Topical review Sonoluminescence: how bubbles glow

Transcription

1 journal of modern optics, 2001, vol. 48, no. 2, Topical review Sonoluminescence: how bubbles glow DOMINIK HAMMER and LOTHAR FROMMHOLD Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA (Received 10 March 2000; revision received 31 March 2000) Abstract. We review recent attempts to elucidate the phenomenon of sonoluminescence in terms of fundamental principles. We focus mainly on the processes which generate the light, but other relevant facts, such as the bubble dynamics, must also be considered for the understanding of the physics involved. Our emphasis is on single bubble sonoluminescence which in recent years has received much attention, but we also look at some of the excellent work on multiple bubble sonoluminescence and its spectral characteristics for clues. The weakly ionized gas models were recently studied most thoroughly and are remarkably successful when combined with a hydrodynamic bubble model, in terms of reproducing observed spectral shapes, intensities, optical pulse widths and the dependencies of these observables on the experimental parameters. Other radiation models, such as proton tunnelling radiation and the con ned electron model, were not combined with hydrodynamic models and/or have freely adjustable parameters so that their relevance to sonoluminescence studies is at present less critically tested. 1. Introduction Large amplitude sound waves in liquids cause cavitation: a cloud of small voids (vapour bubbles) is formed during the tensile phase of sound [1 5]. Cavitation is observed in most liquids, including pure water, salt solutions, the oceans, liquid metals [6], organic solvents and cryogenic liquids [7 9]. Besides strong sound elds, high ow velocities [10] and laser irradiation of liquids [11] cause cavitation. Bubbles generated by sound oscillate with the sound frequency and may survive many oscillations. At some instant during the sonic compression phase the bubbles suddenly collapse to a fraction of their maximum size. The gas inside the bubble is thus compressed and adiabatically heated. Under certain conditions an imploding shock wave may be driven by the collapsing bubble walls, or liquid jets may be generated by a non-spherical collapse, which could further increase the temperature and create a rather destructive environment [4, 10, 12, 13]. The evects of cavitation are often quite striking: erosion of metal surfaces (e.g. severe damage to ship propellers), lithotripsy, ultrasonic cleaning, sonochemistry [14, 15], and the emission of light: sonoluminescence [12, 13, 16], gure 1. Sound energy is converted into light an amazing and yet quite common process. The facts that photons of energies up to several ev are generated and molecular syntheses are possible in such environments, maybe even of molecules that could have been instrumental in creating life on earth, have fascinated Journal of Modern Optics ISSN print/issn online # 2001 Taylor & Francis Ltd

2 240 D. Hammer and L. Frommhold scientists of various backgrounds and laymen alike. Without doubt, the phenomenon of sonoluminescence is intriguing. How do soundwaves, or the voids generated by sound in the liquid, produce such energetic photons? Could existing speculations possibly be true that photons of much higher energy, maybe even in the X-ray region, are generated as well? (Such photons would be diycult to detect because of the absorption characteristics of water [18, 19].) And if so, do these energetic photons prove that temperatures suitable for nuclear fusion exist [20] or could soon be reached? Reports have appeared in journals and newspapers which claim that tabletop fusion based on sonoluminescence is simple and quite practical, using inexpensive equipment [21]. Is the sonoluminescent gas bubble at the moment of greatest compression indeed a micro-model of a star a star in a jar [22] as this has occasionally been suggested? Early reports on sonoluminescence appeared in the 1930s, in connection with ultrasound studies in liquids and medical research [23 26]. Since then an estimated thousand articles have appeared in scienti c journals, books, etc., at a rate of about 90 per year in 1999, that report measurements of sonoluminescent phenomena and theoretical analyses. Interdisciplinary research evorts in physical acoustics, hydrodynamics, spectroscopy, chemistry, medicine, physical electronics, plasma physics, engineering and quantum electrodynamics attempt to shed light on the sonoluminescent phenomena which, however, only now we are beginning to understand. Moreover, movies and newspaper articles have appeared that popularized the perceived mysteries of sonoluminescence. Sonoluminescence is observed as a transient phenomenon, for example, in converging uid ows and in compression bursts in uids. As a quasi-continuous glow it may be seen either as single bubble or multiple bubble sonoluminescence. Until roughly ten years ago, sonoluminescence was synonymous with the latter: a liquid is insonated, for example by mounting a sonic transducer in a water tank, as seen in gure 1. At suyciently high sound intensities a beam of light is observed which consists of a cloud of small, luminescing bubbles. In more recent years, Gaitan et al. discovered single bubble sonoluminescence, an interesting modi cation that makes use of a standing soundwave with one pressure antinode in the container. A single, pulsating bubble may be levitated stably for hours [27]. The remarkably simple apparatus is sketched in gure 2; cylindrical or spherical resonators ( asks) lled with water are typically used. Lowest-order acoustical modes (resonances) are set up, typically at frequencies in the 10 to 50 khz range, depending on the dimensions of the container. Cavitation is observed at the pressure antinode at the centre of the resonator. Bursts of light are observed synchronously with the sound frequency, if the preparation of the water and other experimental conditions (e.g. acoustic pressures, etc.) are favourable. To the unaided eye the bubble looks very stable and a seemingly steady glow may be seen without dimming the room lights much, gure 3, but rapid bubble oscillations actually take place and subnanosecond light bursts occur with clocklike precision. The single bubble arrangement, gure 2, overs signi cant advantages for the study of cavitation and sonoluminescence. Historically, theoretical attempts have always considered single, isolated bubbles, even in the early years when only multiple bubble experiments were known: comparisons of single bubble theory with single bubble experiments are doubtlessly more meaningful. Moreover, the single bubble arrangement permits studies under highly controlled conditions. It

