single-bubble sonoluminescence · pressure, and light intensity (top to bottom) during this fig. 1....

REVIEWS OF MODERN PHYSICS, VOLUME 74, APRIL 2002

Single-bubble sonoluminescence

Michael P. Brenner

Division of Engineering and Applied Sciences, Harvard University, Cambridge,Massachusetts 02138

Sascha Hilgenfeldt and Detlef Lohse*

Department of Applied Physics and J. M. Burgers Centre for Fluid Dynamics,University of Twente, 7500 AE Enschede, The Netherlands

(Published 13 May 2002)

Single-bubble sonoluminescence occurs when an acoustically trapped and periodically driven gasbubble collapses so strongly that the energy focusing at collapse leads to light emission. Detailedexperiments have demonstrated the unique properties of this system: the spectrum of the emitted lighttends to peak in the ultraviolet and depends strongly on the type of gas dissolved in the liquid; smallamounts of trace noble gases or other impurities can dramatically change the amount of lightemission, which is also affected by small changes in other operating parameters (mainly forcingpressure, dissolved gas concentration, and liquid temperature). This article reviews experimental andtheoretical efforts to understand this phenomenon. The currently available information favors adescription of sonoluminescence caused by adiabatic heating of the bubble at collapse, leading topartial ionization of the gas inside the bubble and to thermal emission such as bremsstrahlung. Aftera brief historical review, the authors survey the major areas of research: Section II describes theclassical theory of bubble dynamics, as developed by Rayleigh, Plesset, Prosperetti, and others, whileSec. III describes research on the gas dynamics inside the bubble. Shock waves inside the bubble donot seem to play a prominent role in the process. Section IV discusses the hydrodynamic and chemicalstability of the bubble. Stable single-bubble sonoluminescence requires that the bubble be shapestable and diffusively stable, and, together with an energy focusing condition, this fixes the parameterspace where light emission occurs. Section V describes experiments and models addressing the originof the light emission. The final section presents an overview of what is known, and outlines somedirections for future research.

CONTENTS

I. Introduction 426A. The discovery of single-bubble sonoluminescence 426B. Structure of the review 427C. Historical overview 428

II. Fluid Dynamics of the Flask 435A. Derivation of the Rayleigh-Plesset equation 435B. Extensions of the Rayleigh-Plesset equation 437C. The bubble’s response to weak and strong

driving 438D. The Rayleigh collapse 439E. Comparison to experiments 439F. Sound emission from the bubble 440G. Bjerknes forces 441

III. The Bubble Interior 442A. Full gas dynamics in the bubble 442

1. Inviscid models 4432. Dissipative models 4443. Dissipative models including water vapor 445

B. Simple models 4481. Homogeneous van der Waals gas without

heat and mass exchange 4482. Homogeneous van der Waals gas with heat

and mass exchange 449C. How accurate are the bubble temperatures? 450

IV. The Parameter Range of Single-BubbleSonoluminescence 451

*Electronic address: [email protected]

0034-6861/2002/74(2)/425(60)/$35.00 425

A. The Blake threshold 451B. Diffusive stability 452C. Sonoluminescing bubbles rectify inert gases 454

1. The mechanism 4542. Bubble equilibria with chemical reactions 455

D. Shape stability 4571. Dynamical equations 4572. Parametric instability 4583. Afterbounce instability 4594. Rayleigh-Taylor instability 4605. Parameter dependence of the shape

instabilities 460E. Interplay of diffusive equilibria and shape

instabilities 461F. Other liquids and contaminated liquids 462

V. Sonoluminescence Light Emission 462A. Theories of MBSL: discharge vs hot spot

theories 463B. SBSL: A multitude of theories 463C. Narrowing down the field 464D. The blackbody model and its failure 465E. The SBSL bubble as thermal volume emitter 466

1. Simple model for bubble opacity 4682. Light emission and comparison with

experiment 469F. Modeling uncertainties: additional effects 471

1. Bubble hydrodynamics 4712. Water vapor as emitter and quencher of

light 4713. Further difficulties in modeling the

temperature 4724. Modifications of photon-emission processes 472

©2002 The American Physical Society

426 Brenner, Hilgenfeldt, and Lohse: Single-bubble sonoluminescence

5. Towards a more comprehensive model ofSBSL light emission 472

G. Line emission in SBSL 472VI. Summary and Outlook 474

A. An SBSL bubble through its oscillation cycle 474B. Unanswered questions 475C. Scientific uses and spinoffs 475D. Other applications of bubble dynamics and

cavitation 476E. Multibubble fields: in search of a theory 477

Acknowledgments 477References 477

I. INTRODUCTION

A. The discovery of single-bubble sonoluminescence

Single-bubble sonoluminescence was discovered in1989 by Felipe Gaitan, then a graduate student at theUniversity of Mississippi working with Larry Crum(Gaitan, 1990; Gaitan and Crum, 1990; Gaitan et al.,1992). Crum had seen hints of light emission from asingle bubble in 1985 (Crum and Reynolds, 1985), andGaitan’s objective for his thesis was to search systemati-cally for it. Gaitan was carrying out a set of experimentson the oscillation and collapse of bubbles, using a flaskof liquid lined with transducers tuned to set up an acous-tic standing wave at the resonant frequency at the jar.When the pressure amplitude Pa of the sound waves islarger than the ambient pressure P051 bar, the pressurein the flask becomes negative, putting the liquid undertension. At large enough tension, the liquid breaks apart(cavitation), creating unstable bubble clouds in whichthe bubbles often self-organize into dendritic structures(streamers; see Neppiras, 1980). These cavitation cloudscollapse with enormous force, powerful enough to doserious damage to the surfaces of solid bodies in theirvicinity.

In his search for single-bubble sonoluminescence,Gaitan at some point found a regime with a moderateforcing pressure Pa /P0'1.2–1.4 and with the water de-gassed to around 20% of its saturated concentration ofair. He then observed that ‘‘as the pressure was in-creased, the degassing action of the sound field was re-ducing the number of bubbles, causing the cavitationstreamers to become very thin until only a single bubbleremained. The remaining bubble was approximately 20mm in radius and [ . . . ] was remarkably stable in positionand shape, remained constant in size and seemed to bepulsating in a purely radial mode. With the room lightsdimmed, a greenish luminous spot the size of a pinpointcould be seen with the unaided eye, near the bubble’sposition in the liquid’’ (Gaitan et al., 1992). The experi-ment is shown in Fig. 1, a sketch of a typical experimen-tal setup for single-bubble sonoluminescence in Fig. 2.

At the time of Gaitan’s experiment, all previous workwith light-emitting bubbles involved many unstablebubbles being simultaneously created and destroyed.Using Mie scattering to track the volumetric contrac-tions and expansions of the bubbles (Gaitan, 1990;Gaitan and Crum, 1990; Gaitan et al., 1992) Gaitan and

Rev. Mod. Phys., Vol. 74, No. 2, April 2002

co-workers demonstrated that their setup indeed gener-ated a single bubble, undergoing its oscillations at afixed, stable position at a pressure antinode of the ultra-sound field in the flask. The oscillation frequency f isthat of the sinusoidal driving sound (typically 20–40kHz), but the dynamics of the bubble radius is stronglynonlinear. Once during each oscillation period, thebubble, whose undriven (ambient) radius R0 is typicallyaround 5 mm, collapses very rapidly from its maximumradius Rmax;50 mm to a minimum radius of Rmin;0.5 mm, changing its volume by a factor of 13106

(Barber et al., 1992). Figure 3 shows the radius, forcingpressure, and light intensity (top to bottom) during this

FIG. 1. A sonoluminescing bubble. The dot in the center ofthe jar is the bubble emitting light. From Crum, 1994.

FIG. 2. Sketch of a typical setup for generating sonoluminesc-ing bubbles.

427Brenner, Hilgenfeldt, and Lohse: Single-bubble sonoluminescence

process (Crum, 1994). The bubble expansion caused bythe negative pressure is followed by a violent collapse,during which light is emitted. The process repeats itselfwith extraordinary precision, as demonstrated by mea-surements of the phase of the light emission relative tothe driving.

Light emission from collapsing ultrasound-drivenbubbles had long been dubbed sonoluminescence (SL).Researchers were familiar with the energy-focusingpower of cavitation clouds, and it was therefore not sur-prising when Frenzel and Schultes (1934) demonstratedthat these cavitation clouds emitted a low level of light[slightly earlier, Marinesco and Trillat (1933) had foundindirect evidence when photographic plates fogged in anultrasonic bath]. After all, if the cloud collapses violentlyenough to break the molecular bonds in a solid, causingcavitation damage (Leighton, 1994), there is no reasonwhy photons should not also be emitted. The energy-focusing power of the cavitation cloud was understoodto arise from a singularity occurring when a bubble col-lapses in an ambient liquid (Rayleigh, 1917): inertialforces combined with mass conservation lead to bubble-wall velocities that become supersonic during the col-lapse, causing rapid heating of the bubble interior. Tothe engineering community of the time, the fluid me-chanics of this process were much more interesting thanthe character of the radiation produced. This was for avery practical reason: people wanted to understand howto prevent cavitation damage, or how to harness itsenergy-focusing power. Although historically the lightemission has played a useful role in measuring proper-ties of cavitation [Flint and Suslick (1991b) used thespectrum to measure the temperature in a cavitatingbubble cloud], it was not considered of intrinsic impor-tance until Gaitan’s discovery of what is now known assingle-bubble sonoluminescence (SBSL).

The brightness of Gaitan’s single, isolated bubblecaused great excitement in the scientific community; it isvisible to the naked eye! Though the light emission fromconventional cavitation clouds [now called multibubble

FIG. 3. Radius R(t), driving pressure P(t), and light intensityI(t) from Crum (1994), as measured by Gaitan et al. (1992). Anegative driving pressure causes the bubble to expand; whenthe driving pressure changes sign, the bubble collapses, result-ing in a short pulse of light, marked SL.


sonoluminescence (MBSL); see Kuttruff, 1962; Waltonand Reynolds, 1984; Brennen, 1995] is also visible asdiffuse glowing, in that case no individual, stablebubbles can be identified. The excitement about single-bubble sonoluminescence was driven in large part by aset of experiments by Seth Putterman’s group at UCLAfrom 1991 to 1997, which exposed further peculiarities,making single-bubble sonoluminescence seem very dif-ferent from MBSL (the experiments of the UCLAgroup are reviewed by Barber et al., 1997 and Puttermanand Weninger, 2000). Was new physics (beyond that im-plied by the collapse mechanism of Lord Rayleigh in1917) responsible for this difference? Many people werealso excited by the fact that single-bubble sonolumines-cence appeared to be much more controllable than itsmultibubble counterpart, bringing expectations of bothgood careful scientific studies and the possibility of newtechnologies, including the harnessing of the energy-focusing power of SBSL.

It is natural that the excitement at first caused specu-lation about very exotic conditions inside the bubble,such as extremely high temperatures and pressures.Even Hollywood caught on to the excitement, producinga movie in which the central character created a fusionreactor using a single sonoluminescing bubble. As thefield matured over time and the models were refined,the results became more down to earth; for instance, thecommonly believed maximum temperature at thebubble collapse has been revised downward during a de-cade of research from early estimates of 108 K to themore modest present-day estimates which clusteraround 104 K.

In the years since SBSL was discovered, much hasbeen learned about how and why it occurs. The goal ofthis review is to clarify the basic ideas that have provennecessary for a quantitative understanding of single-bubble sonoluminescence and to present an overview ofthe current state of the field, of what is known and whatis yet to be fully understood.

B. Structure of the review

The structure of this review is as follows: The remain-der of this Introduction presents an overview of the sa-lient historical and experimental facts and qualitativelydescribes the ideas and issues that have been shown tobe important for understanding the phenomenon. Thisoverview will illustrate the enormous variety of physicalprocesses taking place inside this simple experiment,ranging from fluid dynamics, to acoustics, to heat andmass transfer, to chemical reactions, and finally to thelight emission itself. We shall then spend the next foursections following in detail the sequence of events thathappen to a sonoluminescing bubble, beginning with themotion of the flask and liquid and proceeding to thedynamics of the bubble wall and interior. Figure 4 showsthe radius R(t) of the bubble as a function of time dur-ing a single cycle of the driving; the inset blows up theinnermost '60 ns around the cavitation event, where


the bubble temperature rises rapidly due to adiabaticcompression and light is emitted.

Section II reviews classical studies of the hydrody-namics of bubble motion, showing, for example, how toderive the equation for the bubble radius leading to Fig.4, and also discussing the hydrodynamics of Lord Ray-leigh’s cavitation collapse (Fig. 4, inset). Section III de-scribes the fluid dynamics of the bubble’s interior, focus-ing mainly on what happens to the gas during thecavitation event, but also discussing water evaporation,heat transfer, and chemistry. Section IV discusses thephysical processes that fix the ambient size R0 of thebubble, including the diffusive and chemical processes ofmass exchange between bubble and liquid as well as me-chanical stability constraints. Finally, Sec. V discussesthe light emission itself, which occurs when the bubble isin its maximally compressed state. The discussion willemphasize the mechanisms that are consistent with thecurrent experimental data. In the final section, we give abrief summary and present our opinions on the currentstate of the field as well as the areas of activity with thebrightest outlook for future work.

C. Historical overview

After Gaitan’s discovery, the initial goal of researchwas to quantify how much more efficiently a singlebubble focuses energy than a bubble cloud. To addressthis question, Barber and co-workers (Barber andPutterman, 1991; Barber et al., 1992) measured thewidth of the light pulse, by studying the response of a

FIG. 4. Classical bubble dynamics calculation for a drivingpressure amplitude Pa51.2 atm, frequency f526.5 kHz, andambient bubble radius R054.5 mm. One oscillation cycle ofR(t) is shown. The bubble expands to nearly ten times itsambient radius, then collapses extremely quickly, leading toadiabatic heating of the gas inside the bubble. The collapse isfollowed by afterbounces with roughly the eigenfrequency ofthe bubble. The vertical dashed lines and small-print numbersindicate the intervals 1–10 (summarized in Sec. VI) at whichdifferent physical processes are important, which are discussedthroughout the review. The inset shows the innermost 60 nsaround the time t* of maximum compression and highlightsthe bubble radius during Rayleigh cavitation collapse, wherethe light is emitted.


single photomultiplier tube to the sonoluminescent flash.It was concluded that the width of the light pulse wasless than 50 ps. The importance of the measurement wasthat this upper bound for the pulse width was muchsmaller than the time during which the bubble remainedin its most compressed state. Roughly, the time scale ofbubble compression is given by the time it takes a soundwave to travel across the minimum radius of the bubble.With a sound velocity of c;1000 m/s, one obtains aballpark estimate of Rmin /c;1029 s, far in excess of themeasured pulse-width limit. Lord Rayleigh’s cavitationmechanism implies that the energy focusing is coupledto the bubble collapse: this discrepancy suggests that inSBSL the light emission is decoupled from the bubbledynamics.

The gauntlet was thus thrown, and a search for thecorrect mechanism began. An influential early idea [in-troduced independently by Greenspan and Nadim(1993), Wu and Roberts (1993), and Moss et al. (1994)]was that the energy focusing in the bubble was causedby a converging spherical shock. It had been knownsince the seminal work of Guderley (1942) (see alsoLandau and Lifshitz, 1987) that such shocks focus en-ergy, and in the absence of dissipation the temperatureof the gas diverges to infinity. In fact, Jarman (1960) hadalready suggested converging shocks as the source ofmultibubble sonoluminescence. This mechanism neatlysolved the upper-bound problem for the width of thelight pulse (since in this picture the light originates froma much smaller region in the center of the bubble) andproposed an elegant mechanism for energy focusingcompounding Lord Rayleigh’s bubble-collapse mecha-nism. Simulations by Wu and Roberts (1993) had themaximum temperature approaching 108 K, very hot in-deed.

For several years, experimental information accumu-lated about the properties of sonoluminescing bubbles.Hiller et al. (1992, 1994, 1998) measured the spectrum ofa sonoluminescing air bubble in water and demonstratedthat it increases toward the ultraviolet (Fig. 5). The ap-parent peak in some spectra is due to the strong absorp-

FIG. 5. Spectrum of single-bubble sonoluminescence, for wa-ter at 22 °C. The data points are redrawn from Fig. 1 of Hilleret al. (1992). Fits to a blackbody spectrum can be attemptedfor different temperatures, with best results for about 40 000 K(solid line), higher than the 25 000 K suggested by Hiller et al.(1992).


tion of wavelengths below '200 nm by the water in theflask. In sharp contrast to the spectrum of MBSL, single-bubble sonoluminescence shows a smooth continuum,without spectral lines (see Fig. 6). The presence of spec-tral lines points to lower temperatures, since the atomictransitions leading to lines tend to be overwhelmed bycontinuous emission processes at high temperatures. Byfitting the observed spectra to that of a blackbody emit-ter (Fig. 5), Hiller et al. (1992) concluded that the tem-perature of the gas was at least 25 000 K.

Barber et al. (1994) demonstrated that both the lightintensity and amplitude of the oscillations of the bubbledepend sensitively not only on the forcing pressure am-plitude, but also on the concentration of the gas dis-solved in the liquid, the temperature of the liquid, orsmall amounts of surface active impurities (Weningeret al., 1995; Ashokkumar et al., 2000; Toegel, Hilgen-feldt, and Lohse, 2000). As an example, Fig. 2 of Barberet al. (1994) shows the dependence of R(t) and the totallight intensity on the increasing drive level for an airbubble in water. As the forcing is increased, the bubblesize abruptly decreases, and then the light turns on (seeFig 7). For some years, the precise reasons for this sen-sitivity (observed repeatedly in experiments) were diffi-cult to understand, mostly because varying one of theexperimental parameters, such as the water tempera-ture, would tend to change others as well.

Perhaps most surprisingly, Hiller et al. (1994) found asensitive dependence on the type of gas within thebubble: when the air dissolved in the liquid was replacedwith pure nitrogen, the characteristically stable SBSLdisappeared. With a gas composed of 80% nitrogen and20% oxygen, there was still no sonoluminescence. Onlywhen the inert gas argon was added did SBSL lightemission return. Figure 8 shows a plot of the intensity ofsonoluminescence as a function of the percentage of in-ert gas doped in nitrogen. For both argon and xenon, theintensity peaks around 1%, the concentration of argonin air.

FIG. 6. MBSL (thin line) and SBSL (thick line) spectra in a0.1M sodium chloride solution. Each spectrum was normalizedto its highest intensity. Note the prominence (MBSL) and ab-sence (SBSL, see the inset for an enlargement) of the sodiumline near 589 nm. Figure reproduced from Matula et al. (1995).


SBSL can be achieved with a pure noble gas as well,but in a vastly different range of gas concentrations: Inthe original experiment with air, Gaitan (1990) observedstable light emission when degassing using a partial pres-sure of p`

air/P0;0.2–0.4; i.e., the water contained 20–40 % of the air it would contain if in saturation equilib-rium with a pressure of P051 bar. Barber et al. (1995)demonstrated that, when using pure argon gas, the de-gassing has to be 100 times stronger, requiring partialpressures as low as p`

Ar/P0;0.002–0.004 to obtain stableSBSL. The pressures p` are the partial gas pressuresused in experiment when preparing the degassed liquid.

FIG. 7. The ambient bubble radius as a function of forcingpressure Pa for a gas mixture of 5% argon and 95% nitrogenat a pressure overhead of 150 mm. For sonoluminescingbubbles the symbols are filled; for nonglowing bubbles they areopen. Note the abrupt decrease in bubble size right before thesonoluminescence threshold. The figure is a sketch from Fig.38 of Barber et al. (1997). In that paper the ambient radius isobtained from a fit of the Rayleigh-Plesset equation to theR(t) curve. In that fit heat losses are not considered explicitly,but material constants are considered as free parameters.Therefore the values for R0 are only approximate; see the dis-cussion in Sec. II.E.

FIG. 8. Dependence of the sonoluminescence intensity (nor-malized to that of air) in water as a function of the percentage(mole fraction) of noble gas mixed with nitrogen. Two noblegases are shown: xenon (d) and argon (j). Both give maxi-mum light intensity around 1% dissolution, as does helium(not shown). The figure is a sketch from Fig. 22 of Barber et al.(1997).


During this time the predominant belief in the fieldwas that shocks (see, e.g., Barber et al., 1994, 1997) weresomehow important for the energy focusing and lightemission of sonoluminescence. However, there was littleagreement as to the details of how this worked, andmany other physical mechanisms were suggested, includ-ing dielectric breakdown of the gas (Garcia and Le-vanyuk, 1996; Lepoint et al., 1997; Garcia and Hasmy,1998), fracture-induced light emission (Prosperetti,1997), bremsstrahlung (Moss, 1997; Frommhold, 1998),collision-induced emission (Frommhold and Atchley,1994; Frommhold, 1997; Frommhold and Meyer, 1997),and even the quantum-electrodynamical Casimir effect(Eberlein, 1996a, 1996b), an idea pioneered in this con-text by Schwinger (1992).

The difficulty in evaluating these ideas was that theyrequired probing the bubble collapse in greater detailthan was experimentally possible. This led Robert Apfelto pose a ‘‘challenge to theorists’’ in a session on sonolu-minescence at the annual meeting of the Acoustical So-ciety of America in Honolulu in 1996. The challenge wasto make concrete, experimentally testable predictions.Many creative ideas were collected at this meeting, onlya fraction of which still survive today. [One of the earlycasualties includes the acoustic-resonator theory devel-oped by the present authors speculating on energy stor-age in the bubble (Brenner, Hilgenfeldt, et al., 1996).]

Meanwhile, it was equally clear that at least some ofthe experimental facts of sonoluminescence were directconsequences of the classical theory of bubble dynamics,having nothing to do with light emission per se. The timescale of the light emission is so much shorter than acomplete cycle of the acoustic driving that bubble dy-namics goes a long way towards explaining issues ofbubble stability and constraints for driving parameters.Since Lord Rayleigh’s characterization of cavitation col-lapse (Rayleigh, 1917), bubble dynamics had becomewell understood,1 but, although the theory was formallyquite mature, it had never been put to work in the pre-cise regime of single-bubble sonoluminescence.

The application of classical bubble dynamics to SBSLsubstantially clarified the experimental situation. Thefirst contribution in this regard was made in the originalpaper of Gaitan et al. (1992), which demonstrated thatthe radius of the bubble as a function of time observedexperimentally exhibits the same behavior as solutionsto the Rayleigh-Plesset equation (to be derived in Sec.II); subsequently, studies by Lofstedt et al. (1993, 1995)confirmed and elaborated on this conclusion. TheRayleigh-Plesset theory is remarkably simple, and itcaptures many important features of single-bubblesonoluminescence. To practitioners of classical bubble

1This was primarily due to the contributions of Plesset, 1949,1954; Epstein and Plesset, 1950; Plesset and Zwick, 1952;Plesset, 1954; Plesset and Mitchell, 1956; Eller and Flynn, 1964;Eller, 1969; Eller and Crum, 1970; Prosperetti, 1974, 1975,1977a, 1977d; Plesset and Prosperetti, 1977; Prosperetti andLezzi, 1986; Prosperetti et al., 1988.


dynamics, the excellent agreement was particularly sur-prising because this theory has long been known to showlarge quantitative discrepancies even for bubbles thatare more weakly forced than in the case of SBSL (Pros-peretti et al., 1988). In the SBSL parameter regime, theperiodic forcing of the pressure waves in the containerleads to a periodic bubble response, with a cavitationcollapse happening exactly once per cycle [chaotic mo-tion as in Lauterborn (1976) and Lauterborn and Suchla(1984) is notably absent]. The qualitative and even mostquantitative features of bubble oscillations agree withthe experimental observations. The solution also has thecourtesy to predict its own demise: at cavitation collapsethe speed of the bubble wall approaches or surpasses thespeed of sound in the liquid, contradicting one of theessential assumptions of the theory. The total time dur-ing which the bubble wall is supersonic is a tiny fractionof a cycle; the errors that accumulate in this regime donot substantially affect the rest of the cycle.

If the solutions to the Rayleigh-Plesset equation ex-plain the experimental measurements of the bubble ra-dius, then their stability must constrain the parameterspace where SBSL can occur (Brenner et al., 1995; Bren-ner, Hilgenfeldt, et al., 1996; Hilgenfeldt et al., 1996).There are three major instabilities of the bubble thatneed to be avoided: (i) the bubble must not changeshape (shape instabilities; Brenner et al., 1995; Hilgen-feldt et al., 1996); (ii) the average number of gas mol-ecules in the bubble must not increase or decrease overtime (diffusive instability; Brenner, Lohse, et al., 1996;Hilgenfeldt et al., 1996); (iii) the bubble must not beejected from the acoustic trap where it is contained(Bjerknes instability; Cordry, 1995; Akhatov et al., 1997;Matula et al., 1997). All of these constraints must be sat-isfied in a parameter regime where the bubble oscilla-

FIG. 9. Phase diagram in the p`Ar /P0 vs Pa /P0 parameter

space, according to the hydrodynamic/chemical theory ofHilgenfeldt et al. (1996) and Lohse et al. (1997). The drivingfrequency is f533.4 kHz. The three phases represent stableSL, unstable SL, and no SL. The symbols represent measure-ments by Ketterling and Apfel (1998), either stable sonolumi-nescing bubbles (l) or stable, nonsonoluminescing bubbles(s), showing good agreement with the earlier theoretical pre-dictions.


tions become nonlinear enough for sonoluminescence tooccur. The allowable parameter space of SBSL is thusseverely limited to a narrow range of relative gas con-centrations c` /c05p`/P0 and forcing pressure ampli-tudes Pa (see Fig. 9).

While the regime of stable sonoluminescence in argongas is in good agreement with that predicted by the hy-hydrodynamic stability calculations of Hilgenfeldt et al.(1996), Barber et al. (1995) found that the ranges of dis-solved gas concentrations for stable SBSL were lower bya factor of 100 in pure argon gas than in air. Lofstedtet al. (1995) pointed out that a sonoluminescing bubblecannot possibly be in diffusive equilibrium for these pa-rameters and postulated another ‘‘anomalous massflow,’’ whose mechanism would be ‘‘the key to SL in asingle bubble.’’

To account for these discrepancies to classical bubbledynamics, Lohse et al. (1997) proposed that the extramass-ejection mechanism of Lofstedt et al. (1995) is of achemical nature. The gas in the bubble is hot enoughupon collapse to allow for significant dissociation of N2and O2 . The dissociated nitrogen and oxygen, as well assome radicals from dissociated water vapor, will undergochemical reactions, whose products are very soluble inwater and are expelled from the bubble. Only inert, non-reactive gases (such as argon) remain inside according tothis ‘‘argon rectification hypothesis.’’ This idea immedi-ately resolves the apparent discrepancy between themeasured and predicted parameter regimes for stableSBSL in air: if the bubble ends up filled with argon gasonly, then only the argon dissolved in the liquid has tobe in diffusive equilibrium with the bubble. As air con-tains 1% of argon, the effective dissolved gas concentra-

FIG. 10. Phase diagram for air at p` /P050.20 in the R0-Pa

space. The arrows denote whether the ambient radius grows orshrinks at this parameter value. Curve A denotes the equilib-rium for an air bubble; on curve C the bubble contains onlyargon. The intermediate curve B necessarily exists because ofthe topology of the diagram and represents an additionalstable equilibrium. The thin line indicates where the (approxi-mate) threshold temperature of nitrogen dissociation(;9000 K) is reached. From Lohse et al. (1997).


tion for diffusive stability of argon is 100 times smaller,and explains the hundredfold difference between ob-served concentrations for air and argon bubbles. Thephase diagram in the R0-Pa space resulting from that

FIG. 11. Experimental phase diagram in the R0-Pa parameterspace for air at p` /P050.20. The driving frequency is20.6 kHz. Arrows indicate whether the bubbles grow orshrink. Three equilibrium curves A, B, and C can be recog-nized. In between curves B and C there is a ‘‘dissolution is-land.’’ The shaded area shows the shape-stable parameter do-main (see Sec. IV.D). Figure adopted from Holt and Gaitan(1996).

FIG. 12. Experimental phase diagram for air saturated in wa-ter to 20%. Each data point represents the Pa and R0 foundfrom a single R(t) curve and is indicated to be luminescingand/or stable. The curves in the plot are lines of diffusive equi-librium for a given gas concentration c` /c050.2 (solid line)and c` /c050.002 (dashed line). The range of Pa where danc-ing bubbles were observed is indicated, as are regions ofbubble growth (g) and dissolution (d) relative to each equilib-rium curve. The stable no-SL points (d) correspond to a stablechemical equilibrium which would lie above the c` /c050.2curve if plotted. From Ketterling and Apfel (1998).


theory is shown in Fig. 10. In particular, the theory pre-dicts a new stable branch (called ‘‘B’’ in Fig. 10) onwhich mass losses from chemical reactions and growthfrom rectified diffusion just balance.

Experiments by Holt and Gaitan (1996) on bubblestability published contemporaneously with the theoret-ical work indeed showed this extra regime of bubblestability predicted from the argon rectification hypoth-esis (see Fig. 11). Ketterling and Apfel (1998, 2000a,2000b) later showed the stability predictions to be quan-titatively correct. Figure 12 shows experimental mea-surements of a phase diagram in comparison with theo-retical predictions. One consequence of the interplay ofdiffusive and shape instabilities is indicated in this figure:bubbles can ‘‘dance’’ due to the recoil when they un-dergo fragmentation (see Sec. IV.E).

Phase diagrams such as Figs. 9–12 help us to under-stand the limitations of the parameter space for sonolu-minescence, and in particular the crucial role of noblegases for SBSL stability. The same theoretical conceptscould be applied to explain the pronounced increase inthe intensity of emitted light with decreasing water tem-perature (Hilgenfeldt, Lohse, et al., 1998), and thequenching of light due to small concentrations of surfac-tants, both of which were shown to be in agreement withexperiments (Ashokkumar et al., 2000; Matula, 2000;Toegel, Hilgenfeldt, et al., 2000).

