Droplet Pinch-off Research Articles

Mechanism of temporal interface evolution and internal circulations during the droplet formation in a planar slit T-microchannel

The present study has numerically explored the mechanism of interface evolution and internal flow circulations during the droplet formation in two-phase flow through a planar T-microchannel. The two-dimensional unsteady form of the conservative level set equation coupled with Navier–Stokes equations has been solved using the finite element method. The range of parameters include the contact angle (θ) from 120° to 180°, and the flow rate ratio (Qr) from 0.1 to 10 for the low capillary number (Cac≤10−2). The present study indicates that surface wettability plays a crucial role in influencing the temporal evolution of the interface. The internal flow circulation in the droplet is controlled by the axial and radial velocities primarily influenced by shear stress. The newly introduced novel “interface-to-neck ratio” parameter has provided another platform to investigate the pinch-off dynamics of droplets. Moreover, the phenomenon of droplet pinch-off is primarily initiated and driven by the Laplace pressure, defined by three distinct approaches: the pressure difference method, the determination of the minimum local radius of curvature on the rear side, and a calculation of the neck width. The predictive correlations have been established to estimate the droplet characteristics as a function of the flow rate ratio and contact angle. The findings reported have significant implications for the design of droplet dispensing systems that depend on surface wettability as a critical regulating parameter.

Auto-ejection of liquid from a nozzle.

Auto-ejection of liquid is an important process in engineering applications, and is also very complicated since it involves interface moving, deforming, and jet breaking up. In this work, a theoretical velocity of meniscus at nozzle exit is first derived, which can be used to analyze the critical condition for auto-ejection of liquid. Then a consistent and conservative axisymmetric lattice Boltzmann (LB) method is proposed to study the auto-ejection process of liquid jet from a nozzle. We test the LB model by conducting some simulations, and find that the numerical results agree well with the theoretical and experimental data. We further consider the effects of contraction ratio, length ratio, contact angle, and nozzle structure on the auto-ejection, and observe some distinct phenomena during the ejection process, including the deformation of meniscus, capillary necking, and droplet pinch off. Finally, the results reported in the present work may play an instructive role on the design of droplet ejectors and the understanding of jetting dynamics in microgravity environment.

Singular jets during the impingement of compound drops upon lyophilic surfaces

An important phenomenon produced during the impingement of drops upon solid surfaces is the formation of singular jet, which is often followed by the pinch-off of satellite droplets. Great efforts have been made to investigate the jetting dynamics of low-viscosity single-phase drops impact upon sufficiently lyophobic surfaces. However, whether such singular jets can be produced during the impact of compound drops and how the liquid properties and surface wettabilities affect the dynamics have remained largely unexplored. Herein, we perform comparative and systematic experiments on the impact dynamics of single-phase water and silicon oil drops, as well as water-in-oil compound drops on lyophilic substrates. We show that singular jets only occur during the impact of compound drops. The critical values in terms of the Weber number depend on both the viscosity of the silicon oil and the volume ratio of the two liquids composing the compound drops. We also show that the singular jets break up and throw out satellite droplets only when they are considerably fast and thin. Power-law correlations between the jet velocities and the jet radii, between the jet neck radius and time, and between the maximum jet height and the jetting time are obtained. A linear correlation between the radii of the jet droplets and those of the singular jets is also found and analyzed.

Exploring impact, spreading, and bonding dynamics in molten metal deposition for novel drop-on-demand printing

