Review/Synthèse

Single-bubble sonoluminescence:

“bubble, bubble, toil and trouble”1

J. David N. Cheeke

Abstract: This review describes in detail the new field of single-bubble sonoluminescence(SBSL). Increasingly refined studies on classical cavitation led to a pioneering experiment byGaitan and Crum in 1990 in which a single-bubble in water was trapped in a remarkably stableposition by an applied acoustic standing wave. The bubble oscillated reversibly in synchronismwith the sound wave and under certain conditions a catastrophic collapse led to the emission ofa tiny burst of light, known as SBSL. The effect was characterized by an amazingly precisesynchronicity with the sound field, extremely sharp pulses, and a continuous emission spectrumin the visible. SBSL has generated tremendous interest and excitement, mainly due tospeculation that the bubble temperatures at the moment of extreme collapse are very high;temperatures higher than 1 × 106 K have been proposed. The main experimental results aresummarized and the bubble dynamics and SBSL theories are described and evaluated.

Résumé: Nous présentons une revue détaillée du nouveau champ de recherche connu sous lenom de sonoluminescence à une bulle (SBSL). Des études de plus en plus fines en cavitationclassique ont mené à l’expérience clé de Gaitan et Crum en 1990 oû une bulle isolée dans l’eaua été conservée dans une position remarquablement stable par une onde acoustique stationnaire.La bulle oscille en phase avec l’onde sonore et sous certaines conditions il se produit uneffondrement catastrophique de la bulle accompagné d’un signal lumineux très bref connu sousle nom de SBSL. Cet effet est remarquable par son extraordinaire synchronisme avec le champacoustique et son impulsion extrêment brève montrant un spectre continu dans le visible. SBSLgénère beaucoup d’intérêt, surtout parce qu’a été avancée l’hypothèse que la température de labulle pouvait atteindre de très hautes valeurs, jusqu’à 106 K, au moment de son effondrementcatastrophique. Nous résumons ici les principaux faits expérimentaux et analysons lesdifférentes théories pour la dynamique de la bulle et le mécanisme d’émission de lumière.[Traduit par la rédaction]

1. Introduction

Acoustic cavitation may be defined as the excitation and vibration of vapor cavities under acousticstress. The subject first became of interest near the end of the nineteenth century with the introductionof steel propellers by the Royal Navy. Trials on a new destroyer, H.M.S. Daring, revealed that insuffi-cient thrust was developed due to the creation of voids and bubbles. The associated rapid erosion of thepropellers by cavitation was also obviously a cause of concern. Lord Rayleigh, as has happened so oftenin acoustics, was the first to provide a quantitative treatment of the general cavitation phenomenon, in1917. His work is the starting point in our discussion of single-bubble sonoluminescence.

Received July 23, 1996. Accepted December 18, 1996.

J. David N. Cheeke.Physics Department, Concordia University, 1455 de Maisonneuve Blvd. W.,Montreal, QC H3G 1M8, Canada. Telephone: (514) 848-3292; e-mail: cheeke@alcor.concordia.ca

1 With apologies to the Bard.

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Since that time, cavitation effects have proven to be important in a number of scientific and techno-logical processes, such as ultrasonic cleaning, sonochemistry, medical applications, biology, bubblechambers etc. One particular aspect of the phenomenon was observed in 1934, the so called “sonolumi-nescence” where the collapse of individual bubbles was accompanied by the emission of light [1]. Theorigin of this effect has been a mystery until fairly recently. The present review provides a summary ofrecent work on sonoluminescence under very controlled conditions.

Historically, acoustic cavitation effects were produced by high-power ultrasonic horn devices im-mersed in a liquid. The resulting multiple-bubble cavitation regimes are generally classified into twotypes, “transient” and “stable”. Transient cavitation of each bubble lasts only a fraction of a second, andleads to a violent collapse of the bubble. Stable cavitation is a more controlled phenomenon, althoughthe movement of each bubble is unsynchronized with that of its neighbour. The associated light emissionfrom multiple bubbles will be termed multiple-bubble sonoluminescence (MBSL).

A major breakthrough was made in the late nineteen eighties by Gaitan [2] and Gaitan et al. [3] whofound in their work on stable cavitation that it was possible to trap a single-bubble in an acousticlevitation chamber, and have it oscillate stably in the acoustic field almost indefinitely. It thus becamepossible for the first time to study cavitation under highly controlled conditions. The effect was alsoaccompanied by the emission of a tiny burst of light once per acoustic cycle, which Gaitan found to behighly synchronous with the sound wave. This effect, known as single-bubble sonoluminescence(SBSL), has generated tremendous interest and controversy during the last few years, partly because ofthe magic associated with direct conversion of sound waves into light. The present review presents aunified description of this intriguing phenomenon.

This paper is divided up as follows. Section 2 provides a summary of the main principles of bubbledynamics in acoustic fields. A grasp of this is essential to an understanding of how bubbles are trappedand how they oscillate in acoustic fields. Section 3 contains a summary of the main experimental factsavailable on SBSL. The three most important ones, very sharp light pulses [4], picosecond synchronicitybetween light and sound [5], and a continuous emission spectrum [6] constitute the hallmarks of SBSLthat any successful theory must explain. The theories are presented in Sect. 4. Initially, considerableemphasis is placed on the shock-wave model, as this places the detailed bubble dynamics and acousticson a firm, quantitative footing. Some of the current theories on light emission are then presented. Finally,Sect. 5 summarizes the present situation and points to directions in which further work is needed. Therehave been many reviews of MBSL in the pre-SBSL period [7–17]. Since the discovery of SBSL byGaitan [2] and Gaitan et al. [3] there have also been several general accounts of the phenomenon[17–21].

2. Bubble dynamics

Historically, all of the experimental work was done on multiple bubbles, while the theory was for asingle isolated bubble. These theoretical ideas can be directly adapted and extended to the configurationused for SBSL. Some of most useful papers are by Gaitan and co-workers [2, 3], Walton and Reynolds[14], Flynn [22, 23], Lauterborn [24], Neppiras [25], Prosperetti [26], and Wu and Roberts [27].