There was, however, still the nagging problem of thelight emission itself. In contrast to the bubble dynamics,the available experimental information was insufficientto constrain the theories. The breakthrough contributionwas made by Gompf et al. (1997), who measured thewidth of the light pulse using time-correlated single-photon counting (TC-SPC). This technique has a muchhigher resolution for measuring flash widths than asingle photomultiplier tube, because it measures timedelays in arrivals of single photons. The measurement ofthe delay time between the two photons reaching thetwo different photomultiplier tubes is repeated manytimes so that the width of the flash can be reconstructed.Gompf et al. (1997) discovered that the width of thelight pulse is actually of the order of a few hundred pi-coseconds (see Fig. 13), much longer than the previous50-ps upper bound measured by Barber and Putterman(1991). Moreover, since Gompf et al. (1997) could nowresolve the shape of the light pulse, it was possible tostudy the dependence of the width on external param-eters (the forcing pressure and dissolved gas concentra-tion; see Fig. 14).

After this paper was published, at a meeting onsonoluminescence at the University of Chicago, twoother groups announced that they had confirmed itsfindings: Moran and Sweider (1998) and Hiller et al.(1998) also used TC-SPC. At the same time, Gompf’sgroup succeeded in obtaining an independent confirma-tion of the much longer duration of the light pulse usinga streak camera for direct measurement of the pulsewidth (Pecha et al., 1998). A previous attempt by Moran


et al. (1995) employing a streak camera had yielded onlya tentative upper bound for pulse width, which againproved too small.

The increased experimental resolution of TC-SPC andthe subsequent discovery of a long flash width put all ofthe theories of light emission and energy focusing, whichrequired ultrashort flash widths, out of business. More-over, as was emphasized by Gompf et al. (1997) in theirseminal paper, the measurement restored hope that avariant of the simplest possible theory for the light emis-sion might be correct: the cavitation collapse of thebubble is so rapid that heat cannot escape from thebubble. Therefore, the bubble heats up, leading to lightemission. Figure 15 shows the heating as calculated byGompf et al. (1997), by solving a variant of theRayleigh-Plesset equation for the bubble radius and as-suming adiabatic heating (ratio of specific heats G55/3) near the collapse. Although the calculation con-tains some severe approximations, the agreement isquite reasonable.

This idea was buttressed by an earlier numerical simu-lation of Vuong and Szeri (1996), which, when reinter-preted with the new experiments in mind, questionedthe notion that strong shocks are important for single-bubble sonoluminescence. Vuong and Szeri included dis-

FIG. 13. First measurement of SBSL pulse widths. The param-eters were Pa51.2 bars, f520 kHz, and the gas concentrationwas 1.8-mg/l O2 . Both the width in the red and the ultravioletspectral range were measured. The indistinguishable widthsrule out blackbody radiation, but not a thermal emission pro-cess in general. From Gompf et al. (1997).

FIG. 14. Dependence of the full width at half maximum of theSBSL pulse on the driving pressure and the gas concentrationat room temperature. f520 kHz. From Gompf et al. (1997).


sipative effects and showed that the strong shocks pre-dicted by Wu and Roberts (1993) and Moss et al. (1994,1996, 1997) were absent in noble gas bubbles, and werereplaced by gentler inhomogeneities. The predictedmaximum temperatures in the bubble were thereforemuch lower, several 104 K, compared with the 108 Kpreviously announced by Wu and Roberts (1994). More-over, the hot spot was not highly localized in the bubblecenter. These arguments were elaborated upon byVuong et al. (1999); these models are much closer to thesimple picture of adiabatic heating and thermal lightemission than the shock-wave scenario. The tempera-ture profiles and motions of Lagrangian points as com-puted by Vuong and Szeri (1996) are shown in Fig. 16:The characteristic scale over which temperature varies isof the order of the bubble radius.

Since the experimental resolution of the flash, re-searchers have focused on trying to determine whichvariant of the thermal light-emission model is correct. Isthe interior of the bubble uniform? Is the radiationblackbody, bremsstrahlung, or some other process? Isthe bubble optically thin or thick? What physical mecha-nism is suppressing spectral lines? Since experiments arenow able to measure both the shape of the light pulseand the spectrum independently and accurately, it is pos-sible to determine how these quantities depend on ex-perimental parameters like forcing pressure, gas concen-tration, etc. The power of these measurements is thatthey provide severe constraints for theories of SBSLlight emission that did not exist when the pulse widthwas believed to be very short. Moreover, since thebubble dynamics itself is well understood, closer exami-nation of these parameter dependencies makes it pos-sible to focus attention on subtle details of the light-emitting process. Single-bubble sonoluminescence hasthus become a rather sophisticated testing ground forthe ability of mathematical models and numerical simu-lations to explain detailed experimental data from acomplicated physical process.

Although there are still open questions about the de-tails of the light emission, considerable progress hasbeen made. When Gompf et al. (1997) resolved the light

FIG. 15. Calculated shape of temperature pulse using a simplemodel based on the Rayleigh-Plesset equation, assuming thegas temperature and density are uniform throughout the col-lapse. Figure reproduced from Gompf et al. (1997).


pulses, they also made measurements of the dependenceof the width on optical wavelength. Strikingly, such adependence was found to be absent, contradicting asimple blackbody emission model, which demands thatthe width increase with the wavelength.

FIG. 16. Motion and temperature in a bubble shortly beforecollapse: (a) motion history of 20 Lagrangian points inside aR054.5 mm bubble driven at Pa51.3 atm and f526.5 kHz.Strong wavy motion occurs inside the bubble, but no shockwaves develop. (b) Temperature profiles in the bubble for vari-ous times around the bubble collapse. The profiles span a timeinterval of '170 ps near the collapse. The temperature at thecenter increases monotonically, until the maximum tempera-ture is reached at the last snapshot. Note that the temperatureprofile is smooth, without any discontinuity that would bepresent with a shock. From Vuong and Szeri (1996).


A resolution for this conundrum was hinted at in nu-merical simulations by Moss et al. (1994, 1997, 1999),who realized that the temperature-dependent photonabsorption coefficients of the gas must be taken into ac-count. The size of the bubble and thus the size of thelight-emitting region are so small that the bubble isnearly transparent for its own photons: the bubble is avolume emitter, not a surface emitter like an ideal black-body. Among other things, Moss et al. (1999) used thisidea to rationalize the qualitative shape of the emissionspectrum in noble gases.

Hilgenfeldt et al. (1999a, 1999b) used varying absorp-tion coefficients to explain the wavelength-independentpulse widths: Both the absorptivity and emissivity of thebubble drop precipitously directly after collapse for allwavelengths, since they depend exponentially on tem-perature, but only weakly on wavelength. Combiningthis model of thermal radiation with the parameter de-pendencies predicted by the stability constraints on thebubble, they also found agreement with the observedparameter dependencies of the pulse width, number ofphotons per burst, and spectral shape. Hammer andFrommhold (2000a, 2000b) demonstrated that thismodel could be refined with ab initio quantum-mechanical calculations of electron-neutral bremsstrah-lung, further improving the agreement with experi-ments. Examples of their spectra are shown in Fig. 17.

An important aspect of bubble thermodynamics,which has been pointed out by Kamath et al. (1993), Ya-sui (1997b), Colussi and Hoffmann (1999), Moss et al.(1999), Storey and Szeri (2000, 2001); Toegel, Gompf,et al. (2000), Hilgenfeldt et al. (2001), and Puttermanet al. (2001), is the presence of water vapor inside thebubble. Upon bubble expansion, vapor invades thebubble. At collapse, it cannot completely escape (con-dense at the bubble wall) because the diffusion timescale is much slower than the time scale of the collapse.Therefore water vapor is trapped inside the bubble (Sto-rey and Szeri, 2000). It limits the maximum temperature

FIG. 17. Emission spectra from rare gases at room tempera-ture. The dotted lines are calculations based on the theoreticalmodel of Hammer and Frommhold (2000a). The only adjust-able parameters in the comparison are the ambient radii andforcing pressures of the bubbles. From Hammer and Fromm-hold (2001).


in the bubble due to its lower polytropic exponent (com-pared to inert gases) and above all because of the endo-thermic chemical reaction H2O→OH1H, which eats upthe focused energy. Within the model of Storey andSzeri (2000), taking water vapor and its chemical reac-tions into account leads to calculated maximum tem-peratures in the bubble of only around 6000 K. Thisseems to contradict experiments, in that thermal lightemission would be strongly suppressed below the mea-

FIG. 18. Dependence of the spectra of argon SBSL (for apartial pressure of '150 torr at 25 °C) on the forcing pressure.Spectra are shown for five levels of overall brightness. The OHline is vanishing in the thermal bremsstrahlung spectrum withincreasing forcing pressure Pa . From Young et al. (2001).

FIG. 19. Light-emission spectra from moving SBSL bubbles inadiponitrile. The driving pressure amplitude increases frombottom to top, between 1.7 bars and 1.9 bars. The spectral lineat '400 nm corresponds to an excitation of CN. FromDidenko et al. (2000b).


sured values, and is an indication that the modelingoverestimates the amount of water vapor in the bubble.

In two very recent experiments, the signatures (char-acteristic lines) of the liquid or liquid vapor were de-tected in the spectrum, finally closing the gap betweenMBSL and SBSL. In both cases the lines belong to con-stituents of vapor molecules.

Young et al. (2001) discovered spectral lines for SBSLin water by decreasing the driving pressure very close tothe threshold for SBSL. In this regime, the light pulse isso weak that Young et al. (2001) had to collect photonsover several days. Figure 18 shows how, as the forcingpressure is increased, the OH line vanishes behind theenhanced continuum contribution to the spectrum.

Didenko et al. (2000b) found spectral lines of SBSL inorganic fluids (see Fig. 19). These tend to require largerdriving to show SBSL, because the vapor moleculeshave more rotational and vibrational degrees of free-dom, leading to a weaker temperature increase atbubble collapse.

We believe that the observation of spectral lines her-alds a new era of research on single-bubble sonolumi-nescence, one in which it will be possible to use SBSL tostudy chemical reactions. Such studies have long beenconducted for multibubble cavitation, and indeed Sus-lick and collaborators (Suslick et al., 1986; Flint and Sus-lick, 1991b; Didenko et al., 1999) have used the widthsand intensities of spectral lines in multibubble sonolumi-nescence to deduce the temperature of cavitation. Thegreat advantage of using single-bubble sonolumines-cence in these studies is that, in contrast to MBSL, themechanics of SBSL is well understood and character-ized. It thus seems possible that one will be able to useSBSL to carefully study chemical reactions under exoticconditions of high temperatures and extreme densities.

II. FLUID DYNAMICS OF THE FLASK

The very existence of a sonoluminescing bubble de-pends critically on a subtle balance of hydrodynamic andacoustic forces inside the flask. During sonolumines-cence, a diverse array of physical effects influences thisbalance: the pressure becomes low enough that theliquid-air interface vaporizes, and temperatures rise sohigh that the gas inside the bubble emits light. Gas iscontinually exchanged between the bubble and the sur-rounding liquid, causing the number of molecules in thebubble to vary. In a small part of the cycle, the bubble-wall velocity may become supersonic. During all of theseprocesses there is no a priori reason for the shape of thebubble to remain spherical, so this must be accountedfor as well.

Although the equations of motion governing these ef-fects were written in the nineteenth century, it is a tri-umph of twentieth-century applied mathematics that allof them can be accounted for simultaneously in a preciseand controlled way. This is the theory of classical bubbledynamics, started by Lord Rayleigh (1917) during hiswork for the Royal Navy investigating cavitation dam-age of ship propellers. The formalism was substantially


refined and developed by Plesset, Prosperetti, and oth-ers over a span of several decades. A review of earlywork is presented by Plesset and Prosperetti (1997); alater overview is given by Prosperetti (1998). Thepresent section summarizes this theory with a view to-wards its application to experiments on single-bubblesonoluminescence. Our discussion will highlight the va-lidity of the approximations made when the theory isapplied to SBSL, and will also underscore how and whythe theory works when it does. The presentation of thissection was greatly influenced by the excellent recentreview by Prosperetti (1998).

A. Derivation of the Rayleigh-Plesset equation

The ultrasonic forces in the liquid are caused by theoscillating transducers on the container walls, which aretuned to excite an acoustic resonance mode of the con-tainer, often the lowest. The Q factor of a typical flask is'103, so the resonance is quite sharp. Its frequency isabout 20 kHz for a container a few centimeters across,mercifully above the range of human hearing.2 The driv-ing pressure amplitude at the center of the flask isaround Pa'1.2–1.4 bars when SBSL occurs.

The equations governing the sound waves in the liq-uid are the compressible Navier-Stokes equations

r~] tu1u•¹u !52¹p1h¹2u1z¹¹•u , (1)

] tr1¹•~ru !50, (2)

where u is the fluid velocity, r the density, p the pressure(as specified by an equation of state), h the shear viscos-ity, and z the bulk viscosity of the liquid. In writing theseequations, we have assumed that the liquid is isothermaland so have neglected the equation for the fluid tem-perature. As an approximation, the bubble’s extensioncompared to that of the flask and that of the sound waveis neglected, as it is orders of magnitude smaller.

The forces on the bubble depend on where it is lo-cated in the flask. In general there will be both an iso-tropic oscillatory pressure (causing volumetric oscilla-tions) and, in addition, pressure gradients, quadrupolecomponents, etc. In practice, for small bubbles, all thatmatters are the isotropic volumetric oscillations and thepressure gradients, which can create a net translationalforce on the bubble. The translation can vanish only atpressure maxima or minima. We shall see below thatthese forces cause sonoluminescing bubbles to betrapped at a pressure antinode of the sound field.

To compute the magnitude of the forces it is necessaryfirst to characterize the volumetric oscillations, for whichthe sound field around the bubble is purely radial. Thevelocity can then be represented by a potential, with u5¹f . Equations (1) and (2) then become

r@] tf1 12 ~]rf!2#52p , (3)

2Efforts to scale up sonoluminescence have ventured into thelower-frequency regime of audible sound. Bad luck for the ex-perimentalist.


] tr1]rf]rr1r¹2f50. (4)

Note the assumption that the flow field is purely radialand therefore viscous stresses are not important.

To proceed we need to combine Eqs. (3) and (4) intoa single equation for f. Defining the enthalpy dH5dp/r , and using dp5(dp/dr)dr5c2dr (with c thespeed of sound in the liquid) implies

¹2f5F u

c2 ~] tu2]rH !G11c2 ] t

2f , (5)

where u5]rf is the radial velocity field. As long as thefluid velocity is much smaller than c , the square-bracketed terms are negligible. The linear c22] t

2f termis only negligible close to the bubble: at distances on theorder of the sound wavelength away from the bubble,this term will become important.

We would like to solve Eq. (5) for the velocity of thebubble wall dR/dt , caused by the resonant oscillation ofthe container. We proceed in two steps: near the bubblethe velocity potential obeys the Laplace equation, ¹2f50. The solution satisfying the boundary condition atthe bubble wall ]rf(r5R)5R is

f52RR2

r1A~ t !, (6)

where A(t) is a free constant. This free constant is de-termined by matching the solution (6) onto the pressurefield far from the bubble. Neglecting the sound radiatedby the bubble itself, the velocity potential far from thebubble is a standing wave—the acoustic mode that isexcited by the transducer. For our present purposes, wedo not require the entire spatial structure of this mode,but only the field close to the bubble. Since the bubble ismuch smaller than the sound wavelength, this soundfield will be independent of r , so that f5f`(t). Match-ing the near field and the far field implies A5f` . Thepressure in the neighborhood of the bubble is then p52r] tf`5P01P(t), i.e., the sum of the backgroundstatic pressure P051 bar and the sinusoidal driving pres-sure P(t)52Pa sin vt.

The velocity field in the liquid around the bubble thenfollows as

u5RR2

r2 . (7)

We now use this to solve for the dynamics of the bubblewall. To this end, we use the force balance on the bubblesurface, which gives

pg~ t !1Srr@r5R~ t !#5pg~ t !2p@R~ t !#12h]ru~r5R !

5pg~ t !2p@R~ t !#24hR

R52

s

R,

(8)

where Srr is the radial component of the stress tensor inthe liquid, s is the surface tension of the gas-liquid in-terface, and pg is the pressure in the gas, assumed to be


spatially uniform. Evaluating this formula using Eq. (3)for the pressure in the liquid gives

RR132

R251rS pg2P02P~ t !24h

R

R2

2s

R D . (9)

Equation (9) is the celebrated Rayleigh-Plesset equa-tion. The left-hand side of the equation was known toLord Rayleigh (though never written). A historical re-view of the development of this equation is given byPlesset and Prosperetti (1977).

Closing the equation requires knowing the pressure inthe gas. Roughly speaking, when the bubble wall movesslowly with respect to the sound velocity in the gas, thepressure in the gas is uniform throughout the bubble. Inthis regime, how strongly the pressure depends on thebubble volume depends on the heat transfer across thebubble wall (Prosperetti et al., 1988). The pressure-volume relation is given by

pg~ t !5S P012s

R0D ~R0

32h3!g

@R~ t !32h3#g . (10)

Here R0 is the ambient radius of the bubble (i.e., theradius at which an unforced bubble would be in equilib-rium), and h is the van der Waals hard-core radius de-termined by the excluded volume of the gas molecules.

If the heat transfer is fast (relative to the time scale ofthe bubble motion), then the gas in the bubble is main-tained at the temperature of the liquid, and the pressureis determined by an isothermal equation of state withg51. On the other hand, if the bubble wall moves veryquickly relative to the time scale of heat transfer, thenheat will not be able to escape from the bubble, and thebubble will heat (cool) adiabatically on collapse (expan-sion). For a monatomic (noble) gas, this implies that g5G55/3. The dimensionless parameter that distin-guishes between these two regimes is the Peclet number,

Pe5uRuR

xg, (11)

where xg is the thermal diffusivity of the gas.This idea about heat transfer is based on a more care-

ful version of this argument by Kamath et al. (1993) andProsperetti et al. (1998). They showed that the tempera-ture Ts at the bubble surface is basically the water tem-perature: Conservation of energy at the bubble interfacerequires continuity of the heat flux,

Kg]rT5Kl]rTl , (12)

with the thermal conductivities Kg and Kl of gas andliquid. The gradients are estimated via the thermalboundary layer thicknesses dg and d l in and around thebubble,

]rT5Tg2Ts

dg, ]rTl5

Ts2Tl

d l, (13)

where Tg is the temperature at the bubble center. Thediffusion lengths can be estimated with the relevant timescale Dt of the bubble oscillation and the respectivethermal diffusivity x, namely, d;AxDt . With the con-


nection between thermal conductivity and diffusivity, K5xrCp , where Cp is the specific heat per unit mass, oneobtains the final result,

Ts2Tl

Tg2Ts5AxgrgCp ,g

x lrCp ,l. (14)

Since the density and the specific heat of water are somuch larger than the respective values for gas, the right-hand side of Eq. (14) is typically of the order of1023 –1022. Therefore the temperature drop basicallyoccurs inside the bubble, and the temperature at the sur-face basically equals the water temperature.

If the rate of heat transfer is intermediate betweenadiabatic and isothermal, the situation is more compli-cated. Here, a correct calculation requires solving theheat conduction problem throughout the bubble cycleand using the computed temperature in the bubble toevaluate the pressure in the gas (through its equation ofstate). This is quite a difficult task. Over the years, sev-eral methods have been proposed that amount to vary-ing g continuously between the isothermal value and theadiabatic value (Plesset and Prosperetti, 1977; Prosper-etti et al., 1988; Kamath et al., 1993) depending on thePeclet number. This approach can yield quantitativelyincorrect results, as shown by Prosperetti and Hao(1999), in large part because energy dissipation fromthermal processes is neglected.

B. Extensions of the Rayleigh-Plesset equation

So far we have not considered damping of the bubbledynamics by the sound radiated by the bubble itself. Themost complete and elegant derivation of this effect isdue to Lezzi and Prosperetti (1987; Prosperetti andLezzi, 1986).

In arriving at Eq. (9), we asserted that the velocitypotential of the sound field in the liquid far from thebubble is the same as in the absence of the bubble, f5f`(t). The radial sound wave emitted from thebubble introduces a modification,

f5f`~ t !21r

F~ t2r/c !'f`~ t !21r

F~ t !1F~ t !

c, (15)

where we have estimated the velocity potential at smallr . As above, this now must be matched to the near-fieldvelocity potential Eq. (6). The matching yields F(t)5R2R and A(t)5f`1F/c . Substituting this into thepressure jump condition one obtains

r~RR1 32 R2!5@pg2P02P~ t !#24h

R

R22s

1R

1r

c

d2

dt2 ~R2R !. (16)

The sound radiation term is of order R/c times the otherterms in the equation. When the bubble-wall motion isslow it is therefore negligible.


When uRu/c;1, sound radiation is important. For-mally, sound radiation raises the order of the Rayleigh-Plesset equation from second order to third order. Atfirst glance, this seems strange, because physically initialconditions are given for both R and R , but not R . Thediscrepancy arises because Eq. (16) has a spurious un-stable solution which grows exponentially in time. Thisis unphysical; the initial condition on R must be chosento suppress this solution.

As emphasized by Prosperetti et al. (1988; Prosperettiand Hao, 1999), this procedure is inherently impractical,as numerical errors will always excite the spurious solu-tion. A better way to take care of this is to calculate thed2/dt2 (R2R) term using the Rayleigh-Plesset equationitself. A standard way of doing this was invented byKeller and co-workers (Keller and Kolodner, 1956;Keller and Miksis, 1980) and leads to the Keller equa-tion (Prosperetti and Lezzi, 1986; Brennen, 1995)

S 12R

c D rRR132

R2rS 12R

3c D5S 11

R

c D @pg2P02P~ t !#

1R

cpg24h

R

R2

2s

R. (17)

As discussed by Prosperetti et al. (1988; Prosperetti andLezzi, 1986), the precise form of this equation is notunique: There is a one-parameter family of equationsthat can be consistently derived from Eq. (16), namely,

S 12~l11 !R

c D rRR132

R2rS 12~l1 13 !

R

c D5S 11~12l!

R

c D @pg2P02P~ t !#

1R

cpg24h

R

R2

2s

R, (18)

where the parameter value l50 recovers the Kellerequation, and l51 results in the formula used by Her-ring (1941) and Trilling (1952). Introducing higher-orderterms leads to variations like the form derived by Flynn(1975a, 1975b), but Prosperetti and Lezzi (1986) haveshown that the higher order does not, in general, guar-antee higher accuracy of the formula. Other well-knownforms of Rayleigh-Plesset derivatives are compared byLastman and Wentzell (1981, 1982). Prosperetti andLezzi (1986) demonstrate that, for a number of relevantexamples, the Keller equation yields results in closestagreement with full partial differential equation numeri-cal simulations.

An ‘‘odd cousin’’ of Eq. (18) is the Gilmore equation(Gilmore, 1952; Brennen, 1995),


S 12R

C D RR132

R2S 12R

3c D5S 11

R

C D H

r1S 12

R

C D R

C

H

r, (19)

whose derivation relies on the Kirkwood-Bethe approxi-mation (Kirkwood and Bethe, 1942). In Gilmore’s equa-tion, the key quantity is the enthalpy H , and not thepressure. In this approach, the speed of sound C is not aconstant, but depends on H . According to Gompf andPecha (2000; Pecha and Gompf, 2000), this allows one tomodel the increase of the speed of sound with increasingpressure around the bubble, which leads to significantlyreduced Mach numbers at bubble collapse.

The breakdown of the Rayleigh-Plesset variants whenuRu/c approaches unity is reflected in unphysical singu-larities when uRu/c;1 in the major terms of the equa-tions. Since equations with different l lead to similarresults, one solution to this problem is to delete all theprefactors in parentheses containing R/c . We thus arriveat a popular form in the context of sonoluminescence(see, for example, Lofstedt et al., 1995; Barber et al.,1997),

r~RR1 32 R2!5@pg2P02P~ t !#24h

R

R22s

1R

1R

c

d

dt~pg!. (20)

For very strong forcing, these different equations de-viate in the small time interval of bubble collapse,though they are in near-perfect accord for the rest of thedriving cycle. Therefore they can be expected to pro-duce quantitative discrepancies for the properties of thecollapsed bubble (e.g., the minimum radius, maximumgas pressure, etc.). These discrepancies are a principalsource of modeling error for theories of SBSL. Anotheris the treatment of heat exchange via an effective poly-tropic exponent in Eq. (10). Simple refinements for heatexchange have been employed by Yasui (1995), thoughthe only infallible solution is a direct calculation of theheat transfer. This was first carried out in numericalsimulations by Vuong and Szeri (1996) and more re-cently by Moss et al. (1999).

Given these difficulties, it is surprising that solutionsto Rayleigh-Plesset-type equations still provide a quan-titatively accurate representation of the mechanics of asonoluminescing bubble and of many of its accompany-ing effects. Recently, Lin et al. (2001) achieved a betterunderstanding of why finite Mach number corrections toRayleigh-Plesset-type equations are relatively unimpor-tant. They showed that the Rayleigh-Plesset equation isquite accurate even with significant spatial inhomogene-ities in the pressure field inside the bubble. This extendsthe utility of the Rayleigh-Plesset equation into the re-gime where the Mach number for the gas Mg5R/cg(where cg is the speed of sound in the gas) is no longersmall. Lin et al. (2001) show that the relevant conditionis not uMgu,1, but uepu,1, where


ep[RRrgas

Gp~r50,t !, (21)

i.e., what is relevant is the bubble-wall acceleration. Soeven in the sonoluminescence regime, Lin et al. (2001)find excellent agreement when comparing their full gas-dynamical partial differential equation simulations withthe solutions to the Rayleigh-Plesset ordinary differen-tial equation with the assumption of a uniform pressureinside. They also developed an approximation for theinternal pressure field, taking into consideration first-order corrections from pressure inhomogeneity.

In the remainder of this section, we present calcula-tions and experiments on bubble dynamics during acycle of the driving, discussing the various physical ef-fects that are important away from the bubble collapse.Later sections will describe our present knowledge ofthe collapse itself.

C. The bubble’s response to weak and strong driving

First, to give some feeling for solutions to theRayleigh-Plesset equation, we study small oscillations ofthe bubble about its ambient radius R0 . A straightfor-ward calculation (Brennen, 1995) shows that such abubble oscillates at the resonant frequency

2pf05A 1

rR02 S 3gP01~3g21 !

2s

R0D . (22)

A typical sonoluminescing bubble has R0'5 mm, corre-sponding to a resonant frequency of f0'0.5 MHz, muchhigher than the frequency of the driving f'20 kHz.

Figure 20 shows solutions to the modified Rayleigh-Plesset Eq. (20) for a bubble at different forcing pres-sures. At low forcing, the bubble undergoes almost sinu-soidal oscillations of relatively small amplitude, with aperiod equal to that of the external forcing f . Here, theoscillations are essentially ‘‘quasistatic,’’ because theresonant frequency is so much larger than f : the oscilla-

FIG. 20. Solutions to the modified Rayleigh-Plesset Eq. (20) atforcing pressures Pa51.0, 1.1, 1.2, and 1.3 atm. The ambientbubble radius is R052 mm, the frequency f51/Td526.5 kHz.


tory pressure forcing is balanced by the gas pressure(Lofstedt et al., 1993; Hilgenfeldt, Brenner, et al., 1998),with inertia, surface tension, and viscosity playing a neg-ligible role. At a critical pressure around Pa'P0 , suchquasistatic oscillations are no longer possible, resultingin a nonlinear response of the bubble. The critical Padepends slightly on R0 , and is referred to as the (dy-namical) Blake threshold (Blake, 1949; see also Hilgen-feldt, Brenner, et al., 1998). Beyond this threshold,sonoluminescence can occur.

In the SBSL regime, the solution to Eq. (20) in thisregime can be divided into several different stages.

• Expansion: During the negative half-cycle of the driv-ing, the applied tension makes the bubble expand.Since f!f0 , the expansion continues until the appliedpressure becomes positive. The time scale of this re-gime is thus set by the period of the driving pressurewave and is typically '20 ms for sonoluminescenceexperiments. This is sufficient to increase the bubbleradius by as much as a factor of 10.

• Collapse: When the driving changes sign, the ex-panded bubble is ‘‘released’’ and collapses inertiallyover a very short time (;1 ns for SBSL bubbles). Thesolution during collapse is well described by the clas-sical solution of Lord Rayleigh. SBSL light emissionoccurs at the end of the collapse.

• Afterbounces: After the collapse, the bubble spendsthe remaining half of the cycle oscillating about itsambient radius at roughly its resonant frequency f0@f , giving rise to characteristic ‘‘afterbounces.’’

It is worthwhile at this point to comment on the rolesof surface tension and viscosity. The surface tensionterm is dynamically important when it is as large as theexternal forcing pressure, implying that s/R;Pa . Thisoccurs when the bubble radius is smaller than Rs

5s/Pa . For water, this corresponds to a radius of'0.7 mm/(Pa /bar). We shall see below that this lengthscale plays an important role in determining the stabilityof the solutions to the Rayleigh-Plesset equation withrespect to both dissolution and breakup.

Viscous effects are important when the viscous damp-ing time scale is of the order of the time scale of bubblemotion, roughly n/R0

2;f0 , with the kinematic viscosityn5h/r . For water, this does not occur; for more viscousfluids it can be important (Hilgenfeldt, Brenner, et al.,1998).

D. The Rayleigh collapse

Now we turn to the behavior of the bubble radiusnear the collapse. As emphasized above, this is the re-gime in which the Rayleigh-Plesset description is in dan-ger of breaking down. The approach to the collapsedstate, however, can be captured very well by the equa-tion, and is given by a classical solution of Lord Ray-leigh.

