Mixing and spreading of different liquids are omnipresent in nature, life and technology, such as oil pollution on the sea1,2, estuaries3, food processing4, cosmetic and beverage industries5,6, lab-on-a-chip devices7, and polymer processing8. However, the mixing and spreading mechanisms for miscible liquids remain poorly characterized. Here, we show that a fully soluble liquid drop deposited on a liquid surface remains as a static lens without immediately spreading and mixing, and simultaneously a Marangoni-driven convective flow is generated, which are counterintuitive results when two liquids have different surface tensions. To understand the dynamics, we develop a theoretical model to predict the finite spreading time and length scales, the Marangoni-driven convection flow speed, and the finite timescale to establish the quasi-steady state for the Marangoni flow. The fundamental understanding of this solutal Marangoni flow may enable driving bulk flows and constructing an effective drug delivery and surface cleaning approach without causing surface contamination by immiscible chemical species.
When a sessile oil drop is released on top of a water surface, it spreads until a monolayer is achieved9, because the liquids are immiscible, as shown in Fig. 1a. In contrast, if a water drop is placed on a water surface, it shows a cascade of coalescence events and the liquids are rapidly mixed (Fig. 1b)10. In contrast with these two configurations, we captured unexpected mixing and spreading features between fully miscible liquids. When a drop of alcohol—for example, isopropanol (IPA)—is placed on a water surface, it spontaneously generates a Marangoni convective flow along the outward radial direction and we observed that there is a static liquid lens in the middle (Fig. 1c and Supplementary Fig. 1), even though these two liquids are infinitely miscible. Here, we discuss solutal Marangoni effects in fully miscible liquids to explain the finite size lens and the associated flow (more details are provided in Supplementary Videos 1–3).
To visualize the spreading and mixing pattern of a miscible liquid drop, IPA (volume V = 7.2 ± 0.2?μl), placed on a water bath (400?ml deionized (DI) water in an 196-mm-diameter Petri dish with depth H = 14?mm), we used time-resolved particle tracking velocimetry (PTV) and high-speed schlieren measurement techniques (Supplementary Information). For PTV experiments, we seeded polystyrene particles (diameter = 100?μm) in solution and recorded the particle motion from top and side views.
IPA is less dense than water and therefore the sessile drop floats on the surface. The drop initially spreads out and quickly achieves a static central lens with a near constant diameter 2R (see Fig. 1c and Supplementary Figs 5–7) during which the IPA continuously leaks at the boundary (Fig. 1c, Supplementary Fig. 1 and Supplementary Videos 3 and 4). The inset of Fig. 2a shows a top view of a schlieren pattern, that is, an interfacial turbulence structure representing the mass transfer between the phases11,12. Due to the Marangoni-driven mixing, the spreading coefficient S (Fig. 2a) becomes zero13, that is,
where the interfacial tension (γ23) between IPA and water is extremely small compared to other surface tensions13. Beneath this mixing zone, we observed from the side flow patterns in the experiment (Fig. 2b and Supplementary Video 6): circulating vortices are located near the edge of the lens.
To measure the spreading speed over the water surface, we seeded 100?μm hydrophobic tracer particles in the IPA drop and tracked the particle motion from the top. Here, we assumed that the particles follow the spreading behaviour of the IPA liquid (Fig. 2c, d and details in Methods). The PTV method was used to obtain the flow speed (Supplementary Information and Supplementary Video 7). As indicated by the red arrows sketched in Fig. 2c, starting from the middle of the drop, we observe that the interfacial flow speed rapidly increases to a maximum at the edge of the static lens, and then decreases monotonically with distance beyond the drop. While the IPA propagates along the water surface, after finite timescales (τ) to establish a quasi-steady state, the maximum flow speed (U) and finite lens size (R) remain approximately constant (deviations are typically less than 10%). The circulating flow in the bath brings fresh water from the bulk to the interface (see Fig. 2b), which maintains a constant surface tension gradient near the contact line (that is, air, IPA and water meet together), as sketched in Fig. 3a. Moreover, such flows would act to effectively sweep surface-bound contaminants away from the location of the drop (Supplementary Video 9).
