The lifetime $τ$ of an isothermal and purely diffusively dissolving droplet in a host liquid scales as $τ\sim R_0^2$ with its initial radius $R_0$ [Langmuir, Phys. Rev. 12, 368 (1919)]. For a droplet dissolving due to natural convection driven by density differences, its lifetime scales as $τ\sim R_0^{5/4}$ [Dietrich et al., J. Fluid Mech. 794, 45 (2016)]. In this paper we experimentally find and theoretically derive yet another droplet dissolution behavior, resulting in $τ\sim R_0^4$. It occurs when the dissolution dynamics is controlled by local heating of the liquid, leading to a modified solubility and a thermal Marangoni flow around the droplet. The thermal gradient is achieved by plasmonic heating of a gold nanoparticle decorated sample surface, on which a sessile water droplet immersed in water-saturated 1-butanol solution is sitting. The resulting off-wall thermal Marangoni flow and the temperature dependence of the solubility determine the droplet dissolution rate, resulting in a shrinkage $R(t) \sim (τ-t )^{1/4}$ of the droplet radius and thus in $τ\sim R_0^{4}$.
The {\it unsteady} process for thermocapillary droplet migration at large Reynolds and Marangoni numbers has been previously reported by identifying a nonconservative integral thermal flux across the surface in the {\it steady} thermocapillary droplet migration, [Wu and Hu, J. Math. Phys. {\bf 54} 023102, (2013)]. Here we add a thermal source in the droplet to preserve the integral thermal flux across the surface as conservative, so that thermocapillary droplet migration at large Reynolds and Marangoni numbers can reach a {\it quasi-steady} process. Under assumptions of {\it quasi-steady} stat…
Oscillatory droplet dissolution from competing Marangoni and gravitational flows
University of Twente（特温特大学）；Department of Chemical and Materials Engineering, University of Alberta（阿尔伯塔大学化学与材料工程系）；Max Planck Institute for Dynamics and Self-Organization（马克斯·普朗克动力学与自组织研究所）
In this paper, thermocapillary migration of a planar droplet at moderate and large Marangoni numbers is investigated analytically and numerically. By using the dimension-analysis method, the thermal diffusion time scale is determined as the controlling one of the thermocapillary droplet migration system. During this time, the whole thermocapillary migration process is fully developed. By using the front-tracking method, the steady/unsteady states as the terminal ones at moderate/large Marangoni numbers are captured in a longer time scale than the thermal diffusion time scale. In the terminal s…
The steady thermocapillary droplet migration in a uniform temperature gradient combined with a radiation energy source at large Reynolds and Marangoni numbers is studied. To reach a terminal quasi-steady process, the magnitude of the radiation energy source is required to preserve the conservative integral thermal flux across the surface. Under the quasi-steady state assumption, an analytical result for the steady thermocapillary migration of a droplet at large Reynolds and Marangoni numbers is derived by using the method of matched asymptotic expansions. It is shown that the thermocapillary d…
In this paper, the overall steady-state momentum and energy balances in the thermocapillary migration of a droplet at small Reynolds numbers and large Marangoni numbers are investigated to confirm the quasi-steady state assumption of the system. The droplet is assumed to have a slight axisymmetric deformation from a sphere shape. It is shown that under the quasi-steady state assumption, the total momentum of the thermocapillary droplet migration system at small Reynolds numbers is conservative. The general solution of the steady momentum equations can be determined with its parameters dependin…
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…
Experimental observations indicate that chemically active droplets suspended in a surfactant-laden fluid can self-propel spontaneously. The onset of this motion is attributed to a symmetry-breaking Marangoni instability resulting from the nonlinear advective coupling of the distribution of surfactant to the hydrodynamic flow generated by Marangoni stresses at the droplet's surface. Here, we use weakly nonlinear analysis to characterize the self-propulsion near the instability threshold and the influence of the droplet's deformability. We report that in vicinity of the threshold, deformability …
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 transport and aggregation of particles in suspensions is an important process in many physicochemical and industrial processes. In this work, we study the transport of particles in an evaporating binary droplet. Surprisingly, the accumulation of particles occurs not only at the contact line (due to the coffee-stain effect) or at the solid substrate (due to sedimentation), but also at a particular radial position near the liquid-air interface, forming a "ring", which we term as the Marangoni ring. The formation of this ring is primarily attributed to the solutal Marangoni flow triggered by …