Lord Rayleigh (1917) imagined a bubble dynamics forwhich only liquid inertia mattered, with gas pressure,


surface tension, and viscosity all negligible—in otherwords, the collapse of a void. The equation for the wallmotion of the bubble/void is then RR13/2R250 andcan be directly integrated. The solution is of the formR(t)5R0@(t* 2t)/t* #2/5, with the remarkable feature ofa divergent bubble-wall velocity as t approaches thetime t* of total collapse. Lord Rayleigh pointed out thatthis singularity is responsible for cavitation damage, andit is also the central hydrodynamic feature responsiblefor the rapid and strong energy focusing that leads tosonoluminescence.

Clearly, something must stop the velocity from diverg-ing. For the Rayleigh-Plesset Eq. (9) to capture sonolu-minescence, it must contain the physical effect that doesthis. Viscous stresses 4hR/R}(t* 2t)21 and surface ten-sion forces s/R}(t* 2t)22/5 diverge at slower rates thanthe inertial terms @}(t* 2t)26/5# and are therefore tooweak. What about the gas pressure? The collapse rate iseventually so fast that the heat does not have time toescape the bubble. The pressure in the gas then obeysthe adiabatic equation of state, which diverges as pg}R23G}(t* 2t)22 (for a monatomic ideal gas with G55/3), which is stronger than the inertial acceleration.This effect is therefore capable of stopping the collapse.Modifications from the ideal gas law, e.g., van der Waalsforces [see Eq. (10)], do not affect this conclusion.

Although the gas pressure can halt Rayleigh collapse,it turns out that the most strongly divergent term in Eq.(20) is the last one, associated with sound radiation intothe liquid during the last stages of collapse; it diverges as(t* 2t)213/5 (Hilgenfeldt, Brenner, et al., 1998), andoverwhelms the other terms. Up to 50% of the kineticenergy in the collapse may end up as a radiated pressurewave (Gompf and Pecha, 2000).

E. Comparison to experiments

Of course, it is crucial to compare solutions ofRayleigh-Plesset equations to experimental data on thebubble radius as a function of time. However, neitherthe ambient bubble radius R0 nor the driving pressurePa is known a priori. R0 changes through gas diffusion

FIG. 21. Measured R(t) (with Mie scattering, dots) and a fit tothese data based on the Keller equation (solid curve). The thincurve shows the driving pressure P(t). From Matula (1999).


as well as evaporation/condensation of water vapor (seeSec. III), and the (local) driving pressure Pa is very sen-sitive to perturbations of the flask geometry, such asmight be caused by a small hydrophone attempting tomeasure Pa . In addition, the precision of such a hydro-phone is limited to roughly 0.05 bar.

The standard procedure has been to measure R(t)with Mie scattering3 and then to fit the data to Rayleigh-Plesset-type dynamics by adjusting R0 and Pa . A typicaltrace for a sonoluminescing bubble’s radius during acycle of the drive is shown in Fig. 21. The filled circlesrepresent experimental measurements, and the solid lineis a solution to the Keller equation under the assump-tion of isothermal heating (g51). Superimposed as athin line is the applied forcing pressure.

The problem with these fits is that R0 and Pa sensi-tively depend on model details. In particular, if one ad-justs R0 and Pa such that the bubble’s maximum is wellfitted, the afterbounces are always overestimated (seeFig. 21). Better fits can be achieved by allowing moreparameters, e.g., by allowing the material constants suchas the viscosity or the surface tension to vary. Barberet al. (1992), for example, used seven times the usualvalue of the viscosity of water to achieve a fit to theafterbounces. As clarified by Prosperetti and Hao(1999), the larger viscosity effectively parametrizes otherdamping mechanisms not captured in simple Rayleigh-Plesset-type models. In particular, Prosperetti and Hao(1999) included thermal losses, following Prosperetti(1991), reducing the size of the afterbounces. Yasui(1995) had some success by introducing thermal bound-ary layers as well.

Another effect that must be considered when fittingexperimental R(t) curves to Rayleigh-Plesset models isthe invasion of water vapor at bubble maximum. Thisleads to a varying ambient radius R0 over the bubblecycle, being largest at maximum radius. Since manyearly fits of R(t) curves (summarized by Barber et al.,1997) did not consider these effects, the resulting valuesfor R0 and Pa are only approximate.

Mie scattering data near the collapse are also notori-ously difficult to interpret because of the unknown indexof refraction inside the compressed bubble and becausethe bubble radius R becomes of the order of the lightwavelength. The simple proportionality of Mie intensityand R2, valid for larger R , gets lost and the relationeven becomes nonmonotonic (Gompf and Pecha, 2000).Moreover, at collapse, the light is reflected not only fromthe bubble wall, but also from the shock wave emittedfrom the bubble at collapse. This subject will be treatedin the next subsection.

Another light-scattering technique based on differen-tial measurement and polarization (differential lightscattering) has been developed by Vacca et al. (1999) in

3See, for instance, the work of Gaitan, 1990; Barber et al.,1992, 1997; Gaitan et al., 1992; Lentz et al., 1995; Weninger,Barber, and Putterman, 1997; Matula, 1999; Gompf and Pecha,2000; Pecha and Gompf, 2000; Weninger et al., 2000.


order to measure the dynamics of the bubble radius.With this technique a time resolution of up to 0.5 nsaround the Rayleigh collapse has been achieved.

F. Sound emission from the bubble

The Rayleigh-Plesset equation predicts the responsenot only of the bubble radius, but also of the surround-ing liquid. This has been detected by Cordry (1995),Holzfuss, Ruggeberg, and Billo (1998), Matula et al.(1998), Wang et al. (1999), Gompf and Pecha (2000),Pecha and Gompf (2000), and Weninger et al. (2000).Matula et al. (1998) used a piezoelectric hydrophone tomeasure a pressure pulse with fast rise time (5.2 ns) andhigh amplitude (1.7 bars) at a transducer at 1-mm dis-tance from the bubble. Wang et al. (1999) carried out asystematic study of the strength and duration of thepressure pulses as a function of gas concentration, driv-ing pressure, and liquid temperature. They demon-strated that a probe 2.5 mm from the bubble observespressure pulses with rise times varying from 5 to 30 ns asthe driving pressure and dissolved gas concentrationvary. The amplitude of the pressure pulses varies be-tween 1 and 3 bars.

Another study of this type was carried out by Pechaand Gompf (2000; Gompf and Pecha, 2000). They mea-sured pressure amplitudes and rise times consistent withthe other measurements, and were able to measure thepressure pulse much closer (within 50 mm) to thebubble. In addition, using a streak camera and shadow-graph technique, they visualized the shock wave leavingthe bubble (see Fig. 22). Pecha and Gompf (2000) found

FIG. 22. Outgoing shock wave from a collapsing bubble: (a)Streak image of the emitted outgoing shock wave from thecollapsing bubble and (b) an intensity cross section along theline AA8. From Pecha and Gompf (2000).


that the shock velocity in the immediate vicinity of thebubble is as fast as 4000 m/s, much faster than the speedof sound c51430 m/s in water under normal conditions,but in good agreement with the results of Holzfuss, Rug-geberg, and Billo (1998). This high shock speed origi-nates from the strong compression of the fluid aroundthe bubble at collapse. From the nonlinear propagationthe pressure in the vicinity of the bubble can be esti-mated to be in the range 40–60 kbar.

For large enough Pa the presence of shocks in theliquid results from the Rayleigh-Plesset dynamics for thebubble wall, independent of the state of motion of thegas inside the bubble. Comparisons by Wang et al.(1999) between the strength of the measured pulse andthat predicted by the Rayleigh-Plesset equation showthat the strength of the wave in the liquid can be ac-counted for without including the effects of possibleshocks in the gas.

Another interesting effect of the emitted sound radia-tion is that it influences measurements of the bubbleradius by Mie scattering. Gompf and Pecha (2000; Pechaand Gompf, 2000) showed that in the last nanosecondsaround the minimum radius most of the Mie scattering isby the highly compressed water around the bubble (seeFig. 22), not by the bubble surface itself. Neglecting thiseffect leads to an overestimate of the bubble-wall veloc-ity. Taking this effect into account, Gompf and Pecha(2000) found the bubble wall accelerates to about950 m/s, revising previously reported values of1200–1600 m/s by Weninger, Barber, and Putterman(1997; Putterman and Weninger, 2000).

G. Bjerknes forces

All of the calculations above assume that the center ofthe bubble is stationary in space. When neglecting vis-cous effects, the instantaneous force on the bubble isgiven by

Fbubble52E pndS , (23)

where n is the outward normal vector to the bubble sur-face, and p52r] tf is the pressure in the fluid. Multi-plication of Eq. (23) by b, the unit vector in the direc-tion from the origin to the bubble position, gives theforce component in that direction. Using Gauss’s theo-rem and time averaging over a driving period, we obtainthe (primary) Bjerknes force, first described by Bjerknes(1909),

FBj5^b"Fbubble&5^2 43 pR3u¹pu&. (24)

To leading order, we can replace ¹p by ¹p(r50,t) here.While both p and R are periodic, the product occurringin Eq. (24) does not, in general, average to zero. For thecenter of the bubble to be stationary, this force mustvanish. For bubbles at a pressure minimum or maxi-mum, such as in the center of a flask in an SBSL experi-ment, ¹p50, and indeed FBj50. When the bubble isslightly off center, it depends on the relative phase of the


pressure at the center and the bubble-radius dynamics ifthe net effect of FBj is to drive the bubble back to thecenter (stabilizing it), or to drive it further away. Forlinearly oscillating bubbles, it is easy to verify thatbubbles whose resonance frequency f0 is greater thanthe driving frequency f are attracted by pressuremaxima (antinodes) and repelled by pressure minima(nodes). Bubbles with a smaller resonance than drivingfrequency show the opposite behavior. Indeed, for SBSLbubbles f0@f , and they are driven toward the pressureantinode at the center of the flask, where they are drivenmaximally.

A subtle correction to these results originates in thesmall buoyancy force,

Fbuo5rg

T E0

TdV~ t !dt , (25)

which also acts on the bubble (here g is the gravitationalacceleration, Td51/f the period of the driving, and Vthe bubble volume). This upward force must balance thedownward component of the Bjerknes force so that theresulting equilibrium position is not in the center of theflask (z50), but at (Matula et al., 1997)

zequi'rg

kz2Pa

E V~ t !dt

E V~ t !sin~vt !dt, (26)

where kz is the wave number of the standing pressurefield along the direction of gravity. Experiments byMatula et al. (1997) on zequi qualitatively agree withequation Eq. (26). However, the theoretical predictionseems too small by a factor of about 10. Matula (1999)gives evidence that the discrepancy could be connectedwith the back reaction of the bubble on the sound field.

Note that both the acoustic and the buoyancy forcesare fluctuating over one period, leading to small fluctua-tions of the equilibrium position as well. Aspherical,weaker bubble collapses and fainter light emission couldbe a consequence. Matula (2000) presented evidencethat in microgravity, SBSL is somewhat stronger than fornormal gravity, because the bubble collapse is morespherical.

For small driving pressures, the position of an SBSLbubble is stabilized by the Bjerknes forces (see above).But sonoluminescing bubbles are strongly driven, whichleads to variations in the phase shift between driving andbubble dynamics. As pointed out by Cordry (1995),Akhatov et al. (1997), Matula et al. (1997), and Matula(1999), for very large forcing pressure, FBj can becomerepulsive, driving the bubble away from the center ofthe flask, rendering SBSL impossible. The calculationsof Akhatov et al. (1997), Matula et al. (1997), andMatula (1999) demonstrate that this Bjerknes instabilityoccurs above pressure amplitudes of Pa'1.8 bars, al-ready above the upper threshold where single-bubblesonoluminescence usually occurs. Current experimentaldata appear to indicate that shape instabilities limit theupper threshold of sonoluminescence, which is discussed


in detail below. It should be remarked, however, thatthose calculations neglect the back reaction of the bub-ble’s pressure field on the bubble, as well as the effect ofwater vapor, and so might overestimate the Bjerknesthreshold in some situations.

III. THE BUBBLE INTERIOR

One of the key problems in sonoluminescence re-search is that direct measurements of the state of matterinside the bubble are extremely difficult to perform.Practically all information about the conditions insidethe bubble is obtained indirectly. One can measure andmodel the bubble dynamics and then use this as a basisfor inferring the temperatures, pressures, etc. inside thebubble. Or, alternatively, one starts with observations ofthe light emission and uses the spectral information, theintensity, and the widths of the light pulses to deduce theconditions inside. These two approaches to modelingSBSL are sketched in Fig. 23.

The information obtained in these two ways shouldobviously be consistent in a viable theory of sonolumi-nescence. If this consistency condition is fulfilled, how-ever, it is still not clear whether both the hydrodynamicmodel for the interior of the bubble and the model ofthe light emission are correct, as modeling errors couldcompensate each other.

The most crucial variable of the bubble interior forwhich direct measurement is not possible is tempera-ture. As will be discussed in Sec. V, light emission isexpected to depend sensitively on this quantity. In addi-tion, the contents of the bubble are a complicated func-tion of time. Even when starting out with a certain well-defined gas or gas mixture inside the bubble, processesof gas diffusion (Fyrillas and Szeri, 1994), gas rectifica-tion (Lohse et al., 1997), water-vapor condensation andevaporation (Moss et al., 1999), and chemical reactions(Yasui, 1997a; Storey and Szeri, 2000) lead to variationsin composition, both within a cycle (time scales of mi-croseconds) and over many cycles (time scales of sec-onds). All properties of the matter inside the bubble(the equations of state, thermal diffusivity, viscosity, etc.)in turn depend on both gas composition and tempera-ture. Unfortunately, there are few solid data for theseimportant dependencies under the extreme conditionsof sonoluminescence, conditions not approached in any

FIG. 23. The difficulty in modeling SBSL. The bubble tem-perature T(t) is obtained from the radius dynamics R(t) (left),and the spectral radiance Pl(t) is in turn deduced from thetemperature. In contrast to R(t) and Pl(t), the temperaturecannot be measured directly.


other lab experiment, with the possible exception ofshock tubes (Zel’dovich and Raizer, 1966).

A quantitative understanding of single-bubble sonolu-minescence requires that each of these difficulties be ad-dressed step by step. To the present authors, one of theexciting features of modern research on single-bubblesonoluminescence is that it is a testing ground for howwell mathematical models can deal with such a compli-cated situation.

We shall organize our discussion of the state of matterin the bubble’s interior into two parts: in this section, weshall describe the fluid mechanics of the bubble’s interiorand the various attempts to use it to infer bubble tem-peratures at collapse. The goal of this section is to un-derstand both the maximum temperature and the com-position of the bubble. These pieces of information canthen be fed directly into a model of the light emission, adiscussion of which will be deferred to Sec. V. Althoughwe have chosen for reasons of presentation to break upour discussion into these two parts, it should be empha-sized that the research is not at all independent: Modelsof the light emission critically depend on the tempera-tures predicted from hydrodynamic calculations, whilemore sophisticated models of gas dynamics have in turnbeen developed in order to explain properties of thelight emission.

In Sec. III.A, we shall summarize work in which thefull compressible gas-dynamical equations inside thebubble are solved. Over the years (spurred on by moredetailed information about the light emission) the mod-els have incorporated more and more physical effects.The most important modifications of the earliest modelsconcern the inclusion of dissipative and transport pro-cesses, in particular those involving water vapor insidethe bubble.

An alternative approach assumes a (nearly) uniformbubble interior and thus avoids the solution of theNavier-Stokes equations. While less accurate, such mod-eling is computationally inexpensive and allows for thecalculation of temperatures for many more parametercombinations. Several variants of this simpler approachare treated in Sec. III.B.

We briefly mention here that molecular dynamics is athird possibility for modeling the bubble interior. Fol-lowing the motion of the ;1010 molecules or atoms in aSBSL bubble is beyond the capability of present-daycomputers, so that simulations have had to be conductedwith a far smaller number of quasiparticles (Matsumotoet al., 2000; Metten and Lauterborn, 2000), limiting theprospect for quantitative comparison with experiment.One of the main problems of this type of approach isthat, due to the reduced number of particles, the numberof particle collisions is drastically lower than in reality,and therefore it is hard to achieve thermal equilibrium.

A. Full gas dynamics in the bubble

Assuming local equilibrium, the motion of the gas in-side the bubble can be described by the Navier-Stokes


equations and the equations of energy and mass conser-vation (Landau and Lifshitz, 1987),

] trg1] i~rgv i!50, (27)

] t~rgv i!1] j~pgd ij1rgv iv j2t ij!50, (28)

] tE1] i@~E1pg!v i#2] i~v jt ij!2] i~Kg] iT !50. (29)

Velocity components inside the gas are denoted v i ; rgand pg are the gas density and pressure, while E5rge1rgv2/2 is the total energy density, with e the internalenergy per unit mass. T is the gas temperature and Kgits thermal conductivity. The viscous stress tensor isgiven by

t ij5hg~] jv i1] iv j223 d ij]kvk!, (30)

where hg is the gas viscosity and the effects of the sec-ond viscosity have been neglected. These equations haveto be completed with an equation of state, connectingdensity, pressure, and temperature. Depending on thedegree of sophistication, it might also be necessary toinclude the effects of vibrational excitation, dissociation,ionization, and intermolecular potentials. In addition,the material parameters Kg ,hg themselves depend ontemperature and pressure.

Finally, one must impose boundary conditions at themoving bubble wall r5R(t). These can be dealt with intwo ways: either the velocity at the bubble wall is takento be that predicted by the Rayleigh-Plesset equationvr(r ,t)5R(t), or alternatively one could solve the fullfluid-dynamical equations also in the surrounding water.For completeness, boundary conditions for both massand heat exchange must also be formulated.

This problem has been attacked with an increasinglevel of detail, motivated by advances in experiments.We review these efforts in roughly chronological order,grouping them into

• inviscid models (Wu and Roberts, 1993; Moss et al.,1994; Kondic et al., 1995; Chu and Leung, 1997);

• dissipative models (Vuong and Szeri, 1996; Moss et al.,1997; Cheng et al., 1998);

• dissipative models including phase change, in particu-lar that of water vapor (Storey and Szeri, 2000).

All of these approaches treat the bubble as sphericallysymmetric.

1. Inviscid models

Motivated by the measurements of Barber et al.(1992) indicating that the width of the SBSL light pulsewas shorter than 50 ps, early theories focused on theidea that shocks were important for single-bubblesonoluminescence (Greenspan and Nadim, 1993; Wuand Roberts, 1993; Moss et al., 1994). Shock focusingprovides a natural mechanism for producing both ex-tremely high temperatures and a pulse width that ismuch smaller than the time the bubble stays fully col-lapsed.


One of the first numerical solutions of the (spherical)gas-dynamical equations driven by the Rayleigh-Plessetdynamics was done by Wu and Roberts (1993). Themost important approximations of this work were (i) vis-cosity and thermal diffusion are assumed negligible, (ii)no heat or mass exchange takes place between thebubble and the surrounding water, and (iii) a van derWaals equation of state with a polytropic exponent G57/5 is assumed throughout the collapse. For a R054.5 mm bubble driven at Pa51.275 atm and f526.5 kHz, Wu and Roberts (1993, 1994) found aspherical shock wave launching from the wall, focusingto the center, and reflecting outward again. Tempera-tures in excess of 108 K and light pulses of 1.2-ps dura-tion were predicted.

The high temperatures and short pulse widths can beunderstood from the classical analytical solution of theequations of gas dynamics in an imploding sphere byGuderley (1942; see also Landau and Lifshitz, 1987).Guderley neglected viscosity and thermal diffusion, andassumed an ideal gas equation of state. His result showsthat a converging shock wave focuses to the center ofthe sphere with a radius

Rs~ t !;~ t* 2t !a, (31)

with an exponent a'0.6884 for G55/3 and a'0.7172for G57/5. Here t* represents the time at which theshock reaches the bubble center. In the case of G55/3,the temperature at the center of the shock diverges asRs

2b , with b'0.9053. When the shock reaches thebubble center, the temperature is mathematically infi-nite. With a van der Waals equation of state the samesingularity (31) with a slightly different exponent occurs;Wu and Roberts (1994) show that their simulations con-verge onto this solution. Similar calculations were per-formed by Moss et al. (1994) and Kondic et al. (1995).Moss et al. (1994) used a more sophisticated equation ofstate for air inside the bubble, limiting the maximumtemperature through the energy-consuming processes ofdissociation and ionization. They also solved the fullequations for the motion of the water around thebubble.

The unphysical divergence of temperature from Gud-erley’s solution must, of course, be avoided in reality.Evans (1996) noted that the converging spherical shockwave will be susceptible to instabilities in its shape(much like the bubble shape instabilities discussed indetail in Sec. IV). Evans calculated a relatively slow di-vergence of the relative size of the perturbations, ap-proximately scaling as dRs /Rs}r2x0 with x0,1. Heconcluded that very high temperatures would still bepossible inside the bubble, but that the shape instabilitydoes set a limit to the degree of energy focusing.

Dissipative processes also are capable of stopping thetemperature divergence of Guderley’s solution. Bothheat and temperature diffusion potentially disrupt theformation of a shock wave because they counteract thesteepening gradients at the shock front. The questionhere is primarily one of time scales: when a convergingpressure wave travels towards the bubble center, can it


steepen sufficiently quickly to develop into a (strong)shock? Or is dissipation so strong that a front never de-velops before the center is reached? These ideas werefirst touched upon by Vuong and Szeri (1996, describedin some detail below), who demonstrated the difficultyof generating shocks in a model where the transport co-efficients depend on gas density.

In the years since the shock picture was proposed,there has been direct experimental evidence arguingagainst the likelihood of shock formation. Following thework of Ohl et al. (1998; Ohl, 2000) on laser-inducedcavitating bubbles, which also emit light (‘‘single-cavitation bubble luminescence,’’ SCBL), Baghdassarianet al. (1999, 2001) found that highly shape-distortedbubbles are still able to give off considerable lumines-cence. In such bubbles a spherical shock wave cannotexist. Evans (1996) showed that an asphericity of only5% in the bubble wall is sufficient to disrupt the energy-focusing power of a shock.

2. Dissipative models

Kondic et al. (1995) realized the necessity of includingdissipation in the gas dynamics, giving some estimatesfor heat conduction and pointing out radiative transferas an energy-loss mechanism (the latter process turnsout to have a negligible effect on the total-energy bal-ance of the bubble). Going beyond estimates, Vuongand Szeri (1996) included thermal and viscous dissipa-tion in solving the equations of motion of the gas.4 To dothis properly, it is essential to understand how the mate-rial properties (thermal conductivity and viscosity) de-pend on the temperature and on the pressure. At theextreme conditions achieved in the bubble, those depen-dencies are not known, and one has to try either to de-rive them from first principles or to extrapolate approxi-mate relationships known from measurements at lowerpressures and temperatures. Vuong and Szeri (1996) andalso Cheng et al. (1998) do the latter and assume thelinear relation

Kg5Kg~T0!T

T0(32)

between heat conductivity and temperature; the pres-sure dependence is neglected (see also Kamath et al.,1993; Yasui, 1995).

Vuong and Szeri (1996) and Cheng et al. (1998) alsoincluded the heat flux in the water by coupling to theequation

] tTw1u]rTw5x l

1r2 ]r~r2]rTw! (33)

4We remark that viscous effects had been previously consid-ered in bubble dynamics, primarily by Prosperetti (1991) andKamath et al. (1993). As described in Sec. II, these authorsalso drew the important conclusion that the heat transport be-tween the interior of the bubble and the liquid results in abubble with an isothermal boundary for most of the forcingcycle.


for r.R(t); Tw(r ,t) and u(r ,t) are the temperatureand the spherical velocity field in the water, and x l thethermal diffusivity. The radius R(t) is given by theRayleigh-Plesset equation. At the boundary r5R(t),the heat flux out of the bubble and into the fluid mustmatch, and the same is generally assumed for the tem-peratures of liquid and gas.

The main result of Vuong and Szeri’s paper shows thatno shocks occur in argon bubbles. Though some wavystructures can be seen, they do not steepen to a shock, ascan be seen in Fig. 16. The temperature profile in thecollapsing bubble is not dramatically peaked near thecenter, but rather shows slow variations for most of theradius, with a strong decline to the ambient water tem-perature near the wall. Nevertheless, through the nearlyadiabatic compression of the gas in the bubble, very hightemperatures above 105 K are achieved in the simula-tions. The precise values depend on the type of gas em-ployed and on the control parameters, as seen in Table I.However, all values are dramatically lower than in thecalculations without viscosity and thermal conductivitycarried out by Wu and Roberts (1993), Moss et al.(1994), and Kondic et al. (1995).

Cheng et al. (1998) confirmed the results of Vuongand Szeri (1996) for argon and in addition repeated thecalculation for pure nitrogen gas. They found that, fornitrogen with its smaller polytropic exponent G57/5,shocks can develop for strong enough forcing, but theyare limited to a tiny region around the center of thebubble, and the peak power (assuming blackbody radia-tion) is much less than that of argon. In particular, thewidth of the power pulse is in the range <5 ps, muchsmaller than observed by Gompf et al. (1997). Earlier,Moss et al. (1997) had arrived at similar conclusionsabout shock occurrence when neglecting viscosity andnormal heat conduction, but including heat conductionof the ionic and electronic constituents of the ionized gasgenerated in the bubble. Their finding that ‘‘it is moredifficult to generate a shock in Ar than in N2’’ thusseems to be true no matter what the dissipation processinvolved.

One of the main reasons that shocks are suppressed inargon bubbles is that the strong heating already suppliedthrough adiabatic compression (with G55/3) results [viaEq. (32)] in an enormous thermal conduction, which lev-els temperature gradients in the bubble (Vuong and

TABLE I. Maximum temperatures achieved in a R054.5 mmpure argon bubble driven at f526.5 kHz. A dissipative gasdynamics model was used in five runs using different drivingpressure amplitudes Pa and gas species. From Vuong and Szeri(1996).

Run Gas Pa (atm) Tmax (K)

I Ar 1.1 20 000II Ar 1.2 52 000III Ar 1.3 118 000IV He 1.3 43 000V Xe 1.3 202 000


Szeri, 1996), preventing the steepening of the wavy dis-turbances into a shock. In nitrogen (G57/5) this effect isless pronounced and shocks can develop for strong col-lapses (Moss et al., 1997; Cheng et al., 1998).

A more general statement on shock suppression ismade by Lin and Szeri (2001), who attribute the diffi-culty of observing any sign of spherically convergingshocks to the presence of adverse entropy gradients. Inthe case of sonoluminescing bubbles, the sound speed inthe gas increases towards the center of the bubble, de-laying and weakening shock formation.

Vuong and Szeri (1996) also analyzed the dependenceof the peak temperature on the type of inert gas em-ployed. Both MBSL and SBSL intensity increase withincreasing atomic weight of the noble gas, from He toXe. Often, the lower thermal conductivity of xenoncompared to helium has been favored as the main rea-son for higher temperatures and more light emission inXe (Verral and Sehgal, 1988; Greenspan and Nadim,1993; Hickling, 1993, 1994). However, the inert gasesalso vary in diffusivity, ionization potential, and manyother physical properties, and it is important to deter-mine which are crucial in the context of sonolumines-cence. Vuong and Szeri find that contributing to thehigher peak central temperatures in their numericalsimulations with xenon bubbles (see Table I) are notonly the effect of lower thermal conductivity of Xe, butin addition another mechanism also due to the largermass of this inert gas: waves in heavier gases carry moremomentum and travel more slowly, leading to strongercompression lasting for a longer time.

A further noteworthy complication of the gas dynam-ics inside the bubble originates from a weak mass segre-gation effect inside an oscillating bubble. Storey andSzeri (1999) have shown that for a mixture of two gaseswith different masses the heavier one has a slightlyhigher concentration at the bubble’s edge, whereas thelighter one is concentrated towards the bubble’s center.This segregation is caused by the mass dependence ofthe diffusion coefficients. If the bubble’s interior were amixture of inert gases (the example analyzed by Storeyand Szeri, 1999), this effect would have little relevance.But when water vapor is considered (see Sec. III. A. 3),mass segregation can have important effects when com-paring helium bubbles with xenon bubbles (Storey andSzeri, 1999; Yasui, 2001): The water-vapor concentrationin helium bubbles is smaller than that in xenon, since inhelium the mass diffusivity is higher. Such a distributionof vapor tends to diminish the differences in peak tem-peratures between Xe and He (see also Sec. V. F. 3).

3. Dissipative models including water vapor

Over time, dissipative models for the bubble’s interiorhave become more and more complex. An importantstep was the realization that the water vapor inside theSBSL bubble plays a crucial role in regulating the heattransfer to and from the bubble (Kamath et al., 1993;Yasui, 1997a; Colussi and Hoffmann, 1999; Moss et al.,1999; Storey and Szeri, 2000; Toegel, Gompf, et al.,


2000). This effect of liquid vapor has been well known inmultibubble sonoluminescence for decades (Jarman,1959; Flint and Suslick, 1989, 1991a, 1991b). In addition,the water vapor invading a collapsing bubble will un-dergo chemical reactions that also change the tempera-ture (Kamath et al., 1993; Yasui, 1997a; Gong and Hart,1998; Storey and Szeri, 2000).

Initially, attempts to incorporate water vapor were re-stricted to simplified models (e.g., Yasui, 1997a). To ourknowledge, the first full numerical simulation of asonoluminescing bubble with water vapor was per-formed by Moss et al. (1999), building on their earlierwork (Moss et al., 1997), which will be discussed in detailin Sec. V. Moss et al. (1999) kept the amount of watervapor fixed and uniform during the cycle and did nottake chemical reactions into consideration. Thermalconduction is included for the neutral gas inside thebubble, for ions, for electrons, and for the water aroundthe bubble. Due to the high pressures and densities in-side the collapsed bubble, ions, electrons, and neutralparticles equilibrate on a time scale much shorter thanthe other relevant time scales, so that a single tempera-ture field T(r ,t) is sufficient (rather than having differ-ent temperatures for the ions, electrons, etc., as in Mosset al., 1997). The equation of state was obtained from acombination of data and theoretical work by Young andCorey (1995).

