Oil-on-water droplets faceted and stabilized by vortex halos in the subphase

Significance

Oil droplets on water test the principle of surface free energy minimization when volatility or dissolution produce flow-induced viscous shear stresses that take the system out of equilibrium. We find that single droplets adopt polyhedral shapes in shallow water, droplet pairs pulsate in their spacing and shape, multiple oil droplet arrays display active motion, and these droplets are protected from coalescence. Combined experiments, scaling arguments, and linear stability analysis trace these effects to rapid Marangoni flow (“vortex halos”) that shears individual droplets while maintaining them separate from other droplets. The analogy that active emulsion droplets are like an iceberg with submerged halos formed by high-Reynolds-number flow thus presents an approach to control and manipulate this common system.

Abstract

For almost 200 y, the dominant approach to understand oil-on-water droplet shape and stability has been the thermodynamic expectation of minimized energy, yet parallel literature shows the prominence of Marangoni flow, an adaptive gradient of interfacial tension that produces convection rolls in the water. Our experiments, scaling arguments, and linear stability analysis show that the resulting Marangoni-driven high-Reynolds-number flow in shallow water overcomes radial symmetry of droplet shape otherwise enforced by the Laplace pressure. As a consequence, oil-on-water droplets are sheared to become polygons with distinct edges and corners. Moreover, subphase flows beneath individual droplets can inhibit the coalescence of adjacent droplets, leading to rich many-body dynamics that makes them look alive. The phenomenon of a “vortex halo” in the liquid subphase emerges as a hidden variable.

A common undergraduate science experiment (apparently first described by Ben Franklin though not interpreted in that pre-molecular age) is that the size of individual oil molecules can be inferred by depositing an oil droplet on the surface of a pond and measuring its final area, presuming that the film eventually becomes molecularly thin (1). While this is the common thermodynamic expectation for a completely wetting oil droplet (2, 3), volatility of the liquid in the footprint can prevent spreading to be molecularly thin, thus forcing the droplet’s footprint to keep a limited area. Moreover, the volatility of the droplet (4) or its miscibility with the bath liquid can drive complex flows within the droplet or even within the bath (5–11). Multicomponent droplets can have strong and sometimes even chaotic and turbulent internal flows though looking from above the overall droplet remains circular (12, 13). These flows are often assumed to obey the symmetries of the system, i.e., they are presumed to be radially symmetric. Here we consider simple instances, single-component oil droplets at the water–air interface, and demonstrate that strong flow in the bath can break symmetry, even modulating the shape of the droplet to become faceted. Flows act as hidden variables (6) in systems without surfactant.

We further show that in the case of two or more droplets on water, interactions between the subphase flow fields in the bath lead to complex and rich dynamics, making these droplets active. This activity is visually reminiscent of the bulk oil-in-water active emulsions that are driven by the dissolution of the oil droplet and formation of micelles, leading to gradient of surfactant coverage on the oil droplet and therefore a Marangoni flow along their interface (14–17). Those 3D active emulsions (18) demonstrate a rich landscape of behaviors including directional motion (19–21), self-division (22, 23), self-rotation (24), chasing (25, 26), swarming (27), and self-assembly (28–30). These examples in 3D emulsions commonly involve the presence of surfactant. Moreover, these patterns of behavior are usually in the regime of small scales and slow flows where viscous forces dominate over inertia.

Situations of surfactant-free high-Reynolds-number flow are also interesting to consider, as we show in this study by considering the familiar situation where volatile oil droplets are deposited at the air–water interface. The experimental parts of this study involve simple liquids deposited onto water in open air to allow evaporation and observed using a high-speed camera including particle image velocimetry of subphase flow. Our analysis employs the following concepts. The Reynolds number is defined as $\mathrm{R}\mathrm{e}=\frac{\rho Ud}{\mu }$, where ρ is the water density, µ is the viscosity of the water, U is the maximum flow velocity that is experimentally measured, and d is a characteristic length scale taken for this analysis to be depth of the water bath. The Marangoni number is defined as $\mathrm{M}\mathrm{a}=\frac{\mathrm{\Delta }\gamma }{\mu U}$, where the ∆γ is the maximum reduction of the water–air surface tension when the oil droplet is present. The spreading coefficient is defined as S = γ_wa − γ_oa − γ_wo, where surface tensions γ refer to the water-air (wa), oil-air (oa), and water-oil (wo) interfaces, respectively. Typical values are U = 60 to 120 mm/s, d = 1 to 20 mm, and ∆γ = 1.5 to 6 mN/m. In experiments, S is manipulated by mixing into the water small amounts of a second polar liquid.

Results and Discussion

Faceted Single Droplets.

