Contact angle is an important parameter in characterizing the wetting properties of fluids. The most common methods for measuring the contact angle is to measure it directly from the profile curve of a sessile drop, a method with certain inherent drawbacks. Here we describe an alternative method that uses the height and volume of a sessile drop as constraints to construct its profile by numerical integration of its two governing differential equations. The integration yields, self consistently, the average value of the contact angle along the entire contact line as well as the footprint radius of the drop and its crown radius of curvature. As a test case, the new method is used to obtain the contact angle of pure water on two different substrates, Teflon and Lucite. For each substrate, four drops ranging in volume from 10 μl to 40 μl are used. The computed contact angles are consistent across the four different drop sizes for each substrate and are in agreement with typical literature values.
We compare the lateral retention forces on sessile drops (which are drops that are placed on top of a solid surface), pendant drops (which are drops that are placed on the underside of the surface), and inverted sessile drops (which are drops that are first placed on top and then on the underside of the surface by flipping the surface). We have found experimentally that the retention force on a truly pendant drop is always smaller than that on a sessile drop. However, the retention force on an inverted sessile drop is comparable to, and usually larger than, that on a sessile drop. Thus, the re…
Fluid dynamics video of an evaporating sessile drop of capillary size is presented. The corresponding simulation represents the description taking into account jointly time dependent hydrodynamics, vapor diffusion and thermal conduction in an evaporating sessile drop. The fluid convection in the drop is driven by Marangoni forces associated with the temperature dependence of the surface tension. For the first time the evolution of the vortex structure in the drop during an evaporation process is obtained.
The diffuse-interface model (DIM) is a widely used tool for modeling fluid phenomena involving interfaces -- such as, for example, sessile drops (liquid drops on a solid substrate, surrounded by saturated vapor) and liquid ridges (two-dimensional sessile drops). In this work, it is proved that, surprisingly, the DIM does not admit solutions describing static liquid ridges. If, however, the vapor-to-liquid density ratio is small -- as, for example, for water at room temperature -- the ridges can still be observed as quasi-static states, as their evolution is too slow to be distinguishable from …
We address the issues raised in [R. Tadmor et al., Langmuir 2020, 36, 475-476]. In particular, we explain why we did not use Tadmor's theory to explain our results.
When sessile droplets are excited by ultrasonic traveling surface acoustic waves (SAWs), they undergo complex dynamics with both oscillations and translational motion. While the nature of the Rayleigh-Lamb quadrupolar drop oscillations has been identified, their origin and their influence on the drop mobility remains unexplained. Indeed the physics behind this peculiar dynamics is complex with nonlinearities involved both at the excitation level (acoustic streaming and radiation pressure) and in the droplet response (nonlinear oscillations and contact line dynamics). In this paper, we investig…
Capillarity always favors drop fusion. Nevertheless sessile drops from different but completely miscible liquids often do not fuse instantaneously upon contact. Rather, intermediate non-coalescence is observed. Two separate drop bodies, connected by a thin liquid neck move over the substrate. Supported by new experimental data a thin film hydrodynamic analysis of this state is presented. Presumably advective and diffusive volume fluxes in the neck region establish a localized and temporarily stable surface tension gradient. This induces a local surface (Marangoni) flow that stabilizes a travel…
Low power actuation of sessile droplets is of primary interest for portable or hybrid lab-on-a-chip and harmless manipulation of biofluids. In this paper, we show that the acoustic power required to move or deform droplets via surface acoustic waves can be substantially reduced through the forcing of the drops inertio-capillary modes of vibrations. Indeed, harmonic, superharmonic and subharmonic (parametric) excitation of these modes are observed when the high frequency acoustic signal (19.5 MHz) is modulated around Rayleigh-Lamb inertio-capillary frequencies. This resonant behavior results in…
We present a simple experiment that demonstrates how a water drop hanging from a Plexiglas surface (pendant drop) experiences a lateral retention force that is comparable to, and in some cases larger than, the lateral retention force on a drop resting on top of the surface (sessile drop). The experiment also affords a simple demonstration of the Coriolis effect in two dimensions.
The oscillation instability of sessile drops is ubiquitous in surface acoustic wave (SAW)-powered digital microfluidics. Yet, the physics underlying these phenomena has not been elucidated owing to the interplay between hydrodynamics, acoustics and capillarity. We decipher the instability by combining high-speed imaging with pressure measurements. We rationalize the observed behaviour with a model inspired from optomechanics, which couples an intracavity acoustic mode excited by the SAW to a surface deformation eigenmode through amplitude modulation and delayed radiation pressure feedback.