The fixed water-vapor content was fit to one data setof Gaitan and Holt (1999) on the light intensity forknown forcing pressure, frequency, and compression ra-tio (Rmax /R0), and then extrapolated to other forcingpressures, frequencies, and compression ratios. For stan-dard conditions at Tw520 °C and f'20 kHz, the fixedwater-vapor concentration was found to be 30–40 %. Atcollapse, the transfer of vapor out of the bubble is tooslow to keep up with the bubble-wall motion, so thatvapor becomes trapped inside the bubble (Moss et al.,1999). This point of view was later confirmed and inves-tigated further by Storey and Szeri (2000; see below).For the extreme conditions achieved at collapse, fixingthe amount of water inside the bubble is not as crude anapproximation as it may sound.

The main result of Moss et al. (1999) was that the in-clusion of a sufficient quantity of water vapor leads to asmaller (effective) adiabatic exponent g. Heating is thusreduced and shock waves can occur close to the center,in agreement with earlier work by Cheng et al. (1998)for nitrogen bubbles. The shock-wave heating, however,is very inhomogeneous. Although the center of thebubble can be very hot (up to 130 000 K in the examplesgiven), the total radiating volume within the shock frontis so small that there is far less optical emission than foradiabatic heating with a larger G which does not lead toshock waves (Vuong and Szeri, 1996; Cheng et al., 1998).A large part of the compressional energy of the collapseis found to go into internal degrees of freedom ratherthan into heating of the bubble. Moss et al. (1999) arguethat this is one reason why sonoluminescing bubbles arebrighter at low water temperature (Barber et al., 1994),where the water vapor pressure is less.


The model of Moss et al. (1999) does rather well inexplaining the trends in the experimental data of Gaitanand Holt (1999). Moss et al. in particular stress that apure argon bubble leads to very different heating char-acteristics (and thus light emission) when compared toan argon bubble with some water vapor in it; only thelatter is consistent with the data. Calculations withoutwater vapor would predict that a strongly driven bubblein the Gaitan-Holt measurements (R056.0 mm,Rmax564 mm) should emit 100 times as much light as aweakly driven bubble (R052.1 mm,Rmax530 mm). Theexperiments actually show an intensity ratio of only;10, in agreement with simulations including water va-por. The calculated results are extremely sensitively de-pendent on the experimental conditions andparameters—or, likewise, the modeling assumptions. Aparticularly striking example is the dependence on theambient pressure P0 . Reducing it from P051 bar to 0.99bar can lead to 1.6 times more light, according to themodel. The reason for the sensitivity lies not so much inthe hydrodynamics, but in the extremely strong depen-dence of the light intensity on the temperature achievedin the bubble, as we shall discuss in detail in Sec. V. Inthe above example the temperature of the light-emittingregion in the bubble increases by just 5%.

The restriction of Moss et al. to constant water-vaporcontent is relaxed in a remarkable paper by Storey andSzeri (2000), who build on their earlier work on masssegregation in gas bubbles containing a mixture of dif-ferent inert gases (Storey and Szeri, 1999). The transportparameters (thermal diffusion, mass diffusion, and vis-cosity) are calculated from equations based onChapman-Enskog theory (Hirschfelder et al., 1954);thermal dependencies are considered as far as they areknown. The equations of state are of the Soave-Redlich-Kwong type (Gardiner, 1984; Reid et al., 1987). Storeyand Szeri (2000) argue that although the basic physicalmechanisms they uncover are robust, the exact numbersdo depend on (unknown) details of the equation ofstate.

Evaporation and condensation are modeled using akinetic theory (Carey, 1992). The rate of mass transport(per unit area) at which water molecules pervade theinterface is }(pH2O2psat), where pH2O is the partialpressure of water and psat the saturation pressure at thetemperature of the interface. Not all water moleculesthat hit the wall actually stick to it, only a fraction sathereof. This accommodation coefficient is chosen to be0.4, following Yasui (1997a) and Eames et al. (1997).

As an example, Storey and Szeri (2000) study abubble initially consisting of argon, with R054.5 mmand driven at Pa51.2 bars and f526.5 kHz: Includingwater vapor, the maximum temperature is reduced from20 900 K (cf. Vuong and Szeri, 1996) to 9700 K, due tothe lower G. No shock waves are observed under theseconditions. The amount of water in the bubble is notconstant at all, as can be seen from Fig. 24: A largeamount of water evaporates into the bubble during themain expansion when the pressure is low; at this stagevapor is in equilibrium with the vapor pressure. At


bubble maximum, about 90% of the bubble contents iswater. Vapor condenses out of the bubble again at col-lapse, but not completely, since the time scale of thecollapse tdyn5R/uRu becomes much smaller than thetime scale for the transport of water vapor out of thebubble. The vapor transport is a two-step process, con-sisting of diffusion to the wall and condensation, so thatit involves two time scales, one for vapor diffusion in thebubble,

tdif5R2

DH2O~R ,T !'

1DH2O~R0 ,T0!

R03T0

1/2

RT1/2 , (34)

and one for condensation at the wall,

tcon5R

cgA2pGMH2OT0

9sa2M0Tint

, (35)

where cg5AGpg0 /rg0 is the sound velocity of the gas inthe initial state, and MH2O and M0 are the molecularmasses of water and the initial bubble content, respec-

FIG. 24. Importance of water vapor exchange between bubbleand fluid: (a) Bubble dynamics and (b) number of water mol-ecules in the bubble as a function of time for an (initially) R054.5 mm argon bubble driven at Pa51.2 atm and f526.5 kHz. Note the comparison in (b) to the constant num-ber of argon atoms. From Storey and Szeri (2000).


tively. Figure 25 shows the profiles of the water-vapormolar fraction inside the bubble for various times: In theearly part of the collapse, tdyn@tdif ,tcon , so that theprofiles are uniform. The vapor concentration is de-creasing, as water has sufficient time to escape. Later,when tdyn!tdif ,tcon (the three lowest profiles of Fig.25, starting from 15 ns before the collapse), the water-vapor profile is essentially ‘‘frozen’’ within the bubble.Therefore, the bubble consists of '14% water vapor(on a molar basis) through the collapse; the vapor istrapped in the bubble’s interior.

Comparing the time scales tdif and tcon , Storey andSzeri (2000) found that the vapor transport was alwaysdiffusion limited (tdif.tcon). Models with unrealisti-cally low accommodation coefficients sa;0.001, likethat of Colussi et al. (1998), on the other hand, couldwell be condensation limited.

Rather remarkably, when applying their numericalmodel to the more strongly forced bubbles of Moss et al.(1999; see above), Storey and Szeri (2000) find averagewater-vapor concentrations close to what Moss et al. ob-tained from fitting their numerical results to the Gaitanand Holt (1999) data mentioned above, a strong indica-tion that these numbers are in the right ballpark.

In the second part of their paper, Storey and Szeri(2000) consider chemical reactions of the water vapor.The reaction scheme employed originates from Maasand Warnatz (1988) and includes 19 forward and reverseelementary reactions of the nine species Ar, H, H2 , O,O2 , OH, HO2 , H2O, and H2O2 (argon only enters thereactions as a third body). The reaction rates were takenfrom Gardiner (1984) and in the high-pressure limit, forwhich there is ‘‘considerable uncertainty,’’ from Bow-man et al. (1999).

With the chemical reactions included, the maximumtemperature for the standard case (a R054.5 mm argonbubble driven at Pa51.2 bars and f526.5 kHz) de-creases from 9700 K (with water vapor, but without re-

FIG. 25. Profiles of the molar fraction of water vapor vs radiusfor several times prior to the moment of minimum radius(time50, leftmost profile). In order of increasing radius, theother curves correspond to times prior to collapse oft525, 15, 214, 2153, 2299, 2455, and 2843 ns. Calculationfor a R054.5 mm argon bubble driven at Pa51.2 atm and f526.5 kHz. From Storey and Szeri (2000).


actions) to 7000 K. Most of the reactions are endother-mic and therefore eat up the collapse energy, whichotherwise would be used for heating. The reaction path-ways can also be interpreted as additional degrees offreedom, further lowering the effective polytropic expo-nent. Note that the time scales of the chemical reactionsare so fast (because of the high densities) that thermo-chemical equilibrium should prevail in the collapse up tothe point of minimum radius.

How does the bubble temperature depend on theforcing pressure? Figure 26 shows results by Storey andSzeri (2000) for the temperature at the center as a func-tion of the compression ratio Rmax /Rmin . No shocks oc-curred in these calculations, so that the center tempera-ture is a fairly good indication of the overall bubbletemperature. For other parameter combinations (notshown in Fig. 26), Storey and Szeri (2000) do find shocksand very high maximum temperatures, but only right atthe center, corresponding to ;0.1% of the total bubblevolume. A suitably averaged temperature, representa-tive of the actual energy content of the bubble, will bevery similar to those shown in Fig. 26. It is noteworthythat for these shocks the presence of water vapor is nec-essary, as expected from the work of Cheng et al. (1998)and Moss et al. (1999). For a low compression ratio,hardly any vapor invades the bubbles of Fig. 26 and thethree cases (without water vapor, with nonreacting watervapor, and with reacting water vapor) give roughly thesame maximum temperature. The remarkable feature,however, is that the temperature levels off for large com-pression ratios when water vapor is taken into account,asymptoting to '10 000 K without chemical reactionsand '7000 K with chemical reactions included. Thus theinitial power-law increase of the maximum temperaturewith expansion ratio does not extend into the regime ofSBSL temperatures: the effect of the larger expansionratio of stronger forced bubbles is compensated by the

FIG. 26. Dependence of the bubble center temperature on thecompression ratio Rmax /Rmin . s, calculations without watervapor; * , with water vapor, but no reactions; 1, with watervapor, including reactions. Note the logarithmic scale on theaxes. From Storey and Szeri (2000).


presence of more water vapor, thus limiting the heatingat collapse.

This finding has important implications for the questfor ‘‘upscaled’’ sonoluminescence: To learn the limits ofenergy focusing and light emission, researchers havetried to induce more violent bubble collapses by, for ex-ample, applying nonsinusoidal driving pressures (Holz-fuss, Ruggeberg, and Mettin, 1998) or reduced drivingfrequencies as suggested by Hilgenfeldt and Lohse(1999) and Toegel, Gompf, et al. (2000). But the largerexpansion ratios achieved with these techniques lead toa larger water-vapor content of the bubble, which againlimits the heating at collapse. A theoretical study byToegel, Gompf, et al. (2000) shows that these two effectsroughly cancel each other, essentially leading to thesame temperature and the same amount of light at lowerdriving frequencies for otherwise identical bubble pa-rameters. A residual upscaling effect may still be ob-served (as in Barber and Putterman, 1991) due to thepossibility of stabilizing bubbles with larger R0 at lowerdriving frequencies (see Sec. IV.D).

B. Simple models

The previous section has shown that when the com-plex interplay of physical effects inside a sonoluminesc-ing bubble is meticulously included, spatial inhomogene-ities inside the bubble are not very pronounced.Therefore it seems reasonable to try to approximate thebubble’s interior by spatially constant, time-dependentpressure pg(t) and temperature T(t). Clearly, such anapproximation is too crude to capture some of the afore-mentioned effects such as mass segregation (Storey andSzeri, 1999). However, such modeling simplifications areextremely useful tools in exploring the phase space ofsonoluminescence (see Sec. IV). Such a scanning ofphase space is at present not possible for the completemodels discussed in the previous section.

Simple models assuming a uniform bubble interiorhave been developed with increasing detail; here, weshall discuss two types. Just as in the more elaboratemodels, one can either completely neglect heat and masstransfer to and from the bubble’s exterior (Sec. III.B.1)or try to embody these effects (Sec. III.B.2).

1. Homogeneous van der Waals gas without heat and massexchange

The simplest model is to assume an adiabatic equationof state for the bubble interior (Lofstedt et al., 1993;Barber et al., 1997),

pgas~ t !5S P012s

R0D ~R0

32h3!G

$R~ t !32h3%G, (36)

and the corresponding temperature equation

T~ t !5T0

~R032h3!G21

@R~ t !32h3#G21 , (37)


equivalent to Eq. (10) when replacing g by G5Cp /Cv ,the ratio of the specific heats. Equation (36) supple-ments the Rayleigh-Plesset equation and permits its so-lution.

One obvious problem with Eqs. (36) and (37) is thatan adiabatic bubble motion is assumed with no heat ex-changed between the bubble and the exterior. Aspointed out in Sec. II, Peclet number estimates via Eq.(11) show that there is almost unrestricted heat ex-change for most of the oscillation cycle, and the motionis isothermal. Only at bubble collapse is the Peclet num-ber Pe larger than 1. This means that most of the timethe ratio of the specific heat G has to be replaced by anisothermal exponent g51. Near the cavitation collapsethere is a change in the adiabatic value g→G . Roughlyspeaking, this transition will occur when the Peclet num-ber is of order unity.

In linear approximation, Prosperetti (1977c) derived atransition function g „Pe(t)… from the isothermal behav-ior g51 to the adiabatic behavior g5G . Hilgenfeldtet al. (1999a, 1999b) have employed this approach to cal-culate approximately the temperatures and pressures inSBSL bubbles. The Rayleigh-Plesset equation is thensupplemented by a differential version of Eq. (37) withvariable g „Pe(t)…,

T52@g „Pe~ t !…21#3R2R

R32h3 T2~T2Tw0!xg /R2.

(38)The last term contains the ambient water temperatureTw0 and the thermal diffusivity of the gas xg , which athigh densities is assumed to follow Chapman-Enskogtheory (Hirschfelder et al., 1954). Equation (38), to-gether with the Rayleigh-Plesset equation, gives asimple model for bubble radius and bubble temperature.The peak temperatures obtained—see Fig. 27—are com-

FIG. 27. Temperature inside the SBSL bubble within the sim-plified model of Hilgenfeldt et al. (1999a, 1999b). (a) Bubbledynamics and (b) temperature dynamics of R055 mm bubbledriven at 1.3-atm pressure amplitude at f520 kHz. The tem-perature is calculated via Eq. (38); the full width at half maxi-mum of the asymmetric peak is '1 ns.


parable to bubble temperatures resulting from the moresophisticated model by Storey and Szeri (2000). The lat-ter publication reports somewhat lower temperatures,though, because of the inclusion of heat loss and massexchange in the model, to which we turn in the followingsection.

2. Homogeneous van der Waals gas with heat and massexchange

In more sophisticated variants of the model, the heatand mass exchange between the bubble and its exteriorare explicitly taken into account. One of the first modelsof this type was conceived by Yasui (1997a). The ingre-dients of his approach are (i) a Rayleigh-Plesset-typeequation for the bubble radius with a van der Waals gasinside; (ii) a spatially homogeneous bubble interior withtime-dependent pressure and temperature (the materialconstants inside the bubble such as the thermal conduc-tivity are modeled as a function of the temperature); (iii)water-vapor exchange with the bubble’s exterior due tocondensation and evaporation; (iv) heat exchange withthe bubble’s exterior modeled by an energy flux depend-ing on the compression, the temperature gradient, andcondensation/evaporation; (v) a thin layer of wateraround the bubble that can be heated; (vi) 25 chemicalreactions of the water vapor, following Kamath et al.(1993) and using data from Baulch et al. (1972–1976). Itis crucial that the net effect of these reactions be con-sumption of thermal energy, i.e., they must be endother-mic.

The central result of Yasui (1997a) is displayed in Fig.28: At bubble maximum, the bubble consists nearly ex-clusively of water vapor. Even at collapse, the bubblestill retains some of the water (;1% of the total bubblecontents). Because of the invading water vapor and theendothermic chemical reactions, the maximum tempera-ture inside the bubble at typical control parameters isonly around 10 000 K, in agreement with the completemodel of Storey and Szeri (2000).

Models like that of Yasui (1997a) are useful for exam-ining the energy balance at collapse. For standard pa-rameters of single-bubble sonoluminescence (see Fig.28), one finds in the last 120 ps before collapse a reduc-tion of the thermal energy by 1.4 nJ through chemicalreactions and by 0.6 nJ through thermal conduction. Theloss through photon emission is negligible, only about0.2 pJ.

Yasui (1997a) assumed that the transport of massthrough the boundary layer was condensation limited,rather than diffusion limited (mass diffusion was not ex-plicitly modeled, and thus assumed instantaneous).However, Storey and Szeri (2000) later showed thattransport under SBSL conditions is diffusion limited(see Sec. III.A.3). Therefore Toegel, Gompf, et al. (2000;Toegel et al., 2002) took the opposite approach and de-veloped a simple diffusion-limited model for water-vapor exchange between bubble and liquid, using aboundary layer approximation. The diffusive change inthe number of water molecules over time is then


NH2Od 54pR2D]rnur5R'4pR2D

n02n

ldiff, (39)

where n0 corresponds to the equilibrium density of va-por molecules at the wall and n is their actual concen-tration. The diffusion length is obtained through dimen-sional analysis as ldiff5min@(RD/R)1/2,R/p# , where D isthe gas diffusion constant. The cutoff prevents theboundary layer from becoming unphysically large.

The heat flux is treated in complete analogy to theflux of water vapor by Toegel, Gompf, et al. (2000; Toe-gel et al., 2002), i.e.,

Q54pR2xmix

Tw02T

lth, (40)

where Q is the heat content of the bubble, Tw0 the equi-librium (ambient) temperature, and the thermal diffu-sion length is l th5min@(Rxmix /R)1/2,R/p# with the ther-mal diffusivity xmix of the gas mixture. Yasui (1997a)

FIG. 28. Bubble temperature and number of molecules withthe model of Yasui (1997a). The parameters are R055 mm,Pa51.35 bars, f520.6 kHz, P051 atm, and a water tempera-ture of Tw520 °C. From Yasui (1997a).


uses a boundary layer as well, but in addition assumes atemperature jump between the outer edge of thisboundary layer and the liquid. This temperature slip ismore usually associated with low-density systems anddoes not appear in other work like that of Toegel,Gompf, et al. (2000) or Storey and Szeri (2000).

Chemical reactions can also be included in a boundarylayer model (Toegel et al., 2002). The most importantendothermic process is

H2O15.1 eV↔OH1H, (41)

whose inclusion already shows the essential effects ofmore sophisticated reaction schemes. Within the ordi-nary differential equations (ODE) formalism of Toegelet al. (2002), Eq. (41) yields reaction rates in good ac-cord with those of Storey and Szeri (2000).

Together with the first law of thermodynamics and avan der Waals equation of state, the above formalismyields another ODE for the temperature inside thebubble,

CvT5Q2pgV1hwNH2Od 2(

X

]E

]NXNX , (42)

where the sum is over all species X5Ar, H2O, OH, andH. The derivatives ]E/]NX as well as the heat capacityCv take into account rotational and vibrational degreesof freedom in the various molecules and are thus depen-dent on temperature. Therefore Eq. (42) is an implicite-quation for T . Finally, hw is the enthalpy of water mol-ecules near the (cold) bubble wall.

Equation (42) provides closure of the model togetherwith a Rayleigh-Plesset equation variant. Similar equa-tions were discussed by Yasui (1997a), though the ex-plicit form of the terms differs as indicated above, andthe diffusion-limited character of the transport was nottaken into account. Such a set of four first-order equa-tions for R(t), R(t), NH2O(t), and T(t) can be solvedfor various physical parameters, such as forcing pressurePa , ambient radius (without water vapor) R0 , watertemperature Tw0 , and driving frequency f , withoutmuch numerical effort. The scheme can also be easilyextended to more reactions. The results of Toegel,Gompf, et al. (2000) agree well with the full simulationsof Storey and Szeri (2000).

Most models that include chemical reactions insidethe bubble—whether based on partial differential equa-tions like that of Storey and Szeri (2000) or on ODE likethose of Kamath et al. (1993) and Storey and Szeri(2001)—seem to underestimate the temperature insidethe bubble. With the exception of the Yasui (1997a)model (in which water vapor is not prominent becauseits transport is very fast), all of these models seem toimply temperatures substantially below 10 000 K in theSBSL regime. Assuming a thermal light-emissionmechanism (see Sec. V), these temperatures do not pro-duce enough photons to comply with experimental re-sults. Toegel et al. (2002) address this paradox and sug-gest that according to the Le Chatelier–Brown principle,the high densities inside the bubble favor the back reac-


tion H1OH→H2O, in particular because of the finiteexcluded volume of the particles. The energy-consumingwater dissociation is thus reduced, and higher tempera-tures in the bubble are possible (see also Sec. V). Aresult like this suggests that one cannot extrapolate thereaction rates at temperatures and pressures achieved inother laboratory experiments to the unusual regime ofsonoluminescence particle densities. Thus it is advisableto revert to a first-principles statistical physics approachin order to deduce reaction rates under SBSL condi-tions, deriving the laws of mass action directly from thepartition function (Toegel et al., 2002).

Another similar, simple model with both heat andwater-vapor exchange was developed by Storey andSzeri (2001). Here the authors even drop the assumptionof boundary layers for heat and mass transport and relyon the ratios of the relevant time scales for bubble dy-namics, particle diffusion, and heat diffusion. This ap-proach still contains the same essential physics as themodels of Yasui and Toegel et al. Storey and Szeri(2001) quantitatively tested the simple model againsttheir full simulations (Storey and Szeri, 2000). In bothapproaches, they find the same trends (and similar val-ues) for the peak temperature in the bubble, the masspercentage of argon in the bubble, and the number ofreaction products, which again lends credibility to thesimple ODE approach.

C. How accurate are the bubble temperatures?

It was mentioned in the introductory remarks that anunderstanding of the bubble interior and SBSL lightemission hinges on a good understanding of the bubbletemperature. The many models developed so far, withdifferent degrees of sophistication, result in predictionsfor the peak temperature (suitably averaged over thebubble) ranging from 6000 K to slightly above 20 000 K,given the same (typical) driving parameters for an SBSLbubble. While this factor of uncertainty of about 3 issignificant (and leads to widely different results for theensuing light emission), it is reassuring that these modelsshare a number of common traits: (i) They agree thatthe interior of the bubble heats up and becomes at leastas hot as that measured for MBSL bubbles; (ii) watervapor is a major temperature-limiting factor, forming asubstantial part of the bubble contents at collapse; and(iii) the temperature, when averaged over the bubble, isnot likely to rise much above 104 K, ruling out the muchtoo high predictions of earlier attempts at modelingSBSL. We shall see in Sec. V that a thermal origin ofSBSL is deemed very likely today, and that the tempera-ture range presented here does support light emission inthe experimentally observed range of intensities. In-deed, any nonthermal theory of SBSL has to explainfirst why thermal emission, which explains the experi-mental results, is suppressed.

The uncertainties in bubble temperature and the re-sulting predictions for light emission do not allow for adirect comparison between experiment and theory for aspecific combination of parameters (e.g., a bubble with


R055 mm, Pa51.3 bars, and f520 kHz). On top of themodeling uncertainties, the sensitive dependence of lightemission on parameters severely limits the reproducibil-ity of measurements. A comparison to experiment overa whole range of parameter values is much more prom-ising, focusing on the experimentally robust dependenceof bubble dynamics and light emission on various exter-nal parameters. The following section takes this stepfrom single-parameter combinations to an overview ofthe parameter space of SBSL—starting with the ques-tion of under what conditions sonoluminescence can beobserved at all.

IV. THE PARAMETER RANGE OF SINGLE-BUBBLESONOLUMINESCENCE

In the previous section we studied the dynamics of abubble under the action of a driving pressure of ampli-tude Pa , frequency f5v/2p , and with an ambientbubble radius R0 , all fixed to appropriate values forsonoluminescence to occur. The key question now iswhat happens if these and other experimental param-eters are changed—will SBSL still be observed, and ifso, will it be brighter or dimmer? What are the physicalprocesses that determine the limits of the parameter re-gime of sonoluminescence? Experiments have shownthat, apart from the parameters implicit in the Rayleigh-Plesset equation, other quantities of crucial importanceto SBSL are the concentration of gas c` dissolved in theliquid, the temperature of the liquid, the type of liquid,and the type of gas. The goal of this section is to understand these parameter dependencies quantitatively.

Various physical constraints limit the parameter rangein which sonoluminescence can be observed: to emitlight, the bubble must be forced strongly enough for acavitation event to occur during each cycle of the drive;the bubble must not break into pieces, which roughlytranslates into the requirement that viscous processesand surface tension be strong enough to limit the growthof bubble shape instabilities. For the consistent, stablelight emission of SBSL, the number of gas moleculesinside the bubble, averaged over one cycle of oscillation,must neither increase nor decrease. This requirement iswhat sets the ambient radius R0 of the bubble; it in-volves a subtle interplay between diffusive processes ex-

FIG. 29. Minimal bubble radius Rmin /R0 as a function of R0and Pa . The frequency is f526.5 kHz.


changing gas between the bubble interior and the out-side liquid, and chemical reactions. And finally, it isnecessary that the Bjerknes forces holding the bubbletrapped in the flask (see Sec. II.G) be strong enough toensure that the center of the bubble does not move ap-preciably. This section presents the current understand-ing of each of these effects and assesses the extent towhich the theoretical predictions agree with experi-ments.

A. The Blake threshold

Regardless of the exact mechanism of sonolumines-cence, it is abundantly clear that the light results fromenergy focusing during a rapid bubble collapse. There-fore the bubble must be forced strongly enough to in-duce a cavitation event of sufficient violence—in es-sence, the Rayleigh collapse solution of Sec. II.D mustbe fully established. Whether this happens depends onboth the ambient bubble size (mass of gas inside thebubble) and the forcing pressure. Figure 29 shows theminimal radius during a cycle of the drive as a functionof forcing pressure Pa and ambient radius R0 . There isan abrupt transition R0(Pa) where the onset of Rayleighcollapse occurs and the gas inside the bubble getsstrongly compressed, leading to heating. Thereforesonoluminescence can only occur above this threshold.

The functional form of this threshold curve can bededuced from dimensional considerations: other thanthe ambient radius R0 , the parameters in the Rayleigh-Plesset equation are Pa , P0 , s, r, h, and v. Since we aredealing with the transition from gentle oscillation to in-ertial collapse, we do not expect dissipative effects to beimportant and exclude the viscosity h from our consid-erations. From the remaining parameters, two indepen-dent length scales can be derived: the resonant bubblesize R0

res5A3P0r21v22 [cf. Eq. (22)] and the capillarylength scale s/P0 . As we have seen in previous sections,SBSL bubbles are driven far from resonance, so R0

res istoo large to be important here.

The relationship between the critical ambient size andthe pressure must therefore have the form

R0c5

s

PafS Pa

P0D (43)

with a dimensionless function f . A closer look at theRayleigh-Plesset equation allows for the specialization

R0c5C

s

Pa2P0, (44)

where C is a scalar constant. A calculation like this wasfirst performed by Blake (1949) for bubbles under staticpressure. He found C52/3 for isothermal bubble move-ment (cf. Brennen, 1995). The threshold in R0-Pa spaceseparating gently oscillating from violently collapsingbubbles is therefore known as the Blake threshold.More detailed studies for driven bubbles arrive at C54)/9'0.77 (Leighton, 1994; Hilgenfeldt, Brenner,et al., 1998).


Only bubbles larger than R0c in Eq. (44) can show

SBSL, so that the Blake threshold criterion cuts downthe available parameter space considerably. When shockwaves inside the bubble were considered crucial for thelight emission, Hilgenfeldt et al. (1996) suggested replac-ing this criterion with the threshold for strong shock for-mation, expected when the maximum bubble speed dur-ing collapse exceeds the speed of sound in the gas(although shocks can and will form at lower speeds aswell). Because of the abruptness of the Blake threshold,supersonic velocities are reached already at parametercombinations very close to the threshold. Thus thisshock threshold, like any other dynamical criterion forstrongly collapsing bubbles, will effectively yield thesame result as the Blake threshold calculations.

Further constraints set an upper limit to the bubblesize, in order to ensure the bubble’s stability. Let us firstconsider diffusive stability.

B. Diffusive stability

Since bubble dynamics and energy focusing duringcollapse are sensitively dependent on R0 , it is crucial fora stable SBSL bubble to maintain the same ambient ra-dius, i.e., not to exchange any net mass with its sur-roundings. The gas exchange between the bubble andthe liquid is affected by diffusion of gas through the liq-uid and by advection of this gas with the fluid velocity.

The typical model that is used for this process startswith the transport equation for the mass concentrationc(r ,t) (mass/volume) of gas around a spherical bubble:

] tc1u]rc5D1r2 ]r~r2]rc !, (45)

where D is the gas diffusion coefficient in water. Thevelocity u(r ,t)5R2R/r2 of the fluid a distance r fromthe center of the bubble is given by Eq. (6), with R(t)entering through solution of the Rayleigh-Plesset equa-tion. The gas in the bubble is assumed to remain in equi-librium with that in the liquid at the boundary of thebubble wall; hence the gas concentration at the bubblewall is given by Henry’s law,

c~R ,t !5c0pg~R ,t !/P0 . (46)

Finally, the gas concentration far from the bubble isgiven by the ambient concentration c` ,

c~` ,t !5c` . (47)

The mass loss/gain of the bubble is then proportional tothe concentration gradient at the bubble wall,

m54pR2D]rcuR(t) . (48)

This model for the gas exchange is accurate as long asthe Rayleigh-Plesset solution is valid. As was empha-sized above, this may not be generally true very close tothe point of bubble collapse. At collapse, many othereffects could also play an important role and affect thegas transfer, including (a) sound waves in the liquid, (b)breakdown of Henry’s law at the interface because of its


fast motion, (c) chemical reactions inside the collapsedbubble, and (d) phase transformation (boiling) of theliquid surrounding the bubble.

