The evaporation of sessile drops in quiescent air is usually governed by vapour diffusion. For contact angles below $90^\circ$, the evaporative flux from the droplet tends to diverge in the vicinity of the contact line. Therefore, the description of the flow inside an evaporating drop has remained a challenge. Here, we focus on the asymptotic behaviour near the pinned contact line, by analytically solving the Stokes equations in a wedge geometry of arbitrary contact angle. The flow field is described by similarity solutions, with exponents that match the singular boundary condition due to evaporation. We demonstrate that there are three contributions to the flow in a wedge: the evaporative flux, the downward motion of the liquid-air interface and the eigenmode solution which fulfils the homogeneous boundary conditions. Below a critical contact angle of $133.4^\circ$, the evaporative flux solution will dominate, while above this angle the eigenmode solution dominates. We demonstrate that for small contact angles, the velocity field is very accurately described by the lubrication approximation. For larger contact angles, the flow separates into regions where the flow is reversing towards the drop centre.
An evaporating droplet is a dynamic system in which flow is spontaneously generated to minimize the surface energy, dragging particles to the borders and ultimately resulting in the so-called "coffee-stain effect". The situation becomes more complex at the droplet's surface, where surface tension gradients of different nature can compete with each other yielding different scenarios. With careful experiments and with the aid of 3D particle tracking techniques, we are able to show that different types of surfactants turn the droplet's surface either rigid or elastic, which alters the evaporating…
For a small sessile or pendant droplet it is generally assumed that gravity does not play any role once the Bond number is small. This is even assumed for evaporating binary sessile or pendant droplets, in which convective flows can be driven due to selective evaporation of one component and the resulting concentration and thus surface tension differences at the air-liquid interface. However, recent studies have shown that in such droplets gravity indeed can play a role and that natural convection can be the dominant driving mechanism for the flow inside evaporating binary droplets (Edwards et…
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…
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…
This study aims to build a theoretical model to describe the ignition process of spherical spray flame laden with fuel droplets under overall fuel-lean condition. Two characteristic fronts are introduced to reveal where the evaporation starts and ends in the ignition process. The results demonstrate that the flame has been strengthened with extra fuel droplet adding while the minimum ignition energy (MIE) is decreased. The evolution of the two evaporation fronts in the flame kernel and propagating branches has also be revealed coupled with the evaporation zone length.
The flow in an evaporating glycerol-water binary sub-millimeter droplet with Bond number Bo $\ll$ 1 is studied both experimentally and numerically. First, we measure the flow fields near the substrate by micro-PIV for both sessile and pendant droplets during the evaporation process, which surprisingly show opposite radial flow directions -- inward and outward, respectively. This observation clearly reveals that in spite of the small droplet size, gravitational effects play a crucial role in controlling the flow fields in the evaporating droplets. We theoretically analyze that this gravity-driv…
The stationary single vortex Marangoni convection in an axially symmetrical sessile drop of capillary size is considered. The detailed description of the fluid flows is presented for a wide range of contact angles, which takes into account the boundary conditions and the mass balance equation, without explicitly solving the Navier--Stokes equations. The analytical approach developed is compared with the results of numerical simulations and demonstrated to describe reasonably well the single-vortex Marangoni flows. This indicates the substantial role of the boundary conditions in the problem.
The pioneering work of Deegan et al. [Nature 389, (1997)] showed how a drying sessile droplet suspension of particles presents a maximum evaporating flux at its contact line which drags liquid and particles creating the well known coffee stain ring. In this Fluid Dynamics Video, measurements using micro Particle Image Velocimetry and Particle Tracking clearly show an avalanche of particles being dragged in the last moments, for vanishing contact angles and droplet height. This explains the different characteristic packing of the particles in the layers of the ring: the outer one resembles a cr…
Evaporating liquid droplets are omnipresent in nature and technology, such as in inkjet printing, coating, deposition of materials, medical diagnostics, agriculture, food industry, cosmetics, or spills of liquids. While the evaporation of pure liquids, liquids with dispersed particles, or even liquid mixtures has intensively been studied over the last two decades, the evaporation of ternary mixtures of liquids with different volatilities and mutual solubilities has not yet been explored. Here we show that the evaporation of such ternary mixtures can trigger a phase transition and the nucleatio…
We study the problem of evaporating drops contracting to a point. Going back to Maxwell and Langmuir, the existence of a spherical solution for which evaporating drops collapse to a point in a self-similar manner is well established in the physical literature. The diameter of the drop follows the so-called $D^{2}$ law: the second power of the drop-diameter decays linearly in time. In this study we provide a complete mathematical proof of this classical law. We prove that evaporating drops which are initially small perturbations of a sphere collapse to a point and the shape of the drop converge…
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…
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…
The deposition of particles on a surface by an evaporating sessile droplet is important for phenomena as diverse as printing, thin-film deposition and self-assembly. The shape of the final deposit depends on the flows within the droplet during evaporation. These flows are typically determined at the onset of the process by the intrinsic physical, chemical and geometrical properties of the droplet and its environment. Here, we demonstrate deterministic emergence and real-time control of Marangoni flows within the evaporating droplet by an external point-source of vapor. By varying the source lo…
Alkane mixtures are model systems where the influence of surface tension gradients during the spreading and the evaporation of wetting drops can be easily studied. The surface tension gradients are mainly induced by concentration gradients, mass diffusion being a stabilising process. Depending on the relative concentration of the mixture, a rich pattern of behaviours is obtained.