This study delves into the dynamics of molten metal deposition (MMD)-based drop-on-demand (DoD) printing, focusing on the interaction between aluminum droplets and substrates. Nozzle-to-substrate distances below 20 mm prevent droplet pinch-off, while distances exceeding 40 mm result in no droplet-substrate adhesion. Operating within the substrate temperature range of 350–550 °C is crucial, avoiding delamination, and shrinkage pits at lower temperatures, and substrate deformation and heightened oxidation at higher temperatures. Droplet adhesion is impeded at nozzle temperatures below 700 °C. Impact-driven and inviscid, DoD-MMD exhibits spreading outpacing overall solidification, particularly at higher temperatures. Surface tension forces dominate, influencing droplet spreading and leading to underdamped interfacial oscillations. Weak droplet-substrate adherence, facilitated by Van der Waals forces, allows easy droplet detachment, beneficial for successive drop-on-drop deposition. Unique to DoD-MMD, ridges along the droplet periphery act as solidification paths, influenced by thermal contraction and surface tension. The spherical pancake shape of the droplet, characterized by a solidification angle >90°, is elucidated through Weber and Freezing numbers. The final deposited droplet width to initial diameter ratio increases with the droplet temperature and deposition height. In contrast to other metal DoD studies, the spreading factor decreases with a rise in substrate temperature, attributed to intensified oxidation at higher substrate temperatures.

Droplet generation by the oscillation of two spark-generated bubbles near a confined opening

Cavitation bubbles can be both beneficial and detrimental. One of their beneficial applications is to produce droplets as they oscillate near a confined free surface. In this paper, the oscillation of two spark-generated bubbles beneath a confined free surface and the resultant droplet generation are studied numerically using boundary element method. Three different configurations are considered: (i) a vertical cylinder (θ = 0°), (ii) a vertical nozzle (θ = 45°) and (iii) a perforated horizontal flat plate (θ = 90°). The influences of the effective parameters, including the initial pressure and locations of the bubbles, and the configuration geometry on the dynamics of the bubbles, the free surface and the resultant droplet are studied. It was found that by decreasing the inter-bubble distance, the droplet size increases and its pinch-off occurs earlier. In general, the effect of the distance of the upper bubble from the free surface on the droplet behavior is much more significant than that of the inter-bubble distance. Furthermore, through the augmentation of the upper bubble's strength parameter, the droplet size is reduced and the moment of droplet pinch-off is postponed. Finally, the droplet size and its pinch-off time increase from the vertical cylinder to the flat plate case.

Simultaneous Separating, Splitting, Collecting, and Dispensing by Droplet Pinch-Off for Droplet Cell Culture.

Simultaneous separating, splitting, collecting, and dispensing a cell suspension droplet has been demonstrated by aspiration and subsequent droplet pinch-off for use in microfluidic droplet cell culture systems. This method is applied to cell manipulations including aliquots and concentrations of microalgal and mammalian cell suspensions. Especially, medium exchange of spheroid droplets is successfully demonstrated by collecting more than 99% of all culture medium without damaging the spheroids, demonstrating its potential for a 3D cell culture system. Through dimensional analysis and systematic parametric studies, it is found that initial mother droplet size together with aspiration flow rate determines three droplet pinch-off regimes. By observing contact angle changes during aspiration, the difference in the large and the small droplet pinch-off can be quantitatively explained using force balance. It is found that the capillary number plays a significant role in droplet pinch-off, but the Bond number and the Ohnesorge number have minor effects. Since the dispensed droplet size is mainly determined by the capillary number, the dispensed droplet size can be controlled simply by adjusting the aspiration flow rate. It is hoped that this method can contribute to various fields using droplets, such as droplet cell culture and digital microfluidics, beyond the generation of small droplets.

Suppression of droplet pinch-off by early onset of interfacial instability

HypothesisInterfacial instabilities cause undesirable droplet breakage during impact. Such breakage affects many applications, such as printing, spraying, etc. Particle coating over a droplet can significantly change the impact process and stabilize it against breakage. This work investigates the impact dynamics of particle-coated droplets, which mostly remains unexplored. ExperimentsParticle-coated droplets of different mass loading were formed using volume addition. The prepared droplets were impacted on superhydrophobic surfaces, and their dynamics were recorded using a high-speed camera. FindingsWe report an intriguing phenomenon where an interfacial fingering instability helps suppress pinch-off in particle-coated droplets. This island of breakage suppression, where the droplet maintains its intactness upon impact, appears in a regime of Weber numbers where bare droplet breakage is inevitable. The onset of fingering instability in particle-coated droplets is observed at much lower impact energy, around two times less than the bare droplet. The instability is characterized and explained using the rim Bond number. The instability suppresses pinch-off because of the higher losses associated with the formation of stable fingers. Such instability can also be seen in dust/pollen-covered surfaces, making it useful in many applications related to cooling, self-cleaning, anti-icing etc.