First, we shall examine the consequences of mass dif-fusion alone and see what they imply about the sonolu-minescence experiments. The advection of the solute inEq. (45) reflects the conservation of fluid volume: aspherical shell of fluid around the oscillating bubble ispushed in or out when the bubble contracts or expands.For this reason, it is useful to think about the solutionsto the equation in Lagrangian coordinates tracking thevolume changes according to (Plesset and Zwick, 1952)

h~r ,t !5 13 @r32R3~ t !# . (49)

This transformation trick has often been used (e.g.,Eller, 1969 or Brennen, 1995). If one was to neglect dif-fusion, the concentration field outside the bubble wouldbe just c5c(h). However, even in the limit where themass diffusion coefficient is very small, it has importantconsequences (for example, an undriven bubble eventu-ally dissolves by diffusion). The Henry’s law boundarycondition at the bubble wall implies that the gas concen-tration at the bubble wall is time dependent and in gen-eral different from the concentration c` in the bulk.There is therefore a boundary layer near the bubblewall, where the gas concentration relaxes from the valuedictated by Henry’s law to c` . The numerical profile isshown in the paper by Hilgenfeldt et al. (1996).

The concentration gradient in the boundary layer dic-tates the mass flux into the bubble. There are two timescales over which mass transfer occurs: (i) during asingle cycle of driving, gas is pushed into and out of thebubble when the bubble expands and contracts; (ii) overmany cycles of driving, small net gains or losses of masspotentially accumulate to produce significant changes inthe number of molecules in the bubble.

With the coordinate (49), we obtain from Eq. (45)

] tc5D]h~r4]hc !. (50)

Focusing on the boundary layer region close to thebubble (r'R) and redefining the time t[*R4dt yields

]tc5D]hhc , (51)

which is a pure diffusion equation. Following Fyrillasand Szeri (1994), the solution to this equation can bewritten c(h ,t)5c`1cosc(h ,t)1csm(h ,t). Here, cosc isan oscillatory solution, transporting gas back and forthat the frequency of the bubble oscillations, but effectingnegligible net gas transport. The smooth part of the pro-file csm , however, determines the gas exchange overtime scales much longer than the oscillation period. Forthe number of molecules in the bubble to maintain equi-librium over long periods of time, it is necessary that^c(h ,t)&5c` , where ^•& denotes averaging over t. Spe-cializing to the surface of the bubble, this implies (be-cause of Henry’s law)

^c&45c0

E R4pg~R ,t !dt

P0E R4dt[c0

^pg&4

P05c` . (52)


The index 4 indicates that the average is weighted withR4. The mass loss from the bubble over long periods oftime can be calculated via

m524pR02D

^c&42c`

d, (53)

where d is the boundary layer thickness, given by

d5R02S E0

` dh

Td21E @3h1R3~ t !#4/3dtD 21

, (54)

where the time integral spans one oscillation period Td .Because of the dominance of the maximum radius inthe integration (see also below), d can be approximatedas 'R0

2/Rmax (cf. Hilgenfeldt et al., 1996). The timescale of mass diffusion is therefore approximatelyR0

3rg /RmaxDc0;0.1 s, with the gas density rg understandard conditions.

Fyrillas and Szeri (1994) applied the method of mul-tiple scales (see, e.g., Hinch, 1991) to systematically de-rive the above formulas and demonstrate their accuracyto leading order in D(vR0

2)21. As long as the diffusivetime scale is much longer than the bubble oscillationcycle, the approximation is very reliable and can there-fore be used instead of the more cumbersome direct so-lution of Eq. (45).

Equation (53) can be used to study the stability of theequilibrium ambient radius R0* : taking R05R0* 1e andexpanding ^c&(R0) around R0* implies that

e52D

rgd

d^c&4

dR0U

R0*e .

The equilibrium point is therefore stable as long asd^c&4 /dR0.0 or, equivalently, d^p&4 /dR0.0.

Lofstedt et al. (1995) pointed out a useful approxi-mate version of Eq. (52): Since the time average isweighted by R4, and the bubble spends a large fractionof the cycle near the maximum radius, the equilibriumcondition is approximately

FIG. 30. ^pg&4 /P0 as a function of the ambient radius R0 forsmall forcing pressure amplitudes, Pa51.0 atm to Pa

51.4 atm (top to bottom). From Hilgenfeldt et al. (1996).


pg~Rmax!

P05S R0

RmaxD 3

5c`

c0. (55)

What do these results imply for sonoluminescence? Fig-ure 30 plots ^pg&4 as a function of R0 for an argonbubble for various forcing pressure amplitudes Pa . Wefirst examine small pressure, Pa'1.0 atm, and gas con-centrations of, say, c` /c0'0.3. There is an unstableequilibrium at R0

e'6 mm. Smaller bubbles shrink andlarger bubbles grow. The mechanism of growth is calledrectified diffusion: when the bubble is large, the gas con-centration in the bubble is small, and vice versa, so thereis an influx or outflux of material. The net effect is notzero for a nonlinear oscillation because at maximum ra-dius (i) the area for gas exchange is larger, and (ii) thediffusive boundary layer is stretched out (Brennen,1995). Both effects favor net growth of the bubble. Notethat an undriven bubble dissolves because its internalpressure exceeds that in the liquid. Sufficiently strongdriving will overcome this tendency to shrink, and startrectified diffusion.

Turning our attention to higher forcing pressures (Fig.30), we see that the average pressure is quite diminished,because Rmax is very large now [see Eq. (55)]. Thismeans very low gas concentrations (saturation levelsc` /c0) are needed to achieve diffusive equilibrium(around 0.005 for Pa51.3 atm). Moreover, under theseconditions the equilibria become stable, as demonstratedby the positive slopes. In this example, the bubblereaches an equilibrium size of R0* '5 mm and will notgain or lose any further mass.

Figure 31 is a diagram showing the equilibrium bubblestates in the R0-Pa parameter space for three fixed con-centrations (c` /c050.5,0.02,0.002). Stable equilibriahave positive slope ]R0 /]Pauc`

; negative slope repre-sents unstable equilibria. To the right of each linebubbles grow, and to the left they shrink.

FIG. 31. Bifurcation diagrams in the R0-Pa parameter space.The regimes with positive slope are stable. Gas concentrationsare (right) c` /c050.002, (middle) c` /c050.02, and (left)c` /c050.5. To the left of the curves the bubbles shrink andfinally dissolve, to the right of them they grow by rectifieddiffusion. From Hilgenfeldt et al. (1996).


Figure 31 shows that there are no diffusively stablesonoluminescence bubbles for large gas concentrations,where all equilibria are unstable.5 For small concentra-tions c` /c0 the situation is quite different. There arestable equilibria at large Pa and small R0 . Only in thisregion, and for very low gas concentration c` /c0;0.001–0.02 (depending on Pa ,) is the bubble diffu-sively stable, and only then is stable SBSL possible. Thisis again easily appreciated using the approximation (55)of Lofstedt et al. (1995): when the bubble is small andstrongly forced, the collapse ratio is Rmax /R0'10. HenceEq. (55) implies that c` /c0;1023 for the bubble to be inequilibrium, as observed in Fig. 31.

Lofstedt et al. (1995) realized that this requirement ofdiffusive equilibrium for strongly forced bubbles canonly be fulfilled at very small gas concentrations, inagreement with experiments using argon or other inertgas bubbles. However, the finding blatantly disagreeswith the results obtained for air bubbles, where stabilityis achieved at roughly 100 times larger gas concentra-tions. Recall that Gaitan (1990) needed to degas to onlyabout 40% of saturation. If he had had to go to 100timeslower concentrations, he might never have discoveredSBSL. Lofstedt et al. (1995) postulated an ‘‘anomalousmass flow mechanism’’ to resolve this discrepancy. In-deed, in order to keep a constant R0 in a liquid withsuch a high gas concentration, an air bubble would haveto eject mass far above the usual diffusive transport ratein order to balance rectified diffusion.

C. Sonoluminescing bubbles rectify inert gases

1. The mechanism

The stabilizing mass ejection mechanism of sonolumi-nescing bubbles is now believed to be the consequenceof chemical reactions that occur when the bubble is com-pressed. The maximum temperature of the bubble(larger than 10 000 K; see Sec. III) is large enough todestroy both molecular nitrogen and oxygen, so thesemolecules will be largely dissociated in the collapsedbubble. Moreover, as discussed in Sec. III, the high ex-pansion ratio of the bubble results in a substantialamount of water vapor. Chemical reactions between wa-ter vapor and dissociated nitrogen and oxygen are wellknown to atmospheric chemists dealing with acid rain:the reaction products are NO, NH, and ultimatelyHNO3 and NH3 . All of these substances (except NO)are very soluble in water. The idea of the argon rectifi-cation hypothesis (Lohse et al., 1997) is that the reactionproducts are absorbed completely into the water, deplet-

5Strictly speaking, stable equilibria do appear due to the‘‘wiggles’’ in the curve of equilibria that result from resonanceeffects. Brenner, Lohse, et al. (1996) speculated that thewiggles could describe multiple stable equilibria. However,when introducing further damping mechanisms into theRayleigh-Plesset dynamics as done by Hao and Prosperetti(1999a), the wiggles weaken and it is presently not clearwhether there are multiple stable equilibria.


ing the bubble of nitrogen and oxygen and thus estab-lishing an efficient mass-loss mechanism. A sonolumi-nescing air bubble thus rectifies argon, the onlysubstance inside the bubble that does not dissociate,which is contained in air with a concentration of 1%.

This argument immediately explains the discrepanciesbetween the diffusive equilibria of air and argon: onlythe inert gas in the bubble is in diffusive equilibriumwith the bulk liquid. Therefore the relevant parameterfor the stability of sonoluminescing bubbles in equationslike Eq. (52) is the partial pressure (or concentration) ofthe inert gas dissolved in the liquid, and not the partialpressure of air. Defining the argon fraction q as

c`Ar

c05q

c`air

c0,

the stability criteria for air and argon should differ by afactor of q'0.01, the fraction of argon in air. Indeed,experiments show stable sonoluminescence in airbubbles for c` /c0;0.2–0.4 (Gaitan et al., 1992) andstable sonoluminescence in argon for c` /c0;0.002–0.004 (Barber et al., 1995). The hundredfolddifference is quantitatively consistent with the argon rec-tification theory. The earliest measurements confirmingconsequences of argon rectification were by Holt andGaitan (1996), described in detail below.

As pointed out by Lohse and Hilgenfeldt (1997), an-other hint for the chemical activity inside the bubble isthe isotope scrambling found by Hiller and Putterman(1995), who had analyzed sonoluminescence in H2 andD2 gas bubbles, in both normal and heavy water. Thefour respective spectra are grouped according to the liq-uid, not according to the dissolved gas. This suggests thatwater vapor invades the bubble and undergoes chemicalreactions, ultimately leaving H2 for normal water andD2 for heavy water, independent of the type of hydrogenbubble with which one started. Hiller and Putterman’s(1995) hydrogen bubbles were all diffusively unstable.

The most direct verification of argon rectification wasaccomplished in a clever experiment by Matula andCrum (1998). They were able to precisely measure thetransition time over which the light intensity changedwhen the driving conditions were changed. They ob-served (see Fig. 32) that an air bubble that has not pre-viously emitted light reaches maximum sonolumines-cence intensity after a transition time of about 103

driving cycles. By contrast, if a bubble initially emitslight, when the forcing pressure is dropped below thelight-emitting threshold and then immediately raisedback above it, sonoluminescence turns on immediately.As experimental controls, Matula and Crum used pureargon and pure nitrogen bubbles. The transition time forthe nitrogen bubble matched the air bubble that initiallydid not emit light, and the argon bubble matched the airbubble which had sonoluminesced previously. Thistherefore provides direct evidence for the hysteresis thatwould be expected with argon rectification.

Other predictions from the rectification hypothesishave not been experimentally verified to our knowledge,e.g., the suggestion for making stable sonoluminescence


without degassing the liquid, preparing a percentage ofargon in nitrogen so that c`

Ar/c050.003, within the diffu-sive stability window. The ultimate proof of the argonrectification hypothesis would be the detection of thechemical species leaving the bubble. Lohse et al. (1997)suggested detecting a pH change due to the acidic reac-tion products. However, pH meters are probably notsensitive enough to detect an effect even if the experi-ment is run for a long time. Therefore Lepoint et al.(1999) exchanged the water in the experiment forWeissler’s reagent. The oscillating bubble produced per-oxide and chlorine radicals which oxidized iodide to io-dine, giving the distinct blue color of the iodine-starchreaction. Thus chemical reactions in and around a singlebubble were conclusively demonstrated. The bubbletriggered the reaction even at driving pressures belowthe SBSL threshold, as the temperature there can al-ready be sufficiently large. A thread of blue color wasobserved to emerge from the bubble, usually in either anupward or downward direction.

2. Bubble equilibria with chemical reactions

Chemical reactions modify the classical diffusive equi-libria described above. The arguments outlined abovecan only make a quantitative prediction for the extremecases of no chemistry or complete argon rectification. Todescribe the transition between these two stages (for ex-ample, as the driving pressure acting upon an air bubbleis increased), the chemical reaction rates have to bemodeled explicitly. As described in Sec. III, this model-ing can be done to varying degrees of detail, tradingaccuracy for computational speed. Full simulations arestill too expensive computationally to allow a mappingof the parameter space of SBSL. Therefore simple reac-tion models like those of Yasui (1997a), Toegel et al.(Toegel, Gompf, et al., 2000; Toegel et al., 2002), or Sto-rey and Szeri (2001) are often employed.

In such a model, the changes in species numbers aregiven by diffusive transport [see Eq. (39) for water va-por], and by chemical reactions such as Eq. (41). Reac-tion rates under SBSL conditions must be either in-ferred from general principles (cf. Toegel et al., 2002) orextrapolated from lower-temperature reaction data (cf.Kamath et al., 1993; Bernstein et al., 1996; Yasui, 1997b).

As an example, the change in the numbers NAr andNN2

of argon and nitrogen particles in a mixture of thesetwo gases can be written as

NAr54pR2DAr]rc

Arur5R

mAr, (56)

NN25

4pR2DN2]rc

N2ur5R

mN2

2ANN2expS 2

T*

T~ t ! D ,

(57)

where the first term in Eq. (57) represents diffusionand the second dissociation reactions, depending onthe bubble temperature T(t). DAr , DN2

, mAr , and mN2

are the respective diffusion coefficients and molarmasses. For simplicity, Lohse et al. (1997) assumed


that the reaction rates follow a modified Arrheniuslaw, using empirical parameters from Bernstein et al.(1996) appropriate for nitrogen dissociation: A'631019(T0 /T)5/2(r0 /mN2

)(R0 /R)3 cm3/(mol s) gives thetime scale of the reaction; T* '113 000 K is the activa-tion temperature, T0 is ambient temperature, and r0 theequilibrium gas density. This reaction law is verycrude—it neglects backward reactions as well as the ki-netics of the expulsion of reaction products; however, itis sufficient for a demonstrative calculation.

The concentration fields cAr(r ,t),cN2 (r ,t) in Eqs.(56) and (57) separately obey Henry’s law at the bubblewall, using the saturation concentrations c0

Ar ,c0N2 specific

for the gases (c0Ar'3c0

N2). For diffusive stability, bothgas species must fulfill the equilibrium condition (52)separately, with concentrations far from the bubblec`

Ar ,c`N2 given by the percentage j l of argon in the gas

dissolved in the liquid.Requiring the temporal averages of NAr and NN2

to

FIG. 32. (1) The maximum radius and (3) sonolumines-cence intensity of an air bubble plotted over approximately240 consecutive cycles. The ‘‘radius’’ is actually a signal level,proportional to the square of maximum radius (assuming ageometrical optics limit for Mie scattering from a sphere). (a)The bubble is initially below the light-emission threshold (thephotomultiplier tube signal level corresponds to noise). At ap-proximately the 120th acoustic cycle, the drive pressure is rap-idly increased to a value above the SBSL threshold. (b) Forthis case, the bubble is initially in a stable sonoluminescingstate. The drive pressure is then rapidly reduced to a valuebelow the light-emission threshold, and then quickly increasedagain after a time period of approximately 90 ms. In both (a)and (b), the drive amplitudes at the lower and upper values arethe same. From Matula and Crum (1998).


vanish yields the number of gas particles in the bubble atequilibrium and thus the percentage jb of argon insidethe bubble, which is larger than j l because of dissocia-tion reactions removing N2 . Of course, the last term inEq. (57) only contributes when T(t) is large. The chemi-cal reaction rate therefore depends on the detailed spaceand time dependence of the temperature in the bubble.The calculations below use a simple spatially uniformtemperature model inspired by Prosperetti (1977c) asdiscussed in Sec. III.B.1.

The resulting equilibrium radii R0* in the R0-Pa planefor air (j l50.01) at p` /P050.20 are shown in Fig. 10.For small forcing, the temperatures are not high enoughto initiate chemical reactions, so that the curve A of(unstable) equilibria corresponds to those describedabove for diffusion alone. These bubbles either shrink orgrow by rectified diffusion, and for them jb5j l to agood approximation. At high forcing (curve C), the re-actions burn off all the N2 , so that the bubble containspure argon (jb'1); this equilibrium corresponds to the(stable) equilibrium at the argon partial pressurep`

Ar/P050.01p` /P050.002.Note that curves A and C belong to the same experi-

mental system now, and that bubbles of low Pa and largeR0 grow by rectified diffusion, while those with high Paand large R0 shrink due to the mass loss through chemi-cal reactions (arrows in Fig. 10). There necessarily mustbe a region of intermediate forcing pressures where bothprocesses cancel, and an additional equilibrium exists.These equilibria, which prove to be stable, are shown ascurve B in Fig. 10. The onset of appreciable nitrogendissociation (T'9000 K) is depicted as a thin line inthe figure and is quite close to line B. Note that thistemperature is much smaller than T* . The argon frac-tion jb along curve B varies with Pa , but stays consid-erably smaller than 1.

This picture predicts the following sequence of eventsas the forcing pressure is increased. At low Pa , thebubble starts near the unstable equilibrium curve A,where the bubble is growing through rectified diffusionand eventually undergoes shape instabilities (see Sec.IV.D below). If the forcing pressure is turned up fairlyrapidly, the ambient radius will grow in this regime. Atsome point, the forcing Pa will be large enough so thatthe bubble is trapped by the stable equilibrium B. WhenPa is now further increased, the ambient radius shrinksalong the curve B. Upon increasing the driving even fur-ther, fluctuations throw the bubble onto the stable curveC, so the ambient radius grows again. This behavior hadbeen observed previously by Barber et al. (1994); seeFig. 7. A direct measurement was made by Gaitan andHolt (1999), who measured both the maximum bubbleradius and the ambient radius as a function of forcingpressure (see Fig. 33). The breakdown in R0 and Rmaxcaused by the onset of chemical reactions inside thebubble is clearly visible.

Holt and Gaitan (1996) showed that at p` /P050.2there is a relatively large forcing pressure regime Pa'1.2–1.3 atm where bubbles dissolve (cf. Fig. 10). Suchdissolution islands do not exist with pure diffusion (as


shown above; see Fig. 31). Gaitan and Holt (1999) dem-onstrated that the stable equilibria of sonoluminescingbubbles were in excellent agreement with the shape pre-dicted when assuming a much lower gas concentration.These observations agree with the theoretical resultstaking both diffusion and molecular dissociation into ac-count.

In their comprehensive experimental study of bubblestability diagrams, Ketterling and Apfel (1998, 2000a)

FIG. 33. Experimental maximal bubble radius and phase dia-gram for SBSL: (a) the ratio between maximum radius Rmaxand ambient radius R0 ; (b) Rmax as a function of the drivingpressure Pa ; (c) data points in the Pa vs R0 phase space; h,stable glowing bubbles; j, stable nonglowing bubbles; 1 , un-stable, nonglowing bubbles; dashed line, the experimentallyfound shape instability; solid lines, theoretical diffusive equi-libria for various argon concentrations. To compare this figurewith theoretical phase diagrams such as Fig. 10 or other experi-mental results such as Fig. 12, one should exchange abscissaand ordinate. From Gaitan and Holt (1999).


demonstrated that all stable and light-emitting bubblesindeed lie on a positively sloped equilibrium curve in theR0-Pa plane, corresponding to the partial concentrationof argon in the mixture. This is true regardless of howmuch nitrogen is present. A representative phase dia-gram is shown in Fig. 12 for air in water at c` /c050.2.The solid line represents the (unstable) diffusive equilib-rium for the mixture, and the dashed line is the (stable)diffusive equilibrium for c` /c050.002 (the argon in themixture). As the forcing pressure is increased, thebubble follows a stable equilibrium line above the un-stable diffusive equilibrium (corresponding to line B inFig. 10) and then transitions to the lower-concentrationstable equilibrium (where it starts emitting light.) Apure argon bubble at c` /c050.0026 follows its stablediffusive equilibrium (see Fig. 5 of Ketterling and Apfel,1998). No other stable configurations are possible. Onthe other hand, Fig. 4 of Ketterling and Apfel (1998)shows that a pure nitrogen bubble at c` /c050.1 followsa stable equilibrium much like curve B of the air mix-ture, but cannot reach a stable sonoluminescing state athigher Pa . Taken together, these two types of behaviorsynthesize the results in Fig. 12, strongly implying thatthe air bubble is composed mainly of nitrogen alongcurve B, and mainly argon along curve C, in agreementwith the predictions of argon rectification.

Very recently, Simon et al. (2001) confirmed that light-emitting bubbles follow the stable diffusive equilibriumcurves based on argon concentration alone (see Fig. 34).They employ a new experimental technique for measur-ing the parameters of the bubble dynamics (Pa and R0)based on the timing of the light flash in the acousticperiod. They also find that the attainable conditions in-side a sonoluminescing bubble are more extreme at alow partial air pressure of 15 mm Hg as compared to 150mm Hg, because then the bubbles are closer to the

FIG. 34. Experimental data in the phase diagram in the Pa-R0

plane: 1 , with an air concentration of c`air/c050.2; h, 0.15; 1 ,

0.1; j, 0.02. The data lie on the stable diffusive equilibriacurves (solid lines) for argon concentrations of c`

Ar/c050.002,0.0015, 0.001, and 0.0002, confirming the argon rectificationtheory. Experimental data are taken for as large a Pa as pos-sible, to probe the shape instability threshold. From Simonet al. (2001).


Blake threshold (where the most extreme conditions areachieved) and can be driven with larger Pa . This is dueto the bubble shape instabilities treated in the next sub-section.

D. Shape stability

The theoretical diffusive equilibrium curves stretch tofar larger ambient radii R0 than those observed forSBSL bubbles. There must be another requirement lim-iting the ambient size (or the total mass content) of thebubble. This limit is set by the onset of instabilities inthe shape of the oscillating bubble. The analysis of shapestability is a classical problem in bubble dynamics, pio-neered by Plesset (1949), Birkhoff (1954), Plesset andMitchell (1956), Strube (1971), and Prosperetti (1977d).In this section we present the application of these ideasto single-bubble sonoluminescence.

1. Dynamical equations

To analyze the linear stability of the radial solutionR(t), consider a small distortion of the spherical inter-face R(t),

r5R~ t !1an~ t !Yn~u ,f!, (58)

where Yn is a spherical harmonic of degree n . The goalis to determine the dynamics an(t) for each mode. Thederivation of Plesset (1954) follows along the same linesas the derivation of the Rayleigh-Plesset equation,which it recovers to zeroth order in an . A potential flowoutside the bubble is constructed to satisfy the boundarycondition that the velocity at the bubble wall be R1 anYn . This potential is then used in Bernoulli’s law todetermine the pressure in the liquid at the bubble wall.If viscous effects are neglected, applying the pressurejump condition across the interface yields a dynamicalequation for the distortion amplitude an(t),

an1Bn~ t !an2An~ t !an50, (59)

where bn5(n21)(n11)(n12) and

Bn~ t !53R/R , (60)

An~ t !5F ~n21 !R

R2

bns

rR3G . (61)

The stability of the spherical bubble then depends onwhether solutions to Eq. (59) grow or shrink with time.It is already apparent here that Eq. (59) has the form ofa parametrically driven oscillator equation (Hill equa-tion), with the radial dynamics R(t) governing the peri-odic driving.

A more accurate stability analysis requires taking ac-count of viscosity and other dissipative processes. Vis-cosity, treated by Prosperetti (1977d), poses difficultiesbecause viscous stresses produce vorticity in the neigh-borhood of the bubble wall, which spreads convectivelythrough the fluid. Once created, the vorticity acts backon the dynamics of an(t). This interaction is nonlocal intime, and so the problem requires solving integrodiffer-


ential equations for the vorticity in the liquid, coupledwith the shape oscillations. Details can be found in theliterature (Prosperetti, 1977d; Hilgenfeldt et al., 1996;Hao and Prosperetti, 1999b). Here we simply summarizethe results of a simple ‘‘boundary layer’’ approximation,which assumes that the vorticity is localized in a thinregion around the bubble. It was again Prosperetti(1977b) who first realized the usefulness of this approxi-mation. If d is the boundary layer thickness, the prefac-tors of Eq. (59) are modified to

An~ t !5~n21 !R

R2

bns

rR3

12nR

R3 F2bn1n~n21 !~n12 !1

112d

RG ,

(62)

Bn~ t !53R

R1

2n

R2 F2bn1n~n12 !2

112d

RG , (63)

with the kinematic viscosity n of the liquid. The viscouscontribution to An(t) is not important since the ratiobetween the third and the second terms of the right-hand side in Eq. (62) is typically nrR0v/s&1022. How-ever, in the second term of the right-hand side of Eq.(63) it introduces a damping rate which causes exponen-tial damping of shape modulations. The amount ofdamping strongly depends on both the boundary layerthickness d and on n . Brenner et al. (1995) and Hilgen-feldt et al. (1996) choose d to be the minimum of theoscillatory boundary layer thickness An/v and the wave-length of the shape oscillation R/(2n).

The Hill equation (59) is driven by the strongly non-linear Rayleigh-Plesset dynamics R(t). Therefore, incontrast to the monofrequent driving of the prototypicalMathieu equation, instabilities in an can be excited onthe many different time scales of the bubble oscillationdiscussed in Sec. II. In particular, three types of instabili-ties can be distinguished: the parametric instability (overtime scales of the oscillation period), the afterbounceinstability (over time scales of the bubble afterbounces),and the Rayleigh-Taylor instability (over time scales ofthe Rayleigh collapse).

2. Parametric instability

The parametric shape instability acts over the rela-tively long time scale Td52p/v (period of the driving).If the nonspherical perturbations of bubble shape shownet growth over one oscillation period, they will over-whelm the bubble after many periods. This argumentneglects possible (nonlinear) saturation effects not con-tained in the linear approximation (59), which could in-hibit further growth of the perturbations.

In the relevant parameter regime for the parametricinstability, R(t) and thus also An(t) and Bn(t) arestrictly periodic in time with period Td . Thus the stabil-


ity of the Hill equation (59) can be rigorously analyzed(Nayfeh and Mook, 1979). Instability occurs wheneverthe magnitude of the maximum eigenvalue of the Flo-quet transition matrix Fn(Td) of Eq. (59) is larger than1. The Floquet transition matrix is defined as the propa-gator of the perturbation vector over one period,

S an~Td!

an~Td! D5Fn~T !S an~0 !

an~0 ! D . (64)

By numerically computing the eigenvalues of the Flo-quet transition matrix, one can map out the phase dia-gram of parametric stability, i.e., identify parametricallystable and unstable regions.

In the sonoluminescence parameter range of Pa'1.2–1.5 atm and a typical frequency f526.5 kHz, cal-culations with the boundary layer approximation suggestthat parametric instability sets in for ambient radii inexcess of R0

PI'4 –5 mm, with only a weak dependenceon Pa . Refined boundary layer models like those ofProsperetti and Hao (1999) or Augsdorfer et al. (2000)take into account higher-order terms in R/c , heat losses,or the varying gas density in the bubble upon collapse.These models find upper stability bounds for R0 abouthalf a micron larger, because the additional effects resultin less violent oscillation dynamics and smaller values ofthe (destabilizing) bubble acceleration.

For nonsonoluminescing bubbles, stability diagramsof a similar type were first measured by Eller and Crum(1970) and later by Horsburgh (1990). These experimen-tal and theoretical studies examine larger bubble sizeswith smaller forcing pressures for which sonolumines-cence cannot occur. Applying the above shape stabilityanalysis in the regime Pa;0.5–1 atm (Brenner, Hilgen-feldt, and Lohse, 1998; Brenner et al., 1999; Hao andProsperetti, 1999b; Augsdorfer et al., 2000) gives similarthresholds to those found in experiment by Eller andCrum (1970), Horsburgh (1990), and Gaitan and Holt(1998). For Pa&0.9 atm, the n53 mode takes over asthe most unstable surface mode from the n52 mode(Brenner, Hilgenfeldt, and Lohse, 1998; Brenner et al.,1999; Augsdorfer et al., 2000), just as was found in theexperiments by Gaitan and Holt (1998). Overall, largerbubbles can be stabilized at small Pa , and the thresholdin R0 becomes strongly dependent on the driving pres-sure in this regime.

Holt and Gaitan (1996; Gaitan and Holt, 1999) alsomeasured shape instabilities in the sonoluminescence re-gime and close to it. Their experimental results werecompared with theory in the work of Brenner, Dupont,et al. (1998) and Hao and Prosperetti (1999b), with goodagreement in evidence [see Fig. 35(a)]. Similar resultsare found for the higher n modes (see Hao and Prosper-etti, 1999b).

