## Abstract

Based on the theory of F. Gilmore (Gilmore 1952*The growth or collapse of a spherical bubble in a viscous compressible liquid*) for radial oscillations of a bubble in a compressible medium, the sound emission of bubbles in water driven by high-amplitude ultrasound is calculated. The model is augmented to include expressions for a variable polytropic exponent, hardcore and water vapour. Radiated acoustic energies are calculated within a quasi-acoustic approximation and also a shock wave model. Isoenergy lines are shown for driving frequencies of 23.5 kHz and 1 MHz. Together with calculations of stability against surface wave oscillations leading to fragmentation, the physically relevant parameter space for the bubble radii is found. Its upper limit is around 6 μm for the lower frequency driving and 1–3 μm for the higher. The radiated acoustic energy of a single bubble driven in the kilohertz range is calculated to be of the order of 100 nJ per driving period; a bubble driven in the megahertz range reaches two orders of magnitude less. The results for the first have applications in sonoluminescence research. Megahertz frequencies are widely used in wafer cleaning, where radiated sound may be implicated as responsible for the damage of nanometre-sized structures.

## 1. Introduction

It has been known for a long time that bubbles in water oscillating in an ultrasound field emit sound (Minnaert 1933; Noltingk & Neppiras 1950; Leighton 1994). The spectral components of the emitted sound consist of harmonics and subharmonics of the incident sound wave centre frequency and broadband noise (Lauterborn & Holzfuss 1991). The emitted sound of a collapsing bubble may also consist of a pressure pulse or shock wave (Radek 1972; Vogel *et al.* 1986), leading to mechanical damage of nearby water-submersed structures. Recent research on single sonoluminescing bubbles has visualized (Holzfuss *et al.* 1998) and verified backscatter reactions of the bubble dynamics themselves (Rüggeberg *et al.* 1998; Holzfuss *et al.* 2002), leading to spatial dynamics including jumps. Here, the emitted acoustic energy is calculated in the quasi-acoustic approximation (Gilmore 1952) and, following the Kirkwood–Bethe hypothesis (Kirkwood & Bethe 1942), as a shock wave. The relevant parameter space is analysed for bubbles collapsing so violently that sonoluminescence is observed (Gaitan *et al.* 1992; Barber *et al.* 1997; Brenner *et al.* 2002) to determine whether the amount of reflected sound is large enough to induce backreactions and influence the bubble’s dynamical behaviour. Furthermore, results of calculations in the megahertz frequency range are shown. Stimulated by observations that the cleaning of wafers of computer chips that carry structures as small as 50 nm using megahertz frequencies (‘megasonic’) may lead to damage, the emitted sound energy of bubbles that are present during this process is mapped as a function of pressure and ambient radius. This may help to determine whether the emitted sound alone is energetic enough to create defects such as distorted or broken structure elements. Microbubbles driven in the megahertz range are also used as contrast agents in clinical screening (Grossmann *et al.* 1997). Numerical analysis of single pressure pulse excitation of small bubbles (Church 1989) has revealed potential arguments for kidney stone removal through acoustic energy radiation in addition to the impulsive energy of a driving shock wave using lithotripters (Ikeda *et al.*2006*a*,*b*).

To describe a bubble in a driving sound field, one has to find equations for various aspects of this phenomenon. The radial oscillations of the bubble wall are described in §2. The derivation of the Gilmore equation for radial bubble oscillations in a compressible liquid is explicitly described together with some important modifications. The bubble interior is modelled as a mixture of argon and water vapour. The models for acoustic emission are explained in §3, followed by a discussion in §4 and the conclusions in §5.

