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Air bubble bursting phenomenon at the air-water interface monitored by the piezoelectric-acoustic method
Advances in Colloid and Interface Science 272 (2019) 101998
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Air bubble bursting phenomenon at the air-water interface monitored by the piezoelectric-acoustic method Alex Nikolov ⁎, Darsh Wasan Department of Chemical Engineering, Illinois Institute of Technology, Chicago, IL 60616, United States of America
a r t i c l e
i n f o
Article history: 30 July 2019 Available online 16 August 2019 Keywords: Aqueous lamella rupture Air bubble bursting frequency Air/water interface oscillation Prediction
a b s t r a c t When an air bubble arrives at the free interface, the bubble's lamella drains and ruptures. The bubble collapses, and gas vapor is released. The ruptured lamella retreats, and a rim at the edge of the retreating lamella forms. The rim becomes unstable and breaks into fine droplets, leading to the formation of a mist. As the collapsing bubble gas's vapor is released, the collapsing bubble oscillates and a vertical liquid jet erupts; this jet then breaks into a droplet(s). Here, we present a novel approach for monitoring the air bubble bursting frequency at the air-water interface by the piezoelectric-pressure-acoustic technique. The piezoelectric-acoustic technique monitors the lamella's rupture time, the frequency of the oscillation of the collapsing air bubble, and the frequency of the oscillation at the free air/water interface. The aqueous lamella rupture thickness was probed by reflected light interferometry, and the air bubble burst at the air/water interface was monitored with the high-speed photo imaging technique. The data obtained by the three techniques provided essential information for the stages of the air bubble collapse dynamics at the free interface without the presence of a surfactant. The simple model proposed by Rayleigh, Minnaert, and Lighthill (RML) for the oscillation resonance of a single air bubble was applied to calculate the air bubble collapsing frequency. The floating air bubble bursting frequency with an equatorial radius of 0.33 ± 0.05 cm was well predicted using the air bubble resonance frequency model, and was estimated as 1.0 ± 0.3 kHz. The velocity of the ruptured aqueous lamella covering the air bubble was estimated as 1 m/s. This research presents a comprehensive understanding of the phenomenon of the bare air bubble collapse at the free interface. © 2019 Published by Elsevier B.V.
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Monitoring an air bubble film rupture with light reflection interferometry . 2.2. Foam lamella bursting monitored by high-speed photo imaging . . . . . 2.3. Bubble bursting monitored by the piezo-electric pressure transducer . . . 3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Author information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction The presented research is dedicated to the classical work performed by Prof. D. Exerowa and Prof. D. Platikanov on the thinning and stability ⁎ Corresponding author. E-mail address: [email protected] (A. Nikolov).
https://doi.org/10.1016/j.cis.2019.101998 0001-8686/© 2019 Published by Elsevier B.V.
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of common and Newtonian black microscopic free foam films [1], and on the permeability of amphiphile bilayers [2]. They explored the pioneering work of Prof. Scheludko [3] on microscopic free foam thin liquid films and applied it to novel applications. One of us had the opportunity to work with Prof. Exerowa on common and Newtonian black film thickness transitions [4].
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Bubble bursting at the free interface is a common phenomenon and can be observed daily. Bubble bursts are pleasant to observe. When champagne is poured into a glass, the bubbles burst and a fountain of shining, sparkling droplets “dance” over the wine/air interface [5]. The air bubble burst is a phenomenon that needs a better understanding to explain a wide range of observations [6], such as those found in photoluminescence, sonochemistry, cavitation [7], marine life studies [8–10], bio-cell reactors [11,12], biofouling, erosion, bubble jetprinting, enhanced particle spreading [13], and microfluidics. However, monitoring and explaining the bubble bursting dynamics are not easy tasks [14–18]. When an air bubble arrives at the free interface, the bubble's lamella drains and ruptures. The bubble collapses, gas vapor is released, the aqueous lamella retreats, and the rim at the edge of the retreating lamella forms. The rim becomes unstable and breaks into fine droplets, causing a mist to appear. During the time the collapsing bubble's gas vapor is being released, the bottom of the air bubble oscillates and a vertical liquid jet erupts. (Dupre [19], Rayleigh [20], Rayleigh [21], Ranz [22], Taylor [23], Culick [24], Lieberman [25], McEntee and Mysels [26], Pandit and Davison [27], Boulton-Stone and J. Blake [28], Debregeas et al. [29], Muller et al. [30], Lhuissier et al. [31], Bird et al. [32], Lauterborn and Kruz [33], Lhuissier and Villermaux [34], Sabadini et al. [35], Champougny et al. [36], Tammaro et al. [37], Philips et al. [38], and Murano and Okumura [39]). As was pointed out earlier, the air bubble bursting phenomenon is triggered by the rupture of the aqueous lamella (thick film) and its retreat. Here, we discuss the research conducted on the film/lamella stability, the rate of the lamella retreat, and then the air bubble bursting phenomenon in order to propose a model for calculating the frequency of the oscillation of the bursting bubble at the free air/water interface. Dupre and Rayleigh [19,20] were among the first to use photography to monitor the surfactant-stabilized foam lamella rupture triggered by perforation (the punch method). They reported that when a large (e.g., circular) flat foam film (lamella) ruptures by local perforation, the hole rapidly expands under the film tension action. The fluid from the ruptured film builds up a liquid ridge (rim) along the retracted film edge. As the ridge grows in volume, the ridge becomes unstable and breaks up, and a necklace of flying droplets is generated. Dupre and Rayleigh [19,20] calculated the retraction velocity of the film's edge (V) by applying a simple kinetic energy conservation approach. They assumed that the kinetic energy of the foam lamella was equal to the film surface energy that vanished and did not consider the role of the dissipative energies, the internal viscosity, and film ridge instability. For the maximum of the retreating film edge velocity (V), they proposed the following equation: V edge
sffiffiffiffiffiffiffiffi 2 4γ ¼ ρh f
ð1Þ
velocity where γ is the surface tension, hf is the film rupture thickness, and ρ is the fluid density. The unknown parameter is the film rupture thickness. Dupre [19] also calculated the foam film rupture velocity (32 m/s). Ranz [22] investigated the rupture dynamics of an aqueous circular (11 cm) micron-thick flat foam viscous film stabilized with a surfactant and glycerol using a high-speed camera (2720 frames/s) and reported a film edge rupture velocity of 700 cm/s. Ranz [22], like Dupre and Rayleigh [19,20], reported that the ridge film velocity was constant during the ridge film retreatment and 10% lower than predicted by Eq. (1). Ranz [22] applied Dupre and Rayleigh's equation to calculate the film edge retreat velocity of 32 m/s and assumed an equilibrium value of 26 mN/m for the surface tension. It was unrealistic to use the equilibrium value of the surface tension to calculate the retreating film velocity, because for a retreating film with a rupture velocity of 700 cm/s, one has to apply the value of the dynamic surface tension. Ranz reported the formation of bulges along the ridge and noted that “droplets have been torn from the buglers”. He revisited Dupre and Rayleigh's approach to calculate
the film ridge velocity when taking into consideration the role of the ridge and elastic waves; he proposed a smaller value for the film edge velocity:
V edge
sffiffiffiffiffiffiffiffi 2 2γ ¼ ρh f
ð2Þ
velocity The roles of the ridge and the elastic waves were correctly understood as factors that impacted the film edge velocity. Culick [24] revisited the Rayleigh [20] and Ranz [24] analyses on the edge film velocity and considered the energy dissipation due to the liquid rolling at the film's edge. He concluded that Rayleigh's equation overestimated the edge film velocity and confirmed the validation of Eq. (2). However, the actual film rupture thickness of a punched film is not known. The critical thickness of the rupture of an aqueous film stabilized by a surfactant in the presence of glycerol requires an understanding of what triggers the film instability mechanisms. The film instability model (e.g., the theory of the fluctuation of the capillary or perturbation waves on film surfaces triggered by the attractive van der Waals forces) requires a film thickness less than 10 nm. Vrij and Overbeek [40] were the first to validate the model prediction of the film critical thickness. When the film instability is forced by the punch method, an evaluation of the local film rupture thickness is required, and the application of the van der Waals forces film attraction concept to the film stability is invalid. Pandit and Davidson [27] conducted an experimental and theoretical study on the edge film velocity of a ruptured aqueous spherical soap foam film stabilized by a surfactant solution. The foam film rupture was promoted by the punch method. The authors used a video camera observation of 2000 frames/s, and the ridge film velocity was estimated to be 10 m/s. The film mean thickness prior to the rupture was estimated by the electrical conductivity method, and it was estimated to be of the order of 0.3 to 0.9 μm. The authors reported that when the foam film was forced to rupture and the hole was formed, the film liquid retreated at the film edge and a liquid ridge (rim) with a cylindrical shape formed. The authors reported that due to the varicose instability, the ridge (rim) broke up into jets from which droplets of 10−2 cm in size subsequently formed; these droplets then shattered. However, these researchers did not elucidate the mechanism or discuss the relationship between the rim geometry, jet, and droplet size. Bird et al. [32] observed a bubble-bursting cascade phenomenon. An air bubble was created on a pool of river water. The authors used river water because the indigenous surfactants in river water tend to be present in estuaries and oceans. The bubble collapse was monitored with two synchronized high-speed cameras. The bubble was formed both on the ridged surface and on a deep pool of water. When the bubble formed on the ridged flat surface collapsed due to the foam film rupture triggered by the hole nucleation following the film rupture, the film retreated and a ring of small bubbles, each approximately a millimeter in diameter, appeared around the base of the initial bubble. When one of these daughter bubbles ruptured, a ring of even smaller bubbles formed. The foam film of the smaller bubble ruptured and the bubble collapsed; tiny liquid jets formed. When these jets were ejected into the air, they disintegrated into tiny mist droplets. The authors conducted numerical simulations to elucidate the mechanisms of the foam film folding and bubble air entrapment. To perform the numerical simulation, the authors assumed an initially uniform foam thickness for the surfactant-stabilized foam film. Despite the reported observations of the bubble collapsing via a bubble bursting cascade, it is not clear why the authors performed their experiments using different surfactant formulations. The use of different fluid viscosity values for the bubble bursting dynamics is understandable, since the experiments were conducted to slow the film rupture dynamics. They used the Laplace equation and assumed the value of the surface tension was at equilibrium for their model of the air bubble pressure. The authors used the equilibrium surface tension for the bubble bursting