The physical setup is sketched in Fig. 1A: an oil droplet is placed by pipette at the air–water interface in a water bath of controlled depth. In cases of single oil droplets, the initial diameter is in the range millimeter to centimeter. Adding tracer particles into the water bath allowed us to image subphase flow (Movie S1) for a family of oils with differing vapor pressure and solubility (SI Appendix, Table S1). Our surface tension measurements show that the water–air surface tension is lowered by vapor adsorption. Therefore, a gradient in surface tension near the droplet drives solutal Marangoni flow in the water bath away from the droplet (SI Appendix, Table S2).

Fig. 1.

Individual faceted oil-on-water droplets. (A) Schematic diagram showing that oil vapor above an oil-on-water droplet adsorbs preferentially near the drop to produce a surface tension gradient that stimulates convection rolls (“vortex halo”) from Marangoni flow. (B) Flow visualization of suspended graphite particles, illuminated from the side by a green laser, reveals fluid flows with vortices in the liquid phase (the shutter time is 0.3 s). Because the camera is at an oblique angle looking down to improve visualization, the top of this image includes a portion of the water surface. The main portion of this dense droplet is under the water. The organic liquid is carbon tetrachloride (CCl₄) and the water depth is 30 mm with 1% added DMF (N,N-dimethylformamide, volume ratio) to increase the spreading coefficient. (C) Top-down visualization of the lateral extent of the convection flow as indicated by accumulated graphite particles far from the oil shows that the vortex halo develops asymmetry. In this composite image from two experiments, relatively small flow speed (≈60 mm/s for cyclohexane spread on water 4.2 mm deep) gives a ring of graphite particles that is almost circular, but with faster flow (≈80 mm/s for carbon tetrachloride spread on water 5.3 mm deep) the ring of graphite particles is asymmetric. The fluctuating dynamics of ring asymmetry are not addressed in this study. (D) Flow speed at the contact line (left ordinate) is plotted against the spreading coefficient for the four organic liquids identified in this panel for d = 20 mm and initial radius of the oil droplet R = 10 mm. For each speed, the Reynolds number was calculated (right ordinate). Filled symbols labelled by arrows identify data for the organic liquids spread on deionized water without added polar liquid. Open symbols refer to situations when the droplet is circular while the filled symbols refer to situations when the droplet is faceted. The situation where organic solvents (carbon tetrachloride, CCl₄ and cyclohexane, C₆H₁₂) on pure water is labelled by the dashed line.

Looking at the water bath from the side, convection near the oil droplet is accompanied by vortices at positions laterally distant from it. This is rendered visible by adding reflective graphite particles (illuminated by a green laser), as illustrated by the image in Fig. 1B. We analyzed images such as this by particle image velocimetry to determine the speed of water flow at the oil droplet. Meanwhile, looking at the oil droplet from above, we learn that the lateral extent of convection flow as indicated by the accumulation of graphite particles extends farther from the oil droplet, the faster the water flow at the droplet, as illustrated in Fig. 1C and SI Appendix, Fig. S1. Plotting flow speed at the oil droplet as a function of the spreading coefficient for four organic liquids identified in the panel caption, all data fall empirically on a curve that increases with a sigmoidal shape, as shown in Fig. 1D. These data include, in addition to oil on deionized water, situations in which small amounts of a polar liquid were mixed with the water to increase the spreading coefficient, thereby making these droplets spread to be flatter than on pure water. While it is true that evaporation also introduces flows inside the droplet, the maximum flow speed inside the oil droplet is less than 10 mm/s, an order of magnitude slower than flow speed near the contact line (SI Appendix, Fig. S2), so we consider that flow-induced shear stress at the contact line plays the major role. From the speed, we calculate the implied Reynolds numbers. This gave the numbers identified in the right-hand ordinate of Fig. 1D. As a technical point, we explain that spreading coefficient S is selected as the abscissa to account for the fact that thinner droplets (higher S) are expected to be most easily deformed by subphase flow. While it is true that one might also consider surface energy depression Δγ as a variable, as this quantity is largest near the droplet and diminishes moving away from it, and the presence of polar additives increases this quantity and therefore results in larger U, (SI Appendix, Fig. S3A), such analysis appears to be physically unrealistic as this plot extrapolates to finite U when Δγ = 0 (SI Appendix, Fig. S3B). On physical grounds, we consider S to be more meaningful.

The droplets acquire polygonal footprints illustrated in Fig. 2 provided that subphase flow is sufficiently rapid, yet in control experiments we confirm the literature surface tensions of these liquids, showing the absence of surface-active contaminants (SI Appendix, Table S2). This faceting at room temperature is therefore distinct from the previously reported mechanisms, including freezing of a surface layer (31–34) or subtleties of adding surfactants (35). The polygonal footprints appear spontaneously, once the diameter of the petri dish is larger than the vortex halo. There is no other dependence on diameter of the water bath, nor how its perimeter is shaped, circular or rectangular. The dependence on water depth is discussed below.

