Letter pubs.acs.org/Langmuir
Can Water Float on Oil? Chi M. Phan,* Benjamin Allen, Luke B. Peters, Thu N. Le, and Moses O. Tade Department of Chemical Engineering, Curtin University, GPO Box U1987, Perth WA 6845, Australia S Supporting Information *
ABSTRACT: The floatability of water on oil surface was studied. A numerical model was developed from the Young−Laplace equation on three interfaces (water/oil, water/air, and oil/air) to predict the theoretical equilibration conditions. The model was verified successfully with an oil/water system. The stability of the floating droplet depends on the combination of three interface tensions, oil density, and water droplet volume. For practical purposes, however, the equilibrium contact angle has to be greater than 5° so the water droplet can effectively float. This result has significant applications for biodegrading oil wastes.
1. INTRODUCTION The tensile strength of interfaces between air/liquid and immiscible liquid/liquid systems has been the driving force in nature and industrial processes such as wetting/dewetting and coating.1 Despite its small magnitude, interfacial tension can be the dominant force in capillary tubes, microgravity conditions,2 and nanofluids.3 In addition to capillary rise, the surface tension can work against gravity and support a solid body floating in water surface, which has been noticed as early as 350 BC by Aristotle.4 In nature, insects such as water striders rely on surface tension to walk on water. Experimentally, small spherical particles, few millimeters in diameter, have been reported as floating on the air/water interface.5 In contrast to rigid bodies, a fluid droplet at surface of another fluid has three deformable interfaces and variable contact angles. The equilibrium of three interface tensions at the contact line, if it exists, results in an unique combination of three contact angles, which form a Neumann’s triangle.6,7 The interfacial interaction between three fluids has been investigated for an oil droplet spreading on water surface by Langmuir8 and for a fluid droplet at the interface between two fluids by Princen.9,10 More recently, numerical methods were applied to identify the shape of such droplets.11,12 In these instances, the liquid droplet is less dense than the supporting liquid, and gravity always plays a stabilizing role on the system. On the other hand, there are no reports on the “floatability” of a heavier liquid droplet on the surface of a lighter liquid. It would be particularly significant if the phenomena can be applied to the most common system in nature and industrial processes: a water droplet on an oil surface. In this study, we investigate theoretically and experimentally the floatability of a water droplet on an oil surface. This result can lead to a new and advanced mechanism in processing oil/water mixtures, such as biodegrading process of unwanted oils, including vegetable oils, sand oil tailings, and oil spillages. © 2012 American Chemical Society
2. MODELING A new model is developed for a water droplet, with volume Vb, depositing on oil/air interface (Figure 1). The physical properties of the system, three densities (ρa, ρw, and ρo) and three interfacial tensions (γwa, γow, and γoa), are fixed parameters. The aim of the numerical model is to predict the vertical force on water droplet, as a function of droplet shape: F = g[Vbρw + (πr 2h3 − V1)ρa − (πr 2h3 + V2)ρo]
(1)
where g is the gravitational constant, r is the contact line radius, h3 is the height of the “helm”, V1 and V2 are the volumes of air/water and water/oil sections, respectively (V1 + V2 = Vb). In the above equation, the first two terms account for gravity, while the last term accounts for buoyancy. The numerical model first finds V1, V2, r, and h3 from the contact angles. Since the balance of three tension forces along the contact line is zero, the three contact angles in Figure 1 satisfy
γ wasin θ1 − γow sin θ2 + γoa sin θ3 = 0
(2)
γwa cos θ1 − γow cos θ2 − γoa cos θ3 = 0
(3)
Equations 2 and 3 are equivalent to
⎡γ 2 + γ 2 − γ 2⎤ wa ow oq ⎥ θ1 + θ2 = π − arccos⎢⎢ ⎥ 2γwaγow ⎦ ⎣
(4)
⎡γ 2 + γ 2 − γ 2⎤ ow wa ⎥ θ1 + θ3 = arccos⎢ oa ⎥⎦ ⎢⎣ 2γoaγow
(5) 7
One can easily relate eqs 4 and 5 to Neumann’s triangle: two angles in conventional Neumann’s triangle are (θ1 + θ2) and (π − θ1 − θ3). It should be also noted that the required condition for the above of equations (i.e., existence of Neumann’s triangle) is that the sum of any two tensions has to be greater than the third one. Received: December 7, 2011 Revised: January 25, 2012 Published: February 21, 2012 4609
dx.doi.org/10.1021/la204820a | Langmuir 2012, 28, 4609−4613
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Figure 1. Diagram of a water droplet on oil surface. The three interfaces are governed by the Young−Laplace equation, and matched at the contact line, radius r. The three contact angles are arranged according Neumann’s triangle so that the sum of three tensile forces acting on the contact line is zero.