Much of the theoretical description of SBSL dynamics is based directly on the work of Lauterborn,who has conveniently summarized the basic results [24]. In the absence of an acoustic wave the bubbleis acted upon by a net buoyant force and it rises to the surface; in the present work the bubble will alwaysbe considered to be in a standing acoustic wave field, and it will be shown shortly that it will be trappedby such a field at a pressure antinode. Following Lauterborn, we use the bubble model developed byRaleigh, Plesset, Noltingk, Neppiras and Poritsky (the RPNNP model), whose associated nonlineardifferential equation can be written

ρRR⋅⋅ + 3

2ρR

⋅ 2 = pg

R0

R

3Kp

+ pυ − pstat −2σR

−4µR

R⋅ − p(t) (1)

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Where ρ = liquid density,R(t) = instantaneous bubble radius,R0 = equilibrium radius,σ = surface tension,pstat = static pressure (atmospheric plus hydrostatic),pυ = vapour pressure,

µ = shear viscosity

Pg =2ρR0

+ pstat − pυ

Kp = polytropic coefficientFor an acoustic wave

p(t) = −pa sin ωt

where pa is the sound pressure amplitude and ω = 2π f is the angular frequency.The minus sign assures that we start with tension at t = 0. The model requires that p(t) be constant

at the bubble surface, which implies that the wavelength be large compared to the bubble radius. Thiswill always be true for the frequencies considered here, which are much lower than the free-bubbleresonance frequency given by;

f0 = 12πR0√ρ

3Kp

pstat +

2σR0

− pυ

−2σR0

−4µ2

ρR02

1/2

(2)

This equation can be used to determine the resonant radius for a given applied frequency or, alterna-tively, to determine the frequency at which a bubble of radius R0 would resonate. For large air bubblesin water f0R ~ 3 where f0 is in hertz and R in metres. It follows from (2) that a bubble of equilibriumradius equal to 4 µm would resonate at about 1 MHz, while an applied frequency of 20 kHz corresponds

to a resonant radius of about 160 µm.Before looking at solutions of (1), we note that there is an additional process that is normally very

important in describing the growth of bubbles in an ultrasonic field, which is the process of rectifieddiffusion. During the acoustic compression cycle the gas concentration inside the bubble increases, sothat gas tends to diffuse out into the liquid. Contrarywise, during an expansion cycle the gas concentra-tion decreases and gas diffuses back into the bubble. However, these two processes do not compensateeach other, mainly because the surface area is greater during an expansion cycle; hence the bubble tendsto grow with time. What actually happens in a specific instance depends on many parameters, includingthe state of degasification of the liquid. Rectified diffusion will not be a problem in practise in SBSL, assteady-state conditions at constant-equilibrium bubble radius can be maintained experimentally forseveral hours. In principle it is certainly important, but its specific role in SBSL has not yet beenelucidated. Rectified diffusion is reviewed by Neppiras [25] who also gives a very detailed account ofbubble dynamics, including sources of damping.

Lauterborn solved (1) numerically for various special cases, and this work was taken up and extendedby Gaitan and co-workers [2, 3] to describe the SB case. The bubble response curves were first describedby calculating the normalized maximum bubble radius Rmax/R0 as a function of the equilibrium radiusR0. A maximum response was found near resonance, with harmonic peaks at R0/Rres = 1/2, 1/3, 1/4....1/n.A special notation was introduced by Flynn [22] and used by Lauterborn and Gaitan to describe theseresonances, as follows. We define Tr as the period of bubble motion, Tf the period of free bubbleoscillations, and T as the period of the driving frequency. Then Tr

= mT and Tr= nT

f. The various peaks

are then labelled as order n/m, where m = 1 and n = 2, 3, 4... correspond to the harmonics, as alreadyseen, and n = 1 and m = 2, 3,... are called subharmonics. For SBSL we are specifically interested inm = 1 and the harmonics.

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The particular case of interest for SBSL is that of acoustic pressures above 1 atm and small bubblesat acoustic frequencies well below resonance. Then the steady-state bubble radius vs. time curve takeson a characteristic form, as shown in Fig. 1 for pa = 1.275 atm, R0 = 4.5 µm, fa = 26.5 kHz. During oneacoustic period there is a slow expansion during negative acoustic pressures, with the radius increasingup to about 38 µm. As the acoustic pressure goes positive the bubble undergoes contraction, which leadsto a catastrophic collapse. This is followed by several well-defined rebounds at the resonance frequency;since Tf is proportional to R0, smaller bubbles will have a large number of minima and vice versa. Thebubble finally attains the equilibrium radius and the whole process is repeated the following cycle.

Gaitan defined three main parameters to describe the above bubble dynamics:(i) the maximum bubble radius Rmax;(ii) the phase of the collapse f measured in degrees from the beginning of the negative pressure

half cycle, where f = 0; and

(iii) the number of minima M, which includes the main collapse.Finally, we consider the forces at work in single-bubble trapping. Walton and Reynolds [14] describe

the forces on a bubble due to acoustic waves. They find that large bubbles will experience a so-calledBjerknes force towards a pressure node; small bubbles will experience a similar force towards a pressureantinode. This fact is exploited in the acoustic levitation cell, where a standing acoustic wave is set upin the z direction and a small bubble may be trapped slightly above the pressure antinode at the centre(for fundamental resonance in the cell), the Bjerknes force balancing the buoyant force. Cylindrical,rectangular, and spherical levitation cells have been used, the latter using radial modes to trap the bubble.Hiller et al. [28] point out that “because λ >> R0 , the stress exerted on the bubble is spherically uniformregardless of the shape of the resonator.”

Fig. 1. The radius of an air bubble in water trapped in an acoustic field with pa = 1.275 atm, R0 = 4.5 µm, andfa = 26.5 kHz. The calculation is based on the theory of Wu and Roberts [27] for the uniform adiabatic modeland is essentially equivalent to their Fig. 1; calculation done at Concordia University courtesy of Haizhong Lin.

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3. Experiments on SBSL

A schematic diagram of the levitation cell and light-scattering system used in the Ultrasonics Group atConcordia University is shown in Fig. 2. The cell is driven at resonance in the range 17–18 kHz with apressure antinode at the centre. One of the critical aspects of the SBSL experiment is that the watershould be thoroughly degassed; a treatment consisting of heating and pumping with a mechanical pumpfor half an hour is sufficient for a 100 mL volume. The gas to be used to form the bubble is injected atthe centre; if quantitative measurements are to be made, an airtight cell should be used. If the acousticamplitude is sufficiently high the bubble will levitate. Although equilibrium bubble sizes of 10–20 µmwere reported by Gaitan [2], more recent work at these frequencies has been done with equilibrium radiiof about 5 µm.