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…
A finite element model was developed to compute the fluid flow inside a sessile evaporating droplet on hydrophilic substrate in ambient conditions. The evaporation is assumed as quasi-steady and the flow is considered as axisymmetric with a pinned contact line. The Navier-Stokes equations in cylindrical coordinates were solved inside the droplet. Galerkin weight residual approach and velocity pressure formulation was used to discretise the governing equations. Six node triangular mesh and quadratic shape functions were used to obtain higher accuracy solutions. Radial velocity profiles in axial…
The Greek aperitif Ouzo is not only famous for its specific anise-flavored taste, but also for its ability to turn from a transparent miscible liquid to a milky-white colored emulsion when water is added. Recently, it has been shown that this so-called Ouzo effect, i.e. the spontaneous emulsification of oil microdroplets, can also be triggered by the preferential evaporation of ethanol in an evaporating sessile Ouzo drop, leading to an amazingly rich drying process with multiple phase transitions [H. Tan et al., Proc. Natl. Acad. Sci. USA 113(31) (2016) 8642]. Due to the enhanced evaporation n…
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.
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 evaporation of a drop from a liquid subphase is investigated. The two liquids are immiscible, and the contact angles between them are given by the Neumann construction. The evaporation of the drop gives rise to flows in both liquids, which are coupled by the continuity of velocity and shear-stress conditions. We derive self-similar solutions to the velocity fields in both liquids close to the three-phase contact line, where the drop geometry can be approximated by a wedge. We focus on the case where Marangoni stresses are negligible, for which the flow field consists of three contributions…
We study the evolution of the mass of evaporating single microdroplets of sodium do-decyl sulphate (SDS) / diethylene glycol (DEG) mixture. First, we recognise and deconvolute the influence of residual water evaporation [Kolwas et al. Soft Matter 2019;15:1825], which accelerates the composite droplet evaporation, a simple exponential decay of the evaporating droplet surface change rate. This enables us to study the influence of SDS concentration on the composite droplet evaporation. Next, we establish a simple relationship between the average SDS concentration and the droplet evaporation rate …
Hypothesis: Evaporation of surfactant droplets on leaves is complicated due to the complex physical and chemical properties of the leaf surfaces. However, for certain leaf surfaces for which the evaporation process appears to follow the standard constant-contact-radius or constant-contact-angle modes, it should be possible to mimic the droplet evaporation with both a well-chosen synthetic surface and a relatively simple mathematical model. Experiments: Surfactant droplet evaporation experiments were performed on two commercial crop species, wheat and capsicum, along with two synthetic surfaces…
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…
Evaporating salty droplets are ubiquitous in nature, in our home and in the laboratory. Interestingly, the transport processes in such apparently simple systems differ strongly from evaporating "freshwater" droplets since convection is partly inverted due to Marangoni stresses. Such an effect has crucial consequences to the salt crystallization process and to the deposits left behind. In this work we show unprecedented measurements that, not only confirm clearly the patterns of the flow inversion, but also elucidate their impact on the distribution of non-volatile solutes. Contrary to what has…
An analysis of the projectile motion in stagnant air is presented for an evaporating respiratory micro-droplet which has been ejected from the mouth as an isolated droplet. It is assumed that the air resistance is a nonlinear function of the droplet's velocity and that the rate of decrease of the droplet's external surface area depends only on the relative humidity and the ambient temperature. The droplet's initial content is considered to be 98 wt% water, 1 wt% salt and 1 wt% protein. The change of the average density of the droplet due to water evaporation is determined, up to the instant wh…
Phase segregation triggered by selective evaporation can emerge in multicomponent systems, leading to complex physiochemical hydrodynamics. Recently, Li et al. (Phys. Rev. Lett., vol. 120, 2018, 224501) and Kim & Stone (J. Fluid Mech., vol. 850, 2018, pp. 769-783) reported a segregative behavior (i.e., demixing) in an evaporating binary droplet. In this work, by means of experiments and theoretical analysis, we investigate the flow dynamics after the occurrence of the phase segregation. As example, we take the 1,2-hexanediol-water binary droplet system. First, we experimentally reveal the over…
Evaporation-driven particle self-assembly can be used to generate three-dimensional microstructures. We present a new method to create these colloidal microstructures, in which we can control the amount of particles and their packing fraction. To this end, we evaporate colloidal dispersion droplets on a special type of superhydrophobic micro-structured surface, on which the droplet re- mains in Cassie-Baxter state during the entire evaporative process. The remainders of the droplet consist of a massive spherical cluster of the microspheres, with diameters ranging from a few tens up to several …
Building on the recent theoretical work of Wray, Duffy and Wilson [J. Fluid Mech. 884, A45 (2020)] concerning the competitive diffusion-limited evaporation of multiple thin sessile droplets in proximity to each other, we obtain theoretical predictions for the spatially non-uniform densities of the contact-line deposits (often referred to as "coffee stains" or "ring stains") left on the substrate after such droplets containing suspended solid particles have completely evaporated. Neighbouring droplets interact via their vapour fields, which results in a spatially non-uniform "shielding" effect.…