A new set of mathematical identities is presented for axi-symmetric sessile drops on flat and curved substrates. The geometrical parameters, including the apex curvature and height, and the contact radius, are related by the identities. The validity of the identities are checked by various numerical solutions both for flat and curved substrates.
We present a fully analytical solution for the natural oscillation of an inviscid sessile drop of arbitrary contact angle on a horizontal plate for the case for the case of low Bond number, when surface tension dominates gravity. The governing equations are expressed in terms of the toroidal coordinate system which yields solutions involving hypergeometric functions. Resonant frequencies are identified for zonal, sectoral and tesseral vibration modes. The predictions show good agreement with experimental data reported in the literature, with better agreement than the model of \citeauthor{bostw…
The evolution of the liquid bridge formed between two coalescing sessile yield-stress drops is studied experimentally. We find that the height of the bridge evolves similar to a viscous Newtonian fluid, $h_0\sim t$, before arresting at long time prior to minimizing its liquid/gas interfacial energy. We numerically solve for the final arrested profile shape and find it depends on the fluid's yield stress $τ_y$ and coalescence angle $α$, represented by the Bingham number $τ_y h_{drop} / σ$ modified by the drop's height-width aspect ratio. We present a scaling argument for the bridge's temporal e…
Colliding drops are widely encountered in everyday technologies and natural processes, from combustion engines and commodity sprays to raindrops and cloud formation. The outcome of a collision depends on many factors, including the impact velocity and the degree of head-on alignment, in addition to intrinsic properties like surface tension. Yet little is known on the binary impact dynamics of low surface tension oil drops on a low-wetting surface. We experimentally and numerically investigate the dynamics of an oil drop impacting an identical sessile drop sitting on a superamphiphobic surface.…
Inviscid flow within an evaporating sessile drop is analyzed. The field equation, E^2(Psi)=0, is solved for the stream function. The exact analytical solution is obtained for arbitrary contact angle and distribution of evaporative flux along the free boundary. Specific results and computations are presented for evaporation corresponding to both uniform flux and purely diffusive gas phase transport into an infinite ambient. Wetting and non-wetting contact angles are considered with flow patterns in each case being illustrated. The limiting behaviors of small contact angle and droplets of hemisp…
In our fluid dynamics video, we demonstrate our method of visualizing and identifying various mode shapes of mechanically oscillated sessile drops. By placing metal mesh under an oscillating drop and projecting light from below, the drop's shape is visualized by the visually deformed mesh pattern seen in the top view. The observed modes are subsequently identified by their number of layers and sectors. An alternative identification associates them with spherical harmonics, as demonstrated in the tutorial. Clips of various observed modes are presented, followed by a 10-second quiz of mode ident…
Drop evaporation is a simple phenomena but still unclear concerning the mechanisms of evaporation. A common agreement of the scientific community based on experimental and numerical work evidences that most of the evaporation occurs at the triple line. However, the rate of evaporation is still empirically predicted due to the lack of knowledge on the convection cells which develop inside the drop under evaporation. The evaporation of sessile drop is more complicated than it appears due to the coupling by conduction with the heating substrate, the convection and conduction inside the drop and t…
Several dynamical stages of the Marangoni convection of an evaporating sessile drop are obtained. We jointly take into account the hydrodynamics of an evaporating sessile drop, effects of the thermal conduction in the drop and the diffusion of vapor in air. The stages are characterized by different number of vortices in the drop and the spatial location of vortices. During the early stage the array of vortices arises near a surface of the drop and induces a non-monotonic spatial distribution of the temperature over the drop surface. The number of near-surface vortices in the drop is controlled…
Using an identity, directly derived from the Young-Laplace equation, the problem of the equilibrium shape of an axisymmetric sessile drop is reduced to a one-parameter shooting method problem. Based on the method the numerical solutions for drops with Bond number up to 15 are plotted. The agreement between the method and the ADSA-D method as well as the experimental data is tested. A Mathematica code based on the method is presented.