3 241 Sonoluminescence Figure 1. Multiple bubble sonoluminescence in a water tank. To the left the ultrasonic transducer is seen which produces a beam of blue light [17]. Reprinted with permission from Crum, L. A., and Roy, R. A., 1994, Science, 266, 233; # 1994 American Association for the Advancement of Science. sine wave generator and power amplifier T B M Figure 2. Schematic single bubble sonoluminescence apparatus. A standing sound wave is set up by the transducer (T) in a boiling ask lled with water. A microphone (M) and feedback circuit are used to maintain resonance conditions. A bubble (B) is levitated and glows. Frequently a gas handling system is added and the water temperature is controlled. Figure 3. A microscopic view of a single sonoluminescent bubble. The circular, darker outer region near the centre of the photograph re ects the maximal size of the bubble. The light spot at the centre of the dark region indicates the much smaller emission region. Exposure time is 1 s, i.e. here about cycles are superimposed. The dark lines are ducial lines; the outermost vertical lines are 105 mm apart. Reproduced with permission from [28].

4 242 D. Hammer and L. Frommhold also permits the recording of novel experimental data, because of the remarkable stability of the single bubble dynamics. For example, direct measurements of the bubble size as a function of time are thus possible; glass bres may be positioned next to the bubble so that spectra may be recorded with good light gathering power, minimally av ected by the absorption characteristics of the liquid. Even a microscope objective may be moved up close to the bubble, permitting a direct, magni ed view of the bubble, gure 3. The main part of this review, section 2, is devoted to single bubble sonoluminescence. Just as has been the case with most of the research conducted in recent years, we will almost exclusively consider aqueous solutions of rare gases or simple molecular gases. First, a brief overview over the experimental results concerning bubble dynamics is given; for more detailed information on this subject we refer the reader to existing excellent reviews [13, 20, 29]. Following the summary of experimental facts concerning bubble dynamics is a brief, semi-quantitative theoretical analysis of such observations. The second part of section 2 focuses on light emission. Again, a summary of the experimental facts is followed by a general review of the mechanisms that may be responsible for the emission. In section 3 we look at multiple bubble sonoluminescence in aqueous solutions and section 4 concludes the review with a summary and outlook. 2. Single bubble sonoluminescence 2.1. Bubble dynamics: the experimental facts Single bubble cavitation may be observed in an acoustic resonator, e.g. a ask as in gure 2, lled with a liquid. If the sound pressure is too low, a bubble may be injected but it will drift to the water surface and disappear (buoyancy region). With slightly increased sound amplitudes, at the lowest trapping pressures, a seeded bubble oscillates linearly with the sound pressure, but it dissolves slowly in time. If the sound amplitude is further increased, the bubble begins to jump back and forth irregularly ( dancing region ), up to sound pressures of typically º1.2 atm, the threshold of stable single bubble sonoluminescence, gure 4. Depending on the liquid, its temperature, and on other parameters of the experiment, a sonoluminescent region of great stability exists, up to acoustic pressure amplitudes of roughly 1.6 atm. In this region bubble oscillations are highly nonlinear but synchronous with the sound eld. Individual bubbles may be trapped for minutes or even hours. Above the extinction threshold only transient sonoluminescent bubbles are observed [27, 30]. All possible combinations of experimental parameters under which stable sonoluminescence occurs de ne the parameter space of single bubble sonoluminescence; see for example gure 6 in section Besides depending on the driving pressure amplitude Pa, single bubble sonoluminescence experiments depend on the ambient pressure P0, driving frequency a, liquid temperature T1, and the kind of gas dissolved in the water, along with its concentration c1 far from the bubble. Usually the ambient pressure is about 1 atm; the driving frequency is a resonance frequency of the container lled with the liquid ( khz); water temperatures are often controlled from near 08C to about 358C; and gas concentrations should amount to a few percent of the saturation concentration c0 of the given gas in the liquid. Variations of the sound eld, e.g. the controlled additions of harmonics or pulsed insonation, do also

5 243 Sonoluminescence spherical oscillations dancing region stable transient single bubble sonoluminescence sonoluminescence «1.2 «1.6 acoustic pressure [atm] ± relative time 30 (m s) applied pressure (atm) radius (m m) Figure 4. With increasing driving pressure the bubble goes through various unstable and one stable regimes. Stable sonoluminescence occurs with driving pressures between roughly 1.2 and 1.6 atm, depending on the exact experimental conditions [27, 30]. ± 2 Figure 5. Driving pressure (thin sinusoidal line) and bubble radius (dots) measured in a typical single bubble sonoluminescence experiment. The shape of the bubble radius R t was measured by Mie scattering and the theoretical data (solid line) were obtained by solving the Rayleigh Plesset equation (1) repeatedly so that a t of the data is obtained [30]. Reproduced with permission from Matula, T. J., 1999, Phil. Trans. R. Soc. (London), Ser. A, 357, 225. av ect the sonoluminescence [31 33], as do additions of surfactants to the liquid [34]. Observable quantities are the bubble radius R as a function of time; the emitted light pulses their intensity, timing, duration and spectral composition; and the chemical modi cations introduced. Figure 5 shows a representative measurement (dots) of one cycle of the bubble radius R t as a function of time, along with a theoretical t of that curve. The sinusoidal driving pressure variation is also shown. The bubble expands during the tensile (negative) phase of sound and collapses rapidly at a point in the compression phase (with positive sound pressure). After a few afterbounces the process repeats itself. The burst of light occurs at the point of maximum compression of the rst and most violent collapse. The key experimental ndings have been discussed elsewhere [20, 29, 35] or are communicated in more recent work to be quoted below. We give a brief summary here of the experimental facts concerning the bubble dynamics; observations concerning the emission of light are considered in section 2.3. With air-saturated water, no stable sonoluminescence is observed. It is necessary to degas the water. Also, a small rare gas content of the bubbles is necessary for

