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Decorating a Liquid Interface Promotes Splashing Stephane Douezan and Franc-oise Brochard-Wyart* Physico-Chimie Curie, Institut Curie, UMR 168, UPMC, Paris, France ABSTRACT: We report on the role of a surfactant monolayer in the impact of small spheres on an airwater interface. We observe that an air cavity (splashing) is induced above a threshold impact velocity. We explore the dependence of this threshold velocity on the bath surface tension and on the bath viscosity using water ethanol and waterglycerol mixtures, respectively. Interestingly, the threshold velocity for air entrainment is reduced by the presence of a stearic acid monolayer because the hydrophobic tails favor the forced entry of air during the sphere’s impact. More generally, we show that this threshold velocity is determined by the wettability of the sphere by the bath, which can equivalently be controlled by tuning the bath properties (presence of a monolayer) or the sphere surface properties (via a surface treatment). These results provide insight into recently developed vesicle production techniques based on the impact of drops or jets on a lipid layer.
’ INTRODUCTION Encapsulation of biological molecules requires the development of new vesicle production techniques. A main type of such techniques is based on the impact of a drop of the biomolecular solution on an interface decorated with surfactants, which provides the encapsulating layer. Under the right conditions, the impact deforms and pinches off the interface, releasing the desired vesicle. An example of application of this principle is the inverted emulsion technique to produce unilamellar vesicles. This technique consists in impacting water-in-oil emulsion droplets on an oilwater interface.1,2 Another such technique, of great interest for scientific, industrial, and medical applications, impinges a pulsed microfluidic jet onto a lipid bilayer to simultaneously create and load giant unilamellar vesicles (GUVs).3 This technique requires one to carefully tune the bilayer properties in order to form vesicles and avoid the bursting of the lipid bilayer under the action of the jet. Motivated by understanding the impact on a surface layer, we investigate the role of a surfactant monolayer on the impact dynamics by examining the impact of small spheres onto bare and decorated interfaces. Impacts of objects on liquid interfaces have been intensively studied due to their relevance in a broad range of applications. Naval and military applications originally aroused interest to impact phenomena to develop the sea landing of planes or to control the torpedo trajectories.4,5 Geophysics and industrial engineering a few decades ago gave rise to worthwhile studies to understand the mechanisms of crater formation.6 More recently, biology explored the impact mechanisms to understand the ability of small insects to walk on water.7 The wide variety of applications prompted general studies on the liquid entry of spherical objects8 and the resulting cavities.911 Most of the existing studies characterized the cavities resulting from the impact of big objects,12,13 where the Bond number that estimates the relative magnitude of gravitational to capillary forces is larger than unity (Bo = FgR02/γ, where F is the liquid density, g is the r 2011 American Chemical Society
gravitational acceleration, R0 is the sphere radius, and γ is the liquidvapor surface tension). Given the size range of interest for biological applications, we consider here the impact of small spheres. Recently, Aristoff et al.14 examined experimentally the impact of small spheres (low Bo) on water. They theoretically characterized the time evolution of the cavity shape and the cavity collapse using an extension of the method developed to solve the RayleighBesant equation for the collapse of a spherical cavity in an infinite liquid bath.15,16 Here we investigate the criteria to create an air cavity (splashing) when a small solid sphere (Bo , 1) impacts on an airwater interface with a velocity U0. A first condition for air cavity formation is that the inertial pressure FU02 is higher than the Laplace pressure γ/R0. Equivalently, the Weber number We = FU02R0/γ must be larger than unity (We . 1). A second condition is that fluid inertia is not dissipated by viscous friction, which requires that the Reynolds number is larger than unity, Re = FU0R0/η . 1, where η is the fluid dynamic viscosity. Given the parameter range of interest (We . 1 and Re . 1), viscosity and sphere wettability have usually been neglected in the impact description.4,5,17 Contrary to this assumption, Duez et al. demonstrated (for high Bo) that the wettability of the spheres plays a important role in the cavity formation.18 They found a threshold impact velocity for air entrainment, UE, which decreases as the sphere becomes more hydrophobic. A recent study investigated the effect on cavity formation of imparting a spin to the sphere, which proved to be analogous to coating the sphere to render it half hydrophobic and half hydrophilic.19 Here we examine the role of both the sphere and the bath wettability in the creation of air cavities for the case of low Bo impacts on a liquidgas interface. Received: May 5, 2011 Revised: July 12, 2011 Published: July 12, 2011 9955