The Marangoni contraction of sessile droplets occurs when a binary mixture of volatile liquids is placed on a high-energy surface. Although the surface is wetted completely by the mixture and its components, a quasi-stationary non-vanishing contact angle is observed. This seeming contradiction is caused by Marangoni flows that are driven by evaporative depletion of the volatile component near the edge of the droplet. Here we show that the composition of such droplets is governed by Taylor dispersion, a consequence of diffusion and strong internal shear flow. We demonstrate that Taylor dispersi…
Small water droplets or particles located at an oil meniscus typically climb the meniscus due to unbalanced capillary forces. Here, we introduce a size-dependent reversal of this meniscus-climbing behavior, where upon cooling of the underlying substrate, droplets of different sizes concurrently ascend and descend the meniscus. We show that microscopic Marangoni convection cells within the oil meniscus are responsible for this phenomenon. While dynamics of relatively larger water microdroplets are still dominated by unbalanced capillary forces and hence ascend the meniscus, smaller droplets are…
Thermocapillary migration of a planar non-deformable droplet in flow fields with two uniform temperature gradients at moderate and large Marangoni numbers is studied numerically by using the front-tracking method. It is observed that the thermocapillary motion of planar droplets in the uniform temperature gradients is steady at moderate Marangoni numbers, but unsteady at large Marangoni numbers. The instantaneous migration velocity at a fixed migration distance decreases with increasing Marangoni numbers. The simulation results of the thermocapillary droplet migration at large Marangoni number…
Multicomponent liquid drops in a host liquid are very relevant in various technological applications. Their dissolution or growth dynamics is complex. Differences in solubility between the drop components combined with the solutal Marangoni effect and natural convections contribute to this complexity, which can be even further increased in combination with the ouzo effect, i.e., the spontaneous nucleation of microdroplets due to composition-dependent miscibilities in a ternary system. The quantitive understanding of this combined process is important for applications in industry, particularly …
The motion of a viscous droplet in unbounded Poiseuille flow under the combined influence of bulk-insoluble surfactant and linearly varying temperature field aligned in the direction of imposed flow is studied analytically. Neglecting fluid inertia, thermal convection and shape deformation, asymptotic analysis is performed to obtain the velocity of a force-free surfactant-laden droplet. The present study is focused on two limiting situations of surfactant transport: (i) small surface Peclet number, and (ii) high surface Peclet number. Thermocapillary-induced Marangoni stress, strength of which…
In the present study we analytically investigate the deformation and bulk rheology of a dilute emulsion of surfactant-laden droplets suspended in a linear flow. We use an asymptotic approach to predict the effect of surfactant distribution on the deformation of a single droplet as well as the effective shear and extensional viscosity for the dilute emulsion. The non-uniform distribution of surfactants due to the bulk flow results in the generation of a Marangoni stress which affects both the deformation as well as the bulk rheology of the suspension. The present analysis is done for the limiti…
The Marangoni effect refers to fluid flow induced by a gradient in surface tension at a fluid-fluid interface. We determine the full three-dimensional Marangoni flow generated by a non-uniform surface tension profile at the interface of a self-propelled spherical emulsion droplet. For all flow fields inside, outside, and at the interface of the droplet, we give analytical formulas. We also calculate the droplet velocity vector $\mathbf{v}^D$, which describes the swimming kinematics of the droplet, and generalize the squirmer parameter $β$, which distinguishes between different swimmer types ca…
When a water droplet is deposited within a sparsely miscible liquid medium such as certain oils, the droplet surprisingly vanishes, even in a confined geometry. Such a phenomenon has crucial consequences for multiphase flows in which confined nano- and/or picoliter droplets are considered. We report here experiments of microdroplet dissolution in microchannels that reveal an enhancement of the shrinkage of confined water microdroplets in oil due to the permeability of the walls - made of polydimethilsiloxane (PDMS) - and a delay when collective effects are present. The system is first modelled…
We report the generation of directed self-propelled motion of a droplet of aniline oil with a velocity on the order of centimeters per second on an aqueous phase. It is found that, depending on the initial conditions, the droplet shows either circular or beeline motion in a circular Petri dish. The motion of a droplet depends on volume of the droplet and concentration of solution. The velocity decreases when volume of the droplet and concentration of solution increase. Such unique motion is discussed in terms of Marangoni-driven spreading under chemical nonequilibrium. The simulation reproduce…
