A gas bubble trapped in water by an oscillating acoustic field is expected to
either shrink or grow on a diffusive timescale, depending on the forcing
strength and the bubble size. At high ambient gas concentration this has long
been observed in experiments. However, recent sonoluminescence experiments show
that in certain circumstances when the ambient gas concentration is low the
bubble can be stable for days. This paper presents mechanisms leading to
stability which predict parameter dependences in agreement with the
sonoluminescence experiments.Comment: 4 pages, 3 figures on request (2 as .ps files

B. P. Barber

D. F. Gaitan

D. Lohse

Detlef Lohse

E. J. Hinch

H. Frenzel

K. Weninger

L. A. Crum

L. D. Landau

L. Rayleigh

M. J. Moran

M. Plesset

Michael P. Brenner

P. Jarman

R. Hiller

R. Löfstedt

Rodolfo R. Rosales

S. Putterman

Sascha Hilgenfeldt

V. Kamath

W. Lauterborn

English

arXiv

A gas bubble trapped in water by an oscillating acoustic field is expected to either shrink or grow on a diffusive time scale, depending on the forcing strength and the bubble size. At high ambient gas concentration this has long been observed. However, recent sonoluminescence experiments show that when the ambient gas concentration is low the bubble can be stable for days. This paper discusses mechanisms leading to stability

Brenner, Michael P.

Lohse, Detlef

Oxtoby, David

Dupont, Todd F.