The boundary layer approximation has been criticizedby Putterman and Roberts (1998) as underestimatingthe amount of dissipation. These authors claim thatthere is insufficient evidence for the role of shape insta-bilities in limiting sonoluminescence, although Brenner,Dupont, et al. (1998) found that the approximation gives


results in good agreement with experiments. It is there-fore important to ask how the shape stability resultschange when the boundary layer approximation is notmade, and the full integrodifferential equations aresolved. Such a comparison was carried out by Wu andRoberts (1998) and Hao and Prosperetti (1999b). Atypical result for forcing pressure around Pa;1 atm isshown in Figs. 35(a) and 35(b), taken from Hao andProsperetti (1999b): Hardly any difference between theexact result and the boundary layer approximation canbe seen. Wu and Roberts (1998) found similar agree-ment even for larger Pa in the regime of SBSL. Theseresults indicate that the boundary layer approximationof Prosperetti (1997c), Brenner et al. (1995), and Hilgen-feldt et al. (1996) is appropriate for sonoluminescenceexperiments.

FIG. 35. Comparison of boundary layer approximation andfull integrodifferential equation: (a) The dark area is the cal-culated stability region for the n52 mode for an air bubble inwater at f520.6 kHz. The open circles are the data of Holtand Gaitan (1996; Gaitan and Holt, 1999). The calculation em-ployed the boundary layer approximation. (b) Same calcula-tion as above, but based on the full integrodifferential equa-tions, rather than the boundary layer approximation: Hardlyany difference as compared to (a) can be observed. From Haoand Prosperetti (1999).


3. Afterbounce instability

During the afterbounces, the bubble oscillates close toits resonance frequency (see Sec. II) on a time scale t0

51/f0;ArR02/3P0;0.3 ms. It turns out that the charac-

teristic period of shape oscillations about the sphericalbubble is very close to this resonant time scale, namely,ArR0

3/(gbn)'1 ms/Abn (for the n52 mode, Abn'3).This coincidence of time scales is the root cause of theparametric instability (which exhibits maximal growthwhen the time scale of the forcing is of the order of the

FIG. 36. Shape distortion of a bubble. (Upper part): Timedevelopment of the bubble radius R(t) (lower part) and dis-tortion amplitude a2(t) for a R054.4 mm bubble driven atPa51.1 atm. Note the transition from Rayleigh-Taylor (timescale ns) to afterbounce perturbations (time scale ms) duringthe afterbounce part of the bubble dynamics. It can also beseen that the dynamics of the distortion a2(t) has half thefrequency of the forcing radial dynamics R(t), as is typical foran instability of the Mathieu type. From Hilgenfeldt et al.(1996).

FIG. 37. Mie scattering data of the afterbounce dynamics of abubble driven below the luminescence threshold. Parametricinstabilities can be inferred from the occurrence of large scat-tering spikes (near 9 and 10.5 ms in this figure), a consequenceof the strong shape distortions of the bubble. Direct imaging ofthe bubble (Matula, 1999) shows nonspherical bubble shapes.From Matula (1999).


time scale of the natural oscillation frequency). Underthe right circumstances this instability can be so violentthat the bubble is destroyed during the afterbounces of asingle cycle. Examples for which the bubble ‘‘survived’’considerable distortions during the afterbounces areshown in Fig. 36 (theory) and Fig. 37 (experiment), thelatter taken from Matula (1999). The distortion cangrow so much that the bubble breaks apart during theafterbounce period. The growth of instabilities duringthe afterbounce phase has been directly observed byGaitan and Holt (1999).

The afterbounce instabilities must be triggered bynoise. A good way to analyze this dependence is tomodel the thermal noise through coupling a Langevin-type force to the dynamical Eq. (59) for the shape dis-tortion (Augsdorfer et al., 2000), with a magnitude ad-justed to satisfy the fluctuation-dissipation theorem.

4. Rayleigh-Taylor instability

The Rayleigh-Taylor shape instability occurs when alighter fluid is accelerated into a heavier fluid (the clas-sical example is the interface between two layers of liq-uid, the lower one being lighter and with buoyancy asthe accelerating force). For sonoluminescing bubbles,this shape instability acts over the extremely short timescales of the final stages of Rayleigh cavitation collapse.Here the bubble interface decelerates in preparation forthe reexpansion, leading to an extremely large relativeacceleration of the gas with respect to the water in ex-cess of 1012 g. This deceleration occurs for only a shorttime (nanoseconds); it is roughly the time a sound waveof speed cg needs to cross a fully collapsed SBSL bubbleof radius R;h . For the Rayleigh-Taylor instability to beeffective, it must destroy the bubble during this timeperiod. The competition between large magnitude andshort duration of the accelerating force determines thestability threshold. A shock-front-driven variant of theRayleigh-Taylor instability is the Richtmeyer-Meshkovinstability, whose occurrence does not seem likely, sinceno evidence for shock-wave passage through the bubblewall was found (Wang et al., 1999).

It should be emphasized that, in contrast to the para-metric and afterbounce instabilities, the Rayleigh-Taylorinstability acts on such a short time scale that the bubbledynamical approximations cannot be expected to bequantitatively correct. A proper calculation requires afull simulation of the gas-liquid interface dynamics downto the latest stages of the Rayleigh collapse.

The thresholds for the Rayleigh-Taylor shape instabil-ity do not depend on whether the boundary layer ap-proximation is used or not, but instead on the chosendynamics for the bubble radius R(t). Hilgenfeldt et al.(1996) used the modified Rayleigh-Plesset dynamics(20), i.e., without the R/c corrections of the kinetic term[cf. Eq. (18)], and without thermal damping. Thoseterms were included by Prosperetti and Hao (1999), whofound the Rayleigh-Taylor instability greatly suppresseddue to the smaller bubble-wall acceleration R resultingfrom higher-order equations (see Prosperetti and Hao,


1999). The uncertainties in modeling the bubbledynamics—and in particular in the second derivativeR—are substantial enough to make a quantitative de-scription of the Rayleigh-Taylor instability a difficulttask. Another factor that changes the location of theRayleigh-Taylor instability line in the phase diagrams isthe diminishing density contrast between the liquid andthe extremely compressed gas at collapse. Taking thiseffect into account in Eq. (62), Augsdorfer et al. (2000)and Yuan et al. (2001) found further suppression of theRayleigh-Taylor instability.

In conclusion, though experimental results and theorygive many hints that the Rayleigh-Taylor instability setsthe upper threshold of the SBSL regime towards largeforcing pressures Pa , the matter is not yet fully settled.As discussed in Sec. II, the Bjerknes force instabilitymay also play a role. The parametric and afterbounceinstabilities, on the other hand, set well-established lim-its for the parameter space of SBSL towards large R0 .Figure 36 shows that instabilities such as Rayleigh-Taylor and afterbounce can occur simultaneously in thesame bubble, with the perturbation a2(t) growing overboth nanosecond and microsecond time scales.

5. Parameter dependence of the shape instabilities

All calculations up to this point have been describedfor the material parameters of water at 20 °C and driv-ing frequencies around f520 kHz. The shape stabilitythresholds strongly depend on changes in the radial dy-namics brought about, for example, by different liquid

FIG. 38. Parameter space restrictions for sonoluminescing ar-gon bubbles: The M51 curve (long-dashed) characterizes theonset of strong bubble collapse and heating. The bubble growsthanks to rectified diffusion to the right of the diffusive stabil-ity curves (heavy lines, shown for c` /c050.5, 0.02, and 0.002,left to right). The thin solid line marks the onset of the para-metric instability and the short-dashed line combines thethreshold of Rayleigh-Taylor instability and afterbounce insta-bilities. These lines are calculated within the simplified theoryof Hilgenfeldt et al. (1996), which slightly underestimates theshape stability, as discussed in the text.


viscosities or driving frequencies. The dependencies mayact differently for different stabilities: if f is decreased,the parametric instability is suppressed as the stabilizinginfluence of viscosity can act for a longer time to sup-press perturbations. The more violent collapses of low-frequency-driven bubbles, however, favor an earlier on-set of the Rayleigh-Taylor instability. Nevertheless, fornot too high forcing pressures, larger bubbles can bestabilized to show sonoluminescence, possibly emittingbrighter light pulses (‘‘upscaled’’ SBSL). However, asdescribed in Sec. III.A.3, water vapor becomes increas-ingly important at low frequencies and counteracts theupscaling (Yasui, 1997b; Moss et al., 1999; Storey andSzeri, 2000; Toegel, Gompf, et al., 2000).

The water temperature also has an effect on the phasespace of SBSL. As pointed out by Hilgenfeldt, Lohse,and Moss (1998) and Vuong et al. (1998), the increasedviscosity at lower water temperature means that bubblescan be stabilized by cooling the fluid, allowing for muchstronger acoustical driving and thus more light, as in-deed experimentally observed by Barber et al. (1994).

E. Interplay of diffusive equilibria and shape instabilities

The conditions for diffusive equilibrium and shapestability must be fulfilled simultaneously for stable SBSL(Hilgenfeldt et al., 1996). Outside this parameter regime,bubbles do not necessarily perish, but can undergo dy-namical processes like rectified diffusion that can allowfor unsteady sonoluminescence at a weaker level (‘‘un-stable sonoluminescence’’).

When at low forcing pressure Pa'1.1 atm the gasconcentration is sufficiently large (e.g., c` /c0550%),rectified diffusion can overcome the tendency for disso-lution, and growing bubbles are possible if R0 lies abovethe unstable equilibrium line. Rectification then contin-

FIG. 39. Theoretical result from Hilgenfeldt et al. (1996) onthe drift of the phase of light emission fs( t ) for three differentargon concentrations, c` /c050.003 95, c` /c050.0658, andc` /c050.26, corresponding to gas pressures overhead of p`

53 mm Hg, 50 mm Hg, and 200 mm Hg, respectively. Accord-ing to the theory of Hilgenfeldt et al. (1996), the drift in thephase of light emission is a result of bubble growth throughrectified diffusion, which is followed by a pinch-off of a mi-crobubble when the bubble is running into the shape instabil-ity. The figure resembles the corresponding experimental re-sult, Fig. 4 of Barber et al. (1995). Note that for air bubblesstable sonoluminescence (corresponding to a constant phase ofthe light pulse) is achieved for much higher gas concentration,c`

air/c050.2, corresponding to 150 mm Hg [see also Fig. 4 ofBarber et al. (1995)].


ues until shape instabilities limit the growth (see Fig.38). When R0 reaches the boundary for shape instability,a microbubble pinches off, decreasing R0 . If the remain-ing bubble is still large enough, i.e., above the unstableequilibrium line in Fig. 38, the process will repeat. Forthese low forcing pressures, the allowed size of thebubble after the pinch-off is very restricted. If thepinched-off microbubble is too large, the remainingbubble dissolves.

For relatively large forcing pressure (Pa'1.3 atm)the situation is quite different. For low enough gas con-centration (e.g., c` /c050.2% in Fig. 31), bubbles cangrow (or shrink) and approach a stable diffusive equilib-rium. Since Rayleigh collapse occurs at large Pa , stableSBSL occurs here, with a well-defined, stable R0 follow-ing from Pa and the gas concentration. For larger gasconcentrations, large enough bubbles will again grow upto the threshold of parametric shape instability wheremicrobubbles pinch off. Here, in contrast to the smallerPa regime, the remaining bubble is very likely to end upin a regime where it can grow again.

The characteristically slow growth of R0 (over thetime scales of rectified diffusion) and sudden breakdown(at microbubble pinch-off) are reflected in other experi-mentally observable parameters as well, such as thephase of light emission (with respect to the driving). Ex-perimental measurements of the phase are presented inFig. 4 of Barber et al. (1995), showing the pinch-off/growth dynamics. From the diffusive theory outlined inSec. IV.B, Hilgenfeldt et al. (1996) simulated this behav-ior (see Fig. 39), finding good agreement with the ex-perimental result.

The momentum of pinched-off microbubbles alsogives the remaining bubble a recoil. As this repeatsagain and again on the diffusive time scale of ;0.1 s, thebubble seems to ‘‘dance,’’ as originally observed byGaitan (1990) and later by Barber et al. (1995). The re-gime of dancing bubbles is indicated in the experimentalphase diagram Fig. 12. If the bubbles in this regime arelarge enough (close to the instability line), they will alsoemit sonoluminescence light even as they undergo recti-fied diffusion, leading to the same pattern of slow in-crease and sudden breakdown in the light signal. This isknown as (diffusively) unstable SBSL and is generallyfainter than stable SBSL.

For very large driving pressures, the Rayleigh-Taylorshape instability (and possibly the Bjerknes instability)make stable bubble oscillations impossible for small R0 .This sets the upper limit in Pa for the observation ofSBSL bubbles.

After computing phase diagrams like Fig. 38 withmany different dissolved gas concentrations, the resultscan be summarized in a new plot whose variables are theexperimentally controllable parameters c` and Pa . Thisphase diagram in c`-Pa phase space is shown in Fig. 9.The notation in that diagram is as follows: stable SL orunstable SL means that there are glowing bubbles ofcertain ambient radii which are diffusively stable or un-dergoing rectified diffusion, respectively; other, smallerbubbles dissolve. Only a small, crescent-shaped region in


this phase space allows for stable SBSL. Luckily, the nar-row range of extremely small argon concentrations nec-essary for stable SBSL is easily achievable by workingwith gas mixtures such as air, because the molecular con-stituents of air dissociate (Sec. IV.C). Experimentalphase diagrams in the c` and Pa phase space can befound in Simon et al. (2001).

F. Other liquids and contaminated liquids

Other parameter dependencies of SBSL involvechanges in the material parameters of the liquid. It wasalready mentioned in Sec. IV.D.5 that increasing liquidviscosity allows for stable bubbles at larger R0 . Like-wise, a change in surface tension can influence the loca-tion of the shape instability line in the phase diagrams(see Hilgenfeldt, Brenner, et al., 1998). For nonsonolu-minescing bubbles, Asaki and Marston (1997) have ex-perimentally examined the effect of surfactants on bothshape oscillations and dissolution rates; Fyrillas andSzeri (1995, 1996) supply theoretical understanding ofthe surfactant effect.

A change of the liquid in which a bubble oscillates hasprofound consequences on possible SBSL light emission,largely because of the chemical reactions occurring in-side the bubble. In Secs. III.A.3 and III.B.2, we dis-cussed the influence of water vapor invading the bubbleon the bubble temperature and the resulting light emis-sion, finding that more water vapor leads to less light.Moreover, dissociation products of the liquid (water) arecrucial for the radical reactions that remove moleculargases from the bubble (Sec. IV.C).

For quite some time, it was thought that stable SBSLcould be achieved in water only, and even today water isstill considered the most ‘‘friendly’’ liquid for SBSL ex-periments. While dissolved salts (Matula et al., 1995) ormixtures of water and freely miscible liquids (Gaitanet al., 1996) do not prohibit stable SBSL, for years itcould not be observed in other liquids, even though mul-tibubble sonoluminescence in nonaqueous liquids hadbeen known for a long time (see, for example, Suslickand Flint, 1987).

Weninger et al. (1995) showed that weakly emitting,unstable single sonoluminescing bubbles could be ob-served in pure alcohols, while the light of a stable SBSLbubble in water could be turned off by just adding a fewdrops of alcohol to the solution. The latter effect can beunderstood by recognizing that alcohols are surface ac-tive and tend to accumulate at the bubble surface. Atcollapse, they enter the bubble and reduce the heatingbecause of their smaller polytropic exponent and endo-thermic chemical reaction. Alcohols act much like watervapor in this respect, but are much more efficient in re-ducing the temperatures. Evidence for this kind ofmodel has been gathered experimentally and theoreti-cally for both MBSL and SBSL (Ashokkumar et al.,2000; Toegel, Hilgenfeldt, and Lohse, 2000; Grieser andAshokkumar, 2001).

While alcohol contaminations quench the light emis-sion quickly, the bubbles still oscillate in a stable fashion,


indicating that the general mass balance of the diffusive/chemical equilibrium is not severely disrupted. The dif-ference with the Weninger et al. (1995) experiment inpure alcohols thus lies in the solubility of the chemicalreaction products in the surrounding liquid, which is stillalmost pure water in the case of alcohol contaminationexperiments. Didenko et al. (2000b) therefore soughtideal nonaqueous liquids for SBSL, requiring (i) a lowvapor pressure to limit vapor invasion of the bubble and(ii) a high content of O or N heteroatoms to facilitatethe chemical formation of species upon collapse that willreadily dissolve in or react with the liquid phase. Usingliquids such as formamide or adiponitrile, Didenko et al.(2000b) did produce bright SBSL and were the first toobserve molecular spectral lines (see Introduction andSec. V.G). Their bubbles were, however, unstable, in thesense that they were moving on circular or elliptic tra-jectories around the pressure antinode that held themtrapped. Didenko et al. (2000b) call this state movingSBSL. Possibly the chemical reactions (of which the lineemission gives direct evidence) still produce too much‘‘waste’’ unable to dissolve fast enough in the surround-ing liquid, sending the bubble into an unstable state ofrectified diffusion and microbubble pinch-off. Revealingthe mechanism of moving SBSL is an interesting openproblem for future research.

V. SONOLUMINESCENCE LIGHT EMISSION

The previous sections have described how amicrometer-sized bubble in a water-filled flask can un-dergo oscillations of incredible violence, collapse at su-personic speeds, burn molecular gases, and still—underthe right experimental conditions—maintain the stabilityof its spherical shape, showing precise repetitions of thishighly nonlinear dynamics for millions and billions ofdriving cycles. These remarkable properties might nothave been studied in detail without the phenomenonthat gives sonoluminescence its name: the emission ofvisible light induced by insonation with an acousticwave.

In order to release a photon of visible wavelength, anatom, ion, or molecule must be excited a few eV aboveits ground state. A sound wave of 1-atm amplitude, bycontrast, carries an energy density of typically 10211 eVper particle. The required tremendous energy concen-tration of almost 12 orders of magnitude (Barber andPutterman, 1991) is precipitated by the rapid collapse ofthe sonoluminescent bubble, where the layers of watersurrounding the bubble act as a radial piston compress-ing its interior. But is this the whole story? Is the heatingof the gas inside the bubble resulting from the rapidcollapse sufficient to explain the light? Is the light emis-sion completely of thermal origin, or is it necessary toinvoke other physical processes? While researchers havecome up with a plethora of creative ideas concerninglight-emission processes, the results of a number of cru-cial experiments favor the thermal light-emission ap-proaches over others, and rule out some other theoriesoutright.


A. Theories of MBSL: discharge vs hot spot theories

When multibubble sonoluminescence was discoveredin the 1930s by Marinesco and Trillat (1933) and Frenzeland Schultes (1934), different theories for its occurrencewere soon put forth. Almost all light-emission mecha-nisms discussed since then can be classified under one oftwo headings: thermal or electrical processes.

The first attempts at explaining the mechanism behindthe light emission favored electric discharges. Levshinand Rzhevkin (1937) initially brought up the subject ofcharge separation in cavitation bubbles; Harvey (1939)thought of the bubble as a spherical capacitor withcharges at the center and the wall. Upon collapse, thecapacitance decreases and voltage increases until elec-tric breakdown takes place. Frenkel’ (1940) suggestedcharge separation by enhancing charge fluctuations onthe bubble wall. In this latter theory, however, break-down should occur during the expansion phase of thebubble dynamics (Leighton, 1994), whereas the closeproximity of bubble collapse and light pulse has beenfirmly established by the work of Meyer and Kuttruff(1959).

Since a symmetric charge distribution cannot radiatelight, discharge theories in general have to assume thatthe emitting bubble undergoes an asymmetric collapse,and would predict increasing intensity of light emissionas the asymmetry increases (Margulis, 2000). This con-tradicts recent systematic studies of single-bubble cavi-tation luminescence (Ohl et al., 1998; Baghdassarianet al., 1999, 2000, 2001; Ohl, 2000), in which the bubblesare created with strong, focused laser pulses, and inwhich, above a certain collapse asymmetry, light emis-sion ceases altogether. It is also in contradiction with theobservation that single sonoluminescing bubbles, whichcollapse under controlled conditions with high symme-try, emit light of much higher intensity than MBSLbubbles of comparable size driven at comparable levels,with the light of a single bubble easily visible to thenaked eye (Gaitan et al., 1992). Another example inwhich an increase in symmetry leads to (slightly) morelight rather than less is given by the SBSL experimentsunder microgravity performed by Matula (2000).

The other large group of MBSL theories have beencharacterized as ‘‘hot spot’’ models, in which the energyfor the light emission is supplied by thermal energy re-sulting from an adiabatic bubble collapse. Noltingk andNeppiras (1950) were the first to use Rayleigh-Plessetbubble dynamics to deduce bubble internal tempera-tures as high as 10 000 K at collapse of a sphericallysymmetric bubble. Within the hot spot models, whichprocess of light emission will dominate depends on theactual maximum temperatures reached, e.g., recombina-tion of dissociated molecules at lower temperatures(Saksena and Nyborg, 1970), or characteristic molecularradiation due to electronic excitation, in particular of theOH radical (Sehgal et al., 1980). The latter was referredto as chemiluminescence by Suslick (1990) and must notbe confused with secondary chemiluminescence, whichmay occur in the liquid as a result of chemical reactions


of the radical molecules generated in the bubble col-lapse. One example of this is luminol, which emits lightas it reacts with OH. While chemicals such as luminolare useful in detecting cavitation (Negishi, 1961), theiremission is not directly related to sonoluminescence.

In the past decade, Kenneth Suslick and his grouphave amassed impressive evidence in favor of thermalmolecular-emission luminescence in multibubble cavita-tion fields in the context of sonochemistry. From thepresence of clearly identifiable molecular bands and theabsence of other lines associated with discharges, Suslick(1990) deduces the thermal origin of the emission. Iden-tifying line emissions by their location in the spectrum,Suslick et al. (1986) used comparative rate thermometryto evaluate the temperature from the intensity of thedifferent lines. The temperatures obtained are very con-sistent and have been confirmed by different methods ofspectral analysis by Flint and Suslick (1991a) and Mc-Namara et al. (1999). Typical maximum temperatures ofMBSL bubbles are thus determined to vary between;3000 and 6000 K, depending on experimental param-eters (McNamara et al., 1999). Another convincing fea-ture of thermal hot spot theories is that they naturallypredict higher temperatures for collapses of higherspherical symmetry, which is confirmed by studies suchas those of Ohl et al. (1998) and Matula (2000), as men-tioned above.

B. SBSL: A multitude of theories

With the advent of single-bubble sonoluminescence, asimple case study for cavitation and the resulting lightemission was found. It was hoped that all open questionscould be answered by studying this ‘‘hydrogen atom ofcavitation physics.’’ Initially, the discovery of single-bubble sonoluminescence actually increased the confu-sion about sonoluminescence light-emission mecha-nisms. The main reason for the multitude of speculativemodels was the uncertainty whether SBSL was similar toMBSL. SBSL led to more light than a single MBSLbubble, while the spectra did not reveal any structuressuch as lines or bands (Fig. 6). Both of these facts (aswell as the apparently short duration of the light pulse)suggested more extreme temperatures and pressures.

Thus, early theoretical research tried to invent mecha-nisms that occur for more extreme conditions thanMBSL. Some of these theories were quite exotic. Forexample, Schwinger (1992) hinted at the dynamical Ca-simir effect as a potential photon-emission process at thenoninertially moving bubble interface. When a corre-sponding model was developed by Eberlein (1996a,1996b), it soon became clear that, in order to match theobserved light intensities, the bubble-wall speed wouldnot just have to be comparable to the speed of sound,but exceed the speed of light (Unnikrishnan et al., 1996;Lambrecht et al., 1997; Milton and Ng, 1998; Breviket al., 1999). This particular line of research was thusabandoned.

A number of theories placed the location of lightemission in the liquid, rather than inside the bubble. In


this vein, another attempt at an electrical breakdownmodel was made by Garcia et al. (1999). Earlier, Lepointet al. (1997) speculated on sparklike discharges aroundwater jets invading a bubble.

Prosperetti (1997) also invoked an electrical mecha-nism for light emission (fractoluminescence) as a by-product of a fluid-mechanical picture of sonolumines-cence light emission that requires asymmetric collapse ofthe bubble. His idea was based on the fact that Bjerknesforces cause a sonoluminescing bubble to oscillate verti-cally during a bubble cycle, and that such oscillations arewell known to cause asymmetric collapse (jets). The col-lision of the jet with the bubble wall would initiate frac-toluminescence, an effect documented in solid-state ma-terials. Prosperetti argued that, over the small timescales of collapse and jetting, water could ‘‘break’’ aswell. The model made various predictions, but it fell outof favor when it became clear that they do not hold.First, it had predicted that less light should be emittedunder microgravity, where a weaker jet is expected. YetMatula (2000) found the opposite. Second, the details ofProsperetti’s (1997) model specifically rely on the prop-erties of water, while Didenko et al. (2000b) foundbright SBSL in nonaqueous liquids. Finally, it had mean-while become clear that models based on a sphericallysymmetric collapse were able to give a more quantita-tive explanation for the experimental data.

Other SBSL theories focused on a more ordinary ex-planation, the emission of photons due to the high tem-peratures in the bubble, akin to the hot spot theory ofMBSL. In fact, all nonthermal models have to explainwhy their mechanism of emission would not beswamped by thermal radiation. Requiring spherical sym-metry of collapse for intense light emission, thermalmodels of SBSL are favored even by proponents of dis-charge models for MBSL (Margulis, 2000). In an inter-esting twist, Xu et al. (1999) have suggested that electricfields do not directly lead to light emission, but suppressthe Rayleigh-Taylor instability (see Sec. IV) and thusstabilize the bubble. However, the existence of largeenough fields for such stabilization is a subject of debate(Moss, 2000).

Depending on the actual temperatures achieved dur-ing collapse, different excitations become dominant inthe compressed gas, so that ‘‘thermal emission’’ can referto a large variety of different processes. As temperaturesincrease from several hundred to many thousand kelvin,those processes can be, among others, molecular recom-bination (Saksena and Nyborg, 1970), collision-inducedemission (Frommhold and Atchley, 1994), molecularemission (Didenko et al., 2000b), excimers (Hammerand Frommhold, 2001), atomic recombination (Hilgen-feldt et al., 1999b), radiative attachment of ions (Ham-mer and Frommhold, 2001), neutral and ion bremsstrah-lung (Moss et al., 1997; Xu et al., 1998; Hilgenfeldt et al.,1999b), or emission from confined electrons in voids(Bernstein and Zakin, 1995). The uncertainty about theprecise temperatures of SBSL bubbles (see Sec. III)would allow for most of these mechanisms. Generallyspeaking, however, very-low-energy excitations are un-


likely to produce a spectrum with enough visible pho-tons to account for SBSL. On the other hand, the bubblecollapse cannot generate arbitrarily high temperatures,so that very-high-energy excitations are rare and do notproduce a large photon flux either.

Which of these mechanisms actually dominates de-pends critically on accurate measurements and calcula-tions of the temperature inside the bubble. For example,at temperatures above several thousand kelvin, pro-cesses like collision-induced emission lose importance ashigher-energy emission processes such as bremsstrah-lung take over (Frommhold, 1998; Hammer and Fromm-hold, 2000a, 2000b, 2001). Calculations have shown thatthe emission intensity from a body at a temperature of;104 K is not inconsistent with SBSL observations; al-though radiative enhancement effects (e.g., collectiveemission as proposed by Mohanty and Khare, 1998)have not been strictly ruled out, they are not necessaryto explain the experimental data. It should be empha-sized here that although all thermal processes can con-tribute to blackbody radiation, thermal emission doesnot necessarily result in a blackbody spectrum (as elabo-rated upon in Secs. V.D and V.E).

For many years, the most serious argument againstthermal processes was the sonoluminescence pulsewidth. As measured, it was much shorter than the timefor which the bubble is maximally collapsed (;1 ns, seeFig. 27). The need for ultrashort pulses fueled the popu-larity of ‘‘shock-wave’’ models, in which a focusing shockcauses the light-producing region to be much smallerthan the bubble size (see the Introduction). This shockpicture (referred to as the mechanochemical mechanismin Leighton, 1994) was the most popular view of SBSLfor quite some time.

C. Narrowing down the field

The critical event that narrowed the field was the ex-perimental resolution of the light pulse, showing itswidth to be much longer than previously anticipated.Gompf et al. (1997) used time-correlated single-photoncounting, a powerful method from solid-state physicsand biophysics (O’Connor and Phillips, 1984), where itis used to register fluorescence lifetimes of sometimesonly a few picoseconds. Using a modification of the clas-sical setup, Gompf et al. (1997) employed two photomul-tiplier tubes to record the arrival times of single photonsfrom the same SBSL light pulse in both detectors. Fromthe autocorrelation function of the time differences be-tween the registered events, they could reconstruct theshape (the temporal variation of intensity) of the lightpulse and thus measure its duration. Gompf and co-workers confirmed these results using a direct streakcamera measurement (Pecha et al., 1998). Furthermore,both publications demonstrate that higher driving pres-sures lead to light pulses of both higher intensity andlonger width. Their results were confirmed by Moranand Sweider (1998) and Hiller et al. (1998), the latternoting that xenon bubbles, long known to be the bright-est SBSL emitters, also yield the longest pulses, with


widths up to 350 ps and more. Only for bubbles in highlydegassed water can very faint flashes of about 40–60 psduration be found.