## 2. Radial oscillations

### (a) The Gilmore equation

The collapse of a spherical bubble in an infinite volume of water is analysed in the situation where the liquid flow is irrotational, that is, gravity effects and surface oscillations are left aside thus assuming sphericity at all times (Gilmore 1952). Then, the (particle) velocity **u** can be written as a function of a velocity potential *Φ* as (Landau & Lifschitz 1991)
2.1
and the equation of conservation of momentum is
2.2
where *p* is the pressure in the fluid and *ρ* the fluid density. In the viscous term, only the shear viscosity *η* is considered. The conservation of mass requires
2.3
Inserting equation (2.3) into the viscous term in equation (2.2) shows that the latter vanishes for a small viscosity and moderate compressibility as it describes the interaction of two small effects. Integrating equation (2.2) gives
2.4
for vanishing velocity and velocity potential at infinity. The liquid density is a function of pressure only and the function *h*(*p*) is the enthalpy difference between the liquid at pressure *p* and at . For velocities below the sonic velocity, which itself does not differ significantly from the constant ambient value of , the function *Φ* can be described by a spherical diverging sound wave
2.5
where *r* is the distance from the centre and *f* an unspecified function of . Equation (2.4) can then be written as
2.6
where denotes the derivative of *f*. The quantity *r*(*h*+*u*^{2}/2)≡*G*(*r*,*t*) in equation (2.6) is seen to be propagated outwards with velocity . As a more accurate approximation, the Kirkwood–Bethe conjecture suggests (Kirkwood & Bethe 1942; Gilmore 1952; Cole 1965) that the invariant quantity *G* is propagated outwards with the variable characteristic velocity (*c*+*u*), where *c* is the local sound velocity. This assumption leads to
2.7
or in terms of the substantive derivative (particle derivative following fluid motion), D/D*t*=∂/∂*t*+*u*∂/∂*r*,
2.8
Equation (2.8) is expanded by equation (2.6) to
2.9
Again neglecting viscosity/compressibility interaction effects, the momentum relation (2.2) in the spherical case is
2.10
and the continuity equation (2.3)
2.11

with *c*^{2}=d*p*/d*ρ* and equation (2.4) d*p*=*ρ* d*h*. If in equation (2.9) all partial derivatives with respect to *r* are eliminated with the help of equations (2.10) and (2.11), it follows
2.12
The particle derivatives in equation (2.12) can be replaced by time derivatives at the bubble wall, as a bubble wall oscillation happens along a particle path. Then, all variables can be replaced by their values at the bubble wall, denoted by *H*, *R*, , *C*, and rearranged to give the Gilmore equation
2.13

The enthalpy difference *H* is calculated at the bubble wall
2.14
This equation has to be supplemented with an equation of state for water. The modified Tait equation (Kirkwood & Bethe 1942)
2.15
for adiabatic compression is taken, as it fits the behaviour for water up to the very high pressures that occur. *n* and *B* are fitted constants (Holzfuss *et al.* 1998), *n*=7.025, *B*=3046 bars, the indexed variables denote ambient values. The local sound velocity *C* is calculated from equation (2.15)
2.16
The pressure outside the bubble wall is given by (Poritsky 1952)
2.17
where *p*_{g}(*R*) is the gas pressure in the bubble and calculated using a van der Waals hardcore *b* and a polytropic exponent *γ* (Church 1990; Brenner *et al.* 2002)
2.18
The surface tension *σ* leads to an increase in the pressure in the bubble *p*_{g} (equation (2.17)), as does the shear viscosity *η*. At 20^{°}C, the following parameters have been used in calculations (Lemmon *et al.* 1998): surface tension *σ*=0.0725 N m^{−1}, viscosity *η*=0.001 N s m^{−2}, water density *ρ*_{0}=998.23 kg m^{−3} and speed of sound . The hardcore *b* is calculated from the actual gas content and updated continuously (Holzfuss 2008).

In the derivation of the Gilmore equation (2.13), validity of the Kirkwood–Bethe hypothesis (2.7) is required. Disturbances in an incompressible liquid are transported with infinite velocity. For a linear sound wave in a compressible liquid, the motion is taken as a function of with a constant sound velocity. The Kirkwood–Bethe assumption goes one logical step further, namely assuming that disturbances propagate as a function of the sum of local (pressure dependent) velocity and the particle velocity. This assumption is superior to the aforementioned ones in the description of a nonlinear wave propagation as the sound velocity in the conditions reported here is highly pressure dependent.