University of Twente Research Information

VOLUME 76, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 FEBRUARY 199621390637rowt gasw thatusses1158Mechanisms for Stable Single Bubble SonoluminescenceMichael P. Brenner,1 Detlef Lohse,2,3 David Oxtoby,4 and Todd F. Dupont31Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02Fachbereich Physik der Universität Marburg, Renthof 6, 35032 Marburg, Germany3Department of Mathematics, The University of Chicago, Chicago, Illinois 606374Department of Chemistry and James Franck Institute, The University of Chicago, Chicago, Illinois 6(Received 11 October 1995)A gas bubble trapped in water by an oscillating acoustic field is expected to either shrink or gon a diffusive time scale, depending on the forcing strength and the bubble size. At high ambienconcentration this has long been observed. However, recent sonoluminescence experiments showhen the ambient gas concentration is low the bubble can be stable for days. This paper discmechanisms leading to stability.PACS numbers: 78.60.Mq, 42.65.Re, 43.25.+y, 47.40.Nmeshaaesbecdotercivi-]nletri-i-s aLle,yae-teshintal,w.l-toL.oryforal-ebleRecent experiments on sonoluminescence (SL) [1–allow detailed studies of the dynamics of a bubble levitatin a periodic acoustic field. Besides the light emissioitself, one of the greatest mysteries is how the bubbcan exist in a stable state for many billions of cycleMeasurements of the time between successive light flasshow that the total mass of the bubble remains constto high accuracy [1,2,7]. This result contradicts classicnotions about the dynamics of periodically forced bubbleAn unforced bubble of ambient radiusR0 dissolves overa diffusive time scale,t , r0R20yDsc0 2 c`d [8], wherer0 is the ambient gas density in the bubble,D is thediffusion constant of the gas in the liquid,c0 is thesaturated concentration of the gas in the liquid, andc` isthe concentration of gas in the liquid far from the bubblA strongly forced bubble grows by rectified diffusion, afirst discovered by Blake [9,11]. This is because whethe bubble radius is large, the gas pressure in the bubis small, resulting in a strong mass flux into the bubblConversely, when the bubble radius is small there isstrong mass outflux. Since the diffusive time scale is mularger than the short time the bubble spends at small ragas cannot escape from the bubble during compressithe net effect is bubble growth. At a special value of thambient radiusRp0 rectified diffusion and normal diffusionbalance. However, the above arguments suggest thatequilibrium point is unstable; if the ambient radius isinfinitesimally different fromRp0 , the bubble is pushedaway from equilibrium. The classic papers on rectifiediffusion (see, e.g., Eller and Crum [10–12]) verified thqualitative picture described above whenc`yc0 ø 1.There are two controlled parameters in the SL expements: the forcing pressurePa and the gas concentrationc`. The key to the discovery ofstablesingle bubble SL(whose existence completely contradicts the above snario) by Gaitanet al. [1] was that (i) c`yc0 , 1, and(ii) Pa must lie between a lower critical pressure and aupper critical pressure both of which depend onc`. Twodifferent types of stable SL exist: In the first, the amb0031-9007y96y76(7)y1158(4)$06.007]dnle.esntls:.nle.ahii,n;ehisdi-e-n-ent radius remains constant for billions of cycles, as edenced by the constant phasef of the light emission rela-tive to the oscillatory forcing. Barber and Putterman [2showed that the “jiggle” in the phase differs by less tha50 psec from cycle to cycle. In the second type of stabSL, the bubble also persists for long periods althoughf(and henceR0) varies on a diffusive time scale [1,5]. Theambient radius grows until the bubble becomes paramecally unstable [5,13] and microbubbles pinch off. Experments [5] show that this cycle can repeat indefinitely.Transitions between the types of stable SL occur afunction of the gas concentrationc` as well as the forcingpressure. For pure argon bubbles whenc`yc0 is betweenapproximately0.06 and 0.25, f oscillates on a diffusivetime scale. At lower argon concentrationc`yc0 ­ 0.004the phase becomes perfectly stable [5].Many of the dynamical phenomena exhibited by the Sexperiments, including the existence of a stable bubbmight occur independently of the light emission. To studthis question, we analyze the stability of a bubble withsimple model for the dynamics: although the model nglects many effects important for SL (including the lighemission), it exhibits the qualitative features of both typof stable SL. Whenc`yc0 is decreased at high enougforcing pressures, the classical unstable equilibrium poRp0 undergoes a bifurcation and stabilizes [14]. In generthere can beseveralstable equilibria, although far fromequilibrium small bubbles shrink and large bubbles groFor higher Pa the window of stability closes, and thebubble can only survive through rectified diffusion folowed by parametric instability. These results applyany bubble oscillating at small