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We numerically investigate both single and multiple droplet dissolution with droplets consisting of less dense liquid dissolving in a denser host liquid. In this situation, buoyancy can lead to convection and thus plays an important role in the dissolution process. 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 experimental results by Dietrich et al. (J. Fluid Mech., vol. 794, 2016, pp. 45–67). The two distinct regimes, namely the diffusively and the convectively dominated regimes, exhibit different flow morphologies: when $Ra\geqslant 10$, a buoyant plume is clearly visible, which contrasts sharply with the pure diffusion case at low $Ra$. For multiple droplet dissolution, the well-known shielding effect comes into play at low $Ra$, so that the dissolution rate is slower as compared to the single droplet case. However, at high $Ra$, convection becomes more and more dominant so that a collective plume enhances the mass flux, and remarkably the multiple droplets dissolve faster than a single droplet. This has also been found in the experiments by Laghezza et al. (Soft Matt., vol. 12 (26), 2016, pp. 5787–5796). We explain this enhancement by the formation of a single, larger plume rather than several individual plumes. Moreover, there is an optimal $Ra$ at which the enhancement is maximized, because the single plume is narrower at larger $Ra$, which thus hinders the enhancement. Our findings demonstrate a new mechanism in collective droplet dissolution, which is the merging of the plumes, which leads to non-trivial phenomena, contrasting the shielding effect.
The evaporation of multicomponent droplets is relevant to various applications but challenging to study due to the complex physicochemical dynamics. Recently, Li etal. (Phys. Rev. Lett., vol. 120, 2018, 224501) reported evaporation-triggered segregation in 1,2-hexanediol–water binary droplets. In this present work, we added 0.5 wt % silicone oil to the 1,2-hexanediol–water binary solution. This minute silicone oil concentration dramatically modifies the evaporation process, as it triggers an early extraction of the 1,2-hexanediol from the mixture. Surprisingly, we observe that the segregation of 1,2-hexanediol forms plumes, rising up from the rim of the sessile droplet towards the apex during droplet evaporation. By orientating the droplet upside down, i.e. by studying a pendent droplet, the absence of the plumes indicates that the flow structure is induced by buoyancy, which drives a Rayleigh–Taylor instability (i.e. driven by density differences and gravitational acceleration). From micro particle image velocimetry measurement, we further prove that the segregation of the non-volatile component (1,2-hexanediol) hinders the evaporation near the contact line, which leads to a suppression of the Marangoni flow in this region. Hence, on long time scales, gravitational effects, rather than Marangoni flows, play the dominant role in the flow structure. We compare the measurement of the evaporation rate with the diffusion model of Popov (Phys. Rev., vol. 71, 2005, 036313), coupled with Raoult's law and the activity coefficient. This comparison indeed confirms that the silicone-oil-triggered segregation of the non-volatile 1,2-hexanediol significantly delays the evaporation. With an extended diffusion model, in which the influence of the segregation has been implemented, the evaporation can be well described.
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 al., Phys. Rev. Lett., vol. 121, 2018, 184501; Li et al., Phys. Rev. Lett., vol. 122, 2019, 114501). In this study, we derive and validate a quasi-stationary model for the flow inside evaporating binary sessile and pendant droplets, which successfully allows one to predict the prevalence and the intriguing interaction of Rayleigh and/or Marangoni convection on the basis of a phase diagram for the flow field expressed in terms of the Rayleigh and Marangoni numbers.
Micro- and nanodroplets have many important applications such as in drug delivery, liquid–liquid extraction, nanomaterial synthesis and cosmetics. A commonly used method to generate a large number of micro- or nanodroplets in one simple step is solvent exchange (also called nanoprecipitation), in which a good solvent of the droplet phase is displaced by a poor one, generating an oversaturation pulse that leads to droplet nucleation. Despite its crucial importance, the droplet growth resulting from the oversaturation pulse in this ternary system is still poorly understood. We experimentally and theoretically study this growth in Hele-Shaw-like channels by measuring the total volume of the oil droplets that nucleate out of it. In order to prevent the oversaturated oil from exiting the channel, we decorated some of the channels with a porous region in the middle. Solvent exchange is performed with various solution compositions, flow rates and channel geometries, and the measured droplets volume is found to increase with the Péclet number, $Pe$, with an approximate effective power law $V\propto Pe^{0.50}$. A theoretical model is developed to account for this finding. With this model we can indeed explain the $V\propto Pe^{1/2}$ scaling, including the prefactor, which can collapse all data of the ‘porous’ channels onto one universal curve, irrespective of channel geometry and composition of the mixtures. Our work provides a macroscopic approach to this bottom-up method of droplet generation and may guide further studies on oversaturation and nucleation in ternary systems.