We have seen before that the bubble dynamics nearminimum radius occurs on a time scale (‘‘turnaroundtime’’) of ;1 ns (Fig. 27). While the typical pulse widthof 100–200 ps is still much smaller than this value, thegap between the light-emission time scale and the radialdynamics time scale is not nearly as wide as oncethought. Gompf et al. (1997) demonstrate that a simplethermal model can result in a light pulse with the essen-tial characteristics they measured. This made more cred-ible all thermal models with heating processes directlyconnected to bubble dynamics. Conversely, exotic pro-cesses (yielding ultrashort pulses) were no longer neces-sary. It was then that shock-wave theories underwent acloser scrutiny with respect to their consequences forlight emission. In Sec. III, we discussed how calculationsshowed that shock waves for noble gases would be farless intense than for molecular gases (Moss et al., 1997)or might be altogether absent (Vuong et al., 1999). Thenonrobustness of the shock-wave phenomenon makes itan unlikely candidate for the heating necessary to gen-erate thermal SBSL emission. In the work of Bourneand Field (1991), a shock wave was detected in an asym-metrically collapsing, mm-sized air bubble. However, thelight emission detected at collapse did not originatefrom the shock wave, but from the compressed gas. Allthis evidence indicates that, while shock waves may bepresent in collapsing bubbles under certain circum-stances, they do not substantially contribute to SBSLlight emission. This can be because either (i) they aretoo weak, so that they do not focus energy to a scalemuch smaller than the bubble size, or (ii) they can onlyheat a tiny volume near the very center of the bubble tovery high temperatures, and the ensuing light emission ismuch less intense than the thermal radiation from thebulk volume of the bubble.

D. The blackbody model and its failure

The seminal publication of Gompf et al. (1997) pro-vided more valuable information: using wavelength fil-ters, they determined the widths of the light pulse indifferent parts of the optical spectrum. The resultsshowed (Fig. 13) that there was hardly any detectabledifference between the pulse width in the ultravioletpart of the spectrum (300–400-nm wavelength) and inthe red part (590–650 nm). This contradicted thermalmodels favoring blackbody emission. These predict that,since the bubble heats during collapse and cools duringexpansion, it maintains somewhat lower temperaturesfor a longer time than higher temperatures. It is there-fore capable of emission at long optical wavelengths fora longer time than at short wavelengths (the latter re-quiring higher temperatures). Translating typical tem-peratures achieved in models of collapsing bubbles intoblackbody emission shows that the ‘‘red pulse’’ shouldbe about twice as long as the ‘‘ultraviolet pulse’’ (seebelow). Again, Hiller et al. (1998) and Moran and


Sweider (1998) confirmed these findings, although thelatter group did find a slightly longer pulse in the redthan in the UV if cooled water was used.

Adiabatic heating does not necessarily result in black-body radiation. Even though a model of pure blackbodyemission cannot account for the experiments, it is help-ful to outline this simplest of all thermal light-emissiontheories in the context of SBSL to introduce basic con-cepts and to point out where it fails. A blackbody calcu-lation is of unrivaled simplicity because there is no needto specify the thermal emission as resulting from one ofthe many excitation processes listed Sec. V.B—only thetemperature matters.

A blackbody of a given temperature emits a spectrallight intensity (energy per unit time, wavelength inter-val, solid angle, and projected surface area) at wave-length l of

IlPl@T#5

2hc2

l5@exp~hc/lkBT !21#, (65)

the Planck intensity, with the Planck and Boltzmannconstants h and kB , and the speed of light in vacuum c .Experiments measure the spectral radiance (emitted en-ergy per time and wavelength interval), for which wemust integrate over the projected bubble surface and allsolid angles:

PlPl~ t !54p2R~ t !2Il

Pl@T~ t !# . (66)

This formula reflects the fact that a blackbody is a sur-face emitter, i.e., its radiation comes from its surfacealone, as photons originating in deeper layers are ab-

FIG. 40. Blackbody spectra with the indicated temperaturesfitted to the data from Barber et al. (1997) for the noble gases:h, helium; s, neon; 3, argon; 1, krypton; and d, xenon. Thebest-fit temperatures decrease dramatically with increasingmolecular weight of the gas. From Hammer and Frommhold(2001).


sorbed within the blackbody. It can be compared directlywith the measured spectra of sonoluminescence, as theonly quantities that matter are the bubble temperatureT(t) and radius R(t). Figure 40 shows a comparison(Hammer and Frommhold, 2001) of data from Barberet al. (1997) with blackbody calculations. The spectramatch the blackbody law well, as the temperature variesfrom 8000 K to 30 000 K, depending on the inert gas.

The curves in Fig. 40 result from Eqs. (65) or (66) withboth intensity and temperature as free fit parameters. Toput SBSL models to the test, these parameters have tobe linked to the dynamics of bubble radius and tempera-ture resulting from the model. Indeed, since there is nodirect method for measuring the temperature, this is theonly way to correctly deduce it. Unfortunately, we shallsee that small differences in modeling can lead to largedifferences in light intensity.

As outlined in previous sections, there are various ap-proaches to obtaining bubble dynamics and tempera-ture, with varying degrees of sophistication and accu-racy. A simple starting point is to assume a spatiallyhomogeneous temperature T(t) (see Sec. III.B). Specifi-cally, we choose Eq. (38) to supplement the radius dy-namics R(t) derived from the modified Rayleigh-PlessetEq. (20). The time dependence of R ,T results, via Eq.(66), in a dynamics of the emitted power Pl

Pldl in thewavelength interval @l ,l1dl# , i.e., the temporal shapeof the SBSL pulse as viewed through filters of differentcolor.

Experiments observe wavelengths in the detectablerange luv,l,lr , where luv'200 nm is the ultravioletcutoff of the visible spectrum (due to the strong absorp-tion of smaller-wavelength light in water or other liq-uids), and lr'800 nm marks the red end of the visiblespectrum, where the spectral radiance is already quite

FIG. 41. Theoretically calculated light emission from a R055 mm SBSL bubble driven at Pa51.3 atm and f520 kHzunder the assumption of blackbody radiation, using the tem-perature dynamics T(t) from Fig. 27 and Eq. (66): dashedlines, pulses in the ultraviolet (Puv); dotted lines, pulses in thered (Pr) wavelength range; (a) absolute instantaneous powers;(b) pulses normalized to a peak value of 1 for better compari-son of their widths; solid line, pulse integrated over the entirerange of detectable wavelengths (Pd). From Hilgenfeldt et al.(1999b).


small and experimental noise begins to obscure the spec-trum. In experiment, filters of a certain bandwidth Dlare used, e.g., Dl'100 nm by Gompf et al. (1997) andDl'40 nm by Moran and Sweider (1998), or replacedby spectrographic means for higher resolution (Hilleret al., 1998). It is easy to integrate Eq. (66) over thecorresponding wavelength ranges and compare the pre-diction with light pulse measurements such as those inFig. 13.

Figure 41 shows the result of this calculation for theargon bubble whose dynamics and transport behaviorwe have discussed previously (driven at Pa51.3 atm,ambient radius R055 mm). Using R(t),T(t) from Fig.27, the emission intensities are calculated for the ultra-violet wavelength regime (200 nm,l,300 nm), for thered part of the spectrum (700 nm,l,800 nm), and forthe complete detectable spectral range (200 nm,l,800 nm). Figure 41(a) shows the predominance of ul-traviolet emission.

A calculation like this allows for comparison of atleast three characteristic quantities with experimentaldata:

(a) The intensity of the pulse. In this model, it is abouttwo orders-of-magnitude larger than the experi-mental values for these driving parameters.

(b) The duration of the pulses. These are comparableto the turnaround time of bubble dynamics [notsurprisingly, as R(t) is directly translated intoPl

Pl(t)], and is thus at least a factor of 2 longer thanexperiments.

(c) The wavelength dependence of the pulse width.The length of the pulse varies dramatically withwavelength, in direct contradiction to the experi-ments of Gompf et al. (1997). The ‘‘red’’ pulse du-ration is over 1000 ps, and thus more than twicethat of the UV pulse.

Because of the considerable modeling uncertainties inbubble temperature (see Sec. III), the absolute values ofpulse intensity and width produced by this model shouldnot be considered quantitative predictions. Even thoughthe discrepancies with experiment are very large, onecould still imagine that they could be due to defects inthe temperature modeling T(t) or a change in the size ofthe light-emitting region (as in some shock models).However, the large wavelength dependence is an intrin-sic problem with the blackbody model that is indepen-dent of the temperature law or the absolute value oftemperature. Therefore experiments conclusively showthat it is necessary to seek a modification of the black-body model.

E. The SBSL bubble as thermal volume emitter

Which of the assumptions of the blackbody modelfails? One basic prerequisite is that the radiation be ofthermal origin, with T(t) a well-defined temperature,i.e., local thermodynamic equilibrium must hold evenover the short time scales of the bubble dynamics atcollapse, which seems doubtful at first sight. However,


the immense particle densities (n;1028 m23) and hightemperatures (T;104 K) at bubble collapse create anenvironment in which collisions between particles arevery frequent: a simple estimate suggests collision timeswell below a picosecond, so that local thermodynamicequilibrium is still well obeyed during SBSL light emis-sion.

The other crucial assumption of the blackbody pictureis that the bubble is black, i.e., it perfectly absorbs allwavelengths of electromagnetic radiation. This requiresthat the mean free path of photons (kl

21) be muchsmaller than the size of the object. Here kl is the ab-sorption coefficient for photons of wavelength l (whendivided by the gas density, kl is often referred to as theopacity; see Unsold and Baschek, 1991). A blackbody ofextent (radius) R has klR@1. The product tl[2klR isthe (dimensionless) optical thickness of the object at l.

That opacity may play a role for sonoluminescencelight emission had first been noted by Kamath et al.(1993), who speculated that radicals in the outer layersof the bubble ‘‘could resonantly absorb and scatter theradiation coming from the center thus shutting off theobserved light. This effectiveness of resonant scatteringin rendering gases effectively opaque is well known.’’Wu and Roberts (1993) also speculated that the com-pressed bubble might be a thermal volume emitter. In-dependently, Moss et al. (1994, 1997) discussed the pos-sibility that the SBSL bubble might be transparent to itsown photons. Using tabulated absorption coefficients ofdifferent gas species, they concluded that there was onlya small region of opaque gas in the very center of thebubble (induced in their model by a converging shockwave), but that the rest of the bubble volume was char-acterized by tl,1, and was thus optically thin (transpar-ent). Figure 42 shows the line of demarcation betweenthe optically thick and optically thin parts of the bubble.The emission volume of the optically thin shell is muchlarger than that of the optically thick core.

FIG. 42. Calculated power of light emission: dashed line, withplasma thermal conduction; dotted line, without plasma ther-mal conduction; short-dashed line, boundary line of the opti-cally thick region of a nitrogen bubble; heavy solid line, thebubble radius; thin line, the shock front in the model. FromMoss et al. (1997).


Moss et al. (1997) did not explicitly calculate pulses ofemitted light at different wavelengths, and other fea-tures of their model lead to pulses that are much shorterthan observed. However, the idea of a transparentbubble is the only proposition put forth to date that ex-plains the wavelength independence of sonolumines-cence radiation, and it is now widely accepted as a keyingredient to a consistent view of SBSL light emission.Models built on bubble transparency are sometimes re-ferred to as ‘‘weakly ionized gas models’’ (Hammer andFrommhold, 2001), because a low degree of ionizationof the gas is crucial (see below). We prefer the termthermal volume emission, as the radiation from thewhole volume of a transparent body reaches the detec-tor, rather than only the surface emission of a blackbody.

To illustrate that such a model leads to wavelength-independent emission spectra, we revert to the simplemodel discussed below, drawing R(t) and T(t) fromEqs. (20) and (38), respectively. For a volume emitter,the light intensity from a volume element of the bubbleis dependent on the location inside the body (the depths along the ray of vision to the element). Using the lawsof absorption and emission in a medium (see, for ex-ample, Zel’dovich and Raizer, 1966, or Siegel and How-ell, 1972),

Il~s ,t !5IlPl@T~ t !#„12exp$2kl@T~ t !#s%…,

0,s,2R (67)

is the intensity at wavelength l from a depth s , providedthe temperature of the body is spatially uniform. Notethat the source function of the emission is still Il

Pl , be-cause local thermodynamic equilibrium is obeyed, andthat for infinite absorption coefficient kl , the Planckemissivity is recovered.

To obtain the spectral radiance, one must first inte-grate the intensity per surface area and per solid anglefrom Eq. (67) over the projected cross section of thebubble, the thickness s of the medium varying as s52R cos(u) with the angle u from the line of view. Theresult is the emitted power from the whole bubble crosssection per unit solid angle. Assuming isotropy of SBSLradiation, the spectral radiance is then

Pl~ t !dl54p2R2IlPl@T~ t !#S 11

exp~22klR !

klR

1exp~22klR !21

2kl2R2 D dl . (68)

Note that the Planck radiance (66) is now multiplied bya factor whose magnitude varies between 0 and 1, de-pending on the optical thickness tl52klR . As tl→` ,the Planck spectrum is recovered. In the transparentlimit (tl!1) the radiance reduces to

Plthin~ t !dl5

23

tlPlPl@T~ t !#dl , (69)

a spectral power that is much smaller than that forblackbody emission.


1. Simple model for bubble opacity

To proceed further, we need to know how the absorp-tion coefficients kl depend on parameters. In contrast tothe blackbody calculation, this requires identification ofthe most significant physical processes that contribute tophoton absorption in the gas inside a sonoluminescingbubble. Now not only the temperature is relevant, butalso the physics of photon-matter interaction. The origi-nal work of Moss et al. (1997) extrapolated tabulatedvalues for kl to the regime of the sonoluminescence ex-periments. Moss et al. (1999) used a full opacity modelto extend these calculations. Here we summarize asimple model for kl (Hilgenfeldt et al., 1999a, 1999b;Hammer and Frommhold, 2001).

The model assumes that light emission predominantlystems from the ionization of a noble gas, and that nei-ther molecular gases nor liquid vapor play a substantialrole. If it is further assumed that the peak temperature isof order ;10 000 K, the literature on absorption andemission of radiation (e.g., Zel’dovich and Raizer, 1966)suggests that there are three important processes, illus-trated on the left-hand side of Fig. 43: the absorption ofphotons by (i) free electrons near ions (inverse brems-strahlung), (ii) free electrons near neutral atoms (in-verse neutral bremsstrahlung), and (iii) bound excited

FIG. 43. Dominant photon absorption and emission processesat the typical temperatures and densities in a sonoluminescingbubble: left, absorption; right, emission. The three pairs of pro-cesses are (i) electron-ion (inverse) bremsstrahlung, (ii)electron-atom (inverse) bremsstrahlung, and (iii)photoionization/radiative recombination. Note the large gapbetween the ground-state energy (Egs) and the first excitedstate (Ees1) in the term scheme of (iii).


electrons that reach continuum energies after photonabsorption (i.e., photoionization).

All three processes depend critically on the density offree electrons in the gas, which can be obtained from theSaha equation (Zel’dovich and Raizer, 1966; Unsold andBaschek, 1991). At a temperature of T;104 K, a noblegas such as argon is only weakly ionized. This is becausekBT!Eion , the ionization energy of argon being Eion'15.8 eV. Zel’dovich and Raizer (1966) show that thedegree of ionization a5ne /n is then

a@T#5S 2pmekBT

h2 D 3/4S 2u1

nu0D 1/2

expS 2Eion

2kBT D , (70)

where me is the electron mass and u1 ,u0 are the statis-tical weights for the ionic and the neutral ground states,respectively. For argon SBSL bubbles a typically doesnot exceed 1%.

Note that, for the three processes mentioned above,the frequency of the electron-ion reaction (i) is propor-tional to a2, while the two others are proportional to aalone. A small a thus seems to favor (ii) and (iii) asdominant, but the interaction cross section of electronsand ions is much larger than that for electrons and neu-tral atoms, which makes (i) an important factor for pho-ton absorption.

Given a, the absorption coefficients for the processes(i)–(iii) can be calculated (Zel’dovich and Raizer, 1966)for ionized noble gas atoms. The evaluation of kl for(iii) in principle requires knowledge of the completeatomic levels of the noble gas atom. Since the energyrequired for excitation to the first excited level is alreadya large fraction of the ionization energy (Fig. 43), thehigher levels can be roughly approximated as part of acontinuum of a hydrogenlike atom. In this limit, the onlyimportant parameter is the energy E2 of the first excitedlevel or, equivalently, the corresponding photon wave-length l25hc/E2 . Following Zel’dovich and Raizer(1966), one can show that the sum kl

ion of the contribu-tions to kl from (i) and (iii) is

klion@T#5

16p2

3)

e6kBTn

~4p«0!3h4c4 l3

3expS 2Eion2hc/max$l ,l2%

kBT D . (71)

The contribution of electron-neutral atom inversebremsstrahlung (ii) is rarely important in conventionalplasma physics, where the degree of ionization is oftenhigh. For a weakly ionized SBSL gas, with the assump-tions made above, it is often the dominant factor in kl

and can be written as (Zel’dovich and Raizer, 1966;Hilgenfeldt et al., 1999b)

kl0@T#54

e2

4p«0

~2kBT !9/4n3/2

h3/2c3me3/4p3/4 l2

3S ctr1dtr

3kBT D expS 2Eion

2kBT D . (72)


Here, ctr and dtr are coefficients describing the electron-atom transport scattering cross section, extracted fromBrown (1966). The accuracy of Eq. (72) is limited by itsassumptions. Hammer and Frommhold (2000b, 2001)have evaluated the electron-neutral contribution usingan ab initio quantum-mechanical calculation of theelectron-atom scattering problem. At long wavelengthsthese calculations agree with the simple theory pre-sented above, but at short wavelengths the quantum cal-culations are significantly more intense. The two ap-proaches will be compared below.

2. Light emission and comparison with experiment

From Eqs. (71) and (72), the total absorption coeffi-cient kl5kl

01klion is obtained. The dimensionless opti-

cal thickness satisfies tl,1 throughout the collapse forall wavelengths, and tl!1 for the dominant ultravioletpart of the emission. Figure 44 presents the emissionfrom such a bubble according to Eq. (68), using thesame parameters that resulted in the blackbody emissiondisplayed in Fig. 41.

Comparing with the blackbody calculation, it is seen,first, that the total intensity is reduced dramatically (to afew 105 photons per pulse). This is because the smalloptical thickness, by Eq. (69), leads to a drastic reduc-tion of radiance compared to an opaque bubble. Second,the light pulses are now considerably shorter, and finally,the dependence of pulse width on wavelength has al-most disappeared. All of these properties of the lightpulse are now in much better agreement with those ofexperimentally observed SBSL pulses. Both the shorterpulse width and the wavelength independence are con-sequences of the exponential dependence of the degreeof ionization on T in Eq. (70), which carries over to the

FIG. 44. Theoretically calculated light emission from a R055 mm SBSL bubble driven at Pa51.3 atm and f520 kHzunder the assumption of thermal volume emission. We useT(t) from Fig. 27, and Eq. (68) to evaluate the emission. (a)Absolute power; (b) relative power; line styles as in Fig. 41.The strong dependence of the pulse width on wavelength dis-played in that figure is not present here, in agreement withexperiments. The total power in the pulse is now much smaller.From Hilgenfeldt et al. (1999b).


SBSL radiance. Since kBT!Eion , this exponential‘‘switch’’ is extremely sensitive to changes in T . Oncethe temperature has dropped only slightly as the bubblestarts to reexpand, a and kl are reduced severely. Con-sequently, the emission (69) is quenched rapidly (pro-ducing shorter pulse widths), independent of wavelength.

Equation (68) also yields reasonable results for thespectrum of the emission, although the spectra have a‘‘kink’’ due to modeling assumptions of the atomic en-ergy levels. Hammer and Frommhold (2000a) have ex-tended the above calculations, using ab initio quantummechanics. These calculations give a spectral shapemore like the experiments for many parameter combi-nations (Fig. 45), with generally more intense radiationat the UV end of the spectrum, due to more intenseelectron-neutral bremsstrahlung.

It is also useful to examine the predictions of thismodel when the gas is changed from argon to, say, xe-non. The spectrum is now much more intense (in agree-ment with observation) and has a maximum at l2(Xe)'336 nm, quite close to what is measured (Barber et al.,1997). For xenon, with its lower ionization energyEion(Xe)'12.1 eV, the degree of ionization a can be aslarge as 10% (the temperatures obtained in argon and

FIG. 45. Theoretical SBSL spectra for a R055 mm argonbubble driven at Pa51.3 atm and f520 kHz: heavy solid line,total spectrum calculated from first principles; thin solid line,spectrum calculated from the semiclassical model outlined inSec. V.E; heavy and light dashed line, contribution of electron-neutral bremsstrahlung in the two models (more intense in theab initio theory); dotted lines, electron-ion interaction contri-bution, the same for both approaches. From Hammer andFrommhold (2000a).


xenon bubbles are comparable). Consequently, thebubble is less transparent and its light emission moreintense and more similar to those of a blackbody. Asquantitatively shown by Hammer and Frommhold(2000b), xenon spectra display the characteristic kinkedmaximum at l2 in the quantum-mechanical version ofthe theory. This typical shape of the xenon spectrum wasfirst understood by Moss et al. (1999).

Identifying the major contributions to the absorptioncoefficient automatically identifies the main light-emission mechanisms of SBSL. This follows from Kirch-hoff’s law, which states that every absorption processmust balance a corresponding emission process—this iswhy Pl in Eq. (68) grows with increasing absorption.Inverting the absorption events (i)–(iii) identifies SBSLlight emission as a combination of (i) electron-ionbremsstrahlung, (ii) electron-neutral bremsstrahlung,and (iii) radiative recombination (see the right-hand sideof Fig. 43). None of the three processes seems to bedominant over the whole range of the SBSL parameterspace.

FIG. 46. Comparison of the theoretical results from the ther-mal volume emitter model of Hilgenfeldt et al. (1999b) to theexperimental results of Hiller et al. (1998). The figure wasmodified from Hilgenfeldt et al. (1999a) to include the experi-mental results for helium (1). Filled symbols represent experi-ments using xenon (d), argon (m), and air (l) at the satura-tion concentrations displayed in the figure. Open symbols aretheoretical results of the parameter-free theory of Hilgenfeldtet al. (1999b). (a) Higher light intensities and pulse widths ofxenon bubbles; (b) lower-intensity data.


One possible absorption/emission process not dis-cussed here is line emission (bound-bound-state transi-tions of electrons). Two sets of recent experiments havedemonstrated that, in certain parameter regimes, lineemission is important. We discuss these very recent re-sults below in Sec. V.G (see also the Introduction).

We shall now use Eqs. (68), (71), and (72) to trya direct comparison with experimental results. To thisend, a calculation for a single-parameter combination ofPa ,R0 is not useful, as the emission depends sensitivelyon these parameters, and their accuracy from experi-ment is not very high. However, a promising set of datawas presented by Hiller et al. (1998), graphing the widthof the light pulses vs their intensity. Both of these quan-tities can be measured reliably and with good accuracy.The experiments are conducted at a given relative con-centration c` /c0 of gas in the liquid, and the data pointsshown in Fig. 46 are a result of changing Pa through thewhole range of stable SBSL under these conditions. Thesimple character of the light-emission model of Hilgen-feldt et al. (1999a, 1999b) allows one to first calculate thePa ,R0 values for all stable SBSL bubbles at the givenc` /c0 (see, for example, Fig. 31) and then use these tocalculate the light-emission intensities and pulse widths.Like the experimental data, the calculated light intensi-ties were normalized by the intensities for a standardargon bubble. As Pa is increased in experiment, we fol-low the prescribed increase of R0 for stable bubbles andconsequently obtain larger, more strongly drivenbubbles whose light pulses become longer and more in-tense. Without adjusting any fit parameters, the theoret-ical results are in close agreement with the data (Fig. 46)for both xenon and argon gases. Note that the observedclose proximity of the width vs intensity curves for Xeand Ar [Fig. 46(b)] is reproduced, even though the Paand R0 are quite different for the two gases.

The data of Hiller et al. (1998) provide another checkof the dissociation hypothesis discussed in Sec. IV.C.One of the data sets was obtained with strongly de-gassed air, which results in very small pulse widths andintensities [diamonds in Fig. 46(b)]. If the dissociationhypothesis is correct, these bubbles should not be airbubbles at c` /c053%, but argon bubbles at c` /c050.03%. The corresponding calculation indeed repro-duces the experimental results.

The simple theory outlined above cannot reproducethe absolute light-emission intensities for every param-eter combination. This should not be expected, since im-portant factors such as the influence of water vapor onthe bubble temperature are absent from this model.However, this kind of model does yield the correct rela-tive intensities and their dependence on pulse width(Fig. 46), reasonable spectra (Fig. 45), and wavelength-independent pulse widths (Fig. 44). This agreement withexperiment suggests that the essential physics is ad-equately represented and the concept of a bubble as athermal volume emitter is probably correct (cf. also theextensive review of light-emission mechanisms by Ham-mer and Frommhold, 2001).


Even so, it should be emphasized that the good agree-ment between the model and experiment does not de-finitively settle the issue because a number of significanteffects have not been included, in particular (i) the pres-ence of water vapor and chemical reactions inside thebubble and (ii) the additional possibility of light emis-sion from the molecular components of the gas(Didenko et al., 2000b; Young et al., 2001); again, watervapor may play an important role here. These and otherprocesses could act to suppress or enhance the lightemission, and to our knowledge no quantitative theoryhas incorporated all of these influences. In the theoryformulated above, the number of photons is in reason-able agreement with the experiments, but it is entirelypossible that this is because several neglected effectscancel each other out. The following section outlinessome additions to the model that could affect the out-come of a light-emission calculation, and could lead to amore comprehensive model of SBSL radiation.

F. Modeling uncertainties: additional effects

The number of photons per oscillation cycle ofsonoluminescence is the most conspicuous experimentalobservable; however, it is also the hardest to predict, inparticular within a thermal volume emitter theory. Thereason is that the light intensity depends exponentiallyon the temperature in the bubble [see Eq. (70)], whichin turn depends strongly on modeling assumptions. Thesituation is illustrated in Fig. 23. What can be measuredwith good accuracy are the radius R(t) and the radiancePl(t) of SBSL light. Those two quantities are connectedvia the temperature T(t) of the SBSL bubble, which upto now could not be experimentally measured. There-fore modeling is required both for obtaining T(t) fromR(t) and for obtaining Pl(t) from T(t). Both modelingsteps depend on the assumptions entering the respectivetheories, and often the dependence is sensitive. We nowreview some of the most relevant sources of modelinguncertainties.

1. Bubble hydrodynamics

As discussed in Sec. III, any uncertainty in modelingthe hydrodynamics and thus the temperature inside thebubble will be directly reflected in some uncertaintyabout the light emission. Since the work of Plesset andProsperetti (1977), it has been popular to state that thetemperature law is initially isothermal and then switchesabruptly to adiabatic once the bubble accelerates suffi-ciently. There are slightly more sophisticated versions ofthis, e.g., the approximation used by Hilgenfeldt et al.(1999b) in which the polytropic exponent changes con-tinuously. Precisely where the temperature law changesfrom isothermal to adiabatic strongly affects both thewidth of the pulse (since heating commences when thistransition occurs) and the total number of photons emit-ted (since the transition controls the total amount oftime for which heating occurs).

Storey and Szeri (2000) have performed full numericalsimulations showing that these approximations are rea-


sonably reliable for typical parameter combinations.However, to date there has not been a complete study ofthe full parameter space, documenting quantitatively theerror in the simple models for the temperature. Withoutsuch an error estimate (as a function of parameters), it isdifficult to assess how much of the current uncertaintyabout light-emission intensities comes from simple hy-drodynamics.

2. Water vapor as emitter and quencher of light

As an example for the above-mentioned possibility ofcanceling errors of approximations connected with thetwo modeling steps in Fig. 23, we mention the role ofwater vapor in argon bubbles: In Sec. III we have shownhow the consideration of vapor can drastically reducethe temperatures achieved inside the bubble, both be-cause of the reduced polytropic exponent and becauseof the endothermic water dissociation. Because of thelower temperatures, less light should result. On theother hand, the reaction products of water (O and H)have an ionization potential ('13.6 eV for both species)very similar to that of argon, and therefore will also con-tribute to the light emission, increasing the photon out-put again.

SBSL experiments with helium bubbles illustrate thelatter effect: The large ionization energy Eion

He

'24.6 eV, together with the exponential dependence ofthe photon absorption coefficient on the ratio ofEion /kBT , predicts negligible light emission from He,about four orders-of-magnitude dimmer than for Xe.Yet Barber et al. (1997) observed He bubbles to be onlyabout a factor of 10 less bright than Xe (see data in Fig.46). To a lesser degree, this discrepancy occurs for neonas well. The data of Barber et al. (1997) demonstratethat there is a pronounced decrease in SBSL intensityfrom the heavier to the lighter noble gas species in theorder Xe→Kr→Ar, but Ne and He are not much differ-ent (in fact, He spectra are sometimes more intense thanNe spectra).

Moss (1998) pointed out that these observations canbe accounted for when light emission from O and Hradicals generated from dissociated water is considered.These species emit light much more readily than neonand helium, whose ionization energies are considerablyhigher. In a Ne or He bubble, the emission intensity ofthe dissociation products of water vapor overcomes thetiny contribution of the light noble gases and does notvary much between these two gases. Thus, as pointedout by Hilgenfeldt et al. (1999b), the light from Ne andHe bubbles could originate from the vapor in the bubblerather than from the noble gas. The measurement ofOH lines in the spectrum of a sonoluminescing bubbleby Young et al. (2001) (see Sec. V.G below) supports thisview, directly demonstrating light emission from waterdissociation products.

Section III.A.3 detailed the important role of watervapor as a quencher for SBSL light. Through changes inpolytropic exponent and endothermic chemical reac-tions, it can reduce the bubble temperature considerably,


as demonstrated by Yasui (1997a), Storey and Szeri(2000), and others. In bubbles consisting of heaviernoble gases, this effect more than outweighs the role ofwater vapor as an emitter of light. Vazquez and Putter-man (2000) have found a nice demonstration of this ef-fect, in which two SBSL bubbles display almost indistin-guishable R(t) dynamics (very similar R0 and Pa), butdiffer in light intensity by a factor of about 5 due to thedifferent water-vapor pressures at different ambienttemperatures.