Temperature distributions and the corresponding vortex structures in an evaporating sessile droplet are obtained by performing detailed numerical calculations. A Marangoni convection induced by thermal conduction processes in the drop and the substrate is demonstrated to be able to result not only in a single vortex, but also in two or three vortices, depending on the ratio of substrate to fluid thermal conductivities, on the substrate thickness and the contact angle. The "phase diagrams" containing information on the number, orientation and spatial location of the vortices for quasistationary…
A partially-wetting sessile drop is driven by a sinusoidal pressure field that produces capillary waves on the liquid/gas interface. The analysis presented in Part 1 of this series (Bostwick & Steen 2014) is extended by computing response diagrams and phase shifts for the viscous droplet, whose three phase contact-line moves with contact-angle that is a smooth function of the contact line speed. Viscous dissipation is incorporated through the viscous potential flow approximation and the critical Ohnesorge number bounding regions beyond which a given mode becomes over-damped is computed. Davis …
This fluid dynamics video was submitted to the Gallery of Fluid Motion for the 2009 APS Division of Fluid Dynamics Meeting in Minneapolis, Minnesota. Drop evaporation is a simple phenomena but still unclear concerning the mechanisms of evaporation. A common agreement of the scientific community based on experimental and numerical work evidences that most of the evaporation occurs at the triple line. However, the rate of evaporation is still empirically predicted due to the lack of knowledge on the convection cells which develop inside the drop under evaporation. The evaporation of sessile drop…
The dynamics of an oscillated sessile drop of incompressible liquid with the focus on the contact line hysteresis is under theoretical consideration. The solid substrate is subject to transverse oscillations, which are assumed small amplitude and high frequency. The dynamic boundary condition that involves an ambiguous dependence of the contact angle on the contact line velocity is applied: the contact line starts to move only when the deviation of the contact angle exceeds a certain critical value. As a result, the stick-slip dynamics can be observed. The frequency response of surface oscilla…
Using lubrication theory, drying processes of sessile colloidal droplets on a solid substrate are studied. A simple model is proposed to describe temporal dynamics both the shape of the drop and the volume fraction of the colloidal particles inside the drop. The concentration dependence of the viscosity is taken into account. It is shown that the final shapes of the drops depend on both the initial volume fraction of the colloidal particles and the capillary number. The results of our simulations are in a reasonable agreement with the published experimental data. The computations for the drops…
We investigate the retention force factor of sessile droplets of pure (ethanol) and binary (water-ethanol) fluids laden with alumina nanoparticles placed on a critically inclined substrate. It is observed that while the critical angle of an ethanol droplet increases with an increase in nanoparticles concentration, for water-ethanol binary droplets, it reaches to plateau and decreases slightly after 0.6 wt.\% nanoparticle loading. The effect of composition and concentration of nanoparticles on the retention force factor is studied, and correlations are proposed for the retention force factor an…
We present a theoretical study related to a recent experiment on the coalescence of sessile drops. The study deals with the kinetics of relaxation towards equilibrium, under the action of surface tension, of a spheroidal drop on a flat surface. For such a nonspherical drop under partial wetting conditions, the dynamic contact angle varies along the contact line. We propose a new nonlocal approach to the wetting dynamics, where the contact line velocity depends on the geometry of the whole drop. We compare our results to those of the conventional approach in which the contact line velocity depe…
The impact of a sessile droplet with a moving meniscus, as encountered in processes such as dip-coating, generically leads to the entrapment of small air bubbles. Here we experimentally study this process of bubble formation by looking through the liquid using high-speed imaging. Our central finding is that the size of the entrapped bubble crucially depends on the location where coalescence between the drop and the moving meniscus is initiated: (i) at a finite height above the substrate, or (ii) exactly at the contact line. In the first case, we typically find bubble sizes of the order of a fe…
We have proposed and investigated a model of drying colloidal suspension drop placed onto a horizontal substrate in which the sol to gel phase transition occurs. The temporal evolution of volume fraction of the solute and the gel phase dynamics were obtained from numerical simulations. Our model takes into account the fact that some physical quantities are dependent on volume fraction of the colloidal particles.
