arXiv:cond-mat/0305126v1 [cond-mat.soft] 7 May 2003 ubeada niubecaatrzdb ne radius inner in- a arrows of by the characterized Comparison gas; antibubble (left) an a and with transformations. bubble contact possible in the co liquid dicate different a the of of representation figurations Schematic (right) 1: FIG. ε betwsfis eotdb uhsadHge 2,and unusual [2], This Hughes coined and thin Fig.1). Hughes thereafter a by (see was have reported liquid also first a was can enclosing object one gas that of known large However, film generally a [1]. not and subject is this established, it concerns well books text is of bubbles number of physics The iudt eoeabbl n opn ubebecomes bubble popping the a from and bubble extracted a be become to may liquid antiglobule an entities. fluid example, pos- different For in- different the two the between represent of transformations Fig.1 sible that composed in arrows are Note The antibubble terfaces. the antibubble. and pressures, called bubble different object the at symmetric gas its the has separating spherical liquid More- thin of a antiglobule). of layer (so-called constituted bubble liquid known a well the or in over globule) gas consid- (so-called of water is sphere of interface drop a a single either a have we If ered, interfaces. fluid spherical physics. but their and curiosity stability a their mainly about remain were questions antibubbles reports, these en h hcns ftemnrt fluid. minority the of thickness the being ubei hnfimo iudecoigagspocket. gas a enclosing liquid of film thin a is bubble A iue1smaie h ieetcngrtosof configurations different the summarizes 1 Figure Antiglobule ASnmes 01.55+b,82.70.Kj,82.70.Rr numbers: criticPACS respec the with explain evolution pressu to thickness critical allows interaction shell a layers to air surfactant related the depth describes small explai critical law a We A with depth. antibubbles globule. overweighting definite liquid By a stability. surrounding their shell air spherical epeeteprmna netgtoso niube.Su antibubbles. of investigations experimental present We Antibubble Bubble antibubble Globule niube vdne faciia pressure critical a of evidences : Antibubbles RS,Isiu ePyiu 5 nvri´ eLi`ege, de Universit´e B5, Physique de Institut GRASP, yCnet[] Since [3]. Connett by Antibubble R R Bubble ε .obl n N.Vandewalle and S.Dorbolo ε -00L`g,Belgium. Li`ege, B-4000 R n- a ; xeietlcl.W aeosre httearfimis film the air in the that liquid observed the have b´echer We and wire the cell. copper in experimental a liquid film, po- air the electrical thin any connects cm the prevent accross 5 to difference order to tential In zero The surface. from the surface. extends below the zone liquid below of production streamer antibubbles antibubble a a forming presents of feeds 2 when help that Figure formed the sink. are antibubbles With globules by the the and jet surface, cell. liquid the the the liq- on adjusting of liquid the of top clean globules keep the drop b´echer, to we at order is surface a in flow uid in pump continuous a placed a and with is and leaks maintained cell preventing water The for l tank (30 larger detergent). mixture dishwashing liquid ml a 245 with filled completely noa nilbl ilb xlie below. finally explained and be antibubble will an trans- antiglobule symmetric into an The into globule a liquid). of from drop formation small (a globule a metallic right. the The the of on liquid seen floats. the be and globule can liquid liquid vessel antibubbles. incoming a the several between surface, in- into connection The liquid b´echer. up a the breaks from At streamer antibubble an liquid of coming Formation 2: FIG. oe hseprmna eli 0 mtl n a a has and tall cm cm 200 10 x is 15 cell of experimental base This pose. lxgasvse a enseilybitfrorpur- our for built specially been has vessel plexiglass A ldphfratbbl stability. antibubble for depth al opesr.Ti a obndwith combined law This pressure. to t ehsbe on.AmdfidLaplace modified A found. been has re muto at hysn n o ta at pop and sink they salt, of amount o opouete n estudy we and them produce to how n ha nsa udojc sathin a is object fluid unusual an ch 2 h eli pno h o n is and top the on open is cell The . 2

