The struggle of modelling sonoluminescence

Transcription

1 The struggle of modelling sonoluminescence Richard James Wood School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom Abstract. There have been many attempts to model the phenomenon of sonoluminescence but all have fallen short of accurately describing its mechanism. Early theories; the shockwave model, blackbody radiation and adiabatic compression all provide a possible explanation. However, inherent difficulties arise in determining the accuracy of each model when it is compared with experimental data; some fit extremely well under certain conditions, yet when those conditions are changed another model may then more aptly fit the data. The current state of the art are models that attempt to describe the effect quantum mechanically. In this paper we will discuss the basic properties of sonoluminescence and show how early theories do not fully agree with experimental evidence. We will then explore how quantum mechanics may be applied; does it offer a rational approach to the modelling of sonoluminescence? Do experimental results support this interpretation? A final conclusion is given that explains how theories still cannot (as yet) be fully substantiated and the question of the mechanism of sonoluminescence is still an open one.

2 The struggle of modelling sonoluminescence 2 TABLE OF CONTENTS 1. INTRODUCTION 4 Page 2. EXPERIMENTAL BACKGROUND AND MEASUREMENTS Experimental Background Induction Bubble dynamics Light emission Driving pressure of the sound wave Disassociation theory Experimental Measurements Pulse Width Light emission characteristics Emission spectra EARLY THEORIES OF SL The Shockwave Model Motivation Model and support Model discrepancies Blackbody Radiation Motivation Model and support Model discrepancies Adiabatic Compression Motivation Model and support Model discrepancies A QUANTUM MECHANICAL APPROACH TO SL The Casimir effect SL as quantum vacuum radiation Motivation Schwinger s static calculation 24

3 The struggle of modelling sonoluminescence 3 Page Eberlein s dynamic model Model discrepancies Quantum optical heating Motivation Theory Supporting evidence Model discrepancies DISCUSSION 29 ACKNOWLEDGEMENTS 30 REFERENCES 30

4 The struggle of modelling sonoluminescence 4 1. Introduction Single-bubble sonoluminescence, SBSL is a phenomenon that occurs when a bubble in a liquid is subject to a sound wave causing it to slowly expand then rapidly collapse instigating the emission of light. Perhaps the most interesting feature of sonoluminescence, SL and something that makes it the subject of continuous study is the massive energy focussing (and related temperatures) that must be present in order to facilitate such light emission. To release a photon in the visible wavelength an atom, ion or molecule must be excited a few ev above its ground state, a sound wave carries an energy density of typically ev per particle [1]. Therefore, there must be a huge energy concentration of almost 12 orders of magnitude in the SL process. The extremely high temperatures associated with this emission have been said to be in the tens of thousands of Kelvin leading many to consider the possibility of a new energy source. Some going further to speculate that cavitation bubbles may be able to precipitate the release of D-D fusion neutron pulses [2] or as it s more commonly known, bubble fusion [3]. At its origins SL was first discovered by Marinesco et al. [4] when photographic plates were submerged in an insonified liquid and upon exposure were seen to be fogged. Frenzel and Schultes [5] further realised the phenomenon when they showed light emission from cavitation clouds using film that had been developed in the presence of an ultrasound transducer [6]. These observations came to be known as muti-bubble sonoluminescence, MBSL. SBSL was later achieved in 1989 by Felipe Gaitan [7] who was undertaking experiments on the oscillation and collapse of bubbles. He used a flask of liquid lined with transducers tuned to set up an acoustic standing wave at the resonant frequency of the jar [1]. Several authors have made significant contribution to the development of SBSL experimental methods and theory, in particular, Putterman [8-22], Suslick [16,23-29], Milton [30-32] and Matula [33-35]. The mechanism for the light emission, however, is still unknown and there have been several models presented to explain its facilitation. Initially the motion of the bubble and thermodynamics were imparted to a model involving adiabatic heating of the bubbles constituents, which precipitated the formation of an ionised core [1,36]. Another early model was realised when Lord Rayleigh s study of the damage caused to ship propellers by cavitation was related to the phenomenon. The idea of damage to a solid boundary was used to suggest the formation of jets as the bubble collapses that are propelled through the centre of the bubble at supersonic velocities [8,37,38]. At around the same time a model involving the propagation of spherical shockwaves that ionise the gas inside the bubble was also suggested [11,39,40,41]. Upon achieving an associated SBSL emission spectrum, efforts were made to make comparative analyses to known phenomenon. Surface blackbody radiation [9,10], and radiative plasma processes (e.g., bremsstrahlung and ion-electron recombination) [11,17,42,43] were all considered, many to a very good degree of success when the right conditions were applied to the experiment. Quantum vacuum radiation is a more recent development in the attempt to describe the SL mechanism quantum mechanically [44-46]. The idea was conceptualised when it was considered that a fast moving dielectric can precipitate the release of real photons from a quantum vacuum. The calculations for this model have taken various forms, many giving the correct amount of energy associated with that of SL emission. New quantum mechanical ideas involve the coupling of quantised motion of atoms to their electronic states due to an electric field gradient. These characteristics have been said to be inherent to rapid bubble deformations [47]. Several external factors influence SL; the nature of gas inside the bubble, the character and temperature of the surrounding liquid and the driving pressure of the sound wave. The type of the gas