3. Further difficulties in modeling the temperature

Temperature calculations that take into accountchemical reactions suffer from considerable uncertainty,because the reaction rates of water-vapor chemistry arenot well known under the extremely high densities andpressures achieved in the bubble. For example, as dis-cussed in Sec. III.B.2 above, Toegel et al. (2002) took thereduction of the water dissociation rate caused by thehigh densities inside the bubble into consideration andgot about 50% higher temperatures than Storey andSzeri (2000).

Another relevant effect may be the segregation of dif-ferent species inside the bubble, which had been sug-gested by Storey and Szeri (1999). Yasui (2001) put forththis mechanism to account for the relative brightness ofhelium bubbles compared to what would be expected ina thermal volume emitter model with homogeneous dis-tribution of all species. In a bubble with helium and wa-ter vapor, the lighter noble gas is driven towards thebubble center. A higher temperature could then be ex-pected inside the bubble: First, the diffusive equilibriumcondition changes, allowing for higher driving pressures;second, the water vapor accumulates near the (cool)wall of the bubble and therefore does not consume somuch of the collapse energy by its endothermic dissocia-tion; third, the condensation of water out of the bubbleis also facilitated by this mass segregation (in a heliumbubble), further helping to keep temperatures high.

4. Modifications of photon-emission processes

All the effects listed above concern modeling uncer-tainties on the R(t)→T(t) side of Fig. 23. On the otherside, less work has been done to evaluate how muchquantitative errors affect the simplifying assumptions in-troduced by the light-emission formulas (70), (71), and(72). The ab initio calculations of Hammer and Fromm-hold (2000a, 2000b) show encouraging agreement withthis simpler model, but other effects could still play arole, in particular those associated with the extremelyhigh densities in the compressed bubble.

The Saha equation may require corrections due tomany-body effects (Chihara et al., 1999), leading to dif-ferent degrees of ionization and different absorption co-efficients. Also, the term schemes and ionization poten-tials of the noble gas atoms could shift under a highdensity of electrons (Zel’dovich and Raizer, 1966). Ham-mer and Frommhold (2001) pointed out that the lattereffect tends to enhance the light emission, because


electron-electron interactions decrease the ionizationpotential. Thus more atoms get ionized, resulting inmore light.

A quantitative modeling of the magnitude of theseeffects for the case of a sonoluminescing bubble has notyet been attempted. It would be valuable in order toassess where, in the two-step scheme of Fig. 23, the mostimportant modeling errors are introduced.

5. Towards a more comprehensive model of SBSL lightemission

How can we overcome the various modeling uncer-tainties? From our point of view the only way is to testthe models through detailed experiments in a large pa-rameter domain, varying not only the forcing pressureand the gas type and concentration (and therefore alsothe ambient radius), but also the water temperature anddriving frequencies. Indeed, the experiments along theselines by Vazquez and Putterman (2000) at low tempera-ture and those by Toegel, Gompf, et al. (2000) at lowfrequencies have demonstrated the relevance of liquidvapor.

Many more experiments of this type will be necessaryto improve the models connecting R(t) and T(t), on theone hand, and T(t) and Pl(t) on the other hand. Giventhe unusual conditions inside the bubble (in particular,the high pressure at low degree of ionization), this is aresearch direction whose benefits lie not only in explain-ing SBSL, but in elucidating the physical makeup ofmatter in a state that has not been studied in any detailbefore. Understood this way, the SBSL bubble is anideal microlab for high-pressure/high-density hydrody-namics, chemistry, and plasma physics. The focus of thequestions asked should perhaps shift from the nature oflight-emission mechanisms (which, from our point ofview, is now understood quite well) to more application-oriented problems.

If this model is able to capture the essential featuresof SBSL, no exotic ‘‘new physics’’ is necessary to explainthe phenomenon. With classical hydrodynamics, disso-ciation chemistry, thermodynamics, and the theory oflight absorption and emission in hot gases, well-knownresults from diverse fields now seem to be fitting into aconsistent whole.

G. Line emission in SBSL

Line emission in multibubble sonoluminescence haslong been known (Suslick, 1990; Flint and Suslick,1991b); typically it originates from neutral hydroxylradicals, or other molecular species present in the liquid.Until very recently, molecular lines had never been ob-served in single-bubble sonoluminescence. Two explana-tions were typically given for the absence of lines insingle-bubble sonoluminescence: either the lines arebroadened by the immense pressure inside the bubbleand smeared over the whole visible spectrum, or the


continuum radiation is just much more intense than thelines, which are consequently ‘‘swamped’’ by the con-tinuum spectrum.

Very recently, the groups of Suslick (Didenko et al.,2000b) and Kang (Young et al., 2001) have discoveredparameter regimes in which there is line emission fromsingle sonoluminescing bubbles. These were measuredin situations where the bubble tended (i) to be unstable(‘‘moving SBSL,’’ in which the bubble gyrates around apressure antinode of the ultrasound field) and (ii) toemit light rather weakly, close to the sonoluminescencethreshold.

Didenko et al. (2000b) generated SBSL in various or-ganic liquids such as adiponitrile, a liquid with very lowvapor pressure (to keep the effective polytropic expo-nent high) and a chemical structure that results in disso-ciation products easily soluble in the liquid. In these re-spects, adiponitrile and other polar aprotic liquidsresemble water. The choice of liquids by Didenko et al.(2000b) was guided by the theoretical findings that liq-uid vapor can be responsible for quenching SBSL (seeSec. III.A.3) and that the chemistry of dissociation prod-ucts plays an important role in bubble stability (see Sec.IV.C).

Figure 19 shows the observed spectra from movingSBSL bubbles in adiponitrile. Didenko et al. (2000b) at-tribute the observed lines to excited species of CN, oneof the groups that make up adiponitrile. It can be seenthat the line-emission component of the spectra is veryprominent at lower driving pressure amplitudes (andlower total emission intensities) and becomes undetect-able as the driving pressure and the light intensity in-crease, until only the continuum part of the emissionremains. This observation favors the idea of ‘‘swamped’’spectral lines: at lower forcing, with lower temperaturesin the bubble, the thermal bremsstrahlung processes(generating a continuum spectrum) are still weak, andthe easily excited characteristic molecular emission fromCN is strong enough to outshine the continuum. Higherdriving generates higher temperatures and more ioniza-tion in the bubble, leading to bremsstrahlung emissionmuch brighter than the contributions from lines, yieldingthe appearance of a pure continuum as observed forstable SBSL bubbles. While it is not clear to what extentthe instability of the moving SBSL bubble in this experi-ment influences the relative intensity of the spectrallines (as compared to stable SBSL), this work is ex-tremely important because it conclusively demonstratesthat liquid-vapor chemistry occurs inside a single lumi-nescing bubble.

Working independently, Young et al. (2001) foundspectral lines for pure noble gases dissolved in water.The lines were present at low driving pressures, right atthe onset pressure for sonoluminescence. The dissolvedgas concentration was about 20%, which corresponds tounstable sonoluminescence for a pure noble gas, with‘‘dancing’’ bubbles ejecting microbubbles (see Sec.IV.B). For pure argon dissolved in water, stable SBSL isnot possible at the very low light intensities that are ofinterest here. To collect enough photons, Young et al.


(2001) needed to develop methods for collecting reliabledata from single bubbles for up to 5 days; the dimmestbubble they studied had an intensity more than threeorders-of-magnitude smaller than a typical sonolumi-nescing bubble.

For bubbles with very low light intensity, spectralpeaks were seen at both 310 nm and 337 nm (see Fig.18). The 310-nm line is a clear signature of thevibrational/rotational bands in excited OH radicals. Thisexperiment therefore provides the first direct experi-mental indication that there is water vapor in thebubble. The origin of the line at 337 nm is unclear;Young et al. (2001) conjecture that it could be due toanother molecular excitation of OH.

When Young et al. (2001) increased the driving pres-sure, they observed, just as had Didenko et al. (2000b),that the continuum part of the spectrum became stron-ger relative to the line emission. They determined that,although the relative intensity of the lines decreases withincreasing driving, the absolute line intensity still in-creases, again indicating that the reason spectral linesare not typically seen in single-bubble sonoluminescenceis that they are overwhelmed by the continuum. Younget al. (2001) also used different noble gases and showedthat the intensity of the 310-nm line increased with themolecular weight of the noble gas (Fig. 2 of that work),so that xenon had a more intense 310-nm emission thanargon (the 337-nm line, however, seemed to be unaf-fected by the gas species). By lowering the temperatureof the water, Young et al. (2001) observed that the inten-sity of the spectral lines increased. At 5 °C, additionallines appeared.

Another crucial issue was to determine precisely whythe spectral lines were only observed in unstable ormoving single-bubble sonoluminescence. Young et al.(2001) tried to find spectral lines in stable sonoluminesc-ing bubbles, but did not succeed at the high driving pres-sures needed for stable SBSL, where the continuum partof the spectrum is always overwhelming. On the otherhand, growth by rectified diffusion, followed by the ejec-tion of microbubbles due to shape instabilities (Hilgen-feldt et al., 1996) could lead to quantitatively differentwater-vapor content and different chemistry inside anunstable bubble as compared to a stable one. Again, aquantitative model of line emission would settle thisquestion, predicting the range of parameters for whichline emission can outshine the SBSL continuum.

Recent experimental results of Baghdassarian et al.(2001) on the spectra of single-cavitation bubble lumi-nescence may help in developing such a model: Beyonda certain laser energy threshold, OH lines show up in theemitted spectra. The interpretation is that for largeenough bubbles the collapse is aspherical, so that somefluid can enter the bubble. This interpretation of thesingle-cavitation bubbles thus provides a connection tothe moving SBSL (Didenko et al., 2000b), unstableSBSL (Young et al., 2001), and MBSL (Matula et al.,1995) spectra, which can also show lines.


VI. SUMMARY AND OUTLOOK

This review has presented current ideas about single-bubble sonoluminescence, emphasizing the most impor-tant physical processes that govern bubble behavior andlight emission. A combination of different physical ef-fects are important for the dynamics and thermodynam-ics that act over different time scales and, therefore, atdifferent times during the bubble’s oscillation. To sum-marize these effects, the beginning of this section followsan SBSL bubble through one oscillation cycle, indicatingthe most important phenomena at work. The secondpart of this section deals with unanswered questions inSBSL and possible applications of the new insightssonoluminescence research has allowed.

A. An SBSL bubble through its oscillation cycle

In Fig. 4, we have numbered the various parts of theoscillation cycle of a typical SBSL bubble containing ar-gon in water [for definiteness, we use an example fromStorey and Szeri (2000, 2001), with R054.5 mm, drivenat f526.5 kHz and Pa51.2 atm]. The cycle begins, bydefinition, when the driving pressure begins to dip intothe negative half of its cycle, and the bubble is thereforeallowed to expand. The surrounding liquid far from thebubble is degassed to some level and maintained atroom temperature. Consequently, the bubble containsabout 1010 argon atoms and about 23108 water mol-ecules at the outset.

(1) Expansion. The bubble expansion is comparativelyslow and the growth is sustained for almost half a cycle(;15 ms). In this phase, the bubble is in both thermaland mass transfer equilibrium with the liquid. Becauseof the falling pressure inside the bubble, it gains largenumbers of water-vapor molecules (evaporating fromthe wall) and also some gas molecules from the liquid.

(2) Turnaround at maximum radius. The driving pres-sure begins to increase again, and the expansion comesto a halt. At maximum radius (Rmax'7R0), the bubblecontains little more than the initial 1010 argon atoms, buthas collected up to 1011 water molecules.

(3) Rayleigh collapse. As the external pressure in-creases, the inertial collapse of the liquid layers aroundthe bubble begins. Even with the increased number ofmolecules, the internal pressure is still very low, and thecollapse proceeds almost exactly like the classical col-lapse of an empty cavity treated by Lord Rayleigh(1917). The radius decreases quickly (over about 4 ms)from Rmax to a value comparable to R0 . During thiscollapse, water vapor recondenses at the wall and theargon atoms again become the dominant species insidethe bubble.

(4) Decoupling of water vapor. About 50 ns before theminimum radius is reached, the time scale of the bubblecollapse (;R/uRu) becomes smaller than the time scalefor the diffusion of water vapor. The water vapor stillleft inside the bubble is now trapped until the reexpan-sion. Calculations show that about 20% water vaporshould be mixed with the argon. Up to this moment, the


polytropic exponent of the gas mixture has not increasedsignificantly above 1, and the temperature has only risento about 500 K.

(5) Thermal decoupling. Only '30 ns later, the accel-erating bubble wall becomes fast enough that heat canno longer escape the bubble. Until reexpansion, thebubble is now thermally isolated from the liquid as well,and the polytropic exponent g rises quickly to its adia-batic value. The latter is determined by the mixture of80% Ar and 20% H2O to g'1.6. From now on, thetemperature increases rapidly.

(6) Onset of dissociation reactions. Once the tempera-ture exceeds roughly 4000 K, water-vapor moleculesstart dissociating into OH and H radicals. While the dis-sociation products have lower adiabatic exponents thanH2O, this endothermic reaction also consumes much en-ergy and curbs the temperature rise. At this stage, faintmolecular band light emission is a possibility.

(7) Onset of light emission. Despite the temperature-limiting influence of water vapor, about 10 000 K arefinally reached in the bubble about 100 ps before maxi-mum compression. At this temperature, a small fractionof the Ar as well as of the O and H atoms now presentundergo ionization and release free electrons. The elec-trons interact with the ions and neutral atoms, and emis-sion of electromagnetic radiation (thermal bremsstrah-lung and radiative recombination) begins, which spansthe spectrum of visible and ultraviolet wavelengths.Sonoluminescence is observed.

(8) Maximum compression. At this point, the gas den-sity reaches (almost) solid-state values. The decelerationof the bubble wall down to zero speed has begun toenhance random shape perturbations (Rayleigh-Taylorinstability) and leads to massive energy loss throughacoustic wave emission. The temperature and light emis-sion peak, helped by the high densities that prevent fur-ther endothermic dissociation reactions.

(9) Reexpansion. The bubble loses about 90% of itsenergy in the collapse, mostly due to acoustic emission.The reexpansion is much slower than the collapse. TheRayleigh-Taylor instability grows and may overwhelm astrongly driven bubble during this stage. Only a smallincrease in radius and decrease in temperature are suf-ficient to dramatically reduce the photon absorption co-efficient and quench the light emission uniformly for allwavelengths, about 100–200 ps after it has begun. Sub-sequently, reaction chemistry stops and thermal as wellas diffusive equilibria are reestablished.

(10) Afterbounces. The bubble rebounds to a muchsmaller size than the maximum radius before the maincollapse, and for no parameter combination realized sofar is there enough energy left to induce sonolumines-cence during the afterbounces. The afterbounces pro-vide a parametric excitation that can accumulate andrender the bubble shape unstable. The radial motion is,however, damped rapidly until the driving pressure dipsinto its negative cycle once again, and the oscillationstarts anew. Over the whole cycle, shape perturbationsmay have been enhanced (then the bubble is parametri-cally unstable), or a net gain or loss of gas may have


resulted (diffusive instability). In the correct parameterrange, the bubble is stable with respect to both processesand continues to oscillate and emit light in exactly thesame fashion.

The above 10 steps hold for SBSL bubbles in waterwith only argon (or any other noble gas) dissolved. If inaddition molecular gases such as nitrogen and oxygenare dissolved, not only water dissociates at step 6, butalso these gases (at around 7000 K for N2 and O2). Thereaction products subsequently dissolve in water.‘‘Cleaning’’ the bubble from molecular gases in this fash-ion can take thousands of cycles, as shown by Matulaand Crum (1998).

B. Unanswered questions

Possible refinements of this simple picture abound.Detailed simulations of the bubble interior have beencarried out (Moss et al., 1997, 1999; Cheng et al., 1998;Storey and Szeri, 1999), indicating that many effectscould in principle play a role that were not mentioned inthe above summary. Moss et al. (1997) include the pos-sible existence of two temperatures for ions and elec-trons, as the collision rate of the atoms may not be suf-ficient to thermalize the electrons. They emphasize theimportance of this effect for a proper modeling of heatexchange between the ionized gas and the exterior ofthe bubble. In the same work, shock waves have beencomputed, although Vuong and Szeri (1996) and Lin andSzeri (2001) showed that their formation is much de-layed due to the entropy profiles inside the bubble.While it seems clear that shock waves are not necessaryto explain SBSL light emission, they could lead to highertemperatures in an extremely small central region of thebubble.

Detailed models of the water-vapor chemistry in thebubble (Yasui, 1997a; Storey and Szeri, 2000) sufferfrom uncertainties in the reaction rates for even funda-mental processes, as the peculiar conditions inside thebubble have not been probed in other experimental set-ups. Therefore, bubble temperature predictions fromthese reactions carry considerable uncertainty. Morefundamental work is necessary here. A similar uncer-tainty concerns a more quantitative modeling of thebound-free contributions to the light emission, as theterm schemes of the atoms involved could be substan-tially altered by the extreme densities.

Current models of sonoluminescence use simplifiednotions about the final stages of the bubble collapse.There are two classes of results for heat transfer: simplemodels, which extrapolate the point where the heattransfer transitions from isothermal to adiabatic fromlinear theory, and full simulations, which solve for thecomplete heat transfer numerically. The drawback of thelatter approach is that it is not possible to span the entireSBSL parameter space with large numerical simulations,while the drawback of the former is that more approxi-mations have to be made.

The quest for the ideal liquid for SBSL continues.Didenko et al. (2000b) showed that the peculiar combi-


nation of low vapor pressure and benign dissociationchemistry favors certain organic liquids. Attempts byBaghdassarian et al. (2000) to achieve SBSL in liquidnoble gases (e.g., for a helium bubble in liquid argon)have not achieved higher levels of light intensity thanthose for the conventional case. One reason for this isthe necessarily much lower temperature at which thebubble starts. If the compressive heating at comparableexpansion ratios starts at, say, 10 K instead of 300 K, thepeak temperature is automatically penalized by a factorof 30, even though the collapse heating itself may bemore efficient.

Perhaps other liquids or environments can also beused to probe spectral ranges of the light emission out-side the optical regime—the exact shape of the ultravio-let spectrum for light noble gases is still unknown, due tothe strong absorption of those wavelengths in water(and most other liquids). Matula et al. (2001) have re-cently measured the near-infrared part of the spectra ofSBSL and MBSL, confirming the thermal nature of theradiation.

One of the persistent puzzles in the research onMBSL is the origin of spectral lines from metal ionsdissolved in the water (see Fig. 6). Do the metal ions getinto the bubble at its asymmetric collapse? Or is thewater around the bubble somehow playing a role in thelight-emission process? At present, there are no answersto these questions.

We might also briefly mention here the possible effectof a strong magnetic field on sonoluminescing bubbles.While preliminary work by Young et al. (1996) suggestedthat this effect would be very pronounced, later findingsby the same group revealed that the observed magnetic-field effect was mainly on the flask, and not on thebubble itself.

C. Scientific uses and spinoffs

One question arises naturally whenever SBSL is dis-cussed: Is there an application for sonoluminescence? Acommercial use of the light itself seems unlikely, giventhat only a fraction of ;1024 of the energy in a collaps-ing bubble ends up as visual photons. Sonoluminescencemight provide illumination for certain photographs, e.g.,in dentistry where ultrasound-driven devices are used(Leighton, 1994). But in most situations, illuminationcan easily be achieved by other means.

Grieser and Ashokkumar (2001) have recently shownthat sonoluminescence can be used to excite fluorescentmolecules to emit light themselves, often at much higherintensity, a process which those authors dubbed sono-photoluminescence. This form of emission due to exter-nal photoexcitation changes in the same fashion as thesonoluminescence signal, with changes in solution con-tents and solution concentration, and thus provides atrue, amplified representation of the sonoluminescenceintensity. Being closely associated with chemical reac-tions due to cavitation, sonoluminescence light can alsoserve as an indicator for sonochemistry (see below)


when light intensities and chemical reaction yields arecorrelated (Beckett and Hua, 2001).

D. Other applications of bubble dynamics and cavitation

With the consistent and rather simple picture that hasnow emerged, SBSL could well fulfill its promise as anexemplary, well-controlled system of cavitation physics.Sonochemistry, in particular, will profit from it. Insonochemistry, ultrasound (usually of low frequency) isused to enhance, assist, or induce chemical reactionsthat would not occur spontaneously—the ultrasoundworks like a catalyst (Mason and Lorimer, 1988; Suslick,1990; Reisse et al., 1999). Collapsing bubbles generatetemperatures and pressures not easily achieved other-wise, and do so repeatedly over very short time scales. Itis thus possible to induce unique reactions.

While it has been demonstrated that the presence ofbubbles and cavitation is essential for sonochemistry, itis not clear at all where exactly the chemical reactionstake place: in the interior of a collapsing bubble, at theirsurface, in a liquid layer around the bubble, or even atgreater distances, mediated by the diffusion of primaryreaction products from the bubble interior (Suslick,1990). The situation is complicated by the presence ofmany bubbles, inhomogeneities in the sound field, andthe proximity of vessel walls, which all favor asymmetricbubble collapses. The ensuing strong shape deformationand/or fragmentation of the bubbles enhance mixing be-tween bubble exterior and interior, which makes thequestion of the reaction locus even more difficult to de-cide.

On the other hand, even if a large number of bubblespartake in the reaction catalysis, the yields of manysonochemical reactions are still woefully low (Reisseet al., 1999). Some selected ultrasound-assisted processesof chemical technology are now approaching applicationon an industrial scale, e.g., waste water treatment (Rus-sel, 1999). Nevertheless, the reaction efficiency will haveto be enhanced before a more widespread use ofsonochemistry is feasible. One approach to achievinghigher yields is the search for an optimal frequency ofthe driving ultrasound (Beckett and Hua, 2001).

Just like sonochemistry, a great number of other fieldsrely crucially on the energy-focusing powers of collaps-ing bubbles: Materials science makes use not only of thehigh temperatures and pressures, but also of the tremen-dous cooling rate of reexpanding bubbles far in excess of1010 K/s (see Suslick, 1995). These cooling rates are un-matched by any other technique, and allow for the fab-rication of amorphous metal nanoclusters that prove tobe highly effective catalysts, as shown by Suslick andCasadonte (1987).

In ultrasonic cleaning, the shear and pressure forcesaround collapsing bubbles are used to rid material sur-faces of contaminations (see Leighton, 1994). If this sur-face erosion goes too far and affects the material prop-erties, we speak of cavitation damage (Leighton, 1994;Brennen, 1995). Weninger, Cho, et al. (1997) observedthat an isolated bubble sitting on a solid surface can


even emit light. Ever since Lord Rayleigh’s (1917) firststudy of cavitation was motivated by the study of cavi-tation damage on ship propellers, this has been a majordriving force behind research on bubbles. A large indus-try is thus concerned with avoiding violent bubble col-lapses that can do damage by shear as well as with theshock waves they emit, or the thin, fast liquid jetsejected from asymmetrically collapsing bubbles (Blakeet al., 1997; Ohl et al., 1998). Even when the forces arenot too large, cavitation can simply disrupt laminar fluidflow, with serious consequences for applications such asfuel transport through valves and tubes or inkjet print-ing (Dijksman, 1999).

In biology, cavitation can occur spontaneously inwater-transporting vessels in plants, which are often un-der high tension (Holbrook et al., 2001). Another niceapplication of collapsing bubbles occurs for various spe-cies of ‘‘snapping shrimp’’ (alpheus heterochaelis andothers), a shrimp whose most distinctive feature is onegiant claw opposite a normal-sized one. This animal livesin large colonies that generate noise so loud that it dis-turbs submarine communication. It was believed thatthe noise is emitted when the giant claw rapidly closesand its two sides hit each other. However, Versluis et al.(2000) showed with the help of high-speed video andparallel sound detection that the origin of the noise infact is a collapsing cavitation bubble: When rapidly clos-ing the pair of scissors, the shrimp emits a thin water jetat such high speeds that a cavitation bubble develops.When the bubble collapses, sound is emitted in the formof a shock wave that stuns or even kills small prey. Veryrecently Lohse et al. (2001) showed that, at bubble col-lapse, not only sound but also a short light pulse (about104 photons) gets emitted. They called the phenomenon‘‘shrimpoluminescence.’’

Another example in which high water velocities leadto bubble cavitation is the converging fluid flow in aVenturi tube analyzed by Peterson and Anderson (1967)and later by Weininger et al. (1999). The bubbles cavi-tate as they flow through the device, and the resultinglight emission from this device has many features incommon with SBSL, for example, enhancement of emis-sion using xenon instead of argon gas.

Some of the most promising applications lie in thefield of medicine. More than 30 years ago, Gramiak andShah (1968) had already suggested using micrometer-sized bubbles as contrast agents for ultrasound diagnos-tics. The bubbles are of resonant size for the MHz fre-quencies of diagnostic ultrasound, and therefore areextremely potent scatterers with cross sections severalthousand times larger than their geometrical cross sec-tion (see Nishi, 1975). When injected intravenously, thebubbles allow for brighter images and higher contrast.The nonlinearity of bubble dynamics and the violent col-lapses help to increase the bubble response and imprinta distinctive signature onto the emitted sound, making iteasier to distinguish bubble echoes from unwanted tis-sue reflections (de Jong and Hoff, 1993; Hilgenfeldt,Lohse, and Zomack, 1998; Frinking et al., 1999). Yeteven here the potential for various kinds of ‘‘cavitation


damage’’ has to be carefully assessed: mechanical dam-age to living tissue (Barnett, 1986), thermal hazard fromthe absorption of high-frequency sound in tissue andblood (Wu, 1998; Hilgenfeldt et al., 2000), or chemicalhazard from the sonochemical production of radicals in-side the body (Barnett, 1986).

Medicine is also beginning to discover the benefits ofcavitation damage, using bubbles as vehicles for thera-peutic applications as well. It has recently been demon-strated by Tachibana et al. (1999) and others that thedirected delivery of drugs as well as the transfection ofgenes through the wall of living cells is dramatically en-hanced in the presence of ultrasound and microbubbles.Doubtless, one or several of the above-mentioned pro-cesses of cavitation damage are at work here to renderthe cell wall permeable for drugs and genes, but withoutpermanently damaging the cell (in experiment, manycells appear healthy after treatment, as drugs are effec-tive and genes are expressed). The exact mechanism re-mains to be uncovered.

E. Multibubble fields: in search of a theory

All of the applications mentioned in previous para-graphs will be feasible only if many bubbles undergo thedescribed processes to focus the energy of the cavitationcollapse where it is needed. Thus a theoretical descrip-tion of sonochemistry or ultrasound diagnostics cannotrely entirely on what we have learned about singlebubbles studying SBSL. Interactions between bubblesand their emitted sound fields will be very importantabove a certain volume fraction of gas, which may be aslow as 1025 (Marsh et al., 1998). These interactions takeon a plethora of different shapes, such as secondaryBjerknes forces (Brennen, 1995; Mettin et al., 1997),bubble shadowing (Marsh et al., 1998), collective bubblecollapses (Brennen, 1995), or collective bubble transla-tion (streamers; Akhatov et al., 1996). In addition, mul-tibubble applications always have to deal with the inter-action of bubbles with boundaries, be they hard (as inmaterials science) or soft (as in biological and medicalcontexts). With the experimental observation and nu-merical simulation of jet cavitation in bubble collapsesnear a wall by Vogel and Lauterborn (1988a, 1988b),Vogel et al. (1989), Tomita and Shima (1990), Blakeet al. (1997), Philipp and Lauterborn (1997), Lauterbornand Ohl (1998), Blake et al. (1998, 1999), Tong et al.(1999), Brujan et al. (2001a, 2001b), research in thiscomplex area has only just begun.

Modeling the collapses of interacting bubbles in detailis very cumbersome, since the bubble motion is typicallyasymmetric, and the bubble interface can even changeits topology. The liquid jet can penetrate the bubble thatgenerated it, as seen in the famous photograph by Crum(1979). Clearly, a tractable theory cannot model eachindividual bubble in a cloud. The ultimate goal is asimple, effective-medium theory that describes the re-sponse of a bubble field hit by an intense sound wave


(continuous or pulsed) in terms of field variables encod-ing the deposited energy density and its distribution.This goal still seems far ahead.

So for those with applications in mind, the originalcase of multibubble sonoluminescence from the 1930s(Frenzel and Schultes, 1934), deemed unimpressive andnot very interesting, may in fact be the more challengingand rewarding task. Single-bubble sonoluminescencehas taught us more about the astounding forces at workwhen a bubble collapses, and remains a beautiful andunique phenomenon. We now must show that we havelearned enough to take this research program to thenext, more general level.

ACKNOWLEDGMENTS

We thank all the colleagues with whom we had veryfruitful collaborations and discussions over the years.Without them this work would not have been possible.We would like to name Robert Apfel, Bradley Barber,John Blake, Tom Chou, Lawrence Crum, Todd Dupont,Lothar Frommhold, Felipe Gaitan, Bruno Gompf, FranzGrieser, Siegfried Grossmann, Dominik Hammer, GlynnHolt, Joachim Holzfuss, Leo Kadanoff, Woowon Kang,Jeffrey Ketterling, Lou Kondic, Werner Lauterborn,Stefan Luther, Philip Marston, Thomas Matula, WilliamMcNamara, Michael Moran, William Moss, Claus-Dieter Ohl, David Oxtoby, Rainer Pecha, Andrea Pros-peretti, Ruediger Toegel, Gabor Simon, Brian Storey,Kenneth Suslick, Andrew Szeri, Keith Weninger, Kyu-ichi Yasui, and Joseph Young.

M.B. acknowledges support from the National Sci-ence Foundation (NSF) Division of Mathematical Sci-ences and also from an NSF International Travel grantsupporting this collaboration. The work is part of theresearch program of the Stichting voor FundamenteelOnderzoek der Materie (FOM), which is financially sup-ported by the Nederlandse Organisatie voor Weten-schappelijk Onderzoek (NWO).

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single-bubble sonoluminescence · pressure, and light intensity (top to bottom) during this fig. 1....

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