6 244 D. Hammer and L. Frommhold stable light emitting bubbles. Yet, if pure rare gases are used their concentrations have to be much below saturation [20, 36]. In an experiment, it is impossible to choose the equilibrium or ambient bubble radius R0 directly. The bubble radius is determined by equilibrium with respect to div usion and chemical reactions. Recorded bubble radius R t versus time curves are similar to the one shown in gure 5; ratios of maximal and ambient bubble radius, Rmax =R0, and of maximal and minimal radius, Rmax =Rmin, of about one and two orders of magnitude, respectively, are observed, depending mainly on the sound amplitude Pa : the larger Pa, the more violent the collapse, i.e. larger compression ratios are obtained. Both, the equilibrium radius R0 and the maximum radius R max increase as the static pressure P0 is decreased [37]. At lower water temperatures the bubble is more stable and thus may be driven with larger pressure amplitudes Pa [20, 38]. Shock wave emission of imploding bubbles has been observed in the liquid [39 41]. We note that the existence of shockwaves in the liquid is not necessarily related to the existence of shocks in the bubble interior. Strong magnetic elds reduce the sonoluminescence intensities (for constant Pa ), but allow the bubble to be driven at larger acoustic pressures Pa [42]. The addition of a second harmonic increases the light output (by up to 300%), if the amplitudes of the two driving pressures and their relative phase are chosen properly [31]. Single bubble sonoluminescence has been observed in other liquids, e.g. in heavy water [43], water glycerin mixtures, and non-aqueous liquids [44, 45]. For single bubble sonoluminescence, it is widely believed that throughout the collapse the bubble is of a spherical shape. Single cavitation luminescence is suppressed if the spherical symmetry is suy ciently perturbed [41]. However, observations of angular correlations of the emitted light were reported that suggest that some non-sphericity may occur [46, 47] Bubble dynamics: theory Modelling the bubble motion The radial motion of a spherical bubble under the in uence of a time varying external pressure eld Ps t is described by Rayleigh Plesset type equations. One variant of this equation, which accounts for the liquid compressibility by including _ l, was derived by Keller and others rst-order terms in the Mach number R=c [27, 48 53], R_ R 2 R_ R d 3 1 RR R ˆ cl cl cl dt 2 3cl «l µ R_ 2¼l P Ps t P0 4²l 1 : R R The dots denote time derivatives; the symbols «l, cl, ²l, ¼l designate density, speed of sound, viscosity and surface tension of water, respectively; P is the internal bubble pressure, it can be taken to be the sum of the gas pressure in the bubble Pgas and the water vapour pressure Pvap; P0 is the ambient pressure and Ps t ˆ Pa sin!a t the driving sound eld of frequency!a ˆ 2º a. Instead _ l is also of equation (1), a form that neglects the rst-order corrections in R=c frequently used [20, 54]. This form is derived from the Navier Stokes

7 245 Sonoluminescence equation, assuming that the pressure (including surface tension) is constant across the bubble surface and that the liquid at the boundary moves with the velocity of the bubble wall [4]. On the other hand, a form of equation (1) which additionally includes the ev ects of mass transport across the surface is derived in [52]. For a discussion of div erent forms of the Rayleigh Plesset equation see, e.g. [50, 53] Modelling the gas content To solve equation (1) the pressure Pgas in the bubble must be known. In the simplest case one assumes that the pressure is uniform throughout the bubble volume and that the gas is heated adiabatically during the fast collapse. In that case only an equation of state has to be speci ed which connects the pressure with the density in the bubble, which in turn is directly related to the radius R if mass conservation in the bubble is assumed. A simple, reasonable description is an adiabatic van der Waals type equation [20], Pgas ˆ P0 Á R30 H 3 R t 3 H 3! ; 2 1=3 with H ˆ R0 b=vm, b is the excluded volume, vm the speci c molar volume under normal conditions ( K, 1013 hpa), and R0 the ambient radius of the bubble, i.e. its radius in the absence of the driving pressure eld, at P 5 P0 and T ˆ T1. is the adiabatic exponent, the ratio of speci c heats at constant pressure and volume, respectively; for rare gases ˆ 5=3. A slightly more complex approach may be employed since for the most part the bubble motion is slow enough to be considered isothermal. Thus, the value of is interpolated between 1 for the isothermal phase and 5/3 for the rapid collapse, depending on the _ instantaneous values of R t, R t and T t. A variable and also heat div usion from the bubble may be accounted for by using [54] Pgas 4º 3 4º 3 1 R H 3 ˆ R v RT m 3 and 2 T 1Š 3R R T À R; T T T1 ; T_ ˆ R; R; 3 R H3 R2 4 where T is the temperature in the bubble, R is the ideal gas constant and À is the _ T see [54]. coey cient of heat conduction. For details of the function R; R; Extended uniform-bubble adiabatic-heating models (or uniform adiabatic models [55]) also account for mass exchange through the interface and chemical reactions of water molecules in the bubble [56 58], while a model that only considers T to be uniform, but employs a non-uniform pressure eld in the bubble is described in [50]. Without the assumption of a uniform bubble interior, computations become much more involved. In that case the full hydrodynamic partial div erential equations describing the bubble interior must be used [55, 59]:

8 246 D. Hammer and 2 «vr ˆ r2 2 «v 2 ˆ 0; «v 2 E P vr2 ˆ where E ˆ «e v2 =2 is the energy density and «, v, e, P designate density, radial velocity, internal energy and pressure of the gas in the bubble, respectively, all as functions of radial position r and time t. In addition, an equation of state and the internal energy of the gas have to be speci ed [55], Pˆ «RT ; 1 b«e ˆ cv T ˆ 1 b«p : 1 «8 Here cv is the speci c heat at constant volume. More complicated models and equations of state have been considered as well, e.g. in [60 63]. In principle all input parameters of the Rayleigh Plesset bubble model are either known (the material constants cl, «l, ²l, ¼l, b) or can be measuerd in experiment (R 0, P0, Pa, a ). In practice, however, it is diy cult to measure the bubble radius and driving pressure amplitude directly. Mie scattering has therefore been used to measure the function R t on a relative scale. The driving pressure Ps t is also known on a relative scale from the voltage applied to the transducers or from hydrophone measurements. The experimental results are then tted to solutions of some form of equation (1) by adjusting ambient radius R0 and acoustical pressure amplitude Pa to determine their absolute values [20], e.g. the values given in the caption of gure 5. Experimental uncertainties of up to 15% of both parameters are common. However, recently Rmax, and even almost complete R t curves, have been measured photographically [30, 41, 64] Bubble collapse The surprisingly violent bubble collapse may be understood in this way: at low driving pressures the bubble radius follows nearly linearly the acoustical pressure as a function of time. However, with increasing pressure this behaviour changes and once the inertial cavitation threshold [30] is passed, typically at Pa º 1 atm, R t curves, as the one shown in gure 5, are obtained. Mathematically the collapse is a clear consequence of the nonlinearity of equation (1): Neglecting the eœects of one obtains compressibility and solving for R, _2 ˆ 3 R P; R 2R 9 where P, on the right-hand side of equation (1), represents the total pressure. During the collapse of the bubble, R and R_ are both negative and once the R_ term shown explicitly in equation (9) has taken over (which is what happens at the 2=5 inertial cavitation threshold), equation (9) implies a scaling law R t / t t which means that the bubble would reach R ˆ 0 in nite time, at t ˆ t. However, this collapse is stopped once the gas pressure in the bubble is large enough (due to the excluded van der Waals volume) to overcome the inertial term. In other words, if the collapse is strong enough initially, the momentum of the collapsing bubble

9 Sonoluminescence 247 can only be stopped by increasing internal pressure. More detailed discussions of the violent collapse may be found in [53, 65] Bubble stability For a bubble to exist and emit bursts of light for extended periods of time it has to be stable in three ways, namely div usively, hydrodynamically and chemically. These stability criteria determine the parameter space for single bubble sonoluminescence [50, 66 68]. DiV usive stability determines the ambient radius R 0. During the expansion of the bubble, gas dissolved in the water streams into the bubble, while during the compression it is pushed back into the water. For diœusion equilibrium, the amount of gas streaming into the bubble has to be equal to the amount of gas leaving the bubble during every cycle. For small enough gas concentrations a stable equilibrium radius R0 will be attained through diœusion, with R0 depending on the gas concentration c1, the acoustic driving frequency a, and the static and acoustic pressures, P0 and Pa. The larger Pa, the larger R0 will be and the brighter the glow. In turn, for a bubble to be stable at higher acoustic pressures Pa, the concentration of the gas in the water c1 has to be smaller. Typical phase diagrams are shown in gure 6 [50, 66]. DiV usion equilibrium alone is not suy cient. If it were, stable bubbles would exist at any acoustic pressure amplitude Pa and sound frequency a ; no limit on the achievable radiance would thus be set. Yet, an upper limit of the bubble size clearly exists, given by the threshold of hydrodynamic or shape stability. Small deviations from the shape of a sphere with radius R t can be described by expanding them in spherical harmonics Ynm and writing instead of R t, R t an t Ynm ; : 10 Figure 6. Sonoluminescence (SL) phase diagrams of argon bubbles: The left panel shows in which part of the c1 =c0 versus Pa =P0 parameter space which type of sonoluminescence can be expected [66]. The right panel shows curves of div usive equilibria (thick lines) for argon bubbles driven at Hz at div erent dissolved gas concentrations. The solid parts of the curves represent stable equilibria, the dashed parts are unstable. Also indicated are theoretical (above thin free-hand drawn line) and experimental limits (above dashed line) toward shape instabilities [54]). Reproduced with permission from Hilgenfeldt, S., Lohse, D., and Brenner, M. P., 1996, Phys. Fluids 8, 2808, and Hilgenfeldt, S., Grossmann, S., and Lohse, D., 1999, Phys. Fluids 11, 1318, respectively.

