Coalescence of liquid drops by surface tension

Fog deposition and accumulation on smooth and textured hydrophobic surfaces

We investigated the deposition and accumulation of droplets on both smooth substrates and substrates textured with square pillars, which were tens of micrometers in size. After being coated with a hydrophobic monolayer, substrates were placed in an air flow with a sedimenting suspension of micrometer-sized water droplets (i.e., fog). We imaged the accumulation of water and measured the evolution of the mean drop size. On smooth substrates, the deposition process was qualitatively similar to condensation, but differences in length scale revealed a transient regime not reported in condensation experiments. Based on previous simulation results, we defined a time-scale characterizing the transition to steady-state behavior. On textured substrates, square pillars promoted spatial ordering of accumulated drops. Furthermore, texture regulated drop growth: first enhancing coalescence when the mean drop size was smaller than the pillar, and then inhibiting coalescence when drops were comparable to the pillar size. This inhibition led to a monodisperse drop regime, in which drop sizes varied by less than 5%. When these monodisperse drops grew sufficiently large, they coalesced and could either remain suspended on pillars (i.e., Cassie-Baxter state) or wet the substrate (i.e., Wenzel state).

Coalescence preference depends on size inequality

During bubble or droplet coalescence, there is a puzzling tendency for the coalesced bubble or droplet to be preferentially placed closer to the larger of its two parents. We confirm that this preference is a function of parent size ratio by directly visualizing coalescing air bubbles on an oil-water interface and coalescing water droplets immersed in oil. We find that the final position of the coalesced sphere is controlled by surface energy release and is related to the parent size ratio by a power-law relationship. 

The inexorable resistance of inertia determines the initial regime of drop coalescence

Drop coalescence is central to diverse processes involving dispersions of drops in industrial, engineering, and scientific realms. During coalescence, two drops first touch and then merge as the liquid neck connecting them grows from initially microscopic scales to a size comparable to the drop diameters. The curvature of the interface is infinite at the point where the drops first make contact, and the flows that ensue as the two drops coalesce are intimately coupled to this singularity in the dynamics. Conventionally, this process has been thought to have just two dynamical regimes: a viscous and an inertial regime with a cross-over region between them. We use experiments and simulations to reveal that a third regime, one that describes the initial dynamics of coalescence for all drop viscosities, has been missed. An argument based on force balance allows the construction of a new coalescence phase diagram.

Coalescence of two drops on partially wettable substrates

The present work contains the results of the experiments with two tiny drops on partially wettable substrates with contact angles of 10°, 24°, 27°, and 56°, which coalesce in the regime entirely dominated by viscous forces. Both side and bottom views are examined. The results for these three-dimensional coalescence flows are compared with scaling laws and the numerical two-dimensional model developed in the present work.

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.

Perturbed breakup of gas bubbles in water: memory, gas flow, and coalescence

The pinch-off of an air bubble from an underwater nozzle ends in a singularity with a remarkable sensitivity to a variety of perturbations. I report on experiments that break both the axial (i.e., vertical) and azimuthal symmetry of the singularity formation. The density of the inner gas influences the axial asymmetry of the neck near pinch-off. For denser gases, flow through the neck late in collapse changes the pinch-off dynamics. Gas density is also implicated in the formation of satellite bubbles. The azimuthal shape oscillations described by Schmidt et al. can be initiated by anisotropic boundary conditions in the liquid as well as with an asymmetric nozzle shape. I measure the n=3 oscillatory mode and observe the nonlinear, highly three-dimensional outcomes of pinch-off with large azimuthal perturbations. These are consistent with prior theory.

The dynamics of the impact and coalescence of droplets on a solid surface

A simple experimental setup to study the impact and coalescence of deposited droplets is described. Droplet impact and coalescence have been investigated by high-speed particle image velocimetry. Velocity fields near the liquid-substrate interface have been observed for the impact and coalescence of 2.4 mm diameter droplets of glycerol∕water striking a flat transparent substrate in air. The experimental arrangement images the internal flow in the droplets from below the substrate with a high-speed camera and continuous laser illumination. Experimental results are in the form of digital images that are processed by particle image velocimetry and image processing algorithms to obtain velocity fields, droplet geometries, and contact line positions. Experimental results are compared with numerical simulations by the lattice Boltzmann method.