The bubble movement as a function of acoustic driving pressure has been discussed in detail byGaitan and co-workers [2, 3] and Barber [17]. The various stable regimes as functions of acousticamplitude are shown in Fig. 3. At low amplitude, the buoyant force dominates and the bubble rises tothe surface. Above the trapping threshold the bubble will stay near the pressure antinode, automaticallyadjusting its position so that the buoyant force is balanced by the Bjerknes force. The acoustic powermust be above a dissolving threshold, otherwise the bubble will slowly dissolve into the liquid. Abovethe dancing threshold the bubble follows an asymmetric dancing motion. At higher acoustic amplitude,in the range 1.3 < pa < 1.5 bars for the water/glycerine mixture shown, the bubble passes the sonolumi-nescence threshold and begins to glow. Above the upper threshold, the bubble suddenly disappears. Theexact value of the thresholds depends on liquid composition; for example, for pure water, the sonolumi-nating range is typically 1.1 < pa < 1.3 bars. The effect of various experimental parameters such as paand water temperature on the bubble radius R(t) and the SBSL luminescence output has been investi-gated in detail recently by Barber et al. [29]. The envelope and rebounds of the non-sonoluminating

Transducer

Computer

Oscilloscope

x

yz

High VoltagePower Supply

PMT

Function Generator

PowerAmplifier

Fig. 2. Sketch of the simple sonoluminescence experiment set up at Concordia University. The 500 mL flaskis filled with degassed water and bonded to a 50 kHz piezoelectric transducer from Sensor Technology Ltd.That is positioned by x,y,z translation stages. The PMT is Type RCS 8575 TRIUMF N.P.W. Eng. Ltd. and thefunction generator a Wavetek model 29 with a frequency precision of ± 0.01 kHz. The other equipment isstandard and noncritical.

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ACOUSTICPRESSURE

TRANSIENTCAVITATION

SL REGIME

DANCINGREGIME

STABLEOSCILLATIONS

DISSOLVINGREGIME

BUOYANCYREGIME

1.5ATM

1.1ATM

HYSTERESISSL REGIME

1.3ATM

Fig. 3. Various bubble regimes in a stationary acoustic wave field in a water/glycerine mixture. For increasingpressure for this mixture the SL regime occurs for an acoustic pressure from 1.3 to 1.5 atm. Once the SLregime has been attained, there is considerable hysteresis in the SL where the acoustic pressure is subsequentlyreduced, as shown in the figure.

Fig. 4. Simultaneous plots of the sound field (top), bubble radius (middle), and sonoluminescence (bottom) ina glycerin–water sample at PA = 1.2 atm and f = 22.3 kHz. (Taken from ref. 3 with permission of thePublisher and the author.)

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bubble can be fitted quantitatively to the Rayleigh–Plesset equation. For the case of water, once the SLregime is entered, the bubble dynamics are changed dramatically; in particular the mean and maximumamplitudes are sharply reduced and the afterbounces are almost completely suppressed. Although noexplicit calculation has been reported, these dramatic changes at the SL threshold clearly correspond toa very significant conversion of the elastic energy of the bubble into light.

Several authors [18, 19] have pointed out that the parameter space for SBSL is quite restricted andat the same time it is a very robust phenomenon. It is very important to clarify the situation, at the veryleast in order to critically test the various theoretical ideas that have been advanced. At this point thedominant experimental observations on SBSL will be summarized, starting with the most spectacularresults.(i) Sharp pulse widths: perhaps the most spectacular observation, made by the UCLA group,

is that the light flashes are so sharp that up to now it has been impossible to measure theirintrinsic width, an upper limit of 50 ps being assigned [5]. The light emission is visible tothe naked eye and involves 105–107 photons per flash. Comparing the applied acousticenergy to the emitted intensity, this corresponds to energy focussing of the order of 12 ordersof magnitude.

(ii) Synchronicity: Gaitan’s original work clearly showed that the light emission was synchro-nous with the acoustic wave, occurring at the end of the first collapse of the bubble as is seenin Fig. 4. Much of this work involved measuring the phase of the light flash in the acousticcycle, which was found to be remarkable stable. This aspect was further investigated by theUCLA group, who found that the jitter associated with the synchronicity is also in thepicosecond range [5]. It is intriguing to say the least that the absolute synchronicity of thelight pulses is much more precise than the jitter of the exciting acoustic field. Reference 21

Fig. 5. Imploding cavity in a liquid irradiated with ultrasound is captured in a high-speed flashphotomicrograph. The implosion heats the gases inside the cavity to 5500°C. Since this cavity formed near asolid surface, the implosion is asymmetric, expelling a jet of liquid at roughly 400 km h–1. Both the heat andthe jet contribute to a unique chemical environment in the liquid. The diameter of the cavity is about 150 µm.The cavity is spherical at first and then shrinks rapidly. The jet develops opposite the solid surface and movestowards it. (Taken from ref. 31, with permission from the author and L.A. Crum.)

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given in ref. 30 suggests a form of mode-locking in the chamber to explain this stability.Holt et al. [30] show that small variations in the driving frequency lead to period-doubling,chaos, and quasi-periodicity.

(iii) Continuous spectrum: Hiller et al. [6] show that the spectrum of the light emitted from thesingle-bubble is continuous over the visible spectrum, down to the ultraviolet cutoff of water(180 nm). There is no evidence of spectral lines, to a resolution of 1 nm. The intensity of SLincreases by a factor of ten in cooling from 22°C to 10°C and becomes even more heavilyskewed to the UV. The data can be fitted to a 25 000 K black body spectrum at 22°C and agreater than 50 000 K black body spectrum at 10°C.

(iv) Liquid not involved in emission lines for SBSL: Crum [21] has made convincing argumentsin pointing out that there is a fundamental difference between MBSL and SBSL. The caseof MBSL, called sonochemistry, has been studied for many years, particularly by Suslick etal. [31]. They and other workers have shown directly that this case is characterized byasymmetric bubble collapse near walls or other bubbles, as seen in Fig. 5. This leads to animploding jet of liquid into the collapsing bubble. Flint and Suslick [32] measured thetemperature at the centre of the bubble to be approximately 5000 K and to involve charac-teristic emission lines of liquid molecules (Fig. 6). The picture is one of gas and liquidmolecules being dissociated by the high temperatures in the bubble and recombining to giveoff characteristic emission lines. Crum suggests that SBSL is qualitatively a different phe-nomenon (which he calls sonophysics) in which bubble collapse is spherically symmetricand involves only the compression of the vapour in the bubble. This leads to much highertemperatures and pressures and leads to a continuous, blackbody-like spectrum. How ex-actly this spectrum is formed will be deferred until the next section. Several recent control-led spectral measurements of MBSL and SBSL on identical fluids and gases with the samecalibrated spectrometer have been carried out by Matula et al. [33] and are shown in Fig. 7.

Fig. 6. Emission from the ∇ν = 0 manifold of the d3Πg − a3 Πu transition (Swan band) of C2. The dotted lineshows observed sonoluminescence from silicone oil.The boldface line shows the best-fit synthetic spectrum,with Tv = Tr = 5125 K. The thin line shows the difference spectrum. (Reprinted from ref. 32 with permissionof the Publisher and the author.)

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These results tend to confirm the above picture, as spectra on dilute NaCl solutions demon-strate sharp emissions lines for OH* and Na* for MBSL but a very continuous spectrum forSBSL.