enoughc`yc0, regardlessof whether the forcing is strong enough to produce SOnce the bubble enters the SL regime the simple themight break down. For example, Löfstadtet al. suggestthat nondiffusive effects [7] are necessary to accountstable air bubbles at strong forcing pressures. Our cculations of a dynamical model for specific nondiffusiveffects show that they can indeed stabilize an unsta© 1996 The American Physical SocietyVOLUME 76, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 FEBRUARY 1996ttfiuieetraele13].setre)aienteothdfortstheand0]:rwoeequilibrium. The calculations suggest an experimentest to determine whether diffusive or nondiffusive effecdominate the SL experiments.We first set up a formalism for studying the stability othe equilibrium point, following Fyrillas and Szeri [15] andLöfstedtet al. [7]. Let csr , td denote the concentration ofgas dissolved in the liquid a distancer from the center ofthe bubble. For . Rstd, whereRstd is the radius of thebubble,c satisfies a convection diffusion equation≠tc 1R2 ÙRr2≠rc ­ D=2c . (1)The boundary conditions are given by Henry’s lawcsR, td ­ c0PsR, tdyP0 and bycs`, td ­ c`. The concen-tration gradient at the boundary gives the mass lossygainof the bubble ÙM ­ 4pR2D≠rcjRstd.These equations determine the growth of the bubble afunction of time. There are two crucial observations: FirsEller noted that changing coordinates toh ­ sr3 2 R3dy3andt ­Rt R4 dt transfers Eq. (1) to the simpler form≠tc ­ D≠h∑µ1 13hR3∂4y3≠hc∏­ 0 . (2)For the following it is convenient to define thet averageof a functionfstd by k fstdlt ­RfstdRstd4 dtyRRstd4 dt.The second observation [7,15] is that the bubble radchanges over a much faster time scale than the ambradius. Averaging Eq. (2) over the fast time scale givthe dynamics of the ambient radius,r0R20dR0dt­ Dc` 2 kcsRstd, R0stddltR`0 dhyk1 1 3hyR3lt. (3)Equilibrium points satisfykpltP0­c`c0. (4)The equilibrium is stable if the quantityb ­ dkpltydR0is positive.Now we proceed to analyze this model. We calculanumericallykplt as a function ofR0, for different drivingpressures, byt averaging solutionsRstd of the Rayleigh-Plesset (RP) equation. The RP equation [6,16,17] govethe dynamics of an acoustically forced bubble, andgiven byRR̈ 132ÙR2 ­1rwfpsR, td 2 Pstd 2 P0g 1Rrwcw3ddtfpsR, td 2 Pstdg 2 4nÙRR22srwR.(5)We use parameters corresponding to [3,6,18] anbubble in water: the surface tension of the air-watinterface iss ­ 0.073 kgys2, while water has viscosityn ­ 1026 m2ys, densityrw ­ 1000 kgym3, and speed ofsoundcw ­ 1481 mys. The acoustic field is driven viaPstd ­ Pa cossvtd with vy2p ­ 26.4 kHz and externalalss at,sntsensisirrpressureP0 ­ 1 atm. The pressure inside the bubbvaries adiabatically likepsRd , sR3 2 a3d21.4. Herea ­ R0y8.73 is the hard core van der Waals radius.Figure 1 showskpltyP0 for several values ofPa. Forsmall Pa, kplt monotonically decays withR0, signalinga diffusively unstableequilibrium. For example, whenc`yc0 ­ 1 with a forcing amplitude ofPa ­ 0.8 the unsta-ble equilibrium occurs atR0 ø 5 mm. Note that at largeR0 the bubble becomes unstable to shape oscillations [At large Pa, however,kplt develops oscillations as afunction of R0, so for a range ofc`yc0 there areseveralstable equilibrium points. As an example, see the inof Fig. 1: whenc`yc0 ­ 1022 andPa ­ 1.25 atm, thereare stable equilibria (denoted by small dots in the figuat R0 ­ 6.5, 6.8, 7.1, 7.5, 8.0, and 8.5 mm. To furtherverify the existence ofmultiple stable equilibria, we havesolved the full equations (1) and (5) numerically withstandard finite difference scheme. Indeed, the ambradius saturates atdifferent values, depending on thinitial bubble size. These equilibria are approached bfrom above and from below. Details will be publisheelsewhere [19].We now outline the predictions of these calculationsexperiments. A common protocol in the SL experimen[1,3] is to slowly increase the driving pressurePa. Theinitial ambient radius depends on the preparation ofbubble. At low pressures the bubble shrinks, sincec` ,kclt . As the forcing pressure is increased, there iscritical pressure where equilibrium points (both stable aunstable) appear in the parametrically stable region [2calculations forc`yc0 ­ 0.25 indicate this occurs nea1 atm. Above this forcing pressure, the bubble can folloFIG. 1. kpltyP0 as a function ofR0 (in mm) for Pa ­ 0.8,0.9, 1, 1.1, 1.2, and1.25 atm, top to bottom. Equilibrium cor-responds tokpltyP0 ­ c`yc0. The equilibrium is diffusivelystable if the slopeb ­ dkpltydR0 is positive. Inset: An en-largement ofPa ­ 1.25 atm. The straight line corresponds tc`yc0 ­ 1022. The intersection of the straight line with thcurve corresponds to equilibrium points. Whenb . 