Partial Coalescence of Sessile Drops with Different Miscible Liquids

Letter pubs.acs.org/Langmuir

Partial Coalescence of Sessile Drops with Different Miscible Liquids Rodica Borcia* and Michael Bestehorn Lehrstuhl Statistische Physik/Nichtlineare Dynamik, Brandenburgische Technische Universität, Erich-Weinert-Strasse 1, 03046 Cottbus, Germany S Supporting Information *

ABSTRACT: Using computer simulations in three spatial dimensions, we examine the interaction between two deformable drops consisting of two perfectly miscible liquids sitting on a solid substrate under a given contact angle. Driven by capillarity and assisted by Marangoni effects at the droplet interfaces, several distinct coalescence regimes are achieved after the droplets’ collision.

■

INTRODUCTION The impact of a drop on a liquid reservoir having the same body as the drop has long been a problem of interest for physicists owing to its relevance in raindrops dynamics,1,2 airborne salt particles,3,4 and foams and emulsions.5−7 Depending on its impact velocity, one can achieve either total coalescence when the falling droplet goes through the liquid reservoir under gravity effects without rupture or partial coalescence when the drop splits after impact into two or more small droplets, called daughter or satellite droplets (refs 8−16 and references therein). In this Letter, we investigate numerically similar phenomena in a new geometry: two deformable drops with the same density ρ, kinematic viscosity ν, and radius R sitting on a solid substrate under the same contact angle θ. The two drops consist of different but perfectly miscible liquids. The droplet on the left-hand side in Figure 1 has a smaller surface tension, σ1, than the second one, σ2 (with σ2 > σ1). The two sessile drops, surrounded by a gas and connected through a precursor film, are approached until they come into contact with each other. After contact, the two droplets merge into a single drop, driven by capillarity and assisted by solutal Marangoni forces acting along the liquid−vapor interfaces.

Figure 1. Sketch of the system: two interacting liquid droplets on a solid substrate. The two drops consisting of different but perfectly miscible liquids are surrounded by a gas and connected through a precursor film about 20 μm from the solid wall to 50% liquid density. The solid substrate is hydrophilic with contact angles varying between 10 and 45°. The two sessile drops approach each other until their contact lines become close enough to initiate the coalescence process. Because of the difference in surface tension, Marangoni flow occurs from lower to higher surface tension.

■

PROBLEM FORMULATION Depending on the competition among the viscosity, the capillarity, and the Marangoni forces, several distinct coalescence regimes can be achieved. We study these coalescence regimes using a phase field model intensively validated earlier in describing floating liquid droplets with an applied temperature gradient,17 static contact angles,18 and the coalescence of drops in two and three spatial dimensions.19,20 Phase field models adopt a continuum description of multiphase systems: they introduce an order parameter ρ that is nearly constant in every bulk region and varies continuously from one phase to the other with a rapid but smooth variation across interfaces. The position of the interface is controlled by the gradients of ρ. The physical processes such as surface tension phenomena at the droplet interface are incorporated © 2013 American Chemical Society

into the Navier−Stokes equation with the help of the Korteweg stress tensor19,21−23 ⎤ ⎡ ∂ρ ∂ρ 1 Tij = 2⎢ρΔρ + (∇ρ)2 ⎥δij − 2 ⎦ ⎣ 2 ∂xi ∂xj

(1)

where δij is the Kronecker symbol and 2 is a parameter connected to the surface tension coefficient that may be extended by concentration gradients. The numerical code based on the Navier−Stokes equations with the Korteweg stress with negligible inertia terms in the vertical direction and the Received: December 28, 2012 Revised: March 20, 2013 Published: March 21, 2013 4426

dx.doi.org/10.1021/la3050835 | Langmuir 2013, 29, 4426−4429

Langmuir

Letter

Figure 2. Time evolution for various coalescence regimes: (a) total coalescence, Δσ = 0, at 0.15, 0.75, 1, and 1.25 s, respectively; (b) total coalescence, Δσ = 4.2 mN/m, at 0.96, 1, 1.43, and 1.79 s, respectively; (c) partial coalescence, Δσ = 4.6 mN/m, at 0.96, 1, 1.43, and 1.57 s, respectively; and (d) drop separation, Δσ = 6.5 mN/m, at 0.15, 0.5, 0.65, and 1.25 s, respectively (ρ = 1000 kg/m3, ν = 5 × 10−6 m2/s, R = 4 × 10−4 m, θ = 15°, and σ1 = 0.05 N/m). Except for case a, the other configurations shown in the lower row of b−d do not represent the last state of evolution.