(v) Temperature: It has been observed for both MBSL [14] and SBSL [29] that the SL intensityincreases rapidly for decreasing temperature. For SBSL the light emission increases by afactor of 200 in cooling to 1°C from 40°C.

(vi) Enhancement of SBSL by noble gas doping: A surprising result was observed by Hiller etal. [28]. In an attempt to enlarge parameter space they constructed an airtight chamber toinject gases other than air in a controlled fashion. An experiment using “artificial air”composed of an 80/20 N2/O2 mixture gave surprisingly an SL intensity much less than thatfor ambient air. The discrepancy disappeared when 1% argon normally present in ambientair was added to the mixture. A similar “catalytic” effect was found from other noble gases.The origin is not yet understood but this fact must be explained by any successful micro-scopic model of SL.

(vii) Until very recently, SBSL had only been observed in water and water–glycerin mixturesover a limited range [2, 3]. The reason for this is not clear. Recent work in our group andwork published in ref. 34 indicates that the liquid shear viscosity is a key parameter, and thatfor sufficiently high values it is impossible to force the bubble to undergo the catastrophiccollapse necessary to produce SL. Recent work by Weninger et al. [35] using xenon gasbubbles in the temperature range –10°C to 20°C showed the existence of SBSL in a largenumber of liquids, Fig. 8.

(viii) SBSL occurs in a very small window in pa-frequency–radius parameter space (Fig. 9)

Fig. 7. Comparison of the background-subtracted spectra of MBSL and SBSL in a 0.1 M sodium chloridesolution. Each spectrum was normalized to its highest intensity. Absolute radiance comparisons cannot bemade due to differences in light gathering techniques. Note the prominence (in MBSL) and absence (in SBSL)of the sodium emission line near 589 nm. The inset illustrates the absence of structure associated with SBSLaround 589 nm. The signal level in this region is a factor of 3 above background. (Taken from ref. 33 withpermission of the Publisher and the author.)

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although it is a very robust phenomenon within that space [18, 19]. Further work is neededto better define this space for a wide variety of liquid and gas combinations.

(ix) Work on MBSL [14] indicates that SL increases for increasing values of σ2/psat when σ isthe surface tension and psat the saturated vapour pressure of the gas, and for low values ofgas thermal conductivity [36]. This needs to be tested out for SBSL and compared withtheoretical models.

4. Theories of SBSL

4.1. Shock-wave model for bubble dynamics of SBSLWu and Roberts [27] have given the most complete dynamical description of bubble collapse publishedto date. There are compelling reasons for using their model as a guide for the fine time scale bubbledynamics, irrespective of whether the microscopic model they propose for light emission proves to be

Fig. 8. Intensity of sonoluminescence from a single xenon bubble trapped in various fluids as a function oftemperature (normalized to 150 mmHg air in water at room temperature). The xenon is dissolved into thedegassed liquid at a partial pressure of 150 mmHg at room temperature and then the system is sealed. Theseare the largest signals that can be attained for 30 s or longer. The noise level recorded on the lock-in amplifierwith sound on but in the absence of a light-emitting bubble is 10 V (0.0005 when normalized as is the data inthe figure). For 1-pentanol below 1°C, nonlight-emitting bubbles can be sustained. By sweeping the drivelevel, a signal of 1–2 mV (0.05–0.1 normalized) can be attained for about 50 ms from this system. The signalfor an air bubble in water at room temperature is 20 mV. Si oil is Dow Corning 200 fluid (1 cSt viscosity).(Taken from ref. 35 with permission of the Publisher and the author.)

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the correct one or not. Their model is based upon a self-consistent solution of the coupled equations offluid dynamics.

The Wu and Roberts model is based on the solution of the RPNNP equation ((1)) coupled with thefull van der Waals (VDW) equation of state and the conservation equations of fluid dynamics. The lattertwo sets of equations are given as:

∂ρ∂t

+ 1r2

∂∂r

(ρνr2) = 0 (3)

∂∂t

(ρv) + 1r2

∂∂r

(ρv2r2) +∂p

∂r= 0 (4)

∂E

∂t+ 1

r2

∂∂r

[(E + p)vr2] = 0 (5)

p = RT

V − b, e = cvT =

V − b

γ − 1p, S = cv ln

p(V − b)γ + const (6)

Fig. 9. Bubble radius versus time for about one cycle of the imposed sound field as a function of increasingdrive level. The shaded area represents the light-emitting region. The relative intensity of emitted light as afunction of drive level is indicated by the continuous-line ramp. For the unshaded region, the bubble is trappedbut no light is emitted. At drive levels below the unshaded region the bubble dissolves over a long time (~1 s).The lowest amplitude sweep (no bubble present) indicates the noise level. (Taken from ref. 29 with permissionof the Publisher and the author.)

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Fig. 10. The acoustically driven bubble for p′a = 1.275 atm, R0 = 4.5 µm, ωa/2π = 26.5 kHz at 10 instants oftime, ta– tj, during an interval of 0.417 ns, during which two shocks launched by the incoming bubble surfacefocus at 0, at t = t0 < ta + 0.19894 ns and at t < ta = 0.386 ns. Each curve terminates at the right at theappropriate value of R(t). The density, ρ, is shown in kg m–3, radial velocity, v, in km s–1, pressure, p, in Pa,and temperature, T, in Kelvin; entropy, S, is in units of cv. The air is modelled by a van der Waals gas withρ3 = 794 kg m−3 (Note: when two curves in juxtaposition cross one another, one is represented as a broken lineto ease their identification.) (Taken from ref. 27 with permission of the Publisher and the author.)

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Here R is the gas constant, cv = R/(γ − 1) is the specific heat at constant volume, ρ is the gas density,

E = 12

ρν2 + ρe is the total energy density, and v(r,t) is the radial component of the gas velocity, which

obeys v(R,t) + R⋅.

These equations were solved with a Lax–Friedrich shock-capturing scheme with a moving grid of800 points and a temporal resolution of about 4 × 10−4 ps near the principal minimum of the bubbleradius.

Several cases were solved numerically for different pa, R0, and fa; results for one of these will besummarized here. This case corresponds to pa = 1.275 atm, R0 = 4.5 µm, and fa = 26.5 kHz. The bubbleis initially supposed to be in a uniform adiabatic state, giving the solution already shown in Fig. 1. Thisgives Rmin = 0.58 µm at t > 20.490 µs, Rmax = 37.09 µm at t = 16.65 µs.