Figure 2. Flowchart of the numerical model. The model integrates the Young−Laplace equation along the three interfaces, from the initial points to the contact line, and matches three
contact angles (the model flowchart is shown in Figure 2). Since the system is axis-symmetric, the air/water, oil/water, and oil/air interfaces 4610
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Figure 3. Experimental setup for air/water surface tension in the presence of oil. are part of sessile drop, pendant drop, and “submerged helm”,13 respectively. The interfacial configuration of three interfaces can be numerically integrated using the system of first-order ODEs:13 dX = cos ϕ dS dZ = sin ϕ dS
(6)
dϕ sin ϕ + = Z + υ0 dS X where S is the arc length measured from the initial point, ϕ is the tangent angle, and υ0 is the dimensionless curvature at the initial points (S = 0). The three dimensionless quantities (S, Z, X) in eq 6 are the ratios between the dimensional quantities and the specific capillary length at each interface.13 For air/water and oil/water interfaces, the initial points are located at the apex: X = 0, Z = 0, ϕ = 0. For the oil/air interface, the initial point is located near infinity (since X = ∞ at ϕ = π), which can be determined by an asymptotic solution:13
Za = − 0
tan(φa) K 0( 2 X a) 2 K1( 2 X a)
(7)
1
where K , K are the Bessel functions of the second kind of first and second order, respectively; Xa and ϕa are the horizontal coordinate and tangent angle at the reference point. Boucher13 has found that the asymptotic values are acceptable when ϕa ∼ 0.973π. In this study, a higher value, 0.9995π, was selected to ensure the accuracy. In developing the numerical model, we assume h1 and h3 ≪ h2, so that the hydrostatic pressure of water/air and oil/air interfaces on water/oil interface is negligible. Once the three interfaces are matched, V1, V2, r, and h3 are substituted in eq 1 to determine F. Consequently, the model can find equilibrium state, that is, angle θ2* so that F = 0. However, the equilibrium configuration, even if it exists, is not always obtainable in practice. This is a fundamental difference between the studied system and the opposite system of oil droplet at water surface,8,11,12 in which the droplet always returns to equilibrium configuration if disturbed. For a water droplet in oil, the droplet might sink if strongly disturbed and thus equilibrium would not be observed. The significance of the disturbances and stabilization process is discussed in following sections.
Figure 4. Water droplets at different volumes (10, 40, 100, 130, and 160 μL). unsuccessful. The failure of mineral oils can be explained by the low interface tensions of these oils with both water and air. As mentioned above, if γwa + γow < γoa, eqs 2 and 3 have no feasible solutions. In our experiments, a stable contact line was never observed in mineral oils even at 10 μL. On the other hand, vegetable oil can support a water droplet up to 170 μL, with a stable and visible three phase contact line. Consequently, the vegetable oil was selected for further analysis. Oil density was measured by using a DMA 4500 instrument (Anton Paar). The three interfacial tensions were measured by the pendant drop method using PAT1 (SInterface). The air/oil and water/oil tensions were measured using a standard method as described elsewhere.14 For the air/water interface, an oil layer is deposited on the top (as shown in Figure 3), to correctly represent the tension of water/air interface in floating water droplet. All measurements were repeated twice and were reproducible at room temperature (22 °C).
