New suggestion concerning the origin of sonoluminescence

Physica D 216 (2006) 136–156 www.elsevier.com/locate/physd

New suggestion concerning the origin of sonoluminescence Bishwajyoti Dey a , Serge Aubry b,∗ a Department of Physics, University of Pune, Pune 411 007, India b Laboratoire L´eon Brillouin (CEA-CNRS), CEA Saclay, 91191 Gif-sur-Yvette Cedex, France

Available online 10 March 2006

Abstract We suggest a new mechanism where sonoluminescence is produced by the tremendously large adiabatic pressure pulse (shock wave) generated by the close to supersonic (or above) impact of the fluid on the hard core bubble. The light flash is mostly emitted by the fluid surrounding the bubble. More generally, the emission spectrum of any material submitted to a large adiabatic compression is (roughly) globally dilated by some Gr¨uneisen coefficient γ¯ . Temperature simultaneously increases by the same factor, which increases the power of the emitted radiation by a factor γ¯ 4 . A rigorous lower bound for the sound velocity in the compressed region at impact is obtained with purely kinematic arguments only assuming the existence of a non-negative Van der Waals volume for the fluid. For supersonic impacts, the increase of the sound velocity reaches at least one order of magnitude (and, with reasonable assumptions, much more), which yields an estimation of the Gr¨uneisen coefficient γ¯ and indicates it may become very large. Then, during the pressure pulse, the thermal infrared (IR) radiation of the compressed fluid can be extended up to visible–ultraviolet (UV) simultaneously with an intense brightness. The dynamics of collapsing bubbles have been analyzed taking into account fluid compressibility. Shock waves are generated when the bubble, at a minimum radius, suddenly becomes almost incompressible. For impacts close to supersonic (or above), an intense pressure is briefly generated in a sphere which extends beyond the central bubble and which thus mostly contains surrounding fluid compacted to near its Van der Waals volume. This compacted fluid generates an intense emission of UV–visible light which suddenly disappears when the fluid expands from its Van der Waal volume. This situation occurs when the sphere of compacted fluid reaches a critical size of a few minimum bubble radii. Next, this pressure pulse radially propagates through the fluid, initially at highly supersonic velocities, which decay to the normal sound velocity as it simultaneously spreads out. c 2006 Elsevier B.V. All rights reserved.

Keywords: Sonoluminescence; Nonlinear acoustics; Shock waves; Gr¨uneisen coefficient

1. Introduction Sonoluminescence (SL) is a process by which light is emitted from collapsing ultrasound-driven bubbles. There are two main types of SL: multiple bubble sonoluminescence (MBSL) and single bubble sonoluminescence (SBSL). SL is characterized by the emission of very short flashes of broadband UV light that is synchronous with the periodic acoustic driving field. The study of SL gained momentum after the discovery of SBSL by Gaitan [1]. SBSL is much easier to study, since the bubble can be extremely stable and glow for many ∗ Corresponding author. Tel.: +33 33169086128; fax: +33 3169088261.

E-mail addresses: [email protected] (B. Dey), [email protected] (S. Aubry). c 2006 Elsevier B.V. All rights reserved. 0167-2789/$ - see front matter doi:10.1016/j.physd.2005.12.012

minutes, making it possible to study both the bubble and the light that it emits. In contrast, it is very difficult to study MBSL, as individual bubbles last only for a few acoustic cycles and are in constant motion. In the years since SBSL was discovered, a lot of studies have been carried out, both experimental and theoretical, to understand how and why SL occurs. Various mechanisms, such as black-body radiation, Bremsstrahlung radiation from a hot plasma core, electrical mechanisms such as fractoluminescence, the Casimir effect, radiation from quantum tunneling, collision-induced emission, etc., have been proposed as the mechanism for light emission. Although these models go a long way to describing various aspects of SL, recent experiments have exposed serious limitations of the present understanding of SL.

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We do not consider that the processes of black-body, Bremsstrahlung, collision-induced, cooperative emission from a shock wave induced plasma inside the bubble etc. are the mechanisms responsible for SL. This is because, as mentioned below, the observations/predictions of most of these theories involving one or more of these mechanisms do not agree with later experimental findings. In this paper, we propose a new interpretation for SL. Our theory is consistent with the findings of the most recent experiments (see Section 2 below) of SBSL or MBSL, in either aqueous or non-aqueous media. Unlike other models (see Section 3 below) which place the location of light emission only inside the bubble, we believe that, in the case of SL, light is mostly emitted by the fluid surrounding the collapsing bubble. Actually, we suggest that SL is a particularly favorable experimental situation for studying a more general phenomenon that may occur in conditions other than bubble collapse. The emission of brief flashes of light should be observed in any material (solid, fluid, gas, or even cold plasma) providing that a tremendously high adiabatic pressure pulse could be produced. Such pressure pulses are typically obtained at shock waves generated by impacts close to or above supersonic. Under a tremendously large pressure that is applied briefly, the whole frequency spectrum and the temperature of a given material are roughly multiplied by the same large factor γ¯  1. This assumption is fulfilled under the common assumption that the Gr¨uneisen parameter of a mode ν, which is usually defined as γν = −d ln ων /d ln V (where ων is the mode frequency and V is the volume), is the same for all modes. Then, the excitation energies E ν = h¯ ων are simply multiplied by the same factor. Thus, the compressed material becomes hotter and therefore brighter. It emits a flash of thermal radiation over an extended range of frequencies (multiplied by the factor γ¯ compared with its normal spectrum) with a larger intensity (roughly proportional to γ¯ 4 ). At the present stage, the shape and parameters of the pressure pulse cannot be calculated precisely, because the equation of state of materials are unknown at those tremendously large pressure. Nevertheless, simple kinematic arguments allows one to scale the order of magnitude of the Gr¨uneisen shift and the pressure that could be expected from only the knowledge of the impact velocity and the Van der Waals contraction rate. This paper is organized as follows: in Section 2, we mention the recent experiments that provide some evidence that a large part of SL light may be emitted from the liquid surrounding the bubble; in Section 3, we mention the main early theories; in Section 4, we justify and discuss the assumptions of our model; in Section 5, we discuss the modelling of bubble dynamics, which refers to the appendices; in Section 6, we explain why an adiabatic pressure pulse produces light; and finally Section 7 is devoted to concluding remarks. The Appendix describes the technical calculations supporting our arguments: Appendix A recalls the Rayleigh–Plesset theory for bubble dynamics and discusses its limit of validity; Appendix B describes the propagation of the pressure front and the supersonic impact in one dimension; Appendix C discusses the 3D spherical case.

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2. Experimental facts The main experimental results that have been obtained so far from SBSL experiments can be summarized as follows (for details, see the recent reviews [2–5]): (1) The sonoluminiscent bubble oscillates with the oscillation frequency of the sinusoidal driving force (typically 20–40 kHz). The forcing pressure PPa0 ∼ 1.2–1.4, where Pa is the pressure amplitude of the sound wave and P0 is the ambient pressure (1 bar). (2) Once during each oscillation period the bubble collapses very rapidly from its maximum radius Rmax (typically 10R0 , where R0 ∼ 5 µm is the undriven radius of the bubble) to a minimum radius of Rmin (typically R0 /10) and the flash of light is emitted. After light emission, the bubble pulsates freely until it comes to rest, and then the whole process is repeated in following cycles. Both the light intensity and amplitude of oscillation of the bubble depend sensitively not only on the forcing pressure amplitude but also on the concentration and type of the gas dissolved in the liquid. (3) The SL spectrum displays a strong UV spectrum, as well as sensitivity to temperature. SL in colder water makes for much larger light emission. Typically, as the water is cooled from 30 to 0 ◦ C, the intensity of SL increases by a factor of ∼100. (4) The effect of changing the acoustic frequency (typically 20–40 kHz) appears to be comparatively small. (5) Flash width is typically of the order of a few hundred picoseconds and varies with external parameters such as forcing pressure and dissolved concentration of the gas. Flash width is independent of the color of light emitted [6]. (6) The presence of some noble gas increases SL. There is a pronounced decrease in SBSL intensity from the heavier to the lighter noble gas species in the order Xe → Kr → Ar. However, Ne and He spectra are not much different, even though He spectra are sometimes more intense than Ne spectra. (7) Besides light, there is also sound emission in SL. The velocity of the outgoing shock wave in the immediate vicinity of the collapsing bubble (∼4000 m/s) is much faster than the speed of sound (c = 1430 m/s) in water under normal condition. The high shock speed originates from the strong compression of the fluid around the bubble at collapse. The pressure at the surface of the collapsing bubble just before it reaches the minimum size has been inferred from needle hydrophone measurements and computation to be over 6000 bar. (8) Symmetry of the collapsing bubble plays a role. An increase in spherical symmetry of the collapsing bubble under controlled conditions leads to more light. (9) Observation of the isotopic effect SL. Most SL spectra exhibit a peak wavelength below 290 nm. However, in systems such as H2 or D2 dissolved in D2 O, the peak wavelength is observed to be shifted far to the red, with a peak spectral radiance at 400 nm [7]. The authors [7] could not explain the origin of the remarkably large shift,

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especially in view of the small difference in chemical and elastic properties between light and heavy water. Recent Mie scattering experiments [8] show that, in the last nanosecond around the minimum velocities, most of the Mie scattering is by the highly compressed water around the bubble and not at the bubble wall. Yet another experiment by Matula et al. [9] on MBSL showed the spectral lines from metal ions dissolved in water. Since metal ions are non-volatile, the origin of the metal ion lines is a puzzle. One possibility is that the metal ions get into the bubble on its asymmetric collapse; another is that the SL is produced from the water outside the bubble. The most recent experiment on the measurement of OH lines in the spectrum of SBSL by Young et al. [10] and Baghdassarian et al. [11] supports the latter view. The authors could not explain the origin of the OH lines, but suggested that it may be due to instabilities in the collapse of the large bubble. Alternately, the appearance of the OH in the SL spectra may be a signature of SL produced from the water outside the bubble. The needle hydrophone measurements [5] and computation [12] show that the pressure at the surface of the bubble in the final nanosecond before reaching the minimum size is over 6000 bar. Didenko et al. [13] generated SBSL in various liquids (aqueous and non-aqueous) such as adiponitrile. Moreover, similar to the observed OH lines for water, they [13] observed the excited CN lines, one of the groups that make up the adiponitrile. Similarly, other non-aqueous liquids (having an OH group) like ethanol and a mixture of water with hydrogen peroxide or glycerine also show strong SL emission [14]. Of course, for better SL effect, one has to choose a liquid with very low vapor pressure and a chemical structure that results in dissociation products that are easily soluble in the liquid. This is because the liquid vapor can be responsible for quenching SL and, also, the chemistry of the dissociation product plays an important role in the bubble stability [2]. Recent experimental results of SL in high magnetic fields [15] show a strange interplay between acoustic and magnetic fields, with an increased region of stability under the magnetic field.

