Droplet Shape Evolution Research Articles

Dynamics of a droplet on the surfactant-infested free surface of another liquid

The dynamics of liquid droplets surfing over the surfactant-infested free surface of another liquid have been explored experimentally. We analyze the motion of oil droplets that has been initiated through the creation of a surface tension gradient resulting from the deposition of a drop of surfactant at the water surface contained in the petri dish. The experiments reveal that the location of surfactant deposition with respect to the droplet position influences its motion. Due to the presence of a surface tension gradient, the footprint area of the droplet reduces and its shape changes. We have studied the temporal variation in the velocity (|vx|) of the droplets in relation to their proximity to a wall. Based on the evolution of droplet shape and change in droplet velocity, the drop dynamics can be experimentally divided into four distinct zones. Results indicate that in zone-1, |vx| grows with t as |vx|≈tn, where n is between 0.8 and 1.0. The scaling argument shows that in this zone, the surface tension force dominates the drag force, and thereby, |vx| of the droplets increases linearly with t expressed as |vx|∝t. The experimental investigation and the scaling law exhibit a reasonable agreement. In zone-2, |vx| remains more or less constant, as it is postulated that the surface tension force balances the drag force. In zone-3, a decrease in surface tension force results in a deceleration of the droplets. In zone-4, the deceleration becomes more prominent as the droplet approaches the petri dish wall.

Atmospheric corrosion of iron under a single droplet: A new systematic multi-electrochemical approach

Utilizing a dedicated micro-sized three-electrode cell, this study systematically investigates early-stage electrochemical properties and corrosion behavior of pure iron under single droplets. Various volumes and NaCl concentrations were considered during the evaporation-driven shape and concentration evolution of single droplets. The measurements disclosed that reducing the droplet size from 5 µL to 1.5 µL at 0.01 M NaCl concentration, increased noise resistance (Rn) and polarization resistance (Rp) values. However, at 0.1 M and 0.2 M NaCl concentrations, reducing droplet size led to the domination of relatively high chloride ion concentration over oxygen diffusion, resulting in a very low Rn and Rp, and hence enhanced localized corrosion.

A multiphase model for exploring electrochemical Marangoni flow

A multiphase numerical model based on the volume of fluid (VOF) method is proposed to simulate the transient, electrochemically-generated Marangoni flow in a system comprising a NaOH electrolyte and a eutectic gallium–indium (EGaIn) metal droplet. The model incorporates appropriate equations to accurately represent the transport phenomena, including flow, electric potential, and electric current density, within the entire system. The model includes the transient variation in the interfacial tension as a function of electric current density at the interface, leading to the generation of Marangoni flow and enabling the tracking of droplet shape evolution. Notably, the model successfully captures the elongation of the droplet towards the cathode, which is validated through comparison with available experimental data.

Confined behavior of nanodroplets under electric field: A molecular dynamics simulation study

The wetting effect and related applications of nanodroplets have attracted extensive research interest. In this work, we report molecular dynamics simulations of the confinement dynamics of nanodroplets at different droplet-surface interactions, surface separations, and ion concentrations under the electric field. The results show that in the case of a single droplet, there is a critical value of surface wettability for the shape transition of the confined droplet. Below this value, the droplet shrinks into a spherical shape and attaches to the surface. Above this value, the droplet splits into two smaller droplets which locate at the bottom and top surfaces, respectively. For the two-droplet cases, a liquid bridge forms and some ions migrate into the pure water droplet under the electric field. The results reveal that the ion concentration has a significant effect on the shape evolution of droplets. The stable morphology of droplets under the electric field depends on the droplet-surface interactions as well as the ion concentration. We explore the underlying mechanisms causing the morphological transition by analyzing the intermolecular interactions and interfacial characteristics.

Magnetic-Field Mediated Active Propulsion of Ferrofluid Droplets on a Wire.