5 The struggle of modelling sonoluminescence 5 used in SL experiments is known to effect the intensity of light that can be achieved and is important in demonstrating the strong catalytic role of noble gases in the phenomenon [47,48]. More recent developments have shown that bubbles formed from hydrogenic gases in water can also demonstrate stable SL. Putterman et al. [10] observed, dynamical measurements of the bubble motion combined with diffusion calculations suggested that there was indeed hydrogen within the bubbles. Other conditions that effect the properties of SL are the temperature and viscosity of the fluid. A lower surrounding temperature constricts bubble expansion and higher surrounding temperature causes more water vapour to enter the bubble, thus causing less light emission [49]. Viscosity of the fluid influences the form of the translational motion; lower viscosity fluids (water) have circular paths, whereas high viscosity fluids (phosphoric acid) have elliptical paths. In viscous liquids such as glycol, methylformamide, or sulphuric acid it is not possible to trap the bubble in a stable position [50]. There are five sections in this paper. In Section 2 we consider the experimental background and measurements that led to early theories. Section 3 describes the mechanism of these theories and offers supporting evidence for and against their realisation. The application of quantum mechanics to SL is explored in Section 4, with discussion of several new ideas. Section 5 summarises our analyses. 2. Experimental Background and Measurements 2.1. Experimental Background Induction The cavity of the SL experiment may either be a pre-existing bubble or can be generated via the process of cavitation whereby sound waves of high intensity break up the liquid and produce thousands of tiny bubbles. Further application of the sound wave can cause some to dissolve while others will drift to the centre and unite, the associated force causing the bubble to levitate and position itself at the pressure antinode. This trapping of the bubble is caused by the primary Bjerknes force and is a combined effect of the sound wave and (nonlinear) bubble oscillations [51] and may be described as the force arising from the pressure difference (gradient) across the bubble [34]. A small increase in low driving pressure will cause an increase in the Bjerkness force and the bubble will be pulled closer to the antinode, whereas at large driving pressures it will become smaller and the bubble will shift away from the antinode [34] Bubble dynamics Due to the sound wave the liquid will experience cycles of high pressure (compression) and low pressure (rarefaction), once trapped the bubble is also subject to these changes; the low pressure causing the bubble to expand (rectified diffusion), the high pressure forcing the bubble to contract. A complete SBSL cycle is shown in Figure 1 where it is shown that the negative pressure causes the bubble to expand. This continues until the pressure becomes positive and the bubble collapses oscillating about its equilibrium radius.

6 The struggle of modelling sonoluminescence 6 Figure 1. Instantaneous scattered intensity collected from a pulsating bubble. In the geometrical optics limit, the scattered intensity is proportional to the square of the bubble radius. The normalized drive pressure is also shown. The data (non-averaged) fit nicely with the Keller-Miksis nonlinear bubble-dynamics equation (solid line). [33]. To what degree the bubble expands and compresses in the sound wave depends on a number of factors, however the majority of papers [1,8,9,52] are now in agreement that generally an equilibrium bubble radius of R 0 (~4-5μm) leads to a maximum radius of 10R 0 (~40-50μm) and a minimum radius of R 0 /10 (~ μm). Although it should be considered that the type of gas in the bubble will determine the minimum radius (sometimes referred to as the Van der Waals hardcore radius) due to its intrinsic molecular size Light emission During the cycle, light is emitted when the bubble approaches its minimum radius, although exactly when is open to debate and is covered later in this paper Driving pressure of the sound wave The driving pressure (or drive-pressure amplitude to be more precise) of the sound wave is important to SBSL as a higher driving pressure will cause greater non-linearity of the bubble and at lower driving pressures it will slowly dissolve. It is also known that if the pressure is high such that it causes the bubble to become too non spherical light will not be emitted [34,53]. Figure 2 shows bubble radius vs. time for various driving pressures. To relate this to temperature it may be considered that, as the applied acoustic pressure increases, the bubble undergoes a more violent collapse which increases the effective temperature inside the collapsing bubble [22]. Measurements of temperature and pressure for various gases are shown in Figure 3; note that a relatively small increase in driving pressure results in a much higher temperature.

7 The struggle of modelling sonoluminescence 7 Figure 2. The steady-state oscillations of a 5 mm ambient size cavitation bubble in a 25 khz sound field. In (a) and (b), the drive-pressure amplitude (in atmospheres) is given next to the actual curve, if space is available. In (c), the maximum radius is plotted as a function of the drive-pressure amplitude, normalized to the ambient radius. The inertial cavitation threshold is around 1 atm [33]. Figure 3. Maximum temperatures achieved in a R o = 4.5μm bubble driven at f = 26.5 khz. A dissipative gas dynamics model was used in five runs using different driving pressure amplitudes and gas species. From Vuong and Szeri (1996) [1].