3. A short review of early theories There are many theories available in the literature that attempt to deal with the mechanisms of SL, but there are no theories so far that do not suffer from some contradiction with experimental facts and can completely account for all the main experimental properties of SL as summarized above. Many theories, developed on the basis of the experimental results available at the time, later became questionable as new experimental results became available. For example, a theory considering a shock wave focusing a large amount of energy at the bubble center then generating black-body radiation as the cause of SL does not appear any more to be a correct candidate for SL. These shock

wave/plasma Bremsstrahlung theories were motivated by the measurement of Barber et al. [16], which indicated that the width of the SBSL was shorter than 50 ps, as shock focussing provides a natural mechanism for producing both extremely high temperature and small pulse width. Actually, we shall show in Appendix C, that due to high nonlinearities, shock wave focusing is stopped when the fluids approach their Van de Waals volumes. Later, more accurate measurement of the pulse width went against this shock wave theory of light emission. It was discovered by Gompf et al. [6] that the measured width of the light pulse is actually of the order of a few hundred picoseconds, much longer than the 50 ps upper bound measured by Barber et al. [16]. The much longer duration of the light pulse, as discovered by Gompf et al. [6], was later verified by Moran et al. [17] and Hiller et al. [18]. The same set of experiments [6, 18] also proved that the black-body radiation models of SL are also questionable. Gompf et al. [6] and Hiller et al. [18] discovered that there is hardly any variation in the pulse width with the wavelength of the emitted radiation. This contradicted thermal models favoring black-body radiation, which predict a large wavelength dependence of the pulse width (the red pulse duration is more than twice that of the UV pulse). Thermal Bremsstrahlung theories also require an assumption that the temperature inside the bubble is extremely high ∼106 K [19]. But neither an imploding shock nor a plasma or such a high assumed temperature inside the bubble has yet been detected in any SL experiments. Very recently, however, there was a report [20] of observations of SL light emission originating from collisions with high-energy electrons, ions or particles from a hot plasma core. However, the measured temperature from the emission spectra is found to be very low (4000–15 000 K for Ar emission) and therefore inconsistent with any such thermal process. The recent demonstration of the existence of chemical reactions within a single bubble [21] shows that the temperature reached inside the bubble during cavitation may be substantially limited by the endothermic chemical reactions of the molecules inside the bubble. Also, outgoing pulses have been observed [5], but they would be emitted whether or not there is an imploding shock. Even if a shock wave strong enough for generating plasma could develop for large enough forcing, it would be limited to a tiny region around the bubble center and would last too short a time. Another model ( jet theory) invoked an electrical mechanism for light emission (fractoluminescence). This idea [22] requires an asymmetric collapse of the bubble, during which jets of water enter inside the bubble at very high speed. Sonoluminescence is produced by the collision of the jet with the bubble wall, which would initiate fractoluminescence. However, it is now well confirmed experimentally that the best is the bubble spherical symmetry, which favors SL the most. Otherwise, this model relies on specific properties of water and did not find favor with later experiments [13], which showed that bright SBSL is possible even in non-aqueous liquids. Yet another line of research that involved the dynamical Casimir effect as a potential photon-emission process [23,24]

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had to be abandoned later when a calculation based on the actual model showed that the Casimir energy is of the order of a magnitude that is too small to be relevant to SL [25]; in order to match the observed light intensities, the bubble wall speed would exceed the speed of light [26]. Other light emitting mechanism models include radiation from quantum tunneling [27] and the collision-induced emission process [28]. However, the parameter values used for estimating the light emitted in the collision induced model [28] (for example, the assumption that the life time of the radiation is 1 ps) do not agree with the more recent experimental values [6]. It may also be appropriate here to mention other models that place the location of light emission in the surrounding fluid, with mechanisms different to our’s for SL. These are the protontunneling radiation model [27] and the electrical breakdown model of Garcia [29]. Thus, it is clear from these examples that the assumptions/predictions of the existing theories of SL, which were formulated on the basis of experimental results available at the time, do not agree with the new measurement results and that new suggestions would be welcome. 4. Preliminary discussion for a new approach Our theory is also a shock wave theory, but instead of assuming that it focuses a large energy density near the bubble center, generating a high-temperature plasma, we show that this shock wave generates a high-pressure pulse that last much longer and concerns a much larger volume, including the bubble and the surrounding fluid. The fluids inside this radiant sphere do not become a plasma, but nevertheless become sonoluminescent. 4.1. Some very rough physical estimations Let us first rexamine the order of magnitude of the physical quantities (energies, temperature, radial bubble velocities etc.) which could be involved in the real experiments on SL. Let us first consider the bubble temperature. According to the experimental situation, the minimum diameter of the bubble is about 10 times smaller than its diameter at rest, which roughly corresponds to the minimum volume of the gas before it becomes hard core and almost incompressible. Then we assume the most favorable conditions for heating the gas inside the bubble, that is: • the gas in the bubble is a monoatomic gas and is perfect during the whole compression; • there is no heat diffusion in the fluid (the compression is perfectly adiabatic); • the energy absorbed by the dissolution of the high-pressure gas inside the water is negligible (or, equivalently, the gas in the bubble is hardly soluble in water). The ratio Tm /Tr of the bubble temperature Tm at the minimum radius Rm with the temperature Tr ≈ 300 K at equilibrium radius Rr fulfills  1−γ   Vm Rm 3(1−γ ) Tm = = Tr Vr Rr

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where γ = 53 for a monoatomic gas, and we get Tm ∼ 30 000 K ∼ 3 eV. This result is a strict upper bound on the final temperature inside the bubble, assuming that it has uniform temperature. This overestimates the heating of the bubble, because the above assumptions are likely to be not fulfilled very well. Some real experiments and more accurate theoretical calculations, which suggest that the maximum bubble temperature may not exceed 10 000–20 000 K [19,22], are consistent with our estimation. Since the ionization energy of the rare gas (12.13 eV for free xenon atoms and higher for other rare gas) inside the bubble is quite a lot higher than this thermal energy, the gas inside the bubble should not be substantially ionized without an extra mechanism for energy focusing. It cannot become a plasma, as suggested in some papers. Actually, experiments show that a substantial part of the energy injected by the pressure force is not used for heating the inside of the bubble, but is transiently converted into kinetic energy for the fluid (see Appendix A). The order of magnitude of the energy W injected by the external applied pressure during bubble collapse can be roughly estimated. We have W = P1V , where the pressure P ∼ 105 Pa is of the order of the atmospheric pressure, and the 4π 3 3 3 volume variation 1V = 4π 3 (R M − Rm ) ≈ 3 R M , where R M (resp. Rm ) is the maximum (resp. minimum) bubble diameter. According to experiments, we may choose R M = 25 × 10−6 m and Rm = 25 × 10−8 m ∼ 0, which is W ∼ 6.25 × 10−9 J ∼ 40 GeV. This is roughly the scale of the whole energy available for the SL process. In good experimental conditions, the kinetic energy of the fluid may reach an important fraction (typically a half) of this energy. Since the kinetic energy of the water is 2πρ R˙ 2 R 3 as a function of bubble radius R and velocity R˙ (see Appendix A), we obtain that, when this radius is almost minimum (R ≈ Rm = 25 × 10−8 m), R˙ ∼ 6 km/s, which is comparable to the velocities observed in real experiments [30]. Most of this kinetic energy will be released in the shock wave generated by the supersonic impact of the fluid on the hard core bubble. The order of magnitude of the pressure peak due to this shock wave, which is comparatively very high, is estimated in Appendices B and C. Some authors speculatively suggested that the possibility of nuclear fusion [31] could be generated by such imploding bubbles. Such a phenomena would require that a substantial amount of the available energy (∼40 GeV) generated from bubble collapse is focused over quite a small number of atoms. Although, for nearly linear theories and for a perfectly spherical impact, temperature and pressure near the bubble center should diverge, it looks unlikely that energy could be sufficiently well focused (at the atomic scale!) for nuclear fusion to occur. Actually, most the available energy is involved in the compression of a relatively large volume of fluid spreading roughly uniformly much beyond the bubble core and thus does not focus at the center. Our basic argument is to prove, by purely kinematic considerations, that this tremendous pressure peak must be associated with a tremendous hardening of the atomic interactions which may raise the sound velocity in the compressed material by several orders of magnitude (this

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feature holds whatever the temperature and the state of the highly compressed material could become: solid, liquid, or plasma). This only considers that mass is conserved and that the volume of a fluid element cannot become negative. No precise knowledge of the equation of state of the involved fluids is required. We also assume, for simplicity, that this volume decays monotonically as a function of pressure (that is, phase separation is absent). To be more realistic, assuming that a volume element cannot be reduced below a certain Van der Waals minimum (positive) volume will yield more tightened bounds, favoring the occurrence of SL at impacts. Although still at the qualitative level, the theory that we propose is consistent with all the many different experiments on SL undertaken to date. Our theory simply relates SL to the adiabatic compression of standard matter at tremendously high pressure. SL involves the bubble, but also mostly a relatively large volume of the surrounding fluid, which is assumed to remain fluid. 4.2. Optimized conditions for sonoluminescence within our theory We now have to consider the order of magnitude of some essential physical parameters in our theory. We set the assumptions required for our model precisely. We assume that, at the end of bubble collapse, when the bubble radius velocity is a maximum: • The bubble radius shrinks at a very large velocity that is highly supersonic or of the order of the sound velocity in the fluid. • The gas inside the bubble becomes a dense liquid near molecular or atomic close packing, with a compressibility comparable to the compressibility of the surrounding liquid. Then, there is an impact between the bubble hard core and the fluid, which generates a shock wave. We shall show that, very generally, any supersonic impact systematically generates luminescent shock waves. According to our scenario, the brightest SL requires the largest pressure pulse. The largest pressure pulse requires the largest impact velocity and, at the time of impact, the fluids are the most incompressible they can possibly be. For the largest impact velocity, the maximum work provided by the driving pressure during the change of bubble volume should be converted into kinetic energy for the collapsing fluid. This is a problem of fluid mechanics, which involves the intrinsic properties of the fluid and of the gas inside the bubble. It also involves the frequency and amplitude of the applied force. Experimentalists have already found empirically the conditions that realize sharp supersonic impacts. Among the important issue is that the energy losses due to fluid viscosity should be minimized for better impact. In order to minimize the friction forces in the flowing fluid, good bubble sphericity is highly favorable, since the standard dissipative terms (A.3) in the Navier–Stokes equation vanish (at the lowest order) for an incompressible fluid when the flow has perfect spherical symmetry. Breaking sphericity should diminish the

bubble velocity at impact and consequently SL. Thus, more intense SL is favored when spherical bubbles are the most stable. In that respect, small bubbles are more favorable for SL, because their sphericity is better stabilized by surface tension. Bubbles of rare gas are also more favorable. Because the atoms of a rare gas are chemically inert, they interact weakly with water molecules, which make them barely soluble in water with large surface tension. Moreover, since they are barely soluble in water, there is almost no extra energy dissipation of the bubble kinetic energy due to the partial dissolution of the gas in water which could occur during compression. Since solubility increases with temperature, it is also preferable to work with water at the lowest possible temperature, close to 0 ◦ C. It is interesting to note that energy dissipation during bubble collapse has been increased artificially by applying a strong magnetic field to water. It was found that the fast motion of the dipolar water molecules generate a torque on these molecules, which induces energy dissipation. Then, SL may be reduced or suppressed [32]. As mentioned in Section 2 above, all these empirical features seems to be confirmed experimentally in the experiments producing SL. The second condition is again favored by rare gases, which are monoatomic with complete electronic shells. Electronic overlaps between nearest atoms cost a very large amount of energy, which makes these materials barely compressible in the solid or liquid state when the atoms pack near to maximum compactness. The hardness grows for rare gases with increasing mass, i.e. He, Ne, Kr, Xe. However, the situation for water is special, because it is a relatively compressible liquid. The reason for this is that it consists of a soft hydrogen bond network between oxygen atoms, leaving many empty spaces. Indeed, phase transitions have been induced at low temperature from low-density ice to amorphous high-density ice with a density 1.19 at pressures of about 6 kbar ∼ 6 × 108 Pa [33,34], which corresponds to a volume contraction compared to normal water of about 20%. Despite the absence of first order transition in the liquid phase, we may reasonably expect, at the yet more gigantic pressures involved in the shock wave, that the volume of water may shrink by up to 20% (h W = 0.2), corresponding to the minimum Van der Waals volume. Thus, there is an inversion point at relatively low pressure, where the rare gas suddenly becomes much less compressible than water. If this point is crossed during bubble dynamics, when water has already accumulated a large amount of kinetic energy, one gets an impact of water on the hard anvil formed by the compacted rare gas. In summary, within our approach of SL, there are three different problems to analyze with more details for properly describing SL. • The first problem is related to hydrodynamics. How do we produce stable bubbles that reach radial velocities near to the minimum radius which are supersonic and as large as possible? • What is the order of magnitude of the amplitude and the width of the pressure peak generated by the supersonic

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impact of the liquid on the hard core gas in the bubble? How does this pressure pulse propagate and how does it decay? • Why does a high-pressure pulse produce SL? We discuss these three points in the following, although our analysis will only be partial. We essentially focus on qualitative aspects and the physical order of magnitudes. 5. Modelling bubble dynamics The general behavior of the dynamics of a spherical bubble is quite well described within a Rayleigh–Plesset (RP) model, which is studied in details in [35]. We show in Appendix A that this model is nothing but an anharmonic oscillator which is parametrically driven by the external pressure (Duffing oscillator). When the amplitude of the driving pressure is not too large, the bubble radius oscillates smoothly at the driving frequency. The compressibility of the gas in the bubble always remains larger than that of the surrounding fluid. Although the driven bubble oscillator is highly nonlinear, with a motion that is not sine like and may become complex, it does not generate any substantial impact (and shock waves) at the minimum radius. The RP model accurately fits the experimental observations quite well, but then there is no SL. Beyond a certain critical amplitude of the driving pressure, there is a quite abrupt change in the bubble dynamics, because the compressibility of the bubble core becomes of the same order or smaller than those of the surrounding fluid. In that case, the fluid impacts the bubble at the minimum radius and bounces back several times, with smaller and smaller impacts. In that regime, the experimental observations [8] of the bubble dynamics, and especially the bubble radius bouncing, are not well fitted by the RP model. The main reason for the discrepancy between the observations and the RP model is likely to be due to the extra energy dissipation by the shock waves which are generated at each impact and carry out some energy from the bubble oscillator. After these fast oscillations damps, the bubble radius grows smoothly again, while the bubble’s hard core returns to gas. The expansion of the bubble is again well described by an RP model. When the bubble is at maximum radius, the pressures are still rather low everywhere in the fluid (and of the order of the ambient pressure, at most). At this stage, the gas inside the bubble is highly compressible, while the fluid (water) outside the bubble is practically incompressible. The pressure inside the bubble is much lower than the increasing driving pressure, so that the bubble starts to collapse, almost as for a vacuum bubble. The radial velocity of the fluid near the bubble accelerates to very large velocities. As mentioned above, it may reach up to four times the sound velocity in the fluid just before the minimum bubble radius. When the radial velocity becomes sufficiently large, the bubble’s collapse ends with a sharp impact on the hard core gas. The impact produces the shock wave that propagates in the fluid, as currently observed in experiments [36]. The pressure pulse generated at the first impact has been observed travelling radially in the water (or the fluid) with an initial velocity well above supersonic [36].

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SL is observed during the first few hundreds of picoseconds after its creation. The amplitude of this pressure pulse decays rapidly, not only as a function of the radial distance but also because it spreads out. The SL flash rapidly cools down to lower frequencies, while its brightness disappear. The amplitude of the next bounce and the corresponding impacts are much weaker because of the energy dissipation due the shock waves of the previous impacts. No substantial SL is generated. This is qualitatively consistent with our theory, since the first impact is the sharpest and should generate the most SL. The impact occurs when the gas inside the bubble becomes less compressible than the fluid outside the bubble, which is close to its Van der Waals radius. The compressibility of the fluid (initially assumed to be incompressible in the RP model) must be taken into account to describe the bubble impact and the travelling pressure pulse in the surrounding fluid. Because of the spherical symmetry, this pressure pulse propagates radially from the bubble center. Its amplitude decays while it also spreads out. In Appendices B and C we model the impacts in 1D and 3D spherical models, respectively. When the gas inside the bubble becomes compact and less compressible than the surrounding fluid, heating of the bubble slows down sharply because most of the energy injected for compressing compact materials (solid or liquid) then becomes essentially potential energy and produces relatively little heating. The correct model to be treated to describe the impact should be an improved RP model, where both the gas inside the bubble and the fluid around the bubble are considered to be compressible. Such a complex model should be investigated numerically, but we may, however, draw strong conclusions with simple analytical modeling and arguments. We first investigate the generation of a pressure pulse by a supersonic impact in a simplified situation where two identical compressible fluids in a one-dimensional pipe are initially launched one against each other at supersonic velocity (the impact of water on water). This is done just by fixing an initial velocity field to an homogeneous fluid at constant pressure P0 . This problem is treated in Appendix B. We have chosen a simple initial velocity profile where we have an exact solution describing the impact, but nevertheless we think that the essential result should not depend much on the details of the initial conditions. The important physical feature for SL is not the amplitude of the pressure itself but the shift of the sound velocity in the fluid adiabatically submitted to this highpressure pulse. The volume of a fluid element of unit volume at pressure P0 necessarily decays as a function of pressure, with an asymptote at 1 − h W . Since it cannot be negative, we have 0 < h W < 1 (the maximum Van der Waals contraction rate). We assume, for simplicity, that the sound velocity increases monotonically as a function of pressure. It must diverge at infinite pressure. Then, we proved that the sound velocity si in the region of high pressure generated by the impact fulfills the inequality si > Vi / h W (see Eq. (B.19)). This inequality does not require any precise knowledge of the equation of state of the fluid(s). Actually, this minimum bound for the sound velocity si in the high-pressure region generated at impact may be widely

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Fig. 1. Scheme of the conjectured evolution of a spherical bubble impact described by Eq. (C.7). The grey annulus represents the high-pressure zone P > PW when it emits light. The dark zone represents the zone of the pressure peak P < PW which does not emit light. Top left: Just after impact, a region of high and roughly uniform pressure Pi appears, delimited by two fronts that propagate at highly supersonic velocities in opposite directions. Weak UV–visible light starts to be emitted. Top right: The inner front reaches the center. Pressure is much larger than PW in an expanding sphere delimited by the outer front moving at highly supersonic velocity. The whole fluid inside that sphere is adiabatically compressed near to the Van der Waals minimum volume. Intense UV–visible light is emitted. Bottom left: When the radius of the fluid compacted sphere reaches a minimum of a few bubble radii RSL , its expansion slows down. The compressed fluid starts to expand at the center, where pressure P00 decreases. UV–visible light decreases. Bottom right (larger scale): When the sphere radius increases beyond RSL , pressure drops below PW and, especially at the center, P00 ∼ P0 . Light emission stops. A highpressure pulse forms, moves radially and becomes quasi-harmonic when its velocity decays to the normal sound velocity, while its profile ceases to spread and becomes constant. The amplitude of the pressure peak decays as 1/r .

underestimated by this inequality (which is often the case for rigorous bounds). Only in the case of highly compressible diluted gas, where h W ≈ 1, do we have both si > Vi and si ∼ Vi . In contrast, dense fluids are barely compressible, so that the maximum Van de Waals contraction, h w , is small. In the case of highly supersonic impacts, the fluid compression rate approaches the maximum h W , at which the sound velocity diverges (see Fig. 4). Then the sound velocity si in the highpressure region generated at impact should reasonably be expected to be much larger than this rigorous bound. It is then rather obvious that highly supersonic impacts may shift the sound velocity in the compressed region to values which are several order of magnitudes larger than the normal sound velocity, s0 , and which cannot be realized today in any static experiments. The exact solution considered in Appendix B for proving this inequality has the flaw of being one dimensional. It does not correspond to a pressure pulse but to a high-pressure region that extends as a function of time by two fronts moving in opposite directions. Moreover, our initial condition involves an infinite

kinetic energy, which is not realistic for describing a 3D bubble impact where the total kinetic energy of the fluid is finite. In 3D, the high-pressure region cannot extend indefinitely but becomes a pulse. Nevertheless, one may consider that the 1D impact model could describe the very first stage of impact, when the high-pressure region is confined to a thin shell around the sphere of impact. Thus, the 1D bound yields a lower bound for the sound velocity at the impact region and at the time of impact (see Fig. 1). We show in Appendix C that the equation describing an impact within a spherical geometry, considered for the variable u(r, t) = r P(r, t), is formally equivalent to the 1D equation for the pressure P(r, t) but with inhomogeneous nonlinearities that become stronger and stronger near the center. In the opposite case, the effect of the nonlinearities disappears at great radial distance. Unlike the 1D case, a discontinuous pressure step is not an exact solution anymore. We should expect that the shape of an initially sharp front should change as a function of time mostly close to the bubble center. We have not been able to perform an accurate analytical investigation of the evolution of the pressure pulse generated by a radial impact of the fluid on a sphere. A numerical investigation should be performed later. Nevertheless, using the same strong argument of fluid mass conservation as in the 1D case, we can make empirical predictions on the size evolution of the highly compressed region (beyond the Van der Waals pressure) before it spreads out and decays. We found that in the experimental conditions, a highly compacted sphere involving the bubble core and the surrounding fluid near Van der Waals volume should form just after bubble collapse. Its pressure is roughly uniform and non diverging at bubble center. There is emission of light till the radius of this compacted sphere grows up to a maximum size which corresponds to a sphere called radiance sphere with radius RSL which extends over few minimum bubble radius Rm . This result implies that most SL should be generated by the fluid surrounding the bubble inside this radiance sphere. Pressures larger than the Van der Waals pressure will persist in that region only for a few hundred picoseconds. According to our theory, when the pressure in the fluid is larger than its Van der Waals pressure, the fluid should emit a broad band of light with characteristic frequencies that drop from UV to IR (when the pressure decays). Beyond that distance, the pressure peak becomes of the order of, or smaller than, the Van der Waals pressure. Then, this compacted sphere ceases to emit light and begins to behave quasilinearly. First, its pressure drops at the center and the resulting spherical pressure pulse propagates and decays as the inverse distance from bubble center, while it velocity drops to the normal sound velocity in the fluid. This evolution is pictured schematically in Fig. 1. 6. Adiabatic high-pressure pulse generates light emission The previous section, together with the three appendices, was devoted to the demonstration that • Supersonic impacts can be generated by bubble collapse.

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• Then, the pressure pulse (shock wave) that is generated propagates radially, initially at highly supersonic velocity in the fluid surrounding the bubble. It decays abruptly and slows down to the normal sound velocity within a few bubble radii. • The Gr¨uneisen shift of the sound velocity in the fluid at the pressure pulse is very large within a few bubble radii, but drops to one beyond this critical distance. The question to be discussed here is to explain why a large sound velocity shift should produce SL. First, the fluid compression should be considered to be practically adiabatic, considering the scale in time and space of the pressure pulse that emits the light flash. In order to firm up the ideas, an exact spherical solution of the Fourier heat |r|2 equation yields the temperature T (t) = (t+tT0)3/2 exp − 4D(t+t 0) 0 of a hot spot as a function of time t and radius r , where D is the heat diffusion constant and t0 is an arbitrary √ constant. At t = 0, the radius of the hot spot is r0 = 2 Dt0 and the 0 . It is divided by two after temperature at r = 0 is T (0) = T3/2 t0

(22/3

r2

0 a time τ (r0 ) = − 1)t0 = (22/3 − 1) 4D . Then, for a hot spot with an initial radius of r0 = 0.25 µm in water, where D ≈ 1.4 × 10−7 m2 /s, its characteristic time τ to cool down to half the temperature is τ ≈ 1.6 × 10−7 s, which is three orders of magnitude longer than the characteristic time for the SL flash. Thus we may neglect heat diffusion during the bubble impact.

6.1. Discussion about the concept of adiabaticity When an element of fluid is submitted to a travelling pressure pulse, the molecules in it are suddenly compressed and then relaxed within a few hundred picoseconds. Although, for simplicity, we may model the pulse as being delimited by two fronts that are a pure discontinuity (which is not an exact solution of the 3D spherical model), the pressure pulse should correspond at the microscopic level to a slow pressure variation, with a characteristic time much longer than the fast IR vibrations of the molecules, which means that they are submitted to an adiabatic compression. We assume that the energies that are involved in bubble collapse are insufficient for substantial ionization of the fluid (except perhaps within a negligible volume near the bubble center, which we may discard). This assumption requires a large gap between the electronic excitations and the other atomic modes, that is, the fluids involved in bubble collapse are good insulators. We consider that the fluid remains standard insulating condensed matter, where the atoms interact with effective Born–Oppenheimer potentials and where the electronic excitations are not directly involved. During an adiabatic compression, a part of the energy provided to the material becomes potential energy for the atomic interaction, and another part becomes kinetic energy for the atoms. These are very opposite situations. For example, the energy released during the adiabatic compression of a perfect monoatomic gas is totally converted into the atomic

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kinetic energy, since there is no atomic potential. This increase in the kinetic energy is directly related to the increase in temperature, which is thus a maximum. For a polyatomic gas, a given compression energy produces less heating, since there are intramolecular potentials that generate intramolecular modes and capture part of the injected energy. The opposite situation is obtained for a solid or any dense material. In the limiting case, where this material is at 0 K, all the compression energy essentially goes into the potential energy of the atomic interactions (elastic interactions) with no heating at all. From the thermodynamical point of view, any material may be described very generally by a collection of quantum levels α at energy P E α , which completely defines its partition function Z(β) = α e−β E α where the temperature is T = 1/(k B β). At finite temperature, each quantum level α is occupied with the probability f α = e−β E α /Z. These energy levels E i obviously depend on the atomic interactions and consequently on the adiabatic change in volumes. However, we have to refine the concept of adiabaticity. In thermodynamics, a compression is said to be adiabatic: • when there is no exchange of heat; • when the volume variation is sufficiently slow, in order that the system can be considered to always be at thermal equilibrium — its temperature varies as a function of volume. We call this concept thermodynamical adiabaticity, which should be distinguished from the concept of dynamical adiabaticity, well known in the theory of dynamical systems, which we recall now. If we consider our system as only a large classical dynamical system with external parameters (in our case, the volume v), and if we assume for simplicity that this system is integrable,1 it can be described by a set of action-angle variables {Ii , θi } with a Hamiltonian H ({Ii }, v) which only depends on the actions and the external parameters. Then, any trajectory is a quasiperiodic orbit involving a set of frequencies ωi ({Ii }, v) = ∂ H/∂ Ii . When model parameters vary slowly as a function of time (referring to those periods of oscillations), it is well known that the quasiperiodic orbits evolve slowly at constant actions Ii so that their frequencies ωi ({Ii }, v) change. If Ii are initially chosen at thermal equilibrium, this system may become out of equilibrium after a dynamically adiabatic transformation. Actually, this action conservation law has a simple analog in the quantum case. When the system is initially in the quantum state α with energy E α , it remains in that state when the external parameters vary slowly.2 The consequence is that the occupation number of each quantum state is unchanged during a dynamically adiabatic transformation. Assuming that this quantum system is initially at thermal equilibrium at 1 For example, it is commonly assumed that the small amplitude dynamics of systems close to the ground state can be described by harmonic phonons. 2 Within the standard semiclassical quantization, which consists of quantizing the actions Ii = pi h¯ as integer multiple pi of h¯ , the conservation of actions Ii means the conservation of the quantum numbers α = { pi }.

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temperature T = 1/(k B β), each quantum state α is occupied with probability f α . When the volume varies slowly, the occupation probability f α of each quantum state must remain constant, despite the energy level E α changing into E α0 . But then the system is a priori no longer at thermal equilibrium at some new temperature T 0 , because these probabilities f α should be determined from the new distribution of quantum levels {E α0 } instead of the old distribution. However, if there exists β 0 such that f α = e−β E α /Z = 0 0 e−β E α /Z, this system is again at thermal equilibrium at the new temperature T 0 = 1/(k B β 0 ). This condition requires E α0 T0 = = γ¯ Eα T

(1)

i.e. that the energy changes of all the excitations scale with the same factor γ¯ . In principle, we have dynamical adiabaticity for very fast volume variation (but not too fast, at the atomic scale) when thermalization does not occur locally. We have thermodynamical adiabaticy for medium-fast volume variation. The transformation is isothermal when the volume variation is extremely slow. All the intermediate situations are possible. SL should occur only when volume variation is fast enough to have either dynamical or thermodynamical adiabaticity. We ignore which regime applies at the bubble impact that generates SL, but this question is not essential at the present stage. In any case, no SL should be expected in the isothermal situation when a tremendously large (quasi-static) pressure is applied to the material. 6.2. Spectral rescaling and light emission The variation of a quantum energy level E α with respect to volume v is usually characterized by a Gr¨uneisen coefficient which a priori depends on the quantum excited state α: γα = −

d ln E α d ln v

(2)

which yields, for a volume change from v to v 0 , E0 ln α = − Eα

v0

Z v

γα (ζ ) dζ. ζ

(3)

With the assumption that γα is independent of α, the second member of Eq. (3) is independent of α, and thus condition (1) is fulfilled. Then, dynamical adiabaticity and thermodynamical adiabaticity are strictly equivalent concepts only when the Gr¨uneisen coefficient γα is independent of the considered modes. There are many simple physical models where the energies of the excited states scale perfectly as a function of volume. For example, this property is found to be exact for a perfect monoatomic gas viewed as non-interacting quantum particles in a box. In that case, the quantum energies E α vary proportionally to v −2/3 . Another example concerns periodic monoatomic crystals, where the phonon spectrum depends only of one atomic parameter supposed to depend on volume. Then the

whole phonon spectrum is just rescaled with a constant factor when volume varies. However, there are minor exceptions. Some vibration modes in ice (or water?) are known to exhibit anomalies in their Gr¨uneisen coefficients over a pressure range that is relatively low on the scale of pressures involved in supersonic impacts [34]. This is due to the presence of soft modes associated with structural phase transitions concerning the hydrogen bond network. However, it is worthwhile recalling that the kinetic energy carried by a single molecule at impact is of the order of several eV and is thus quite a lot larger than the hydrogen bond energy (a fraction of an eV). In a first approach, we may thus discard the role of possible phase transitions and the small associated anomalies of some Gr¨uneisen coefficients. In any case, this remark does not concern other liquids without phase transitions and where SL nevertheless also exists. If the Gr¨uneisen coefficients are not mode independent and if the pressure variation occurring at the passage of the pressure pulse is relatively fast, we should only have dynamical adiabaticity. The pressure variation should be sufficiently slow to have thermodynamical adiabaticity in order that thermalization (independently) occurs in each fluid element. In any case, we should expect SL, although the spectral emission could be slightly modified according to the assumption which is made. The supersonic impact of the bubble brings each element of the fluid up to a pressure beyond the Van der Waals pressure inside a sphere with radius RSL extending over a few bubble radii (see Appendix C and Fig. 1). The pressure peak depends on the distance from the bubble center and lasts a time tSL on the order of a few hundred picoseconds. As a consequence, the energies of the quantum states of the fluid elements will also vary. We have seen above that the assumption that the Gr¨uneisen coefficient γα does not depend on the mode α is not a severe approximation that could change our global predictions about SL qualitatively. Then, the order of magnitude of the Gr¨uneisen shift γ¯ can simply be obtained from the frequency shift of the acoustic modes or, equivalently, the sound velocities γ¯ ∼ si /s0 . We have proved above that the sound velocity si at the pressure peak created at supersonic bubble impact is necessarily larger, by one or several orders of magnitude, than the normal sound velocity s0 . Consequently, γ¯ should range from at least 10 to perhaps 100 or more, depending on the strength of the impact. The important fact is that the whole frequency spectrum of this fluid element is roughly multiplied by the same factor, as well as its temperature. According to the standard laws of black-body radiation, the typical frequency of its black-body spectrum is extended by a factor γ¯ (Wien’s law) and its global intensity is increased by γ¯ 4 (Stefan’s law). However, the Planck model for black-body radiation assumes a simple continuous frequency spectrum extending up to infinity, which does not take into account the real frequency spectrum of the material. It is well known that the vibrational spectrum of any insulating material at normal atmospheric pressure is limited up to IR frequencies not exceeding ∼0.5 eV. Above that threshold, there is a large

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Fig. 2. Scheme of intensity versus frequency, showing the change in light emission of the fluid under adiabatic pressure. Before impact, the thermal emission (dark grey area) of the cold fluid is mostly due to the vibrational spectrum in the IR range. There is also a negligible electronic contribution in the UV range. The fluid is transparent in a visible frequency gap between these two contributions. At impact, the pressure and temperature of the fluid elements near the bubble center simultaneously increase during a short instant (a fraction of a nanosecond) and drop down. During that instant, the thermal IR spectrum of the fluid (light grey) is dilated by a large Gr¨uneisen coefficient ∗γ up to UV frequencies, while its intensity is drastically magnified by a factor ∗γ 4 . The emitted light is screened by the surrounding cold fluid, except for the frequencies in the transparency gap. The electronic contribution remains negligible, since practically no electronic excitations were initially present in the cold fluid.

frequency gap between the vibrational IR frequencies and the much higher UV electronic excitation energies. In that gap, the insulating fluid at normal pressure is transparent and cannot emit any black-body radiation. When the fluid element is brought simultaneously to high pressure and high temperature, its IR spectrum is dilated upwards and covers the transparency gap (see the scheme in Fig. 2). Then, the compressed fluid emits in the frequency gap where the fluid at normal pressure is transparent, which allows the emitted light not to be absorbed by the surrounding fluid at normal pressure and to be observable far from the bubble center. Since it is not absolutely granted that the fluid submitted to the pressure pulse is at thermal equilibrium, we note, to be more rigorous, that if the compression is only dynamically adiabatic, the radiation power emitted by each mode α at frequency ωα is nevertheless proportional both to its occupation probability f α and to its frequency at power 4 ωα4 , according to Fermi’s golden rule [37]. If we assume that the dipole of this mode is not substantially changed by the compression, f α should not be changed, while its frequency is multiplied by γα . As a result, the power emitted by this mode is multiplied by γα4 and is shifted at frequency γα ωα . In the case γα = γ¯ independent of α, the global power of the emitted spectrum is multiplied by γ¯ 4 , which is the same result as those given by Stefan’s law for black-body radiation. The spectrum of emission and its intensity could differ substantially from those due to purely thermal radiation when γα is not independent of α, but nevertheless the results

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will be qualitatively the same concerning an intense emission of light. To explain the (almost) frequency independence of the pulse width, various early theories have been proposed, such as temperature-dependent photon absorption of the gas [38], and weak dependence of absorbtion and emission coefficients on wavelength but strong (exponential) dependence on temperature [39] etc. Our explanation of the frequency independence of the pulse width is simply related to the sharp pressure dependence of the Gr¨uneisen parameter γ¯ as a function of pressure due to the fact the potential V(v) should reasonably be assumed to diverge very sharply at v = −h W when the fluid is very close to its Van der Waals volume. When the pressure increases beyond PW , the curve F(−P) becomes very flat and close to its asymptote (see Fig. 4), which implies that the ratio between the sound velocity in the high-pressure region and the corresponding front velocity departs from unity and diverges sharply (take, as an example, atomic interactions described at short distance by a LennardJones potential). Then, although the space-time variation front velocities and the associated pressures may be relatively smooth (at the scale of the experiment), the Gr¨uneisen coefficient γ¯ that simultaneously controls the change of temperature and the emission spectrum of the fluid element falls abruptly to unity at the crossover pressure PW . Consequently, the light emission of this fluid element switches off abruptly almost simultaneously for all observable frequencies as soon its pressure decays below PW . However, this result is only valid for observable frequencies that are not too large (and belonging to the transparency gap of the surrounding normal fluid). Then, when the radius of the sphere of the compacted fluid expands beyond RSL , the highly compacted fluid starts to relax from the Van der Waals volume and the light emission progressively switches off shell by shell, but nevertheless the light emission decays quasi-identically for all frequencies. However, our model may be refined, because light emission does not really switch off instantaneously. The consequence is that the lower frequencies should exhibit a slightly prolonged tail, as mentioned in certain experiments [17]. In our model, we assume that the characteristic time of the pressure pulse tSL is sufficiently short so that each fluid element cannot emit enough thermal radiation (the SL light flash) to cool down by itself. Its temperature is essentially driven by its pressure, which depends on time and distance from the bubble center. As a result, the emitted SL light is a superposition of the many different emission spectra of the fluid elements (which depends on their pressure profile versus time) in the rather welldefined radiance sphere with radius RSL . This energy for light emission is borrowed from part of the compression energy, a part being used for shifting to higher energy the occupied quantum excitations of the initial fluid at normal pressure and temperature. 7. Concluding remarks Here we have not proposed a complete quantitative theory of SL but suggested the basic principles that should be taken into account for a new approach to this puzzling problem.

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Our basic argument is that we proved first that the Gr¨uneisen shift involved in supersonic impacts must be very large. Although the models that we investigated for supporting our theory are incompletely solved (especially concerning their numerical analysis), or simplified (homogeneous fluid), they should easily be extendable and should yield similar results. We claim, more generally, that light flash emission is an ubiquitous phenomenon for shock waves when they are sufficiently intense (like those that are generated by supersonic or close-to-supersonic impacts). The important parameter that controls the light emission is the Gr¨uneisen parameter which is determined by the instantaneous pressure. The mechanism of light production is that the excitation spectrum of a fluid element adiabatically submitted to a very intense and brief pressure pulse is globally shifted to higher frequencies by a large factor (the Gr¨uneisen coefficient), while its temperature is simultaneously increased by the same factor. Then, a brief and hot thermal light flash is emitted simultaneously. Concerning bubble SL, unlike other existing models in the literature which place the location of light emission essentially inside the bubble, we propose that the light is mostly emitted from the liquid surrounding the collapsing bubble. We discarded the effect of the formal divergence of pressure at the bubble center, which could generate a plasma for several reasons. The first reason is that it requires almost perfect bubble sphericity, which is unlikely to ever be realized in experiment. The second reason is that, even assuming perfect enough sphericity, the focusing time of this pressure peak calculated by some authors [40] ranges on the scale of just picoseconds, which is shorter by at least two orders of magnitude than the width of the observed light flash of SL. A third reason is that our investigation in Appendix C, suggests the absence of pressure divergency near bubble center because of the tremendously high nonlinearities which tend to uniformize the pressure when the fluid approaches its Van der Waals volume. SL emitted by collapsing bubbles is the superposition of the thermal emission of many fluid elements with a spectrum and intensities that depend on their distance to the bubble center and on time. Thus, SL is not simple black-body radiation of hot matter at a well-defined temperature. We found that, schematically, SL is essentially emitted by a radiance sphere with a radius of a few minimum bubble radii, where the fluids are highly compacted close to their minimum Van der Waals volume during a time tSL which we estimated to be compatible with the observations. However, many questions remain to debate, but nevertheless we have a clue for interpreting the most puzzling features unexplained to date. Why is water the best fluid for producing intense SL? We think that this question is essentially a problem of bubble hydrodynamics. Since SL is generated by supersonic impacts, it requires that the collapsing bubble velocity reaches at least close to supersonic velocities (referred to the surrounding fluid). Then, an essential condition is the absence of energy dissipation which could slow down the bubble collapse and generate only a soft impact. For that purpose, it is more favorable that the collapsing bubble preserves its sphericity as well as possible

until the end of the collapse and does not develop any aspherical instabilities (we noted in Appendix A that dissipation is smaller in that case). Our models were assumed to be Hamiltonian without energy dissipation, which corresponds to the ideal situation for SL. However, in real situations, there are many possible sources of energy dissipation that could reduce the sharpness of the bubble impact and consequently suppress SL. For example, it should be avoided that the bubble contains gases or liquid vapor that are too soluble under pressure.3 Many parameters concerning the surrounding fluid and the gas inside the bubble are involved in producing sharp and stable bubble collapses generating supersonic impacts and SL. At the present stage, these parameters are not controllable in detail. In any case, up to now, experiments have shown that cold degassed water with rare gas bubbles is the most efficient system for producing bright SL. According to our theory, the brightness of SL is controlled by the maximum Gr¨uneisen parameter γ¯ which could be generated at the bubble impact. It depends sharply on the impact velocity but, moreover, there is a threshold impact velocity below which there is practically no SL. Beyond this threshold but close to it, the coefficient γ¯ is still not very large and close to unity. The emission spectrum of the fluid is insufficiently dilated to overlap completely the transparency gap of the normal fluid. Moreover, the temperature is relatively low. This is the regime of dim SL. The emitted light flash is weakly intense, with a frequency at maximum intensity shifted to the red. This is also a regime that is especially sensitive to isotopic effect, which may sharply modify the emission spectrum of the normal fluid. For example, the substitution of light water (H2 O) by heavy water (D2 O) reduces the cut-off IR frequency by a factor of √ 2, because the highest frequency modes, which are proton or deuteron vibrations, are proportional to the inverse square roots of their masses. The transparency gap is also extended on the IR side, while the UV side does not change much. The result is that, for the same bubbles and the same impact velocity (neglecting the minor effect of different fluid densities), the whole SL spectrum is roughly shifted toward the red in the same proportion and with a lower intensity, as observed in real experiments (see [7]). For larger impact velocities, γ¯ becomes larger, and the emission spectrum of the compressed fluid completely overlaps the transparency gap of the normal fluid. The consequence is that SL is brighter, with a maximum intensity at a frequency shifted close to the upper UV edge of the absorption gap. Then, in equivalent conditions for both light and heavy water, there is no important change in the SL spectrum, but nevertheless with an important diminution of SL intensity. In contrast, the same experiments [7] show only a small change due to isotopic substitution concerning the rare gas inside the bubble, which confirms that their contribution is minor for SL according to our theory. 3 We may note that there is systematically some energy dissipation of the bubble oscillator energy by the propagating shock wave at the bubble impact, but this does not affect the precursive light flash emission, which also yields a minor contribution to the dissipation of the bubble kinetic energy.

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If the pressure peak of the shock wave could become sufficiently large to make the fluid inside (or outside) the bubble metallic, then electronic excitations could be involved for generating SL (plasma?). Unfortunately, the experiments performed up to today have shown that bubble sphericity becomes unstable when the amplitude of the driving pressure increases too much [31]. As a consequence, the impact at bubble collapse is more complex and most likely not sharp enough for generating SL. Acknowledgements Both of us are indebted to J. Teixeira from Laboratoire Lon Brillouin (LLB), CEA Saclay, France for useful discussions about the amazing properties of water. One of the authors (BD) would like to thank LLB for kind hospitality during his visits there. He also thanks the DST (India) for financial assistance through a research grant. S.A. acknowledges T. Dauxois and all the organizers of this 60th birthday symposium for the honor they gave him. Appendix A. Rayleigh–Plesset model We show that a spherical bubble of gas in an incompressible fluid submitted to an external pressure is nothing but an anharmonic oscillator driven by an external force. This model yields that the bubble radius reaches its maximum radial velocity when the bubble pressure becomes equal to the applied pressure and is still low. This model is valid up to this point, but cannot properly describe the following impact of the fluid on the hard core gas. A.1. Assumptions We make the following assumptions: • The flow is radial and the bubble is spherical. The dynamics of the bubble are characterized by the time dependence of its radius R(t). • The fluid outside the bubble is incompressible and its pressure far from the bubble center is Pa (t), the external pressure. • The gas inside the bubble is compressible. It is assumed to remain in adiabatic equilibrium when the radius of the bubble varies. As a result, there is no energy dissipation. • The energy of the gas inside the bubble, Φ(V ), is supposed to be a certain function of its volume V = (4π/3)R 3 , only. The pressure inside the bubble is Pi = −dΦ/dV . The bubble is in static equilibrium with the external pressure Pa when Φ(V ) + Pa V is a minimum, which implies that Pi = Pa . We may assume that the energy Φ(V ) is a function of the volume, assuming that the gas compresses adiabatically with a Van der Waals radius h. Φ(V ) = K /(V − h)γ −1 . The surface tension energy σ S = 4π σ R 2 = (36π )1/3 σ V 2/3 can be added to the bulk energy terms. Any more accurate potential can be chosen. The physically important property is that potential Φ(V ) diverges to +∞ for small V or V → h (Van der Waals volume).

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A.2. Rayleigh–Plesset equation for the bubble dynamics The field of radial velocity v(r ) = v(r ).r/r in the incompressible fluid outside the bubble fulfills the condition ˙ ∇.v(r ) = 0, which that implies v(r ) = k/r 2 . Since v(R) = R, it comes out as R˙ R 2 . (A.1) r2 The kinetic energy of the fluid outside the bubble is then Z ∞ Z ∞ dr 1 2 ρv (r )4πr 2 dr = 2πρ R˙ 2 R 4 EK = 2 2 R r R

v(r ) =

= 2πρ R˙ 2 R 3 .

(A.2)

We neglect the kinetic energy of the gas inside the bubble. For a bubble with radius R = 0.5 × 10−6 m and velocity R˙ = 6 km/s (which is four times the sound velocity 1.5 km/s in water), the kinetic energy of the fluid is E K = 2π × 103 × 62 × 106 × (0.5)3 × 10−18 J ≈ 3 × 10−8 J ≈ 2 × 1011 eV. Note that the kinetic energy of a single water molecule moving at 6 km/s is ≈3.4 eV. The viscosity force F = η∇ 2 v + (η + λ)∇(∇.v)

(A.3)

is zero, since ∇.v = 0, as the fluid outside the bubble is incompressible, and ∇ 2 v = 0, as the velocity field is radial. There might exist viscosity forces at highest order, but we shall neglect it.4 Thus, there is no energy dissipation and the total energy is conserved (assuming that the external pressure Pa is constant). We have   4π 3 4π 3 2 3 ˙ H = 2πρ R R + Φ R + Pa . R . (A.4) 3 3 With the new variable  5/6 4π u(t) = V 5/6 = R 5/2 3

(A.5)

the total energy defines the Hamiltonian 1 M u˙ 2 + W (u; Pa ) 2 of a particle with mass   12 3 2/3 M= ρ 25 4π H=

(A.6)

(A.7)

in the potential W (u; Pa ) = Φ(u 6/5 ) + Pa .u 6/5 .

(A.8)

This potential behaves as Pa .u 6/5 for large u, while it becomes very steep and diverges for u → u W , where u W = 4 If the velocity field is non-radial, it dissipates energy into heat in the water, which necessarily reduces the kinetic energy of the water at impact.

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Consequently, when u(t) reaches the value u c (Pa ) that corresponds to the minimum versus i of the potential W (u; Pa ), we have W 0 (u c (Pa ); Pa ) = u¨ = 0, which implies that P(R) = Pa . Next, the acceleration changes sign, so we conclude that the pressure in the bubble is equal to the external pressure when the velocity u˙ is negative with maximum modulus. A.4. Bubble dynamics

Fig. 3. Bubble potential W (u; Pa ) versus u at different pressure Pa . During the pressure variation that generates SL, the minimum u c (Pa ) of the curve varies by a factor 105 from the smallest to the largest bubble size.

h 5/6 corresponds to the hard core (Van der Waals) volume of the gas inside the bubble (see Fig. 3). The minimum at u = u c (Pa ) of potential W (u; Pa ) versus u yields the equilibrium radius of the bubble at this pressure. We have W 0 (u c (Pa ); Pa ) = 0. At small pressure, the minimum of this potential is obtained for large u c (Pa ), while for large Pa the minimum is obtained for u = u c (Pa ) → u W close to the hard core radius. The equation for the bubble dynamics is simply described by the equation M u¨ + W 0 (u; Pa ) = 0.

(A.9)

A.3. Pressure field The Navier–Stokes equations are obtained from the Newton law, which equates the force (−∇ P(r) + F(r) + F(r)) dr on a small volume element dr to the product of its mass and its acceleration ρ (˙v + ∇v.v). F(r) is the friction force and F(r) is the force generated by an external field. In our case, there is no external field and the friction force is also zero for a radial bubble oscillation with an incompressible fluid. Then, using the spherical symmetry, we get    ˙ 2  ∂v d( R R ) 1 ∂P R˙ 2 R 4 = ρ v˙ + v =ρ − − 2 . ∂r ∂r dt r 2 r5 This equation can be integrated, which yields   ¨ 2 1 R˙ 2 R 4 R R + 2R R˙ 2 − + Pa . P(r, t) = ρ r 2 r4

(A.10)

The pressure at the bubble surface is, using Eqs. (A.9), (A.5) and (A.7),   3 2ρ d2 R 5/2 P(R, t) = ρ R¨ R + R˙ 2 + Pa = Pa + √ 2 5 R dt 2  2/3 u¨ 2ρ 3 = Pa + 5 4π u 1/5 = Pa −

25 W 0 (u; Pa ) . 12 u 1/5

(A.11)

When the external pressure Pa (t) varies as a function of time, this anharmonic oscillator (A.6) is parametrically driven. Potential W (u; Pa ) also depends on time, so that there is no energy conservation. Then Eq. (A.9) becomes non-integrable when Pa (t) is time dependent, and consequently this undamped model should exhibit chaotic trajectories in addition to periodic and quasi-periodic trajectories corresponding to KAM tori. A small damping should favor the existence of attractors. It is observed experimentally that, in good conditions for SL, the radial motion of the bubble is periodic at the frequency of the driving ultrasound. Although numerical simulation of Eq. (A.9) should be performed for a more complete analysis, we can already make some qualitative remarks concerning the possibility of generating a sharp impact. We initially consider the bubble at its maximum size when the external pressure Pa (t) is a minimum. If Pa (t) increases very slowly from its minimum value Pm to its maximum value PM , the motion of the bubble is adiabatic. Its radius is practically the bubble equilibrium position u c (Pa (t)) at the corresponding pressure. Then, the pressure remains practically uniform everywhere and equal to Pa (t). No impact would be generated. Actually, the characteristic frequency ω B (Pa ) of the bubble oscillator goes to zero at small pressure Pa and large radius u c (Pa ), and in contrast diverges at large pressure Pa and small bubble radius. If the amplitude of the driving pressure becomes too large, the bubble motion cannot be adiabatic anymore. Indeed, if the maximum radius of the bubble is too large, when the bubble is at this maximum radius the variation in the time-periodic driving pressure Pa (t) becomes too fast, so that the response of the bubble oscillator u(t) cannot follow the adiabatic solution at that bubble size region. It will be delayed at u(t) > u c (Pa (t)) and then the bubble radius will be submitted to a (negative) acceleration, W 0 (u; Pa )/M. It is then intuitive that, from that point, the following conditions will favor the sharpest bubble impacts. • The pressure Pa (t) increases very quickly up to its maximum value PM . Then, the acceleration of the bubble oscillator becomes a maximum, and a maximum amount of potential energy can be transformed into kinetic energy at u = u c (PM ) (see Fig. 3). • The radius u c (PM ) of the bubble at equilibrium at maximum pressure PM should be close to its Van der Waals radius in order that potential W (u, PM ) becomes very steep for u < u c (PM ) and behaves like a hard wall. This condition requires that PM is large enough.