The dissolution process of small (initial (equivalent) radius $R_{0}<1$ mm) long-chain alcohol (of various types) sessile droplets in water is studied, disentangling diffusive and convective contributions. The latter can arise for high solubilities of the alcohol, as the density of the alcohol–water mixture is then considerably less than that of pure water, giving rise to buoyancy-driven convection. The convective flow around the droplets is measured, using micro-particle image velocimetry (${\rm\mu}$PIV) and the schlieren technique. When non-dimensionalizing the system, we find a universal $Sh\sim Ra^{1/4}$ scaling relation for all alcohols (of different solubilities) and all droplets in the convective regime. Here $Sh$ is the Sherwood number (dimensionless mass flux) and $Ra$ is the Rayleigh number (dimensionless density difference between clean and alcohol-saturated water). This scaling implies the scaling relation ${\it\tau}_{c}\propto R_{0}^{5/4}$ of the convective dissolution time ${\it\tau}_{c}$, which is found to agree with experimental data. We show that in the convective regime the plume Reynolds number (the dimensionless velocity) of the detaching alcohol-saturated plume follows $Re_{p}\sim Sc^{-1}Ra^{5/8}$, which is confirmed by the ${\rm\mu}$PIV data. Here, $Sc$ is the Schmidt number. The convective regime exists when $Ra>Ra_{t}$, where $Ra_{t}=12$ is the transition $Ra$ number as extracted from the data. For $Ra\leqslant Ra_{t}$ and smaller, convective transport is progressively overtaken by diffusion and the above scaling relations break down.
The dissolution of a multicomponent nanodrop in a sparingly miscible liquid is studied by molecular dynamics (MD) simulations. We studied both binary and ternary systems, in which nanodroplets are formed from one and two components, respectively. Whereas for a single-component droplet the dissolution can easily be calculated, the situation is more complicated for a multicomponent drop, as the interface concentrations of the drop constituents depend on the drop composition, which changes with time. In this study, the variation of the interface concentration with the drop composition is determined from independent ‘numerical experiments’, which are then used in the theoretical model for the dissolution dynamics of a multicomponent drop. The MD simulations reveal that when the interaction strengths between the drop constituents and the surrounding bulk liquid are significantly different, the concentration of the more soluble component near the drop interface may become larger than in the drop bulk. This effect is the larger the smaller the drop radius. While the present study is limited to binary and ternary systems, the same method can be easily extended to a larger number of components.
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 convection 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 quantitative understanding of this combined process is important for applications in industry, particularly for modern liquid–liquid microextraction processes. In this work, as a model system, we experimentally and theoretically explore water–ethanol drops dissolving in anethole oil. During the dissolution, we observed two types of microdroplet nucleation, namely water microdroplet nucleation in the surrounding oil at drop mid-height, and oil microdroplet nucleation in the aqueous drop, again at mid-height. The nucleated oil microdroplets are driven by Marangoni flows inside the aqueous drop and evolve into microdroplet rings. A one-dimensional multiphase and multicomponent diffusion model in combination with thermodynamic equilibrium theory is proposed to predict the behaviour of spontaneous emulsification, i.e. microdroplet nucleation, that is triggered by diffusion. A scale analysis together with experimental investigations of the fluid dynamics of the system reveals that both the solutal Marangoni flow inside the drop and the buoyancy-driven flow in the host liquid influence the diffusion-triggered emulsification process. Our work provides a physical understanding of the microdroplet nucleation by dissolution of a multicomponent drop in a host liquid.
The dissolution of a single droplet, containing a mixture of oils, in water is experimentally studied. The oils in the droplet varied in terms of their solubility in water and their hydrophobicity. We demonstrate that the polarity of the droplet constituents strongly influences the dissolution dynamics. A binary-component droplet, containing two polar components (one soluble the other insoluble) exhibits a retarded dissolution as compared to a droplet containing only the soluble component. We argue that in this case the mixture in the droplet can be assumed homogeneous, leading to a smaller effective contact area of the soluble liquid in the droplet with the bulk water, and thus delayed dissolution. On the other hand, it is shown that this is not the case when a polar, soluble component is mixed with an insoluble non-polar component, in which case segregation between the different liquids inside the droplet occurs, leading to Marangoni flows and superspreading of the droplet. The segregation is confirmed by volumetric measurements and by the use of a solvatochromic dye in combination with confocal microscopy, which clearly showed that during dissolution local concentration differences inside the droplet developed.