through precursor film dominates for hydrophilic rigid boundaries (low contact angles).

conservation equations for the total and the partial masses uses a finite-difference method with a mesh of 630 points × 600 points × 200 points under periodic boundary conditions in the horizontal plane. The characteristic thickness of the diffuse interface d is proportional to 2 1/2.18 For the numerical results results presented in this Letter, we have 2 = 3.75 × 10−7 N and d = 1.25 × 10−6 m. A good convergence of the numerical code is achieved for meshes with more than 100 points in one direction. In this way, one assures more than 10 lattice points in the diffuse interface. Figure 2 presents the interaction between two sessile drops with ρ = 1000 kg/m3, ν = 5 × 10−6 m2/s, R = 4 × 10−4 m, θ = 15°, and σ1 = 0.05 N/m for different surface tension gradients Δσ = σ2 − σ1 (i.e., various aqueous mixtures of different nonvolatile diols and carbon acids24−26). The two droplets are connected through a precursor film about 20 μm from the solid wall to 50% liquid density. The droplets are situated in a gas atmosphere with ρv = 1 kg/m3. Diffusion phenomena in the liquid phase are also taken into account with a diffusion coefficient of D = 10−9 m2/s. Being a continuous model, evaporation is naturally included in the phase field formalism. That means in our problem that two kinds of interactions between both droplets are possible, via bridging film and via the gas phase, but the mass transport

■

RESULTS AND DISCUSSION

The coalescence process is initiated through the precursor film between the two liquid drops and is favored by capillarity effects. For the coalescence of identical drops, without Marangoni effects at the droplet interface, the fusion is perfectly symmetric. After collision, the combined drop relaxes from an elliptical to a spherical shape (Figure 2a). For the coalescence assisted by the Marangoni effect, the pushing forces along the droplet interface deform the fused drop. A momentum transfer appears in the longitudinal and transverse directions from the droplet with lower surface tension (left) to the droplet with higher surface tension (right). After fusion, the merging drop is significantly distorted, but no separation of small liquid portions occurs (Figure 2b). When the surface tension gradients between the interacting drops increases, the collision is strong enough to produce the rupture of the mother droplet and to leave behind a secondary droplet (Figure 2c). By analogy to the situations reported in refs 8−16, we will call the case displayed in panel c partial coalescence, whereas both cases 4427

dx.doi.org/10.1021/la3050835 | Langmuir 2013, 29, 4426−4429

Langmuir

Letter

illustrated in Figure 2a,b correspond to the total coalescence regime. Increasing in the following surface tension gradients, the Marangoni-driven flows create vortices inside the droplet with a lower surface tension.27 The return flows near the substrate in the drop with lower surface tension push this droplet to the left. In this way, the two drops move in opposite directions until a complete breakdown of the separating film between the droplets is achieved. Therefore, the droplets separate before coalescing, as illustrated in the time series in Figure 2d. This case will be called the drop separation regime. The full dynamics of the situations presented Figure 2 can be followed through movies available as additional supporting material (the lengths in the movies are scaled to 4d). Because the phase field simulations are time-consuming, the drop behavior is followed numerically only in the first 3 to 4 s from the beginning of the coalescence process, which is not enough time for the establishment of the final equilibrium state, except for the case displayed in Figure 2a. To fuse sessile drops having different surface tensions, the main control parameters describing delayed coalescence have been recently identified in refs 24 and 25 and have been numerically reproduced in ref 20. These parameters are the Ohnesorge number, which describes the ratio between the capillary time (ρR3/σ1)1/2 and the viscous time (R2/ν), Oh = ν

ρ σ1R

the reduced surface tension difference Rσ =

σ2 − σ1 Δσ = σ1 σ1

Figure 3. Phase diagrams (Rσ, Oh) for (a) θ = 15 and (b) θ = 30°.