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dynamics; however, they needed to use the dynamic surface tension, or at least, discuss its role. The authors' proposed numerical simulation did not explain the physics of the bubble bursting cascade and the nature of the oscillation at the free surface during and after the bubble collapse. Rayleigh [21] studied the pressure development vs. time in a liquid during the collapse of a spherical cavity. Rayleigh [21], and later Plesset [41], proposed an equation that was a simple approach to explain the various aspects of the cavity. Their approach was based on a second– order differential equation to elucidate the time evolution of bubble radius r(t). The Rayleigh-Plesset (RP) equation is expressed as r 3α 2γ r_ 3 0 ρ r €r þ r_ 2 ¼ ½pν −p∞ ðt Þ þ pg0 − −4μ l r 4 r r
3α 3 2 r0 1 1 r_ ρ r €r þ r_ −1 pg0 þ 2γ − −4μ ð4Þ ¼ −δp sinωt þ 4 r0 r r r where δp is (r-ro) and ω is the period of oscillation; ω = 2πf, and f is the frequency of oscillation. The simplified equation for the forced collapse of a millimeter-sized bubble's harmonic frequency of oscillation (f) when neglecting the role of the surface forces (capillary) and viscous forces is 1 2πr
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3α P go ρ
(5) based on a thermodynamic approach. They treated the bubble as an adiabatic harmonic oscillator and calculated the bubble's kinetic energy. The role of the heat capacity ratio was considered, and the effects of the surface tension and viscosity were neglected. A simplified version of Eq. (5) was applied to predict the submerged gas bubble frequency of oscillation. For a bubble in water, the oscillation frequency at one atmosphere is
f ≈
3 ms−1 r
ð6Þ
ð3Þ
where r_ and €r are the first and second order derivatives of the bubble radius vs. time, ρ is the gas density, μl is the kinematic viscosity, ro is the initial bubble radius, pν is the gas partial pressure, and pg0 is the initial condition of the gas inside the bubble. pν is the vapor pressure, which depends upon the temperature. p∞ is the atmospheric pressure. α is the ratio of the heat capacity of the enclosed gas. During cavitation, the bubble is subject to oscillations at its natural frequency. The term on the left side of the equation represents the variation in the kinetic energy of the liquid body multiplied by the gas density ρ. The first term on the right side represents the pressure forces acting on the liquid, and it is the driving term. The second term considers the ratio of the heat capacity α of the gas inside the bubble, and it is the contribution of the noncondensable gas. Its derivation is based on the assumption that the mass of the non-condensable gas inside the bubble remains constant. The third term is the contribution of the surface tension forces (capillary), and it is important only for small, micron-sized bubbles. The fourth term is the dissipation rate due to the fluid viscosity. More information about the derivation and the physics of Eq. (3) can be found in the literature of Franc and Michel [42]. The air bubble in a liquid is a possible oscillator because of the elastic behavior of the non-condensable gas that the bubble contains and the inertia of the liquid [43]. The oscillation behavior of a bubble and its resonance frequency can be computed from the Rayleigh-Plesset equation (Eq. (3)). In this study, we are interested in the relationship between the air bubble size at the air/water interface and the bursting frequency. The modified RP equation was proposed by Franc [43] to characterize the acoustic response (natural resonance frequency, f) that is associated with the bubble's cavitation collapse:
f ≈
3
ð5Þ
where α is the gas specific heat ratio and Pgo is the gas pressure inside the bubble. Recently, researchers have reported that there are oscillation sounds produced by an entrapped air bubble [38]. The authors used a microphone and hydrophone to simultaneously monitor the airborne and underwater sound frequency with time when a 4.0 mm diameter drop impacted the air/water interface and the air bubble was generated. The two recorded sound frequencies were in good agreement. The authors reported the measured millimeter bubble oscillation frequency and the predicted bubble oscillation frequency (f) were in good agreement with Eq. (5). Minnaert [44] and Leigton [45] also proposed Eq.
Eqs. (5) and (6) predict the strong frequency dependence vs. bubble radius. The calculated frequency of the resonance oscillation is of the order of 1 kHz for an air bubble with a radius of 0.3 cm. The following is true for a submicron-sized bubble (it includes the role of the surface tension and accounts for a negligible viscosity):
f ¼
1 2πr
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3αP 0 4γ þ ρr ρ
ð7Þ
Lhuissier and Villermaux [34] studied the air bubble film rupture to understand how an aerosol is produced on the surface of the sea. They investigated the air bubble bursting evolution of a pressurized air bubble with a size between 0.01 cm and 100 cm at the air/water interface. A single bubble was created on the tip of a capillary by an injection of air. The capillary tip was immersed in tap water. The authors reported that the bubble film rupture was due to a nucleus hole at the bubble's foot (the film meniscus region), and it was associated with the film fluid's conventional motions. The hole quickly expanded, and a rim formed at the edge of the ruptured film. The rim became unstable by the Rayleigh–Taylor mechanism and fragmented into droplets. The authors were unable to present an explanation for why the nucleation of the hole resulted in a micron-thick film. Lauterborn and Kurt [33] presented a comprehensive review of the current state of the knowledge of the physics of bubble oscillations. The article did not highlight the air bubble bursting (collapse) oscillation frequency at the free air/ water interface. In summary, we presented above a brief review of the research conducted on the foam film rupture velocity and air bubble collapse triggered by the film's forced rupture, where the soap film thickness is unknown. High-speed photography and video techniques provided a visual observation of the film bursting rate and were limited to monitoring the bubble collapse frequency (acoustic). Studies were conducted using surfactants as foam film stabilizers and glycerol as a film viscous controller. The role of the dynamic surface tension on the retreated soap film velocity was not considered in the model. The goals of the presented research are to monitor and explain the air bubble collapse at the air/water free interface, the lamella's (thick film) rupture thickness that triggers the gas-vapor release, the bursting bubble oscillation, and the periodical wave propagation at the free interface. To achieve these goals, a novel application of the piezoelectricpressure-acoustic technique (PEPAT) was applied to monitor the air bubble bursts and their oscillations at the air/water interface. The PEPAT technique simultaneously monitored the air bubble lamella's rupture time, the bursting air bubble oscillation frequency, and the periodic waves that propagated at the free interface. The aqueous lamella's rupture thickness was monitored by reflected light interferometry, and the bubble bursts at the air/water interface were monitored with the high-speed photo imaging technique. The data obtained by the three techniques provided essential information for the stages of the air bubble collapse dynamics. The experiments were conducted using Millipore water without the presence of a surfactant.
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Fig. 1. The drainage vs. time of the spherical-shaped aqueous lamella of the floating air bubble at the air/water interface observed in white reflected light. A. The lamella's plumes exhibit color at 1–3 s. B. The lamella's plumes exhibit color at 3–7 s before the lamella ruptures and the bubble collapses.
2. Experimental To study the bubble bursting, three complimentary techniques were applied: aqueous film reflected light interferometry, high-speed photo imaging, and monitoring the bursting air bubble frequency of oscillation by the piezo-electric transducer - hydro-acoustic device. 2.1. Monitoring an air bubble film rupture with light reflection interferometry Millipore water was used to study the air bubble bursting at the air/ water interface. A petri dish was filled with Millipore water until a convex-shaped meniscus was formed. An air bubble from a clean plastic pipette was carefully injected under the water, and a floating bubble with a contact radius size of 0.25–0.30 cm was formed at the air/water interface. The bubble quickly arrived at the apex of the concave meniscus, and it burst after 4–7 s. The bubble's hemispherical lamella drainage was observed in white reflected light. The observation was conducted at a 45° angle with a digital still camera. As the lamella of the floating bubble drained, colored convectional plumes flowed up from the lamellameniscus region to the upper part of the lamella. Before the lamella burst, the color of the convectional plumes changed from blue-green to magenta-yellow, as shown in Fig. 1. The analyses of the lamella color revealed the estimated lamella critical thickness of rupture was between the 200 and 300 nm. Since the aqueous lamella ruptured at such a high thickness, the conventional theory based on the Londonvan der Waals film surface attraction forces cannot be used as an
explanation. The probable explanation of the aqueous lamella rupture at this high thickness is that it was due to gas-film diffusion [2] and/or the lamella/meniscus resonance instability. The bubble's lamella rupture mechanism requires more study. 2.2. Foam lamella bursting monitored by high-speed photo imaging The foam lamella bursting was monitored using a digital still photo video camera (Canon G15) attached to an optical bench. The camera can take high-speed photos (up to 1/4000 s) and video clips (29 frames/s) with a resolution of 120 Kb per frame. The sequence of high-speed photos in Fig. 2 depicts the bubble's aqueous lamella rupture, the bubble collapse, and the triggered oscillatory wave's propagation at the air/water interface. 2.3. Bubble bursting monitored by the piezo-electric pressure transducer The bubble bursting was monitored by the PEPAT. Previously, the PEPAT was applied by us to monitor the coalescence time of a water droplet at the oil/water interface [46]. A glass cell (petri dish) was designed to monitor the air bubble bursting acoustic-frequency oscillation. The PEPAT acted as a hydrophone and was attached to the inner central bottom part of the circular glass petri dish (diameter of 6.0 cm and height of 1.8 cm), as shown in Fig. 3. The cell was placed in a sound-shielded chamber to prevent noise interface from outside. The PEPAT pressure sensitivity was 40 μV/Pa, and it had a capacity of 27 nF. The PEPAT was connected to a silver-plated,
Fig. 2. The sequence of high-speed photos depicting the stages of the bubble bursting and the collapsing phenomenon at the air/water interface. [A] After the thick aqueous lamella covering the bubble ruptures, the lamella retreats and a rim forms at its edge. The rim becomes unstable and bursts into a jet of droplets. [B] The bubble's gas-vapor pressure is released and triggers the formation of a water crater at the air/water interface. [C] The air/water interface oscillates. [D] A water jet is expelled up from the crater [cavity]. The phenomenon of the cavity collapse leading to the liquid jet expulsion is known as the Worthington jet instability. The jet becomes unstable and breaks into droplets. [E] The interface oscillation triggers surface waves. [F, J and G] Depict the surface waves periodically propagate from the center of the petri dish to its rim and back to the center until they are dampened. See the attached video clip 1.