Viscous to inertial crossover in liquid drop coalescence

Using an electrical method and high-speed imaging, we probe drop coalescence down to 10 ns after the drops touch. By varying the liquid viscosity over two decades, we conclude that, at a sufficiently low approach velocity where deformation is not present, the drops coalesce with an unexpectedly late crossover time between a regime dominated by viscous and one dominated by inertial effects. We argue that the late crossover, not accounted for in the theory, can be explained by an appropriate choice of length scales present in the flow geometry.

Charge interaction between particle-laden fluid interfaces

Experiments are described where two oil/water interfaces laden with charged particles move at close proximity relative to one another. The particles on one of the interfaces were observed to be attracted toward the point of closest approach, forming a denser particle monolayer, while the particles on the opposite interface were repelled away from this point, forming a particle depletion zone. Such particle attraction/repulsion was observed even if one of the interfaces was free of particles. This phenomenon can be explained by the electrostatic interaction between the two interfaces, which causes surface charges (charged particles and ions) to redistribute in order to satisfy surface electric equipotential at each interface. In a forced particle oscillation experiment, we demonstrated the control of charged particle positions on the interface by manipulating charge interaction between interfaces.

Self-propelled dropwise condensate on superhydrophobic surfaces

In conventional dropwise condensation on a hydrophobic surface, the condensate drops must be removed by external forces for continuous operation. This Letter reports continuous dropwise condensation spontaneously occurring on a superhydrophobic surface without any external forces. The spontaneous drop removal results from the surface energy released upon drop coalescence, which leads to a surprising out-of-plane jumping motion of the coalesced drops at a speed as high as 1 m/s. The jumping follows an inertial-capillary scaling and gives rise to a micrometric average diameter at steady state.

Modeling the coalescence of sessile droplets

This paper proposes a simple scenario to describe the coalescence of sessile droplets. This scenario predicts a power-law growth of the bridge between the droplets. The exponent of this power law depends on the driving mechanism for the spreading of each droplet. To validate this simple idea, the coalescence is simulated numerically and a basic experiment is performed. The fluid dynamics problem is formulated in the lubrication approximation framework and the governing equations are solved in the commercial finite element software COMSOL. Although a direct comparison of the numerical results with experiment is difficult because of the sensitivity of the coalescence to the initial and operating conditions, the key features of the event are qualitatively captured by the simulation and the characteristic time scale of the dynamics recovered. The experiment consists of inducing coalescence by pumping a droplet through a substrate which grows and ultimately coalesces with another droplet resting on the substrate. The coalescence was recorded using high-speed imaging and also confirmed the power-law growth of the neck.

Satellite Formation during Coalescence of Unequal Size Drops

The coalescence of a drop with a flat liquid surface pinches off a satellite from its top, in the well-known coalescence cascade, whereas the coalescence of two equally sized drops does not appear to leave such a satellite. Herein we perform experiments to identify the critical diameter ratio of two drops, above which a satellite is produced during their coalescence. We find that the critical parent ratio is as small as 1.55, but grows monotonically with the Ohnesorge number. The daughter size is typically about 50% of the mother drop. However, we have identified novel pinch-off dynamics close to the critical size ratio, where the satellite does not fully separate, but rather goes directly into a second stage of the coalescence cascade, thus generating a much smaller satellite droplet.

Coalescence of low-viscosity fluids in air

An electrical method is used to study the early stages of coalescence of two low-viscosity drops. A drop of aqueous NaCl solution is suspended in air above a second drop of the same solution, which is grown until the drops touch. At that point a rapidly widening bridge forms between them. By measuring the resistance and capacitance of the system during this coalescence event, one can obtain information about the time dependence of the characteristic bridge radius and its characteristic height. At early times, a new asymptotic regime is observed that is inconsistent with previous theoretical predictions. The measurements at several drop radii and approach velocities are consistent with a model in which the two liquids coalesce with a slightly deformed interface.

Coalescence of bubbles covered by particles

The interaction between two bubbles coated with glass particles in the presence of a cationic surfactant (cetyltrimethylammonium bromide, CTAB) was studied experimentally. The time taken for two bubbles to coalesce was determined as a function of the fractional coverage of the surface by particles. The results suggested that the coalescence time increases with the bubble surface coverage. Interestingly, it was found that although the particles did not have any physical role in film rupture at low surface coverage, they still added resistance to film drainage. For particle-loaded bubbles, the initial resistance was due to the lateral capillary interactions between particles on the interface, which hold the particles firmly together. The coalescence dynamics of bubbles was also observed to be affected by the presence of attached particles.