In the literature, a qualitatively similar flow pattern of a radially outward velocity profile has been reported—for example, the spreading of injected soluble surfactant14 or continuous injection of partially (or fully) miscible liquid on water15,16,17,18—although the quantitative features (power laws) are distinct owing to the different physics of the spreading miscible fluids studied here. For example, for the soluble surfactant spreading case, the velocity profile shows a power-law behaviour14, u ∼ r−1/3 where Δγ is constant at the leading edge. For the miscible liquids spreading case, an understanding of the spreading and mixing mechanism is still lacking, although particular mixing features were captured19,20. Therefore, to understand this case, we performed many different experiments and report results for 12 different combinations of miscible liquids (Table 1) and by adding different surfactants—sodium dodecyl sulfate (SDS) and cetyl trimethyl ammonium bromide (CTAB)—into water. We measured the interfacial flow speed profile during the quasi-steady regime. As shown in Fig. 3c and d, respectively, the maximum Marangoni convective flow speed (U) is proportional to the surface tension difference between the two liquids (Δγ = γ13 − γ12) and inversely proportional to the dynamic viscosity of the liquid bath. To investigate the velocity profile (Fig. 3e), we plotted u/U versus r/R for all of the tested cases (Fig. 3f). The fully miscible case without surfactants shows u/U ∼ (r/R)−1/2 (see the dashed line in Fig. 3f). Δγ at the leading front beyond the static lens decreases along the r-direction as a result of mixing between the ejected alcoholic liquid (IPA, ethanol and tert-butanol) and the liquid bath (water, water–glycerol mixtures and methanol) while spreading.
For the surfactant-contaminated liquid bath, the surface tension profile along the interface is expected to change as sketched in Fig. 3b, as a consequence of a Marangoni-driven circulating flow and non-uniform surfactant distribution. Below the critical micelle concentration (cmc), the surfactant concentrations in the adsorbed layer (near the air–water interface) and in the bulk are different21. The circulating flow delivers bulk liquid that has a low surfactant concentration (relatively high surface tension) compared to that in the adsorbed layer, and hence two opposite Marangoni effects occur in a replenishment zone, as indicated by the brown arrows sketched in Fig. 3b. Thus, the resulting flow speed rapidly decays along the r-direction, as compared to a pure system (Fig. 3e, f and Supplementary Video 10).
We observe that the maximum Marangoni convective flow speed occurs in the replenishment zone (r ≈ R) (Fig. 3e and Supplementary Videos 3 and 4). The location of the replenishment zone (R) and the maximum flow speed (U) are influenced by multiple variables of the system including, for example, viscosities (μ[b, d]), densities (ρ[b, d]), surface tensions (γ[12,13]), droplet volume (V), and the diffusion coefficient (D) between two miscible liquids, where the subscripts ‘b’ and ‘d’ indicate the bath and drop liquid, respectively.
To understand the transport mechanism, we identified the dominant physical mechanisms and performed a scaling analysis to predict finite R and U as well as the time to establish the quasi-steady state. In this problem, viscous effects are negligible for the initial spreading of the deposited drop because the Reynolds number is much larger than unity, Re = ρUR/μ = O(103). Also, we observed that a finite droplet radius (R) is quickly developed in 40–60?ms (Fig. 1c; see more details in Supplementary Fig. 6). Compared to the whole experimental time period (1–2?s), the time to develop a static lens shape is much shorter than the other important timescales. Gravity is not important because of the small Bond number, Bo = ΔρgRh/Δγ = O(10−3), where Δρ = ρb − ρd and h is the thickness of the liquid lens (Fig. 2a), which is a function of time and space during the initial spreading (see Supplementary Fig. 7 and Supplementary Video 11). Eventually, the lens becomes nearly flat in O(10?ms). Thus, for the spreading mechanism, we can estimate the order of magnitude of the dominant forces. At the early times, the driving force/volume for radial spreading is mainly due to the surface tension difference (Δγ/ℓh) and the retarding force/volume is dominated by inertial effects (ρdℓ/t2), where ℓ is the time-dependent spreading radius (see Fig. 1c and Supplementary Fig. 6)22. Therefore, a balance between these two effects gives