A. Nikolov, D. Wasan / Advances in Colloid and Interface Science 272 (2019) 101998
Fig. 3. Photo of the glass cell with PEPAT.
high-frequency, low-noise FR 50 Ω coaxial cable. The other side of the cable was connected with a gold-plated NBC male adaptor. The NBC male adaptor was coupled to a digital storage oscilloscope (Singlent SDS1052DL, 50 mHz). The piezo electrode of the PEPAT was shielded with a grounded copper foil to avoid electrical interference. The PEPAT cell was tested for the pressure frequency response using an LED 1000 lm strobe light passed to a 546 nm filter. The test was conducted when the strobe light simultaneously radiated both the PEPAT cell and the silicon phototransistor. The data for the frequency and intensity response depicted by the PEPAT and silicon photo transistor are presented in Fig. 4. The two devices, the photo transistor and the PEPAT, correctly depicted the strobe light frequency of 9.38 Hz. The photo transition correctly depicts the strobe light square wave profile. The PEPAT depicted a sawtooth wave profile instead of a square wave profile. For the sine wave type, both devices correctly depict the wave profile and frequency. Monitoring the PEPAT signal in terms of the pressure (Pa) instead of mV requires additional calibration. The bursting bubble is a complex phenomenon, and we would expect more than one acoustic signal at the same time to simultaneously be released during the various stages (i.e., during the lamella rupture, its retreatment, and rim instability, vapor release, and air/water oscillations). The PEPAT cell responses to the simultaneous monitoring of the two signals with different frequencies and amplitudes were tested. The
Fig. 4. The data for the LED strobe light frequency and intensity response from the photo transistor and PEPAT cell are presented. Both correctly depict the strobe light frequency of 9.38 Hz. The square wave type is the photo transistor's signal response, and the sawtooth wave type is the PEPAT signal's response.
5
cell with the PEPAT was simultaneously radiated with two sinusoidal acoustic waves at different frequencies, 9.17 Hz and 180 Hz. The monitored interference signal of the two sinusoidal waves with a frequency ratio of 20 is presented in Fig. 4. Millipore water was used to monitor the bursting bubble and the dynamics at the air/water interface. The PEPAT cell was cleaned with acetone and then rinsed several times with Millipore water. To test the clean petri dish for water contamination, it was filled with Millipore water and allowed to rest for 1 h. Then, air from a clean plastic syringe was slowly injected under the air/water interface. The air bubble arrived at the air/water interface and floated; it had a contact radius of 0.25– 0.30 cm and burst after 5–7 s. The air bubble burst at the central part of the cell and the PEPAT picked up the signal. The air bubble bursting response depicted by PEPAT in mV vs. time in ms (monitored with high-speed photos) is presented in Fig. 5. A schematic view of the air bubble bursting at the air/water interface is presented in Fig. 5A. The type of the surface waves (capillary or gravitational) was evaluated based on the following criteria: λ ¼ 2π
γ ρg
1=2 ð8Þ
where λ is the wavelength. If λ is larger than the term on the right side, it is the gravitational waves, and when λ is smaller than the right term, it is the capillary waves. The wavelength was estimated from the video frame presented in Fig. 8. The measured wavelength was λmes = 0.45 ± 0.02 cm. The calculated wavelength based on Eq. (8) was λcal = 1.7 cm. λcal N λmes, which indicated that the capillary waves dominated the gravity waves. To test the PEPAT's response to measure the wave's travel time, and respectively, the wave velocity, the data presented in Fig. 8 for the wave travel time were used to calculate the wave velocity. The wave travel length of 6.0 cm was passed at time (t) 240–250 ms (see Fig. 6). The calculated wave velocity c = 25–26 cm/s. The calculated value of the wave speed at the air/water interface at 25 °C was in a good agreement with the surface wave velocity at the air/water interface in deep water: c = 23 cm/s at 25 °C when the ratio is 2d N λ, where d = 1.8 cm is the water depth in the petri dish. The analyses of the data for the aqueous lamella bursts and the bubble collapse are presented in the discussion section. In summary, the PEPAT cell was designed to monitor the bubble bursting phenomenon and the stages of the bursting bubble (i.e., the lamella's rupture time, lamella's rim instability, gas-vapor release,
Fig. 5. The PEPAT monitored wave spectrum when the cell is simultaneously radiated with two acoustic signals with different frequencies and amplitudes (9.17 Hz, with a mean amplitude of 0.22 mV, and 180 Hz, with a mean amplitude of 0.05 mV). The resulting wave spectrum vs. time is a superposition of the two waves.
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Fig. 6. Acoustic spectrum of the bursting of a floating bubble at the air/water interface with a contact size of 0.30 ± 0.05 cm depicted by the PEPAT in mV vs. time and shown in highspeed photos. [A] The air bubble arrives at the air/water interface and triggers harmonic wave oscillations of 6–7 Hz. [B] The bubble's lamella ruptures and triggers the collapsing bubble to oscillate with a harmonic wave frequency of 0.8–1.7 kHz [see the inset]. The pressure of the bursting [collapsing] bubble also oscillates and is scaled in millivolts. The water crater forms at the air/water interface. [C] The surface waves propagate from the central part of the petri dish [with a diameter of 6.0 cm] to its rim and periodically go forward and backward with the period of 240–250 ms; the waves' oscillations are damped after 10–15 s. The frequency of the surface wave oscillation at the center of the petri dish is 40–50 Hz.
(also see Nikolov et al. [47]). The bubble's collapse also causes the gasvapor to be released and results in the pressure gradient dp/dt. The pressure gradient drives the surface of the collapsing bubble to oscillate with its natural frequency, as shown in Fig. 7 B. The bubble oscillation triggers the water jet, as shown in Fig. 7C, and its break-up into droplets (see the video clip 1). The bubble collapse simultaneously triggers the oscillation of the surface of the bubble and the bubble's meniscus surface in contact with the free surface in a millisecond of time (see Fig. 6 inset). Both oscillate with a different frequency due to their mass differences. The challenge here is this: Can we simultaneously monitor the oscillation with two different frequencies vs. time? The bubble mass and the mass of the lamella/meniscus are different. The resistant viscous force of the collapsing air bubble and the resistant viscous force of the bubble lamella/meniscus are different, so the oscillation dampening time will not be the same. The dampening will reduce the amplitude of the oscillation, and it is not expected to have an effect on the frequency of the oscillation. Modeling the rim instability of the retreating bursting aqueous lamella and predicting its frequency of oscillation and break-up into jets of droplets are challenging tasks, and they are beyond the scope of this study. Here, we discuss the literature approaches to modeling the collapsing air bubble oscillations and how they apply to predict the bubble oscillation at the air/water interface. We also consider the Rayleigh-Plesset equation on cavitation and Franc's [43] analysis of the frequency of the forced harmonic bubble oscillation during cavitation. The bubble's frequency of oscillation was modeled as a simple harmonic oscillation using a second-order linear differential equation [45]: 2
bubble's bursting frequency of oscillation, and wave's surface propagation). 3. Discussion The air bubble bursting phenomenon at the interface is a complex phenomenon, and it is still not well understood [34,6]. The rupture of the aqueous lamella covering the air bubble triggers the bursting process and forces the collapsing bubble to oscillate. In the experiment section, we discussed the stages of the bubble bursting and the wave propagation at the air/water interface (see Figs. 2, 6, 7 and 8). The bubble's collapse causes two events to occur simultaneously during a few milliseconds: the ruptured lamella retreats and the air bubble gas–vapor pressure released drives the bubble's surface to oscillate at its natural frequency, as shown in Fig. 7B. The fast lamella retreat causes the formation of a rim at the lamella wedge, as shown in Fig. 7A. The rim becomes unstable and bursts into a jet of flying droplets, as shown in Fig. 7B. The rim break-up is due to the Rayleigh type of instability
d v dv m 2 þb þ kv ¼ 0 dt dt
ð9Þ
where v is the bubble's volume subject of pulsing; v = V(t) − V0, where m, b, and k are the inertial, dissipation (e.g., the viscous effect), and stiffness (e.g., the spring) constants. The inertial constant m represents the pressure required to give the liquid a volume acceleration, and it takes into consideration that the displaced mass is equal to three times the mass of the fluid displaced by the sphere Strasberg 1933 [47]. The stiffness constant is the pressure change inside the bubble when it is deformed. For a bubble containing gas vapor at pressure P0 with a specific heat ratio α, the stiffness k ≈ −p/dv = αP0/V0. For a spherical bubble with a radius R0 in a liquid of density ρ, the optimum value of the radial resonance frequency (ωopt) is given by ωopt ¼ ½k=m
1=2
ð10Þ
Fig. 7. Schematic View of the Air Bubble Bursting at the Air/Water Interface. The sequence of sketches depicting the stages of the bubble bursting and the collapsing phenomenon at the air/ water interface. [A] After the aqueous lamella covering the bubble ruptures, the lamella retreats and a rim forms at its edge. [B] The rim becomes unstable and bursts into a jet of droplets. The bubble's vapor pressure is released and triggers the formation of a water crater at the air/water interface. The air/water interface oscillates. [C] A water jet is expelled up from the crater [cavity]. The jet becomes unstable and breaks into droplets. The bubble collapse triggers surface waves along the air/water interface.
A. Nikolov, D. Wasan / Advances in Colloid and Interface Science 272 (2019) 101998
Fig. 8. The high-speed video frame depicts the harmonic surface wave's propagation at the air/water interface after the bubble's collapse. The periodic dark patterns mark the position of the wave's minima.
