Physics of beer tapping

Javier Rodrıguez-Rodrıguez1, Almudena Casado-Chacon1 and Daniel Fuster2

1 Fluid Mechanics Group, Carlos III University of Madrid, SPAIN2 CNRS (UMR 7190), Universite Pierre et Marie Curie, Institut Jean le Rond d’Alembert, FRANCE

The popular bar prank known in colloquial English as beer tapping consists in hitting the topof a beer bottle with a solid object, usually another bottle, to trigger the foaming over of theformer within a few seconds. Despite the trick being known for long time, to the best of ourknowledge, the phenomenon still lacks scientific explanation. Although it seems natural to thinkthat shock-induced cavitation enhances the diffusion of CO2 from the supersaturated bulk liquid intothe bubbles by breaking them up, the subtle mechanism by which this happens remains unknown.Here we show that the overall foaming-over process can be divided into three stages where differentphysical phenomena take place in different time-scales, namely: bubble-collapse (or cavitation) stage,diffusion-driven stage and buoyancy-driven stage. In the bubble-collapse stage, the impact generatesa train of expansion-compression waves in the liquid that leads to the fragmentation of pre-existinggas cavities. Upon bubble fragmentation, the sudden increase of the interface-area-to-volume ratioenhances mass transfer significantly, which makes the bubble volume grow by a large factor untilCO2 is locally depleted. At that point buoyancy takes over, making the bubble clouds rise andeventually form buoyant vortex rings whose volume grows fast due to the feedback between thebuoyancy-induced rising speed and the advection-enhanced CO2 transport from the bulk liquid tothe bubble. The physics behind this explosive process sheds insight into the dynamics of geologicalphenomena such as limnic eruptions.

Understanding the formation of foam in a supersatu-rated carbonated liquid after an impact on the containerinvolves a careful physical description of a number of pro-cesses of great interest in several areas of Physics andChemistry. In order of appearance in this problem: prop-agation of strong pressure waves in bubbly liquids, bub-ble collapse and fragmentation, gas-liquid diffusive masstransfer and the dynamics of bubble-laden plumes andvortex rings. All these phenomena are observed, for in-stance, in the explosive formation of foam occurring ina beer bottle when it is tapped on its mouth, an effectknown as beer tapping. In this letter, we will use thiseffect as a convenient system to quantitatively describethe interaction between the processes mentioned abovethat ultimately leads to the explosive formation of foamthat occurs in gas-driven eruptions [1]. As a consequenceof the broad range of phenomena taking part in the over-all process, the better understanding of the foam formingprocess in supersaturated liquids finds application in var-ious fields of natural sciences and technology where sim-ilar gas-driven eruptions occur. The dynamics of limnic[1, 2] or explosive volcanic [3, 4] eruptions and the forma-tion of flavour-releasing aerosols by bursting Champagnebubbles [5] are just a few examples. Mott and Woods[6] have triggered a chain reaction in a stably-stratifiedtank containing a deep layer of CO2-saturated lemonadeand a shallower layer of fresh water by spilling a grav-ity current of salt grains along the bottom of the tank.This example shows that the dynamics of CO2 bubbles indaily-life liquids can be used to explain complex naturalphenomena such as limnic eruptions.

To understand how the processes described above in-teract to lead to the foaming-up of beer, we have carriedout an experimental investigation impacting commercial

beer bottles under well-controlled repeatable conditions(see Supplementary Material). In particular, bubbleshave been generated at a fixed location far from the bot-tle walls by focusing a laser pulse into the bulk liquid. Inthis way, we avoid the variability in the formation of bub-bles upon the impact caused by the arbitrary distributionof nucleation sites [7], thus ensuring that a bubble with aknown initial size is always present in the measurementvolume. By recording the evolution of these gas bub-bles with a high-speed camera and the liquid pressuretemporal evolution with a hydrophone we provide quali-tative and quantitative analyses of the various processesthat develop during the foam formation. Thus, we dividethe overall foaming-over process into well-differentiatedstages controlled by different physical mechanisms. Moreimportantly, we show experimental evidence supportingthe explanation given for each step of the outgassing pro-cess.