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’ FORMATION OF AN AIR CAVITY: THRESHOLD VELOCITY A glass sphere (density Fsphere = 2.5 103 kg m3 ; radius R0 = 0.75 mm) is dropped from a height H0 above the fluid surface of a tank (length: 4 cm; width: 4 cm; depth: 10 cm) filled with pure water at room temperature. The sphere hits the surface with a velocity U0 ≈ (2gH0)1/2. We image the sphere crossing of the airwater interface for increasing impact velocities using a Vision Research Phantom V4.2 high-speed video camera (8130 frames/s). We analyze the images using ImageJ software (National Institutes of Health, USA), and we measure the precise impact speed. At low velocity, the sphere becomes completely surrounded by water without any air entrainment (Figure 1A). Upon impact, a cavity forms and then collapses when the contact line approaches the apex of the sphere, at a depth approximately equal to the capillary length (lc ≈ 2.7 mm).14 As U0 increases and becomes larger than a threshold velocity UE, cavity pinch-off occurs. The impact
generates an air cavity that deepens for a few sphere diameters and finally pinches off at a depth of the order of the capillary length, entraining a volume of air in the liquid phase (Figure 1B). Further increases in U0 give rise to cavities that pinch off at larger depths and will not be discussed here. We characterize the extension and lifetime of the cavity for impact velocities larger than the air entrainment threshold, U0 > UE. For various velocities, we measure the time τ for the cavity to pinch off and the cavity extension H at the pinch-off. The cavity size H scales as H = U0τ. Figure 2 shows the cavity size H = U0τ normalized by the sphere radius R0 as a function of We. We observe that τ increases with We. The pinch-off time can be theoretically predicted by balancing kinematics and Laplace
Table 1. Relevant Dimensionless Groups and Their Characteristic Valuesa dimensionless group
a
definition
range
Weber number
We = FU02R0/γ
10110
Bond number
Bo = FgR02/γ
102
Reynolds number
Re = FU0R0/η
1103
The sphere radius is R0 = 0.75 mm. The impact velocities are varied between U0 = 0.05 and 1.50 m s1.
Figure 2. Dimensionless cavity extension H/R0 = U0τ/R0 is represented as a function of We1/2. The dashed line is the theoretical prediction obtained by balancing kinematics and Laplace pressures on the air cylinder (eq 1).
Figure 1. Chronophotography of the water entry of spheres at low Bo (R0 = 0.75 mm and Δt = 0.5 ms between two images). The resulting cavity can be controlled by either the impact velocity or the wetting conditions of the impact. (A) When a hydrophilic sphere enters the airwater interface with a velocity U0 = 0.72 m s1, no air cavity is created (U0 < UE). (B) When the same hydrophilic sphere enters the airwater interface with a velocity U0 = 1.30 m s1, an air cavity is formed (U0 > UE). (C) When a hydrophobic sphere enters the airwater interface with a velocity U0 = 0.72 m s1 (identical to (A)), an air cavity is formed (U0 > UE). (D) When a hydrophilic sphere enters the airwater interface decorated by a stearic acid monolayer (surface concentration Cs = 2.7 107 g cm2) at U0 = 0.72 m s1 (identical to (A)), an air cavity is formed (U0 > UE). 9956
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Table 2. Depence of UE on the Sphere Radius R0 in Pure Water
Figure 3. (A) Schematic of the dependency of the triple line velocity V on θd. The triple line is unstable above a critical velocity UE. (B) Proposed mechanism of an air cavity formation: When the sphere enters water with a velocity U0 > UE, the forced entry of air leads to air entrainment.
pressures on the air cylinder. If R0 is the cavity size and R0/τ is the closure velocity, this balance yields FR02/τ2 ∼ γ/R0. Thus, the time to pinch off scales as !1=2 FR0 3 ð1Þ τ∼ γ In water where F = 103 kg m3, γ = 72 mN m1, and R0 = 0.75 mm, we obtain τ ≈ 2.5 ms, which is consistent with the time measured experimentally. Using eq 1, the resulting cavity extension scales as (H/R0)2 ∼ We. The dashed line in Figure 2 stands for the predicted pinch-off time. This simple scaling is in good agreement with our experimental data. We now discuss the role of a surface film on the cavity formation. The threshold velocity UE to create a cavity can be interpreted in the framework of wetting dynamics, as shown by Duez et al.18 To create a cavity, air has to be entrained by the sphere as illustrated in Figure 3. This situation is an inverse case of the problem of forced wetting.20 The forced entry of air will occur above a critical speed analogous to the threshold speed of forced wetting of a nonwettable plate by a liquid.21,22 This critical condition is derived from the balance between the viscous and air friction contributions on one hand and the Young force on the other hand. This can be written (in scaling) ηV η V þ air ∼ γðθd 2 θE 2 Þ θd π θd
ð2Þ
where V is the triple line velocity, θd is the dynamic contact angle, and θE is the static contact angle. The dependence of V on θd obtained from eq 2 is represented in Figure 3A. The instability of the triple line corresponds to air entraiment. Figure 3 A exhibits a critical value to create an air cavity given by the maximum velocity. The threshold velocity for air entrainment UE has been
R0 (mm)
UE (m s1)
0.3
0.83 ( 0.03
0.5 0.75
0.82 ( 0.02 0.83 ( 0.02
Figure 4. Dependence of the threshold velocity UE for a hydrophilic glass sphere on the ratio γ/η. The dashed line is the theoretical prediction UE ≈ ξγ/η. From the fit, we obtain ξ ≈ 103.