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• After impact, the decay of the pressure from PM to Pm should be as smooth as possible in order that the bubble could reach its maximum radius almost adiabatically. It turns out that the ideal shape of Pa (t) in a given period and amplitude is more like a sawtooth periodic function of time. However, a sine time-periodic function is still efficient if a good compromise is chosen for its frequency. First, it is clear that, for generating bubble impact, the maximum pressure PM should be large enough. Second, the frequency of the driving pressure should have a range in the frequency interval visited by the bubble frequency ω B (P) in order to be non-adiabatic enough during the bubble collapse and adiabatic enough when it expands. Note that, a priori, the maximum velocity for u does not correspond to those of the radial bubble velocity R˙ because u ∝ R 5/2 . However, since R ≈ Rm is close to its minimum radius 5/2 3/2 Rm , we may linearly expand u ∝ Rm + 25 Rm (R − Rm ) + · · ·, which yields that u¨ = 0 and R¨ = 0 occurs practically at the same time ti . In real experiments on SL, the maximum radial velocity R˙ has been observed close to the minimum bubble radius Rm to be about 6 km/s, which is four times larger than the sound velocity in water at rest [30]. At that point, the pressure field in the fluid is, from Eq. (A.10) (for r > R),   R3 R R˙ 2 1 − 3 + Pa . P(r, ti ) = ρ 2r r This pressure field decays as a function of r but, since ρ R˙ 2 ≈ 103 × 62 × 106 = 36 GPa, it reaches the order of magnitude of 100 kB (atmospheric pressure), where water is not incompressible anymore. Beyond this point at time ti , the RP model does not hold, because the pressure in the fluid close to the bubble will become tremendously large, so it cannot be assumed to be incompressible. The impact generates a pressure pulse in the fluid propagating radially from the bubble center (which also dissipates energy away from the bubble center). We now model the system after time ti by an impact model where the fluid is compressible. It is initially at a uniform pressure, with a field of initial velocities roughly corresponding to those generated by the bubble collapse. We first consider this model in one dimension, which is simpler and already yields interesting information.

149

Since, at this time, the fluid pressure is still relatively small compared with those involved in the pressure pulse, we assume a uniform pressure in the fluid. We simplify the model by assuming that the impact occurs from water on water. This is equivalent to saying that the highly compressed gas inside the bubble has the same density and the same compressibility as water when impact occurs. We first consider this model in one dimension, which is simpler and already yields essential information. We show that we can bound from below the sound velocity in the region submitted to the pressure pulse from only the knowledge of the impact velocity. B.1. Dynamical equation We consider a fluid moving in a rigid one-dimensional pipe with a strictly constant section (assumed to be the unit surface). The fluid is initially at uniform pressure P0 . When the fluid is moving, the element of fluid initially at x is at x + (x, t) at time t. If ρ is the mass of a unit volume of fluid at the initial pressure P0 , the kinetic energy of the fluid is then Z 1 2 ρ ˙ dx. K = (B.1) 2 V(v) is the potential energy of an element of fluid as a function of its volume 1 + v (supposed to be the unit volume) in the initial state (v = 0) at pressure P0 (we have V 0 = −P0 ). We assume that V(v) is defined for v > −h W and becomes infinite for v → −h W , where 1 − h W is the minimum volume (Van der Waals hard core). Moreover, we assume that V(v) is a convex function of v and that its second derivative V 00 (v) (hardness) is a monotonically decreasing function of v. The relative change in volume of a small fluid element initially at x is v(x, t) = ∇(x, t) = ∂∂x . Thus, the potential energy of the fluid is Z   ∂ W = V dx. (B.2) ∂x The action of this Hamiltonian system is (dissipation is neglected)   Z Z  1 2 ∂ ρ ˙ − V dtdx. (B.3) A= K −W = 2 ∂x

Appendix B. Impact in one dimension

Then, with vanishing variation,

Our purpose is to obtain some order of magnitude for important physical parameters involved in the impact without accurate calculations, from which we can argue that there is substantial light emission. Actually, we do not estimate the amplitude of the pressure pulse that depends explicitly on the equation of state of the fluid, which is unknown in situations at tremendous pressure. After the bubble radius reaches maximum velocity at time ti , which we now consider as the origin of time, we model the impact that occurs immediately afterwards by a different model where the fluid is compressible.

δA = 0 yields the nonlinear dynamical equation   ∂. 0 ∂ ρ ¨ − V = 0. ∂x ∂x

(B.4)

It is convenient to consider the pressure P(x, t) = −V 0 ( ∂∂x ) as variable instead of (x, t) and to define the inverse function F(ξ ) of V 0 which fulfills ξ = V 0 (F(ξ )).

(B.5)

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Fig. 5. Scheme of the evolution of a Hugoniot solution of Eq. (B.6). An arbitrary initial pulse-shaped pressure profile P(x, t) versus x propagates to the right side. Each isobar propagates toward the right at a velocity s(P) Eq. (B.8) which is a monotonically increasing function of pressure P. The front side becomes steeper, while the backside flattens. Since P(x, t) must remain univalued, a pressure discontinuity must appear within a finite time. Actually, the solution should leave the Hugoniot class, because the pressure profile is not a univalued function anymore. Its further evolution is unknown in general. Fig. 4. Sketch of the variation of F(−P) versus −P. F(−P) and its derivative F 0 (−P) are monotonically increasing functions of −P with an asymptote at −h W infinite pressure P. The slopes at −P0 and −Pi are proportional to the inverse square of the sound velocities s02 and si2 , respectively. The slope of the cord between −P0 and −Pi is proportional to the inverse square velocity ci2 of a front between two regions at the corresponding pressures. PW is the crossover pressure between the linear regime and the Van der Waals regime. For Pi  PW , this figure shows that s0  ci  si .

−F(−P) is the static contraction rate of the fluid at pressure −P referred to the fluid at pressure P0 (a unit volume of fluid initially at pressure P0 becomes 1 + F(−P) at pressure P). Because of the assumptions made on potential V, F(ξ ) is a convex monotonically increasing function of ξ with an asymptote F(−∞) = −h W < 0, where 1−h W is the minimum Van der Waals volume of an initial unit volume. A typical variation of F(ξ ) is shown in Fig. 4. Then ∂∂x = F(−P). We note that F(−P0 ) = 0, since 0 V = −P0 . Eq. (B.4) becomes ρ

∂ 2 F(−P) ∂ 2 P + =0 ∂t 2 ∂x2

(B.6)

where function F(ξ ) is some monotonically increasing function related to the equation of state of the fluid. It has an asymptote at −∞: F(−∞) = −h W , as shown in Fig. 4. Eq. (B.6) has trivial solutions with arbitrary uniform pressure P(x, t) = Pa . Small pressure fluctuations Pa + δ P(x, t) fulfill the linearized equation ∂ 2δ P ∂ 2δ P =0 −ρ F (−Pa ) 2 + ∂t ∂x2 0

(B.7)

which has general solutions with the form δ P(x, t) = f (x − s(Pa )t) + g(x + s(Pa )t), where f and g are arbitrary functions and the sound velocity is defined as s(P) = √

1 . ρ F 0 (−P)

(B.8)

It is convenient to define a cross-over pressure PW that we call the Van der Waals pressure, at which the sound velocity starts to diverge. This situation is obtained when F(−P) bends to its asymptote. Strong nonlinear effects will be manifested when the pressure goes beyond this Van der Waals pressure PW while, for smaller pressures, the harmonic features remains

qualitatively good. We may choose (see Fig. 4) PW − P0 = h W /F 0 (−P0 ) = ρh W s02

(B.9)

where s0 = s(P0 ). For water, where s0 ≈ 1.5 × m/s, ρ = 103 kg/m3 and choosing h W = 0.2, we obtain PW − P0 ∼ 0.45 GPa ∼ 4.5 kbar. 103

B.2. Rankine characteristics For this equation in 1D, it is convenient to consider, instead of the function P(x, t) of space x and time t, the motion of the isobars defined as the spatial location x(P, t) at time t of the point where the pressure is equal to P. x(P, t) is defined by the implicit equation P(x(P, t), t) = P. Then, Eq. (B.6) becomes, after some tedious calculations,   ∂x ∂x 2 ∂2x ρ F 00 (−P) − ∂ P ∂t ∂ P2  2 2 ∂x ∂ x ∂ x ∂ x ∂2x + ρ F 0 (−P) − 2 ∂t ∂t ∂ P ∂ P∂t ∂ P2  2 2 ! ∂x ∂ x + = 0. (B.10) ∂P ∂t 2 This equation has special travelling solutions that are obtained from the ansatz ∂x (P, t) = σ s(P) (B.11) ∂t where σ = ±1 determines the direction of propagation of the pressure pulse and s(P) is the sound velocity (B.8) in the fluid at uniform pressure P. Then, Eq. (B.11) yields x(P, t) = σ s(P)t + x(P, 0), which are special solutions of Eq. (B.10). These solutions are called Hugoniot solutions. Consequently, if we consider a travelling pressure profile which is initially P(x, 0) (see Fig. 5), each isobar at pressure P that is located at x(P, t) moves uniformly either to the left side or the right right at a velocity s(P), which is only a function of P. Since V(v) has been assumed to be a convex function of v with a Van der Waals hard core at −h W , V 0 (v) is monotonically increasing as well as its inverse function F(ξ ), which fulfils F(−∞) = −h W . Consequently, the sound

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of Eq. (B.6) which is discontinuous with respect to x at xd (t). The limit pressures at x → xd (0) are P+ (t) for x > xd (0) and P− (t) for x < xd (0). Vanishing (only) the components of the Dirac derivatives in Eq. (B.6) yields the instantaneous front velocity

Fig. 6. Scheme of a one-dimensional impact between two fluids moving in opposite directions initially at pressure P0 and velocity Vi . A region of high pressure Pi grows from the impact line. The two fronts between the lowpressure and high-pressure regions move at velocity ci in opposite directions. The sound velocities in those regions are s0 and si , functions of the pressure P0 and Pi , respectively.

velocity s(P) monotonically increases as a function of the pressure and diverges at infinite pressure. Since an isobar at larger pressure moves faster, it is clear that any travelling pressure profile has to develop a singularity within a finite time; see Fig. 5. Since for a given x only one pressure can be defined, this singularity should appear as a pressure discontinuity.

Actually, Eq. (B.6) has exact discontinuous step solutions (considering the time and space derivatives as generalized functions) with the form (B.12)

where xd (t) = ±c(P− , P+ )t + xd (0). Y (x) is the Heaviside function (Y (x) = 0 for x < 0 and Y (x) = 1 for x ≥ 0) and the velocity is s P+ − P− c(P− ; P+ ) = . (B.13) −ρ(F(−P+ ) − F(−P− )) It corresponds to a travelling pressure discontinuity at xd (t) = ±c(P+ ; P− )t + xd (0) between two regions with constant pressure P− for x < xd (t) and P+ for x > xd (t). Since function F(ξ ) is monotonically increasing (see Fig. 4), we have the important inequality for P+ < P− : s(P+ ) < c(P+ ; P− ) < s(P− ).

P+ − P− = c2 (P+ (t); P− (t)). (B.15) ρ(F(−P+ ) − F(−P− ))

The velocity of the discontinuity is determined by the pressures P+ (t) and P− (t) on both sides, which are also time dependent. Consequently, in general, the pressures at both sides of the discontinuity are not constant, so its velocity is not uniform. B.4. Impact solution We choose initial conditions where the fluid is initially at uniform pressure P0 and moves at velocity −Vi for x > 0 and the opposite velocity Vi for x < 0 (see Fig. 6). Vi is the impact velocity. Thus, we have (x, 0) = 0 and ˙ (x, 0) = −Vi sign(x). An exact continuous solution of Eq. (B.4) is (x, t) = F(−Pi )((ci t + x)Y (ci t + x)

B.3. Propagation of a front

Pd (x, t) = P− + (P+ − P− )Y (x − xd (t))

x˙d2 (t) =

(B.14)

Assuming that P+ < P− in order to fix the ideas, this solution is stable when x˙d (t) = +c(P+ ; P− ) and unstable when x˙d (t) = +c(P+ ; P− ). In the stable case, if we split this discontinuity into two nearby discontinuities, according to the scheme shown Fig. 5, the second discontinuity travels faster than the first one and merges with it within a finite time. In the unstable case, the distance between these discontinuities always increases. Actually, the width of the discontinuity front spontaneously increases under perturbations. We get the important conclusion that the front is stable only when it propagates from a high-pressure region to a lower-pressure region. A front propagating backwards is unstable and spontaneously thickens. More generally, if P(x, 0) is an initial pressure profile that is discontinuous at time 0 at xd (0), it generates a solution P(x, t)

− (ci t − x)Y (ci t − x) − x) = −(−x, t)

(B.16)

where Pi is the pressure generated at the impact to be determined and ci = c(Pi , P0 ) is the front velocity between regions at pressures Pi and P0 , respectively.5 We have (x, 0) = 0 and ˙ (x, t) = F(−Pi )ci (Y (ci t + x) − Y (ci t − x)). which fulfills the required initial condition at t = 0 when Vi = −F(−Pi )ci .

(B.17)

Since we have ∂∂x = F(−Pi ) (Y (ci t + x) + Y (ci t − x) − 1), it comes out that ∂∂x = F(−Pi ) and P(x, t) = Pi for −ci t < x < ci t and ∂∂x = 0 and P(x, t) = 0 elsewhere. We have P(x, t) = P0 + (Pi − P0 )(Y (ci t + x) + Y (ci t − x) − 1) (B.18) which yields the pressure evolution versus time and space for a symmetric impact of the fluid at the origin at time 0 and at velocity Vi . Since ci = c(Pi , P0 ) is also a function of Pi , Eq. (B.17) determines the amplitude of the pressure Pi generated by the impact as a function of the impact velocity Vi . Actually, Eq. (B.17) is nothing but a kinematic equation that expresses the mass conservation of the fluid. The volume of a large mass of fluid defined by −L < x < L, with L large, shrinks per unit time by 2Vi . Considering that a unit volume element initially at pressure P0 shrinks its volume to 1 + F(−Pi ) when its pressure becomes Pi , this volume also shrinks per unit time by −2ci F(−Pi ), where ci is the front velocity. These two results have to be equal, which yields Eq. (B.17). 5 However, let us note that the apparent location of the front is not x (t) = F ±ci t but x F +(x F , t) = ±(ci −Vi )t. The front velocity ci −Vi > 0 appearing in a real experiment would be smaller, because it moves with respect to the fluid moving in the opposite direction at velocity Vi .

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For a weak impact, that is, when Vi  s0 (much smaller than the sound velocity), Pi is also small and we get at lowest order ci ≈ s0 and Pi − P0 ≈ V 00 (0) Vs0i = ρs0 Vi . When the impact is supersonic (Vi > s0 ), Pi becomes tremendously large, as we show now. As represented in Fig. 4, function F(ξ ) is asymptotic to −h W , which represents the maximum Van der Waals contraction of the fluid. Consequently, 0 < −F(−Pi ) < h W < 1. If the fluid is barely compressible, that is, the fluid compactness is close to the Van der Waals volume, then F(−Pi ) is small. Then Eq. (B.17) implies that the front velocity ci is bounded as Vi / h W  ci . In any case, even assuming the fluid volume could shrink to zero, which is unrealistic (h W = 1), the front velocity is always larger than the impact velocity Vi < ci . In the limit of large pressure Pi , −F(−Pi ) ≈ h W , this inequality √ becomes an equality. We have ci = Vi / h W . Since ci = (Pi − P0 )/(−ρ F(−Pi )), we get Pi − P0 = h W ρci2 = ρVi2 / h W . We note for consistency that we should have Pi  PW , where PW is the Van der Waals pressure (B.9). This situation is fulfilled when Vi  h W s0 . Thus, for barely compressible materials where h W is small, we already get supersonic fronts for subsonic impacts. For understanding SL, it is essential to get information about the sound velocity si in the compressed region at pressure Pi defined by Eq. (B.8). We have the following sequence of inequalities (see Fig. 4), which is always strictly fulfilled: Vi Vi < ∼< ci < si . hW

(B.19)

When the pressure Pi goes beyond the Van der Waals pressure, it is clear (see Fig. 4) that ci  si . The sound velocity in the high-pressure region si may become larger than ci by one (or several) orders of magnitude. For highly supersonic impact velocity Vi  s0 , the sound velocity in the high-pressure region should be shifted up to many more orders of magnitude above normal sound velocity. For example, if we assume an impact velocity Vi = 2 s0 only twice the sound velocity s0 and a maximum Van der Waals contraction h W = 0.2 (−20%), then the front velocity ci will be strictly larger than 10 s0 , that is, one order of magnitude above the normal sound velocity. The sound velocity in the compressed region is strictly larger than this front velocity. Since the compression at impact brings the fluid very close to its minimum volume (F(−Pi ) ≈ −h W ), very close to the asymptote, the slope of F(−Pi ) at Pi is almost horizontal (see Fig. 4), which suggests that si could be larger than ci by one (or several) orders of magnitude. Thus, assuming si ∼ 100s0 is not physically unreasonable. The impact pressure Pi is given by Eq. (B.13). For highly supersonic impacts, −F(−Pi ) ∼ h W which yields ci = √ (Pi − P0 )/(ρh W ). Since h W ci ∼ Vi , we obtain Pi ∼

ρ 2 V hW i

(B.20)

for highly nonlinear impacts. For example, for a supersonic impact with water at velocity Vi = 2 s0 ∼ 3 × 103 m/s, h w = 0.2 and ρ = 103 kg/m3 , we could obtain Pi ∼ 0.45 × 1011 Pa ∼ 0.45 Mbar (and then Pi  PW ∼ 4.5 kbar).

Fig. 7. One-dimensional scheme showing an incoming front at pressure Pi and velocity ci (top) crossing an interface (dashed line) between two different media (top). An intermediate pressure region at pressure Pm appears delimited by two fronts that propagate in opposite directions from the interface at velocity cr for the reflected front and ct for the the transmitted front. The fluid contraction rate is h i = −F− (Pi ), h r = −F− (−Pm ) for the regions in the left medium at pressure Pi and Pm , respectively. It is h t = −F+ (−Pm ) in the region in the left medium at pressure Pm . The middle scheme corresponds to a situation where the right medium is less compressible (‘harder’) than the left medium (then Pm > Pi ). The bottom scheme corresponds to the opposite situation where the right medium is softer (Pm < Pi ). Then the front moving backwards is unstable and flattens.

B.5. Inhomogeneous media When the medium is not uniformly homogeneous, the propagation of pressure fronts may be sharply affected. It is pedagogical to consider the simple example of inhomogeneity where the nonlinear equation Eq. (B.6) is modified for describing front propagation through the interface at x = 0 (see Fig. 7) between two different media at the left and at the right. This situation may model the behavior of a pressure front penetrating inside the compacted rare gas bubble, but should also give some intuition concerning the propagation of a spherical 3D front, which we will show to be described by an inhomogeneous 1D model. An exact solution can be found analytically for that model which describes the scattering of a single front (with the form (B.12) at negative x) at the interface. This front splits into two fronts propagating in opposite directions in the different media, as shown Fig. 7. One considers two different contraction rates and densities F− (−P) and ρ− for x < 0 in the left medium and F+ (−P) and ρ+ for x < 0 in the right medium.

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Since the global rate of contraction is determined by the fluid velocity at infinity, which is unchanged because of the scattering at the interface, it is invariant before and after scattering, which directly yields the equation h i ci = (h r − h i )cr + h t ct where h i = −F− (−Pi ), h r = −F− (−Pi ) and h t = −F+ (−P p m ) and the front velocities arepfrom Eq. (B.13), ci = (Pi − P0 )/(−ρ− F− (−Pi )), cpr = (Pm − Pi )/(−ρ− (F− (−Pm )) − F− (−Pi )) and ct = (Pm − P0 )/(−ρ+ F+ (−Pm )). Actually, this equation determines the pressure Pm as a function of Pi , since the parameters are only functions of the involved √ pressures Pi , Pm and P0 . Since we have h i ci = √1ρ− (Pi − P0 )h i and, with σm = r − h i )cr √ = sign(Pm − Pi ), we have (h√ σm √1ρ− (Pm − Pi )(h r − h i ) and h t ct = √1ρ+ (Pm − P0 )h t , then this equation becomes p p (Pi − P0 )h i − σm (Pm − Pi )(h r − h i ) r p ρ− = (Pm − P0 )h t . (B.21) ρ+ If one assume two identical media, it comes out readily that Pm = Pi is the expected solution. By definition, we say that the left fluid is less compressible than the right fluid at pressure P when −ρ+ F+ (−P) < −ρ− F− (−P)

(B.22)

is fulfilled. We assume for simplicity that this inequality is fulfilled for the whole range of pressures P in the interval [Pi , Pm ]. Then, it can be proved readily that an overpressure (Pm > Pi and σm = +1) is generated at the interface when the pressure front comes from the left fluid, which is more compressible than the right fluid. Then the front propagation are both stable. In the opposite case, when the left fluid is less compressible than that on the right side, we have a depression (Pm < Pi and σm = −1) at the interface. However, in that case, the front propagating backwards to the left side is unstable and should smooth out (see Fig. 5), while that propagating forwards to the right side remains stable. For example, if we assume that the right medium is totally incompressible (h t = 0), we obtain Pm − P0 = (Pi − P0 ) hi hr h r −h i > Pi − P0 and cr = h r −h i ci > ci , since h i < h r < h −W . If, moreover, we assume that the pressure front in the left medium involves a contraction rate h i close to its maximum i Van der Waals contraction h +,W , then hr h−h must be very large, i which implies that the relative increase in pressure Pm and also in the front velocity cr can be very large. In conclusion, when a pressure front propagates through an interface towards a less compressible medium, it develops a region with a larger pressure at the interface. This region is delimited by two fronts that propagate faster than the initial front in both directions. When the medium is more compressible, it is the opposite; the pressure is smaller at the interface. The front penetrates the soft medium with a smaller amplitude and propagates slower. On the other side, the backward front smooths out.

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Appendix C. Spherical bubble impact For simplicity, we again study the simpler situation of an impact of water on water, which mimics the impact of the fluid onto the spherical hard core bubble. We assume, as in 1D, that the pressure P0 is initially uniform in the fluid but that the fluid outside the sphere with radius r > R is initially moving radially with a large velocity toward the bubble center, according to Eq. (A.1), and while the fluid inside the sphere is at rest inside. We neglect the fluid viscosity and thus assume that the system is Hamiltonian. We also assume that the solution with perfect spherical symmetry is stable under non-spherical perturbations. C.1. Dynamical equation An element of fluid at the radial distance 0 < r in the steady fluid at pressure P0 moves radially at r + (r, t) at time t (Euler coordinates). The volume contraction rate v(r, t) fulfills the inequality −1 < v(r, t) = ∂ ∂r , since the volume 1 + v of a unit volume of fluid at pressure P0 is never negative. If ρ is the mass of this unit volume of fluid, the total kinetic energy of the fluid is then different to the 1D case (B.1): Z ∞ 1 2 ρ ˙ 4πr 2 dr. (C.1) K = 2 0 Potential V(v) is the potential energy of a unit volume of fluid initially at pressure P0 as a function of its volume 1 + v. It is assumed to have the same properties as in the 1D case; V(v) becomes infinite for v → −1 or v → −h W (maximum Van der Waals contraction rate). Again we assume that V(v) is a convex function of v with monotone decreasing second derivatives. The volume contraction rate of a small element initially at 2 radial distance r is v(r, t) = ∇(r, t) = r 2 ∂r∂r  . Then, the potential energy of the fluid becomes formally different from the 1D case (B.2):  Z ∞  1 ∂r 2  W = 4πr 2 dr. (C.2) V r 2 ∂r 0 The action of this Hamiltonian system is   Z Z ∞ 1 2 1 ∂r 2  A = K − W = dt ρ ˙ − V 4πr 2 dr. 2 r 2 ∂r 0 (C.3) Vanishing the variation δ A yields the corresponding nonlinear dynamical equation of the fluid   ∂. 0 1 ∂r 2  ρ ¨ − V = 0. (C.4) ∂r r 2 ∂r As in 1D, it may be more convenient to write an equation for the pressure instead of the displacements, P(r, t) = −V 0 (v(r, t)) (we have P0 = −V 0 (0)) or   1 ∂r 2  0 . (C.5) P(r, t) = −V r 2 ∂r

