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 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…
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.
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.
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 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…
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…
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. …
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 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…
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…
We study the equilibrium solutions of a sessile drop on top of a horizontal substrate when it is partially covered by another inmiscible liquid, so that part of the drop is in contact with a third fluid (typically, air). The shapes of the interfaces are obtained by numerical integration of the differential equations resulting from the application of the Neumann's and Young's laws at the three fluids junction (triple point) and the two fluids--solid contact line (at the substrate), as well as no--flow boundary conditions at the wall of the container under axial symmetry. The formulation shows t…
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…
Non-coalescence of sessile drops from different but miscible liquids: Hydrodynamic analysis of the twin drop contour as self stabilizing, traveling wave
The so-called coffee stain effect has been intensively studied over the past decades, but most of the studies are focused on sessile droplets. In this paper, we analyse the origin of the difference between the deposition of suspended particles in a sessile drop and in an axisymmetric drop deposited on a fibre. First, we model the shape of a drop on a fibre and its evaporative flux with some approximations to derive analytical calculations. Then, for pinned contact lines, we solve the hydrodynamics equations in the liquid phase under the lubrication approximation to determine the flow velocity …
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.
The present study experimentally and numerically investigates the evaporation and resultant patterns of dried deposits of aqueous colloidal sessile droplets, when the droplets are initially elevated to a high temperature before being placed on a substrate held at ambient temperature. The system is then released for natural evaporation without applying any external perturbation. Infrared thermography and optical profilometry were used as essential tools for interfacial temperature measurements and quantification of the coffee-ring dimensions, respectively. Initially, a significant temperature g…
Stokes flows in a sessile hemispherical drop due to evaporation and surface tension gradient
Shubnikov Institute of Crystallography, Kurchatov Complex of Crystallography and Photonics of NRC‘Kurchatov Institute’（舒布尼科夫晶体学研究所，俄罗斯国家研究中心库尔恰托夫研究所晶体学与光子学库尔恰托夫综合体）
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.…
The inter-connection between the elasticity of a dielectric film and the wetting of a sessile drop on the same, under an applied electrical voltage, remains unaddressed. Here, we report the electrowetting-on-dielectric (EWOD) behaviour of sessile drops on dielectric elastomer films of varying elasticities- from an apparently rigid dielectric film to a soft, deformable dielectric film. Our results reveal that the elasticity of the underlying dielectric film provides an additional control over the droplet electrowetting behaviour, which may be best addressed from free energy based consideration,…
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 …
Effects of substrate temperature, substrate wettability and particles concentration are experimentally investigated for evaporation of a sessile water droplet containing colloidal particles. Time-varying droplet shapes and temperature of the liquid-gas interface are measured using high-speed visualization and infrared thermography, respectively. The motion of the particles inside the evaporating droplet is qualitatively visualized by an optical microscope and profile of final particle deposit is measured by an optical profilometer. On a non-heated hydrophilic substrate, a ring-like deposit for…
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…
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…
Water droplet freezing is a common phenomenon in our daily life. In both natural scenarios and industrial production, different surface inclinations bring distinctive deformation and freezing dynamics to frozen droplets. We explore the freezing of pendent and sessile droplets at different Bond number regimes. The effect of gravity on the droplet freezing process is analyzed by considering droplet morphology, freezing front dynamics, and freezing time. It is found that gravity can significantly influence droplet freezing processes via shaping the initial droplet, resulting in the flattening or …
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…
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…
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…
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…
We investigate deposit patterns and associated morphology formed after the evaporation of an aqueous droplet containing mono- and bi-dispersed colloidal particles. In particular, the combined effect of substrate heating and particle diameter is investigated. We employ high-speed visualization, optical microscopy and scanning electron microscopy to characterize the evaporating droplets, particle motion, and deposit morphology, respectively. In the context of mono-dispersed colloidal particles, an inner deposit and a typical ring form for smaller and larger particles, respectively, on a nonheate…
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…
We present an experimental and theoretical description of the kinetics of coalescence of two water drops on a plane solid surface. The case of partial wetting is considered. The drops are in an atmosphere of nitrogen saturated with water where they grow by condensation and eventually touch each other and coalesce. A new convex composite drop is rapidly formed that then exponentially and slowly relaxes to an equilibrium hemispherical cap. The characteristic relaxation time is proportional to the drop radius R * at final equilibrium. This relaxation time appears to be nearly 10 7 times larger th…