10 248 D. Hammer and L. Frommhold If one of the modes an grows with time, this mode and thus the bubble become unstable [50, 66, 69]. Theory and experiment place the stability limit for R 0 somewhere below 10 mm, depending on the experimental parameters. The combined div usion and shape stability criteria explain why at low water temperatures more sonoluminescent light is observed: by lowering the temperature, the material constants of water, especially viscosity, gas solubility and vapour pressure, change such that the bubble is stable at higher driving pressures [70], which in turn means more light. The special importance of gas solubilities has been considered in [71]. Chemical stability addresses the question of bubble content: only gases that remain in the bubble for many cycles contribute to stable single bubble sonoluminescence. However, molecular gases such as air, O2, N2, etc., dissociate at bubble temperatures of several 1000 K; the resulting radicals react with each other, and with water vapour, to form compounds such as NO, OH and NH, which dissolve easily in water to form, e.g. H 2 O2, HNO2 and HNO3. This process (`recti cation ) leads to rare gas accumulation in the bubble [67, 68] as has been veri ed experimentally [72, 73]. Recti cation also explains previously not understood observations: (i) Stable sonoluminescence is observed only for small argon but comparatively large air concentrations in water: since only rare gases survive in the bubble, only their partial pressure counts; because ordinary air contains only about 1% argon, sonoluminescence is observed at much larger ratios c1 =c0 for air than for pure argon bubbles. (ii) A small amount of rare gas dissolved in the water is essential for single bubble sonoluminescence: because of recti cation, all molecular gases simply disappear from the bubble over time and the small argon content of air is really responsible for the glow of sonoluminescent air bubbles. The chemical recti cation hypothesis also predicts a stable equilibrium where the gas loss of the bubble due to reactions is balanced by a gas in ux during the expansion [68]. This (non-luminescent) equilibrium has been observed experimentally [72] The light pulse The nal step of a theoretical treatment is concerned with the emission of the light. For example, for the vacuum radiation models (section 2.5.1) light emission is directly computable from the results of the hydrodynamical calculations, speci cally from the R t curves. Also, in the case of the radiation models which assume an optically thick black body emitter, the emission as a function of time is directly related to the temperature, equation (4). However, some models of the emission processes suggest that the sonoluminescent bubble is not optically thick and thus cannot be treated as a black body. Instead, radiative processes well known from ionized gas and plasma studies have to be considered [54, 58, 74]. In order to calculate the emitted intensities as a function of time, the degree of ionization of the gas in the bubble has to be known as a function of temperature and density in the bubble. Under the assumption of thermal equilibrium it is given by the Saha equation [75 77], see section Figure 7 shows the driving pressure, bubble radius and temperature in the bubble as a function of time in a uniform adiabatic model [54]. The time scale of bubble motion is given by the driving frequency (here for a typical sound

11 249 Sonoluminescence 1.5 R(t) Pa T [K] 0 ±.5 20 T(t) time [1 acoustic cycle] Pa [atm] 40 R [m m] 60 ± ± 1.5 Figure 7. Bubble radius (upper panel, solid curve), driving pressure (dotted curve) and temperature in the bubble (lower panel) as a function of time, measured in fractions of one period, 1 ˆ 1/ s. Calculations according to an adiabatic-heating uniform-bubble model [54] for an argon bubble with R0 ˆ 5 mm driven with Pa ˆ 1.3 atm at Hz R(t) 15 T(t) 4 10 I(t) cycle time T [kk] R [m m] Figure 8. Bubble radius (dashed), temperature (dotted) and light emission (solid curve) of the bubble as a function of time, measured again in fractions of one period, 1 ˆ 1/ s. Same calculations and parameters as above, on an expanded time scale, near the moment of the minimal radius. The peak intensity Imax is 0:002 W and the integrated intensity I t dt per pulse is 0.2 pj. frequency of a ˆ Hz, it is 50 ms). On this scale the temperature peaks very sharply. Figure 8 shows radius and temperature of the bubble near the minimum radius on a 2500-fold expanded time scale. One sees that the temperature pulse has a width of about units, which corresponds to 1 ns. But even in this plot, the light pulse is very narrow. The reason is that the light emission, which is here assumed to be due to bremsstrahlung processes, is proportional to or 2, see below, and is an exponential in 1=T, equation (13). The calculated pulse width in this case amounts to 160 ps, a value consistent with observations Upscaling The observation that the addition of a second harmonic to the drive signal can increase the light output of sonoluminescence [31] has recently been interpreted in the context of the rare gas recti cation model [78]. Excellent quantitative agreement between theory and experiment was obtained, thus providing further evidence for the assumption of rare gas recti cation. Also based on the rare gas recti cation model, as another method for upscaling sonoluminescence and increasing the light emission, it was suggested to reduce the frequency of the

12 250 D. Hammer and L. Frommhold driving pressure [79]. This would give the bubble more time to expand, thus achieving a larger expansion ratio and a more violent collapse. No experimental investigations have been reported so far Shockwaves Early calculations carried out assuming a non-uniform bubble interior (equations (5) (7)) [55, 59] are consistent with the existence of strong shocks in the bubble. They are launched by the bubble walls imploding at supersonic speeds; such speeds have actually been measured [80]. Other calculations [61, 81 84], using div erent equations of motion and a div erent equation of state, obtain similar results. The main ev ect of shocks is to provide additional heating to the region near the bubble centre, with temperatures reaching 105 to 107 K for very brief time intervals. Actually, in inviscid theories, at the exact bubble centre the temperature should be in nite nite temperatures are the result of nite grid sizes in the numerical integration of, e.g. equations (5) (7) [83]. At the highest temperatures a dense plasma core would exist which would cause extreme electrostatic phenomena to occur that would av ect bubble dynamics and light output [85]. However, the main reason why shocks were thought to be essential was that the duration of the light pulse was previously believed to be signi cantly less than 50 ps [20]; it seems that no other mechanism allows for such short light pulses. Yet under most conditions of interest here, the light pulses are now known to be almost an order of magnitude longer than previously believed [86 89]. Consequently, a need for shocks hardly exists anymore [54, 58, 60, 62, 63]. Moreover, the observation of sonoluminescence from aspherical bubbles [41, 90 92] seems to weaken the imploding shock-wave models, which require a spherically symmetric collapse. Furthermore, newer calculations show that even the non-uniform models do not necessarily develop shocks [60, 63, 83]. It is suggested that rather the bubble launches a spherically convergent compression wave whose re ection at the bubble centre does produce a brief pressure peak [93]. However, that wave never becomes the powerful shock wave previously assumed, because of an adverse gradient in the sound speed caused by heat transfer. The peak temperature in the bubble lasts for a time set by the dynamics of the compression wave, which is typically 100± 300 ps, in agreement with light pulse duration measurements Spectra: experimental facts Spectra of sonoluminescing bubbles have been recorded for some time, see for example the summaries that have appeared in books [12] and review articles [20, 29]. The early work was concerned mostly with what was believed to be air bubbles ; or bubbles of diatomic gases, such as nitrogen and oxygen. It was later noticed that a trace of noble gas is apparently required for emission [36]; pure nitrogen or arti cial air (e.g. a mixture of nitrogen and oxygen, without the argon content of natural air) showed very weak or no sonoluminescence. The present thinking is that molecular gases, such as nitrogen and oxygen, are recti ed, i.e. converted chemically to non-gaseous compounds which then disappear from the bubble interior [67, 68, 72, 73]; see section 2.2. It is therefore reasonable to just focus on the sonoluminescence of rare gas bubbles. Of course, there will always be some other molecules mixed in with the rare gases, certainly water vapour,