When two sessile drops of the same liquid touch, they merge into one drop, driven by capillarity. However, the coalescence can be delayed, or even completely stalled for a substantial period of time, when the two drops have different surface tensions, despite being perfectly miscible. A temporary state of non-coalescence arises, during which the drops move on their substrate, only connected by a thin neck between them. Existing literature covers pure liquids and mixtures with low surface activities. In this paper, we focus on the case of large surface activities, using aqueous surfactant solut…
Theoretical description and numerical simulation of an evaporating sessile drop are developed. We jointly take into account the hydrodynamics of an evaporating sessile drop, effects of the thermal conduction in the drop and the diffusion of vapor in air. A shape of the rotationally symmetric drop is determined within the quasistationary approximation. Nonstationary effects in the diffusion of the vapor are also taken into account. Simulation results agree well with the data of evaporation rate measurements for the toluene drop. Marangoni forces associated with the temperature dependence of the…
We investigate numerically quasi-steady internal flows in an axially symmetrical evaporating sessile droplet depending on the ratio of substrate to fluid thermal conductivities, fluid volatility, contact angle and droplet size. Temperature distributions and vortex structures are obtained for droplets of 1-hexanol, 1-butanol and ethanol. To this purpose, the hydrodynamics of an evaporating sessile drop, effects of the thermal conduction in the droplet and substrate and diffusion of vapor in air have been jointly taken into account. The equations have been solved by finite element method using A…
The evaporation of sessile droplets on a flat surface involves a complex interplay between phase change, diffusion, advection and surface forces. In an attempt to significantly reduce the complexity of the problem and to make it manageable, we propose a simple model hinged on a surface free energy-based relaxation dynamics of the droplet shape, a diffusive evaporation model and a contact line pinning mechanism governed by a yield stress. Our model reproduces the known dynamics of droplet shape relaxation and of droplet evaporation, both in the absence and in the presence of contact line pinnin…
During coalescence of liquid drops contacting a solid, the liquid sweeps wetted and solid-projected areas. The extent of sweeping dictates the performance of devices such as self-cleaning surfaces, anti-frost coatings, water harvesters, and dropwise condensers. For these applications, weakly- and non-wetting solid substrates are preferred as they enhance drop dynamical behavior. Accordingly, our coalescence studies here are restricted to drops with contact angle 90° $\le θ_{0} \le$ 180°. Binary sessile drop coalescence is the focus, with volume of fluid simulations employed as the primary tool…
The investigation of evaporating liquid drop deposited onto a flat surface is of great importance for physical, engineering and medical applications. Novel analytical expressions are proposed to calculate the evaporation rate of sessile drop (mass loss per unit surface area per unit time) and total evaporation rate (mass loss per unit time). To obtain these results, the H.M. Macdonald's solution for a flat wedge was transformed by method of inversion in a sphere originally developed by J.C. Maxwell in the Treatise on Electricity and Magnetism with further derivation the solution for a lens bas…
Coalescence of sessile polymeric fluid drops on a partially wettable substrate exhibits a transition from inertial to viscoelastic regime at concentration ratio $c/c^* \sim 1$. Our findings unveil that the temporal evolution of the growing bridge height follows a power-law behaviour $t^b$, such that the coefficient $b$ continuously decreases from 2/3 in the inertial regime ($c/c^*<1$) to an asymptotic value of 1/2 in the visco-elastic regime ($c/c^*>1$). To account for fluid elasticity and characteristic time-scale in the viscoelastic regime, a modified thin film equation under lubrication app…
Natural or industrial flows of a fluid often involve droplets or bubbles of another fluid, pinned by physical or chemical impurities or by the roughness of the bounding walls. Here we study numerically one drop pinned on a circular hydrophilic patch, on an oscillating incline whose angle is proportional to $\sin(ω\,t)$. The resulting deformation of the drop is measured by the displacement of its center of mass, which behaves like a driven over-damped linear oscillator with amplitude $A(ω)$ and phase lag $φ(ω)$. The phase lag is ${\cal O}(ω)$ at small $ω$ like a linear oscillator, but the ampli…
Oscillation of sessile drops is important to many applications. In the present study, the natural oscillation of a sessile drop on flat surfaces with free contact lines (FCL) is investigated through numerical and theoretical analysis. The FCL condition represents a limit of contact line mobility, i.e. the contact angle remains constant when the contact line moves. In the numerical simulation, the interfaces are captured by the volume-of-fluid method and the contact angle at the boundary is specified using the height-function method. The oscillation frequencies for sessile drops with FCL are ma…