The chain of events that ultimately leads to thefoaming-up of beer is triggered by a sudden impact onthe top of the bottle, which generates a compression wavethat propagates through the glass towards the bottom aspredicted by the classical theory of impact on solids [8].When the wave reaches the base of the bottle, it is par-tially transmitted to the liquid as an expansion wave thattravels towards the free surface, where it bounces backas a compression wave. A simple model for the impactproblem [8] implies that the stiffer the bottle, the moreefficient is the transmission of the expansion wave to theliquid. Thus, the shock is more efficiently transmittedinto the liquid in the case of a glass container than in asofter bottle (e.g. plastic), although the expansion waveis still generated, albeit its smaller amplitude. The trainof waves transmitted to the liquid bounce back and forth

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several times until it damps out. Figure 1 shows snap-shots of the first instants after the impact to illustrate theeffects of the first expansion-compression cycle. It can beseen how bubbles start to expand first near the bottomwhereas, at approximately t ≈ 124µs, those located nearthe free surface begin to shrink. The train of rarefaction-compression waves drives the fragmentation of most ofthe existing gas pockets during the first wave cycles. Fig-ure 2 illustrates a typical time evolution of the bubble ra-dius measured in our experiments along with snapshotsshowing the bubble at different relevant instants of theexpansion-collapse process (see also Movies S1 and S2 inSupplementary Material). This first stage of the over-all foaming-up process lasts of the order of the acoustictime of the liquid volume, usually tac = 2H/c ≈ 0.2 ms,assuming a typical liquid height H ≈ 10 cm and a speedof sound around c ≈ 1000 m/s. Notice that, since wavespropagating in bubbly liquids are strongly damped, theintensity of successive rebounded waves decays rapidlythus they are less likely to cause bubble collapse.

Similarly to what happens in the generation of medicalultrasound contrast agents through sonication (ref. [9])or, albeit in a more violent way, in sonoluminescence [10],it seems reasonable to attribute the break-up of the bub-bles to a Rayleigh-Taylor instability [10]. The number offragments, N , resulting upon the break up of a bubblecannot be measured due to the high void fraction of theresulting bubble cloud (Fig. 2d). Instead, an estimationof this number is obtained using the model of Brennen[11], based on the ideas put forward by several authors[12, 13]. Following this model (see Supplementary Mate-rial), the most unstable mode is given by

nm =1

3

((7 + 3Γm)

1/2 − 2), (1)

with Γm = ρR2R/σ, evaluated at the instant when theradius, R, is minimum, ρ the fluid density and σ theliquid-gas surface tension. The size of the fragments isexpected to be of the order of R/nm, thus the number offragments generated is N ≈ n3m. For the typical bubblesizes and pressure wave amplitudes used in these exper-iments, we find a most unstable mode of the order ofnm ≈ 102 and a number of fragments N ≈ 106.

As a consequence of the fast bubble collapse and break-up, right after the implosion the total gas-liquid interfa-cial area increases by a factor of the order of N1/3. Thissudden increase of the interfacial area leads to a secondstage where the clouds of bubble fragments grow rapidlyas a result of the diffusion of carbonic gas into the newlycreated cavities. This stage can be modelled using theclassical theory of bubble growth in supersaturated me-dia (Ref. [14]). Under the reasonable assumption thatthe cloud grows as the sum of its components, this theorystates that the cloud size, Lc, follows

Lc = L0 + αN1/3F

(∆C

ρg

)√κt

π, (2)

where ∆C is the difference between the concentration of

t = 0 µs t = 31 µs t = 62 µs

t = 93 µs t = 124 µs t = 155 µs

t = 0 µs t = 31 µs t = 62 µs

t = 93 µs t = 124 µs t = 155 µs

FIG. 1. Sequence of images corresponding to the instantsright after the pressure wave starts to propagate through thebeer. To ensure a continuous bubble cloud, a metal disk wasplaced at the bottom of the bottle in this experiment, whateffectively introduces a large number of nucleation sites. Scalebar: 10 mm.

