid bridge and surface texture

Whether droplet jumping velocity is enhanced or weakened depends on the impact position of liquid bridge.

Symmetric drop coalescence on an under-liquid substrate

We have derived a modified one-dimensional lubrication equation to describe the early coalescence behavior of a symmetric sessile drop coalescence for under-liquid substrates, which takes into account the viscosities of both the drop and the surrounding medium. We found a time scale, which governs the process, and there exists a crossover time between the universal scaling of the bridge height growth h^{*}∼t^{*} (valid for both under-liquid and air) and a much slower bridge growth h^{*}∼t^{*}^{0.24} occurring at a later time. It is also found that the drop coalescence bridge profile has a self-similarity, which breaks up much earlier for under-liquid substrates as opposed to symmetric coalescence in air.

Coalescence of bubbles and drops in an outer fluid

When two liquid drops touch, a microscopic connecting liquid bridge forms and rapidly grows as the two drops merge into one. Whereas coalescence has been thoroughly studied when drops coalesce in vacuum or air, many important situations involve coalescence in a dense surrounding fluid, such as oil coalescence in brine. Here we study the merging of gas bubbles and liquid drops in an external fluid. Our data indicate that the flows occur over much larger length scales in the outer fluid than inside the drops themselves. Thus, we find that the asymptotic early regime is always dominated by the viscosity of the drops, independent of the external fluid. A phase diagram showing the crossovers into the different possible late-time dynamics identifies a dimensionless number that signifies when the external viscosity can be important.

Coalescence preference in densely packed microbubbles

A bubble merged from two parent bubbles with different size tends to be placed closer to the larger parent. This phenomenon is known as the coalescence preference. Here we demonstrate that the coalescence preference can be blocked inside a densely packed cluster of bubbles. We utilized high-speed high-resolution X-ray microscopy to clearly visualize individual coalescence events inside densely packed microbubbles with a local packing fraction of ~40%. The surface energy release theory predicts an exponent of 5 in a relation between the relative coalescence position and the parent size ratio, whereas our observation for coalescence in densely packed microbubbles shows a different exponent of 2. We believe that this result would be important to understand the reality of coalescence dynamics in a variety of packing situations of soft matter.

Coalescence and breakup of oppositely charged droplets

The coalescence process of oppositely charged drops for different electrical conductivities of liquids is presented. When the electrical conductivity was relatively low, oppositely charged drops failed to coalesce under sufficiently high electrical fields and capillary ripples were formed on the surfaces of droplets after rebound. For a high electrically conductive liquid, it was found that a crown profile of drop fission always appeared on the top surface of negatively charged drops after the two charged drops contacted and bounced off. Furthermore, we report here, for the first time, the newly found phenomenon and argue that the break up might be caused by Rayleigh instability, a form of Coulomb fission. The different mobility of positive and negative ions is the underlying mechanism that explains why the break up always happened on the negative side of charged drops.

Dynamics of liquid drops coalescing in the inertial regime

We examine the dynamics of two coalescing liquid drops in the "inertial regime," where the effects of viscosity are negligible and the propagation of the front of the bridge connecting the drops can be considered as "local." The solution fully computed in the framework of classical fluid mechanics allows this regime to be identified, and the accuracy of the approximating scaling laws proposed to describe the propagation of the bridge to be established. It is shown that the scaling law known for this regime has a very limited region of accuracy, and, as a result, in describing experimental data it has frequently been applied outside its limits of applicability. The origin of the scaling law's shortcoming appears to be the fact that it accounts for the capillary pressure due only to the longitudinal curvature of the free surface as the driving force for the process. To address this deficiency, the scaling law is extended to account for both the longitudinal and azimuthal curvatures at the bridge front, which, fortuitously, still results in an explicit analytic expression for the front's propagation speed. This expression is shown to offer an excellent approximation for both the fully computed solution and for experimental data from a range of flow configurations for a remarkably large proportion of the coalescence process. The derived formula allows one to predict the speed at which drops coalesce for the duration of the inertial regime, which should be useful for the analysis of experimental data.

Approach and coalescence of liquid drops in air

The coalescence of liquid drops has conventionally been thought to have just two regimes when the drops are brought together slowly in vacuum or air: a viscous regime corresponding to the Stokes-flow limit and a later inertially dominated regime. Recent work [Proc. Natl. Acad. Sci. 109, 6857 (2012)] found that the Stokes-flow limit cannot be reached in the early moments of coalescence, because the inertia of the drops cannot be neglected then. Instead, the drops are described by an "inertially limited viscous" regime, where surface tension, inertia, and viscous forces all balance. The dynamics continue in this regime until either viscosity or inertia dominate on their own. I use an ultrafast electrical method and high-speed imaging to provide a detailed description of coalescence near the moment of contact for drops that approach at low speed and coalesce as undeformed spheres. These measurements support a description of coalescence having three regimes. Signatures both before and after contact identify a threshold approach speed for deformation of the drops by the ambient gas.

Coalescence of Pickering emulsion droplets induced by an electric field

Combining high-speed photography with electric current measurement, we investigate the electrocoalescence of Pickering emulsion droplets. Under a high enough electric field, the originally stable droplets coalesce via two distinct approaches: normal coalescence and abnormal coalescence. In the normal coalescence, a liquid bridge grows continuously and merges two droplets together, similar to the classical picture. In the abnormal coalescence, however, the bridge fails to grow indefinitely; instead, it breaks up spontaneously due to the geometric constraint from particle shells. Such connecting-then-breaking cycles repeat multiple times, until a stable connection is established. In depth analysis indicates that the defect size in particle shells determines the exact merging behaviors: when the defect size is larger than a critical size around the particle diameter, normal coalescence will show up, while abnormal coalescence will appear for coatings with smaller defects.