For a spherical air bubble floating at the air/water interface with the radius R0 and water density ρ, the bubble radial resonance frequency is ωopt ¼
1 3 α P 0 1=2 R0 ρ
ð11Þ
P0 is the pressure inside the bubble. Eq. (11) predicts the 1/Ro dependence for the bubble radial resonance frequency of oscillation. The frequency of oscillation is higher for small air bubbles. The period of ω : For the experiments conducted at the atmooscillation is 1/τ = υ ¼ 2π 5 spheric pressure of 10 Pa and the specific heat ratio of water gas vapor
A
7
at 25 C0, α = 1.33. To calculate the bubble bursting frequency of the oscillation, we need information about the bubble radius Ro, or if the bubble is not spherical and has a deformed shape (for the gravity effect on the bubble shape, see Fig. 7A), we require a correction factor to calculate the bursting frequency of oscillation ωopt. Strasberg [48] proposed a correction factor for ωopt of bubbles with non-spherical shapes, e.g., bubbles with an elongated shape like an ellipsoid, the ratio Rb/Req. Rb is the bubble radius at the bubble bottom, and Req is the bubble radius at the equator (see Fig. 7a). To calculate ωopt, we used the photo presented in Fig. 7A to measure Rb, and Req. The effect of gravity on the bubble shape is given by the ratio of the capillary length [γ/ρg]1/2 to the bubble radius at the equator [Req]; this value ≈ 1. As seen from the photo, gravity contributes to the bubble shape Fig. 9. For an air floating bubble with an equatorial radius Req = 0.33 ± 0.05 cm and Rb/Req = 1.5, the ωopt (the correction factor) was 1.006. The predicted resonance frequency for the bubble bursting was obtained: 0.98 ± 0.15 kHz. This value is in good agreement with the experimentally measured value of 0.8–1.7 kHz. The scatter of the measured frequency contributed to the experimental reproducibility of the bubble size. The rate of the aqueous lamella's bursting was also studied. The area of the aqueous lamella covering the air bubble was calculated. The shape of the aqueous lamella covering the air bubble (the cap) was part of a sphere with a radius Rf (see Fig. 7B). Rf and Sf of the lamella's area were calculated with the following equations:
r 2f ¼ h f 2R f −h f and S f ¼ π 2R f h f þ r 2f
ð12Þ
The bubble lamella/meniscus contact radius rf and the height of the lamella cap hf are required to calculate Sf. Experimentally, the bubble lamella/meniscus radius rf and the cap height hf are measurable parameters (see the bubble sketch presented in Fig. 7B). The experimentally measured parameters, rf = 0.30 ± 0.05 cm and hf = 0.02 cm, were used to calculate Rf = 0.57 cm. The area of the aqueous lamella covering the air bubble was Sf = 0.142 cm2. The effective radius Reff, which corresponds to a spherical lamella, was calculated as Reff = 0.213 cm. The data for Reff = 0.213 cm and the time of the lamella bursting are presented in the Fig. 6 inset (2.5–3 ms); they were used to calculate the lamella bursting velocity of 1 m/s. The estimated aqueous lamella bursting velocity is lower than that reported in the literature for the bursting of a large free aqueous lamella stabilized with a surfactant (8–10 m/s). The explanation for this result could be that the aqueous lamella covering the air bubble burst at a thickness of 200–300 nm, while a film stabilized by a surfactant is expected to be thinner (e.g., 100 nm, a thin film). The mass per unit area of the aqueous lamella without the surfactant is larger than the film stabilized with the surfactant. 4. Conclusions
B Fig. 9. A. Photo of the air/bubble at the air/water interface B. Sketch of an air bubble at the air/water interface and the geometrical parameters used to calculate the air bubble's optimum value of its radial resonance frequency and velocity of the aqueous lamella burst.
This research presents a comprehensive understanding of the phenomenon of the air bubble collapse at the free air/water interface without the presence of surfactants. The stages of the bubble burst vs. time were monitored using a novel piezoelectric-pressure-acoustic technique (PEPAT). The bubble's burst and collapse were also monitored by high-speed photos and video. The critical lamella thickness of the ruptured lamella was monitored using white light reflected interferometry. The bubble's lamella rupture at a thickness higher than 200 nm contributed to the gas-lamella diffusion destabilization effect. The bubble collapsed, the gas–vapor released drove the bubble surface to oscillate, and a water jet was expelled and broke up into droplet/s. The bubble burst also triggered the free surface to oscillate. For the first time, the floating air bubble collapsing oscillation frequency at the air/ water interface was monitored and modeled using the bubble harmonic oscillation approach. Despite the similarities between the air bubble bursting at the free interface and the air bubble cavity collapse on a solid, which both can be modeled with the natural harmonic oscillation
8
A. Nikolov, D. Wasan / Advances in Colloid and Interface Science 272 (2019) 101998
resonance approach proposed by the Rayleigh-Minnaert-Lighthill (RML) model, there are some differences. The bubble collapse simultaneously triggers the oscillation of the surface of the bubble and oscillation of the bubble's meniscus surface in contact with the free surface; it also causes the propagation of the surface waves. Both oscillate with a different frequency due to their mass differences. The measured collapsing frequency of oscillation of floating an air bubble with an equatorial radius of 0.33 ± 0.05 cm was 0.8–1.7 kHz, and its bubble meniscusfree interface oscillates of 40–50 Hz. The amplitude of the oscillation frequency of the collapsing floating air bubble at the air/water interface and the oscillation of the air-free interface are presented in mv. As shown in Fig. 6, both oscillations had different amplitudes: the bursting bubble amplitude of oscillation is 0.6 mv, and the free interface amplitude of oscillation is 0.2 mv. In the future, our intention is to calibrate the PEPAT to measure the amplitude of the oscillating pressure in Pa and also to conduct the systematic study of the air bubble bursting frequency vs. air bubble size. The Rayleigh-Minnaert-Lighthill model prediction for the natural resonance frequency of an air bubble collapse with an equatorial radius of 0.33 cm was 1.0 kHz. The measured and the predicted air bubble collapse frequency are in good agreement. The velocity of the ruptured aqueous lamella covering the air bubble was estimated as 1 m/s. Supplementary data to this article can be found online at https://doi. org/10.1016/j.cis.2019.101998. Author information The authors declare no competing financial interests. Readers are welcome to comment on the version of the paper. Correspondence of and request for materials should be addressed to A.N. ([email protected]). References [1] Exerowa D, Kashchiev D, Platikanov D. Stability and permeability of amphiphile bilayers. Adv Colloid Interface Sci 1992;40:201–56. [2] Platikanov D, Exerowa D. Willy-nilly thin liquid films prevail over foams. Curr Opin Colloid Interface Sci 2015;20:79–80. [3] Scheludko A. Thin liquid films. Adv Colloid Interface Sci 1967;1:391–464. [4] Exerowa D, Nikolov A, Zacharieva M. Common black and Newton film formation. J Colloid Interface Sci 1981;81:419–29. [5] Liger-Belair G, Seon T, Antkowaik A. Collection of collapsing bubble driven phenomenon found in champagne glasses. Bubble Sci Eng Technol 2012;4:21–34. [6] Lohse D. Bubble puzzles. Physics Today 2003:36–41 [S-0031-9228-0302-020-9]. [7] Brennen C. Cavitation and bubble dynamics. Oxford Press; 1995 [ch. 2]. [8] Resch F, Darrozes S, Afeti G. Marine liquid aerosol production from air bubble. J Geophys Res 1986;91:1019–29. [9] Resch F, Afeti G. Submicron film drop production by bubbles in seawater. J Geophys Res 1992;97:3679–83. [10] Spiel D. The size of the jet drops produced by air bubble bursting on sea-and freshwater surfaces. Tellus 1994;46B:325–38. [11] Michaels J, Nowak J, Mallik A, Koczo K, Wasan D, Papoutsakis E. Analysis of cell-tobubble attachment in sparged bioreactors in the presence of cell protecting additives. Biotechnol Bioeng 1995;20)47:407–19. [12] Wall P, McRae OV, Natarajan Johnson C, Antoniou Bird J. Scientific reports. Quantifying the potential for bursting bubbles to damage suspended cells; 2017. https://doi. org/10.1038/s41598-17-14531-5. [13] Nikolov A, Wasan D. Particles driven up the wall by bursting bubbles. Langmuir 2008;24:9933–6.