3. EXPERIMENTS In order to verify the theory, different oils including both mineral and vegetable oils were tried. Deionized water (Millipore, 18 MΩ) was used as heavier liquid. Water droplet was deposited in oil interface by a micropipet of 10 μL. If the droplet floats, additional volume of 10 μL was added until it sinks. Among tested oils, a commercial vegetable oil has a suitable combination of three surface tensions to support water droplets, whereas pure mineral oils (hexane, octane, and decane) were 4611
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volume. On the other hand, θ2 increased slightly as predicted by the model (Figure 5). Hence, the droplet was able to balance the volumetric increment by deforming the water/oil interface (i.e., increasing the curvature υ02). It can be seen that the model predicts the experimental data very well. The variation and discrepancy can be contributed to errors in contact angle measurement,15 which depends on the accuracy of contact line position, and the model assumption of h1 ≪ h2 (the ratio ∼1% to 5%). Maximum Volume for Stabilized Droplet. From experimental observation, it is evident that the water droplet becomes less stable with increasing volume, and there is a maximum limit beyond which the droplet will sink regardless of deposition process. Theoretically, on contrast, an equilibrium configuration is possible for a very large droplet as the water/oil interface can further deform to accommodate additional volume (Figure 5). However, additional volume significantly decreases θ1* as shown in Figure 6. The discrepancy between modeling and experimental maximum volumes can be explained by considering disturbance and stabilization process. When disturbed, the system tends to return to equilibrium configuration by bouncing vertically as observed. During stabilization process, the total force expectedly acts in the opposite direction of displacement. However, if the system is disturbed strongly, for example, θ1 = 0°, the geometry is fundamentally changed and thus cannot return to the equilibrium configuration. As shown in Figure 1, smaller θ1* means less capacity to bounce around. Practically, the results indicated that minimum θ1* for stabilized droplet is ∼5° (Figure 5), which corresponds to droplet of 170 μL.
Table 1. Physical Properties of the Water/Oil System interface tension (mN/m) oil density (kg/m3)
water/air
oil/air
water/oil
915.8
44.9
21.7
31.8
4. RESULTS AND DISCUSSION Droplet Stability. Although stable, the water droplet was disturbed slightly (bouncing once or twice) when additional volume was deposited. Typically, the droplet is stabilized within 100 ms after addition. At higher volume, ∼80 μL, the droplet can sink during the deposition process. Images of stable water droplets of different size are shown in Figure 4. A movie is also supplied in the Supporting Information. When water was gently deposited, a stable droplet can be observed up to 170 μL. Carefully watching the movie, one can see that the droplet is more stable (i.e., sustaining disturbance without sinking) at smaller volumes. Physical Properties. The surface tensions and density of the oil were measured and tabulated in Table 1. The density of the oil is similar to other vegetable oils (which comprise fatty acids). In this case, the natural surfactants from the oil are water-soluble and reduce the air/water interface tension correspondingly (surface tension of pure water is ∼72 mN/m). Contact Angle. The experimental contact angles were measured by snake-based method.15 Since θ1 and θ3 were very small (Figure 4), only θ2 was used to validate the model. Using the physical properties in Table 1 as input, the model can predict contact angles and compare to experimental data. In Figure 4, it can be seen that h2 increased with increasing
Figure 5. Modeled and experimental values of θ2. The maximum volume of a stable droplet is 170 μL.