Fig. 2.

Top-down images of faceted cyclohexane droplets during evaporation. (A–G) photo image sequence at times identified in each image (A–F) and corresponding number of edges (G) of a cyclohexane droplet with initial volume 40 μL spread on a water bath 4.2 mm deep. The water is mixed with 15% DMF (N,N-dimethylformamide, volume ratio) to lessen the spreading coefficient. Immediately after deposition onto the water surface the droplet is nearly circular with fluctuations of shape, and then evolves to be faceted with a discrete number of edges. The number of edges decreases stepwise as the droplet evaporates. In the final stages of evaporation, the droplet becomes circular again, then vanishes.

For droplets in the range mm to cm in diameter, between 9 and 4 facets are seen soon after deposition onto the water surface; this number decreases stepwise as evaporation continues causing the droplet diameter to shrink, as illustrated in Movie S2. But when the water bath is covered to produce a saturated vapor, faceting disappears (Movie S3), confirming our hypothesis that solutal Marangoni flows cause the faceting. The thinnest droplets are most easily deformed by subphase flow (Movie S4). These are the droplets with the largest spreading coefficient S. Mixing water with small amounts of a second polar liquid (ethanol or N,N-dimethylformamide, DMF) to control S as shown in SI Appendix, Fig. S4, the largest values of S and accordingly the thinnest droplets are so fragile that they fragment, while only circular footprints are observed for the thicker droplets associated with S < 0. We remark parenthetically, though it goes beyond the scope of this paper, that subphase flow also can shear the thinnest droplets to produce fingering at their contact line. Microscopic fingering can occur at the edges of faceted droplets (SI Appendix, Fig. S4). It is also observed for droplets whose shape is circular macroscopically.

It is natural to wonder about thermal Marangoni flows, as we determined using an infrared camera that evaporation cools the surface by 1 to 2 °C (Materials and Methods). Temperatures are lower near the droplet. In principle, this should create thermal Marangoni flow toward the droplet, from regions of lower surface tension toward those of higher surface tension, but this is never seen in our PIV experiments. In the present problem, radially outward solutal Marangoni flows in the bath dominate. Within individual droplets, likewise one can have thermal Marangoni flows along the droplet–air interface from the apex to the edge, but again the strength of these flows should be weaker than flow generated within the bath, which in turn shears the droplet. Parenthetically, we remark that this is not straightforward to test experimentally; while one might imagine using glycerol or some other solute to raise the viscosity of water to facilitate experiments, doing so would reduce the water-air surface tension, and also higher viscosity would encourage fingering. This analysis suggests that thermal Marangoni flows and temperature effects are not expected to play major roles in understanding this problem.

Scaling and linear stability analysis. Based on the hypothesis of solutal Marangoni-driven flow in the bath, we can estimate the flow velocity. The Marangoni stress on the water-air interface scales as $\frac{\mathrm{\Delta }\gamma }{R}$, where the surface tension change due to oil vapor $\mathrm{\Delta }\gamma$ and the radius of the droplet $R$ set the scale over which the oil vapor concentration decays due to diffusion. This driving force is balanced by the tangential viscous stress at the water–air interface, which scales as $\frac{\mu U}{\delta }$, where $\mu$ is the water viscosity, $U$ is the flow velocity, and $\delta$ is the boundary layer thickness at the water–air interface and we have assumed $\delta \ll d$ where the d is the water bath depth. This boundary layer is established by diffusion of momentum and its thickness scales as $\delta \sim \sqrt{\nu t}$, where $\nu =\frac{\mu }{\rho }$ is the kinematic viscosity of water and $t\sim \frac{R}{U}$ is the characteristic convection timescale, leading (for U = 100 mm/s) to $\mathrm{\delta }\approx 0.2 \mathrm{mm}$, which is much smaller than the typical water bath depth in our experiments, for which $d>1 \mathrm{mm}$. Therefore, we have the following balance

$$\frac{\mathrm{\Delta }\gamma }{R}\sim \frac{\mu U}{\sqrt{\nu t}},$$ [1]

which can be rearranged to obtain an estimate for the flow velocity in the bath

$$U \sim {\left(\frac{(\mathrm{\Delta }{\gamma )}^2}{\rho \mu R}\right)}^{\frac{1}{3}},$$ [2]

where we obtain $U\approx 120 \mathrm{m}\mathrm{m}/\mathrm{s}$ for typical experimental values of $R\approx 5 \mathrm{m}\mathrm{m}$, $\mathrm{\Delta }\gamma \approx 3 \mathrm{m}\mathrm{N}/\mathrm{m}$, $\mu \approx 0.001 Pa.s$ and ρ ≈ 1,000 kg/m³. This velocity estimate is close to the experimental measurements in the case of faceted droplets. For cyclohexane, $\begin{array}{c}U \approx 86 \mathrm{m}\mathrm{m}/\mathrm{s} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{\Delta }\gamma \approx 3.8 \mathrm{m}\mathrm{N}/\mathrm{m}.\hfill \end{array}$ For carbon tetrachloride, $\begin{array}{c}U \approx 84 \mathrm{m}\mathrm{m}/\mathrm{s} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{\Delta }\gamma \approx 4.2 \mathrm{m}\mathrm{N}/\mathrm{m}.\hfill \end{array}$