8 The struggle of modelling sonoluminescence Disassociation Theory An interesting phenomenon occurs when a diatomic gas is mixed with a noble gas; the SL character is that of the noble gas only [8]. This was initially not understood, Löfsted et al. [22] stated that there was an as yet unidentified mass ejection mechanism in air bubbles which is the key to SL in a single bubble. Further research determined that reactive diatomic gases disassociate in the hot bubble and form a soluble species, thus when air is dissolved in water a strongly forced bubble is completely filled with argon [54,55,56,57] while its other constituents are ejected and dissolve into the water. More recent developments have shown that bubbles formed from hydrogenic gases in water can also demonstrate stable SL. Putterman et al. observed that dynamical measurements of the bubble motion combined with diffusion calculations suggested that there was indeed hydrogen within the bubbles [10] Experimental Measurements Pulse width The duration of the light pulse emitted is perhaps the most important measurement, it is shown later that it is fundamental in disproving a number of purported theories and is essential to the understanding of the mechanism of SL. The width of the pulse may be measured by a photomultiplier tube (PMT) and can be related to the intensity of the flash by considering that brighter flashes give pulses that are broader in time as it takes time for the built up charge to drain from the PMT [58]. Initial pulse width measurements were found to be no greater than 50ps [11,12,13,39] leading to speculation of massive energy focussing and emission, however more accurate measurements were needed to substantiate these high energy assumptions. The breakthrough came from Gompf et al. [42] who measured the width of the light pulse using time-correlated single photon counting (TC-SPC). This proved to be much more accurate than a single PMT because it measures time delays in arrivals of single photons [1]; the time delay is measured and repeated many times so that the width of the flash can be reconstructed. It gave the result that the pulse widths can be of the order of a few hundred picoseconds and is not limited to less than 50ps as previously thought. The experimental configuration used by Gompf et al. and an example of a pulse width achieved is shown below in Figure 4.

9 The struggle of modelling sonoluminescence 9 Figure 4. (a) Experimental setup for time-correlated single photon counting (TC-SPC). PMT: photomultiplier tube; CFD: constant fraction discriminator; F: optical filter; TAC: time-to-amplitude converter; MCA: multichannel analyzer. (b) SL pulse after subtraction of the reflections and deconvolution. The pulse shape is slightly asymmetric and its FWHM at 1.2 atm driving pressure and a gas concentration of 1.8 mg/l O 2 at 22 o C is 138 ps (±10 ps) [42] Light emission characteristics There are approximately one million photons emitted per flash, and the time averaged total power emitted is between 30 and 100mW [48]. However, the intensity of light emission has a strong dependence on the nature of the gas dissolved in the liquid; a more conductive gas for example will lower the maximum temperature reached during SL thereby lowering the flash intensity [59]. This is shown in Figure 5.

10 The struggle of modelling sonoluminescence 10 Figure 5. Sonoluminescence luminous intensity versus thermal conductivity for the rare gases [60]. Since the flash intensity is directly linked to the pulse width i.e. lower flash intensities giving smaller pulse widths, then pulse width measurements can be dependent on the gas involved. If, for example, the water is degassed so that it contains only 3% of its usual content of dissolved air the flash lasts for 40ps, but if the water contains dissolved xenon, the flash can last for as long as 350ps [8]. Early papers suggested that light emission occurred at (or very close to ~5ns) the minimum radius [8,14,44] while more recent experiments have shown that it can vary from this position by up to 133ns [57]. Another consideration is the periodicity of the flash, initially it was thought that the flash per cycle happened periodically with clock-like precision [10,15,44] but again this has been shown to vary up to 100ns, this variation is often referred to as flash-to-flash jitter [58] Emission Spectra The spectral emission during SL is extremely important in the determination of its mechanism. However, there are inherent comparative difficulties in that the typical SBSL spectrum for water was seen very early on to be a featureless continuum whose intensity increases smoothly from near-ir into the near-uv devoid of any lines or bands [14,23]. Figure 6 shows the SL spectra for water saturated with various gases. Of worthy note is its strong relation to Figure 5 in that Xenon gives the greatest radiance followed by Krypton, Argon, Neon and Helium.

11 The struggle of modelling sonoluminescence 11 Figure 6. The spectrum of sonoluminescence for various gases at 293 K [61]. By analysis and comparison of SL spectra to known spectral patterns of bremsstrahlung or blackbody radiation, temperatures (averaged over the bubble) have been estimated to be K [9,10,13,27]. Some papers suggest an extremely hot ionised core reaching temperatures of up to 10 6 K [17]. It is of interest to note the absence of SL spectra below 200nm, this can be attributed to the absorption coefficient of water increasing rapidly below that wavelength i.e. wavelengths below 200nm are readily absorbed by the surrounding water and emission in that range is not seen [39]. Line emission during MBSL had been documented since the early 1990 s but was not seen in SBSL until around 10 years later. The absence of lines was thought to be due to the high pressure conditions that broadened them or that the continuum radiation was much more intense and they effectively became swamped [1]. Flannigan and Suslick [24] were among the first to observe spectral line emission from ions during SBSL in concentrated sulphuric acid. It was an extremely important observation as it provided experimental evidence of the generation of a plasma during SBSL. They also noted that by using sulphuric acid (a low vapour pressure fluid) flashes could be 2700 times brighter than the standard room temperature argon bubble in water [25]. 3. Early Theories of SL There have been several theories proposed as models for SL, the more popular of these at their conception were the shockwave, blackbody radiation and adiabatic compression models. We will now describe their basis and present supporting and opposing experimental evidence for each The Shockwave Model Motivation A model was required to account for the observed short pulse width times and reported high temperatures that could be associated with an ionised plasma, one proposition was the existence of a shockwave. Light emission from shockwaves had been known for some time; Bradley [62] in his 1962 book, Shock Waves in Chemistry and Physics states; the passage of a strong shock wave