carbonic gas in the bulk liquid and the saturation value, κits diffusivity, ρg the density of the gas inside the bubbles,α a dimensionless constant and F (x) a known function(Supplementary Material). Taking the estimated num-ber of fragments generated during the collapse of a sin-gle bubble, N ≈ 106, we expect the radius of the bubblecloud to grow about 100 times faster than a single bubblewith the same volume than the cloud. In fact, this mag-nitude represents an upper bound, since those fragmentsat the center of the cloud will grow more slowly, due totheir limited access to CO2. Initially, the growth ratescales roughly as t1/2 albeit exhibiting some oscillationscaused by cycles of expansions and compressions that arenot yet attenuated (see the stage labelled as “diffusion-driven” in Fig. 3, blue squares in the upper panel). Thisdiffusion-driven stage ends when carbon dioxide is locallydepleted and thus the cloud’s size significantly moderatesits growth.

3

0.00 0.05 0.10 0.15 0.20 0.25

Time [ms]

0.0

0.1

0.2

0.3

0.4

0.5

Rad

ius

[mm

]

(a)

(b)

(c)

(d)

SimulationExperiment

a b c d

FIG. 2. A bubble of initial radius R0 ≈ 180 µm, induced by alaser pulse nearly a second before the impact, grows after thepassage of the expansion wave reaching the bubble location atinstant (a). After the instant of maximum radius (b), the bub-ble collapses at some point between frame (c) and the previousone, turning into a bubble cloud. Panel (d) shows the bubblecloud 0.1 ms after instant (c). The Rayleigh-Plesset equationhas been integrated numerically (red solid line, see Supple-mentary Material) for a bubble subjected to a pressure pulsewith an amplitude, pA = 100 kPa, and period, T = 0.24 ms,measured at the bubble’s location with a hydrophone. Noticethat, after the implosion, t ≈ 0.22 ms, the Rayleigh-Plessetequation no longer describes the behavior of the bubble sinceit is only valid for a single bubble, not for a bubble cloud.(see Movies S1 and S2).

In the example of figure 3, this occurs at about t ≈ 10ms (stage labelled as “depletion”). To avoid the noiseintroduced by the acoustic waves at short times we haveperformed additional experiments by focusing a high-energy laser pulse inside the liquid to trigger the for-mation of a bubble cloud by laser-induced cavitation [15](see Supplementary Material). In the focal region, thelaser generates a dense bubble cloud that initially growsas the square root of time, which is consistent with apurely diffusive growth (Fig. 3, red squares in the upperpanel).

The rapidly growing bubble clusters act as buoyancysources that lead to the formation of bubble-laden buoy-

ant vortex rings in time scales of order tg ∼ (L/g)1/2

(Fig. 3c), very much like a localized release of heat formsa thermal [16]. As the vortices rise through the liquid,the advection due to their self-induced velocity and themixing caused by their vortical motion contribute to en-hance the transport of CO2 to the bubbles.

10�1 100 101 102 103

10�1

100

Clo

udsi

ze,L

c�

L0

[mm

]

� t1/2

� t2

diffusion-driven depletion buoyancy-driven

10�1 100 101 102 103

Time [ms]

10�1

100

101

Clo

udto

phe

ight

,Y�

Y0

[mm

]

� t

(a) (b) (c)

a b c

FIG. 3. Upper plot: Time evolution of the size of a bubblecloud, Lc, after the shock-induced collapse of an already exist-ing bubble (blue) and after a laser-induced bubble implosion(red). Three different stages have been marked qualitatively:diffusion driven, depletion and buoyancy-driven. Black solidlines depict the scaling laws Lc ∼ t1/2 (diffusion-driven) andLc ∼ t2 (buoyancy-driven). Lower plot: Time evolution ofthe top of the bubble cloud corresponding to the same casesof the upper plot. Notice how, at long times, the plume ap-proaches a steady rising velocity. Letters denote the instantscorresponding to the images in the lower pannels (scale bar:1 mm). (See movie S3)

In turn, this results in a growth-rate faster than thatfound for pure diffusion, namely t1/2. Indeed, the cloud’ssize grows roughly as t2 during this stage (Fig. 3, upperplot). Moreover, as a consequence of the continuous gen-eration of gas volume inside the vortex, the velocity ap-proaches a constant value (Fig. 3, lower plot) instead of

4

decreasing as happens in buoyant vortex rings induced bythe release of a fixed amount of buoyancy. For instance,in thermals originated by the sudden release of a fixedamount of heat, the velocity decays as t−1/2 due to theentrainment of colder fluid [16, 17].