Faceting of the droplet shape implies loss of radial symmetry. While we expect the flow field to be radially symmetric and pointing away from the droplet, our flow visualization in the bath shows a 3D flow field when the droplet becomes faceted (Movie S4). We therefore hypothesize that the flow field in the bath breaks symmetry, developing recirculating 3D vortices, which shear the droplet and make it faceted. To test this hypothesis, we performed a linear stability analysis of the flow field in the bath. We consider a purely radial flow field and introduce infinitesimal 3D perturbations to it. If these perturbations grow in time, the flow becomes unstable and fully 3D.

We assume that the vapor field around the evaporating droplet is diffusion-limited and quasi-steady, decaying as $1/r$ with $r$ the distance from the center of the droplet. Taking $d$ as the characteristic length scale, $U=\nu /d$ as the characteristic velocity, $\rho U^2$ as the characteristic pressure, and $d^2/\nu$ as the characteristic timescale, the nondimensional Navier–Stokes equations can be written as

$$\frac{\partial \mathit{u}}{\partial t}+\mathit{u}·\mathrm{\nabla }\mathit{u}=-\mathrm{\nabla }p + {\mathrm{\nabla }}^2\mathit{u},$$ [3]

$$\mathrm{\nabla }·\mathit{u}=0,$$ [4]

where $\mathit{u}=(u,v,w)$ are the velocity components in cylindrical coordinates $(r,\theta ,z)$ subject to shear stress boundary conditions at the water–vapor interface:

$$\frac{\partial u}{\partial z}=\mathrm{Ma}\frac{\mathit{R}/\mathit{d}}{{\mathit{r}}^2},$$ [5]

$$\frac{\partial v}{\partial z}=0,$$ [6]

where Ma (defined earlier), proportional to the surface tension change from the presence of oil vapor, indicates the strength of the Marangoni-driven flow.

Decomposing the velocity and pressure components as the sum of the radial base state and infinitesimal 3D perturbations, we can write

$$u\left(r, \theta , z, t\right)=u_0\left(r,z\right)+\widehat{u}\left(r,z\right)e^{\lambda t+i m \theta },$$ [7]

$$v\left(r, \theta , z, t\right)=0+\widehat{v}\left(r,z\right)e^{ \lambda t+i m \theta },$$ [8]

$$w\left(r, \theta , z, t\right)=w_0\left(r,z\right)+\widehat{w}\left(r,z\right)e^{\lambda t+i m \theta },$$ [9]

$$p\left(r, \theta , z, t\right)=p_0\left(r,z\right)+\widehat{p}\left(r,z\right)e^{ \lambda t+i m \theta },$$ [10]

where the subscript zero indicates the purely radial base flow field, and the caret symbol denotes the infinitesimally small perturbation variables. Here, we assume that the perturbations are periodic in the azimuthal direction, θ, with a periodicity that is determined by the mode number m. Substituting these variables in the Navier–Stokes equations, we then numerically obtain the growth rate, $\lambda =\sigma +i \omega$. A positive growth rate indicates that perturbations grow in time exponentially. The mode with the largest positive growth rate is called the most unstable mode denoted by $m_c$.

Our numerical linear stability analysis of these equations shows that the flow in the bath remains stable when the Marangoni flow is weak, while beyond a critical Ma number the flow becomes unstable and develops non-radial recirculation as shown in Fig. 3. For the range of Ma numbers we have investigated, $m_c=4$ is always the most unstable mode. Moreover, the larger the Marangoni number, the larger the growth rate, indicating that perturbations grow more rapidly. This analysis agrees with our hypothesis that the flow in the bath breaks its symmetry when the Marangoni flow becomes strong enough. It is consistent with our experimental measurements of the flow field in the bath, showing that the transition from circular to faceted droplet shapes coincides with the velocity at the contact line exceeding a critical value (Fig. 1D). We note, however, that while our analysis above reveals the 3D nature of the flow field in the bath, it can only provide the necessary, but not sufficient conditions for the modulation of the droplet shape.

Fig. 3.