25

20

15

10

# of poppings 5

0 0 50 100 150 200 FIG. 3: Antibubble located just below the liquid surface. The h (cm) diameter of the antibubble is 1.5 cm. Interference fringes simi- lar to Newton’s rings are emphasized. The dark circumference FIG. 4: Distribution of the critical depths for three experi- is due to some total reflections. mental sessions (circles, triangles and squares). Solid curves represent gaussian fits of the data. more stable when this wire is present. This point is im- 30 portant and will be discussed below. Antibubbles have been produced with diameters up to 1.6 cm. It was im- 20 possible to create larger antibubbles with our setup and 10 physical reasons for that upper limit are given below. The mechanism of antibubble production can be de- 0 composed as follows. A globule is first created at the sur- -10 face of the liquid. The incoming liquid flow which feeds |h-| (cm) the globule has a certain velocity and the globule sinks. -20 The flow forms a streamer inside the liquid. Rayleigh- -30 Plateau instabilities appears along this streamer (see Fig. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 2) [5]. The streamer breaks up and forms antibubbles. d (cm) Once formed, the antibubbles rise in the liquid. In- deed, they are dragged towards the surface by buoyant FIG. 5: Relative critical depths h− as a function of d forces. Since the air volume contained in an antibubble is their diameter . quite small, the motion is slow and the antibubbles keep a spherical shape. When reaching the surface, the antibub- bles touch the air/liquid interface where they stay for a a critical depth h=120 cm wathever their size and what- long time (a few minutes) before naturally popping due ever their falling velocity. Other experiments have been to some aging. Such an antibubble ‘glued’ just below the performed the next days. Figure 4 presents three exper- liquid surface is shown in Fig.3. By popping antibubbles iments performed at different days (circles, triangles and with a needle and by comparing the diameters of the an- squares); the preparation of the liquid and the antibub- tibubble and the resulting air spherical pocket (antiglob- ble production process had been the same. About 60 ule), we estimated the average thickness of the air film to antibubbles of different sizes have been sunk during each be ε = 3µm. This distance is quite small but similar to serie. The distribution of the critical depths has been re- the film thickness of bubbles. In the case of bubbles, the ported in Fig.4 and fitted by gaussian laws (solid lines). liquid film leads to interferences and subsequent horizon- From one day to another, the mean critical depth hhi tal colored fringes appear. Colored interference fringes changes, namely 120 cm, 116 cm, and 148 cm for the cir- can be seen in antibubbles as well (Fig.3). Their shape cles, triangles and squares respectively. Those variations similar to that found in Newton rings problem can be ob- have been attributed to the variations of the atmospheric served [4]. The thick black circumference is due to total pressure. Let us remind that a typical daily variation of reflections of light on the first diopter liquid-air shell. the atmospheric pressure (about 20 hPa) corresponds to In order to avoid floating antibubbles, we added a small a depth of 20 cm in water. Thus, the stability of an an- amount of salt in some antibubbles. Such ‘heavy antibub- tibubble seem to depend on the external pressure. bles’ reach rapidly a constant free falling velocity. The The limit speed of a free falling object in a fluid is largest antibubbles sink with a velocity around 10 cm/s mainly determined by its diameter d and to the viscos- while the smallest ones (diameter 0.5 cm) fall at 2 cm/s. ity. One can wonder whether a relation exists between In the first series of experiments, we surprisingly observed the antibubble diameter and its critical depth. In order that the great majority of the antibubbles breaks around to better compare all series of data, the relative critical 3