Ultrafast x-ray phase-contrast imaging of the initial coalescence phase of two water droplets

We report an ultrafast x-ray phase-contrast imaging study of the early merging dynamics of two water drops in air. Owing to the edge-enhancement capability, the high penetrability, and the unprecedented temporal and spatial resolutions offered by this new x-ray technique, the coalescence singularity of two water drops was revisited. A finite initial contact radius was identified and the evolvement of the trapped toroidal air bubble was studied for the first time. Despite the existence of this finite initial contact radius, the subsequent meniscus radius followed power laws which agree with theoretical predictions for the inviscid regime.

Coalescence in low-viscosity liquids

The expected universal dynamics associated with the initial stage of droplet coalescence are difficult to study visually due to the rapid motion of the liquid and the awkward viewing geometry. Here we employ an electrical method to study the coalescence of two low-viscosity droplets at early times. We measure the growth dynamics of the bridge connecting the two droplets and observe a new asymptotic regime inconsistent with previous theoretical predictions. The measurements are consistent with a model in which the two liquids coalesce with a slightly deformed interface.

Role of dimensionality and axisymmetry in fluid pinch-off and coalescence

We present data on the pinch-off and coalescence of thin liquid alkane lenses floating on water. Pinch-off in quasi-2D lenses is distinctly different from pinch-off in axisymmetric 3D drops and involves a cascade of satellite droplets which extends to micron length scales. In contrast, coalescence of lenses is qualitatively similar to coalescence of 3D drops. Coalescence is predicted to involve entrainment of the exterior fluid as the droplets merge. This reentrant folding is obscured in 3D droplets but is clearly visible in coalescence of thin lenses.

Morphology and dynamics of droplet coalescence on a surface

The coalescence of a pair of droplets on a surface is investigated experimentally with images from detailed flow visualisations revealing the morphology of the process. It is found that they merge and evolve to a final state with a footprint that is peanut like in shape, with bulges along the longer sides resulting from the effects of inertia during spreading. The associated dynamics involve a subtle interplay between (i) the motion of the wetting process due to relaxation of the contact angle and (ii) a rapid rise in free-surface height above the point where coalescence began due to negative pressure generated by curvature. During the early stages of the motion, a traveling wave propagates from the point of initial contact up the side of each droplet as liquid is drawn into the neck region, and only when it reaches the apex of each do their heights start to decrease. A further feature of the rapid rise in height of the neck region is that the free surface there overshoots significantly its final equilibrium position; it reaches a height greater than that of the starting droplets, producing a self-excited oscillation that persists long after the system reaches its final morphological state in relation to its footprint.

Condensation and wetting transitions on microstructured ultra-hydrophobic surfaces

On rough surfaces, two distinct wetting modes can appear. These two states are usually described by the theories of Cassie (drops suspended on top of roughness features) and Wenzel (drops impaled on roughness features). Whereas the wetting transition from the Cassie to the Wenzel state has been relatively well studied both experimentally and theoretically, the question of whether metastable Wenzel drops exist and how they transition to the Cassie state has remained open. In this work, we study the wetting behavior of microstructured post surfaces coated with a hydrophobic fluoropolymer. Through condensation, the formation of metastable Wenzel droplets is induced. We show that under certain conditions drops can transition from the Wenzel to the Cassie state.

Critical parameters for the partial coalescence of a droplet

The partial coalescence of a droplet onto a planar liquid-liquid interface is investigated experimentally by tuning the viscosities of both liquids. The problem mainly depends on four dimensionless parameters: The Bond number (gravity vs surface tension), the Ohnesorge numbers (viscosity in both fluids vs surface tension), and the density relative difference. The ratio between the daughter droplet size and the mother droplet size is investigated as a function of these dimensionless numbers. Global quantities such as the available surface energy of the droplet have been measured during the coalescence. The capillary waves propagation and damping are studied in detail. The relation between these waves and the partial coalescence is discussed. Additional viscous mechanisms are proposed in order to explain the asymmetric role played by both viscosities.