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Function F(ξ ) is defined, as in the 1D case, as the inverse function of V 0 (see Fig. 4). Then, multiplying both members of Eq. (C.4) by r 2 and next differentiating with respect to r , we obtain ∂ 2. 1 ∂ 2. F(−P(r, t)) + (r P(r, t)) = 0 r ∂r 2 ∂t 2

are Dirac derivatives, Dirac functions, and standard functions or r and t, which should be vanished separately. Then, vanishing only the components of the Dirac derivatives yields that the pressure discontinuity necessarily travels at velocity

(C.6)

r˙d2 =

which is defined for r ≥ 0. It is now more convenient to define the new variable u(r, t) = r P(r, t) which fulfills

=

ρ

ρ

∂ 2 . h  u i ∂ 2 u rF − + 2 = 0. r ∂t 2 ∂r

(C.7)

Since the pressure is a radial function P(r, t) = P(−r, t), we have to search only for antisymmetric solutions u(r, t) = −u(−r, t) of Eq. (C.7). The advantage of this equation is that it is formally similar to Eq. (B.6), except that the uniform nonlinear function F(ξ ) is replaced by a spatially dependent nonlinear function G(ξ, r ) = r F( ξr ). Actually, this change only affects the nonlinear part, since G(ξ, r ) = κξ is independent of r in the case F(ξ ) = κξ is linear. If we assume u(r, t) bounded for large r , then ur goes to zero when r → +∞. Consequently, F(ξ ) ≈ ξ/(ρs02 ) can be linearized for large r , which yields the planewave equation −

2 ∂ 2u 2∂ u + s = 0. 0 ∂t 2 ∂r 2

(C.8)

Since u(r, t) = −u(−r, t), this linear equation (C.8) has exact solutions with the general form u(r, t) = f (s0 t −r )− f (s0 t +r ), where f (x) is an arbitrary function. If f (x) is pulse shaped with a single peak, it comes out that this pulse propagates radially far from bubble center at the sound velocity s0 of the fluid at rest. It keeps its shape over time, although the corresponding pressure field P(r, t) = f (s0rt−r ) decays with inverse distance from bubble center. The role of the nonlinearities necessarily becomes essential for the pulse propagation near the bubble center. The equation that yields the Hugoniot characteristics analogous to Eq. (B.10) can be written formally but, because of the spatial inhomogeneity of the nonlinearities, it does not have simple solutions with the form (B.11). Another consequence is that discontinuous front solutions similar to (B.12) are no more exact solutions of Eq. (C.7). The last section of the previous Appendix B has shown that a spatial inhomogeneity in a 1D model at a discontinuous interface breaks a single front in two fronts. In the 3D model, where the spatial inhomogeneity varies continuously, the situation is surely more complex. Some numerical investigations would help, but have not been performed up to now. Nevertheless, we may obtain some analytical information about the initial evolution of some given u profiles. We consider for example, an initial profile u(r, 0) that is discontinuous at r = rd (0). The corresponding solution u(r, t) is assumed to be discontinuous at r = rd (t), and there are two different limits for u(r, t) when r → rd (t) which are u + (t) = rd (t)P+ (t) for r > rd (t) and u − (t) = rd (t)P− (t) for r < rd (0). By substitution of such a form in Eq. (C.7), one gets different components which

u+ − u− 

−ρrd F(− urd+ ) − F(− urd− )



P+ − P− = c2 (P+ , P− ) −ρ (F(−P+ ) − F(−P− ))

(C.9)

which may be either positive or negative and is the same velocity as in the 1D case (B.15). However, this velocity depends on the values u − (t) and u + (t) of u(r, t) at the discontinuity for which the evolution is unknown. We now propose an empirical scenario for a spherical impact which, however, should require some numerical confirmations. The initial conditions that we propose for our impact model are obtained from the RP velocity field (A.1), which yields V R2

(r, 0) = 0 and ˙ (r, 0) = − rr 2 m sign(r ) for |r | > Rm , where ˙ Vr = R(0) is the velocity of the bubble radius at the time of impact. We assume that the bubble core is initially immobile, ˙ (r, 0) = 0 for |r | < Rm . Then we have initially P(r, 0) = P0 and u(r, 0) = P0r . For simplicity, we assume that P0 = 0 and thus u(r, 0) = 0.6 Other initial conditions could be chosen to be more realistic, but we think that the general scenario would be similar, except for the very short initial stage of the impact. C.2. Scenario of bubble collapse We may take the risk to draw some conjectures about the pulse shape evolution (see Fig. 1) according to Eq. (C.7) without performing any numerical tests, which are left for later works. We propose only a qualitative scheme about the coarse evolution of the pressure in the fluid neglecting fluctuations. During a short time after the impact of the fluid on the sphere, this impact is planar and may be described by the onedimensional model described in the previous Appendix B. Then we should consider Vi = Vr /2, since only the fluid on one side of the impact surface is moving. This impact generates a pulse initially at pressure Pi ∼ hρW Vi2 (B.20). Just after impact (see Fig. 1 top left), the solution of Eq. (C.7) is such that u(r, t) = 0 for all r except in a narrow shell delimited by two fronts at the inner radius r I (t) and the outer radius r O (t), where u(r, t) = u i is constant. We initially have r I (0) = r O (0) = Rm at time zero and −˙r I (0) = r˙O (0) = ci , where ci is defined from Eq. (B.17) by Vi = Vr /2 = −F(−Pi )ci at time 0. The two u-fronts appear and propagate in opposite directions at velocities c I = −˙r I (t) and c O = r˙O (t). These two fronts propagate from high-pressure to low-pressure regions so that, according to the 1D model, they may be considered as initially stable. However, the spatial nonlinear inhomogeneity drastically modifies the propagation of the pressure fronts which should not remain square shaped. 6 There is no lack of generality in assuming P = 0, which can be obtained 0 by shifting the origin of pressures at P0 . The equations are the same, except 0 that potential Vs (ξ ) = V(ξ ) − V (0)ξ replaces potential V(ξ ) everywhere.

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We have ∂∂r 2f = ξr 3 F 00 ( ξr ) > 0. Since F(ξ ) has been assumed to be a convex function of ξ , function G(ξ, r ) is also a convex function of r for arbitrary fixed ξ . Then ∂∂rf = F( ξr ) − ( ξr )F 0 ( ξr ) is monotonically increasing with respect to r . Since ∂∂rf (ξ, +∞) = 0, this derivative is strictly negative, which implies that, for fixed ξ , G(ξ, r ) is a monotonically decreasing function of r . Then, when r1 < r2 , we have the inequality G(ξ, r1 ) > G(ξ, r2 ), which is valid for any ξ . We conclude that this inhomogeneous medium becomes less compressible at smaller r referred to variable u, according to definition (B.22) (in our case, mass density ρ of the medium at rest is constant). We say that the system is less u-compressible at smaller r . According to Appendix B.5 which shows some front behaviors in 1D when the compressibility is spatially discontinuous, we expect that the u-pulse should not remain square-shaped. Complex pressure oscillations should be generated in the high pressure region behind the two fronts in the highly compressed region. In any case, an extremely short time is needed for the pressure pulse to reach the bubble center. If we assume, for example, average front velocities c I of the order of 10–20 s0 ∼ 25 × 103 m/s, then a typical time for the inner front to reach the center is about 10 ps (for a 0.5 µm diameter bubble); see Fig. 1 top right. At this stage, the fluid is compressed close to its Van der Waals volume in a full sphere at a pressure that is much beyond PW . Since u(0, t) = 0 at the center, a naive scheme would be that the inner front simply reflects at the center and moves backwards, while its amplitude and velocity decays again. However, this scheme is surely wrong, because we have shown in the 1D case that a front moving backwards is unstable and thickens (see Fig. 5). Another effect is that, when the direction of propagation of this front reverses, it should interact with its own tail generated beforehand (see Fig. 7 middle). The pressure evolution in the highly compressed region should be highly chaotic. Actually, in the Van der Waals regime when the pressure becomes tremendously large, the sound velocity becomes larger than the front velocities by many order of magnitudes. Thus, we should expect that the pressure fluctuations in that highly compressed region delimited by this relatively immobile fronts, tend to dissipate extremely fast into heat so that the pressure should be roughly uniform in that region. This argument for fast pressure uniformization which holds essentially in the Van der Waals regime, forbids any pressure divergency at bubble center. We may describe this regime with a highly compacted sphere of fluid by assuming the existence of a rather well-defined sphere with radius r O (t), where the fluid is compacted close to its Van der Waals volume, which persists (see Fig. 1 bottom left) for some time (this assumption does not mean that the pressure inside is uniform but only that it is larger than PW ) and such that the fluid outside is close to the normal pressure P0 . This compact sphere regime lasts a certain time tSL , which we can simply estimate by using the same argument of mass conservation as in the 1D case (Appendix B). The volume of fluid (at rest) per unit time entering from far away in a 2 large sphere with radius r is 4π Rr 2 Vr , and Vr are the bubble 2

2

m

155

radius and velocities at the time 0 of impact. The volume 2 r˙ . of fluid (at rest) that is compacted per unit time is 4πr O O Since the contraction rate is close to the maximum Van der 2 2 r˙ , Waals contraction h W , we should have 4π Rr 2 Vr ≈ h W 4πr O O m which yields r˙O = c O =

2 Vr Rm 2 hw r O

(C.10)

3 (t) ≈ R 3 (1+ 3Vr t). The which yields the time dependence r O m Rm h w outer front velocity r˙O decays as a function of its distance r O 2 ! Since this velocity is related from the bubble center as 1/r O to the pressure discontinuity from Pi0 to P0 according to Eq. q q Pi −P0 i −P0 ≈ (B.13), we get r˙O = ρ PF(−P ) ρh W . Consequently, for i small r O , while the whole fluid in the sphere is compacted at its Van der Waals volume, the pressure discontinuity behaves as

Pi0 − P0 ∝

1 . 4 rO

(C.11)

Our assumption of a compacted sphere is consistent while the front velocity r˙O is larger than the sound velocity s0 (which is the minimum velocity of a pressure front), which yields q

r O < RSL = Rm s0Vhrw . Thus, if we choose a Van der Waals contraction rate h W = 0.2 and an impact bubble velocity Vr = 4s0 , it comes out that RSL ≈ 4.5Rm , where Rm is the minimum bubble radius at impact. We call the sphere with radius RSL the radiance sphere, because most light will be emitted from that region. The consequence is that, under realistic experimental conditions, a substantial volume of the fluid around the bubble, and not only the inner part, is highly compressed up to its to Van der Waals volume and emits light according to the explanation developed in Section 6. The time duration tSL where this outer front is in the supersonic regime, that is, the life time of the compacted sphere, can be estimated. Integrating Eq. (C.10) with respect to time t r3

yields RO3 = 1 + 3 RmVrh w t. Thus, for having r O = RSL ≈ 4.5Rm , where the impact radius is Rm = 0.25 × 10−6 m and Vr = 4s0 ≈ 6×103 m, it comes out that tSL ≈ 0.5×10−9 s ∼ 500 ps, which is typically the order of duration of the SL light flash. When r O becomes larger RSL , the outer front velocity r˙O is bounded from below by the sound velocity s0 . Then, the fluid in the compacted sphere starts to relax to a larger volume. The volume of the fluid most likely expands first from the bubble center, where the pressure decays to smaller pressures P00 smaller than PW (see Fig. 1 bottom left). A pressure pulse starts to form that is initially more extended spatially (∼RSL ) than the bubble size Rm at impact. The effect of nonlinearities and their inhomogeneity still spreads the pressure pulse to larger width. At this point, the pressure drops in the compacted sphere to smaller values than the Van der Waals pressure PW defined in (B.9). Next, the pressure pulse move radially away from the bubble center almost at sound velocity s0 with an almost constant profile (see Fig. 1 bottom right). The amplitude of the pressure peak now decays as 1/r , as expected for harmonic waves. Measurements of the velocity and the relatively large

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spatial width of the acoustic pulse observed far from the bubble center qualitatively confirms this scenario [36]. Emission of light (SL) will occur from the fluid compressed to pressures much larger than PW (see Section 6), that is, only at the initial stage of the pressure pulse formation in the highly compressed sphere. It is worthwhile noting now that it has been observed that the pressure pulse velocity is much larger than the normal sound velocity when it is observed just after impact. Pulse velocities up to 4 km/s were observed in water [2,36] with a spatial width of several µm. This velocity was found to decrease rapidly to the normal sound velocity away from the bubble center. References [1] [2] [3] [4]

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