Dynamics of Droplet Pinch-Off at Emulsified Oil-Water Interfaces: Interplay between Interfacial Viscoelasticity and Capillary Forces.

The presence of submicrometer structures at liquid-fluid interfaces modifies the properties of many science and technological systems by lowering the interfacial tension, creating tangential Marangoni stresses, and/or inducing surface viscoelasticity. Here we experimentally study the break-up of a liquid filament of a silica nanoparticle dispersion in a background oil phase that contains surfactant assemblies. Although self-similar power-law pinch-off is well documented for threads of Newtonian fluids, we report that when a viscoelastic layer is formed insitu at the interface, the pinch-off dynamics follows an exponential decay. Recently, such exponential neck thinning was found theoretically when surface viscous effects were taken into account. We introduce a simple approach to calculate the effective relaxation time of viscoelastic interfaces and estimate the thickness of the interfacial layer and the viscoelastic properties of liquid-fluid interfaces, where the direct measurement of interfacial rheology is not possible.

Effects of surface wettability and flow rates on the interface evolution and droplet pinch-off mechanism in the cross-flow microfluidic systems

In this study, the effect of surface wettability on the two-phase immiscible fluids flow and dynamics of droplet pinch-off in a T-junction microchannel has been numerically investigated using the finite element method. A conservative level set method (CLSM) has been adopted to capture the interface topology in the squeezing regime (Cac<10-2) for wide flow rate ratio (1/10⩽Qr⩽10) and contact angle (120°⩽θ⩽180°). This study has revealed that the wettability is a dominant factor in determining the hydrodynamic features of the droplet. Based on the instantaneous phase flow profiles, the droplet formation stages are classified as: initial, filling, squeezing, pinch-off and stable droplet. Wettability effects are insignificant in the filling stage. However, the hydrophobic effects are more vital in the squeezing and pinch-off stages. In general, it is shown that engineering parameters have complex dependence on the dimensionless parameters (Cac,Qr,θ). Capturing the instantaneous interface evolution has revealed that the droplet shape is sensitive to the contact angle. Interface shape profiles transform from convex into concave immediately for hydrophobic conditions (120°⩽θ⩽150°) whereas slowly for the super hydrophobic conditions (150°<θ⩽180°). In contrast to the literature, the pressure in the dispersed phase is not constant, but it is an anti-phase with the pressure in the continuous phase. Maximum pressure in the continuous phase, and neck width of the interface are complex function of the governing conditions (Cac,Qr,θ). Comparison of the filling and pinch-off time based on the pressure and phase profiles has brought new insights that the droplet pinch-off mechanism can be elucidated by installing the pressure sensors even without the flow visualization and phase profiles. The interface curvature adopts a flattened to a more concave shape when the Laplace pressure varies from a smaller to higher value. The interface neck width (2r) shows an increasing trend up to a threshold value and then decreases linearly with the contact angle.

Role of surfactant-induced Marangoni effects in droplet dynamics on a solid surface in shear flow

We address the dynamics of a surfactant-laden droplet on a solid surface in simple shear flow numerically. Our analysis uses the front-tracking method to take surfactant transport into account. The interfacial tension and the slip coefficient, both of which depend heavily on the surfactant concentration, are fully integrated into the generalized Navier boundary condition to model the moving contact lines. Accurate prediction of droplet motion indicates that the surfactant can change droplet behavior drastically. Surfactant-induced effects, such as interfacial tension reduction, the Marangoni stress, and wettability alternation, are investigated for various capillary numbers, surface wettabilities, elasticity numbers, and surface Péclet numbers. Deformation and motion of a sliding droplet are enhanced by the Marangoni effect, which is associated with an interfacial tension gradient. When the capillary number reaches a critical value, the sliding-to-detachment and detachment-to-pinch-off transitions occur. Both transitions can be triggered and accelerated by a surfactant, especially when convection is dominant. As a result, the critical capillary number decreases, but exhibits a non-monotonic relationship with the elasticity number and Péclet number. The mechanisms that underlie the effect of Marangoni stress are discussed by analyzing the distributions of the surfactant concentration and the hydrodynamic forces exerted on the droplet. Accumulation of surfactants near the receding contact line reverses the local concentration gradient, attempts to change its direction along the interface, and delays droplet detachment. Furthermore, the strong surfactant dilution reduces both the surfactant concentration and the interfacial tension gradient, and thereby increasing the critical value for droplet pinch-off.