The nonadiabatic solutions were studied in detail in a 0.417 ns range around the minimum. The resultsare shown in Fig. 10 for 10 instants of time ta to tj in this range. Basically the model corresponds to thecreation of a spherically converging shock front focussed on the centre. As seen in Fig. 10 the first shockforms at r ~ 0.3 µm at t ~ tb. This shock is reflected at t = td and expands to meet the incoming bubbleat t = te and is reflected and refocussed on to the centre where it arrives at t ~ ti. The process is then

Fig. 11. Details of the solutions shown in Fig. 10 in a period of 0.7 ns duration during which the shocks focusat 0 and light is emitted from the bubble. In panel (a), the bubble radius, R(t), is shown as a function of time t

(continuous-line curve); the locations, R = R(t), of the two shocks are also shown (broken-line curves) fromthe instants at which they are formed up until they strike the surface of the bubble. (Taken from ref. 27 withpermission of the Publisher and the author.)

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repeated. The time variations for the bubble radius and the shock front and for other parameters areshown explicitly in Fig. 11. In particular the model predicts extremely high temperatures at the centreof the bubble at the moment of shock focussing and the emission of a very sharp flash of light of theorder of 1.2 ps long. In this model the air at the centre is ionized, resulting in a plasma that emits light.

Other findings of this very detailed theoretical study are as follows.(1) Rather surprisingly, for such a robust system, the simulated results for the bubble dynamics

show an “extraordinary sensitivity of the system to small changes in the conditions ofexcitation”.

(2) Variation of key parameters predicts that the SL should vanish for acoustic frequenciesabove 40 kHz. The predicted Sl intensity is enhanced at lower frequencies and even highertemperatures at the centre are predicted.

While the theory of Wu and Roberts is by far the most comprehensive and the most successful inproviding a quantitative model for bubble dynamics based on a firm formal foundation, it is far fromcomplete. An important aspect that should be formally included is that of thermal diffusion. Greenspanand Nadim [37] point out this leads to smoothing out of the shock and diminishing its strength. In viewof the known effect of the thermal conductivity of the gas [14], thermal diffusion effects should beincluded in the model and quantified. These aspects have recently been developed in refs. 38 and 39,where it is shown that inclusion of the loss terms tempers the singularity, as would be expected. Inaddition, a more realistic equation of state should be used in a more definitive version of the theory. Theimportant thing is, however, that despite its limitations, the model does predict bubble dynamics that areconsistent with the experimental observations.

4.2. Survey of proposed models for SBSLThese theories can be divided into two groups. Firstly the older, more qualitative ideas that wereadvanced on the basis of imprecise experimental data before the work on stabilized SL. These werefollowed by theories more microscopic in nature, although in some cases these are also highly specula-tive. Good summaries of the older theories have been given by Gaitan and co-workers [2, 3] and Waltonand Reynolds [14]. The pre-SBSL theories can be summarized as follows.

4.2.1. Early theoriesFollowing the discovery of SL by Frenzel and Schultes [1], Zimakov [40] proposed the first explanationfor the phenomenon, namely, that it was caused by an electrical discharge between the vapour cavitiesand the glass wall of the container. The first formal theory was put forward by Chambers [41]. At thetime liquids were thought to have a quasi-crystalline structure similar to solids, and in this tribolumines-cent model it was proposed that SL was similar to the emission of light by many crystals when they arecrushed. Levsin and Rzevkin [42] suggested that SL was due to an electric discharge associated withliquid rupture. This idea was extended by Harvey in 1939 [43] with the balloelectric theory, which wasbased on the collection of electric charge at the liquid–vapour interface, leading to an electrical dischargeupon compression of the bubble. An alternative electrical model, the electrical microdischarge theory,was presented by Frenkel in 1940 [44]. The model involved statistical fluctuations of charge on thesurface of nonspherical cavities, leading to electrical discharge, this time during the expansion phase ofthe bubble.

Other models followed in quick succession. In the mechanochemical theory in 1939, Weyl andMarboe [45] proposed that molecules were fractured during expansion of the bubble, their radiativerecombination giving rise to SL. Griffing [46] ascribed the effect to chemiluminescence; the hightemperatures caused by the cavity collapse were supposed to give rise to oxidising agents such as H2O2,which would then dissolve in the surrounding liquid, causing chemiluminescent reactions. In 1950Neppiras and Noltink [47] advanced the hot-spot theory in which the adiabatically compressed gas gaverise to black-body radiation. Jarman [48] proposed a variant of the hot-spot model by postulating thatshock waves were formed inside the bubble and that these lead to SL. Hickling [49] incorporated the

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thermal conductivity of the gas in the hot-spot model and was able to explain SL results for several gases.Finally, further variants of the electrical discharge model were proposed by Degrois and Balso [50] andby Margulis [51] but these efforts proved unfruitful.

Most of these early theories were forcefully qualitative and speculative in nature, mainly due to thelack of systematic experimental results obtained in controlled conditions. Most of them were rejectedover time as more experiments were carried out; in this context the resurrection of Jarman’s shock wavemodel is somewhat ironic, as it was rejected by the opinion of the day more than 30 years before its nowincreasing acceptance. Nevertheless several of these models directly sowed the seeds for further experi-mental and theoretical work, particularly the hot-spot and chemiluminescent models. The hot-spottheory pointed to the need to attempt a fit to a black-body spectrum and generated serious interest in themeasurement of experimental spectra. Gunther et al. [52] showed that the spectra for water–xenonmixtures could be fitted to a black-body spectrum at 600 K for the spectral range 300–700 nm. Further-more, in a related study Gunther et al. [53] showed that SL was emitted as sharp flashes lasting about1/50 of a period and with the same frequency as the acoustic wave, setting the stage for later studies ofSBSL.

4.2.2. Temperature determination from spectral measurements

The early studies on SL spectra and ideas on chemiluminescence lead to a series of systematic studiesby Suslick and co-workers during the 1980s, which culminated in the first quantitative and systematicstudies of SL and an experimental determination of the temperature at the centre of the bubble for MBSLsystems. In 1989 Flint and Suslick [54] carried out a series of methodical experiments on MBSL for thevarious molecular species, demonstrating that the observed spectra are consistent with the excitation ofvarious vapour species inside the bubble. Coupled with parallel work described below, this work showedthat MBSL is fundamentally a thermal chemiluminescent effect and is not due to electrical discharge orother competing models described earlier.

A quantitative determination of the MBSL emission temperature was made for the first time by Flintand Suslick [32] in 1991. The ro-vibronic spectra of excited states of C2 from silicone oil were identifiedand compared with synthetic spectra generated by the Speir method, which is the standard approach foradding a set of overlapping, dense, spectral lines, each of the form:

I , ν4 AS exp

−hc

k

G

Tv

+

F

Tr

(7)

where ν = energy of the transition in cm–1, A = the Franck–Condon factor for the vibrational transition,S = line strength, G = energy of the vibrational state, F = energy of the rotational state above thevibrational state, Tv = vibrational temperature, Tr = rotational temperature.