0 (thesolid dots in the figure) the equilibrium is stable.1159VOLUME 76, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 FEBRUARY 1996di,tanitnwu.edris]ta)leove.ivtobdtoiobth(iodesebthioyinessi-ent-de-og-irthe-deleryhct]:urehe.ein-lera-u-ansisctsgestinoflbyden:tothree different scenarios, depending on its ambient raR0: (a) If R0 is smaller than all equilibrium radii, thebubble shrinks. (b) IfR0 is near a stable equilibriumthe bubble is attracted to it and thus maintains a consambient radius thereafter. (c) IfR0 is larger than all theequilibrium radii, the bubble grows by rectified diffusioThe particular scenario that occurs depends on the inR0 and so could vary from experiment to experimeAlso note that the system is hysteretic: the sequencestates occurring when the forcing pressure increasesnot, in general, be repeated when the forcing pressdecreases. In the experiments of Barberet al. [3], thebubble initially follows (c), growing by rectified diffusionWhen the ambient radius becomes large the bubblparametrically unstable; the bubble can decrease its raby pinching off a microbubble. As the forcing pressuis further increased, the bubble continues diffusive growfollowed by microbubble pinchoff; the ambient radiusthus controlled by the parametric instability line [13Eventually, for even largerPa, microbubble pinchoff issevere enough to place the ambient radius near a sequilibrium, after which the bubble follows scenario (bNote that this transition from a “jiggling” bubble to a stabbubble should always be accompanied by a discontinujump in the ambient radius. Such jumps are obserby Barberet al. [3] for air bubbles near the onset of SLForc`yc0 ­ 0.25 the stable state persists untilPa ø 1.15;abovePa ­ 1.15 within the simple model the equilibriumpoint destabilizes. The bubble must return to diffusgrowth followed by microbubble pinching to survive. Aeven largerPa all parametrically stable bubbles shrink, sthe continued existence of a stable bubble is impossiThe sequence of events predicted by the simplified mois qualitatively similar to those in the SL experiments.The stable equilibrium points are ultimately dueoscillations inkplt as a function ofR0, which arise fromresonances in the Rayleigh-Plesset equation. Oscillateven occur in the maximum radius as a function ofR0,so that in some situations adding more gas to a bubdecreases its maximum size. A comparison withMathieu equation is instructive: If the eigenfrequencythe RP equation this depends onR0) is an integer or halfinteger fraction of the forcing frequency, the amplitudethe oscillations is anomalously large. A detailed studyin progress [19].Quantitative agreement between the simple moand the SL experiments requires accounting for sevneglected effects. These include realistic heat tran[6,21,22] and equations of state for the gas [23], as was spatial variations of the pressure within the bub[23,24]. To illustrate the dependence of solutions ofRP equation on material parameters, consider a3.3 mmbubble at forcing pressurePa ­ 1.3 atm. Upon changingg ­ 1 (isothermal) tog ­ 1.4 (adiabatic) the maximumradiusRmax changes by 20%; setting the surface tensto zero changesRmax by 50%; changing the fluid viscosit1160usnt.ialt.ofillreisiuseth.ble.usdele.elnsleenfiselralferllleenfrom water to 0.07 cm2ysec changesRmax by 30%.Increasing the forcing pressure by 0.05 atm (the errorexperiments [3]) increasesRmax by 20%. The uncertaintyin the effective values of all the aforementioned quantitiduring SL translates into uncertainties in predicted potions for the stable equilibria. However, our qualitativconclusions are robust, occurring throughout the relevarange of parameter space.Multiple stable equilibria exist for a range of forcingpressures for anyc`yc0 ø 1; although these states genrally occur at high forcing, the bubble oscillations neenot be strong enough to produce light [25]. What paramter regime corresponds to SL in the simplified model? Nestablished criterion exists, though experiments [3,26] sugest that the collapse ratioRmaxyR0 is the relevant parame-ter. Forc`yc0 ­ 0.2, the stable equilibria have collapseratios RmaxyR0 ø 3; for c`yc0 ­ 0.004, RmaxyR0 ø 7.The formerc`yc0 corresponds to the strongest SL for abubbles; the latter corresponds to the new phase (i.e.,low c` phase) of SL recently discovered [5] for argon bubbles. The largest discrepancy between this simple moand SL experiments is its inability to explain why thstrongest light emission occurs at a much largerc`yc0 inair bubbles than argon bubbles.This discrepancy is the basic reason for Löfstadtet al.’sspeculation that “nondiffusive” effects may be necessato explain stable SL in air. We have verified througdynamical calculations that a simple nondiffusive effecan stabilize an otherwise unstable equilibrium point [27the increase of the mass diffusion constant [D in Eq. (3)]with temperature and pressure. When the bubble pressand temperature is high, the diffusion constant near tbubble wall islarger than the diffusion constant in the bulkliquid, leading to additional mass outflux from the bubbleThis increase in the interfacial diffusion constant can bstudied with a simple model: Whenever the pressureside the bubble exceeds a critical pressurepthres we discon-tinuously increase the diffusion constant near the bubbwall by a factorfthres. The diffusion