Effects of topological changes in microchannel geometries on the hydrodynamic formation and breakup of all-aqueous droplets

Microfluidic gravity-driven droplet generation in aqueous two-phase systems (ATPS) has recently emerged as an effective technique used to passively create all-biocompatible environments. The hydrodynamic formation and breakup of all-aqueous viscous threads in flow-focusing microfluidic devices were experimentally studied by applying different intersection angles (θ). Four typical flow regimes are observed over a range of 0.2–2 kPa, namely, transitional, jetting, threading, and tubing. The flow rate ratio based on the specifications of the entire device was calculated, and the flow regimes were mapped based on the capillary number of the flows. Scaling laws that describe the flow behavior are then put forth based on physical arguments. In the jetting regime, droplet size and jet width are measured and correlated with flow rate ratio. The critical thread length before droplet pinch-off is determined by flow rate ratio and the capillary number of any of the phases. In the transitional regime, droplet length is affected by flow rate ratio and the capillary number of the continuous phase. In addition, through variations in θ, the influence of altering the geometry of the device on the flow regimes was considered. The findings presented in this article provide insight into the intricate process of droplet generation in passive ATPS microfluidics.

Kinematics of the viscous filament during the droplet breakup in air

The dripping regime in the vicinity of the fluid droplet breakup is analyzed using the correlation between experiments and numerics. The evolutions of filament’s neck and its corresponding thinning velocity are described using the logistic functions. Three flow regions are observed as the relative time decreases: (1) a monotonous increase of the neck’s thinning velocity, where inertia and capillarity are balanced, (2) a transition domain characterized by the equilibrium between inertia, capillarity, and viscous forces, where the thinning velocity varies non-monotonically with the relative time and (3) the final droplet pinch-off, where velocity decreases or oscillates around a constant value. The distributions of the upzeta-coefficient (parameter related to the non-dimensional second invariant of the velocity gradient) on the filament’s surface and droplet’s profile characterize the kinematics at the interface. The regions dominated by extension, where pure elongation is located at upzeta cong 1, are determined. One main result of this study is the confirmation that distribution of the upzeta-coefficient is a relevant parameter to analyze and to quantify the breakup process. This result has the potential of developing novel techniques and more precise procedures in determining the interfacial rheology of viscous and complex fluids.

Transition to the viscoelastic regime in the thinning of polymer solutions.

In this study, we investigate the transition between the Newtonian and the viscoelastic regimes during the pinch-off of droplets of dilute polymer solutions and discuss its link to the coil-stretch transition. The detachment of a drop from a nozzle is associated with the formation of a liquid neck that causes the divergence of the local stress in a vanishingly small region. If the liquid is a polymer solution, this increasing stress progressively unwinds the polymer chains, up to a point where the resulting increase in the viscosity slows down drastically the thinning. This threshold to a viscoelastic behavior corresponds to a macroscopic strain rate. In the present study, we characterize the variations of with respect to the polymer concentration and molar weight, to the solvent viscosity, and to the nozzle size, i.e., the weight of the drop. We provide empirical scaling laws for these variations. We also analyze the thinning dynamics at the transition and show that it follows a self-similar dynamics controlled by the time scale c-1. This characteristic time is different and always shorter than the relaxation time of the polymer.