Figure 6. Total acting force as function of θ1 for three different droplet sizes. 4612
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(5) Extrand, C. W.; Moon, S. I. Using the Flotation of a Single Sphere to Measure and Model Capillary Forces. Langmuir 2009, 25, 6239−6244. (6) Buff, F. P.; Saltsburg, H. Curved Fluid Interfaces. II. The Generalized Neumann Formula. J. Chem. Phys. 1957, 26, 23−31. (7) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, UK, 1982. (8) Langmuir, I. Oil Lenses on Water and the Nature of Monomolecular Expanded Films. J. Chem. Phys. 1933, 1, 756−776. (9) Princen, H. M. Shape of a fluid drop at a liquid-liquid interface. J. Colloid Sci. 1963, 18, 178−195. (10) Princen, H. M.; Mason, S. G. Shape of a fluid drop at a fluidliquid interface. II. Theory for three-phase systems. J. Colloid Sci. 1965, 20, 246−266. (11) Elcrat, A.; Neel, R.; Siegel, D. Equilibrium Configurations for a Floating Drop. J. Math. Fluid Mech. 2004, 6, 405−429. (12) Burton, J. C.; Huisman, F. M.; Alison, P.; Rogerson, D.; Taborek, P. Experimental and Numerical Investigation of the Equilibrium Geometry of Liquid Lenses. Langmuir 2010, 26, 15316−15324. (13) Boucher, E. A. Capillary phenomena: properties of systems with fluid/fluid interfaces. Rep. Prog. Phys. 1980, 43, 497−546. (14) Phan, C. M.; Nguyen, A. V.; Evans, G. M. Dynamic adsorption of sodium dodecylbenzene sulphonate and dowfroth 250 onto the air− water interface. Miner. Eng. 2005, 18, 599−603. (15) Stalder, A. F.; Kulik, G.; Sage, D.; Barbieri, L.; Hoffmann, P. A snake-based approach to accurate determination of both contact points and contact angles. Colloids Surf., A 2006, 286, 92−103. (16) Schramm, L. L. Surfactants: Fundamentals and Applications in the Petroleum Industry; Cambridge University Press: Cambridge, UK, 2000.
Finally, it is also noteworthy that the interface tensions affect the maximum volume through two separate mechanisms: (i) equilibrium contact angles (determined by ratios between three interface tensions7); and (ii) the shapes of interfaces and thus V2,V1, and h3 (determined by the absolute values of interfacial tension).
5. CONCLUSIONS We demonstrated theoretically and experimentally that water droplets can float on oil surface due to a combination of interfacial tensions and buoyancy, up to a certain volume. Our numerical model was developed from Young−Laplace equation on three interfaces (water/oil, water/air, and oil/air) to calculate the theoretical equilibrium. The model compared well to experimental data of a water/vegetable oil system. The stability of the floating droplet depends on the combination of three interface tensions, oil density and droplet volume. For practical purpose, however, the equilibrium contact angle, θ1*, should be greater than 5° so water droplet can effectively float. One potential application of the phenomena is employing small water droplets on immiscible oil (such as fatty waste, oil sands tailings,16 or oil spillages floating in the ocean) to facilitate aerobic biodegradation. The efficiency of oil biodegradation depends on the level of dissolved oxygen, water/oil interfacial area, and bacterial/nutrient availability. Since most vegetable and mineral oils are lighter than water, these compounds tend to cover water surface and prevent atmospheric oxygen from dissolving into water. Aqueous biodegradation of these compounds requires continuous mixing and aeration. Similarly, dispersants are used to disperse oil spillages in ocean to increase the oil/water contact area. The usage of dispersants remains controversial due to uncontained ecological impacts. Small water droplets stabilizing on oil surface, such as in this study, can have large water/oil contact area as well as concentrated/selected bacterial population. In addition, these surface droplets can maintain high level of dissolved oxygen due to direct exposure to the air. With a wide range of available surfactants, the three interface tensions can be easily manipulated to stabilize water droplet in any oils.
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ASSOCIATED CONTENT
* Supporting Information S
Movie of the droplet experiment. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
(1) de Gennes, P. G. Wetting: statics and dynamics. Rev. Mod. Phys. 1985, 57, 827−863. (2) Concus, P.; Finn, R.; Weislogel, M. Measurement of critical contact angle in a microgravity space experiment. Exp. Fluids 2000, 28, 197−205. (3) Wasan, D. T.; Nikolov, A. D. Spreading of nanofluids on solids. Nature 2003, 423, 156−159. (4) Finn, R. The contact angle in capillarity. Phys. Fluids 2006, 18, 047102. 4613
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