Droplet actuation on wires or fibers is highly relevant to digital microfluidics applications. Several different passive or active actuation mechanisms have been studied, including capillary pressure, thermal gradient, tilting, acoustics, and electrowetting, to name a few. However, the lack of precise control and accurate droplet positioning during its actuation on a wire limits the use of earlier methods. On the other hand, using a nonuniform magnetic field to actuate a droplet on a micrometer-size wire helps overcome the above limitations. In this context, we elucidate the motion of microliter-size ferrofluid droplets, smaller than the capillary length, in the presence of a permanent magnet using a simple yet robust experimental setup, wherein the clamshell shape droplets hang on a wire at a predetermined distance from the magnet. Beyond a critical volume, the droplet starts moving toward the magnet while exhibiting shape evolution continuously. Using a high-speed videography technique, we uncover the intricate relationship between the magnetowetting, favoring continuous shape evolution of droplets of different sizes, and their actuation dynamics (velocity) on a wire, which has scarcely been explored. The most significant contribution of this study is the detailed evaluation of the shape factor (k), relating the shape evolution of the droplet to its velocity through the contact angle hysteresis force. A theoretical framework has also been proposed, comprising different forces in action, which is found to set forth a good match with the experimental results. The results from this study are expected to open up new avenues in digital microfluidics, specifically in drug delivery, ferrobotics, and droplet logic gates, to name a few.

Flattening of a hollow droplet impacting a solid surface

The interaction of a hollow droplet impacting a solid surface occurs in several applications, including controllable biomedicine and thermal spray coating. Understanding the physics of the hollow droplet spreading is the key to maintaining the mass transfer process in all relevant applications. In this work, a comprehensive experimental, numerical and theoretical study is performed on water hollow droplets impacting a rigid surface to better understand the flattening process of a hollow droplet. In the numerical part, compressible Navier–Stokes equations are solved using the volume of fluid (VOF) method in a two-dimensional (2-D)-axisymmetric model. The comparison of simulation results with the experimental photographs shows that the numerical solution can correctly predict the hollow droplet shape evolution. The results show that the spreading diameter and height of the counter-jet formed after the hollow droplet impact grow with impact velocity. Investigating the size and location of the entrapped bubble shows an optimum bubble size that facilitates the hollow droplet flattening. It is also shown that the ripples on splats produced by the hollow droplets with a larger bubble size are higher than those of small bubbles. In the end, a theoretical model is developed to analyse the maximum spreading diameter of the hollow droplet impact analytically. Its prediction is in good agreement with the experimental and numerical results.

Small deformation theory for a magnetic droplet in a rotating field

A three-dimensional small deformation theory is developed to examine the motion of a magnetic droplet in a uniform rotating magnetic field. The equations describing the droplet's shape evolution are derived using two different approaches—a phenomenological equation for the tensor describing the anisotropy of the droplet and the hydrodynamic solution using the perturbation theory. We get a system of ordinary differential equations for the parameters describing the droplet's shape, which we further analyze for the particular case when the droplet's elongation is in the plane of the rotating field. The qualitative behavior of this system is governed by a single dimensionless quantity τω—the product of the characteristic relaxation time of small perturbations and the angular frequency of the rotating magnetic field. Values of τω determine whether the droplet's equilibrium will be closer to an oblate or a prolate shape, as well as whether its shape will undergo oscillations as it settles to this equilibrium. We show that for small deformations, the droplet pseudo-rotates in the rotating magnetic field—its long axis follows the field, which is reminiscent of a rotation; nevertheless, the torque exerted on the surrounding fluid is zero. We compare the analytic results with boundary element simulation to determine their accuracy and the limits of the small deformation theory.

Quantifying the interacting mechanisms in shape evolution of sessile volatile droplets

Non-uniform distribution of interfacial mass flux across an evaporating droplet will subsequently induce a temperature gradient across the liquid-air interface, and result in thermal Marangoni stress that reforms the flow field inside the droplet. Recent study (Shiri, et al. Phys. Rev. Lett. 2021, Yang, et al. Langmuir 2022) confirmed the role of thermal Marangoni effect on the shape of evaporating single component droplets on completely wetting substrates. Nevertheless, a comprehensive evaluation of the interacting mechanisms is still lacking due to the intrinsic limitation of experimental techniques to decompose the different influence factors. To this end, we formulate a mathematical model to simulate the evaporation of volatile droplets on completely wetting substrates based on the lubrication theory. With this model, we elucidate the interacting physical mechanisms that governs the droplet kinetics during phase change, which includes the capillary effect, the thermal Marangoni effect, the interface motion due to evaporation, as well as the removal of energy barriers by precursor film. A phase diagram is summarized in terms of the Knudsen number and liquid-to-solid Biot number, which quantifies the prevalence of the interacting effects.