Vertical magnetic field aided droplet-impact- magnetohydrodynamics of ferrofluids

In this present article, we have investigated the magnetohydrodynamics of ferrofluid droplets impacting on different wettability surfaces, in the presence of a vertical magnetic field. The spreading dynamics was studied for a wide spectrum of magnetic Bond number (Bom), Hartmann number (Ha) and Weber number (We). In absence of any magnetic field, the droplets exhibited secondary droplet pinch-off during rebound. In the presence of magnetic field, the field modulated Rayleigh-Plateau instability delays the droplet pinch off as Bom increases. For a fixed We, the rebound of the droplet was suppressed on a superhydrophobic (SH) surface for varying the magnetic field strength (manifested through Bom). An analytical model based on the principle of conservation of energy was formulated to explain the magnetic field modulated droplet pinch-off. We have also investigated the influence of Bom on the temporal spreading dynamics of different Ha ferrofluid droplets impacting on hydrophilic and SH surfaces. With an increase in Bom, the magneto-visco-capillarity of high Ha droplets inhibits the capillary waves and motion of the contact line after impingement onto hydrophilic surface, compared to low Ha ferrofluid drops. Instead, the droplet rim fragments into secondary droplets during retraction event, leading to the onset of rim instability for low Ha ferrofluid drops with an increase in the magnetic field strength. We have also proposed a theoretical formulation based on energy conservation principle to predict the experimentally measured maximum spreading diameters under influence of magnetic field. The findings may hold significance in ferrofluid based droplet microfluidics systems with magnetic control or actuation.

Ferrofluid droplet breakup process and neck evolution under steady and pulse-width modulated magnetic fields

Numerous applications in engineering and biotechnology have attracted the attention of many researchers to the analysis of underlying physical phenomena during the droplet pinch-off. In this study, the neck evolution during the formation of a ferrofluid droplet from a capillary is investigated under two types of magnetic field for a drop-on-demand system. The two types are steady and Pulse-Width Modulated (PWM) magnetic fields. First, under steady magnetic field, the necking process is studied for different values of magnetic Bond number and various angles between magnetic coil centerline and gravity. Subsequently, self-similar behavior in the vicinity of the detachment moment is observed. In the last stage of droplet pinch-off, a scaling law is fitted to the variation of minimum neck diameter versus remaining time to the breakup. The scaling exponent is obtained to be 0.67 which is in good agreement with that for the ferrofluid droplet formation using a syringe pump reported in previous studies. Second, under PWM magnetic field, the necking process during the detachment of a ferrofluid droplet is explored and compared with that in the presence of the steady magnetic field. A new pattern of minimum neck diameter variation with remaining time to the breakup is observed for the first time. The trend of variation can be divided into two regions which are called “slow-going” and “fast-going” necking. Two mechanisms are introduced which explain the transition from the slow-going to fast-going necking region. The effect of magnetic Bond number and dimensionless applied frequency on the neck evolution is studied. In the proximity of the detachment moment, a universal behavior for pinch-off process is observed and the exponent of the fitted scaling law is equal to 0.84. Finally, the variation of relative neck position under the PWM magnetic field is compared with that in the presence of steady magnetic field.

Droplet detachment and pinch-off of bidisperse particulate suspensions.

When a droplet is generated, the ligament connecting the drop to the nozzle thins down and eventually pinches off. Adding solid particles to the liquid phase leads to a more complex dynamic, notably by increasing the shear viscosity. Moreover, it introduces an additional length scale to the system, the diameter of the particles, which eventually becomes comparable to the diameter of the ligament. In this paper, we experimentally investigate the thinning and pinch-off of drops of suspensions with two different sizes of particles. We characterize the thinning for different particle size ratios and different proportions of small particles. Long before the pinch-off, the thinning rate is that of an equivalent liquid whose viscosity is that of the suspension. Later, when the ligament thickness approaches the size of the large particles, the thinning accelerates and leads to an early pinch-off. We explain how the bidisperse particle size distribution lowers the viscosity by making the packing more efficient, which speeds up the thinning. This result can be used to predict the dynamics of droplet formation with bidisperse suspensions.