13 251 Sonoluminescence 10 Xe Kr radiance [10± 11 W/nm] 20 Ar Ne He wavelength [nm] Figure 9. Single bubble sonoluminescence spectra of rare gas bubbles; the water was at room temperature; 3 torr of rare gases were dissolved in the water. For helium bubbles, two measurements are shown: the lower one is for the rare isotope 3 He. No corrections for the transmission through the water and the suprasil container were applied [20, 36]. Reprinted from Barber, B. P., Hiller, R. A., LO fstedt, R., Putterman, S. J., and Weninger, K. R., 1997, De ning the unknowns of sonoluminescence. Phys. Rep., 281, 65; # 1997, with permission from Elsevier Science. possibly its reaction products such as O, H, OH, [56] and perhaps some contaminants, which are likely to somehow av ect the observations. The rare gas spectra shown in gure 9 were recorded at room temperature [20, 36]. No ev ort was made to correct for the non-uniform transmission of the water and the fused quartz (suprasil) container. Such a correction is not necessary for wavelengths from about 300 to 800 nm, where the transmission of the media is uniform. However, at smaller wavelengths the transmission is said to fall ov by roughly 10% between 300 and 260 nm, and more steeply, to about 52% of the 400 nm value at 200 nm [20]. Water absorbs virtually at all wavelengths below 200 nm and above 800 nm [18, 19]. We note that another spectral window of water exists in the microwave region. However, measurements of the microwave intensities only gave an upper limit of sonoluminescent emission intensities of 1 nw in a 1 GHz bandwidth at 2 GHz [94]. The rare gas spectra are continua that fall ov with increasing wavelength. The heavier rare gases show a broad maximum, but the helium spectra seem to peak at wavelengths shorter than 200 nm. No spectral signatures, such as line or band structures, are discernible in the spectra of single bubble sonoluminescence. Studies conducted with higher resolution than employed for the recording of gure 9 did not reveal any line features either. Note that there is a clear isotopic div erence between the spectra of 3 He and 4 He bubbles, gure 9. Spectra recorded with a water temperature near freezing look super cially much like the room temperature spectra, gure 9, but the intensities are increased by an order of magnitude [20, 87] as may be seen in section , gure 13; moreover, the broad maxima are shifted slightly to shorter wavelengths [20, 36]. The number of photons emitted during the sonoluminescence pulse increases with a decrease of the static pressure P0 [37].

14 252 D. Hammer and L. Frommhold Figure 10. Comparison of the widths of emission pulses, measured in the ultraviolet and the red regions of the spectrum, from [86]. Observed light pulses are normalized to peak intensities ˆ 100. The pulses are not deconvoluted and thus appear less asymmetric than they actually are. Reproduced with permission. Time-correlated single photon counting techniques have recently been applied to a measurement of the temporal shape of the sonoluminescent light bursts [86 89]. In this way the pulse shapes could be measured for the rst time. The pulse widths at room temperature increase from about 40 ps to more than 400 ps [87], with typical values of 100 to 200 ps. The pulse shape has been described as an asymmetric gaussian, with a decay time that is slightly longer than the rise time. In general, the pulse width increases with increasing driving pressures [86 89]. Rare gas bubbles of the same intensity have the same pulse width [87], regardless of the rare gas used; also air bubbles at the same driving pressure show very similar pulse widths, independent of the water temperature [88]. In general no wavelength dependence of the pulse widths is observed, gure 10, with the exception of certain air bubbles at 38C [89], where the red pulse is slightly longer than the blue pulse. Spectra of single bubble sonoluminescence in heavy water have also been reported [43, 95, 96], and in liquids other than water [20, 29, 45]. The light bursts of bubbles in liquids other than water are generally weak, but the spectra are not very div erent from the examples shown above. Sonoluminescence in cryogenic liquids has not been observed [8] Spectra: hot-spot theories While a general consensus exists that some form of the Rayleigh Plesset equation, combined with either a uniform or non-uniform bubble interior, gives a fairly accurate representation of the bubble dynamics, no such consensus exists concerning the processes that are responsible for the emitted radiation. In his articulate 1996 status report on sonoluminescence, Glanz [97] remarks that... ideas about what could be producing the light have proliferated, and the list now includes imploding shock waves, jets of liquid crashing into the bubble wall, tiny electric sparks, and radiation torn from the background uctuations of empty space. The faint glow of sonoluminescence seemed mysterious. The featureless continuum of the spectra, gure 9, perhaps suggests that the light comes from some sort of a hot spot. Temperature estimates of the sonoluminescence sources have been attempted almost since their discovery; temperature estimates from many thousands, up to millions of kelvin have been

15 Sonoluminescence 253 reported over the years [20, 55, 59, 74, 81, 82, 84]. These vastly div erent temperature estimates were often based on similar if not identical experimental facts. The radiative processes that generate the light in the sonoluminescence environment would of course vary widely between such temperature limits. Under equilibrium conditions, at temperatures of several thousand kelvin, the degree of ionization for the rare gases is typically very small, ½ 0:01, so that emission would have to come largely from neutral neutral collisions. At somewhat higher temperatures, when the degree of single ionization is still signi cantly smaller than unity, ½ 1, electron neutral and electron ion bremsstrahlung would be among the principal sources of radiation. At the highest temperatures mentioned, the electron ion bremsstrahlung of the dense, multiply ionized plasma would generate the light. It is thus not surprising that several hot-spot radiation models exist that div er in the relative emphasis given to the diverse radiative processes. Moreover, the sonoluminescence environment is very dense and more or less substantial corrections to such radiation models may be needed to account for this fact; special liquid state radiation models have been proposed that div er radically from the concept of pairwise interactions which other models employ. Below, in sections through 2.4.4, we will examine the various hot-spot models that have been advanced. Alternative models will be discussed separately in section Black body model The emissivity I Pl of a black body of temperature T at wavelengths between and d is given by I Pl d ˆ 2hc2 1 d : 5 exp hc= kb T 1 11 For wavelengths between and d, the power emitted by a spherical source of radius R may then be written as 2 2 Pl PPl d ˆ 4º R I d : 12 In gure 11 we compare black body spectra with the spectral pro les of sonoluminescence, gure 9. For this purpose, we have shifted the experimental pro les vertically so that every measured pro le is next to a Planck curve of similar shape. (Absolute intensities are thus temporarily ignored.) We note that Planck s radiation law mimics the measured pro les reasonably well, at a temperature that is characteristic for every gas. Considering absolute intensities next, we note that for argon the absolute intensity suggested by the t of the spectral pro les is reasonably consistent with the measurements. However, for the other rare gases inconsistencies of the intensities of the tted and measured pro les of up to roughly an order of magnitude in either direction result in this way. For example, according to the t, gure 11, helium bubbles should be the brightest and xenon the dimmest, the reverse of what the measurements, gure 9, actually demonstrate. (All bubbles are of roughly the same size.) Moreover, recent measurements of the pulse widths [54, 86] point out other problems of the black body model of sonoluminescence. On the one hand, the black body model, if combined with uniform bubble dynamics,

16 254 D. Hammer and L. Frommhold 1000 nm± 1 ] 100 [ 10 7 W m± MK (Planck) 40,00 0K 20,0 00 K 14,0 00 K I Pl l 1 1M 2M K K 0.4 MK 0.2 MK 0.1 MK 60,00 0K ,000 K 8,500 K 7,000 K wavelength [nm] 700 Figure 11. Black body emissivity, equation (11), at temperatures from 7000 to K, in the spectral window of liquid water (solid lines), compared with the measured pro les of gure 9: helium (&); neon (*); argon ( ); krypton ( ); and xenon (.). overestimates the pulse widths, suggesting values near 500 ps, instead of the representative ps observed [54]. More signi cantly, the model predicts a substantial pulse width variation with wavelengths, gure 10. Since during the heating process the bubble interior is at lower temperatures for a longer time, compared to the moment of peak temperature, the light pulses observed with a red lter should be of longer duration than the ultraviolet pulses [54], in contradiction with experiment [86 89]. In other words, the black body model is not consistent with certain well established observations and should be discarded. In conclusion we note that with the assumption of a black body source, the nature of the emission process need not be speci ed: Planck s law only requires temperature as input, along with the assumption of an optically dense source. Alternative radiation models must, however, be speci c with regard to the light generating mechanisms involved. No such mechanism has been described yet which causes strong enough absorption so that signi cant parts of the bubble are optically thick. However, according to a sophisticated computer model [62], a high opacity of the bubble core, which consists of a dense, highly ionized plasma, is expected. A similar conclusion was reached elsewhere [81]. Yet, even with an optically thick core most of the emitted light is produced in an optically thin halo [62], with halo temperatures signi cantly below the core temperatures. The radiation processes in the halo then are most likely the same as those that have been suggested by a number of authors for uniform bubble models with weakly ionized environments [54, 58, 74, ] to be discussed next.

17 Sonoluminescence Weakly ionized gas model In sections through we examine alternative models of the sonoluminescent source that, like the black body model, may still be called hot spots. Here, speci c emission processes are considered that are familiar from spectroscopic studies of compressed, hot gases [101, 102] in a weakly ionized environment, with temperatures of up to a few tens of thousand kelvin [75, ] but not hundreds of thousands kelvin. This excludes the multiply ionized, dense plasma light sources [55, 77, 81, 105, 107]. In view of the small dimensions of the sonoluminescent source and the low degree of ionization of these models, the question of optical density must be addressed and suitable corrections be employed if the hot spot sources to be considered are neither optically thin nor black Electron concentrations. An important parameter of all spectroscopic models is the electron concentration in a source. Whereas in hot environments radiative processes exist that generate continuous spectra if free electrons are not present, even a small degree of ionization will signi cantly enhance emission; the concentration of free electrons must therefore be known. The degree of ionization as a function of (time- and possibly spacedependent) temperature T and density n, is given by the Saha equation, which, assuming only single ionization of one species, reads u1 1 m e kb T 3=2 Eion 2 ˆ2 exp : u0 n 2º h2 kb T 1 13 Here, me designates the electron mass, kb is Boltzmann s constant and Eion is the rst ionization potential. The ratio of the statistical weights of the ionized state u1 and the neutral state u0 is often set to 2u1 =u0 ˆ 1, according to the hydrogen-like atom model [75] Radiative collisions a brief overview. of the type We consider radiative collisions A B E! A B h! 14 and associative processes ( half collisions ), such as A B E! AB h!: 15 Here A and B may be neutral atoms or molecules, possibly in an excited rotovibrational or electronic state. Alternatively, in weakly ionized environments, A may be an electron or ion interacting with a neutral particle or ion B; E is the h! change of translational, rotovibrational or electronic energy in the collision and stands for the photon emitted in the process Electron ion bremsstrahlung. Early models of sonoluminescent light emission assumed that the principal radiation source is electron ion bremsstrahlung from a high temperature plasma [55, 59]. Given an optically