As discussed above, when S ≈ 0 due to mixing, the drop stops spreading and takes the shape of a static lens, that is, ℓ(t) → R. How long does it take to establish this condition? As long as the drop spreads, a diffusion boundary layer of thickness δD will develop between the two liquids. When δD ≈ h at the leading edge, we assumed that the spreading stops although the drop thickness decreases (Supplementary Fig. 6) due to liquid leakage caused by the Marangoni flow at the edge of the static lens (r ≈ R). We take h ≈ V/(πR2) for a lens shape of the drop and δD ≈ (Dt)1/2, where D is the diffusion coefficient between the two liquids. To establish the quasi-static lens shape, the required time (τD) is approximately h2/D, where the Stokes–Einstein model was used for the diffusion coefficient (see Methods). Then, the required time is rescaled as τD ≈ V2/(π2R4D). Based on this idea, we use equation (2) to estimate the finite radius of the liquid lens on a fully miscible liquid bath:
After establishing the quasi-static lens shape, a quasi-steady Marangoni flow along the interface is observed. At the rim of the lens (r ≈ R), the balance between Marangoni convection and diffusion results in a viscous boundary layer of thickness δν ≈ (ντν)1/2, where ν(= μb/ρb) is the kinematic viscosity and the relevant timescale τν ≈ R/U, where we next estimate the flow speed U. The Marangoni stress that acts over the spreading distance is balanced with the viscous stresses, that is, Δγ/R ∼ μbU/δν. Using these results and equation (3), we obtain the maximum Marangoni convection flow speed
and the required time to establish the quasi-steady state for the flow
Experimental and theoretical results are compared in Fig. 4. To check the theoretical model, we performed extensive experiments by varying the droplet size, liquid system, viscosity, density, diffusion coefficient, and surface tension, as summarized in Table 2. The theoretical predictions are in good agreement with experimental results.
We have focused on the flows and spreading mechanisms of a fully soluble droplet on a liquid bath, which displays finite diffusion time and length scales at the interface of the two liquids. The surface tension difference triggers the Marangoni flow, U ≈ O (0.1?m?s−1), and it mixes the two materials (Supplementary Videos 3–7), which also apply to dynamics for surfactant-mediated delivery of medications for chronic lung disease23 and eye disease24. Although surfactants are a good material to deliver chemicals effectively25, the surfactants typically remain near the liquid–air interface21. However, as described here, a miscible solute causing solutal Marangoni flows will mix with a bulk liquid and does not significantly change the surface property. Nevertheless, the solutal Marangoni flow can not only deliver materials but also clean liquid surfaces without surface contamination (Supplementary Videos 7 and 9). For example, the deposited drop of solute spreads over an area of 30?cm2 in 200?ms. This fundamental study for Marangoni flow phenomena is thus expected to affect material dissolution, transport and cleaning in a myriad of applications.
400?ml DI water is filled in a Petri dish (196?mm diameter, Himedia Lab.). The depth (H) of the liquid bath is about 14?mm and the volume of the deposited drop is varied from 0.5?μl to 7.2?μl. The 7.2?μl droplet is generated at the tip of a needle (inner diameter ≈ 0.3?mm) and the liquid is supplied by a syringe pump (Harvard PHD 2000). The needle tip is fixed 4?mm above the water surface. For the smaller droplets (<7?μl), we carefully deposited liquid using a micropipette (0.5–10 μl, Eppendorf Research). To minimize the inertial and evaporation effects, we used a micropipette to control the droplet volume.
DI water (resistivity = 18.2?MΩ?cm, Milli-Q Millipore) has density ρ = 0.999?g?cm−3, viscosity μ = 1?mPa?s, and surface tension γ = 72.0?mN?m−1. The deposited droplets are isopropanol (purity = 99.5%, BDH, USA) (ρ = 0.785?g?cm−3, μ = 1.1?mPa?s, and γ = 21.2?mN?m−1), ethanol (anhydrous, 99.5%, Sigma-Aldrich) (ρ = 0.789?g?cm−3, μ = 1.07?mPa?s, and γ = 22.1?mN?m−1), methanol (99%, Sigma-Aldrich) (ρ = 0.792?g?cm−3, μ = 0.53?mPa?s, and γ = 23.7?mN?m−1), tert-butanol (anhydrous, 99.5%, Sigma-Aldrich) (ρ = 0.775?g?cm−3, μ = 3.35?mPa?s, and γ = 20.1?mN?m−1), and 5 cSt silicone oil (Sigma-Aldrich) (ρ = 0.913?g?cm−3, μ = 4.5?mPa?s, and γ = 21.2?mN?m−1).