12 The struggle of modelling sonoluminescence 12 through a gas is always accompanied by the appearance of a burst of visible radiation. Also related were the extremely high temperatures observed from shocks that may exceed 15000K [62]. This led to the consideration of a shockwave as a viable mechanism of SL Model and support The mechanism of the shockwave model is a shock created by the rapid collapse of the bubble that compresses and heats its contents such that the gas becomes an ionised plasma, this is the source of emission from which bremsstrahlung radiation may be observed. A more detailed description would consider that the shocks strength increases as it approaches the bubbles centre and therefore more heating occurs at that point than the boundary. The hotter regions become ionised and a twocomponent plasma is formed of ions and electrons; the emission of light is an energy cascade from the ions, to the electrons, to the photons [63]. Evidence for the presence of an ionised plasma was given by Flannigan and Suslick [24] who observed spectral lines from SBSL in concentrated sulphuric acid, this is shown in Figure 7. Figure 7. SBSL spectra from 85% H 2 SO 4 with 50 torr Xe and P a = 1.7 bar. (a) SBSL spectrum showing emission from Xe + as well as Xe and O + 2. (b) Enlarged region of the SBSL spectrum from 440 to 580 nm [24]. Comparisons of spectra from SL experiments to that of bremsstrahlung were found to be an extremely close fit, as shown in Figure 8. It should be noted that this spectrum would require that the plasma be of the order of 10 5 K and that the shockwave reach a radius of 0.1μm.

13 The struggle of modelling sonoluminescence 13 Figure 8. Spectrum of sonoluminescence showing that most of the emitted light is ultraviolet. As pointed out by Paul H. Roberts and Cheng-Chin Wu of the University of California at Los Angeles, the signal corresponds closely with bremsstrahlung radiation - that is, light emitted by a plasma at kelvins. Graph by Jared Schneidman [18]. The idea of converging shocks within the bubble was first suggested by Jarman [64] as a source of MBSL. The SBSL model was conceptualised by Greenspan and Nadim [40] following observation from Barber et al. [12] that the pulse width was less than 50ps. Through mathematical modelling they suggested that as the bubble oscillates a spherical shock forms at regular time intervals during the cycle. Stating, SL may be associated with the high temperatures that a converging spherical shock wave produces near the bubble centre, (this) is consistent with most of the experimental observations of sustained luminescence from oscillating gas bubbles [40]. Others including Wu and Roberts [39] and Moss et al. [65] used the same model to interpret experimental results to some degree of accuracy. Experimental data obtained by Matula [35] seemed to correlate with shockwave theory when an acoustic pulse within 1μs after the flash was detected, this was suggested to be due to the rebound of a shock from the centre of the bubble [58]. Figure 9 shows acoustic emissions originating at or near the local bubble minima.

14 The struggle of modelling sonoluminescence 14 Figure 9. (a) A single-shot R(t) curve as measured using our light-scattering system, with the corresponding acoustic signature. (b) A detailed view of the boxed area in (a). The acoustic data is shown here shifted in time equal to the time necessary for sound to travel from the bubble to the transducer, about ms [35] Model discrepancies Vuong and Szeri [66] were among the first to question the notion that strong shocks are important for SBSL when they gave a numerical simulation including dissipative effects and showed that strong shocks were absent in noble gas bubbles [1]. They also showed much lower temperatures than were first thought to be present in the bubble and that the hot spot was not highly localised in the bubble centre as would be the case for shockwave ionisation. Figure 10 shows motion and temperature in a bubble shortly before collapse. It is important to note it shows a smooth temperature profile; if a shock were present there would be some evidence of discontinuity.

15 The struggle of modelling sonoluminescence 15 Figure 10. Motion and temperature in a bubble shortly before collapse: (a) motion history of 20 Lagrangian points inside a R 0 = 54.5μm bubble driven at P a = 1.3 atm and f =26.5 khz. Strong wavy motion occurs inside the bubble, but no shock waves develop. (b) Temperature profiles in the bubble for various times around the bubble collapse. The profiles span a time interval of 170 ps near the collapse. The temperature at the centre increases monotonically, until the maximum temperature is reached at the last snapshot. Note that the temperature profile is smooth, without any discontinuity that would be present with a shock. From Vuong and Szeri (1996) [1,66].

16 The struggle of modelling sonoluminescence 16 The suggestion of non-central locale of the hot-spot has been supported more recently with images from Putterman [19] showing SL under the effects of an applied laser, Figure 11. Figure 11. (Colour online) Sample images and their false colour versions of laser pulse interaction with extremely dim bubbles showing random locations of hot spot. Laser is incoming from the right side.[19] An important attribute of the shockwave model is that the light flash must occur less than one tenth of a nanosecond before minimum radius; this is because it is not until that moment in time that the bubble wall velocity is large enough to create a thermal shock. Brennan [58] showed that SL flashes can occur hundreds of nanoseconds before the time of the minimum bubble radius, see Figure 12; the trace is from two adjacent bubbles with the smaller producing SL. It can be seen that the SL flash begins and ends before the minimum radius is reached. Figure 12. Data published by the UCLA group [24] for 1 MHz SL. There were two adjacent bubbles resulting in two traces. The smaller bubble produced the sonoluminescence, and the bottom of the three traces shows the SL flashes [58].