We study unsteady internal flows in a sessile droplet of capillary size evaporating in constant contact line mode on a heated substrate. Three-dimensional simulations of internal flows in evaporating droplets of ethanol and silicone oil have been carried out. For describing the Marangoni flows we find it necessary to account for the diffusion of vapor in air, the thermal conduction in all three phases and thermal radiation. The equations have been solved numerically by finite element method using ANSYS Fluent. As a result of the simulations, the nonstationary behavior of Bénard-Marangoni (BM) …
The analogy between evaporating surface droplets in air to dissolving long-chain alcohol droplets in water is worked out. We show that next to the three known modi for surface droplet evaporation or dissolution (constant contact angle mode, constant contact radius mode, and stick-slide mode), a fourth mode exists for small droplets on supposedly smooth substrates, namely the stick-jump mode: intermittent contact line pinning causes the droplet to switch between sticking and jumping during the dissolution. We present experimental data and compare them to theory to predict the dissolution time i…
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…
Multiphase flows are challenging systems to study, not only from a fundamental point of view, but also from a practical one due to the difficulties in visualizing phases with similar refraction indices. An additional challenge arises when the multiphase flow to be observed occurs in the sub-millimetric range. A common solution in experimental fluid dynamics is the addition of a color dye for contrast enhancement. However, added dyes can act as surface-active agents and significantly influence the observed phenomena. In the recently reported Marangoni bursting phenomenon, where a binary droplet…
The dissolution of liquid nanodroplets is a crucial step in many applied processes, such as separation and dispersion in food industry, crystal formation of pharmaceutical products, concentrating and analysis in medical diagnosis, and drug delivery in aerosols. In this work, using both experiments and numerical simulations, we \textit{quantitatively} study the dissolution dynamics of femtoliter surface droplets in a highly ordered array under a uniform flow. Our results show that the dissolution of femoliter droplets strongly depends on their spatial positions relative to the flow direction, d…
We experimentally investigate the dissolution of microscale sessile alcohol droplets in water under the influence of impermeable vertical confinement. The introduction of confinement suppresses the mass transport from the droplet to bulk medium in comparison with the non-confined counterpart. Along with a buoyant plume, flow visualization reveals that the dissolution of a confined droplet is hindered by a newly identified mechanism - levitated toroidal vortex. The morphological changes in the flow due to the vortex-induced impediment alters the dissolution rate, resulting in enhancement of dro…
Bringing the interfaces of two miscible fluids into contact naturally generates strong gradients in surface tension. Here we investigate such a Marangoni-driven flow by continuously supplying isopropyl alcohol (IPA) on a film of water, using micron-sized droplets of IPA-water mixtures. These droplets create a localized depression in surface tension that leads to the opening of a circular and thin region in the water film. At the edge of the thin region, there is a rim growing and collecting the water of the film. We find that the spreading radius scales as $r \sim t^{1/2}$. This result can be …
Underground carbon dioxide ($CO_2$) sequestration is considered to be one of the main methods to mitigate greenhouse gas (GHG) emissions. In this technology, pure $CO_2$ is injected into an underground geological formation and since it is less dense than residual fluids, there is always a risk of leakage to the surface. To increase security of underground $CO_2$ disposal, ex-situ dissolution can be implemented. When $CO_2$ is dissolved in brine before injection, it significantly reduces the risks of leakage. In this approach, pure $CO_2$ is dissolved on the surface before injection. Surface di…
Active oil droplets in a liquid are believed to repel due to the Marangoni effect, while buoyancy effects caused by the density difference between the droplets, diffusing product, and ambient fluid are usually overlooked. Recent experiments have observed active droplet clustering phenomena due to buoyancy-driven convection (Kruger et al. Eur. Phys. J. E, vol. 39, 2016, pp.1-9). In this study, we numerically analyze the buoyancy effect in addition to Marangoni flow, characterized by Peclet number $Pe$. The buoyancy effects originate from (i) the density difference between the droplet and the am…
Droplets on a liquid-immersed solid surface are key elements in many applications, such as high-throughput chemical analysis and droplet-templated porous materials. Such surface droplets dissolve when the surrounding liquid is undersaturated and the dissolution process is usually treated analogous to a sessile droplet evaporating in air. Typically, theoretical models predict the mass loss rate of dissolving droplets as a function of droplet geometrical factors (radius, constant angle), and droplet material properties (diffusion constant and densities), where the thickness of the surrounding li…