Conversely, in the bubble-laden plumes studied here,the feedback between buoyancy-driven rising motion andgas-volume generation results in a nearly constant speed.This behavior is similar to that found in the the so-called autocatalytic vortex rings or plumes [18] wherebuoyancy is continuously produced by a chemical reac-tion that yields products less dense than the reactants.These buoyancy-driven chemically-reacting flows appear,for instance, in the combustion of flame balls in micro-gravity conditions. Interestingly, they are also relevant insome explosion scenarios for Type Ia supernovae [19, 20].

The analogy between the bubble-laden plumes ob-served here and the autocatalytic plumes described inthe literature extends also to their morphology. Panel cin Fig. 3 shows one of these bubble-laden plumes when itis well developed (see also Movie S3). The plume consistsof a vortex with a nearly spherical cap with a thin conduitthat ascends more slowly, features observed in the plumesdriven by autocatalytic chemical reactions [18, 20].

It should be pointed out that, among all the stages ofthe foaming-up process, this one is the most effective interms of the amount of liquid outgassing as a result ofits self-accelerating nature. This stage starts at times ofthe order of tens of milliseconds and concludes when theplumes reach the size of the liquid volume, usually of theorder of a second.

Remarkably, the behavior of the bubble-laden vortexrings during the diffusion-driven and buoyancy-drivenstages is independent of the mechanism used to gener-ate the initial bubble cloud. Figure 3 depicts the evolu-tion of the bubble cluster size and velocity of a bubblecluster originated by laser-induced cavitation. The sizeand velocity follow the same scaling laws as the vortexcreated by the pressure-induced bubble implosion. Thissuggests that similar explosive CO2 outgassing processesdriven by the formation of these bubbly plumes may be

initiated by other physical mechanisms generating densebubble clouds such as the introduction of new bubblenucleation sites [21, 24] or a sudden change on the satu-ration conditions occurring either globally [4] or locally[2, 24]. In fact, our observations suggest that, once theplume is initiated, its dynamics do not seem to dependon the particular initiation mechanism. Thus, one of themain conclusions of this study is that the dynamics ofthese bubble-laden self-accelerating plumes moving in su-persaturated media may partly explain the explosive be-havior of systems such as limnic and explosive volcaniceruptions where current models typically neglect the roleof these autocatalytic structures [1, 3, 4, 6, 24].

Finally, two side effects induced by the developmentof the bubbly plumes must be mentioned here attend-ing to their relevance in the global degassing process inthe case of the bottle. Firstly, due to the finite size ofthe container, a global recirculating motion is generatedthat drags bubbles from near the free surface deep intothe bulk liquid, thus increasing their residence time inthe flow and allowing them to grow for longer times.Secondly, the flow induced inside the bottle speeds-upalso the growth of gas cavities attached to the walls[22, 23] through the enhancement in the transport of car-bon dioxide towards these cavities, that otherwise wouldonly grow by diffusion. A very similar effect is probablybehind the long time scales involved in limnic eruptions.In these phenomena, bubbly plumes form that keep en-training CO2-saturated water from the bottom of the lakeuntil it is almost depleted of this gas due to the globaloverturning flow that they induce in the lake [6]. Alto-gether, the chain of effects described in this letter leads tothe fast appearance of foam that has granted beer tappingits popularity.

We would like to thank Dr. Maylis Landeau for heruseful comments on the potential applications of thepresent study. The authors are indebted to Profs. Nor-man Riley, Jose Manuel Gordillo, Alejandro Sevilla andCarlos Martınez-Bazan for their insightful comments.We acknowledge the support of the Spanish Ministry ofEconomy and Competitiveness through grant DPI2011-28356-C03-02.