Flow Obstacles Minimize Surface Turbulence in Tilt Casting

Here we explore mold obstacles as means of attenuating flow and minimizing surface turbulence in tilt casting. Surface turbulence refers surface undulations and droplet pinch off that increase the surface area of the flow, which become defects as metal freezes. The Weber number that depends strongly on velocity governs the onset of surface turbulence, however, the critical Weber number for tilt casting with large vertical drops between crucible and mold remains unexplored. Here we explore this critical Weber number to find that ∼3 prevents surface turbulence and also explore the influence of staggered posts along the edge of the mold to find that they substantially decrease the local velocity to minimize surface turbulence. For the configurations explored, the staggered posts outperformed a side runner.

Droplet generation via oscillation of a transient bubble inside a funnel-like chamber

A new microdroplet generation technique based on the bubble oscillation in a funnel-like chamber is developed and investigated numerically using combined BEM-finite difference method. In this technique, the droplet is generated and ejected from the top opening of the chamber due to the oscillation of a transient bubble inside. The water in the chamber is exposed to the atmospheric air from the top side and is connected to a liquid reservoir from the bottom side. The bubble may be generated by an electrical spark, a laser beam or a high-intensity focused ultrasound (HIFU). The effects of the bubble-free surface distance (standoff distance) on the bubble and the ensuing droplet characteristics are evaluated. It was found that the when the bubble is initiated at the narrow part of the chamber it will be affected by the chamber's lateral wall so that parts of its surface that are closer to the wall takes the wall shape. As the bubble gets closer to the free surface, the wall effects are attenuated. In addition, the droplet size and pinch off time depend on the standoff distance so that as the standoff distance increases the droplet size decreases but its pinch off time increases.

A Compact Eulerian Representation of Axisymmetric Inviscid Vortex Sheet Dynamics

AbstractA classical problem in fluid mechanics is the motion of an axisymmetric vortex sheet evolving under the action of surface tension, surrounded by an inviscid fluid. Lagrangian descriptions of these dynamics are well‐known, involving complex nonlocal expressions for the radial and longitudinal velocities in terms of elliptic integrals. Here we use these prior results to arrive at a remarkably compact and exact Eulerian evolution equation for the sheet radius r(z, t) in an explicit flux form associated with the conservation of enclosed volume. The flux appears as an integral involving the pairwise mutual induction formula for vortex loop pairs first derived by Helmholtz and Maxwell. We show how the well‐known linear stability results for cylindrical vortex sheets in the presence of surface tension and streaming flows [A. M. Sterling and C. A. Sleicher, J. Fluid Mech. 68, 477 (1975)] can be obtained directly from this formulation. Furthermore, the inviscid limit of the empirical model of Eggers and Dupont [J. Fluid Mech. 262 205 (1994); SIAM J. Appl. Math. 60, 1997 (2000)], which has served as the basis for understanding singularity formation in droplet pinchoff, is derived within the present formalism as the leading‐order term in an asymptotic analysis for long slender axisymmetric vortex sheets and should provide the starting point for a rigorous analysis of singularity formation. © 2019 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, Inc.

Numerical simulation of the liquid-metal jet formation and droplet pinch-off in the cathode spot of a vacuum arc

A two-dimensional axisymmetric model has been developed to describe the formation of liquid-metal jet and the droplet pinch-off. These processes occur during the extrusion of the melt from the crater by the pressure of the cathode spot plasma of a vacuum arc. The jet formation has been numerically simulated for a copper cathode in the “inertial” mode of the melt splashing until the first droplet pinch-off. In this case, a jet with a longitudinal velocity gradient is formed. This gradient decreases the diameter of the jet and causes its elongation, resulting in droplet pinch-off. It has been shown that the mechanism of the droplet pinch-off is based on Rayleigh-Plateau instability. The droplet pinch-off time decreases with increasing jet velocity and increases for droplets of larger diameter. The simulation predicted the electrical explosion of the droplet-jet neck at the current density on the droplet surface ≥ 107 A·cm−2.