Study on dynamic behavior of water droplet impacting on super-hydrophobic surface with micro-pillar structures by VOF method

Super-hydrophobic surfaces have been widely developed due to their unique properties, such as adhesion reduction and corrosion inhibition. It is of great significance to explore droplet dynamic process when impacting on super-hydrophobic surface, which can contribute to understanding the mechanism of super-hydrophobicity, and designing self-cleaning surfaces with ultralow adhesion and excellent corrosion resistance. Herein, three-dimensional (3D) numerical model was developed to study the droplet dynamic behaviors on super-hydrophobic NiTi surface. The volume of fluid (VOF) method was carried out to track the multi-phase flow interface and then predict droplet shape evolution during the spreading, retraction, and rebounding stages. In particular, the compression- and stretch-rebounding behaviors were investigated in detail. Based on the results of numerical simulation, the emphasis was placed on the relationship between droplet shape transformation and internal velocity/pressure distribution, revealing the mechanism of droplet morphological evolution during the impingement process. Then the experimental investigations were conducted on the planar and super-hydrophobic NiTi surfaces at different impact velocities to further verify the simulation models. It is indicated that the falling droplet would directly deposit on the planar surface, while it could fully rebound off the pillar-patterned super-hydrophobic surface. Moreover, the motion mechanism of droplet impacting on super-hydrophobic surface was elaborated, which can provide an essential instruction to effectively reduce liquid adhesion and improve anticorrosion ability for the NiTi material.

Modeling vorticity stretching of viscoelastic droplets during shearing flow

In the present work, we focus on the large deformation of a viscoelastic droplet suspended in a Newtonian matrix. We use the constrained volume model as a basic theoretical framework for the description of droplet shape evolution, and we account for the viscoelasticity of the droplet phase using the single-mode Giesekus model. The velocity gradient term describing flow inside the droplet is modified by the viscoelastic stresses, and the resulting model—we herein call the non-Newtonian constrained volume model—is calibrated by matching its predictions to those of the model of Yu et al. [J. Rheol. 48, 417–438 (2004)] at small deformation and slow flow conditions. The non-Newtonian constrained volume model is then examined under conditions of large deformation and/or fast flow, and its predictions are compared against experimental results. The new model shows the growth of the droplet's width in the vorticity direction at conditions of large capillary number, moderate-to-large viscosity ratio, and small elastocapillary number.

A method to measure vapor concentration of droplet evaporation based on background oriented Schlieren

The direct visualization of vapor distribution and quantitative measurement of vapor concentration plays an important role in studying droplet evaporation. In this study, we propose a method to measure the vapor concentration field based on the modified background oriented Schlieren, inverse Radon transform, high-speed imaging, and image processing. The method is demonstrated for the measurement of the transient concentration field during the impact of volatile droplets on solid surfaces. The results show that the vapor distribution is remarkably different from the diffusion-limited model because of the strong convection. The complex structure of the vapor cloud is the result of the interplay among fluid inertia, fluid evaporation, vapor diffusion, vapor convection, and droplet shape evolution.

Shape evolution of drops on surfaces of different wettability gradients

Passive droplet manipulation on open surfaces can be achieved by creating a wettability gradient on surfaces, which is essential in the fabrication of low cost biological and biochemical chips. We performed 3D numerical simulations to analyze the droplet motion on a broad range of wettability gradient surfaces. We found that the droplet shape evolves with time to maintain a minimum energy state, and the surface energy of the droplet is identical at a particular non-dimensional time (t∗) for different wettability gradient surfaces. Although the droplet is at various locations at a fixed t∗, the shape of the droplet is found to be identical. The physics behind this interesting phenomenon of identical droplet shape formation is explored. A co-relation for t∗ is proposed to get the dependency of t∗ on various geometrical parameters and fluid properties. Three distinct regimes of the droplet identical shape on different wettability gradient surfaces are shown using a regime plot. Along with the identical droplet shape phenomena, the detailed understanding of the dynamics of the droplet shape evolution on different wettability gradient surfaces gives an insight for better open surface passive manipulation.

An Analytical Two-Dimensional Linearized Droplet Shape Model for Combined Tangential and Normal Body Forces

In view of emerging research on forced wetting under complex applied forces, a simple model for a droplet shape evolution is developed here. In particular, the model refers to droplet spreading under quasisteady conditions. The corresponding linearized two-dimensional Young–Laplace equation is solved analytically resulting in a system of two equations that relates the droplet shape features to each other. Despite its simplicity, the final model produces a wealth of droplet behaviors when combined with the physical requirement that the contact angle should be within a particular range of values. Indicative results of the droplet behavior under several forces scenarios are examined here exhibiting why the present model is useful for designing experimental campaigns on forced spreading.