[14] Kientzler C, Arnos A, Blanchard D, Woodcock A. Photographic investigation of the projection of droplets by bubbles bursting at a water surface. Tellus 1954;6:1–7. [15] Knelman F, Domborowski N, Newitt D. Mechanism of the bursting bubble. Nature 1954;172:261. [16] Clanet C, Lashers J. Transition from dripping to jetting. J Fluid Mech 1999;383: 307–26. [17] Duchemmin L, Popinet S, Josserand C, Zaleski S. Jet formation in bubble bursting at a free surface. Phys Fluids 2002;14:3000–8. [18] Sangeeth K, Putheveettil B, Hopfionger E. Jet formation from bubble collapse at a free surface. , vol. XXIIIBeijing China: ICTAM; 19–24 August. 2012. [19] Duprè MA. Sixième memoire sur la theorie mécanique de la chaleur. Ann Chim Phys 1867;4)11:194–220. [20] Rayleigh Lord. Some applications of photography. Nature 1891;44:249–54. [21] Rayleigh L. On the pressure developed in a liquid during the collapse of a spherical cavity. Dublin Phil Mag J Science 1917;34:94–8. [22] Ranz W. Some experiments on the dynamics of liquid films. J Appl Phys 1959;30: 1950–5. [23] Taylor G. The dynamics of thin sheets of fluid III disintegration of fluid sheets. Proc Royal Soc London Ser A 1959;253:313–21. [24] Culick F. Comments on a ruptured soap film. J Appl Phys 1960;31:1128–9. [25] Liebman I, Corry J, Perlee H. Rupture mechanism of a liquid film. Science 1968;161: 373–5. [26] McEntee W, Mysels K. Bursting of soap films I an experimental study. J Phys Chem 1969;73:3018–28. [27] Pandit A, Davison J. Hydrodynamics of the rupture of thin liquid films. J Fluid Mech 1990;212:11–24. [28] Boulton-Stone J, Blake J. Gas bubbles bursting at a free surface. J Fluid Mech 1993; 254:437–66. [29] Debrégeas G, Gennes PG, Brochard–Wyart F. The life and death of bare viscous bubbles. Science 1998;279:1704–7. [30] Muller F, Kornek U, Stannarius R. Experimental study of the bursting of inviscid bubbles. Phys Rev E 2007;75 [:06502-1-4]. [31] Lhuissier H, Villermaux E. J Fluid Mech Soap Films Burst Like Flapping Flags 2009; 103:05501–1-4. [32] Bird J, Ruiter R, Courbin L, Stone H. Daughter bubble cascades produced by folding of ruptured thin films. Nature 2010;465:759–62. [33] Lauterborn W, Kurz T. Physics of bubble oscillations. Rep Prog Phys 2010;73:106501 [88 pp]. [34] Lhuissier H, Villermaux E. Bursting bubble aerosols. J Fluid Mech 2012;696:5–44. [35] Sabadini E, Ungarato R, Miranda P. The elasticity of soap bubbles containing wormlike micelles. Langmuir 2014;30:727–32. [36] Champougny L, Roche M, Drenckhan W, Rio E. Life and death of not so “bare” bubbles. Soft Matter 2018. https://doi.org/10.1039/C6SM00178E. [37] Tammaro D, Pasquino R, Villone M, D'Avito G, Ferraro V, Di Maio E, et al. Elasticity in bubble rupture. Langmuir 2018;34:5646–54. [38] Phillips S, Agarwal A, Jordan P. The sound produced by a dripping tap is driven by resonant oscillations of an entrapped air bubble. Sci Rep 2018;8:9515. [39] Murano M, Okumura K. Bursting dynamics of viscous film without circular symmetry the effect of confinement. Phys Rev Fluid 2018. https://doi.org/10.1103/ PhysRevFluids.3.031601. [40] Vrij A, Overbeek J. Rupture of thin liquid films due to spontaneous fluctuations in thikness. J Am Chem Soc 1968;90(12):3074–8. [41] Plesset M. The dynamics of cavitation bubble. ASME J App Mech 1949;16:228–31. [42] Franc J-P, Michel J-M. Fluid mechanics and its applications: fundamentals of cavitation. Kluwer Acad; 2004. [43] Franc J-P. International center for mechanical sci. Courses and lecture 496. In: d’Agostino L, editor. Springer fluid dynamics of cavitation and cavitating turbopumsThe Rayleigh-Plesset Equation: A Simple and Powerful Tool to Understand Various Aspects of Cavity; 2007. p. 1–41. [44] Minneart M. On musical air-bubbles and the sounds of running water. Phil Mag 1933;16:536–7. [45] Leighton T. The acoustic bubble. Academic Press; 1994 [Ch. 3]. [46] Chatterjee J, Nikolov A, Wasan D. Study of drop-interface coalescence using piezoimaging. Ind Eng Chem Res 1996;35:2933–8. [47] Nikolov A, Wasan D, Lee J. Tears of wine: the dance of the droplets. Adv Colloid Interface Sci 2018;256:94–100. [48] Strasberg M. The pulsation frequency of nonspherical gas bubbles in liquids. J Acoustical Soc American 1933;25:536–7.
Contents lists available at ScienceDirect
Advances in Colloid and Interface Science journal homepage: www.elsevier.com/locate/cis
Historical perspective
Air bubble bursting phenomenon at the air-water interface monitored by the piezoelectric-acoustic method Alex Nikolov ⁎, Darsh Wasan Department of Chemical Engineering, Illinois Institute of Technology, Chicago, IL 60616, United States of America
a r t i c l e
i n f o
Article history: 30 July 2019 Available online 16 August 2019 Keywords: Aqueous lamella rupture Air bubble bursting frequency Air/water interface oscillation Prediction
a b s t r a c t When an air bubble arrives at the free interface, the bubble's lamella drains and ruptures. The bubble collapses, and gas vapor is released. The ruptured lamella retreats, and a rim at the edge of the retreating lamella forms. The rim becomes unstable and breaks into fine droplets, leading to the formation of a mist. As the collapsing bubble gas's vapor is released, the collapsing bubble oscillates and a vertical liquid jet erupts; this jet then breaks into a droplet(s). Here, we present a novel approach for monitoring the air bubble bursting frequency at the air-water interface by the piezoelectric-pressure-acoustic technique. The piezoelectric-acoustic technique monitors the lamella's rupture time, the frequency of the oscillation of the collapsing air bubble, and the frequency of the oscillation at the free air/water interface. The aqueous lamella rupture thickness was probed by reflected light interferometry, and the air bubble burst at the air/water interface was monitored with the high-speed photo imaging technique. The data obtained by the three techniques provided essential information for the stages of the air bubble collapse dynamics at the free interface without the presence of a surfactant. The simple model proposed by Rayleigh, Minnaert, and Lighthill (RML) for the oscillation resonance of a single air bubble was applied to calculate the air bubble collapsing frequency. The floating air bubble bursting frequency with an equatorial radius of 0.33 ± 0.05 cm was well predicted using the air bubble resonance frequency model, and was estimated as 1.0 ± 0.3 kHz. The velocity of the ruptured aqueous lamella covering the air bubble was estimated as 1 m/s. This research presents a comprehensive understanding of the phenomenon of the bare air bubble collapse at the free interface. © 2019 Published by Elsevier B.V.
Contents 1. 2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Monitoring an air bubble film rupture with light reflection interferometry . 2.2. Foam lamella bursting monitored by high-speed photo imaging . . . . . 2.3. Bubble bursting monitored by the piezo-electric pressure transducer . . . 3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Author information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction The presented research is dedicated to the classical work performed by Prof. D. Exerowa and Prof. D. Platikanov on the thinning and stability ⁎ Corresponding author. E-mail address: [email protected] (A. Nikolov).
https://doi.org/10.1016/j.cis.2019.101998 0001-8686/© 2019 Published by Elsevier B.V.
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of common and Newtonian black microscopic free foam films [1], and on the permeability of amphiphile bilayers [2]. They explored the pioneering work of Prof. Scheludko [3] on microscopic free foam thin liquid films and applied it to novel applications. One of us had the opportunity to work with Prof. Exerowa on common and Newtonian black film thickness transitions [4].
2
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Bubble bursting at the free interface is a common phenomenon and can be observed daily. Bubble bursts are pleasant to observe. When champagne is poured into a glass, the bubbles burst and a fountain of shining, sparkling droplets “dance” over the wine/air interface [5]. The air bubble burst is a phenomenon that needs a better understanding to explain a wide range of observations [6], such as those found in photoluminescence, sonochemistry, cavitation [7], marine life studies [8–10], bio-cell reactors [11,12], biofouling, erosion, bubble jetprinting, enhanced particle spreading [13], and microfluidics. However, monitoring and explaining the bubble bursting dynamics are not easy tasks [14–18]. When an air bubble arrives at the free interface, the bubble's lamella drains and ruptures. The bubble collapses, gas vapor is released, the aqueous lamella retreats, and the rim at the edge of the retreating lamella forms. The rim becomes unstable and breaks into fine droplets, causing a mist to appear. During the time the collapsing bubble's gas vapor is being released, the bottom of the air bubble oscillates and a vertical liquid jet erupts. (Dupre [19], Rayleigh [20], Rayleigh [21], Ranz [22], Taylor [23], Culick [24], Lieberman [25], McEntee and Mysels [26], Pandit and Davison [27], Boulton-Stone and J. Blake [28], Debregeas et al. [29], Muller et al. [30], Lhuissier et al. [31], Bird et al. [32], Lauterborn and Kruz [33], Lhuissier and Villermaux [34], Sabadini et al. [35], Champougny et al. [36], Tammaro et al. [37], Philips et al. [38], and Murano and Okumura [39]). As was pointed out earlier, the air bubble bursting phenomenon is triggered by the rupture of the aqueous lamella (thick film) and its retreat. Here, we discuss the research conducted on the film/lamella stability, the rate of the lamella retreat, and then the air bubble bursting phenomenon in order to propose a model for calculating the frequency of the oscillation of the bursting bubble at the free air/water interface. Dupre and Rayleigh [19,20] were among the first to use photography to monitor the surfactant-stabilized foam lamella rupture triggered by perforation (the punch method). They reported that when a large (e.g., circular) flat foam film (lamella) ruptures by local perforation, the hole rapidly expands under the film tension action. The fluid from the ruptured film builds up a liquid ridge (rim) along the retracted film edge. As the ridge grows in volume, the ridge becomes unstable and breaks up, and a necklace of flying droplets is generated. Dupre and Rayleigh [19,20] calculated the retraction velocity of the film's edge (V) by applying a simple kinetic energy conservation approach. They assumed that the kinetic energy of the foam lamella was equal to the film surface energy that vanished and did not consider the role of the dissipative energies, the internal viscosity, and film ridge instability. For the maximum of the retreating film edge velocity (V), they proposed the following equation: V edge
sffiffiffiffiffiffiffiffi 2 4γ ¼ ρh f
ð1Þ
velocity where γ is the surface tension, hf is the film rupture thickness, and ρ is the fluid density. The unknown parameter is the film rupture thickness. Dupre [19] also calculated the foam film rupture velocity (32 m/s). Ranz [22] investigated the rupture dynamics of an aqueous circular (11 cm) micron-thick flat foam viscous film stabilized with a surfactant and glycerol using a high-speed camera (2720 frames/s) and reported a film edge rupture velocity of 700 cm/s. Ranz [22], like Dupre and Rayleigh [19,20], reported that the ridge film velocity was constant during the ridge film retreatment and 10% lower than predicted by Eq. (1). Ranz [22] applied Dupre and Rayleigh's equation to calculate the film edge retreat velocity of 32 m/s and assumed an equilibrium value of 26 mN/m for the surface tension. It was unrealistic to use the equilibrium value of the surface tension to calculate the retreating film velocity, because for a retreating film with a rupture velocity of 700 cm/s, one has to apply the value of the dynamic surface tension. Ranz reported the formation of bulges along the ridge and noted that “droplets have been torn from the buglers”. He revisited Dupre and Rayleigh's approach to calculate