Equation (2.13) has been derived in the case of constant ambient pressure . Here, the equation is solved in the case where the bubble is driven by a periodic pressure signal. Driving is done by setting for the lower limit in the integral in equation (2.4). It is expected that the equation still describes all features up to high Mach numbers: the period of the driving pressure is longer by orders of magnitude than the time scale of bubble collapse. At lower values of the bubble wall pressure, where changes of are reflected in *H*(*p*) (equation (2.4)), the wall velocities are small and the prefactors involving are vanishing. In the case of high Mach numbers and fast time scales around the collapse, the pressure is effectively quasi-static. Also during that time, the pressure exceeds the ambient pressure by up to five orders of magnitude, resulting in a vanishing difference in *H*(*p*) when compared with a static lower limit in the integral in equation (2.4).

Different models for radial bubble oscillations are used in the sonoluminescence literature. The Rayleigh–Taylor equation (Rayleigh 1917) augmented with an expression for viscosity, surface tension, an incident sound wave (Rayleigh/Plesset/Noltingk/Neppiras/Poritsky) (Lauterborn 1976) and radiation damping (Löfstedt *et al.* 1993) is frequently used. An equation with a more complete modelling of radiation damping is the one from Keller & Miksis (1980) (see also Brennen 1995)
2.19
It uses prefactors almost like the Gilmore equation (2.13), but only with a constant ambient speed of sound and the enthalpy *H* (equation (2.14)) is calculated only to first order. The Gilmore equation is found to model radial oscillations reasonably well even at very high pressures and bubble wall speeds up to (Gilmore 1952).

### (b) Polytropic exponent

The value of the polytropic exponent *γ* in equation (2.18) is set between 1 (=isothermal) and the adiabatic exponent of the gas according to an instantaneous Péclet number (Plesset & Prosperetti 1977; Prosperetti 1977; Lohse & Hilgenfeldt 1997)
2.20
reflecting thermal conduction at the involved time scales. *κ* is the thermal diffusivity of the gas, *γ* is calculated as , where the simple function *F* scales and interpolates published tabulated values of the logarithm of the Péclet number to values in [0,1] and *f* is the degrees of freedom of the actual gas content. To avoid a change to isothermal behaviour around collapse time when the bubble wall velocity vanishes, its value in equation (2.20) is effectively kept at its maximum value during positive bubble wall accelerations (around *R*<*R*_{0}). It equals the real bubble wall velocity during the rest of the cycle.

*b* and *γ* in equation (2.18) are updated during calculations to reflect the actual gas content. The value of the thermal diffusivity *κ* is variable, as it depends on the varying density *ρ*_{g} of the gas
2.21
where *k* is the thermal conductivity and *c*_{p} the specific heat at constant pressure. The value of *κ* is scaled with the ratio of ambient gas density to actual density.

The density in the bubble is calculated by
2.22
and
2.23
where *p*_{g} is the pressure in the bubble and *R*_{gas}=8.3143 J mol^{−1} K^{−1} is the gas constant. The van der Waals hardcore *b* is calculated as an average from the tabulated hardcore values and the number of moles of the *i* different gases, *i*=Ar,H_{2}O. is the molar mass averaged over all gases and *n*_{total} the total number of moles. *b* and are updated continuously.

The temperature *T*_{B} is taken to be uniform within the bubble. It is calculated via the adiabatic compression of a van der Waals gas by
2.24
with the ambient liquid temperature *T*_{0}. Other more complex approaches exist that include thermal conduction and a temperature jump across the bubble interface (Yasui 1995, 1997).