Linear stability analysis of the Navier–Stokes equations predicts the critical Marangoni number at which flow develops non-radical recirculation. (A) Real part of the perturbation growth rate plotted against azimuthal mode number, m. Positive numbers indicate instability. Stationary modes have a real growth rate while rotational modes are complex, signifying that unstable modes rotate in the azimuthal direction. (B) The most unstable mode, m_c = 4, with arrows showing the perturbation velocity field, with maximum and minimum values of its radial component indicated in the color bar by red and blue, respectively, and contours showing expected structure of flow. The ${2m}_c$ recirculating flow cells suggest the droplet should develop four facets as flow is toward the edges and away from the corners. (C) The growth rate of mode 4 instability is plotted against Marangoni number, Ma. (D) For mode 4, this map shows the spatial distribution of the radial velocity field with the color map indicated. By assumption the vapor field around the evaporating droplet is established by diffusion and is quasi-steady, while the surface tension of water is reduced by the presence of oil vapor. Simulation conditions: water depth, 5 mm; droplet radius, 5mm; container radius, 10 mm; Δγ = 5 mN/m.

We can use scaling arguments to get a sense of the droplet deformation due to shear stresses from the 3D flow field in the bath. Continuity implies that the azimuthal and radial components of the flow field in the recirculating cells have similar magnitude, and both should scale as $U$, the scaling for which was obtained in Eq. (2). In our experiments, the water bath depth $d$ is typically the smallest geometric length scale in the problem, which would therefore set the length scale for shear stress τ exerted on the droplet

$$\tau \sim \frac{\mu U}{d}.$$ [11]

This shear stress is balanced by the surface tension of the droplet that tends to resist any deformation

$$\frac{\mu U}{d} \sim \gamma \mathrm{\Delta }\kappa ,$$ [12]

where $\mathrm{\Delta }\kappa$ represents the change in the curvature of the droplet. Together with Eq. (11), we therefore arrive at an expression for the droplet deformation

$$\mathrm{\Delta }\kappa \sim \frac{1}{d}{\left(Oh\frac{\mathrm{\Delta }\gamma }{\gamma }\right)}^{\frac{2}{3}},$$ [13]

where $Oh = \mu /\surd \left(\rho R\gamma \right)$ is the Ohnesorge number. This expression implies that for given fluid properties and magnitude of surface tension change $\mathrm{\Delta }\gamma$, the oil droplet becomes more deformed in shallower baths, i.e., smaller $d$ values. This prediction is qualitatively consistent with our experimental observations, which show no droplet deformation in deep water baths.

The existence of different regimes is related to the flow of water in the bath because the latter exerts shear stress on the droplet and drives flow in the droplet. This can be understood from a simple argument based on relative shear stress at the edge of these droplets, which scales as $\mu U/\left(R\theta \right)$. Increasing the spreading coefficient should lead to a decrease in θ, which indicates an increase in the shear stress, which is inversely proportional to θ. Therefore, overall the transition from circular to faceted to fragmented when the spreading coefficient increases can be understood as an increase in the shear stress experienced by the droplet near its edge, assuming the same strength of Marangoni flow in the bath. In parallel, both the flow velocity U and the spreading coefficient S increase as the vapor-induced surface tension difference increases (Fig. 1). Water depth appears to be the key variable, not diameter of the petri dish. When (keeping the water depth constant at 4.2 mm) diameter of the petri dish was increased to 20 cm, which much exceeds size of the vortex halo, the position of the vortex halo was unchanged, as shown in SI Appendix, Fig. S5.

We present an experimental state diagram in Fig. 4: the zone of faceted droplets in a phase space of spreading coefficient and depth of the water bath is in reasonable agreement with our arguments that with appropriate spreading coefficient and water depth, faceting is a general phenomenon. Flow visualization shows that the vortex halo sweeps tracer particles to millimeter-sized distances away from the droplet with almost constant flow speed as the size of the droplet varies (SI Appendix, Fig. S6). A subtlety is that when the oil density exceeds that of water the number of facets first increases, then decreases, probably reflecting less and less penetration below the interface as evaporation proceeds (SI Appendix, Fig. S7). The number of facets decreases monotonically otherwise.

Fig. 4.

State diagram summarizing when faceting is observed. Mapping spreading coefficient against water depth, zones of faceted and circular-shaped droplets are shaded pink and green respectively. We remark parenthetically, though it goes beyond the scope of this paper, that subphase flow also can shear the thinnest droplets to produce microscopic fingering at the contact line; fingering can occur at the edges of faceted droplets (SI Appendix, Fig. S4) and also is observed for droplets whose macroscopic shape is circular. Zones of “droplets fragment” and “oil displaces water” are shaded yellow and blue, respectively. The organic liquids are cyclohexane (black squares, 5 to 20% DMF in water), carbon tetrachloride (red circles, 0.5 to 2% DMF in water), chloroform (blue up-triangles, 1 to 5% ethanol in water), dichloromethane (dark green down-triangles, 3 to 5% ethanol in water), and 1,2-dichloroethane (purple diamonds, 2 to 5% ethanol in water). Though data is shown in the droplets fragment zone only for chloroform, all the other oil droplets also fragment when their spreading coefficients are larger than 12.5 mN/m.