established in the limits of a static contact angle θE . π/2:18 γ UE ¼ R ðπ θE Þ3 ð3Þ η The numerical prefactor R is obtained from the experimental measures of UE when a hydrophobic sphere enters in pure water. We obtain R ≈ 103. This defines a condition to form a cavity. A cavity will be created if the impact velocity is above the critical speed, U0 > UE (assuming that We > 1). In the limit of small θE, the threshold velocity is given by UE ≈ ξγ/η, where ξ is a numerical prefactor of the order of 103. It is remarkable that UE depends on viscosity even at high Reynolds number. This is due to the fact that at the local scale of the contact line, the Reynolds number becomes smaller than unity. Besides, UE is independent of the sphere size. We verify this hypothesis by measuring UE in pure water for spheres of various radii R0 (R0 = 0.3, 0.5, and 0.75 mm). In all cases, we measure a threshold velocity about UE = 0.83 m s1 (Table 2). Hereafter, we consider spheres with a constant radius R0 = 0.75 mm. We examine the dependence of the threshold velocity for a small wetting glass sphere (low Bo) on the ratio γ/η (Figure 4). To tune the surface tension γ we use waterethanol mixtures (between 0 and 100 wt % in ethanol). To tune the viscosity η, we use waterglycerol mixtures (between 0 and 20 wt % in glycerol). For these fluids, the contact angle on the sphere is always below 20°. We vary γ between 22 and 72 mN m1 and η between 1 and 1.75 mPa 3 s. It is noted that larger viscosities affect not only the threshold velocity but also the shape of the cavity.23 The dashed line in Figure 4 is a linear prediction UE ∼ γ/η. The good agreement between this linear prediction and the data indicates that the linear dependence of UE on γ/η established in the case of high Bo impacts18 is still valid for low Bo impacts.
’ ROLE OF WETTABILITY The threshold velocity to form an air cavity (eq 3) depends upon the wettability of the sphere at the liquidair interface. 9957
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adjusted either by a surface treatment of the sphere or a coating of the liquid bath. To understand the role of the stearic acid monolayer on the immersion dynamics when a sphere goes through the interface, we consider quasi-static impacts. Small spheres are dropped off close to the surface so that they impact onto the surface at a very small velocity. As the inertial pressure is lower than the Laplace pressure, the spheres are maintained at the airwater interface (We < 1). Gravity is negligible for sphere sizes R0 smaller than the capillary length lc. Capillary effects dominate and the bead will sit onto the surface at various depths (Figure 5A) adjusting its contact angle θE so that the liquid surface remains flat as for a sphere in a zero-gravity environment. The distance H between the bottom pole of the sphere of radius R0 and the surface is H ¼ R0 ð1 þ cos θE Þ
ð4Þ
Figure 5B,C schematizes the variation of H depending on the spreading coefficient S. This coefficient measures the difference of interfacial tensions between dry and wet surfaces and is defined by S ¼ γS=O ðγS=L þ γÞ ¼ γðcos θE 1Þ Figure 5. (A) Prediction of the floating position of a sphere deposited on a water surface. When adding surfactant on the surface (Cs increases), the hydrophilic sphere moves up above the surface (B) whereas the hydrophobic one moves down (C).
We investigate the role of both the surface state of the sphere and the composition of the bath surface layer in the case of low Bo impacts on an airwater interface. We consider the impact on a pure water bath of two small glass spheres, identical in every aspect except their surface state, with the same velocity U0 = 0.72 m s1. One sphere is immersed during 40 min in a “piranha” solution (1 vol H2O2, 2 vol H2SO4) and then rinsed using ultrapure water to have a very hydrophilic surface (θE ≈ 20°). The second one is coated with silane chains by grafting octadecyltrichlorosilane (Sigma-Aldrich, Saint Louis, MO) for 10 h at ambient temperature in n-hexadecane. After silanization, the beads are rinsed with chloroform and dried. The resulting spheres have a very hydrophobic surface (θE ≈ 100°110°). We observe (Figure 1A) that the hydrophilic sphere enters the pure water bath without creating any cavity. On the contrary, the impact of the hydrophobic sphere leads to the formation of an air pocket (Figure 1C). Therefore, the sphere surface treatment plays a role in the threshold velocity for low Bo impacts as first shown by Duez et al. for large spheres.18 Next, we examine the influence of the bath surface state. A liquid monolayer of stearic acid (Sigma-Aldrich, Saint Louis, MO) is deposited on the interface between air and water. Since the polar group of stearic acid is oriented toward the water phase and the hydrophobic tail toward the air, the surface of the tank is made hydrophobic by the monolayer. By impacting a highly hydrophilic sphere at the same velocity U0 = 0.72 m s1, we observe (Figure 1D) that the sphere enters the airwater interface covered with stearic acid forming an air pocket, similarly to the case of a sphere coated with hydrophobic groups impacting on a pure water bath. Thus, we observe an equivalent effect of treating the glass sphere surface and of coating the surface of the water. It shows that the role of wettability in the threshold velocity can be
ð5Þ
where γS/O, γS/L, and γ are respectively the solidair, solid liquid, and liquidair interfacial tensions, and θE is the static contact angle. Thus, S is controlled by both γS/O and γ and can be adjusted by changing either the solid surface or the coating of the bath. Experimentally, spheres made hydrophilic by surface treatment sit under the interface in pure water (Figure 5B). When adding stearic acid at increasing concentration, S becomes negative and the contact angle θE increases. The hydrophilic sphere moves up above the interface, as shown in Figure 6A. For an identical sphere coated with hydrophobic molecules, we observe the opposite tendency (Figure 5C). In pure water, the hydrophobic sphere sits at the highest position, and it moves down as the concentration of surfactant increases as seen in experiments (Figure 6B). For each monolayer, we measure the depth of the sphere H and the surface tension γ as a function of the surface concentration of stearic acid Cs. From these measurements, we extract the values of the spreading parameter S (Figure 6). The addition of stearic acid to the bath makes less favorable its interaction with the hydrophilic sphere but more favorable with the hydrophobic sphere. As the interaction between the sphere and the bath becomes less favorable, the spreading parameter decreases and the static contact angle θE increases. The static contact angle of the hydrophilic sphere, which is small in pure water, considerably increases with the concentration of stearic acid at the liquid interface. The resulting position of the hydrophilic sphere for Cs = 2.5 107 g cm2 is comparable to that of the hydrophobic sphere on pure water. In conclusion, the presence of a surfactant monolayer modifies the resulting static contact angle θE in a manner equivalent to modifying the surface coating of the sphere. We can now interpret the observations of Figure 1. Figure 1A shows the water entry of a hydrophilic sphere (for which the contact angle θE is small) at an impact velocity smaller than the threshold speed for air entrainment (U0 = 0.72 m s1 < UE = 0.83 m s1). Thus, no air cavity is formed. Figure 1B shows the same hydrophilic sphere impacting at a larger velocity (U0 = 1.3 m s1 > UE = 0.83 m s1) and thus forming a cavity. Figure 1C shows 9958
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Figure 6. Floating depth H = R0(1 + cos θE) when small beads (R0 = 0.75 mm) are deposited on a water surface decorated with stearic acid depends on the surfactant surface concentration Cs. (A) A hydrophilic sphere sits under the airwater interface. As the surface concentration Cs in stearic acid increases, the sphere rises. (B) A hydrophobic sphere floats at the airwater interface. As the surface concentration Cs in stearic acid increases, the sphere moves lower.
Figure 7. Regime diagram indicating the dependence of the observed cavity on the impact velocity (U0) and static wetting angle (θE) for various surface concentrations of stearic acid Cs: Cs = 0 g cm2 (2, Δ), Cs = 2.04 107 g cm2 (b, O), Cs = 2.17 107 g cm2 (9, 0), and Cs = 2.46 107 g cm2 ([, ]). Full and hollow markers correspond to cases with and without cavity formation, respectively. The solid line is the theoretical prediction for the threshold velocity UE to form an air cavity for an hydrophilic sphere (θE < π/2) in pure water.18 Above the line (gray shaded) a cavity is formed, below no cavity. The presence of the surfactant monolayer shifts this theoretical limit (dashed line). The threshold velocity to form a cavity is decreased.
a hydrophobic sphere impacting at the same speed as the hydrophilic sphere in Figure 1A, U0 = 0.72 m s1. The higher contact angle θE of the hydrophobic sphere yields a lower UE (eq 3), and thus the hydrophobic sphere forms an air cavity. Figure 1D shows the same hydrophilic sphere as in Figure 1A impacting at the
same velocity. In this case, however, a surfactant monolayer has been deposited on the bath, thus increasing θE and decreasing UE, so that U0 = 0.72 m s1 > UE and a cavity is formed. The same glass sphere dropped at a velocity U0 can be under the threshold velocity UE while impacting a bare airwater interface (Figure 1A) and above UE while entering the interface decorated with stearic acid (Figure 1D). Figure 7 shows the likelihood of cavity formation as a function of impact velocity and static wetting angle for various surface concentrations Cs of stearic acid. When surfactant is deposited on the surface, the hydrophilic sphere becomes less wettable and UE decreases. This indicates that the theoretical limit (UE) to form an air cavity when a small sphere impacts a liquid interface is shifted by the presence of a surfactant monolayer.
’ CONCLUSION The critical velocity for air entrainment depends on the wettability properties of the body impacting on the surface and can be changed either by surface treatment of the sphere or of the bath surface. Surface tension and viscosity are both relevant parameters to modify the critical velocity, which therefore can be optimized according to the needs of the application. The results presented here can be applied to predict the conditions to create a vesicle in recent vesicle production techniques.1,3 ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. 9959
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’ ACKNOWLEDGMENT The authors thank D. Gonzalez-Rodriguez for a careful reading of the manuscript. ’ REFERENCES (1) Pautot, S.; Frisken, B. J.; Weitz, D. A. Langmuir 2003, 19, 2870–2879. (2) Pautot, S.; Frisken, B. J.; Weitz, D. A. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 10718–10721. (3) Stachowiak, J. C.; Richmond, D. L.; Li, T. H.; Liu, A. P.; Parekh, S. H.; Fletcher, D. A. Proc. Natl. Acad. Sci. U.S.A. 2008, 105, 4697–4702. (4) von Karman, T. NACA Technical Note 321, 1929. (5) Wagner, H. Z. Angew. Math. Mech. 1932, 12, 193–215. (6) Melosh, H. J.; Ivanov, B. A. Annu. Rev. Earth Planet. Sci. 1999, 27, 385–415. (7) Bush, J. W. M.; Hu, D. L. Annu. Rev. Fluid Mech. 2006, 38, 339–369. (8) Korobkin, A. A.; Pukhnachov, V. V. Annu. Rev. Fluid Mech. 1988, 20, 159–185. (9) Worthington, A. M.; Cole, R. S. Philos. Trans. R. Soc. London, A 1897, 189, 137–148. (10) Mallock, A. Philos. Trans. R. Soc. London, A 1918, 95, 138–143. (11) Bell, G. E. Philos. Mag. J. Sci. 1924, 48, 753–765. (12) Birkhoff, G.; Zarantonello, E. H. In Jets, Wakes, and Cavities. In Applied Mathematics and Mechanics; Frenkiel, F. N., Ed.; Academic Press: New York, 1957. (13) Duclaux, V.; Caille, F.; Duez, C.; Ybert, C.; Bocquet, L.; Clanet, C. J. Fluid Mech. 2007, 591, 1–19. (14) Aristoff, J. M.; Bush, J. W. M. J. Fluid Mech. 2009, 619, 45–79. (15) Rayleigh, L. Philos. Mag. 1917, 34, 94–98. (16) Besant, W. H. A Treatise on Hydrostatics and Hydrodynamics; Cambridge University Press: New York, 1859. (17) Oliver, J. M. Water entry and related problems. Ph.D. Thesis, 2002. (18) Duez, C.; Ibert, C.; Clanet, C.; Bocquet, L. Nature Phys. 2007, 3, 180–183. (19) Truscott, T. T.; Techet, A. H. Phys. Fluids 2009, 21, 121703. (20) de Gennes, P.-G.; Brochard-Wyart, F.; Quere, D. Capillarity and Wetting phenomena: Drops, Bubbles, Pearls, Waves; Springer: Berlin, 2004. (21) Landau, L. D.; Levich, B. Acta Physicochim. USSR 1942, 17, 42–54. (22) Derjaguin, B. V. Acta Physicochim. USSR 1943, 20, 349–352. (23) Goff, A. L. Figures d’impact: tunnels, vases, spirales et bambous. Ph.D. Thesis, Universite Pierre et Marie Curie, 2009.
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