[1] Zhang, Y. Z. and Kling, G. W. Dynamics of lake erup-tions and possible ocean eruptions. Annu. Rev. EarthPlanet. Sci., 34, 293-324 (2006)

[2] Zhang, Y. Dynamics of CO2-driven lake eruptions. Na-ture 379, 57-59 (1996).

[3] Cashman, K. V. and Sparks, R. S. J. How volcanoeswork: a 25 year perspective. Geol Soc. Am. Bull. 125,664-690 (2013)

[4] Mangan, M., Mastin, L. and Sisson, T. Gas evolu-tion in eruptive conduits: combining insights from hightemperature and pressure decompression experimentswith steady-state flow modeling. J. Volcanol. Geother-mal Res., 129, 23-36 (2004)

[5] Liger-Belair, G. et al. Proc. Natl. Acad. Sci. USA 106,16545-16549 (2009).

[6] Mott RW and Woods AW. A model of overturn of CO2

lakes triggered by bottom mixing. J. Volcanol. Geother-mal Res. 192 151-158 (2010).

[7] Brennen, C. E., Cavitation and bubble dynamics. OxfordUniversity Press, Oxford (1995).

[8] Goldsmith, W., Impact: The Theory and Physical Be-haviour of Colliding Solids. Dover Publications, NewYork (2001).

[9] Feinstein, S. B. et al. Two-dimensional contrast echocar-diography. I. In vitro development and quantitative anal-ysis of echo contrast agents. J. Am. Coll. Cardiol. 3, 14-

5

20 (1984).[10] Brenner, M. P., Hilgenfeldt, S. & Lohse D. Single-bubble

sonoluminiscence. Rev. Mod. Phys. 74, 425-484 (2002).[11] Brennen, C. E. Fission of collapsing cavitation bubbles.

J. Fluid Mech. 472, 153-166 (2002).[12] Prosperetti, A & Seminara, G. Linear stability of a grow-

ing or collapsing bubble in a slightly viscous liquid. Phys.Fluids 21, 1465-1470 (1978).

[13] Ioss, P. L. & Rossi, M. Stability of a compressed gas bub-ble in a viscous fluid. Phys. Fluids A 1, 915-923 (1989).

[14] Epstein, P. S. & Plesset, M. S. On the stability of gasbubbles in liquid-gas solutions. J. Chem. Phys. 18, 1505-1509 (1950).

[15] Ohl, C. D., Linday, O. & Lauterborn, W. Luminescencefrom spherically and aspherically collapsing laser inducedbubbles. Phys. Rev. Lett. 80, 393-396 (1998).

[16] Turner, J. S. Buoyant vortex rings. Proc. R. Soc. Lond.A 239, 61-75 (1957).

[17] Morton, B. R. Weak thermal vortex rings. J. Fluid Mech.9, 107-118 (1960).

[18] Rogers, M. C. & Morris, S. W. Buoyant Plumes and Vor-tex Rings in an Autocatalytic Chemical Reaction. Phys.Rev. Lett. 95, 024505 (2005).

[19] Vladimirova, N. Model flames in the Boussinesq limit:Rising bubbles. Comb. Theor. Modelling 11, 377-400(2007).

[20] Rogers, M. C. & Morris, S. W. The heads and tails ofbuoyant autocatalytic balls. Chaos 22, 037110 (2012).

[21] Coffey, T. S. Diet coke and mentos: what is really behindthis physical reaction? Am. J. Phys. 76, 551-557 (2008)

[22] Jones, S. F., Evans, G. M. & Galvin, K. P. Bubble nu-cleation from gas cavities - a review. Adv. Colloid Interf.Sci. 80, 27-50 (1999).

[23] Liger-Belair, G., Voisin, C. & Jeandet, P. Modeling non-classical heterogeneous bubble nucleation from cellulosefibers: application to bubbling in carbonated beverages.J. Phys. Chem. B 109, 14573-14580 (2005).