Spreading of respiratory diseases, such as COVID-19, from contaminated surfaces is dependent on the drying time of the deposited droplets containing the virus. The evaporation rate depends on environmental conditions, such as ambient temperature and relative humidity and physical properties (e.g., droplet volume, contact angle and composition). The respiratory droplets contain salt (NaCl), protein (mucin), and surfactant (dipalmitoylphosphatidylcholine) in addition to water, which are expected to influence the evaporation in a big way. A diffusion-based theoretical model for estimating the dry…
We study here the microscopic deformations of elastic lamellae constituting a superhydrophobic substrate under different wetting conditions of a sessile droplet using electrowetting. The deformation profiles of the lamellae are experimentally evaluated using confocal microscopy. These experimental results are then explained using a variational principle formalism within the framework of linear elasticity. We show that the local deformation profile of a lamella is mainly controlled by the net horizontal component of the capillary forces acting on its top due to the pinned droplet contact line. …
We numerically and experimentally investigate evaporation of a sessile droplet on a heated substrate. We develop a finite element (FE) model in two-dimensional axisymmetric coordinates to solve coupled transport of heat in the droplet and substrate, and of the mass of liquid vapor in surrounding ambient while assuming diffusion-limited, quasi-steady evaporation of the droplet. The two-way coupling is implemented using an iterative scheme and under-relaxation is used to ensure numerical stability. The FE model is validated against the published spatial profile of the evaporation mass flux and t…
We study the effects of vertical sinusoidal vibrations on a liquid droplet with a low surface tension (ethanol) deposited on a solid substrate. In a precise range of amplitudes and frequencies, the drop exhibits a dramatic worm-like shape instability with a strong symmetry breaking, comparable to the one observed by Pucci et al. (Phys. Rev. Lett., 106 (2011) 024503) on a vibrated floating lens. However, the geometry of our system is much simpler since it does not involve the oscillation and deformation of a liquid-liquid-air contact line. We show that the Faraday waves appearing on the surface…
The article reports droplet evaporation kinetics on inclined substrates. Comprehensive experimental and theoretical analyses of the droplet evaporation behaviour for different substrate declination, wettability and temperatures have been presented. Sessile droplets with substrate declination exhibit distorted shape and evaporate at different rates compared to droplets on the same horizontal substrate and is characterized by more often changes in regimes of evaporation. The slip stick and jump stick modes are prominent during evaporation. For droplets on inclined substrates, the evaporative flu…
The evaporation of a binary sessile ethanol-water droplet on an inclined substrate is studied experimentally just below the critical sliding angles for different substrate temperatures. A customized goniometer equipped with a CMOS camera and an infrared (IR) camera is used. The droplet is observed to remain pinned in the advancing side during the evaporation process, while the receding side contracts. The asymmetry in the advancing and receding contact angles of the droplet on inclined substrate results in complex thermo-solutal Marangoni convection that is captured through IR images. The drop…
The directional motion of sessile drops can be induced by slanted mechanical vibrations of the substrate. As previously evidenced \cite{Brunet07,Brunet09,Noblin09}, the mechanical vibrations induce drop deformations which combine axisymmetric and antisymmetric modes. In this paper, we establish quantitative trends from experiments conducted within a large range of parameters, namely the amplitude $A$ and frequency $f$ of the forcing, the liquid viscosity $η$ and the angle between the substrate and the forcing axis $α$. These experiments are carried out on weak-pinning substrates. For most para…
Line tension from dual-geometry sessile droplet measurements: Combining contact-angle size-dependence data for axisymmetric and cylindrical droplets to determine the line tension
Exact analytical solutions are derived for the Stokes flows within evaporating sessile drops of spherical and cylindrical cap shapes. The results are valid for arbitrary contact angle. Solutions are obtained for arbitrary evaporative flux distributions along the free surface as long as the flux is bounded at the contact line. The field equations, E^4(Psi)=0 and Del^4(Phi)=0, are solved for the spherical and cylindrical cap cases, respectively. Specific results and computations are presented for evaporation corresponding to uniform flux and to purely diffusive gas phase transport into an infini…
We study a spontaneous relaxation dynamics of arbitrarily shaped liquid drops on solid surfaces in the partial wetting regime. It is assumed that the energy dissipated near the contact line is much larger than that in the bulk of the fluid. We have shown rigorously in the case of quasi-static relaxation using the standard mechanical description of dissipative system dynamics that the introduction of a dissipation term proportional to the contact line length leads to the well known local relation between the contact line velocity and the dynamic contact angle at every point of an arbitrary cont…