The systematic spectra were fitted to experiment using three adjustable parameters; Tv, Tr, and thespectrometer aperture. For thermal equilibrium (Tv = Tr) this procedure gave a best fit forTSL = 5075 ± 156 K. The identification of well-known spectral lines and the ability to fit them usingstandard theoretical procedures gives strong credibility to these experimentally deduced temperatures.

4.2.3. Microscopic theories for SBSL

With growing speculation about increasingly higher temperatures at the centre of collapsing bubbles,there has been a steady increase in the number of models to explain such high temperatures and (or) theobserved spectra. At this time, all of these models suffer from the same drawback; each model can (andhas been!) “fitted” to experimental data to deduce a temperature at the centre. However, there have beenno independent measurements of the temperature, and one clearly cannot verify both the model and thetemperature in the same operation. The principal microscopic models will now be considered in histori-cal order. For convenience, the adaptation of the MBSL work of Suslick et al. will be left to the end, as

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a comprehensive account of the latest work involving an explicit synthesis of MBSL and SBSL has beenpublished very recently by members of that group.

4.2.3.1. Black-body spectrum: While a specific model was not proposed, the original measurementsof the spectrum of SBSL were fitted to a black-body spectrum by the authors of ref. 6. They found thatthe spectrum was continuous over the range 190–700 nm, with a UV cutoff due to the sample chamber.The data could be fitted to source temperatures in the range 25 000–50 000 K.

While a black-body model could be seen as the natural successor to the hot-spot model of Neltingkand Neppiras, no specific microscopic model has been advanced for it. As pointed out by several authorsthere would be difficulties with such a model. Some of these are developed by Eberlein [55], who pointsout that the energy emitted below the 180 nm wavelength for such temperatures should lead to observ-able emissions in the liquid near the bubble. Certainly none have been reported, but by the same tokenit is not clear as to what efforts have been made experimentally in this direction. Furthermore, Eberleinclaims that equilibrium times of the order of nanoseconds are needed for the atomic transitions involvedin black-body radiation, and this is much longer than the observed picosecond time scale. In the absenceof an explicit physical model, we do not consider this mechanism any further at this stage.

4.2.3.2. Bremstrahlung model: The shock wave model of Wu and Roberts [27] integrates a mechani-cal and a thermodynamic model to describe the collapsing bubble. With the assumptions used, veryhigh tempertures at the centre are predicted. They then propose that the air in the bubble is fully ionizedby shock compression, which gives rise to Bremstrahlung. The mechanism is based firmly on standardBremstrahlung theory. They calculate the degree of ionization q, assuming that air has atomic mass14.4 and is singly ionized with ionization potential 14.5 eV, that thermodynamic equilibrium is attainedand that the Saha formula applies. They then use the standard beam power emission formula to obtain;

dPBr = 1.57 ×10−40 q2N2 AT−1/2λ−2 exp

−A

λT

dλ W m−3 (8)

where A = 1.44 × 10−2 mK and N , 4.16 × 1025ρm−3 is the number density of atoms.The Bremstrahlung model does indeed predict several of the observed properties. It predicts a

continuous spectrum with the right order of magnitude intensity (of the order of 10–12 J per flash) andthe correct time scale, which comes naturally out of the shock-wave model. The latter also appears toprovide the correct functional dependence for pa, f , and atomic mass.

The Bremstrahlung model has several shortcomings. As presented, it is a simplistic adaptation of aknown model, and no attempt is made to examine the microscopic processes associated with the par-ticular environment inside the bubble. Again, Eberlein criticizes the model in that the intensity predictedbelow 180 nm would be so high as to not have observable macroscopic (presumably radiative) conse-quences in the liquid surrounding the bubble. The temperatures required are an order of magnitudehigher than those that have been measured inside bubbles, and there simply does not exist any concreteexperimental evidence for the existence of such high temperatures. Finally, the theory is incomplete inthat the VDW equation of state is certainly not applicable at such high temperatures and pressures, andradiative, thermal, and mass transport at the interface have not been included.

The Bremstrahlung model cannot be ruled out or confirmed at this stage; the precise theoreticalpredictions must be quantified and more relevant experimental evidence must be available before adefinite conclusion can be reached.

4.2.3.3. Collision-induced emission (CIE): This model, by Frommheld and Atchley [56] is an adap-tation of well-established work on shock-wave effects in rare gases at high temperatures. The basic ideais simple; pairs of interacting molecules with spherical symmetry (e.g., N2–N2, N2–Ar, etc) produceinteraction-induced dipoles during their short collision times. At high temperatures, re-emission of

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energy from vibrational bands occurs; “the resulting spectra are the Fourier Transforms of the dipolefunction” [56], which is highly diffuse due to the short collision times. The total power emitted per unitvolume is

I (ω ; T) =4ω4

3c3 NL2 ρ1 ρ2Vge (ω ;T) (9)

Vge (ω ;T) = ∑s, s′

Ps (T) VGe ωs, s′ − ω ; T, s, s′

(10)

Synthetic spectra have been calculated for the temperature range 6 000–10 000 K. Good order ofmagnitude and wavelength dependence is obtained for temperatures of the order of 13 000 K. Thedynamics of the model are based on the results of Wu and Roberts [27]; they assume 1 ps emission timesand an acoustic period of about 100 s, which gives approximately 107 photons per burst, in reasonableagreement with experiment.

This model has a strong basis from two standpoints, namely, that it does not require temperaturesinordinately higher than those observed in MBSL experiments and that it is based on a well-definedmicroscopic mechanism. The model has been criticised [55] in that “it contains too many indeterminatepoints and adjustable parameters to permit a judgement on its tenability”. Objectively, the model ispromising, but we will need systematic predictions for experimental verification before being able topronounce on its validity.

4.2.3.4. Quantum radiation theory: This approach is based on the Casimir effect [57]. In its simplestformulation, this involves the attraction of two conducting or insulating plates in vacuo due to fluctu-ating dipoles; the spatial correlation leads to a net attraction.

Schwinger [57–63] first suggested that SBSL was related to the dynamic Casimir effect [64], whichencompasses the idea that a dielectric medium that is accelerated emits light. For SBSL, this correspondsto the collapse of dielectric material into a vacuum. Eberlein [55] has generalized this work, basing thetheory on the Unruh effect [65], which predicts radiation by noninertially moving mirrors. The basicidea is that a stationary or inertial mirror perpetually excites virtual two-photon states, corresponding toa fluctuating radiation pressure. For inertial mirrors, the radiation pressure on the left- and right-handedsides of the mirror balance, therefore there is no net radiation pressure on the mirror. For noninertialmirrors the radiation introduced by local fluctuating dipoles is different on the left and right-hand sides,so the mirror (here the surface of the bubble) experiences a net friction force. If the bubble experiencesconsiderable acceleration (its motion being highly nonlinear, the bubble can collapse and re-expand) theemission of the photons increases. The force of friction is the result of momentum loss by radiation dueto virtual photon pairs transforming into real photon pairs. Eberlein proposes this mechanism as thephysical basis for SBSL.