[24] Woods, A. W. Turbulent Plumes in Nature. Annu. Rev.Fluid Mech. 42, 391-412 (2010).

6

SUPPLEMENTARY MATERIAL

Experimental techniques:

In a first set of experiments, the pressure fluctuationsinduced in the liquid upon the impact were recorded witha hydrophone. In these experiments, the bottle was filledwith deionized water, to delay the formation of cavita-tion bubbles at the hydrophone’s surface. In a secondset of experiments, actual beer was used (CO2 concen-tration: 5.0±0.4 g/l, measurement provided by the man-ufacturer). The evolution of bubbles existing in the liquidpreviously to the impact was recorded with a high-speedcamera. To ensure that a bubble is present at the cam-era’s focal point, bubbles were generated by focusing alow-energy YaG laser pulse inside the bulk liquid witha convergent lens before the impact. In this way, weavoid the variability in the location and size of the bub-ble formed upon the impact due to the randomness of thedistribution of nucleation sites found in all liquids [7]. Inthese two experimental sets, the bottle was impacted bydropping a brass weight (90 g.) in a controlled and re-peatable manner from a height of 25 mm. (see Fig. 4).Moreover, the bottle is held by the neck using a rigidclamp that prevents the net motion of the bottle afterthe impact and keeps the bottom above the ground, thuspreventing the partial transmission of the wave to thebench.

Rail

Beer Bubble

Clamp

Weight

Pole

Lens

Laser pulse

FIG. 4. Sketch of the experimental setup. The bottle is heldin place with a clamp around its neck that prevents its motionafter the impact. To ensure a good repeatability, the weightfalls along a rail that guarantees that its lower side remainshorizontal during the falling. The YaG laser pulse used togenerate both the individual bubbles (low-energy pulse) andthe bubble clouds (high-energy pulse) is also depicted.

Finally, a third kind of experiment was performedwhere, instead of hitting the bottle mechanically, ahigh-energy laser pulse was focused inside the beer toinduce cavitation, similarly as described in [15], thuscreating a bubble cloud that serves as a seed to eventu-ally trigger the formation of a bubble-laden plume. Inall the sets, the projected area of the bubble cluster, A,was measured using custom-made software. The clustersize, Lc, was then calculated as the diameter of a circlewith the same area, namely Lc =

√4A/π.

Estimation of the number of bubble fragments:Upon bubble collapse, the high void fraction of the result-ing cloud of fragments precludes the application of anymeasurement technique to directly determine the numberof those fragments. To get at least an order of magnitudeestimation of that number, we apply the model proposedby Brennen [11]. Briefly, this model uses the equation forthe amplitude, a(t), of a spherical harmonic perturbationof order n > 1,

a+3

RR a−

[(n− 1)

R

R− (n− 1)(n+ 1)(n+ 2)

σ

ρR3

]a = 0,

(3)where ρ is the liquid density and σ the liquid-gas sur-face tension. Notice that, to determine the evolution ofthe perturbations, the time-history of the bubble radius,R(t), is needed. As a further simplification, since thegrowth of the surface instabilities is very fast comparedto the bubble radial dynamics, both the radius, R, and itsacceleration, R, are considered constant during the finalinstants of the break-up process. Thus, it is now possi-ble to calculate the fastest-growing mode, nm, i.e. thatmaximizes the expression in brackets, [...], in equation 3:

nm =1

3

((7 + 3Γm)

1/2 − 2), (4)

where Γm = ρR2R/σ, evaluated using the minimumbubble radius, Rmin and the acceleration at the time atwhich that radius is reached. Consequently, Brennen’smodel allows us to compute the fastest-growing modeusing only values of the radius and its acceleration atthe collapse time.