Two Oil Droplets in Proximity.

Oil droplets in proximity should coalesce from the thermodynamic perspective as this would lower the surface energy, but this is not what we observe except for droplets so large that capillary attraction dominates (cm-sized or larger). Instead, millimeter-sized droplets attract but stop at a large, millimeter-sized fixed separation each with its own vortex halo. The sketch in Fig. 5A, comparing shallow and deeper water, leads to expecting the steady-state separation to be larger in deeper water because in this situation the vortex halo is larger.

Fig. 5.

Two droplets in proximity. (A) Schematic sketch of two volatile oil droplets of similar size with different water depth, each of them stimulating its own subphase flow (“vortex halo”). The better-developed vortex halo in deeper water leads to expecting greater droplet-droplet spacing. (B) Viewed from above with graphite particles added to visualize the flow, a visible boundary separates vortex flow near each oil droplet. (C) Zones of coalescence (labeled as black squares with blue shadow) and no-coalescence (labeled as red circles with yellow shadow) are identified in this plot of relative surface tension depression Δγ/γ against a coalescence criterion, V/l_c³, suggested by scaling arguments. (D) Schematic sketch of two droplets viewed from above with outward Marangoni flow indicated by blue arrows and droplet–droplet interactions between vortex halos depicted in red. (E) Image of droplets on water with fixed water depth of 4.2 mm (spreading coefficient S = 2.0 mN/m, relative surface tension depression Δγ/γ = 0.04. (F) as in panel e except in the presence of 1% DMF (S = 3.4 mN/m, relative surface tension depression Δγ/γ = 0.09. (G) Droplet–droplet spacing, D, plotted against time during evaporation (same system as E and F). In all panels, the water depth is 4.2 mm and the droplets are chlorobenzene.

Scaling analysis. Modeling the droplets as circular disks with radius, R (36), we can estimate the attraction force between them as $\frac{\gamma }{D}{\left(\frac{R}{l_c}\right)}^{\frac{5}{2}}{\theta }^2{l}_c^2$, where D is the spacing between the droplets, $l_c=\sqrt{\frac{\gamma }{\rho g}}$ is the capillary length scale, $\gamma$ is the water–air surface tension, and $\theta$ is the oil droplet contact angle, and $g$ is the gravitational constant. The Marangoni force exerted on the droplet scales as $\frac{\mathrm{\Delta }\gamma R^2}{D}$. For the Marangoni stresses to prevent the coalescence of the two droplets, we therefore need to have $\frac{\mathrm{\Delta }\gamma }{\gamma }>{\theta }^2{\left(\frac{V}{l_c^3}\right)}^{\frac{1}{6}}$, where $V\sim R^3$ is the droplet volume.

To estimate the equilibrium spacing between two neighboring droplets, we balance the shear stress on the droplet due to the solutal Marangoni flow with capillary attraction. The Marangoni-driven force pulling the droplets apart due to the flow on their far side scales as $\left(\frac{\mu U}{d}\right)R^2$, where the velocity $U$ follows the scaling of Eq. (10) and the capillary attraction force scales as $\frac{\gamma }{D}{\left(\frac{R}{l_c}\right)}^{\frac{5}{2}}{\theta }^2{l}_c^2$ as described above. Balancing these forces and taking the typical experimental parameters used above with $R\approx 1 \mathrm{m}\mathrm{m}$, we obtain the equilibrium spacing between two neighboring droplets $D\approx 1 \mathrm{m}\mathrm{m}$ for a water bath depth of $\approx 2.5 \mathrm{m}\mathrm{m}$, and $D\approx 1.5 \mathrm{m}\mathrm{m}$ for a depth of $\approx 4 \mathrm{m}\mathrm{m}$, which are of the same order of magnitude as the droplet spacings observed in our experiments.

State diagram. The state diagram (Fig. 5C) shows that the zones of coalescence, and failure to coalesce during the steady-state evaporating situation, are in reasonable agreement with the scaling argument that balances capillary effects (attractive) and flow effects (repulsive). Indeed, the droplet–droplet spacing (D) increases with spreading coefficient S but subsequently remains constant (Fig. 5 D–G). The stable separation, corresponding to repulsive contact between vortex halos of the individual particles, increases with water depth (SI Appendix, Fig. S8) as suggested by the heuristic sketch in Fig. 5 A and B. Experiments in Movie S5 confirm subphase flow in distinct regions near each droplet, supporting the physical interpretation that vortex halos cushion the droplets against coalescence which is otherwise expected due to the “cheerios effect,” i.e., the deformation of the water-air meniscus that brings the droplets together.