17 The struggle of modelling sonoluminescence 17 Further experimental analysis by Brennan showed a distribution of flash times about means of 133 ± 11ns, 130±12ns and 35±7ns before the minimum radius for argon, krypton and xenon respectively [58]. By using fluid dynamic models he was also able to produce evidence showing that the bubble wall velocity would be subsonic at the moment of SL which he considered, would be far too slow to create a shockwave plasma [58], that is, the formation of a shock front would not be possible. This is shown in Figure 13 where the bubble wall velocity at the moment of SL is shown to be 29m/s [58]. Figure 13. Another fit to our laser scattering data. The results of the fit tell us that this bubble has an equilibrium radius of about 8.5μm, and an acoustic drive of about 1.38atm. The calibration tells us that the bubble wall velocity at the moment of SL is about 29m/s. [58] It is important to note that Brennan does not dismiss the notion of supersonic wall velocities, only that they do not relate to the phenomenon of SL due to the timing of the SL flash; supersonic bubble wall velocities may occur near the minimum radius, but more often than not the SL flash occurs 100 ns before the minimum radius, so that any shockwaves that may occur are created after SL and unrelated to the cause of SL [58]. Another factor is the temperatures required for bremsstrahlung; to be the dominant light emission mechanism it must be higher than 9000K. However several models have predicted temperatures much lower than those thought to be required for a weakly ionised gas [56]. Figure 14 shows the peak temperature at collapse using a basic sonochemical model, it can be seen that the maximum temperature reached is of the order of 7000K.

18 The struggle of modelling sonoluminescence 18 Figure 14. Bubble temperature as the pressure amplitude is varied, the figure shows the peak temperature at collapse [56]. Further evidence against the shockwave model stemmed from the requirement that the SBSL bubble be accurately spherical [41]. Non-sphericity of a SBSL bubble was shown by Weninger, Putterman and Barber [20] where the ellipcity deviated by about 20% (Figure 15). Figure 15. Correlation of light intensity between two phototubes subtending an angle θ AB with respect to a sonoluminescing bubble. The solid line corresponds to an SL bubble whose flash to flash intensity has a large variation. The flash to flash fluctuations for the dotted line are much less and are furthermore consistent with Poisson statistics. Note the appearance of a negative correlation at 90. The maximum dipole that we have observed is about ten parts per thousand. If the dipole is due to refraction of light at the gas fluid interface of the bubble, then the ellipticity of the bubble in the state with large fluctuations is about 20% [20]. Baghdassarian et al. [67] added to the argument when he found that highly shape distorted bubbles were able to emit considerable luminescence, concluding that, a spherical shockwave cannot exist.

19 The struggle of modelling sonoluminescence 19 A related subject is the SL from dusty bubbles which form when the surrounding medium is contaminated with dust particles, it is seen that the bubble deforms and undergoes shape instability and yet (albeit with reduced intensity) SL still is able to occur [68]. More recent papers have shown that a transparent plasma does not fit the majority of experimental data and it must be either considered extremely opaque or composite of a two region system of opaque inner core and outer transparent (dilute) plasma [21,69]. Application of a laser to a SL bubble showed (via absorption characteristics) the SL emitting region to be extremely opaque and that it must be considered 1000 times more opaque than what follows from the Saha equation of statistical mechanics in the ideal plasma limit [19]. Although this does not contradict the formation of a shock it does support the notion that dilute plasma models of sonoluminescence are not valid [19] Blackbody Radiation Motivation A natural course of action when considering the spectra of SL was to make comparisons to known spectra, with the knowledge of high associated temperatures and suggested plasma processes [11,15,18,39,43,63] it seemed logical to test its relation to the known spectrum of a blackbody Model and support Early interpretation of the emission spectra from SL showed that it correlated very closely with that of a blackbody. Vasquez et al. [9] showed that a xenon bubble in water driven by a 40 khz standing sound wave radiates as an 8000K ideal Planck blackbody, this is shown in Figure 16. Figure 17 shows a bubble formed from Hydrogen (and water vapour) that is quantitatively fit to a blackbody spectrum, in its presented paper it is stated that (the fit) is so good that one is tempted to conclude that SL, at least for H 2, is due to blackbody radiation [10]. Figure 16. Spectrum of Sonoluminescence from bubbles of helium (150 torr) and xenon (3 torr) in water (23 o C) driven at 42 khz. Resolution is 12 nm FWHM. The solid lines are blackbody fits at 8000K (xenon) and 20400K (helium). Using measured flash widths of 100 ps (helium) and 200 ps (xenon) gives emission from a surface of radius 0.1 microns (helium) and 0.4 microns (xenon). The dashed line is a bremsstrahlung spectrum at infinite temperature [9].