We numerically investigate both single and multiple droplet dissolution with droplets consisting of lighter liquid dissolving in a denser host liquid. The significance of buoyancy is quantified by the Rayleigh number Ra which is the buoyancy force over the viscous damping force. In this study, Ra spans almost four decades from 0.1 to 400. We focus on how the mass flux, characterized by the Sherwood number Sh, and the flow morphologies depend on Ra. For single droplet dissolution, we first show the transition of the Sh(Ra) scaling from a constant value to $Sh\sim Ra^{1/4}$, which confirms the e…
A Sharp and Conservative VOF Method for Multicomponent Liquid--Gas Mass Transfer: Bubble Dissolution and Droplet Evaporation
School of Engineering Science, University of Chinese Academy of Sciences（中国科学院大学工程科学学院）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University（西安交通大学航空航天学院机械结构强度与振动国家重点实验室）
The migration characteristics of a suspended deformable droplet in a parallel plate microchannel is studied, both analytically and numerically, under the combined influence of a constant temperature gradient in the transverse direction and an imposed pressure driven flow. Any predefined transverse position in the micro channel can be attained by the droplet depending on the applied temperature gradient in the cross-stream direction or how small the droplet is with respect to the channel width. For the analytical solution, an asymptotic approach is used, where we neglect any effect of inertia o…
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…
In this work, we study the effect of insoluble surfactants on the three-dimensional rim-driven retraction dynamics of thin water sheets in air. We employ an interface-tracking/level-set method to ensure the full coupling between the surfactant-induced Marangoni-stresses, interfacial diffusion, and inertia. Our findings are contrasted with the (Newtonian) dynamics of a liquid sheet edge finding that the surfactant concentration can prevent, or delay, the breakup of the rim. Our simulations use the fastest growing Rayleigh-Plateau instability to drive droplet detachment from the fluid sheet (rim…
We study self-propelled dynamics of a droplet due to a Marangoni effect and chemical reactions in a binary fluid with a dilute third component of chemical product which affects the interfacial energy of a droplet. The equation for the migration velocity of the center of mass of a droplet is derived in the limit of an infinitesimally thin inter- face. We found that there is a bifurcation from a motionless state to a propagating state of droplet by changing the strength of the Marangoni effect.
Recently, an active microswimmer was constructed where a micron-sized droplet of bromine water was placed into a surfactant-laden oil phase. Due to a bromination reaction of the surfactant at the interface, the surface tension locally increases and becomes non-uniform. This drives a Marangoni flow which propels the squirming droplet forward. We develop a diffusion-advection-reaction equation for the order parameter of the surfactant mixture at the droplet interface using a mixing free energy. Numerical solutions reveal a stable swimming regime above a critical Marangoni number M but also stopp…
The motion of a surfactant-laden viscous droplet in the presence of background non-isothermal Poiseuille flow is studied analytically and numerically. Specifically, the effect of interfacial Marangoni stress due to non-uniform distribution of surfactants and temperature at the droplet interface on the velocity and direction of motion of the droplet along the centerline of imposed Poiseuille flow is investigated in the presence of linearly varying temperature field. In the absence of thermal convection, fluid inertia and shape deformation, the interfacial transport of bulk-insoluble surfactants…
Colloidal crystals exhibit structural color without any color pigment due to the crystals' periodic nanostructure, which can interfere with visible light. This crystal structure is iridescent as the resulting color changes with the viewing or illumination angle, which limits its use for printing or displays. To eliminate the iridescent property, it is important to make the packing of the colloidal nanoparticles disordered. Here, we introduce a drop-casting method where a droplet of a water- ethanol mixture containing monodisperse polymer-coated silica nanoparticles creates a relatively uniform…
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…
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…
The physicochemical hydrodynamics of bubbles and droplets out of equilibrium, in particular with phase transitions, displays surprisingly rich and often counterintuitive phenomena. Here we experimentally and theoretically study the nucleation and early evolution of plasmonic bubbles in a binary liquid consisting of water and ethanol. Remarkably, the submillimeter plasmonic bubble is found to be periodically attracted to and repelled from the nanoparticle-decorated substrate, with frequencies of around a few kHz. We identify the competition between solutal and thermal Marangoni forces as origin…