Eliminating parasitic currents in the lattice Boltzmann equation method for nonideal gases

A formulation of the intermolecular force in the nonideal-gas lattice Boltzmann equation method is examined. Discretization errors in the computation of the intermolecular force cause parasitic currents. These currents can be eliminated to roundoff if the potential form of the intermolecular force is used with compact isotropic discretization. Numerical tests confirm the elimination of the parasitic currents.

Contact Line Dynamics in the Late-Stage Coalescence of Diethylene Glycol Drops

We investigated the contact line dynamics of a composite drop formed as a result of the coalescence during the condensation of two diethylene glycol (DEG) drops at -4 degrees C on a silicon surface. The composite drop relaxes exponentially toward equilibrium with a typical relaxation time, tc, which depends on the equilibrium radius, R, of the composite drop. The value of tc is found to be in the range of 10-100 s for R approximately 1-4 microm. The relaxation dynamics is found to be larger by 6 orders of magnitude than that predicted by bulk hydrodynamics because of high dissipation in the contact line vicinity. Similar to low viscous liquids (water), this high dissipation can be attributed to an Arrhenius factor resulting from the phase change in the contact line vicinity and to the influence of surface defects that pin the contact line.

Hydrodynamics of droplet coalescence

We study droplet coalescence in a molecular system with a variable viscosity and a colloid-polymer mixture with an ultralow surface tension. When either the viscosity is large or the surface tension is small enough, we observe that the opening of the liquid bridge initially proceeds at a constant speed set by the capillary velocity. In the first system we show that inertial effects become dominant at a Reynolds number of about 1.5+/- 0.5 and the neck then grows as the square root of time. In the second system we show that decreasing the surface tension by a factor of 10(5) opens the way to a more complete understanding of the hydrodynamics involved.

Coalescence of viscous liquid drops

We report on studies of the early stage of coalescence of two liquid drops. The drops were high viscosity silicon oil immersed in a water-alcohol mixture of the same density in order to eliminate the effects of gravity. The viscosity was sufficiently large that measurements could be made under the conditions of Stokes flow. Measurements were made of the radius of the neck between the drops as a function of the time from the onset of coalescence, and the results compared with theoretical predictions.

Coalescence of crystalline drops

We present the first experimental analysis of drop coalescence in a case where the dynamics is not governed by viscous dissipation in the bulk nor by the inertia of the fluid flow, only by the geometry and mobility of surfaces. We found such a situation in the physics of 3He crystals near 0.32 K where the latent heat of crystallization vanishes. Two crystalline drops of 3He coalesce if their crystalline orientations are identical: a neck forms after the contact at time t=0, and the shape evolves towards that of one convex crystal by local growth and melting in a fraction of a second. We have found that the neck radius initially increases as t(1/3), as predicted by Maris. This behavior is also expected for superfluid drops. It is clearly distinguished from the logarithmic behavior and from the t(1/2) power law which have been predicted by Eggers et al. in more usual situations.

Reflectivity-based evaluation of the coalescence of two condensing drops and shape evolution of the coalesced drop

Image analyzing interferometry is used to study the details of the evolving shapes and coalescence of two condensing drops of 2-propanol on a quartz surface. The measured thickness profiles give fundamental insights into the transport processes within the drops before and after coalescence and the evolution of the coalesced drop from asymmetric to symmetric shape. The results indicate that the constant value of the adsorbed film thickness between the drops and profiles of the local thickness, slope angle, curvature, and curvature gradient govern the pressure fields in the coalescing drops. The shape evolution after coalescence is found to be driven by the capillary forces within the drop. Using the experimental data, we find that the calculations of the average shear stress for the fluid flow between the drops, the decrease in the interfacial excess energy, and the positions of the center of mass of the drops explain the physics of the coalescence phenomenon. However, the flow field is found to be complex because the pressure field indicates that there are complicated flows within the drop.

Analysis of the early stage of coalescence of helium drops

We analyze the growth of the neck that forms between two liquid drops that have come into contact. The analysis is for a fluid in which the velocity of each point on the surface is proportional to the local curvature and directed normal to the interface. For this system, we show that the radius of the neck is proportional to t(1/3), where the time t is measured from the moment at which coalescence commences. We are able to find a simple expression for the shape of the interface in the vicinity of the neck.