The evaporation of dispersed, liquid droplets in jet-sprays occurs in several industrial applications and in natural phenomena. Despite the relevance of the problem, a satisfactory comprehension of the mechanisms involved has not still be achieved because of the wide range of turbulent scales and the huge number of droplets involved. In this context, we address a DNS of a turbulent jet spray at relatively high Reynolds number, i.e. Re=10,000. We focus on the effect of the jet Re on the evaporation process and the preferential segregation of droplets, comparing the outcomes also with a DNS at l…
We investigate evaporation of a sessile droplet on a non-wetted surface in the framework of diffusion-limited and quasi-steady evaporation. We extend previous models and numerically solve Laplace equation for the diffusion of liquid vapor in ambient. We propose a unified, simple and accurate expression of the evaporation mass flux valid for 90^o < theta < 180^o, where theta is the equilibrium contact angle. In addition, using the derived expression of the evaporation mass flux, we propose a simple and accurate expression of the evaporation mass rate for a non-wetted surface, which does not exh…
The evaporation of droplets occurs in a large variety of natural and technological processes such as medical diagnostics, agriculture, food industry, printing, and catalytic reactions. We study the different droplet morphologies adopted by an evaporating droplet on a surface with an elliptical patch with a different contact angle. We perform experiments to observe these morphologies and use numerical calculations to predict the effects of the patched surfaces. We observe that tuning the geometry of the patches offers control over the shape of the droplet. In the experiments, the drops of vario…
A model is developed for describing the transport of charged colloidal particles in an evaporating sessile droplet on the electrified metal substrate in the presence of a solvent flow. The model takes into account the electric charge of colloidal particles and small ions produced by electrolytic dissociation of the active groups on the colloidal particles and solvent molecules. We employ a system of self-consistent Poisson and Nernst--Planck equations for electric potential and average concentrations of colloidal particles and ions with the appropriate boundary conditions. The fluid dynamics, …
The shape of deposits obtained from drying drops containing colloidal particles matters for technologies such as inkjet printing, microelectronics and bioassay manufacturing. In this work, the formation of deposits during the drying of nanoliter drops containing colloidal particles is investigated experimentally with microscopy and profilometry, and theoretically with an inhouse finite-element code. The system studied involves aqueous drops containing titania nanoparticles evaporating on a glass substrate. Deposit shapes from spotted drops at different pH values are measured using a laser prof…
On the role of water activity on the formation of a protein-rich coffee ring in an evaporating multicomponent drop
Carlos III University of Madrid（马德里卡洛斯三世大学）；Centro Nacional de Biotecnología, Consejo Superior de Investigaciones Científicas CSIC（西班牙国家生物技术中心，西班牙高等科学研究理事会）
Ring-shaped deposits can be often found after a droplet evaporates on a substrate. If the fluid in the droplet is a pure liquid and its contact line remains pinned during the process, the mechanism behind such ring-shaped deposition is the well-known coffee-stain effect. However, adding small amounts of salt to such a droplet can change the internal flow dramatically and consequently change the deposition mechanism. Due to an increase of surface tension in the contact line region, a Marangoni flow arises which is directed from the apex of the droplet towards the contact line. As a result, part…
The effect of evaporation on droplet sedimentation times is crucial for estimating the risk of infection from virus-containing airborne droplets. For droplet radii in the range 100 nm < R < 60 μm, evaporation can be described in the stagnant-flow approximation and is diffusion limited. Analytical equations are presented for the droplet evaporation rate, the time-dependent droplet size and the sedimentation time, including the significant effect of evaporation cooling. Evaporation makes the time for large droplets to sediment much longer and thus significantly increases the viral air load. Usin…
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…
Smoothed particle hydrodynamics (SPH) method has been increasingly used for simulating fluid flows, however its ability to simulate evaporating flow requires significant improvements. This paper proposes an SPH method for evaporating multiphase flows. The present SPH method can simulate the heat and mass transfers across the liquid-gas interfaces. The conservation equations of mass, momentum and energy were reformulated based on SPH, then were used to govern the fluid flow and heat transfer in both the liquid and gas phases. The continuity equation of the vapor species was employed to simulate…
A theoretical framework is established to model the evaporation from continuously fed droplets, promising tools in the thermal management of high heat flux electronics. Using the framework, a comprehensive model is developed for a hemispherical water droplet resting on a heated flat substrate incorporating all of the relevant transport mechanisms: buoyant and thermocapillary convection inside the droplet and diffusive and convective transport of vapor in the gas domain. At the interface, mass, momentum, and thermal coupling of the phases are also made accounting for all pertinent physical aspe…