the film ridge velocity when taking into consideration the role of the ridge and elastic waves; he proposed a smaller value for the film edge velocity:
V edge
sffiffiffiffiffiffiffiffi 2 2γ ¼ ρh f
ð2Þ
velocity The roles of the ridge and the elastic waves were correctly understood as factors that impacted the film edge velocity. Culick [24] revisited the Rayleigh [20] and Ranz [24] analyses on the edge film velocity and considered the energy dissipation due to the liquid rolling at the film's edge. He concluded that Rayleigh's equation overestimated the edge film velocity and confirmed the validation of Eq. (2). However, the actual film rupture thickness of a punched film is not known. The critical thickness of the rupture of an aqueous film stabilized by a surfactant in the presence of glycerol requires an understanding of what triggers the film instability mechanisms. The film instability model (e.g., the theory of the fluctuation of the capillary or perturbation waves on film surfaces triggered by the attractive van der Waals forces) requires a film thickness less than 10 nm. Vrij and Overbeek [40] were the first to validate the model prediction of the film critical thickness. When the film instability is forced by the punch method, an evaluation of the local film rupture thickness is required, and the application of the van der Waals forces film attraction concept to the film stability is invalid. Pandit and Davidson [27] conducted an experimental and theoretical study on the edge film velocity of a ruptured aqueous spherical soap foam film stabilized by a surfactant solution. The foam film rupture was promoted by the punch method. The authors used a video camera observation of 2000 frames/s, and the ridge film velocity was estimated to be 10 m/s. The film mean thickness prior to the rupture was estimated by the electrical conductivity method, and it was estimated to be of the order of 0.3 to 0.9 μm. The authors reported that when the foam film was forced to rupture and the hole was formed, the film liquid retreated at the film edge and a liquid ridge (rim) with a cylindrical shape formed. The authors reported that due to the varicose instability, the ridge (rim) broke up into jets from which droplets of 10−2 cm in size subsequently formed; these droplets then shattered. However, these researchers did not elucidate the mechanism or discuss the relationship between the rim geometry, jet, and droplet size. Bird et al. [32] observed a bubble-bursting cascade phenomenon. An air bubble was created on a pool of river water. The authors used river water because the indigenous surfactants in river water tend to be present in estuaries and oceans. The bubble collapse was monitored with two synchronized high-speed cameras. The bubble was formed both on the ridged surface and on a deep pool of water. When the bubble formed on the ridged flat surface collapsed due to the foam film rupture triggered by the hole nucleation following the film rupture, the film retreated and a ring of small bubbles, each approximately a millimeter in diameter, appeared around the base of the initial bubble. When one of these daughter bubbles ruptured, a ring of even smaller bubbles formed. The foam film of the smaller bubble ruptured and the bubble collapsed; tiny liquid jets formed. When these jets were ejected into the air, they disintegrated into tiny mist droplets. The authors conducted numerical simulations to elucidate the mechanisms of the foam film folding and bubble air entrapment. To perform the numerical simulation, the authors assumed an initially uniform foam thickness for the surfactant-stabilized foam film. Despite the reported observations of the bubble collapsing via a bubble bursting cascade, it is not clear why the authors performed their experiments using different surfactant formulations. The use of different fluid viscosity values for the bubble bursting dynamics is understandable, since the experiments were conducted to slow the film rupture dynamics. They used the Laplace equation and assumed the value of the surface tension was at equilibrium for their model of the air bubble pressure. The authors used the equilibrium surface tension for the bubble bursting
A. Nikolov, D. Wasan / Advances in Colloid and Interface Science 272 (2019) 101998
dynamics; however, they needed to use the dynamic surface tension, or at least, discuss its role. The authors' proposed numerical simulation did not explain the physics of the bubble bursting cascade and the nature of the oscillation at the free surface during and after the bubble collapse. Rayleigh [21] studied the pressure development vs. time in a liquid during the collapse of a spherical cavity. Rayleigh [21], and later Plesset [41], proposed an equation that was a simple approach to explain the various aspects of the cavity. Their approach was based on a second– order differential equation to elucidate the time evolution of bubble radius r(t). The Rayleigh-Plesset (RP) equation is expressed as r 3α 2γ r_ 3 0 ρ r €r þ r_ 2 ¼ ½pν −p∞ ðt Þ þ pg0 − −4μ l r 4 r r
3α 3 2 r0 1 1 r_ ρ r €r þ r_ −1 pg0 þ 2γ − −4μ ð4Þ ¼ −δp sinωt þ 4 r0 r r r where δp is (r-ro) and ω is the period of oscillation; ω = 2πf, and f is the frequency of oscillation. The simplified equation for the forced collapse of a millimeter-sized bubble's harmonic frequency of oscillation (f) when neglecting the role of the surface forces (capillary) and viscous forces is 1 2πr
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3α P go ρ
(5) based on a thermodynamic approach. They treated the bubble as an adiabatic harmonic oscillator and calculated the bubble's kinetic energy. The role of the heat capacity ratio was considered, and the effects of the surface tension and viscosity were neglected. A simplified version of Eq. (5) was applied to predict the submerged gas bubble frequency of oscillation. For a bubble in water, the oscillation frequency at one atmosphere is
f ≈
3 ms−1 r
ð6Þ
ð3Þ
where r_ and €r are the first and second order derivatives of the bubble radius vs. time, ρ is the gas density, μl is the kinematic viscosity, ro is the initial bubble radius, pν is the gas partial pressure, and pg0 is the initial condition of the gas inside the bubble. pν is the vapor pressure, which depends upon the temperature. p∞ is the atmospheric pressure. α is the ratio of the heat capacity of the enclosed gas. During cavitation, the bubble is subject to oscillations at its natural frequency. The term on the left side of the equation represents the variation in the kinetic energy of the liquid body multiplied by the gas density ρ. The first term on the right side represents the pressure forces acting on the liquid, and it is the driving term. The second term considers the ratio of the heat capacity α of the gas inside the bubble, and it is the contribution of the noncondensable gas. Its derivation is based on the assumption that the mass of the non-condensable gas inside the bubble remains constant. The third term is the contribution of the surface tension forces (capillary), and it is important only for small, micron-sized bubbles. The fourth term is the dissipation rate due to the fluid viscosity. More information about the derivation and the physics of Eq. (3) can be found in the literature of Franc and Michel [42]. The air bubble in a liquid is a possible oscillator because of the elastic behavior of the non-condensable gas that the bubble contains and the inertia of the liquid [43]. The oscillation behavior of a bubble and its resonance frequency can be computed from the Rayleigh-Plesset equation (Eq. (3)). In this study, we are interested in the relationship between the air bubble size at the air/water interface and the bursting frequency. The modified RP equation was proposed by Franc [43] to characterize the acoustic response (natural resonance frequency, f) that is associated with the bubble's cavitation collapse:
f ≈
3
ð5Þ
where α is the gas specific heat ratio and Pgo is the gas pressure inside the bubble. Recently, researchers have reported that there are oscillation sounds produced by an entrapped air bubble [38]. The authors used a microphone and hydrophone to simultaneously monitor the airborne and underwater sound frequency with time when a 4.0 mm diameter drop impacted the air/water interface and the air bubble was generated. The two recorded sound frequencies were in good agreement. The authors reported the measured millimeter bubble oscillation frequency and the predicted bubble oscillation frequency (f) were in good agreement with Eq. (5). Minnaert [44] and Leigton [45] also proposed Eq.
Eqs. (5) and (6) predict the strong frequency dependence vs. bubble radius. The calculated frequency of the resonance oscillation is of the order of 1 kHz for an air bubble with a radius of 0.3 cm. The following is true for a submicron-sized bubble (it includes the role of the surface tension and accounts for a negligible viscosity):
f ¼
1 2πr
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3αP 0 4γ þ ρr ρ
ð7Þ
Lhuissier and Villermaux [34] studied the air bubble film rupture to understand how an aerosol is produced on the surface of the sea. They investigated the air bubble bursting evolution of a pressurized air bubble with a size between 0.01 cm and 100 cm at the air/water interface. A single bubble was created on the tip of a capillary by an injection of air. The capillary tip was immersed in tap water. The authors reported that the bubble film rupture was due to a nucleus hole at the bubble's foot (the film meniscus region), and it was associated with the film fluid's conventional motions. The hole quickly expanded, and a rim formed at the edge of the ruptured film. The rim became unstable by the Rayleigh–Taylor mechanism and fragmented into droplets. The authors were unable to present an explanation for why the nucleation of the hole resulted in a micron-thick film. Lauterborn and Kurt [33] presented a comprehensive review of the current state of the knowledge of the physics of bubble oscillations. The article did not highlight the air bubble bursting (collapse) oscillation frequency at the free air/ water interface. In summary, we presented above a brief review of the research conducted on the foam film rupture velocity and air bubble collapse triggered by the film's forced rupture, where the soap film thickness is unknown. High-speed photography and video techniques provided a visual observation of the film bursting rate and were limited to monitoring the bubble collapse frequency (acoustic). Studies were conducted using surfactants as foam film stabilizers and glycerol as a film viscous controller. The role of the dynamic surface tension on the retreated soap film velocity was not considered in the model. The goals of the presented research are to monitor and explain the air bubble collapse at the air/water free interface, the lamella's (thick film) rupture thickness that triggers the gas-vapor release, the bursting bubble oscillation, and the periodical wave propagation at the free interface. To achieve these goals, a novel application of the piezoelectricpressure-acoustic technique (PEPAT) was applied to monitor the air bubble bursts and their oscillations at the air/water interface. The PEPAT technique simultaneously monitored the air bubble lamella's rupture time, the bursting air bubble oscillation frequency, and the periodic waves that propagated at the free interface. The aqueous lamella's rupture thickness was monitored by reflected light interferometry, and the bubble bursts at the air/water interface were monitored with the high-speed photo imaging technique. The data obtained by the three techniques provided essential information for the stages of the air bubble collapse dynamics. The experiments were conducted using Millipore water without the presence of a surfactant.
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Fig. 1. The drainage vs. time of the spherical-shaped aqueous lamella of the floating air bubble at the air/water interface observed in white reflected light. A. The lamella's plumes exhibit color at 1–3 s. B. The lamella's plumes exhibit color at 3–7 s before the lamella ruptures and the bubble collapses.