### (c) Water vapour

Evaporation and condensation of water molecules at the bubble wall (Kamath *et al.* 1993; Yasui 1997; Storey & Szeri 2000, 2002; Brenner *et al.* 2002) is included in the model for the bubble dynamics, as experimental results (Toegel *et al.* 2000; Vazquez & Putterman 2000) stress the importance of a decrease in the polytropic exponent induced by water vapour at bubble collapse. A simple Hertz–Knudsen model (Hertz 1882; Knudsen 1950) for the change of moles *n*_{H2O} of water vapour in the bubble is
2.25
*α* is the constant evaporation coefficient (also called accommodation coefficient or sticking probability),
2.26
is the average velocity of molecules of a Maxwell–Boltzmann distribution, *ρ*_{g,H2O} is the density of water vapour of molar weight *M*_{H2O} in the bubble, *ρ*^{sat}_{g,H2O}=0.0173 kg m^{−3} is the saturated vapour density at 20^{°}C and *R*_{gas}=8.3143 J mol^{−1} K^{−1} is the gas constant. The bubble surface temperature is taken as *T*_{s}=*T*_{0}. The density of water vapour *ρ*_{g,H2O} depends on the bubble dynamics and is calculated along with the bubble equation with the help of equation (2.22). The simple model (2.25) takes the temperature distributions in the bubble and liquid as fixed and does not capture all effects occurring during evaporation, as more complex treatments would do (Harvie & Fletcher 2001; Bond & Struchtrup 2004; Hauke *et al.* 2007). *Γ* is a correction factor for non-equilibrium conditions induced by mass motion of vapour and bubble wall movement (Schrage correction) (Schrage 1953; Yasui 1995; Storey & Szeri 2000)
2.27
and
2.28
where *Ω* is a ratio of velocities, is the bubble wall velocity, *v*_{H2O} is the vapour velocity and *c*_{peak} is the velocity belonging to the peak of the Maxwell–Boltzmann velocity distribution.
2.29
The change of mass per unit time and unit area *j* can be expressed as
2.30

and when inserted into equation (2.28) leads to
2.31
for a spherical bubble volume. For small values of *Ω*, the approximation can be made (Schrage 1953), leading to an effective constant evaporation coefficient of *α*_{eff}=2*α*/(2−*α*) in equation (2.25) and an effective correction factor *Γ*_{eff} of unity. Calculations show that *Γ* varies by as much as 20 per cent around unity during the collapses (figure 1). However, numerical results (figure 2) in the observed parameter range show that almost no notable difference exists in the amount of water vapour at collapse time between calculations using a constant value of *α*_{eff} and setting *Γ*_{eff}=1, and calculations using equation (2.25) with the variable expressions (2.27) and (2.31) and a smaller value of the evaporation coefficient *α*=2*α*_{eff}/(2+*α*_{eff}). Therefore, a constant value of *α*_{eff}=0.4 (recommended in Eames *et al.* (1997)) without the Schrage correction is taken in the following calculations.

### (d) Surface oscillations

As seen in experiments (Gaitan & Holt 1999; Levinsen *et al.* 2003) and numerical calculations, higher order oscillations of the bubble surface also play a role in a part of the parameter space (Holzfuss 2008). The instabilities that occur limit the available state space. In the case of high-amplitude driven bubbles, growing surface oscillations lead to destruction, whereby the bubbles split or lose part of their volume and serve as seed for a new bubble (Holzfuss 2005) (recycling bubbles). A surface instability is calculated here as an upper bound for the physically relevant parameter space set by pressure and ambient bubble radius. Several instabilities have been identified setting a barrier to sonoluminescing bubbles. The theory is based on equations for surface oscillations (Prosperetti 1977) which are driven by radial bubble oscillations. A further advance of the theory (Hilgenfeldt *et al.* 1996; Brenner *et al.* 2002) showed applications to sonoluminescing bubbles. As the main upper limit for increasing driving amplitude and radius, the parametric instability has been identified. Closely following Holzfuss (2008), the lowest threshold for unbounded growth of an *n*=2 surface oscillation is calculated.

## 3. Acoustic emission

### (a) Emitted sound energy in the quasi-acoustic approximation

The rate at which energy crosses a spherical surface of varying radius *r* and in a non-viscous fluid is the rate of work done by the pressure
3.1
It is also a good approximation for moderately viscous and compressible media (Gilmore 1952). The rate can be separated into
3.2
where the first term would vanish upon integration over one period as the fluid motion is periodic in an acoustic situation. The remaining term is called the ‘wave-energy’ term (Gilmore 1952)
3.3
In the quasi-acoustic approximation, *p* is found by approximating equation (2.6) to first order with and *u* by combining equations (2.1) and (2.5)
3.4
3.4
Using equations (3.4) and (3.6) in equation (3.3) and dropping all terms vanishing for large distances *r* leaves
3.6
Together with equation (3.4) and with the pressure in the liquid at the bubble wall, the rate of acoustic energy is
3.7
The radiated acoustic energy is calculated by an integration over a period of the driving pressure.

#### (i) Emission at 23.5 kHz

As an example, figure 3 shows lines of equal acoustic energy radiated by a driven argon–vapour bubble in the sonoluminescing region. The driving frequency is 23.5 and 20 kHz. In the parameter space spanned by driving pressure and equilibrium radius (calculated as the effective radius with no driving applied), it is seen that the emitted acoustic energy increases while increasing both parameters. The upper threshold is set by the parametric instability. The acoustic energy emitted by a typical bubble during one cycle may reach a value around 100 nJ, as calculated by this quasi-acoustic approximation. It is seen that the isoenergy lines run in parallel very close to each other for higher values of the driving pressure. This is happening near the threshold for inertial or transient bubble collapse (Leighton 1994; Brenner *et al.* 2002), where the size of the maximal radius exceeds the equilibrium radius greatly and a sharp collapse is observed. In the limiting case of static pressures, the threshold for unlimited bubble growth is called the Blake threshold (Blake 1949). The parameter region shown in figure 3 is the typical region where sonoluminescence of stably oscillating bubbles may be observed. It has been seen in experiments (Holzfuss *et al.* 2002) that the emitted sound sets up a rather complex acoustic environment which interacts with the bubble.

The calculations can be compared with experimental data. Pressure profiles from sonoluminescing bubbles have been measured experimentally (Wang *et al.* 1999) and pulse pressure and full width at half maximum (FWHM) have been given as a function of driving pressure at 20 kHz driving frequency. Using an approximation of a shock with a time-dependent pulse having an amplitude of zero before its arrival, a Gaussian-like pressure drop-off with the same *P*_{max} and *T*_{FWHM} as a measured shock (data from fig. 5 in Wang *et al.* 1999), the shock wave energy can be estimated analytically. Using equation (3.9) and calculating its integral as and , one obtains the shock energy. This has been done for the published average values and also, using the spread of the experimental data, for the averaged lowest and highest values. The highest maximum reached is approximately 79 nJ at 1.3 bar. This is in agreement with the energies calculated in figure 3. Using the present model and the sound energy deduced from the experiment, the spread of ambient radii of a bubble driven with the specified parameters and emitting the same acoustic energy is calculated and shown in figure 3. As sonoluminescing bubbles have ambient radii below the threshold of instability, one can see that data points have been taken closely below the line of parametric instability at the specified pressure. Very good agreement of the predicted range of emitted energy with the experimentally measured emitted energy is seen.

#### (ii) Emission at 1 MHz

As an example of an argon–vapour bubble driven with very high frequencies, figure 4 shows isoenergy lines of the acoustic radiation of a bubble driven at 1 MHz. The global picture is more ragged than the one for lower driving frequency. The reason for this is that in this parameter region, where the bubble has a linear eigenfrequency (Leighton 1994) of the order of the driving frequency, different resonances and subharmonic oscillations exist. This may result in a different acoustic energy emitted during subsequent cycles. Therefore, only the maximum collected during six consecutive cycles is shown. The overall trend is the same as in figure 3. Higher energies are found for larger driving amplitudes and larger ambient radii. The general trend is not valid outside the parameter space scanned, as the position of different resonances may become very complex. This is seen at the topmost right parameter values, where islands of different resonances appear. As in figure 3, isoenergy lines running close to each other mark the threshold for an inertial collapse. To generate a comparable picture of the parameter space, the driving pressure has to be increased by a factor of 3. The absolute values of emitted acoustic energy obtained per driving period are lower by two orders of magnitude than the previously analysed case.

Also shown are lines marking the onset of parametric instability. Two thresholds are shown. As subharmonic oscillations exist with a waveform spanning several periods of the driving sound, it seems that surface oscillations are not growing during each driving cycle. They may not lead to destruction as the disturbances may be damped during the remaining oscillations. The lower threshold line (long dashed curve) marks the occurrence of a surface wave amplification rate larger than 1 during one of six consecutive driving periods. At the upper threshold (dashed line), the sum of the logarithms of all successive amplification rates is larger than zero. Here, the surface oscillations are amplified on average, leading to bubble break-up. Thus, only bubbles smaller than the upper threshold exist; some may exist above the lower threshold. However, for higher driving amplitudes, the emitted energy does not change much above the lower threshold.

### (b) Shock waves

In the case when all velocities are not considerably less than the sonic velocity, the calculations in the quasi-acoustic approximation are strictly speaking not valid. They lead only to approximate values. In this case, the pressure and velocity fields outside the bubble have to be calculated using the method of characteristics.

The propagation of the pressure pulse in the liquid is calculated by using the Kirkwood–Bethe hypothesis (Ivany 1965; Knapp *et al.* 1970); the invariant quan- tity (equation (2.6)) propagates with the characteristic velocity *c*+*u*, where *c* and *u* are the local sound and particle velocities in the liquid. By summation of the momentum (2.10) and continuity equation (2.11), with the definitions of *G* and *h* and the rate of change of any variable along a characteristic d/d*t*|_{char} = ∂/∂*t* + (*c* + *u*)∂/∂*r*, the time evolution can be calculated.

The particle velocity and pressure along the characteristics are determined (Lee *et al.* 1997; Rüggeberg 2000; Byun & Kwak 2004) by solving the Gilmore equation (2.13) together with
3.8
Solving the bubble equation (2.13) gives the initial values *R*, , *H* and for each characteristic. Crossing characteristics in *r*−*t* space imply the generation of a shock. The exact position of the shock front can be obtained by equalization of the particle velocities in the multiple valued *u*−*t* curves (Rudenko & Soluyan 1977; Rüggeberg 2000) as the characteristics cross at a shock front.

Figure 5 shows the typical behaviour of a shock front as it is shed from the surface of the bubble around collapse time. The shock front travels with supersonic speed outwards, slowing down until it reaches the value of the ambient sound speed around 1 mm from the centre. In that range also, the pressure in the shock wave falls off more rapidly than the expected 1/*r* law for the non-shock case. The shock pressure drop-off spans up to five orders of magnitude. The energy of the shock calculated from
3.9
drops within a 1 mm range, after which it stays constant as expected. The fall-off describes the increased energy dissipation for a shock wave which is accounted for in the model by the equalization of the particle velocities at the shock front (Holzfuss *et al.* 1998; Rüggeberg 2000). The calculations for figure 5 do not include water vapour effects.

A calculation of the radiated sound energy in quasi-acoustic approximation for this set of parameters results in a value of *E*_{s}≈146 nJ. Compared with the shock wave energy at *r*=1 mm (*E*_{s}≈50 nJ), the quasi-acoustic approximation overestimates the energy by a factor of 3. It cannot account for the dissipative losses of the radiated sound as a shock wave.

## 4. Discussion

In the model, purely radial oscillations have been considered. The initial conditions are those of an irrotational flow, curl *u*=0 everywhere under any condition at all times. However, vorticity is treated in connection with the surface oscillations. Any vorticity created during a real process limits the maximally radiated energy. Stimulated by good experimental and numerical agreement in sonoluminescence research, the Gilmore model has been augmented with an equation for a varying polytropic exponent of the compressed vapour–argon bubble to model the influences of thermal damping at the involved time scales and a varying hardcore reflecting changes in the gas content. Furthermore, it has been seen in experiments and numerical calculations that evaporation and condensation of water vapour through the liquid–gas interface plays an important role; a model of this effect is included. The equations presented here enable the first calculation to be made of sound and shock wave radiation in a compressible medium using these recent modifications.

Some sources of energy consumption have been neglected in the calculations: the enthalpy needed for evaporation and condensation of vapour during an oscillation period can be calculated to be at most of the order of two to three magnitudes below the shock wave energies. The enthalpy for chemical dissociation of all water vapour molecules during main collapse (approx. 0.5 nJ) would typically be two to three orders of magnitude less than the typical acoustic energies.

Energy radiation by surface waves is not included in the calculations. Surface waves are developing near the border of parametric instability, where they lead to the destruction of the bubble. In the linear theory, the radiated pressure of an oscillation of order *n* decays rapidly as *r*^{−(n+1)} (Longuet-Higgins 1989). In a nonlinear theory for a distorted large bubble, a 1/*r* monopole radiation has been shown to exist with twice the frequency of the shape oscillation and pressure fluctuations of the order of the ambient pressure (Longuet-Higgins 1990). In the case here, this should lead to an emission with the same frequency as the radial afterbounces following the main collapse.

The high acoustic energies radiated by an ultrasonically driven bubble in the kilohertz range are used in ultrasonic cleaning devices. Depending on the distance of a collapsing bubble to an interface, peak pressures up to 10^{4}–10^{5} bars are possible. When experimenting with single bubbles in high-Q resonators, as in sonoluminescence research, a reflection of the pressure wave and a refocus near the bubble can create backreactions on the bubble itself. Owing to the two orders of magnitude smaller calculated values of the radiated acoustic energy, mechanical damage of nano-scale electronic components seems less likely with ultrasonic cleaning at megahertz frequencies. Still, pressures may be of the order of 10^{3}–10^{4} bars. The range of driving pressures analysed here is the typical operating range of megasonic cleaning devices (Kapila *et al.* 2006; Holsteyns *et al.* 2008; Minsier & Proost 2008; Ahn *et al.* 2009; Keswani *et al.* 2009). An uncontrolled radiation by cavitation bubble fields will result in mechanical failure, when the high peak energies of the radiated sound containing high-frequency components drive unwanted mechanical oscillations. Radiated shock pulses may also stimulate the growth of submicron-sized bubbles. While the exact mechanisms for damage formation are still debated, mechanical modifications and/or modifications of the acoustical field are actively being probed.

Damage owing to bubble dynamics near interfaces has also been attributed to asymmetric bubble collapse. Jet formation and secondary collapses together with shock wave shedding have been analysed by a large number of authors (Shima *et al.* 1981; Vogel & Lauterborn 1989; Ohl *et al.* 1999; Vogel 2001). Owing to the loss of spherical symmetry by altered flow conditions in the vicinity of a collapsing bubble, damage done by shock pressure and water hammer pressure has been observed. However, these observations have mostly been made with laser-induced, single large cavitation bubbles. It is unclear whether the above dynamical phenomena are also seen in bubbles of the sizes analysed here or whether they are suppressed by the surface tension acting on micrometre-sized ultrasonically driven bubbles, and the shock wave emission is the main reason for damage.

## 5. Conclusions

Calculations for the emitted acoustic energy of single radially driven bubbles have been presented. The physically valid parameter space has been calculated by considerations of surface stability. With the quasi-acoustic approximation, the radiated acoustic energies of bubbles driven at 23.5 kHz are of the order of 100 nJ per driving period; bubbles driven at 1 MHz reach two orders of magnitude less. The average power for emission of a single bubble is of the order of 10^{−3} W in both cases. Owing to the short pulse lengths of the order of 100 ps, the peak powers are of the order of 10^{1}–10^{2} W for a bubble driven at 23.5 kHz and two orders of magnitude less for a 1 MHz bubble. A bubble growing by rectified diffusion will reach a parametric instability threshold, which has been calculated for both cases. After break-up, bubbles may grow again. This mechanism sets the physically relevant parameter space. A model for the analysis of a shock wave being shed from a bubble during collapse shows that parts of the acoustic energy are dissipated while the shock travels outwards. The order of magnitude of the radiated energy is, however, preserved. A comparison with experimental data shows good agreement with numerical calculations. The results show that the sound radiated by a radially collapsing bubble may be implicated as being responsible for the interaction with nearby mechanical structures, which will fail if they cannot withstand the calculated acoustic energies.

## Footnotes

- Received November 10, 2009.
- Accepted December 10, 2009.

- © 2010 The Royal Society