Nonetheless, the short duration of the collapse stage im-pedes recording the process with enough time resolutionto estimate either the minimum bubble radius or theacceleration during this stage (see Figs. 2c and 2d).Therefore, to obtain these parameters to apply Bren-nen’s model [11], the evolution of the bubble radius hasbeen computed numerically by integrating the Rayleigh-Plesset equation:

ρRR+3

2ρR2−

(p0 +

2σ

R0

)(R0

R

)3γ

+

+p0 +2σ

R+

4µR

R= −pw(t),

(5)

7

where µ = 10−3 kg/(m·s) is the liquid viscosity, γ = 1.304is the heat capacity ratio of the gas (CO2) and pw(t)is the pressure pulse induced in the system, given bypw(t) = −pA sin (2πt/T ), with amplitude pA = 100 kPaand period T = 0.24 ms. These values were measuredwith a hydrophone at the bubble’s location. Moreover,the following values have been used: the CO2-water sur-face tension, σ = 0.0434 N/m, the water density, ρ = 103

kg/m3, and the ambient pressure, p0 = 105 Pa.

Figure 2 shows the good agreement between thenumerical and the experimental results before the breakup, justifying the usage of the numerical parametersto estimate the number of fragments resulting from it.For instance, from the experiment shown in figure 2, weobtain nm ≈ 144. This value ranges between 120-160 inall our experiments, with a maximum different betweenthe maximum radius calculated numerically and themeasurement always within 10% or less.

Modelling the growth rate of a bubble cloud: Toget an estimation of the growth rate of the bubble clouddue to the diffusion of CO2 before buoyancy becomesimportant (diffusion-stage), we assume that the cluster’svolume grows as the sum of the volume of its components.The time evolution of the radius of a fragment, Rf(t), isgiven be the Epstein-Plesset equation (Ref. [14]),

dRf

dt=κ∆C

ρg

(1

Rf+

1√πκt

), (6)

where κ is the diffusivity of CO2 in water, ρg the densityof the gas inside the bubble and ∆C the difference be-tween the concentration of CO2 far away from the cloudand that at the bubble’s surface. Neglecting the capil-lary pressure, the solution to the equation is well approx-imated by

Rf(t) ≈ Rf,0 + F

(∆C

ρg

)√κt

π, (7)

with Rf,0 the initial radius and F (x) =

x(

1 +√

1 + 2π/x)

(ref. [14]). Notice that the

second term in this expression corresponds to thesolution to Eq. (7) at long times. Here we have addedthe first term, the initial radius, to obtain an expressionthat yields small relative errors for both long and shorttimes. Assuming that the N fragments are equally sized,

the volume of the cloud can be written as

L3c = α3N

[Rf,0 + F

(∆C

ρg

)√κt

π

]3, (8)

where the dimensionless constant α accounts for the factthat the cluster contains some amount of liquid separat-ing the bubbles. Taking the cube root of the expression,and denoting the initial cluster size by L0 = αN1/3Rf,0,we get

Lc = L0 + αN1/3F

(∆C

ρg

)√κt

π. (9)

Finally, we would like to comment on the applicabilityof the Epstein-Plesset equation to this problem. Thisequation describes the growth of an isolated bubble,whereas in this case bubbles belong to a dense cluster.Therefore we expect that, at some point, the CO2 mustbe depleted, with only those bubbles closer to the clouds’surface being still able to grow. This is consistent withthe behavior observed in figure 3b, where the growthrate of the cluster decreases notably at about 10 ms.

MOVIE LABELS:

Movie S1. Bubble fragmentation 1: fragmenta-tion of a bubble as a result of the passage of the train ofexpansion-compression waves. This is the same bubbleshown in Fig. 2. Scale bar: 500 µm.

Movie S2. Bubble fragmentation 2: the fragmen-tation of a gas bubble is shown, together with a plot ofits instantaneous radius. Notice how, after the secondcompression, the bubble breaks-up and forms a cloud ofbubble fragments.

Movie S3. Bubble plume development: thismovie shows the evolution of a bubble cloud duringthe buoyancy-driven stage. Notice the large increase ofthe foam volume. This plume corresponds to the bluesymbols of Fig. 3. Scale bar: 1 mm.

Movie S4. Global view of the bottle after theimpact: the movie shows the development and growthof the bubble plumes after hitting the bottle. To makemore evident the process, the formation of bubbles hasbeen forced by placing a metal disk at the bottom ofthe bottle, what effectively introduces a large number ofnucleation sites at the bottom.