30 d < 0.85 cm 25 d > 0.85 cm

20

15

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# of antibubbles 5

0 -30 -20 -10 0 10 20 30 |h-| (cm) FIG. 7: Sketch and picture of the antibubble lift. The an- FIG. 6: Distributions of relative critical depths reported for tibubble are trapped under the air bell. The pressure is in- big (d> 0.85 cm) and small (d< 0.85 cm) antibubbles. creased with an arbitrary speed by placing the lift at a definite depth in the column. depths defined as h − hhi are plotted versus the sizes of the antibbuble in Fig.5. The distribution of the results is cause the compressibility of the central globule is neglige- isotropic around the mean diameter of 0.85 cm. Further- able with respect to the compressibility of the spherical more, the distribution of relative critical depths is plotted air shell. That means that the Laplace law has to be for the large (> 0.85 cm) and the small (< 0.85 cm) an- modified for antibubbles. Let R being the radius of the tibubbles in Fig.6. Gaussian distributions are centered liquid globule and ε being the thickness of the spherical around 0. Thus, the stability of an antibubble seems to air shell. The Laplace equations for both interfaces, i.e. be size independent. starting from the center liquid-air and then air-liquid, are An antibubble lift has been also built (Fig.7). This ∆p1 = 2γ/R and ∆p2 = 2γ/(R + ε) respectively. If one system is directly inspired from the diver bell. The an- considers any external pressure variation, ε becomes the tibubble is trapped under the bell and could be placed relevant parameter. Indeed, the only way to equilibrate at some definite depth. The advantage is twofold since both work and surface energy in an antibubble is to ex- no additional salt is needed and the speed is controlled pand or reduce the thickness ε of the air film while the whatever their size. The critical depth has been found to radius R of the central globule remains constant. be the same as before. Thus, the stability of an antibubble Only the possible pressure difference ∆p2 accross the seems to be independent of the free falling velocity. external air/liquid interface contributes to free energy The well-known Laplace law applies for bubbles and variations. In that case, antibubble equilibrium leading links the radius R of the bubble to the difference of air to a modified Laplace law pressure ∆p across the thin film of liquid. Let us con- 2γ sider a surface element dA of the soap film. Two forces ε = − R (2) ∆p2 apply on this surface. The first one is due to the differ- ence of pressure across the film and the second one comes The origin of this pressure is the curvature of the equiv- from the surface tension γ. As the bubble grow, both sur- alent antiglobule of radius R + ε. face and volume variations have to be taken into account. Assuming that the ideal gas law applies, pV is a con- One has a free energy variation dF = −(∆p)dV +2γdA. stant. The internal pressure p is given by the sum of The factor 2 in the surface term comes from both gas- three terms : (i) the atmospheric pressure p0, (ii) the liquid and liquid-gas interfaces. Bubble equilibrium reads hydrostatic pressure ρgh and (iii) the pressure ∆p due to dF = 0. When both volume and surface variations are the surfactant membrane given by Eq.(2). The relation expressed as a function of R and dR, the equilibrium between the pressure in the air shell and ε is thus given condition reads by 4γ R = (1) 2γ p0 + ρgh + ε = c (3) ∆p  R  that one can find in every physics textbook [5]. The 2 since the volume of the air shell is 4πR ε. The parameter Laplace equation is important because it contains the c is a constant. In a scope of comparison, the equation main ingredients which control the physics of a wide relative to bubbles reads range of liquid systems: from droplets to emulsions and foams [1]. 4γ 3 p0 + R = c (4) For an antibubble, the situation is quite different be-  R  4 where the relevant variable is the bubble radius R. mechanism is the following : an increase of the exter- Using Eq.(3), the variation of ε with depth can be esti- nal pressure decreases the thickness of the air shell, this mated. A 10% change of ε is found when the antibubble implies an increase of the Van der Waals interactions be- sinks to a depth of 150 cm in the column. This varia- tween the layers until this force becomes so high that the tion is enough to observe a displacement of interference air film breaks up. fringes during the free falling and is enough to cause the In summary, our experimental results support the popping at this depth. This calculation gives an order modified Laplace law that governs the air shell thick- of magnitude of the thickness variation of the air shell. ness. The popping of antibubbles is mainly controlled by Nevertheless, Eq.(2) surestimates ε. Indeed, an impor- ε and molecular interactions along the air film. As the tant effect has to be taken into account as far as two Bubble Antibubble layers are in interaction : the disjoining pressure pdis. This term should be added to the sum of the different Liquid pressure in the Eq.(3). In Fig.8, a soap film is represented on the left. The Liquid surfactant molecules are composed of a hydrophilic head and a hydrophobic tail. To form a film, the surfactant molecules take place along the liquid film with the head Liquid towards the liquid. During the film thinning, the elec- trostatic repulsion increases between hydrophobic heads. Liquid film Air film This force is counterbalanced by Van der Waals attrac- tive forces. This creates the so-called disjoining pressure FIG. 8: Molecular structure of a soap film as found in bubble that can be positive or negative according to the thick- (left) and of an air film from an antibubble (right). ness of the film [1, 6]. The relevant parameters are the thickness of the film, the nature of the hydrophobic heads (here sodium ion) and the medium between the layers. On the other hand, in Fig. 8 (right), an air film is pressure difference increases, ε decreases to a certain crit- represented. In this case, the tails of the surfactant ical value. This value correspond to the smallest possible molecules are in opposition along the air shell. That distance between two surfactant layers. Below this value, means that the interaction forces are different from that Van der Waals forces dominate and the air film collapses. of a soap film. For an air film, the Van der Waals inter- Our study is the basis for further experimental works on action between the tails of surfactant molecules becomes antibubbles, antifoams, and other exotic fluid entities. more important in front of the electrostatic contribution SD would like to thank FNRS for financial support. of the heads. Moreover, the medium enclosed between This work has been also supported by the contract ARC the layers is air. The dielectric constant χ of the air is 02/07-293. equal to 1, at least 80 times lower than in the soap liq- uid. No electrostatical screening effects are expected in the air shell since no electrolytes are present there. This explains the high sensitivity of antibubbles to any dif- ference of potential between the liquid contained in the [1] D. Weaire and S. Hutzler, The physics of foams, (Claren- globule and the surrounding liquid. don Press, Oxford, 1999). 129 The stability of the antibubble is given by the balance [2] W. Hughes and A.R. Hughes, Nature , 59 (1932). 3 [3] C.L. Stong, Scientific American, 230, 116-120 (April between Van der Waals attractive forces (in 1/ε [6]) and 2 1974). electrostatic repulsion forces (in 1/(R + ε) ). The inter- [4] I. Newton, Optics, (Smith & Walford, London, 1704). action between both surfactant layers is attractive below [5] P.G. de Gennes, F. Brochard-Wyart and D. Qu´er´e, a certain value of the interlayer distance, i.e. a critical Gouttes, bulles, perles et ondes, (Belin, Paris, 2002). thickness εc. Below εc, the equilibrium cannot be main- [6] V. Bergeron, Langmuir 13, 3474 (1997); R. Klitzing and tained anymore and the air film collapses. The popping H. M¨ohwald, Langmuir 11, 3554 (1995).