and the contact angle θ. Using the same control parameters, Figure 3 plots phase diagrams for the regimes of coalescence illustrated in Figure 2. Rσ is varied from 0 to 0.3 by 2 × 10−3 and refined near the borders of different coalescence regimes until a precision of 2 × 10−4 is achieved. As in ref 20, we remain in the range of long viscous times compared to the capillary times (i.e., small Ohnesorge numbers, Oh < 0.1). In this way, strong surface tension gradients are maintained for some time, creating local (Korteweg) stresses that significantly influence the dynamics of the problem. For θ = 15° (Figure 2), a single daughter droplet is observed after the drops collide. At higher contact angle, the contacting neck between droplets becomes thicker, permitting a faster mixing of the miscible components. A faster mixing will favor a rapid reduction of the surface tension gradients between the coalescing drops. Thus, at higher contact angles, the entire fusion dynamics is faster and the separation of the droplets occurs at much higher reduced surface tension differences Rσ (Rσ ≥ 0.184). This allows us to obtain regimes of partial coalescence with one, two, and three daughter droplets (Figure 4). For the case indicated in Figure 4 (Oh = 0.035), the minimal reduced surface tension difference necessary to obtain a secondary drop is Rσ = 0.098, that necessary to obtain two satellite drops is Rσ = 0.1376, and that necessary to obtain three daughter droplets is Rσ = 0.1798. The curves plotted in Figure 3 show a rather poor influence on the Ohnesorge number. The Ohnesorge number is predominantly influenced by viscosity. The viscosity has been used as a parameter in previous work,25 where it was also found

Figure 4. Splitting of one, two, and three satellite droplets for θ = 30°: (a) Δσ = 5 mN/m (Rσ = 0.1) at 0.82 s; (b) Δσ = 7 mN/m (Rσ = 0.14) at 0.71 s; and (c) Δσ = 9 mN/m (Rσ = 0.18) at 0.79 s (ρ = 1000 kg/ m3, ν = 5 × 10−6 m2/s, R = 4 × 10−4 m, and σ1 = 0.05 N/m).

that the viscosity plays a minor role in the coalescence of drops with different but perfectly miscible liquids.

■

SUMMARY We have calculated phase diagrams illustrating different regimes of coalescence (Rσ, Oh) for different contact angles θ, which give predictions of the drop behavior along the solid substrates, control of various interfacial effects, manipulations of tiny droplets in microfluidic and eventually nanofluidic devices without a power supply, and the design of drops or cleaning surfaces. Our computer simulations confirm the experimental results from ref 25, i.e., the most important parameters determining the droplets’ behavior are the reduced surface tension difference Rσ and the contact angle θ. 4428

dx.doi.org/10.1021/la3050835 | Langmuir 2013, 29, 4426−4429

Langmuir

■

Letter

(18) Borcia, R.; Borcia, I. D.; Bestehorn, M. Drops on an Arbitrarily Wetting Substrate: A Phase Field Description. Phys. Rev. E 2008, 78, 066307. (19) Borcia, R.; Menzel, S.; Bestehorn, M.; Karpitschka, S.; Riegler, H. Delayed Coalescence of Droplets with Miscible Liquids: Lubrication and Phase Field Theories. Eur. Phys. J. E 2011, 34, 24. (20) Borcia, R.; Bestehorn, M. On the Coalescence of Sessile Drops with Miscible Liquids. Eur. Phys. J. E 2011, 34, 81. (21) Anderson, D. M.; McFadden, G. B.; Wheeler, A. A. DiffuseInterface Methods in Fluid Mechanics. Ann. Rev. Fluid Mech. 1998, 30, 139−165. (22) Pismen, L. M.; Pomeau, Y. Disjoining Potential and Spreading of Thin Liquid Layers in the Diffuse-Interface Model Coupled to Hydrodynamics. Phys. Rev. E 2000, 62, 2480−2492. (23) Zoltowski, B.; Chekanov, Y.; Masere, J.; Pojman, J. A.; Volpert, V. Evidence for the Existence of an Effective Interfacial Tension between Miscible Fluids. 2. Dodecyl Acrylate−Poly(dodecyl acrylate) in a Spinning Drop Tensiometer. Langmuir 2007, 23, 5522−5531. (24) Riegler, H.; Lazar, P. Delayed Coalescence Behavior of Droplets with Completely Miscible Liquids. Langmuir 2008, 24, 6395−6398. (25) Karpitschka, S.; Riegler, H. Quantitative Experimental Study on the Transition between Fast and Delayed Coalescence of Sessile Droplets with Different but Completely Miscible Liquids. Langmuir 2010, 26, 11823−11829. (26) Karpitschka, S.; Riegler, H. Noncoalescence of Sessile Drops from Different but Miscible Liquids: Hydrodynamic Analysis of the Twin Drop Contour as a Self-Stabilizing Traveling Wave. Phys. Rev. Lett. 2012, 109, 066103. (27) Borcia, R.; Bestehorn, M. Different Behaviors of Delayed Fusion Between Drops with Miscible Liquids. Phys. Rev. E 2010, 82, 036312.