Influence of the viscosity and charge mobility on the shape deformation of critically charged droplets

In this work, we model and simulate the shape evolution of critically charged droplets, from the initial spherical shape to the charge emission and back to the spherical shape. The shape deformation is described using the viscous correction for viscous potential flow model, which is a potential flow approximation of the Navier-Stokes equation for incompressible Newtonian fluids. The simulated shapes are compared to snapshots of experimentally observed drop deformations. We highlight the influence of the dimensionless viscosity and charge carrier mobility of the liquid on the shape evolution of droplets and discuss the observed trends. We give an explanation as to why the observed deformation pathways of positively and negatively charged pure water droplets differ and give a hint as to why negatively charged water droplets emit more charge during charge breakup than positively charged ones.

Sessile droplets shape response to complex body forces

The majority of studies on forced wetting of sessile droplets refers to application of a steadily increasing normal or tangential force or to a specific combination of these forces arising from tilting the substrate. The above constitute well-defined test conditions but are not representative of what is encountered in most industrial applications. To approach realistic industrial conditions and so also expand existing wetting theories, an evaluation of droplet shape deformation under the influence of a complicated evolution of body forces, is performed herein. To this aim, two sets of experiments are carried out in Kerberos device creating force fields by combination of gravitational and centrifugal forces: i) by oscillations of the tilting angle at constant rotation speed and ii) by alternating small step-increases of the rotation speed and the tilting angle, one after the other, so as to trail the symmetric side profile (SSP) curve of a droplet. The latter is done in three or six steps following two different paths: increasing first either the rotation speed or the tilting angle. The aim of the experiments is, on one hand, to explore droplet deformation under cycles of increasing/decreasing tangential forces and, on the other hand, to analyze the effect of residual tangential forces on droplet shape. Several features related to the droplet shape evolution are investigated such as contact angles, side and top contour profiles and droplet 3-dimensional shape. The resulting droplet profiles are analyzed using numerical solutions of the Young-Laplace equation. It is found that the 3-dimensional Young Laplace equation can describe very accurately the experimental profiles.

Dynamics of droplets impacting hydrophilic surfaces decorated with a hydrophobic strip

Mixed-wettability surfaces have been extensively adopted to manipulate impact dynamics of droplets. In this work, a three-dimensional numerical model is developed to investigate the splitting behaviors of droplets when they impact a hydrophobic strip decorated on a hydrophilic background surface. The volume of fluid (VOF) method incorporated with mesh-refinement technique and dynamic contact angle model is employed to capture the shape evolution of droplets accurately. Dynamic behaviors of impact droplets are analyzed for various strip widths, strip hydrophobicities, and surface hydrophilicities at three typical Weber numbers. The results show that the droplet can be successfully split into two symmetric small droplets for all the three Weber numbers, so long as the strip width and wettability contrast between the strip and background surface are properly designed. The non-axisymmetric spreading and retraction of the film triggered by the strip lead to a shorter spreading and a faster retraction on the hydrophobic strip than the hydrophilic surface, which causes the formation of a liquid bridge on the strip and reduces the viscous dissipation on the strip. Thus, the non-axisymmetric spreading and retraction dynamics as well as the reduced viscous dissipation are responsible for the droplet splitting. A simplified theoretical model is developed to predict the splitting or no splitting behavior on the basis of energy conservation principle, which agrees with the present simulations well. Finally, the simulated results are assembled in phase diagrams to obtain the criterion for the droplet splitting from the critical strip width and the critical wettability contrast between the strip and background surface.

Shape Evolution of Droplets Growing on Linear Microgrooves.

Anisotropic spreading of liquids and elongated droplet shapes are often encountered on surfaces decorated with a periodic micropattern of linear surface topographies. Numerical calculations and wetting experiments show that the shape evolution of droplets that are slowly growing on a surface with parallel grooves can be grouped into two distinct morphological regimes. In the first regime, the liquid of the growing droplet spreads only into the direction parallel to the grooves. In the second regime, the three-phase contact line advances also perpendicular to the grooves, whereas the growing droplets approach a scale-invariant shape. Here, we demonstrate that shapes of droplets in contact with a large number of linear grooves are identical to the shapes of droplets confined to a plane chemical stripe, where this mapping of shapes is solely based on the knowledge of the cross section of the linear grooves and the material contact angle. The spectrum of interfacial shapes on the chemical stripe can be exploited to predict the particular growth mode and the asymptotic value of the base eccentricity in the limit of droplets covering a large number of grooves. The proposed model shows an excellent agreement with experimentally observed base eccentricities for droplets on grooves of various cross sections. The universality of the model is underlined by the accurate match with available literature data for droplet eccentricities on parallel chemical stripes.