20 The struggle of modelling sonoluminescence 20 Figure 17. Spectrum of a sonoluminescing bubble formed from a mixture of hydrogen and water at 0 o C (A) and 20 o C (B). The solid lines are fits to Planck s formula for 6644 and 6200 K. The radius of the surface of emission is also fit to 0.25μm at 20 o C and 0.2μm at 0 o C. The inset displays a linear scale [10] Model discrepancies There are however some inconsistencies with the blackbody theory. Vasquez et al. explains that if the spectrum from H 2 bubbles is taken to mean that it is a blackbody then there are 8 orders of magnitude discrepancy with the standard calculation of the mean free path of light when these blackbody parameters are applied to its interior [10]. Also there is an intrinsic difficulty in fitting the spectrum of SL to that of a blackbody, due to absorption of wavelengths below 200nm; the fits are often of very limited accuracy [23]. Perhaps the strongest argument against a SL blackbody emitter is that a blackbody s pulse width increases with wavelength, whereas for SL this is not observed. Figure 18 shows the pulse widths are identical in the red and ultraviolet regions of the spectrum, this fact alone would seem to exclude blackbody radiation as a mechanism for SL.

21 The struggle of modelling sonoluminescence 21 Figure 18. First measurement of SBSL pulse widths. The parameters were P a = 1.2 bars, f = 20 khz, and the gas concentration was 1.8-mg/l O 2. Both the width in the red and the ultraviolet spectral range were measured. The indistinguishable widths rule out blackbody radiation, but not a thermal emission process in general. [1] A blackbody also has the characteristic of being a surface emitter but for SL the bubble compresses to such a small size that it effectively becomes transparent to its own photons. Therefore, the radiation from the whole volume of a transparent body reaches the detector, rather than only surface emission like that of a blackbody [1] Adiabatic Compression Motivation Adiabatic compression (and associated heating) had been an early prediction in the SL process and may be considered a more original model as it developed from early equations of bubble dynamics from Rayleigh and traditional thermodynamics. With the realisation that more recent models were insufficient in describing SL accurately theory went back to its thermodynamic roots and concentrated on the mechanism of adiabatic heating. In particular, it was the experimental results from modern methods for accurately measuring the pulse width that caused many to reconsider this theory. Gompf et al. for example noted that the newer measurements of the pulse width were of the order of the time the bubble spends in its collapsed state, which is exactly what would be expected from simple adiabatic heating of the gas [29] Model and support The Rayleigh-Plesset (RP) equation (a continuation of Rayleigh s work by Plesset) offers a very good approximation of how the radius of a gas filled bubble changes with the pressure changes in the surrounding fluid [58]. We state it simply as: Where R is the bubble radius (the over-dots signify first (velocity) and second (acceleration) derivatives of the interface with respect to time), ρ is the density of liquid, P 0 is the pressure far from the bubble, P g is the pressure in the bubble, σ is the surface tension at the interface, μ is the sheer

22 The struggle of modelling sonoluminescence 22 viscosity of the liquid and c is the speed of sound in liquid. However it should be noted that it does not include damping effects (of acoustic radiation), it approximates the surrounding fluid to be incompressible and it requires several approximations for numerical solution. The bubble undergoes a rapid collapse causing adiabatic heating; it occurs so quickly that heat cannot escape. As a result the bubble heats causing ionisation of the gas and light emission via bremsstrahlung. It should be noted, however, that the process is not purely adiabatic; it is thought that the temperature law is initially isothermal and then switches abruptly to adiabatic once the bubble accelerates sufficiently [1]. Calculations using fluid mechanics and the van der Waals thermodynamic equation of state for adiabatic compression have been found to deliver temperatures that are sufficiently high to produce light from SL via excited electronic states of atom [36]. Figure 19 shows how the temperature rise in adiabatic compression of an ideal gas in a bubble is greater for a noble gas than one of diatomic molecules. Figure 20 shows how a non-ideal gas can deliver much higher temperatures than that of an ideal gas. Figure 19. For the adiabatic compression of an ideal gas in a bubble from radius R i R f < R i, the temperature rise is described by T i T f >T i. For the case of noble gas atoms (γ = 5/3) the temperature rise is much more steep than the case of tumbling diatomic molecules (γ = 7/5) [36].

23 The struggle of modelling sonoluminescence 23 Figure 20. In the adiabatic compression of a fluid within a spherical bubble from radius R i to radius R f, the temperature rise from T i to T f is plotted. The temperature rise from room temperature when interactions are included (as in the van der Waals theory) is large compared with the ideal gas case but only if the compression is substantial [36]. When the bubble tends to its minimum radius there is an increase in the number of excited atoms so much so that the temperature increase is great enough to cause emission of light, the relation of the number of excited atoms to the bubble radius is shown in Figure 21. Figure 21. As a bubble containing pure argon is adiabatically compressed from an initial radius R i at room temperature and pressure to a final radius R f < R i, the concentration x of electronically excited argon atoms sharply increases [36]. The mean pulse width times from experiment using TC-SPC were also found to correlate with the liquid-to-gas phase transition time predicted by an adiabatic van der Waals model. As were liquid to