2. Experimental To study the bubble bursting, three complimentary techniques were applied: aqueous film reflected light interferometry, high-speed photo imaging, and monitoring the bursting air bubble frequency of oscillation by the piezo-electric transducer - hydro-acoustic device. 2.1. Monitoring an air bubble film rupture with light reflection interferometry Millipore water was used to study the air bubble bursting at the air/ water interface. A petri dish was filled with Millipore water until a convex-shaped meniscus was formed. An air bubble from a clean plastic pipette was carefully injected under the water, and a floating bubble with a contact radius size of 0.25–0.30 cm was formed at the air/water interface. The bubble quickly arrived at the apex of the concave meniscus, and it burst after 4–7 s. The bubble's hemispherical lamella drainage was observed in white reflected light. The observation was conducted at a 45° angle with a digital still camera. As the lamella of the floating bubble drained, colored convectional plumes flowed up from the lamellameniscus region to the upper part of the lamella. Before the lamella burst, the color of the convectional plumes changed from blue-green to magenta-yellow, as shown in Fig. 1. The analyses of the lamella color revealed the estimated lamella critical thickness of rupture was between the 200 and 300 nm. Since the aqueous lamella ruptured at such a high thickness, the conventional theory based on the Londonvan der Waals film surface attraction forces cannot be used as an
explanation. The probable explanation of the aqueous lamella rupture at this high thickness is that it was due to gas-film diffusion [2] and/or the lamella/meniscus resonance instability. The bubble's lamella rupture mechanism requires more study. 2.2. Foam lamella bursting monitored by high-speed photo imaging The foam lamella bursting was monitored using a digital still photo video camera (Canon G15) attached to an optical bench. The camera can take high-speed photos (up to 1/4000 s) and video clips (29 frames/s) with a resolution of 120 Kb per frame. The sequence of high-speed photos in Fig. 2 depicts the bubble's aqueous lamella rupture, the bubble collapse, and the triggered oscillatory wave's propagation at the air/water interface. 2.3. Bubble bursting monitored by the piezo-electric pressure transducer The bubble bursting was monitored by the PEPAT. Previously, the PEPAT was applied by us to monitor the coalescence time of a water droplet at the oil/water interface [46]. A glass cell (petri dish) was designed to monitor the air bubble bursting acoustic-frequency oscillation. The PEPAT acted as a hydrophone and was attached to the inner central bottom part of the circular glass petri dish (diameter of 6.0 cm and height of 1.8 cm), as shown in Fig. 3. The cell was placed in a sound-shielded chamber to prevent noise interface from outside. The PEPAT pressure sensitivity was 40 μV/Pa, and it had a capacity of 27 nF. The PEPAT was connected to a silver-plated,
Fig. 2. The sequence of high-speed photos depicting the stages of the bubble bursting and the collapsing phenomenon at the air/water interface. [A] After the thick aqueous lamella covering the bubble ruptures, the lamella retreats and a rim forms at its edge. The rim becomes unstable and bursts into a jet of droplets. [B] The bubble's gas-vapor pressure is released and triggers the formation of a water crater at the air/water interface. [C] The air/water interface oscillates. [D] A water jet is expelled up from the crater [cavity]. The phenomenon of the cavity collapse leading to the liquid jet expulsion is known as the Worthington jet instability. The jet becomes unstable and breaks into droplets. [E] The interface oscillation triggers surface waves. [F, J and G] Depict the surface waves periodically propagate from the center of the petri dish to its rim and back to the center until they are dampened. See the attached video clip 1.
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Fig. 3. Photo of the glass cell with PEPAT.
high-frequency, low-noise FR 50 Ω coaxial cable. The other side of the cable was connected with a gold-plated NBC male adaptor. The NBC male adaptor was coupled to a digital storage oscilloscope (Singlent SDS1052DL, 50 mHz). The piezo electrode of the PEPAT was shielded with a grounded copper foil to avoid electrical interference. The PEPAT cell was tested for the pressure frequency response using an LED 1000 lm strobe light passed to a 546 nm filter. The test was conducted when the strobe light simultaneously radiated both the PEPAT cell and the silicon phototransistor. The data for the frequency and intensity response depicted by the PEPAT and silicon photo transistor are presented in Fig. 4. The two devices, the photo transistor and the PEPAT, correctly depicted the strobe light frequency of 9.38 Hz. The photo transition correctly depicts the strobe light square wave profile. The PEPAT depicted a sawtooth wave profile instead of a square wave profile. For the sine wave type, both devices correctly depict the wave profile and frequency. Monitoring the PEPAT signal in terms of the pressure (Pa) instead of mV requires additional calibration. The bursting bubble is a complex phenomenon, and we would expect more than one acoustic signal at the same time to simultaneously be released during the various stages (i.e., during the lamella rupture, its retreatment, and rim instability, vapor release, and air/water oscillations). The PEPAT cell responses to the simultaneous monitoring of the two signals with different frequencies and amplitudes were tested. The
Fig. 4. The data for the LED strobe light frequency and intensity response from the photo transistor and PEPAT cell are presented. Both correctly depict the strobe light frequency of 9.38 Hz. The square wave type is the photo transistor's signal response, and the sawtooth wave type is the PEPAT signal's response.
5
cell with the PEPAT was simultaneously radiated with two sinusoidal acoustic waves at different frequencies, 9.17 Hz and 180 Hz. The monitored interference signal of the two sinusoidal waves with a frequency ratio of 20 is presented in Fig. 4. Millipore water was used to monitor the bursting bubble and the dynamics at the air/water interface. The PEPAT cell was cleaned with acetone and then rinsed several times with Millipore water. To test the clean petri dish for water contamination, it was filled with Millipore water and allowed to rest for 1 h. Then, air from a clean plastic syringe was slowly injected under the air/water interface. The air bubble arrived at the air/water interface and floated; it had a contact radius of 0.25– 0.30 cm and burst after 5–7 s. The air bubble burst at the central part of the cell and the PEPAT picked up the signal. The air bubble bursting response depicted by PEPAT in mV vs. time in ms (monitored with high-speed photos) is presented in Fig. 5. A schematic view of the air bubble bursting at the air/water interface is presented in Fig. 5A. The type of the surface waves (capillary or gravitational) was evaluated based on the following criteria: λ ¼ 2π
γ ρg
1=2 ð8Þ
where λ is the wavelength. If λ is larger than the term on the right side, it is the gravitational waves, and when λ is smaller than the right term, it is the capillary waves. The wavelength was estimated from the video frame presented in Fig. 8. The measured wavelength was λmes = 0.45 ± 0.02 cm. The calculated wavelength based on Eq. (8) was λcal = 1.7 cm. λcal N λmes, which indicated that the capillary waves dominated the gravity waves. To test the PEPAT's response to measure the wave's travel time, and respectively, the wave velocity, the data presented in Fig. 8 for the wave travel time were used to calculate the wave velocity. The wave travel length of 6.0 cm was passed at time (t) 240–250 ms (see Fig. 6). The calculated wave velocity c = 25–26 cm/s. The calculated value of the wave speed at the air/water interface at 25 °C was in a good agreement with the surface wave velocity at the air/water interface in deep water: c = 23 cm/s at 25 °C when the ratio is 2d N λ, where d = 1.8 cm is the water depth in the petri dish. The analyses of the data for the aqueous lamella bursts and the bubble collapse are presented in the discussion section. In summary, the PEPAT cell was designed to monitor the bubble bursting phenomenon and the stages of the bursting bubble (i.e., the lamella's rupture time, lamella's rim instability, gas-vapor release,
Fig. 5. The PEPAT monitored wave spectrum when the cell is simultaneously radiated with two acoustic signals with different frequencies and amplitudes (9.17 Hz, with a mean amplitude of 0.22 mV, and 180 Hz, with a mean amplitude of 0.05 mV). The resulting wave spectrum vs. time is a superposition of the two waves.
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Fig. 6. Acoustic spectrum of the bursting of a floating bubble at the air/water interface with a contact size of 0.30 ± 0.05 cm depicted by the PEPAT in mV vs. time and shown in highspeed photos. [A] The air bubble arrives at the air/water interface and triggers harmonic wave oscillations of 6–7 Hz. [B] The bubble's lamella ruptures and triggers the collapsing bubble to oscillate with a harmonic wave frequency of 0.8–1.7 kHz [see the inset]. The pressure of the bursting [collapsing] bubble also oscillates and is scaled in millivolts. The water crater forms at the air/water interface. [C] The surface waves propagate from the central part of the petri dish [with a diameter of 6.0 cm] to its rim and periodically go forward and backward with the period of 240–250 ms; the waves' oscillations are damped after 10–15 s. The frequency of the surface wave oscillation at the center of the petri dish is 40–50 Hz.
(also see Nikolov et al. [47]). The bubble's collapse also causes the gasvapor to be released and results in the pressure gradient dp/dt. The pressure gradient drives the surface of the collapsing bubble to oscillate with its natural frequency, as shown in Fig. 7 B. The bubble oscillation triggers the water jet, as shown in Fig. 7C, and its break-up into droplets (see the video clip 1). The bubble collapse simultaneously triggers the oscillation of the surface of the bubble and the bubble's meniscus surface in contact with the free surface in a millisecond of time (see Fig. 6 inset). Both oscillate with a different frequency due to their mass differences. The challenge here is this: Can we simultaneously monitor the oscillation with two different frequencies vs. time? The bubble mass and the mass of the lamella/meniscus are different. The resistant viscous force of the collapsing air bubble and the resistant viscous force of the bubble lamella/meniscus are different, so the oscillation dampening time will not be the same. The dampening will reduce the amplitude of the oscillation, and it is not expected to have an effect on the frequency of the oscillation. Modeling the rim instability of the retreating bursting aqueous lamella and predicting its frequency of oscillation and break-up into jets of droplets are challenging tasks, and they are beyond the scope of this study. Here, we discuss the literature approaches to modeling the collapsing air bubble oscillations and how they apply to predict the bubble oscillation at the air/water interface. We also consider the Rayleigh-Plesset equation on cavitation and Franc's [43] analysis of the frequency of the forced harmonic bubble oscillation during cavitation. The bubble's frequency of oscillation was modeled as a simple harmonic oscillation using a second-order linear differential equation [45]: 2
bubble's bursting frequency of oscillation, and wave's surface propagation). 3. Discussion The air bubble bursting phenomenon at the interface is a complex phenomenon, and it is still not well understood [34,6]. The rupture of the aqueous lamella covering the air bubble triggers the bursting process and forces the collapsing bubble to oscillate. In the experiment section, we discussed the stages of the bubble bursting and the wave propagation at the air/water interface (see Figs. 2, 6, 7 and 8). The bubble's collapse causes two events to occur simultaneously during a few milliseconds: the ruptured lamella retreats and the air bubble gas–vapor pressure released drives the bubble's surface to oscillate at its natural frequency, as shown in Fig. 7B. The fast lamella retreat causes the formation of a rim at the lamella wedge, as shown in Fig. 7A. The rim becomes unstable and bursts into a jet of flying droplets, as shown in Fig. 7B. The rim break-up is due to the Rayleigh type of instability
d v dv m 2 þb þ kv ¼ 0 dt dt
ð9Þ
where v is the bubble's volume subject of pulsing; v = V(t) − V0, where m, b, and k are the inertial, dissipation (e.g., the viscous effect), and stiffness (e.g., the spring) constants. The inertial constant m represents the pressure required to give the liquid a volume acceleration, and it takes into consideration that the displaced mass is equal to three times the mass of the fluid displaced by the sphere Strasberg 1933 [47]. The stiffness constant is the pressure change inside the bubble when it is deformed. For a bubble containing gas vapor at pressure P0 with a specific heat ratio α, the stiffness k ≈ −p/dv = αP0/V0. For a spherical bubble with a radius R0 in a liquid of density ρ, the optimum value of the radial resonance frequency (ωopt) is given by ωopt ¼ ½k=m
1=2
ð10Þ
Fig. 7. Schematic View of the Air Bubble Bursting at the Air/Water Interface. The sequence of sketches depicting the stages of the bubble bursting and the collapsing phenomenon at the air/ water interface. [A] After the aqueous lamella covering the bubble ruptures, the lamella retreats and a rim forms at its edge. [B] The rim becomes unstable and bursts into a jet of droplets. The bubble's vapor pressure is released and triggers the formation of a water crater at the air/water interface. The air/water interface oscillates. [C] A water jet is expelled up from the crater [cavity]. The jet becomes unstable and breaks into droplets. The bubble collapse triggers surface waves along the air/water interface.