ASSOCIATED CONTENT

S Supporting Information *

The full dynamics of the situations presented Figure 2. This material is available free of charge via the Internet at http:// pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was partially supported by the Deutsche Forschungsgemeinschaft (DFG) under the project ”Dynamics of interfaces between drops with miscible liquids”.

■

REFERENCES

(1) Szakall, M.; Mitra, S. K.; Diehl, K.; Borrmann, S. Shapes and Oscillations of Falling Raindrops - A Review. Atmos. Res. 2010, 97, 416−425. (2) Testik, F. Y.; Barros, A. P.; Bliven, L. F. Toward a Physical Characterization of Raindrop Collision Outcome Regimes. J. Atmos. Sci. 2011, 68, 1097−1113. (3) Gantt, B.; Meskhidze, N. The Physical and Chemical Characteristics of Marine Organic Aerosols: A Review. Atmos. Chem. Phys. Discuss. 2012, 12, 21779−20122. (4) Quennehen, B.; Schwarzenboeck, A.; Matsuki, A.; Burkhart, J. F.; Stohl, A.; Ancellet, G.; Law, K. S. Anthropogenic and Forest Fire Pollution Aerosol Transported to the Arctic: Observations from the POLARCAT - France Spring Campaign. Atmos. Chem. Phys. 2012, 12, 6437−6457. (5) Bhakta, A.; Ruckenstein, E. Decay of Standing Foams: Drainage, Coalescence and Collapse. Adv. Colloid Interface Sci. 1997, 70, 1−124. (6) Murakami, R.; Moriyama, H.; Yamamoto, M.; Binks, B. P.; Rocher, A. Particle Stabilization of Oil-in-Water-in-Air Materials: Powdered Emulsions. Adv. Mater. 2012, 24, 767−771. (7) Sameh, H.; Wafa, E.; Simeh, B.; Fernando, L.-C. Influence of Diffusive Transport on the Structural Evolution of W/O/W Emulsions. Langmuir 2012, 28, 17597−17608. (8) Blanchette, F.; Bigioni, T. P. Partial Coalescence of Drops at Liquid Interfaces. Nat. Phys. 2006, 2, 254−257. (9) Vandewalle, N.; Terwagne, D.; Mulleners, K.; Gilet, T.; Dorbolo, S. Dancing Droplets onto Liquid Surfaces. Phys. Fluids 2006, 18, 091106. (10) Gilet, T.; Mulleners, K.; Lecomte, J. P.; Vandewalle, N.; Dorbolo, S. Critical Parameters for the Partial Coalescence of a Droplet. Phys. Rev. E 2007, 75, 036303. (11) Gilet, T.; Vandewalle, N.; Dorbolo, S. Controlling the Partial Coalescence of a Droplet on a Vertically Vibrated Bath. Phys. Rev. E 2007, 76, 035302. (12) Dorbolo, S.; Terwagne, D.; Vandewalle, N.; Gilet, T. Resonant and Rolling Droplet. New J. Phys. 2008, 10, 113021. (13) Gilet, T.; Terwagne, D.; Vandewalle, N.; Dorbolo, S. Dynamics of a Bouncing Droplet onto a Vertically Vibrated Interface. Phys. Rev. Lett. 2008, 100, 167802. (14) Terwagne, D.; Gilet, T.; Vandewalle, N.; Dorbolo, S. Metastable Bouncing Droplets. Phys. Fluids 2009, 21, 054103. (15) Blanchette, F.; Bigioni, T. P. Dynamics of Drop Coalescence at Fluid Interfaces. J. Fluid Mech. 2009, 620, 333−352. (16) Ling, W. Y. L.; Neidl, A.; Ng, T. W. Effect of Ruptering Encapsulated Bubble in Inducing the Detachment of a Drop. Langmuir 2012, 28, 17656−17665. (17) Borcia, R.; Bestehorn, M. Phase-Field Simulations for Drops and Bubbles. Phys. Rev. E 2007, 75, 056309. 4429

dx.doi.org/10.1021/la3050835 | Langmuir 2013, 29, 4426−4429