24 The struggle of modelling sonoluminescence 24 gas phase transitions 20 to 100ns before the minimum radius with respect to observed average SL times [58] Model discrepancies A similar problem to the blackbody radiation model presents itself as there are inherent difficulties in supporting adiabatic theory due to flash width inconsistencies. The pulse width for Hydrogen is longer in the UV which suggests that it is not following an adiabatic process; such processes would yield a longer emission in the red portion of the spectrum [10]. Also, as with the shockwave model, an ionised plasma that produces bremsstrahlung requires much higher temperatures than have been observed during several SL experiments. 4. A Quantum Mechanical Approach to SL More recent papers have endeavoured to describe the mechanism of SL using quantum mechanics. In this section we will explain the bases of these ideas, consider the derived equations, determine if results match those obtained experimentally and where applicable present resolute counter arguments The Casimir effect The Casimir effect is a phenomenon that lies within quantum field theory, whereby it is known that a vacuum is not an empty space but has fluctuating electromagnetic (EM) waves and (therefore) energy. If two mirrors are placed in the vacuum it may be considered that some of the waves will fit while others will not, when the gap between the mirrors is smaller than the wavelength of the EM waves they are excluded from this space and the vacuum pressure inside the gap is then less than outside, this forces the mirrors together [70]. This is known as the Casimir effect. If the mirrors were to move rapidly it is said that some of the EM waves become real (photons), this is the dynamical Casimir effect. It is of worth to note that these photons are always pair-produced from the vacuum in two mode squeezed states, not in coherent states [71] with only one of the photon from the pair observed SL as quantum vacuum radiation Motivation Inspiration for the relation between quantum vacuum radiation and SL was made by Schwinger and conceptually it seems viable. Indeed, light emission in SL has, just like the Casimir effect, has its origin in the interaction of the vacuum fluctuations of the quantised magnetic field with a dielectric medium [71] Schwinger s static calculation Schwinger related SL to quantum vacuum radiation [46] and although he proposed a dynamical Casimir effect, it is at best quasi-static [72]. This may be realised when considering his calculations are the difference between the static Casimir energy of the bubble at maximum and minimum radius. The static Casimir energy of a dielectric bubble (inside) in a dielectric background (outside) may be written:

25 The struggle of modelling sonoluminescence 25 Where V is the volume of the bubble, ω(k) is the dispersion relation inside (ω inside ) and outside (ω outside ) the bubble, integrated over the wavenumber k. Thus, the Casimir energy can be interpreted as a difference in the zero point energies due to the dispersion relations inside and outside the bubble [72]. Schwinger estimated that the energy emitted during collapse was approximately equal to the change in this Casimir energy. Indeed, several papers have produced calculations in agreement with Schwinger finding that it is roughly the right energy budget to drive SL [48,72] Eberlein s dynamic model Eberlein [44,45] followed on from Schwinger to propose a more dynamic model by considering the production of photons due to a change in refractive index in the space between minimum and maximum bubble radius [72] i.e. a time dependant refractive index. This was realised by considering the medium to be an assembly of dipoles that the zero point fluctuations of the EM field can be said to induce, orientate and excite [44]. Thus, an accelerating boundary (the dielectric) moves and the fluctuations are no longer balanced, resulting in the conversion of virtual photons into real photons. It differed from Schwinger s model (which used sudden approximation) in that an adiabatic approximation was used in the calculations. This method of approximation assumes the variable parameter (in this case the frequency dependence of the refractive index) changes very slowly and therefore can effectively remain in the same state from where it started. Eberlein introduces a velocity-dependant perturbation [48] to the EM Hamiltonian to analyse the bubble boundary, given by: Where H ɛ is the Hamiltonian for a stationary dielectric, ΔH is a motional correction (or a velocitydependant perturbation), D is the electric displacement vector, B is the magnetic field at the bubble surface, β is the velocity of the bubble surface and ɛ is the dielectric constant. Since a spectrometer measures the single (observed) photon spectrum the angle integrated spectral density radiated during one acoustic cycle is used: Where, ω is the frequency of the photon, dt is the integral over ime for one cycle, the intergral of Ic kk (t)i 2 is the probability of creating a photon pair in the mode Ik,k from the initial vacuum state at time t, dω k is the integral over the solid angle. P(w) is a function of the radius of the bubble surface with respect to time R(t) and to estimate the spectral density a model for the bubble radius as a function of time must be used:

26 The struggle of modelling sonoluminescence 26 Where R 0 is the initial radius, R min is the minimum radius and γ describes the time scale of the collapse and expansion process. Assuming that the bubble radius is much greater than the wavelength of emitted light the following is derived: Where n is the refractive index of water (~1.3). This result was fundamental because it demonstrates the same ω dependence of blackbody radiation. By equating the exponent in (6) to ħω/kt and using a time of 1 fs for the cycle it corresponds to a temperature of around 4000K [45]. The total energy radiated during one acoustic cycle may also be derived: Eberlein points out at this stage that, the dissipative force acting on a moving dielectric (thus) behaves like R 2 β (4) (t). A final result in S.I. units may be written: Using accepted values for R 0 (~10 µm) and R min (~0.5µm) and n~1.3 a value of J for γ~1fs is obtained which Eberlein states, corresponds roughly to the experimentally observed amount of energy per burst [45] Model discrepancies The issue with Schwinger s model is that relating the power produced to the change in volume of the bubble (static Casimir calculations) would yield a greater photon production at maximum rate of change of volume [11] which experimentally is at the maximum radius [48]. Also if one were to assume that changes in refractive index were responsible for the conversion of Casimir energy into real photons then it would require it to be a femtosecond change. The collapse from R max to R min is known to require approximately 10 ns, which is far too long a timescale to allow for a sudden approximation [48]. Calculations by different authors could not account for the high energy requirements of Eberleins model, in fact it soon became clear that, in order to match the observed light intensities, the bubblewall speed would not just have to be comparable to the speed of sound, but exceed the speed of light [1,31,32,73,74]. This was primarily due to the turn-around time used by Eberlein that did not match experimental measurements and was suggested far too short to be compatible with the adiabatic approximation [48]. Milton et al. [30] in particular found that if the formula for the Casimir energy of a bubble in water is used a large Casimir energy of 13Mev, and something like 3 million photons would (have to) be liberated if the bubble collapsed. Thus, light intensities from SL in comparison to those calculated turned out to be five orders of magnitude too small. Unnikrishnan, in his Comment on Sonoluminescence as Quantum Vacuum Radiation [73] made several statements regarding the inadequacy of quantum vacuum radiation to describe SL, these related to its modelling which ignored gas traces in the trapped air, intensity dependence of ambient temperature and also the temperature dependence of the spectra. Nor could the absence of SL in non-

27 The struggle of modelling sonoluminescence 27 noble gases could not be explained through the dynamic Casimir effect, he concluded that; there could be no change in the refractive index or in the dynamics of the interface significant enough to affect the expressions derived in the vacuum radiation models [73] Quantum optical heating The cooling of trapped ions may offer an explanation as to the heating process involved in SL. The ion trapping process involves the confinement of a single ion (in free space) such that its motion becomes quantised. Cooling of the trapped ion can be achieved with the use of a laser that is detuned to the electronic transition frequency of the ion; therefore the ion can only absorb energy when it is moving towards the laser via a Doppler shift. In the process of photon scattering the total energy of atom and field is conserved, but in the exchange of energy and momentum between atom and radiation, mechanical energy of the atom may be dissipated through spontaneous emission [75]. Therefore the laser imparts a change in momentum to the ion (absorption) but if the ion (in its excited state) spontaneously emits a photon it experienced a further change in momentum but in a random direction (emission). The net result is a deceleration, and repeated over many times, may be considered as cooling. A development of this is resolved sideband cooling that uses a time dependant electric field, which creates a trapping harmonic potential well. By considering that the system may consist of several internal vibrational levelled states cooling of ions beyond the Doppler limit can be achieved by appropriate tuning of the laser. Simply put, it may be considered that the motion is cooled when transitions to the lower energy states are more probable [75] Motivation Beige et al. [47] proposed that the extreme confinement of ions during the SL process could be related to ion trap experiments. Suggesting that, as with ion cooling, the heating during SL could take place in the form of coupling of quantised motion of atoms to their electronic states, and that this is due to an electric field gradient inherent to rapid bubble deformations [47] Theory The basis of the theory is very strong atomic confinement such that the motion becomes quantised, and an increase in the mean phonon number per atom. Figure 22 shows the level configurations and possible transitions between an atom in its ground state and first excited state.

28 The struggle of modelling sonoluminescence 28 Figure 22. Level configuration of a single atom-phonon system indicating the immediately relevant transitions, if the atom is initially in its ground state 0 and possesses exactly m phonons. Ω denotes the coupling constant for phonon conserving transitions to the excited atomic state 1, while Λ is due to the electric field gradient inside the bubble and establishes a coupling between the electronic and the motional states of the atom. Moreover, Γ is the spontaneous photon decay rate of level 1.[47] Since the rate of transition scales differently i.e., + 1 there will be more atoms in the 1,m+1 state. With a large spontaneous decay rate there is a transition back to ground state, since there are more atoms in the 1,m+1 state than the 1,m-1 state, the net effect is an increase in the mean phonon number per atom i.e. heating [47]. The prerequisite of an electric field in this model comes from the coupling of the quantised motion to the electronic states of the noble gas atom (as shown in the above figure) which is similarly observed in ion trap experiments where the coupling may be manipulated by light fields (lasers) Supporting Evidence By assuming the presence of a inhomogeneous electric field and deriving the effective rate equations for the time evolution of a single atom-phonon system [47], Beige et al. showed that the phonon frequency in SL experiments is of the same order of magnitude as ion trap experiments. An important observation to the calculations is that ion heating does not require a large electric field but rather an electric field gradient; this would indicate that intrinsic inhomogeneous electric fields across the bubble under deformations could be a good proponent of this model. Indirect support may be considered from Brennan who proposes a process involving adiabatic cooling of the bubble that creates an excited cold condensate; it is said to emit light as it breaks apart as the bubble collapses [58]. Although the models are on the surface unrelated, his model does require the presence of some electric field or interaction and with that also suggests a coupling mechanism. This is described as a transfer of energy; when the gas expands freely and cools, heat from the centre of mass must be conserved and therefore transferred to electronic excitations which hold the atoms in a bound cluster [58]. A test of the quantum optical heating hypothesis would be the application of an external laser field, which would provide the electric field and create a coupling as predicted in this theory. If correct the