For the case of fully miscible liquids, the diffusion coefficient is estimated by using the Stokes–Einstein model: D = kBT/(6πμbb), where kB is Boltzmann’s constant (1.38 × 10−23?J?K−1), T is the absolute temperature (298?K), μb is the dynamic viscosity of the liquid bath, and b is the radius of the molecule of the deposited liquid.
To study the effect of the liquid bath viscosity, glycerol (99.7%, BDH) was mixed with DI water. The concentrations of glycerol in the water solution were 25 and 50?wt%. The 25?wt% glycerol in the water mixture has ρ = 1.08?g?cm−3, μ = 2.02?mPa?s, and γ = 70.9?mN?m−1. The 50?wt% glycerol in the water mixture has ρ = 1.13?g?cm−3, μ = 5.35?mPa?s, and γ = 68.0?mN?m−1.
The surfactant used was cationic (cetyl trimethyl ammonium bromide; CTAB, 99%, Amresco) or anionic (sodium dodecyl sulfate; SDS, 98.5%, Sigma-Aldrich). The concentrations of surfactant solutions used for the bath in units of the cmc were 0.10 (γ = 69.1?mN?m−1), 0.53 (γ = 47.5?mN?m−1), and 0.95?cmc (γ = 41.4?mN?m−1) for SDS and 0.27 (γ = 57.4?mN?m−1), 0.48 (γ = 46.8?mN?m−1), 0.68 (γ = 43.8?mN?m−1), and 0.96?cmc (γ = 37.9?mN?m−1) for CTAB, where the critical micelle concentrations of SDS and CTAB are 8.2?mM and 1?mM, respectively.
To visualize the flow pattern, we used hydrophobic polystyrene tracer particles (density ρp = 1.05?g?cm−3 and diameter dp = 100?μm, Thermo Scientific) for the top-view measurements. In this study, the particle concentration is typically very low, that is, NI ≪ 1 and Ns ≪ 1, where NI and Ns are the typical image density and particle source density, respectively (see more details in the Supplementary Information)26. For the side-view experiments, we seeded the particles in DI water after plasma treatment to modify the surface condition of the particle from hydrophobic to hydrophilic. For particle tracking velocimetry measurements, we assumed the particles closely follow the flow because the Stokes number (St = τp/τf) is much smaller than unity, where τp = (ρpdp2)/(18μ) = O(10−4) is the particle response time and τf = (R/U) = O(10−1) is the timescale of fluid motion. Here, R is the radius of the static liquid lens and U is the maximum Marangoni convective flow speed, which are obtained from the particle tracking velocimetry method.
For the top-view experiments, we used a high-speed CMOS camera (Phantom V7.3) having a pixel resolution of 800 × 600 and a 8-bit dynamic range at a frame rate of 4,000?fps. For side views, a high-speed CMOS camera (Phantom v9.1) is used, which has a pixel resolution of 1,634 × 400 and a 8-bit dynamic range at a frame rate of 400 fps. To capture the particle images in a specific plane, a thin light sheet of 1?mm is illuminated in the middle of the drop and the light sheet is generated by using a laser (wavelength = 520?nm, power = 50?mW, Coherent BioRay Laser) and cylindrical optics. For the details of the experimental set-ups, see Supplementary Information.
The physical properties were measured at T = 298?K. We used a pendant droplet method to measure the surface tension values, which were computed by using an in-house Matlab code that is based on the algorithm of Rotenberg and colleagues27. This code was validated by comparing with experimental results from a conventional goniometer (Theta Lite, Biolin Scientific). To measure the viscosities of all liquids, we used a rheometer (Anton-Paar MCR 301) with a CP50-1?geometry and a sandblasted cylinder system (CC27 geometry). Weights were measured by a Mettler Toledo XS105 scale.
The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
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H.K. thanks H.-Y. Kim for helpful discussions. O.S. thanks the Natural Sciences and Engineering Research Council of Canada (NSERC) for a postdoctoral fellowship. K.M. thanks the Justus and Louise van Effen Research Grant from TU Delft and Dr. Hendrik Muller Foundation for a Visiting Student Research Collaborator (VSRC) programme in Princeton University. We thank J. Nunes, A. Perazzo and S. Suin for helpful discussions.
The authors declare no competing financial interests.
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Kim, H., Muller, K., Shardt, O. et al. Solutal Marangoni flows of miscible liquids drive transport without surface contamination. Nature Phys 13, 1105–1110 (2017). https://doi.org/10.1038/nphys4214
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