For this mechanism to qualitatively explain SBSL there must be a polarizability difference betweenthe mirrors and the intervening medium. There must also be extremely high accelerations and decelera-tions during a short time interval. In principle, both of these are provided by the air bubble (refractiveindex 1.0) in water (refractive index x = 1.3); Eberlein claims that this refractive index discontinuity issufficient if the bubble wall is moving sufficiently fast and presents a formal quantum mechanical

calculation of the 3(ω) and the total radiated energy 0. The result for 0 is

0 = 1.163(n2 − 1)

2

512n2

h

c4 γ5R0

2 − Rmin2

2

(11)

Eberlein models the time-dependent bubble radius R(t) empirically as

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R2(t) = R02 −

R0

2 − Rmin2

1

(t / γ)2 + 1

where γ is an adjustable parameter that describes the time scale of the collapse. A shorter γ means a fasterturnaround of the velocity at minimum radius and a more violent collapse. Using R0 = 10 µm, Rmin = 0.5µm, with n = 1.3, Eberlein obtains 0 = 2.2 × 10−16 J for γ = 1 fs. Finally,

3 (ω) = 1.16(n2 − 1)

2

64n2h

c4 γR0

2 − Rmin2

2

ω3 e−2γω (12)

Eberlein claims a number of successes in comparing this model with experiment, including order ofmagnitude agreement for the magnitude of 0, a black-body-like spectrum for the visible region, strongdependence of the SBSL spectra on the gas that saturates the water, and the absence of radiation in theUV. Regarding the latter, the real part of the refractive index is 1 below 180 nm, so that no radiation ispredicted in this spectral region.

At first encounter, such an abstract mechanism based on the quantum theory of radiation wouldappear unlikely as the explanation for such a simple macroscopic laboratory experiment. However, themodel has a well-defined physical basis that is amenable to quantitative analysis, and so can be evaluatedobjectively. As presented, the model has a major problem in that it would require time scales of the orderof 10–15 s near the bubble minimum.2 Eberlein uses a trial function with a free parameter to justify this,but this procedure is not convincing in view of the much longer time scales (of the order of 1 to 100 ps)obtained from the quantitative first-principles treatment of bubble dynamics developed by Wu andRoberts [27]. It would be useful to use a more realistic equation of state in the shock region of the Wuand Roberts model and use these results to calculate an R(t) to incorporate directly in Eberlein’s theoryto obtain a more objective assessment of the latter.

4.2.3.5. Confined electron model: This recent work by Bernstein and Zakin [66] is based on knowneffects of excess electrons in fluids [67–72]. It is proposed that in SBSL excess electrons are producedby ionization of atomic and molecular species due to the high temperatures attained inside the collapsingbubble. It is claimed that these electrons are confined in microscopic voids formed during the finalstages of bubble collapse. The problem is then treated as a standard quantum mechanical calculation ofa particle in a box; it is found that the energy level spacings correspond to transitions in the visible andultraviolet spectral regions. The levels are blurred due to variation in void sizes, which gives rise to acontinuous emission spectrum.

The detailed calculations involved a hard-sphere-based model for the fluid structure and the thermo-dynamics. An electron in a parellelipiped was used to calculate synthetic emission spectra. Three keyvariables were identified: mean void size, void size variance, and temperature. Monte Carlo calculationswere made for a hard sphere fluid to obtain the relation between mean void size and void-size variancewith reduced density. For the thermodynamics, an adiabatic description of the collapse process wasadopted, using the Carnahan–Starling equation of state for hard-sphere fluids.

The results obtained for rare gas bubbles (Xe, Ar, and He) were in good agreement with SBSLspectra. The calculated emitted power is a very strong function of the reduced density, which gives riseto a very short emission time scale. As radiation lifetimes are of the order of 3 ns, statistically there willbe a significant number of them in the 10–100 ps range. The bubble dynamics of Wu and Roberts arealso incorporated in the theory. Thus the theory successfully accommodates time scales of the order of10–100 ps. Emission temperatures of the order of 20 000–60 000 K are predicted. Small voids will emitat higher frequencies, and it is found that the very smallest voids are associated with emission below 175nm. The model predicts that the ionization potential will be a driving factor for SBSL. The basic model

2 W.G. Unruh, Private Communication

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predicts a high emitted power, 300 mW, about 100 times greater than that observed experimentally, sothat only a very small degree of ionization, of the order of 0.1% is needed for agreement with experiment.

4.2.3.6. Synthesis of molecular and continuum SL spectra: Bernstein et al. [73] have recently pub-lished a fascinating synthesis of the molecular and electronic emission models that encompasses bothMBSL and SBSL. The main idea is that molecular emitters dominate at lower bubble temperatures,while electronic emission as described in the previous section is dominant at higher temperatures. Thebasic description of the fluid is the same as that given in ref. 66, namely, that of an adiabatic collapsewith a Carnahan–Starling equation of state for dense fluids. For simplicity, the bubble dynamics aredescribed by a simple model dR/dt = constant; it is claimed that this does not play a critical role, and itis clear that more complete bubble dynamics could easily be incorporated into the model. The chemistryof the problem is provided by identifying the various chemical pathways that control molecular emis-sions; these include unimolecular decomposition and collisional dissociation. The chemiluminescentreactions fall into the categories of bimolecular reactions or atom–atom recombinations. Additionalreactions are provided by radioactive decay, collisional quenching, and collisional dissociation. Specificmechanisms have been assigned to C2, CN, and OH molecular emissions that have been studiedexperimentally.

The overall picture of this description is to divide the bubble collapse into two stages. The first stagecorresponds to the early part of the collapse, where the temperatures are relatively low, in the range3 000–8 000 K. This is characterized by the dissociation of single bonds, such as C—H, C—C, andO—H. The dissociation occurs over a significant temperature and (or) time interval, as it is controlledby chemical kinetics. Since the process depends on the temporal overlap of two transient species, theactual temperature range of emission will be fairly narrow for different emitters. Chemiluminescentreactions involving multiple bonded species, such as triply bonded nitrogen, will occur at significantlyhigher temperatures, of the order of 25 000 K, due to the higher dissociation energies involved. Insummary, this region is characterized by two main features, which are that different molecular emitterswill be characterized by different temperatures and that molecular emission of various types is expectedto fall in the range 3 000–25 000 K. For example, experimentally, CN emission occurs over a broadrange of cavitation collapse temperatures from 5 000–15 000 K, while the accompanying C2 emissionis characterized by a single temperature of the order of 5100 K. Atomic emission is relatively rare andrestricted to special cases; it is not included in this discussion.