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Fig. 8. The high-speed video frame depicts the harmonic surface wave's propagation at the air/water interface after the bubble's collapse. The periodic dark patterns mark the position of the wave's minima.
For a spherical air bubble floating at the air/water interface with the radius R0 and water density ρ, the bubble radial resonance frequency is ωopt ¼
1 3 α P 0 1=2 R0 ρ
ð11Þ
P0 is the pressure inside the bubble. Eq. (11) predicts the 1/Ro dependence for the bubble radial resonance frequency of oscillation. The frequency of oscillation is higher for small air bubbles. The period of ω : For the experiments conducted at the atmooscillation is 1/τ = υ ¼ 2π 5 spheric pressure of 10 Pa and the specific heat ratio of water gas vapor
A
7
at 25 C0, α = 1.33. To calculate the bubble bursting frequency of the oscillation, we need information about the bubble radius Ro, or if the bubble is not spherical and has a deformed shape (for the gravity effect on the bubble shape, see Fig. 7A), we require a correction factor to calculate the bursting frequency of oscillation ωopt. Strasberg [48] proposed a correction factor for ωopt of bubbles with non-spherical shapes, e.g., bubbles with an elongated shape like an ellipsoid, the ratio Rb/Req. Rb is the bubble radius at the bubble bottom, and Req is the bubble radius at the equator (see Fig. 7a). To calculate ωopt, we used the photo presented in Fig. 7A to measure Rb, and Req. The effect of gravity on the bubble shape is given by the ratio of the capillary length [γ/ρg]1/2 to the bubble radius at the equator [Req]; this value ≈ 1. As seen from the photo, gravity contributes to the bubble shape Fig. 9. For an air floating bubble with an equatorial radius Req = 0.33 ± 0.05 cm and Rb/Req = 1.5, the ωopt (the correction factor) was 1.006. The predicted resonance frequency for the bubble bursting was obtained: 0.98 ± 0.15 kHz. This value is in good agreement with the experimentally measured value of 0.8–1.7 kHz. The scatter of the measured frequency contributed to the experimental reproducibility of the bubble size. The rate of the aqueous lamella's bursting was also studied. The area of the aqueous lamella covering the air bubble was calculated. The shape of the aqueous lamella covering the air bubble (the cap) was part of a sphere with a radius Rf (see Fig. 7B). Rf and Sf of the lamella's area were calculated with the following equations:
r 2f ¼ h f 2R f −h f and S f ¼ π 2R f h f þ r 2f
ð12Þ
The bubble lamella/meniscus contact radius rf and the height of the lamella cap hf are required to calculate Sf. Experimentally, the bubble lamella/meniscus radius rf and the cap height hf are measurable parameters (see the bubble sketch presented in Fig. 7B). The experimentally measured parameters, rf = 0.30 ± 0.05 cm and hf = 0.02 cm, were used to calculate Rf = 0.57 cm. The area of the aqueous lamella covering the air bubble was Sf = 0.142 cm2. The effective radius Reff, which corresponds to a spherical lamella, was calculated as Reff = 0.213 cm. The data for Reff = 0.213 cm and the time of the lamella bursting are presented in the Fig. 6 inset (2.5–3 ms); they were used to calculate the lamella bursting velocity of 1 m/s. The estimated aqueous lamella bursting velocity is lower than that reported in the literature for the bursting of a large free aqueous lamella stabilized with a surfactant (8–10 m/s). The explanation for this result could be that the aqueous lamella covering the air bubble burst at a thickness of 200–300 nm, while a film stabilized by a surfactant is expected to be thinner (e.g., 100 nm, a thin film). The mass per unit area of the aqueous lamella without the surfactant is larger than the film stabilized with the surfactant. 4. Conclusions
B Fig. 9. A. Photo of the air/bubble at the air/water interface B. Sketch of an air bubble at the air/water interface and the geometrical parameters used to calculate the air bubble's optimum value of its radial resonance frequency and velocity of the aqueous lamella burst.
This research presents a comprehensive understanding of the phenomenon of the air bubble collapse at the free air/water interface without the presence of surfactants. The stages of the bubble burst vs. time were monitored using a novel piezoelectric-pressure-acoustic technique (PEPAT). The bubble's burst and collapse were also monitored by high-speed photos and video. The critical lamella thickness of the ruptured lamella was monitored using white light reflected interferometry. The bubble's lamella rupture at a thickness higher than 200 nm contributed to the gas-lamella diffusion destabilization effect. The bubble collapsed, the gas–vapor released drove the bubble surface to oscillate, and a water jet was expelled and broke up into droplet/s. The bubble burst also triggered the free surface to oscillate. For the first time, the floating air bubble collapsing oscillation frequency at the air/ water interface was monitored and modeled using the bubble harmonic oscillation approach. Despite the similarities between the air bubble bursting at the free interface and the air bubble cavity collapse on a solid, which both can be modeled with the natural harmonic oscillation
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resonance approach proposed by the Rayleigh-Minnaert-Lighthill (RML) model, there are some differences. The bubble collapse simultaneously triggers the oscillation of the surface of the bubble and oscillation of the bubble's meniscus surface in contact with the free surface; it also causes the propagation of the surface waves. Both oscillate with a different frequency due to their mass differences. The measured collapsing frequency of oscillation of floating an air bubble with an equatorial radius of 0.33 ± 0.05 cm was 0.8–1.7 kHz, and its bubble meniscusfree interface oscillates of 40–50 Hz. The amplitude of the oscillation frequency of the collapsing floating air bubble at the air/water interface and the oscillation of the air-free interface are presented in mv. As shown in Fig. 6, both oscillations had different amplitudes: the bursting bubble amplitude of oscillation is 0.6 mv, and the free interface amplitude of oscillation is 0.2 mv. In the future, our intention is to calibrate the PEPAT to measure the amplitude of the oscillating pressure in Pa and also to conduct the systematic study of the air bubble bursting frequency vs. air bubble size. The Rayleigh-Minnaert-Lighthill model prediction for the natural resonance frequency of an air bubble collapse with an equatorial radius of 0.33 cm was 1.0 kHz. The measured and the predicted air bubble collapse frequency are in good agreement. The velocity of the ruptured aqueous lamella covering the air bubble was estimated as 1 m/s. Supplementary data to this article can be found online at https://doi. org/10.1016/j.cis.2019.101998. Author information The authors declare no competing financial interests. Readers are welcome to comment on the version of the paper. Correspondence of and request for materials should be addressed to A.N. ([email protected]). References [1] Exerowa D, Kashchiev D, Platikanov D. Stability and permeability of amphiphile bilayers. Adv Colloid Interface Sci 1992;40:201–56. [2] Platikanov D, Exerowa D. Willy-nilly thin liquid films prevail over foams. Curr Opin Colloid Interface Sci 2015;20:79–80. [3] Scheludko A. Thin liquid films. Adv Colloid Interface Sci 1967;1:391–464. [4] Exerowa D, Nikolov A, Zacharieva M. Common black and Newton film formation. J Colloid Interface Sci 1981;81:419–29. [5] Liger-Belair G, Seon T, Antkowaik A. Collection of collapsing bubble driven phenomenon found in champagne glasses. Bubble Sci Eng Technol 2012;4:21–34. [6] Lohse D. Bubble puzzles. Physics Today 2003:36–41 [S-0031-9228-0302-020-9]. [7] Brennen C. Cavitation and bubble dynamics. Oxford Press; 1995 [ch. 2]. [8] Resch F, Darrozes S, Afeti G. Marine liquid aerosol production from air bubble. J Geophys Res 1986;91:1019–29. [9] Resch F, Afeti G. Submicron film drop production by bubbles in seawater. J Geophys Res 1992;97:3679–83. [10] Spiel D. The size of the jet drops produced by air bubble bursting on sea-and freshwater surfaces. Tellus 1994;46B:325–38. [11] Michaels J, Nowak J, Mallik A, Koczo K, Wasan D, Papoutsakis E. Analysis of cell-tobubble attachment in sparged bioreactors in the presence of cell protecting additives. Biotechnol Bioeng 1995;20)47:407–19. [12] Wall P, McRae OV, Natarajan Johnson C, Antoniou Bird J. Scientific reports. Quantifying the potential for bursting bubbles to damage suspended cells; 2017. https://doi. org/10.1038/s41598-17-14531-5. [13] Nikolov A, Wasan D. Particles driven up the wall by bursting bubbles. Langmuir 2008;24:9933–6.
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