The shape evolution of liquid droplets in miscible environments

Miscible liquids often come into contact with one another in natural and technological situations, commonly as a drop of one liquid present in a second, miscible liquid. The shape of a liquid droplet present in a miscible environment evolves spontaneously in time, in a distinctly different fashion than drops present in immiscible environments, which have been reported previously. We consider drops of two classical types, pendant and sessile, in building upon our prior work with miscible systems. Here we present experimental findings of the shape evolution of pendant drops along with an expanded study of the spreading of sessile drops in miscible environments. We develop scalings considering the diffusion of mass to group volumetric data of the evolving pendant drops and the diffusion of momentum to group leading-edge radial data of the spreading sessile drops. These treatments are effective in obtaining single responses for the measurements of each type of droplet, where the volume of a pendant drop diminishes exponentially in time and the leading-edge radius of a sessile drop grows following a power law of $t^{1/2}$ at long times. A complementary numerical approach to compute the concentration and velocity fields of these systems using a simplified set of governing equations is paired with our experimental findings.

Modeling of Droplet Impact onto Polarized and Nonpolarized Dielectric Surfaces.

This paper is concerned with simulation of the droplet impact on a dielectric surface, referred to as the dynamic electrowetting-on-dielectric (DEWOD). In particular, we seek to shed more light on the fundamental processes occurring during the impact of an electrically conducting droplet onto a dielectric surface with and without an applied voltage. The liquid in the droplet is an ionic conductor (a leaky dielectric). This work employs an approach based on Cahn-Hilliard-Navier-Stokes (CHNS) modeling. The simulations are validated by predicting the equilibrium contact angle, droplet oscillations, and charge density estimation. Then, four cases of droplet impact are studied, namely, the impact onto a surface with no voltage applied and the impacts onto the surfaces with 2, 4, and 6 kV applied. The modeling results of water droplet impact allow for direct comparison with the experimental results reported by Lee et al. [ Langmuir 2013, 29, 7758]. The results reveal the electric field, the body forces acting on the droplet, the velocity and pressure fields inside and outside the droplet, as well as the free charge density and the electric energy density. The model predicts the droplet shape evolution (e.g., the spreading distance over the surface and the rebound height) under different conditions that are consistent with the experimental observations. Thus, our findings provide new qualitative and quantitative insights into the droplet manipulation that can be used in novel applications of the DEWOD phenomenon.

Dynamic Surface Wetting and Heat Transfer in a Droplet-Particle System of Less Than Unity Size Ratio.

Dynamic surface wetting of particles in contact with droplet is a complex phenomenon ubiquitously encountered in many multiphase systems of industrial importance. In this study, we address this aspect by investigating impact behavior of a water droplet (diameter = 2.9 ± 0.1 mm) in the Weber number (We) range from ~4 to 104 on a stationary spherical brass particle (diameter = 10 mm) with and without heat transfer using a combination of high speed imaging and computational fluid dynamics (CFD) modeling approach. In cold state interactions (20°C), droplet exhibited oscillatory interfacial motion comprising periodic spreading and recoiling motion. Interactions involving heat transfer were studied in film boiling regime (350°C) and two outcomes were noted—droplet rebound and disintegration. A coupled Level Set and Volume of Fluid (VOF) approach based multiphase CFD model was utilized to predict the dynamic spread ratio and transient evolution of droplet shape during the interaction. To capture the complex contact line motion realistically, a continuous time varying profile of experimentally measured dynamic contact angles was used as a wall boundary condition for the cold interactions which provided good agreement with experimentally measured droplet spread ratio. In film boiling regime, droplet spread ratio was correlated to impact Weber number and a power law trend was obtained. Rebound and disintegration outcomes were characterized by the droplet-particle contact time. For simulating interactions in film boiling regime, a constant contact angle in the limit of super-hydrophobic surface was implemented in the CFD model to account for the apparent non-wetting effect due to vapor film formation at the contact area. A sensitivity analysis was performed involving three different contact angle boundary conditions (θs = 150, 160, and 170°) to represent the surface hydrophobicity. CFD model predicted interaction outcomes and droplet spread ratios were in reasonable agreement with the experiment at different impact Weber numbers. Increase in spherical surface heat flux and corresponding rise in droplet temperature at different impact Weber numbers were also quantified which showed an increasing trend up to a critical Weber number for droplet disintegration.