Whereas single oil-on-water droplets are stationary after they self-adjust to the center of a water bath, systems composed of two droplets or more display patterns of nontrivial mobility and dynamic self-organization. Here we illustrate this lively behavior without attempting to model it. First, consider pairs of droplets of similar size (Fig. 6). We observe them to usually orbit each other with occasional reversal of the rotation direction, probably reflecting hydrodynamic torques when droplets approach with coupling of the vortex halos that deform the droplets at points of closest approach. Shrinking with time, both the contact angle and the size of the droplet adjust, leading to a competition between shear flow and the surface tension of the oil droplet (Fig. 6 A and B). There usually comes a time when both the contact angle and the size are small enough to be perturbed under the shear of the vortex halo, after which the smaller droplet ceases to have a circular footprint and pulsates rhythmically, suggesting an instability in the flow stagnation point, and after this time the droplet-droplet spacing oscillates accordingly (Fig. 6 C and D). Rotation and pulsation when two droplets are in proximity are also seen in 3D emulsions containing surfactant, for different reasons (26, 30, 37, 38). In this surfactant-free 2D system the size of the accompanying vortex halo is confirmed to be roughly mm in width for centimeter-sized particles, as seen by flow visualization, and increases with water depth. The rhythmical process evokes a dance as time proceeds (Movie S6), but the droplets coalesce following the expected thermodynamic behavior when we cover the water dish to inhibit evaporation (Movie S7).

Fig. 6.

Persistent motion of two droplets in proximity. (A) Droplets of similar size orbit around each other (the image in this illustration is chlorobenzene). The inset defines the axes plotted in the next panel. (B) A 3D plot of droplet position versus time quantifies orbit speed during this stage of evaporation. (C) After the droplets shrink to 30% of the original size in diameter they cease to circle and the smaller one pulsates while both remain in place. (D) Plots the center-of-mass spacing (L) and of the inter-droplet spacing (D) against time shows rhythmic oscillation with frequency ≈10 Hz. In these experiments, water depth = 4.2 mm.

Now consider coupling between vortex halos of multiple droplets. First three, then four large carbon tetrachloride droplets are immobilized by allowing these high-density liquids to sediment to the point of impinging the bottom of a shallow water bath. To visualize near-surface flows, graphite particles are added. One sees a triangular coupled fluid region for the three-droplet case (Fig. 7 A–C) and a rectangular coupled fluid region for the four-droplet case (Fig. 7 D–F), whose detailed shape fluctuates despite the overall triangular or rectangular shape. The case of three particles shows a vortex whose position fluctuates and whose rotation direction switches from time to time. The case of four oil droplets generates multiple vortices whose rotation direction also fluctuates (Movie S8). On first thought it might be tempting to visualize these patterns of chaotic mixing using tracer dyes, but with reflection one realizes that the presence of dyes would modify surface tension. In situations involving an even larger number of volatile oil droplets, the data are consistent in showing that these droplets fail to coalesce. Their rich patterns of collective translational mobility, which qualitatively resemble behavior already known for organic droplets on aqueous surfactant solutions (27–29, 38–40), are beyond the scope of this study.

Fig. 7.

Flow between multiple droplets. (A–F) schematic illustration and photo images of three large (about 500 µL) carbon tetrachloride droplets (A–C) and four large carbon tetrachloride droplets (D–F) which are fixed by sedimentation of this high-density liquid, in which graphite particles are suspended, to the bottom of a shallow water bath. The arrows in A and D represent convection away from the droplets on the surface. Three typical patterns of Marangoni flow are illustrated to show representative fluctuations; these patterns change roughly three times per second but are not periodic. The region (A and B) with coupled flows between droplets is triangular for the three-droplet case and rectangular for the four-droplet case. The water depth is 4.2 mm.

Summary and Outlook

Imaging the shapes of oil-on-water droplets while they evaporate shows that their shapes lack the symmetry anticipated from minimization of energy that is depicted in textbooks to govern this situation. Instead, their shapes, fluctuations, and droplet-droplet interactions are coupled to convective motions in the bulk subphase. In particular, we identify polyhedral shapes that mirror loss of radial symmetry in the subphase flow. The recognition that although a droplet may appear to be quiescent when viewed from the air, when in fact there exists churning flow beneath the water surface, exposes the vortex halo concept as a hidden variable in understanding this physical situation. Moreover, the perspective of considering the interactions between subphase flows on individual droplets allows one to understand why multiple-droplet situations involve significant translational mobility on the air surface while single-droplet situations do not.