constant in the bulkliquid remains constant, since high pressures and tempetures are localized near the bubble wall. Numerical simlations of the full equations [19] with and without thiseffect demonstrate that the unstable equilibrium point cbe stabilized for a wide range ofpthres andfthres. The po-sition of the equilibrium point is shifted to larger radii, apredicted by Löfstadtet al. [7].A central question for both theory and experimentdetermining the parameter ranges where diffusive effealone produce a stable bubble. The present results sugthat the qualitative features of the bubble dynamicsSL experiments also arise within classical theoriesbubble dynamics. Answering definitively whether noveeffects are needed to explain stable SL is complicatedmodeling uncertainties. The present calculations proviqualitative criteria to assist in answering this questioalthough both diffusive and nondiffusive effects can leadVOLUME 76, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 FEBRUARY 1996teahntecGenxua,ereridPov..th:tsu-seusctstable equilibria, only the diffusive effects lead to discreequilibria. In order to determine which effect dominatthe SL experiments, we suggest that experiments sefor the discretization of the ambient radius in both ligemitting and nonlight emitting bubbles.We thank S. Grossmann, S. Hilgenfeldt, and L. Kadaoff for helpful discussions. This work has been supporby the DOE, the MRSEC Program of the National ScienFoundation at the University of Chicago, and the DFthrough SFB 185.Note added.—During the review process, we becamaware of a conference proceeding [28] by Crum aCordry who reached similar conclusions about the eistence and possible importance of multiple stable eqlibrium points. These authors also presented preliminexperimental evidence for the discrete equilibria.[1] D. F. Gaitan, L. A. Crum, R. A. Roy, and C. C. ChurchJ. Acoust. Soc. Am.91, 3166 (1992).[2] B. P. Barber and S. J. Putterman, Nature (London)352,318 (1991).[3] B. P. Barberet al., Phys. Rev. Lett.72, 1380 (1994).[4] R. Hiller, K. Weninger, S. J. Putterman, and B. P. BarbScience266, 248 (1994).[5] B. P. Barber, K. Weninger, R. Löfstedt, and S. J. Puttman, Phys. Rev. Lett.74, 5276 (1995).[6] R. Löfstedt, B. P. Barber, and S. J. Putterman, Phys. FluA 5, 2911 (1993).[7] R. Löfstedt, K. Weninger, S. J. Putterman, and B.Barber, Phys. Rev. E51, 4400 (1995).[8] P. S. Epstein and M. S. Plesset, J. Chem. Phys.18, 1505(1950).[9] F. G. Blake, Harvard University Acoustic Research Labratory Technical Memorandum12, 1 (1949).[10] A. Eller, J. Acoust. Soc. Am.46, 1246 (1969).esrcht-ded-i-ry,-s.-[11] A. Eller and L. A. Crum, J. Acoust. Soc. Am. Suppl.47,762 (1970).[12] L. A. Crum, J. Acoust. Soc. Am.68, 203 (1980).[13] M. P. Brenner, D. Lohse, and T. F. Dupont, Phys. ReLett. 75, 954 (1995); A. I. Eller and L. A. Crum, J. AcoustSoc. Am. Suppl.47, 762 (1970).[14] The existence of stable diffusive equilibria in differenparameter regimes was previously noted by ChurcC. Church, J. Acoust. Soc. Am.83, 210 (1988).[15] M. M. Fyrillas and A. J. Szeri, J. Fluid Mech.277, 381(1994).[16] Lord Rayleigh, Philos. Mag.34, 94 (1917); M. Plesset,J. Appl. Mech.16, 277 (1949).[17] W. Lauterborn, J. Acoust. Soc. Am.59, 283 (1976);A. Prosperetti, Q. Appl. Math.34, 339 (1977).[18] B. P. Barber and S. J. Putterman, Phys. Rev. Lett.69, 3839(1992).[19] M. P. Brenneret al. (to be published).[20] We are grateful to S. Hilgenfeldt for incisive commenon the detailed nature of the bifurcations.[21] V. Vuong and A. J. Szeri, “Sonoluminescence and Diffsive Transport” (to be published).[22] M. J. Miksis and L. Ting, J. Acoust. Soc. Am.76, 897(1984).[23] W. Moss, D. Clarke, J. White, and D. Young, Phys. Fluid6, 2979 (1994).[24] H. P. Greenspan and A. Nadim, Phys. Fluids A5, 1065(1993); C. C. Wu and P. H. Roberts, Phys. Rev. Lett.70,3424 (1993); L. Kondic, J. I. Gersten, and C. Yuan (to bpublished).[25] We are grateful to Professor S. Putterman for makingaware of this extremely important point.[26] S. Putterman (private communication).[27] It should be emphasized that every nondiffusive effedoes not stabilize an unstable equilibrium point.[28] L. A. Crum and S. Cordry, inBubble Dynamics andInterface Phenomena,edited by J. Blakeet al. (KluwerAcademic Publishers, Dordrecht, 1994), p. 287.1161

Mechanisms for stable sonoluminescence

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Single bubble sonoluminescence is understood in terms of a shock focusing towards the bubble center. We present a mechanism for significantly enhancing the effect of shock focusing, arising from the storage of energy in the acoustic modes of the gas. The modes with strongest coupling are not spherically symmetric. The storage of acoustic energy gives a framework for understanding how light intensities depend so strongly on ambient gases and liquids and suggests that the light intensities of successive flashes are highly correlate

Mechanisms for Stable Sonoluminescence

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