The second and final stage of collapse is characterized by temperatures sufficiently high that bothhost and dissociatively formed rare-gas atoms are ionized. The electrons so produced are confined tomicroscopic voids as discussed in the previous section, giving rise to a continuum emission due to thevoid size distribution. There is no upper temperature limit for this mechanism in practical terms.

The above ideas are used by the authors to give a unified description of MBSL and SBSL. Experi-mentally, SBSL is characterized by continuum emission, while MBSL exhibits both continuum andmolecular band emissions. The difference can be explained by the fact the electrons are very efficientemitters and molecules are not. In SBSL the bubble is repetitively driven into the electronic emissionregime once per cycle, so it is the electronic continuum emission that dominates. In contrast, in MBSL,due to the bubble-size distribution, only a small fraction of the bubbles will be driven into extremecollapse. On average, the majority of bubbles in MBSL will thus only attain the single or multiple bondregimes, which explains the dominance of this type of emission for this case.

5. Conclusion

From the preceding, it seems reasonably clear that there are at least two separate problems to beaddressed in SBSL. The first is the issue of bubble dynamics, which will put a constraint on the timescales involved. The approach adopted by Wu and Roberts [27] would seem to be the most promising;there are, of course, many improvements that must be made for the model to become more complete and

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more rigorous, such as the equation of state and inclusion of mass, thermal, and radiative transfer, butin principle it would seem that these could easily be incorporated in the model. The second major pointis the microscopic origin of the light emission, and despite impressive progress on both theoretical andexperimental fronts, this question has not yet been resolved conclusively. The link between the twoaspects mentioned above is the thorny question as to the temperature, if a unique temperature can indeedbe defined at the bubble centre at the moment of collapse.

There has been much speculation on the questions of bubble temperature and exotic applications, butthere has also been a great deal of very good theoretical work based on well-defined and solidly basedmodels. Despite the very excellent experimental work that has been carried out by a small number offirst class research groups, more experimental results will be needed before the theories can be tested ina critical manner. It would seem essential to have alternative methods of temperature measurement atthe bubble centre at the moment of collapse. Despite impressive improvement in the size of availableparameter space for the phenomenon [35], the dependence on material properties should be elaborated;for example, for the gas, dependence on gaseous species, molecular weight, density, thermal conductiv-ity, bubble size etc. For the liquid, the influence of parameters such as viscosity, density, sound velocityetc. should be clarified. Finally, the dependence on experimental measurement parameters such asfrequency, temperature, and pressure has not yet been fully established.

A question that inevitably arises in SBSL is the meaning of the concept of temperature in a very smallvolume during an extremely short time period. As has already been pointed out, the time duration of theflash is shorter than that needed to establish thermodynamic equilibrium, so that the thermodynamic orblack-body temperatures would not seem appropriate to describe the system. Rather, as in the work ofSuslick et al, it would seem most appropriate to assign different temperatures to different molecular,atomic, or electronic emitters. Thus the temperatures in the bubble would depend on the property beingmeasured, and there would be no overall uniform temperature per se.

The story of SBSL is far from over; in many respects it is just beginning. In some ways, thedevelopment of the subject is reminiscent of the early work on the nature of optical sources in general.Historical studies were limited to the use of incoherent sources (the analogue of MBSL), which wascertainly useful in elucidating some of the more prominent characteristics of the nature of light and lightemission. With progressively refined methods, leading finally to the development of coherent sources(the analogue of SBSL), it was possible to make direct contact between theory and experiment viamicroscopic mechanisms. Following years of speculation of a very general nature on SL, SBSL seemsto be entering into this phase right now, and hopefully, microscopic models will soon be directly testedby increasingly refined experiments.

Excitement around SBSL will undoubtedly continue until there are more established views on thebasic mechanisms, bubble temperatures, and the like, at which point possible applications will becomemore clear. Already, SBSL is being used in the Sudbury Neutrino Observatory neutrino project as ameans of calibrating very fast photo-multiplier tubes, exploiting the sharp pulse width and synchronicityof SBSL. But even when the excitement dies down, as it must inevitably, two dominant features of SBSLwill always remain. The first is that this is predominately an interdisciplinary area: chemists, physicists,perhaps soon biologists, experimentalists, and theoreticians are all needed in a cohesive team effort tocrack a hard problem like this one, and develop subsequent applications. Secondly, it is very comfortingin this day and age of ever more complex and expensive equipment needed to probe into increasinglyspecialized problems, that there still exist such marvelous and intriguing experiments that rival the verybest in sensitivity and sophistication and which provide as theoretically challenging problems as onewould care to look for; and all of this for a $200 experiment that could be set up in any undergraduatephysics laboratory!

Acknowledgements

Work at Concordia on SBSL was initiated by the award of a Seagram grant for innovation, which is

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gratefully acknowledged. The work was also supported by the Natural Sciences and Engineering Re-search Council of Canada. Thanks are given to Guy Quirion, Manas Dan, and Zuoqing Wang for theirencouragement and support and particularly to Manas Dan, Haizhong Lin, and Xing Li for help with thefigures. Helpful discussions with Professors Ken Suslick and Bill Unruh are gratefully acknowledged.

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(1995).34. M.P. Brenner, D. Lohse, and T.F. DuPont. Phys. Rev. Lett. 75, 954 (1995).35. K. Weninger, R. Hiller, B.P. Barber, D. Lacoste, and S.J. Putterman. J. Phys. Chem. 99(39), 14195 (1995).36. R. Hickling. Phys. Rev. Lett. 73, 2853 (1994).37. H.P. Greenspan and A. Nadim. Phys. Fluids A, 5, 1065 (1993).38. L. Kondic, J.I. Gersten, and C. Yuan. Phys. Rev. E: Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top.

52(5), 4976 (1995).39. H.Y. Kwak and H. Yang. J. Phys. Soc. Jpn. 64(6), 1980 (1995)40. P. Zimakov. C.R. Acad. Sci. USSR. 3, 452 (1934).41. L.A. Chambers. Phys. Rev. 49, 881 (1936).42. V.L. Levsin and S.N. Rzevkin. C.R. (Dokl) Acad. Sci. URSS. 16, 399 (1937).43. E.N. Harvey. J. Am. Chem. Soc. 61, 2392 (1939).44. J. Frenkel. Acta. Phys. Chim. URSS. 12, 317 (1940). In Russian.45. W.A. Weyl and E.C. Marboe. Research, 2, 19 (1949).46. V. Griffing. J. Chem. Phys. 18, 997 (1950).47. E.A. Neppiras and B.E. Noltingk. Proc. Phys. Soc. B, 64, 1032 (1951).48. P.D. Jarman. J. Acoust. Soc. Am. 32, 1459 (1960).

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