18 256 D. Hammer and L. Frommhold thin source, the emitted power J e i per unit volume, per wavelength interval d, is approximated by [75] 1=2 2 6 Zeff e hc 32º 2º J e i ˆ n 16 d exp d ; 3 3kB Tme kb T 4º 0 3 m e c2 2 where e is the electron charge; 0 is the dielectric constant; c is the speed of light; and Zeff is the ev ective charge of the ions; here Zeff ˆ 1. This expression, multiplied with the high-temperature limit Gaunt factor g ˆ 2 31=2 =º º 1:10 [77], a quantum mechanical correction, has been used by a number of authors [55, 58, 59, 81]. To later take into account the ev ects of optical density we need the ev ective absorption coey cient µe i br, where ev ective means that this absorption coey cient includes stimulated emission. It is quite generally related to emission by KirchhoV s law [75], J d ˆ 4ºµ I Pl d ; 17 with I Pl from equation (11). The absorption coey cient for electron ion bremsstrahlung thus becomes 1=2 2 6 Zeff e hc 4 2º ˆ n 18 µe i 1 exp : 3 3m e kb T kb T 4º 0 3 m e hc Recombination radiation. In addition to the free free radiative transitions of electron ion pairs, free-to-bound radiative transitions have also been accounted for [54, 58]. Convenient estimates of this contribution have been made. We consider again the absorption of radiation due to the inverse process, radiative ionization of a neutral atom from a given electronic excitation level j. In a hydrogen-like atom approximation where the energy of the jth level is given by Ej ˆ Eion =j 2 ( j ˆ 1 being the ground state), the absorption coey cient for this process µre becomes [75] X e me n 3 hc Eion jej j 64º4 Zeff 1 ˆ µre 1 exp exp ; 19 j3 kb T 3 31=2 4º 0 5 h6 c4 kb T jˆj which may be rewritten according to 1=2 2 6 Zeff e 4 2º ˆ µre 2 n m e kb T 4º 0 me hc 1 hc jej j 2Eion X 1 1 exp exp ; kb T jˆj j3 kb T kb T 20 where the summation starts with the lowest energy level j for which a photon of wavelength has suy cient energy to cause ionization, i.e. for which jej j < hc=. Since the absorption coey cients of electron ion bremsstrahlung and bound free transitions are both proportional to 2 n2, a coey cient µion for the total ionic re contribution, µe i µ, can be de ned. A further simpli cation can be made, since the atomic level structure of the rare gases that are of interest here shows a large gap between the ground state ( j ˆ 1) and the rst excited state ( j ˆ 2, its energy corresponding to a wavelength 2 ˆ hc=e2 ), while all other levels are fairly close

19 Sonoluminescence 257 together: for j 2 a continuum of states is assumed such that the summation in equation (20) can be replaced by an integration [54, 75], 1=2 2 6 Zeff e 4 2º ˆ µion 2 n mkB T 4º 0 me hc4 hc hc=max f ; 2 g 1 exp 21 exp : kb T kb T If in a given situation absorption by electron ion interactions prevails, a broad maximum appears in the spectrum at a wavelength ˆ 2. Since this wavelength roughly corresponds to the observed maxima [20], gure 9, it has been suggested [54] that these two facts may be related. However, in the interesting temperature range for sonoluminescence, the electron neutral spectra considered below also show maxima at about the same wavelengths [74] Electron neutral atom bremsstrahlung. In addition to the bremsstrahlung emitted in electron ion collisions, in weakly ionized gases another bremsstrahlung process must be taken into account: electron neutral atom bremsstrahlung. The mechanisms are similarly ey cient in generating light over a wide frequency band, but the electron neutral atom process will generally be more important at lower temperatures when the degree of ionization is small: its intensity is proportional to the degree of ionization, while that of electron ion bremsstrahlung is proportional to 2. The spectra of electron rare gas atom collisions are well known, both from experiment [ ] and theory [108, ]. The electron atom bremsstrahlung spectra may simply be approximated by [74, 118], 1 hc 3 e n 2 ˆ n 22 µ exp 1 ¼ff v ; vf v 4ºv2 dv; 8ºc kb T u 1=2 with u ˆ 2hc= me. The electron kinetic energy is v ˆ me v2 /2; for the electrons a Maxwellian velocity distribution function f v is assumed; and the radiative cross-section for free free transitions ¼ff is expressed in terms of the measured transport scattering cross-section ¼tr, according to [118], hc hc 1=2 4 ~ 23 ¼ff ; ˆ 2 1 ¼tr ; 3º me c2 with the ne structure constant. ~ Alternatively, a similar approximation [75], combined with a linear t of the measured transport cross-section, ¼tr º ctr dtr, has also been used [54, 119, 120] to obtain a long-wavelength approximation hc= kb T < 1 of the electron neutral bremsstrahlung absorption coey cient, ˆ µe n 1=2 8 21=2 e2 kb T ctr 3kB T dtr 2 n2 : 3 4ºe0 ºme c2 3=2 24 Equations (22) and (24) give similar results if the linear t of the transport crosssection ¼tr is used; both expressions can be shown analytically to go to exactly the same limit for! 1. Instead of the approximations, equations (24) or (22), an ab initio calculation of the electron neutral bremsstrahlung spectra based on the quantum mechanical