The Greek aperitif Ouzo is not only famous for its specific anise-flavoured taste, but also for its ability to turn from a transparent miscible liquid to a milky-white coloured 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 (Tan et al., Proc. Natl Acad. Sci. USA, vol. 113 (31), 2016, pp. 8642–8647). Due to the enhanced evaporation near the contact line, the nucleation of oil droplets starts at the rim which results in an oil ring encircling the drop. Furthermore, the oil droplets are advected through the Ouzo drop by a fast solutal Marangoni flow. In this article, we investigate the evaporation of mixture droplets in more detail, by successively increasing the mixture complexity from pure water over a binary water–ethanol mixture to the ternary Ouzo mixture (water, ethanol and anise oil). In particular, axisymmetric and full three-dimensional finite element method simulations have been performed on these droplets to discuss thermal effects and the complicated flow in the droplet driven by an interplay of preferential evaporation, evaporative cooling and solutal and thermal Marangoni flow. By using image analysis techniques and micro-particle-image-velocimetry measurements, we are able to compare the numerically predicted volume evolutions and velocity fields with experimental data. The Ouzo droplet is furthermore investigated by confocal microscopy. It is shown that the oil ring predominantly emerges due to coalescence.
$Ra$, 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 experimental results by Dietrich et al. (J. Fluid Mech., vol. 794, 2016, pp. 45–67). The two distinct regimes, namely the diffusively and the convectively dominated regimes, exhibit different flow morphologies: when 
$Ra\geqslant 10$, a buoyant plume is clearly visible, which contrasts sharply with the pure diffusion case at low 
$Ra$. For multiple droplet dissolution, the well-known shielding effect comes into play at low 
$Ra$, so that the dissolution rate is slower as compared to the single droplet case. However, at high 
$Ra$, convection becomes more and more dominant so that a collective plume enhances the mass flux, and remarkably the multiple droplets dissolve faster than a single droplet. This has also been found in the experiments by Laghezza et al. (Soft Matt., vol. 12 (26), 2016, pp. 5787–5796). We explain this enhancement by the formation of a single, larger plume rather than several individual plumes. Moreover, there is an optimal 
$Ra$ at which the enhancement is maximized, because the single plume is narrower at larger 
$Ra$, which thus hinders the enhancement. Our findings demonstrate a new mechanism in collective droplet dissolution, which is the merging of the plumes, which leads to non-trivial phenomena, contrasting the shielding effect.
$Pe$, with an approximate effective power law 
$V\propto Pe^{0.50}$. A theoretical model is developed to account for this finding. With this model we can indeed explain the 
$V\propto Pe^{1/2}$ scaling, including the prefactor, which can collapse all data of the ‘porous’ channels onto one universal curve, irrespective of channel geometry and composition of the mixtures. Our work provides a macroscopic approach to this bottom-up method of droplet generation and may guide further studies on oversaturation and nucleation in ternary systems.
$R_{0}<1$ mm) long-chain alcohol (of various types) sessile droplets in water is studied, disentangling diffusive and convective contributions. The latter can arise for high solubilities of the alcohol, as the density of the alcohol–water mixture is then considerably less than that of pure water, giving rise to buoyancy-driven convection. The convective flow around the droplets is measured, using micro-particle image velocimetry (
${\rm\mu}$PIV) and the schlieren technique. When non-dimensionalizing the system, we find a universal 
$Sh\sim Ra^{1/4}$ scaling relation for all alcohols (of different solubilities) and all droplets in the convective regime. Here 
$Sh$ is the Sherwood number (dimensionless mass flux) and 
$Ra$ is the Rayleigh number (dimensionless density difference between clean and alcohol-saturated water). This scaling implies the scaling relation 
${\it\tau}_{c}\propto R_{0}^{5/4}$ of the convective dissolution time 
${\it\tau}_{c}$, which is found to agree with experimental data. We show that in the convective regime the plume Reynolds number (the dimensionless velocity) of the detaching alcohol-saturated plume follows 
$Re_{p}\sim Sc^{-1}Ra^{5/8}$, which is confirmed by the 
${\rm\mu}$PIV data. Here, 
$Sc$ is the Schmidt number. The convective regime exists when 
$Ra>Ra_{t}$, where 
$Ra_{t}=12$ is the transition 
$Ra$ number as extracted from the data. For 
$Ra\leqslant Ra_{t}$ and smaller, convective transport is progressively overtaken by diffusion and the above scaling relations break down.