These systems are “active”—the large energy source provided by the enthalpy of vaporization and solubilization promotes active motion over extended periods of time. Recognizing that the existing understanding of active motion is often presented in the context of theoretical models and specially-designed model experimental systems (41, 42) to observe related but subtly modified effects in these systems underscores this concept’s rich application to the naturally-occurring world. Communication between active agents via “chemical signaling” has found recent interest also regarding emulsions in three dimensions (43). In the two-dimensional active emulsions that constitute the subject of this paper, the concept of chemical signaling takes the form of subphase flows.

Materials and Methods

The SI Appendix contains additional information about materials, droplet deposition, surface tension measurement, optical imaging, assessment of evaporation-induced cooling, and particle imaging velocimetry.

Materials.

Organic liquids were purchased in the grade of the highest available purity, usually from several sources to confirm consistency, and used as received. Droplets were usually deposited onto a cylindrical petri dish but control experiments confirmed no dependence on shape provided that the diameter of the petri dish was larger than of the vortex halo.

Numerical Simulation.

Numerical solutions of the Navier–Stokes equation were performed using the COMSOL Multiphysics solver (5.4). The vapor field was governed by the Laplace equation (diffusion dominated limit) and the fluid flow in the water bath by the Navier–Stokes equations. The governing equations were solved using the finite-element method on a fine triangular mesh with linear elements using the direct solver algorithm. Time stepping was done using the backward differentiation formulae. For the linear stability analysis, the base state was obtained assuming axisymmetry in a cylindrical coordinate system. The eigenvalue solver was then used to obtain the growth rate corresponding to the different azimuthal modes.

Statistics.

For directly measured data, such as surface/interfacial tensions, flow velocities, and distances, the uncertainty is considered to be the SD of the data set. The average sample size is 3 for surface/interfacial tension, 5 for flow velocities, and 3 for distances between droplets. Regarding calculations from the measured quantities, such as spreading coefficients, uncertainty is estimated by the law of propagation of uncertainty based on the SD of the measured quantities.

Supplementary Material

Appendix 01 (PDF)

Movie S1.

This video shows the vortex halo surrounding illustrative oil-on-water volatile droplets, viewed edge-on with nano-sized graphite particles added to image the water flow. The oils, carbon tetrachloride (CCl₄), 1,2-dichloroethane (1,2-DCE), and cyclohexane, were viewed in a water bath 30 mm deep after depositing a droplet of volume 200 μL. Scale bar: 10 mm.

Movie S2.

This video shows faceted shapes of single oil-on-water volatile droplets, viewed top-down. The oils, cyclohexane, carbon tetrachloride (CCl₄), and dichloroethane (1,2- DCE), were deposited in a water bath 4.2 mm or 2.5 mm deep with initial droplet volumes of 200 or 40 μL. Scale bar: 10 mm for larger volume and 2 mm for smaller volume.

Movie S3.

This video shows control experiments in which oil-on-water droplets lose faceting when the sample cell is sealed to prevent a surface tension gradient, but regain faceting reversibly when the sample cell is opened later. Illustrated for cyclohexane. Scale bar: 10 mm.

Movie S4.

This video shows the vortex halo surrounding illustrative oil-on-water droplets (soluble and insoluble), viewed top-down with nano-sized graphite particles added to image the water flow. Each was deposited on a water bath 4.2 mm or 6.3 mm deep with initial droplet volumes of 100 μL to 300 μL, fixed at the center of the water bath, then nano-sized graphite particles were added carefully in the water. Scale bar: 10 mm.

Movie S5.

This video shows vortex halos surrounding two oil-on-water droplets, viewed top-down with nano-sized graphite particles added to image the water flow. The oil droplets of carbon tetrachloride (CCl₄) were deposited with a volume of 300 μL on a water bath 5.5 mm deep and sank to the bottom owing to density mismatch, pinning them at initial separations of 25 mm and 9 mm for the examples in this video. Scale bar: 20 mm.

Movie S6.

This video shows the interaction of two oil-on-water droplets, viewed top-down. The oils, chlorobenzene, carbon tetrachloride (CCl₄), and cyclohexane, were deposited with volumes of 5 μL or 1 μL on a water bath 2.5 mm deep. Note that droplets with an initial volume more than 5 uL generally rotate or swirl but rotation switches to oscillation with decreasing volume, and the droplet-droplet separation remains constant while the droplets evaporate and shrink. Scale bar: 2 mm.

Movie S7.

This video shows oil-on-water droplets (chlorobenzene) remain separate in the open air, but after the water bath is covered to inhibit evaporation, they coalesce as expected because this minimizes their surface energy. Scale bar: 10 mm.

Movie S8.

This video shows the evolving patterns of Marangoni flows between multiple oil-on-water droplets (carbon tetrachloride). Scale bar: 3 mm.

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix.