Capillary-Driven Flower-Shaped Structures around Bubbles

Langmuir 2003, 19, 5771-5779

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Capillary-Driven Flower-Shaped Structures around Bubbles Collapsing in a Bubble Raft at the Surface of a Liquid of Low Viscosity Ge´rard Liger-Belair* and Philippe Jeandet Laboratoire d’Œnologie, UFR Sciences Exactes, UPRES EA 2069, BP 1039, 51687 Reims, Cedex 2, France Received February 19, 2003. In Final Form: May 16, 2003 By using a classical photo camera and a high-speed video camera, snapshots and time sequences of the dynamics of champagne bubbles collapsing close to each other in a bubble raft composed of quite monodisperse millimetric bubbles were made, thus completing two recent works (Liger-Belair et al. in C. R. Acad. Sci. Paris Se´ rie 4 2001, 2, 775-780 and Liger-Belair in Ann. Phys. Fr. 2002, 27 (4), 1-106). Bubble caps of bubbles adjacent to a collapsing one were found to be strikingly stretched toward the lowest part of the cavity left by the central bursting bubble. Orders of magnitude of shear stresses developed in the adjacent deformed bubble caps were indirectly estimated. Our results strongly suggest, in the thin film of adjacent bubble caps, stresses higher than those observed around a single millimetric collapsing bubble. High-speed time sequences also proved that bubble caps in touch with collapsing bubbles were never found to rupture, thus causing in turn a chain reaction. As in the case of single collapsing cavities, it was also observed that a tiny daughter bubble, approximately 10 times smaller than the initial central bursting bubble, was entrapped during the collapsing process of the central cavity.

1. Introduction As was summarized by Detlef Lohse in a very recent review article,1 bubbles are very common in our everyday life. They occupy an important role in many natural as well as industrial processes (in physics, chemical and mechanical engineering, oceanography, geophysics, technology, and even medicine). Nevertheless, their behavior is often surprising and, in many cases, still not fully understood. In the present research article, we go back on an unexpected observation recently conducted with bubbles collapsing in the bubble raft at the free surface of a glass poured with champagne.2,3 Since the first pioneering photographic investigation published in Nature, precisely 50 years ago,4 numerous experimental, numerical, and theoretical studies have been conducted with single bubbles collapsing at a free surface (see for examples refs 4-22). This phenomenon becomes more and more understood and is now thoroughly * Corresponding author. Telephone: (33) 3 26 91 86 14. Fax: (33) 3 26 91 33 40. E-mail: [email protected]. (1) Lohse, D. Phys. Today 2003, 56 (2), 36-39. (2) Liger-Belair, G.; Robillard, B.; Vignes-Adler, M.; Jeandet, P. C. R. Acad. Sci., Ser. IV 2001, 2, 775-780. (3) Liger-Belair, G. Ann. Phys. (Paris) 2002, 27 (4), 1-106. (4) Woodcock, A.; Kientzler, C.; Arons, A.; Blanchard, D. Nature 1953, 172, 1144-1145. (5) Kientzler, C.; Arons, A.; Blanchard, D.; Woodcock, A. Tellus 1954, 6, 1-7. (6) Blanchard, D. Prog. Oceanogr. 1963, 1, 77-202. (7) MacIntyre, F. J. Geophys. Res. 1972, 77, 5211-5228. (8) Wu, J. Science 1981, 212, 324-326. (9) Blanchard, D.; Syzdek, L. Appl. Environ. Microbiol. 1982, 43, 1001-1005. (10) Resch, F.; Darrozes, J.; Afeti, G. J. Geophys. Res. 1986, 91, 10191029. (11) Blanchard, D. J. Geophys. Res. 1989, 94, 10999-11002. (12) Blanchard, D. Tellus, Ser. B 1990, 42, 200-205. (13) Dekker, H.; de Leeuw, G. J. Geophys. Res. 1993, 98, 1022310232. (14) Spiel, D. Tellus, Ser. B 1994, 46, 325-338. (15) Spiel, D. J. Geophys. Res. 1995, 100, 4995-5006. (16) Spiel, D. J. Geophys. Res. 1997, 102, 5815-5821. (17) Rossodivita, A.; Andreussi, P. J. Geophys. Res. 1999, 104, 3005930066.

modeled.18-22 In very short, the open cavity left after the disintegration of a bubble cap will collapse inward, leading to the upward projection of the often so-called Worthington jet.23 As a result of the well-known Rayleigh-Plateau instability, this liquid jet will break up into a few droplets called jet drops.24,25 It has also been experimentally and numerically observed that a tiny air bubble of about 1/10 of the radius of the parent bursting bubble may be entrapped during the collapsing process. Bubble entrapment, nevertheless, depends on the parent bubble size and on some physicochemical properties of the liquid medium.20,21 But despite the large body of research concerned with collapsing bubble dynamics, by using classical high-speed macrophotography techniques, the close-up observation of bubbles collapsing at the free surface of a glass poured with champagne, nevertheless, recently revealed an unexplored and visually appealing phenomenon. Actually, while trying to get the famous Worthington jets arising when champagne bubbles collapse at the liquid surface, snapshots of bubbles collapsing close to each other were taken by accident. Clusters of bubbles strongly deformed toward a bubble-free central area were observed, leading to unexpected and visually appealing flower-shaped structures.2,3 As surprising as it may seem, no results concerning the collateral effects on adjoining bubbles of bubbles collapsing in a bubble monolayer had been (18) Boulton-Stone, J. M.; Blake, J. R. J. Fluid Mech. 1993, 254, 437-466. (19) Boulton-Stone, J. M. J. Fluid Mech. 1995, 302, 231-257. (20) Duchemin, L. Quelques proble`mes fortement nonline´aires de surface libre et leur re´solution nume´rique. Ph.D. Thesis, Universite´ d’Aix-Marseille II, Marseille, France, 2001. (21) Duchemin, L.; Popinet, S.; Josserand, C.; Zaleski, S. Phys. Fluids 2002, 14, 3000-3008. (22) Georgescu, S.-C.; Achard, J.-L.; Canot, E. Eur. J. Mech., B 2002, 21, 265-280. (23) Worthington, A. M. A Study of Splashes; Longmann & Green: London, 1908. Reprinted: Macmillan, New York, 1963. (24) Plateau, J. Statique expe´ rimentale et the´ orique des liquides soumis aux seules forces mole´ culaires; Gauthier-Villars: Paris, 1873. (25) Lord Rayleigh. Proc. London Math. Soc. 1878, 10, 4-13.

10.1021/la034290j CCC: $25.00 © 2003 American Chemical Society Published on Web 06/12/2003

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Liger-Belair and Jeandet

reported before. Actually, effervescence in a sparkling beverage ideally lends to a preliminary work with bubbles collapsing in a bubble monolayer. For a few seconds after pouring, the free surface of the liquid is completely covered with a monolayer in constant renewal, composed of quite monodisperse millimetric bubbles organized in a hexagonal pattern and collapsing close to each other. Here, in addition to classical macrophotographic snapshots, details of the dynamics of the collateral effects on adjoining bubbles of bubbles collapsing in a bubble monolayer are presented by use of a high-speed video camera. Our results on such deformations experienced by bubbles in touch with collapsing ones are interpreted as consequences of violent capillary-driven fluxes that develop in the thin films of bubble caps adjacent to the empty cavity left by a central bubble cap that has just ruptured. 2. Experimental Section 2.1. Material. A standard commercial champagne wine was poured in a classical crystal flute first rinsed using distilled water and then air-dried. Some physicochemical parameters of champagne were already determined at 20 °C with a sample of champagne first degassed.26 The static surface tension of champagne γ was found to be on the order of 47 mN m-1, its density F was measured and found to be 998 kg m-3, and its dynamic viscosity η was found to be 1.66 × 10-3 kg m-1 s-1. Champagne bubbles appear by nonclassical heterogeneous nucleation, referred to as type IV nucleation in a recent review article by Jones et al.27 Actually, contrary to a generally accepted idea, bubble nucleation sites are not located on irregularities of the glass itself. Most of the nucleation sites are located on tiny hollow and roughly cylinder-like particles (cellulose-fibers-made structures) able to entrap gas pockets during the pouring process.28-30 Gas pockets entrapped inside immerged particles are responsible for the repetitive and clockwork production of bubbles. For a few seconds after pouring, equilibrium establishes between bubbles nucleated on the glass wall and bubbles bursting at the free surface. The free surface of the liquid is covered with a more or less extended monolayer composed of quite millimetric bubbles collapsing close to each others. 2.2. Setup. Recently, to put in evidence the structural analogy between a single bubble collapse and a drop impact, we used a homemade photographic device able to freeze the different stages of a drop impact.31,32 Unfortunately, in the present situation, because the split-second timing and the right place of a bubblebursting event are obviously unpredictable, such a device was of no use. Snapshots and video sequences of the bubble raft in constant renewal were taken at random during the very few seconds after pouring. A Reflex-type photo camera fitted with bellows and with a 50:1.8 objective was used to get classical macrophotographs of collapsing events. To produce enough illumination to enable the use of a small diaphragm aperture, a strong flashlight (synchronized with the diaphragm aperture) was placed a few centimeters above the free surface. To get a better contrast between the liquid and the bubble caps, the outer surface of the flute wall was painted in black. Because the flashlight duration (on the order of 100 µs) of the photo camera is significantly lower than the characteristic time scale of bubble collapse for millimetric bursting bubbles4-22 (on the order of 2 ms), instantaneous snapshots of the collapsing process are expected. The dynamics of (26) Liger-Belair, G.; Marchal, R.; Robillard, B.; Dambrouck, T.; Maujean, A.; Vignes-Adler, M.; Jeandet, P. Langmuir 2000, 16, 18891895. (27) Jones, S. F.; Evans, G. M.; Galvin, K. P. Adv. Colloid Interface Sci. 1999, 80, 27-50. (28) Liger-Belair, G.; Vignes-Adler, M.; Voisin, C.; Robillard, B.; Jeandet, P. Langmuir 2002, 18, 1294-1301. (29) Liger-Belair, G.; Marchal, R.; Jeandet, P. Am. J. Enol. Vitic. 2002, 53, 151-153. (30) Liger-Belair, G. Sci. Am. 2003, 288 (1), 68-73. (31) Liger-Belair, G.; Lemaresquier, H.; Robillard, B.; Duteurtre, B.; Jeandet, P. Am. J. Enol. Vitic. 2001, 52, 88-92. (32) Liger-Belair, G.; Jeandet, P. Europhys. News 2002, 33 (1), 1014.

Figure 1. Scheme of the whole system, which enabled us to get instantaneous snapshots and time sequences of the dynamics of bubbles collapsing in the bubble raft, a few seconds after champagne was poured into the flute. Bubble-nucleation process illustration reprinted with permission from Liger-Belair. Copyright 2002 EDP Sciences. collapsing cavities was also captured, with a high-speed digital video camera (Speedcam+, Vannier Photelec, Antony, France) able to film up to 1500 frames/s and fitted with a microscope objective (Mitutoyo, M Plan Apo 5, Japan). The flute poured with champagne was placed on a cold lighting table. Bursting events in the bubble monolayer in constant renewal were captured from the upside. A scheme of the whole setup is displayed in Figure 1.

3. Results 3.1. Instantaneous Snapshots. As a result of the extreme briefness of a bursting event, very few photographs froze snapshots of the collapsing process in a champagne bubble raft in constant renewal. The photograph displayed in Figure 2, for example, was taken a few seconds after champagne was poured into the flute. Clusters of bubbles (circled with white) can be observed, where bubbles are strongly deformed toward a bubblefree central area, leading to unexpected and visually appealing flower-shaped structures. Figure 3 is an enlarged detail of Figure 2, where two collapsing events appear simultaneously (during the very brief flashlight’s duration). To get an idea of the three-dimensional structure of the phenomenon, we also tried to freeze oblique views of the collapsing process. Two are displayed in Figure 4. Clearly, bubbles adjacent to the bubble-free central area are literally sucked toward the now-empty cavity left by the central bubble that has just ruptured, thus suggesting

Bubbles at a Low-Viscosity Liquid’s Surface

Figure 2. Top view of the bubble raft a few seconds after pouring, frozen by the flashlight of low duration synchronized with the photo camera diaphragm aperture. Clusters of bubbles (circled with white) can be observed where bubbles are strongly deformed toward a bubble-free central area.

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Figure 4. Two oblique views of the flower-shaped structures, frozen in the bubble raft in constant renewal a few seconds after pouring: global view (A); detail (B).

bubble-free area. Then, neighboring bubbles oscillate during a few milliseconds and progressively recover their initial hemispherical shape. It can also be noted that a tiny daughter bubble, approximately 10 times smaller than the initial central bubble, has been entrapped during the collapsing process of the central cavity (as is clearly seen in frames 5 and 6 of Figure 5). Bubble entrapment during the collapsing process was already experimentally and numerically observed with single millimetric collapsing bubbles,20,21,33,34 including champagne bubbles.31 This bubble entrapment process can also be observed as drops impact on liquid surfaces.35-39 4. Discussion

Figure 3. Enlarged detail of the double flower-shaped structure that appears in Figure 2. In the center of the picture, two bubbles are simultaneously deformed toward the two bubble-free central areas.

violent stresses in the thin films of neighboring deformed bubble caps. Such a behavior first appeared counterintuitive to us. Paradoxically, adjacent bubble caps are sucked and not blown up by bursting bubbles, contrary to what could have been expected at first glance. 3.2. High-Speed Time Sequences. A few time sequences of the whole process have also been captured with the high-speed video camera. One is presented in Figure 5. Between frame 1 and frame 2, the bubble pointed at with the black arrow has disappeared. In frame 2, neighboring bubbles are literally sucked toward this now

4.1. Bubbles’ Shapes at the Liquid Surface Before Collapse. Champagne bubbles, which traveled approximately 10 cm between their nucleation sites and the liquid surface, reach the free surface with a radius R on the order of 0.5 mm (around 100 times larger than their initial radius when released from the nucleation site). It should be noted that the decrease of hydrostatic pressure between a nucleation site in the bottom of the flute and the free surface is completely negligible (around 1% of the atmospheric pressure P0). It causes the bubble volume to grow only about 1% and, therefore, plays no role in the bubble growth, as was observed during ascent,3,26,40 contrary to what was written in a very popular book on the physics of everyday phenomena.41 Actually, bubble growth during ascent is caused by a continuous diffusion of dissolved carbon dioxide molecules through the bubble’s gas/liquid interface.28 (33) Longuet-Higgins, M. S. J. Fluid Mech. 1983, 127, 103-121. (34) Zeff, B. W.; Kleber, B.; Fineberg, J.; Lathrop, D. P. Nature 2000, 403, 401-404. (35) Oguz, H. N.; Prosperetti, A. J. Fluid Mech. 1990, 203, 143-179. (36) Longuet-Higgins, M. S. J. Fluid Mech. 1990, 214, 395-410. (37) Prosperetti, A.; Oguz, H. N. Annu. Rev. Fluid Mech. 1993, 25, 577-602. (38) Rein, M. J. Fluid Mech. 1996, 306, 145-165. (39) Morton, D.; Rudman, M.; Jong-Leng, L. Phys. Fluids 2000, 12, 747-763. (40) Liger-Belair, G.; Jeandet, P. Langmuir 2003, 19, 801-808. (41) Berkes, I. La Physique de Tous les Jours; Vuibert: Paris, 2001.

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Figure 5. Time sequence illustrating the dynamics of adjoining bubbles in touch with a collapsing one. Because the whole process was filmed at 1500 Hz, the time interval between each frame is around 667 µs. From frame 4, in the center of the empty cavity left by the collapsing bubble, a tiny-air-bubble entrapment is observed.

At the free surface, the shape of a bubble results from a balance between two opposing effects: the buoyancy FB, on the order of FgπR3, which tends to make it emerge from the free surface, and the capillary force FC inside the hemispherical thin liquid film, on the order of (γ/R)πR2 ) γπR, which tends to maintain the bubble below the surface. γ/R is the excess of pressure, due to the Laplace law, inside the curved thin liquid film of the bubble cap, and πR2 is the order of magnitude of the emerged bubble cap’s area. Comparing these two opposing forces comes down to comparing the bubble radius, R ≈ 0.5 mm, with the capillary length of the liquid medium, κ -1 ) (γ/Fg)1/2 ≈ 2 mm (where γ, F, and g are respectively the liquid surface tension, the liquid density, and the acceleration due to gravity). With the champagne bubbles’ radii being significantly smaller than the capillary length, gravity will be neglected in front of capillary effects. Consequently, like a tiny iceberg, a bubble in the monolayer only slightly emerges from the liquid surface, with most of its volume remaining below the free surface. Let us denote R1 and R2, the radii of curvature of the emerged bubble cap and the immerged bubble volume, respectively. Therefore, because the thin emerging bubble cap possesses two interfaces whereas the immerged part of the bubble possesses only one, the excess of pressure

inside a bubble, ∆P, may be written as follows (according to the Laplace law):

∆P ) 4γ/R1 ) 2γ/R2 w R1 ) 2R2

(1)

Consequently, the radius of curvature of the emerged bubble cap is approximately twice that of the submerged bubble volume. Furthermore, because most of the bubble volume remains below the liquid surface, R2 ≈ R and, consequently, R1 ≈ 2R. Obviously, the transition between these two zones of different curvature does not appear so abruptly. A complete numerical study of the shape taken by a bubble trapped at a gas/liquid interface was recently conducted by Duchemin et al., in the limit of small and large bubble radii.20 The radius of the emerged bubble cap, r, as schematized in Figure 6, can also be estimated by equaling the bubble buoyancy with the capillary pressure inside the thin film as

Fg(4/3)πR3 ≈ (γ/R)πr2 w r ≈ (x4/3)R2/κ-1 ≈ 0.14 mm (2) Finally, the order of magnitude of the angle between the z axis and the end of the bubble cap may also be easily determined as ψ ≈ sin-1(r/2R) ≈ 8°. Figure 6 presents and


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Figure 6. Oblique view of the bubble raft composed of quite millimetric bubbles organized in a hexagonal pattern, each bubble being surrounded by an arrangement of six neighbors (A); schematic transversal representation of five aligned bubbles in touch in the monolayer (B).

presumes the different geometrical parameters associated with bubbles trapped near the free surface in a bubble raft composed of quite millimetric bubbles. The above reasoning does not take into account interactions between adjoining bubbles (no dimple effects, for example). 4.2. Dynamics of a Bubble Cap’s Aperture. At the free surface, because the bubbles’ radii are significantly smaller than the capillary length, the liquid films of bubble caps progressively get thinner as a result of capillary drainage. When the liquid film of a bubble cap reaches a critical thickness, it becomes fragile and finally ruptures. Since the probable pioneering work of Lord Rayleigh in the late 19th century,42 the rupture of thin films has been widely experimentally, theoretically, and numerically investigated. It was found that a hole appears in the film, surrounded by a rim that collects the liquid and propagates very quickly, driven by surface tension forces. Balancing inertia with surface tension, Culick proposed the following velocity for the growing hole in a thin rupturing liquid sheet:43

uCulick ≈ x2γ/Fe

Figure 7. In a rupturing spherical bubble cap, schematic representation of the growing hole surrounded by a rim that collects the liquid and propagates, driven by surface tension forces.

In the case of a spherical cap with a radius of curvature 2R, rupturing from its axis of symmetry (the z axis) as schematized in Figure 7, the velocity u of the propagating liquid rim (of mass M) is ruled by the following equation:

(3)

where e is the thickness of the liquid film. The latter expression has already been experimentally confirmed numerous times for the bursting of low-viscosity thin liquid films (see for examples the pioneering work by McEntee and Mysels44 and also that by Pandit and Davidson,45 where beautiful snapshots of the disintegration of soap bubbles are presented). (42) Lord Rayleigh. Proc. R. Inst. G. B. 1891, 13, 261. Reproduced in Scientific Papers; Dover: New York, 1964; Vol. 3. (43) Culick, F. E. J. Appl. Phys. 1960, 31, 1128. (44) McEntee, W. R.; Mysels, K. J. J. Phys. Chem. 1969, 73, 30183028. (45) Pandit, A. B.; Davidson, J. F. J. Fluid Mech. 1990, 212, 11-21.

d Mu ) 2γ2R sin θ dφ dt

(4)

By replacing in the latter equation the mass of the liquid rim by its expression, M ) Fe(2R)2(1 - cos θ) dφ, and the velocity u by 2R(dθ/dt) and by developing, one obtains the following differential equation:

[

Fe(2R)2

( )]

d2θ dθ 2 (1 - cos θ) + sin θ ) 2γ sin θ (5) 2 dt dt

Finally, by considering only the constant velocity solution (d2θ/dt2 ) 0), one obtains the following expression u for the growing hole:

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dθ u ) 2R ) dt

x2γFe ) u

Culick


(6)

which is the same as that derived by Culick. The thickness e of a millimetric champagne bubble cap was already experimentally determined by the classical micro-interferometric technique46 and found to be on the order of 10-6-10-7 m. As a result, by replacing in eq 6 the known values of γ, F (≈103 kg m-3), and e, u is expected to be on the order of 10 m s-1 for a rupturing champagne liquid film. Finally, the characteristic time scale τ of a millimetric bubble cap’s disintegration should be around

τ ≈ r/u ≈ 1.4 × 10-4/10 ≈ 10 µs

(7)

4.3. Estimation of the Shear Stresses During a Bubble Cap’s Aperture. To estimate the shear stress during the bubble cap’s disintegration, we will use an alternative approach, inspired from Clarkson et al.,47 based on the energy dissipation rate. The energy dissipation rate is the rate at which energy is dissipated per unit mass of liquid:

)

1 dED m dt

(8)

where m is the mass of the liquid film (m ≈ Feπr2). Half of the total surface free energy ET of the bubble cap (where ET ) 2γπr2, due to the gas/liquid interfaces) is dissipated by viscous effects into the liquid rim around the growing hole, the other half being converted into kinetic energy.43-45 Therefore, ED ≈ ET/2. As a result, during the bubble cap’s disintegration phase, the energy dissipation rate can be approached as

≈

γ 1 γπr2 ≈ 2 τ eFτ Feπr

(9)

As was presented in ref 47, we retrieved the turbulence theory to estimate the shear stress σ in the liquid rim from the energy dissipation rate by use of the following relationship:48

σ ≈ η(/ν)1/2

(10)

where ν is the kinematic viscosity, defined as ν ) η/F. Finally, combining eqs 9 and 10 yields to the following expression for the shear stress encountered as the emerging bubble cap disintegrates:

σ ≈ (γη/eτ)1/2

(11)

By replacing in eq 11 the known values of γ, η, e, and τ in the present situation, shear stresses in the propagating liquid rim are expected to be bounded by between 3 × 103 N m-2 and 9 × 103 N m-2, depending on the film thickness. 4.4. Situation After a Bubble Cap’s Aperture. During the very brief initial phase of a bubble cap’s aperture driven by surface tension forces, which happens on a time scale around 10 µs, the submerged part of the bubble remains frozen in the interface. (46) Sene´e, J.; Robillard, B.; Vignes-Adler, M. Food Hydrocolloids 1999, 13, 15-26. (47) Clarkson, J. R.; Cui, Z. F.; Darton, R. C. J. Colloid Interface Sci. 1999, 215, 323-332. (48) Schlichting, H. Boundary Layer Theory; McGraw-Hill: New York, 1979.

Figure 8. Schematic transversal representation of the situation, frozen after the disintegration of the central bubble cap.

In the case of single collapsing champagne bubbles, immediately after the disintegration of the bubble cap, the nearly millimetric bulk shape has been captured by classical macrophotography and with the use of a very brief flashlight.31 It is, nevertheless, much more difficult to visualize it for bubbles collapsing in a bubble raft as a result of the neighboring bubbles. A transversal scheme of the situation, frozen immediately after the rupture of a central bubble-cap film, is displayed in Figure 8. It was, nevertheless, assumed in Figure 8 that the liquid rim of the central rupturing bubble cap implied negligible effects when impacting neighboring bubbles, which is not necessarily the case. After the disintegration of a bubble cap, the hexagonal symmetry around adjoining bubbles and, therefore, also the symmetry in the field of capillary pressure around them are suddenly broken. Capillary pressure gradients all around the now-empty cavity are detailed in Figure 8. The +/- signs indicate a pressure above/below the atmospheric pressure P0. Finally, inertia and gravity being neglected, the full Navier-Stokes equation applied to the fluid in the thin liquid film drawn by capillary pressure gradients reduces itself to a simple balance between the capillary pressure gradients and the viscous dissipation as follows:

η(∆v b)s ) (∇P B)s

(12)

where v is the velocity in the thin liquid film of adjacent bubble caps, s being the axial coordinate that follows the bubble cap’s curvature and along which the fluid inside the thin film is displaced. The asymmetry in the capillary pressure gradient distribution around a bubble cap adjacent to an empty cavity is supposed to be the main driving force of the violent sucking process experienced by a bubble cap in touch with a bursting bubble. Actually, as a result of higher capillary pressure gradients, the liquid flows that develop in the half of the bubble cap close to the open cavity are, thus, expected to be higher than those that develop in the rest of the bubble cap. It ensues a violent stretching of adjoining bubble caps toward the now-empty cavity, which is clearly visible in Figures 3 and 5. 4.5. Estimation of Stresses in the Thin Film of Adjacent Bubble Caps During the Stretching Process. During this sudden stretching process, adjacent


bubble cap’s areas significantly increase. A systematic image analysis of numerous time sequences similar to that displayed in Figure 5 demonstrated an average increase ∆A of around 15% of a bubble cap’s area adjacent to a central collapsing bubble. The free energy of such a system is supposed to be mainly stored as surface free energy. Consequently, the density of free energy per unit of volume in the liquid film of a distorted bubble cap (one petal of the flower-shaped structure) can be evaluated as follows:

(∆EV)

bubblecap

≈

2γ∆A 0.15 × 2γ ≈ ≈ 104-105 J m-3 ) Ae e 104-105 N m-2 (13)

where ∆E is the corresponding surface free energy in excess during the stretching process, V is the volume of the thin liquid film of the emerging bubble cap, A is the emerging bubble-cap area, and e is the thickness of the thin liquid film of the bubble cap (on the order of 10-610-7 m). Therefore, the liquid flows induced by the capillary pressure gradients are responsible for a density of energy per unit of volume dimensionally equivalent to shear stresses on the order of 104-105 N m-2 (depending on the film thickness). By comparison, in previous studies, numerical models led to stresses on the order of (only) 103 N m-2 in the boundary layer around single millimetric collapsing bubbles.18,20 Therefore, stresses in the bubble caps of bubbles adjacent to collapsing cavities appear to be at least 1 order of magnitude higher than those observed around single collapsing cavities and also even higher than those developed by viscous dissipation as a bubble cap disintegrates (see eq 11). Intuitively, this is finally not so surprising. Actually, after a bubble cap’s aperture, the now-empty cavity has to collapse to recover the horizontality of the liquid surface. The potential energy of such an unstable situation is the sum of two contributions: first, the gravitational energy associated with the buoyancy of the open cavity (on the order of FgπR4) and second, the difference in the surface free energy between a hemispherical open cavity of radius R and a flat circular surface of radius R (on the order of γπR2). The ratio of the gravitational energy to the surface free energy is called the Bond number, Bo ) FgR2/γ. In the present situation, Bo being on the order of 5 × 10-2, the main source of energy is undoubtedly the surface free energy. Now, it should be noted that the same driving potential energy is responsible for the single-cavity collapse and for the collapse of the cavity surrounded by neighboring bubble caps. But, because the volume of liquid displaced in the thin film of adjoining bubble caps is much less than that of the boundary layer drawn around a single collapsing cavity, it logically induces higher energy dissipation rates per unit of volume and, therefore, higher strains in the “petals” of the flower-shaped structure. Finally, while absorbing the energy released during a bubble collapse, as so many tiny air bags would do, adjoining bubble caps store this energy in the thin liquid film of emerging bubble caps, leading finally to stresses much higher than those observed in the boundary layer around single millimetric collapsing bubbles. In addition to purely physicochemical reasons, biological reasons are also readily found for the investigation of such flower-shaped structures around bubbles collapsing in a bubble raft. Actually, in the biological industry, animal cells cultivated in bioreactors were shown to be seriously damaged or even killed by the bursting of gas bubbles

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used to aerate the culture medium.49,50 It has even been suggested that structural deformations of adjacent tissues are induced by bubble collapse during laser-induced angioplasty.51,52 Kunas and Papoutsakis evaluated the critical shear stresses needed to cause irreversible damages to classical animal cells.50 They found critical lethal stresses around 103-104 N m-2. Therefore, bubbles bursting in a bubble raft should be potentially more dangerous for microorganisms or biological tissues trapped in the thin film of these “fast-stretched bubble caps”. It should also be mentioned that, despite this violent sucking process, bubbles adjacent to bursting ones were never found to rupture and collapse in turn, thus causing a chain reaction. This observation differs from the results of experiments recently conducted on the collapsing process of aqueous foams. Actually, avalanches of popping bubbles were put evidenced during the coarsening of twoand three-dimensional aqueous foams,53-55 whereas in the present study, bursting events appear to be spatially and temporally noncorrelated. 4.6. And What About the Worthington Jet? Because we are dealing with bubbles collapsing at a free liquid/gas interface, we are logically tempted to wonder about the dynamics of the famous Worthington jet. Does it exist, as in the single-collapsing-bubble case, or does the roughly hexagonal neighboring bubble pattern around a collapsing cavity strongly modify and even prevent its formation? The length scale l of the upward jet radius is around 100 µm. Therefore, the corresponding Bond number is expected to be around Bo ) Fgl2/γ ≈ 2 × 10-3 , 1. The upward jet velocity U being on the order of several meters per second,11-22 the Reynolds number of the jet is on the order of Re ) FUl/η ≈ 102 . 1. Consequently, the RayleighPlateau instability, which develops in the jet, results from a balance between the inertia (F dU/dt ≈ FU/τR) and the surface tension gradients (∇P ≈ γ/l2), τR being the characteristic time scale for the instability to develop. Finally, by equaling the two latter expressions, the characteristic time scale τR for the Rayleigh-Plateau instability to develop in the jet is on the order of

τR ≈ FUl2/γ

(14)

As can be seen in eq 14, changes in the upward jet velocity U and in its radius l, in comparison with the singlebubble-collapse case, could finally strongly affect the dynamics of the Rayleigh-Plateau instability, which breaks the jet into droplets. Once more, intuitively, differences between the dynamics of the Worthington jet could be expected. Actually, the bulk shapes of bubbles adjacent to the empty cavity strongly change the geometry of the system beneath the free surface and, therefore, undoubtedly modify the converging liquid flows all around the cavity. It thus probably modifies, in turn, the dynamics of the upward liquid jet. Actually, a few snapshots froze the formation of a Worthington jet for collapsing bubbles being surrounded by a hexagonal pattern of six neighboring bubbles. (49) Handa, A.; Emery, A. N.; Spier, R. E. Dev. Biol. Stand. 1987, 66, 241-253. (50) Kunas, K. T.; Papoutsakis, E. T. Biotechnol. Bioeng. 1990, 36, 476-483. (51) Van Leeuwen, T. G.; Meertens, J. H.; Velema, E.; Post, M. J.; Borst, C. Circulation 1993, 87, 1258-1263. (52) Brujan, E. A. Europhys. Lett. 2000, 50, 175-181. (53) Mu¨ller, W.; di Meglio, J.-M. J. Phys.: Condens. Matter 1999, 11, L209-L215. (54) Vandewalle, N.; Lentz, J. F.; Dorbolo, S.; Brisbois, F. Phys. Rev. Lett. 2001, 86, 179-182. (55) Vandewalle, N.; Lentz, J. F. Phys. Rev. E 2001, 64, 021507.

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developed in the boundary layer around a single millimetric collapsing bubble. It seems self-consistent with the idea that, in the bubble raft, the same driving potential energy draws a volume of liquid much less important than that of the boundary layer drawn around a single collapsing cavity. But, because it is clear that we are still a long way from the full understanding of the collapse of a bubble surrounded by neighboring bubbles in touch, further experimental investigations with this visually appealing process should be conducted now. It is worth noting that the influence of surface-active molecules in this process was not taken into account. Experiments with different surfactant solutions under well-controlled and calibrated bubbling rates are planned. A model that includes the role played by surfactants adsorbed at the bubbles’ interfaces on the stretching process is also under construction. Moreover, as a result of the complex hydrodynamical processes involved, for both the deformation of adjoining bubble caps and the formation of the Worthington jet, numerical studies are also expected. Finally, we would like to end this paper by paying homage to Dr. Harold Edgerton, the 20th century master of stop-action photography. He revealed to the general public the beauty of “high-speed events” and inspired generations of young scientists. Edgerton (1903-1990) devoted his entire career to recording what the unaided eye cannot see to reveal the laws of nature.56,57 “The experience of seeing the unseen has provided me with insights and questions my entire life,” said Harold Edgerton. His most famous snapshot, the coronet made by an impacting drop of milk, has become an icon at the crossroads between pure science and modern art.

Figure 9. Comparison, as it divides into jet drops due to the Rayleigh-Plateau instability, between the Worthington jet that follows the collapse of a bubble in the bubble raft (A) and that which follows a single bubble collapse (B; reprinted from the American Journal of Enology and Viticulture. Copyright 2001 by the American Society for Enology and Viticulture.).31

Figure 9 compares a liquid jet issued from a bubble collapsing in a bubble raft (Figure 9a) with a liquid jet issued from the collapse of a single champagne bubble31 (Figure 9b). The jet issued from the single collapsing bubble seems to be better “developed” than that issued from the bubble collapsing in the monolayer. The same tendency has been noticed on the few snapshots that froze the Worthington jet in a bubble raft. Nevertheless, too few snapshots froze the Worthington jet in the bubble raft. Further experimental investigations are to be conducted to enable us to conclude about the similitude and differences of the dynamics of the jet following a single bubble collapse.

Acknowledgment. Thanks are due to the Europol’Agro Institute, the Association Recherche Oenologie Champagne Universite´, Champagne Moe¨t & Chandon, Pommery, and Verrerie Cristallerie d’Arques for supporting our research and to Miche`le Vignes-Adler for valuable discussions. Nomenclature A Bo e ET ED g l m M

5. Conclusions and Prospects Through the present article, we would like to bring our contribution to the field of high-speed events that have long been unwatched by the human eye. Close-ups of the dynamics of millimetric bubbles neighboring a collapsing one were done. Our results suggest shear stresses in the bubble caps of adjoining bubbles higher than those (56) Edgerton, H.; Killian, J. R. Flash!: Seeing the Unseen by Ultrahigh-Speed Photography; Cushman and Flint: Boston, Hale, 1939. (57) Jussim, E. Stopping Time: The Photographs of Harold Edgerton; H. N. Abrams, Inc.: New York, 1987.

P P0 r R R1 R2 Re t

bubble cap’s area, m2 Bond number, Bo ) FgR2/γ thickness of the liquid film in the bubble cap, m surface free energy of the emerging bubble cap, J energy dissipated by viscous effects in the liquid rim that surrounds the growing hole of a rupturing bubble cap, J gravity acceleration, ≈9.8 m s-2 length scale of the upward liquid jet radius, ≈100 µm mass of the thin liquid film that constitutes the emerging bubble cap, kg mass of the liquid rim that surrounds the growing hole of a rupturing bubble cap, kg pressure, N m-2 atmospheric pressure, ≈105 N m-2 radius of the emerged bubble cap (see Figure 6), m bubble radius, m radius of curvature of the emerged bubble cap, m radius of curvature of the immerged bubble volume, m Reynolds number, 2FRU/η time, s

Bubbles at a Low-Viscosity Liquid’s Surface u U v V γ η ν F

velocity of aperture of a rupturing bubble cap driven by surface tension forces, m s-1 velocity of the upward Worthington jet, m s-1 velocity in the thin liquid film of the emerging bubble cap, m s-1 volume of the bubble cap, m3 energy dissipation rate, J kg-1 s-1 liquid surface tension, ≈47 mN m-1 liquid dynamic viscosity, ≈1.66 × 10-3 kg m-1 s-1 liquid kinematic viscosity, η/F, m2 s-1 liquid density, ≈103 kg m-3

Langmuir, Vol. 19, No. 14, 2003 5779 τ τR σ κ-1 ψ

characteristic time scale required for the disintegration of the emerging bubble cap, ≈10 µs characteristic time scale required for the Rayleigh-Plateau instability to develop in the upward Worthington jet shear stresses, N m-2 capillary length of the liquid medium, ≈2 mm angle between the hemispherical bubble cap and the z axis, see Figure 6 LA034290J

Capillary-Driven Flower-Shaped Structures around Bubbles

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