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ARCHIVES OF ACOUSTICS

33, 4, 609–617 (2008)

ACOUSTIC CAVITATION AND BUBBLE DYNAMICS

Werner LAUTERBORN, Thomas KURZ, Robert METTIN, Philipp KOCH,

Dennis KRÖNINGER, Daniel SCHANZ

Universität Göttingen

Drittes Physikalisches Institut

Friedrich-Hund-Platz 1, 37077 Göttingen, Germany

e-mail: lb@physik3.gwdg.de

(received June 15, 2008; accepted October 28, 2008)

Acoustic cavitation is investigated experimentally and theoretically starting with a single

bubble in an acoustic field. The single bubble is trapped in an acoustic resonator and observed

by high-speed imaging and acoustic measurements following its transient dynamics to a possi-

ble steady state after generation by laser light. Numerical calculations are done to explore the

regions of survival for acoustically driven bubbles with respect to dissolution or growth, sur-

face oscillations and positional stability in a standing sound field. The interior gas dynamics

is explored via molecular dynamics simulations. A glance is thrown at the complex dynamics

of multi-bubble systems.

Keywords: acoustic cavitation, bubbles.

1. Introduction

Intense sound fields in liquids generate cavities or bubbles that start to oscillate and

move around in the liquid, a phenomenon called acoustic cavitation [2, 11, 13]. The

basic element to start an investigation with is the single, spherical bubble in an infinite

liquid subject to a sinusoidal sound field. The best experimental approximation so far

is realized in the acoustic bubble trap as mainly used for sonoluminescence studies [5],

and in bubbles generated by focused laser light [9]. Both cases have been combined to

investigate transient single bubble dynamics and its sonoluminescence in an acoustic

field [8]. Theoretical bubble models have been conceived with different states of so-

phistication to include different influences and phenomena as, for instance, liquid com-

pressibility, heat and mass transfer across the bubble wall or stability of the surface.

Noteworthy is the strong collapse of a bubble at high sound pressure amplitudes leading

to shock wave and light emission and high speed liquid jet formation under appropri-

ate conditions [11]. The interior of a bubble is usually treated as homogeneous with

respect to temperature and pressure. However, this may not be true in the last phase of

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a strong collapse [21]. This problem is not only tackled by continuum mechanics via

partial differential equations but also in a molecular dynamics approach [10, 15].

The next step in bubble dynamics investigations is given by bubble interaction.

It leads to complicated forms of attraction and repulsion already in the case of two

spherical, interacting bubbles. A survey of bubble structures as they appear in acoustic

cavitation is given in [16]. A particle model has been formulated to address the problem

of multi-bubble systems in an ultrasonic field as a way to cope with acoustic cavitation

bubble ensembles [17, 19]. This paper is a kind of progress report in the field of bubble

dynamics in acoustic fields.

2. Single bubble dynamics

From the many models to describe the oscillation of bubbles in an acoustic field the

model of KELLER and MIKSIS [6] is adopted:

?

1 −

˙R

Cl

?

R¨R +3

2

?

1 −

˙R

3Cl

?

˙R2=

?

1 +

˙R

Cl

?

p

ρl

+

R

ρlCl

dp

dt.

(1)

Here, R is the bubble radius, Clthe sound velocity of the liquid at the bubble wall,

ρlthe density of the liquid, and p the difference between external and internal pressure

terms at the bubble wall, as given in the following equation.

p =

?

pstat+2σ

Rn

??Rn

R

?3κ

− pstat−2σ

R−4µ

R

˙R − ps.

(2)

The quantities introduced are: pstatthe static ambient pressure, σ the surface tension,

Rnthe radius of the bubble at rest, κ the polytropic exponent of the gas inside the

bubble, and µ the dynamic viscosity. psis an additional external pressure, for instance

the pressure pulse of a lithotripter.

The Keller-Miksis bubble equation has been solved for various parameter configu-

rations with a sinusoidal sound field ps:

ps= ˆ pAsin(2πνt).

(3)

Figure 1 gives a typical set of steady-state bubble oscillations with increasing sound

pressure amplitude from ˆ pA= 100 kPa to 200 kPa in steps of 20 kPa at a frequency of

ν = 20 kHz for a bubble with a radius at rest of Rn= 5 µm. After a long expansion

phase a steep collapse occurs with some afterbounces until the cycle repeats. Further-

more, the maximum of the bubble oscillation shifts from the tension phase of the driving

sound field to the pressure phase leading to a shift in the stability position of the bubble

in a standing sound field [1, 14].

For a more thorough desription of the oscillation properties of a bubble condensed

forms of diagrams have been developed to combine special features as in bifurcation

diagramsorinphasediagrams[12,18].Figure2givesaphasediagramoftheperiodicity

of the oscillation with respect to the forcing sound oscillation for bubbles in the range

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Fig. 1. Typical steady-state bubble oscillations with steep collapse and afterbounces. Shift of the relative

phase of the bubble oscillation with respect to the phase of the sound field with increasing sound pressure

amplitude; ˆ pA = 100...200 kPa, ν = 20 kHz, Rn = 5 µm, isothermal.

of 10 µm to 100 µm and for driving pressure amplitudes from 0 to 300 kPa for a sound

field frequency of ν = 20 kHz. It is seen that specific ranges with repeated structures

of interlaced periodicity appear that are similar for other and presumably all bubble

models [18] and even a larger class of nonlinear oscillators [20].

Periode

Rn [µm]

pA [kPa]

1

4

8

12

16

> 16

020 406080 100

300

200

100

0

Fig. 2. Phase diagram giving the periodicity of the bubble oscillations with respect to the driving sound

oscillation; ν = 20 kHz, isothermal.

The bubble model can be extended to include diffusion processes [3, 4]. Then bub-

bles may dissolve or even grow due to a process called rectified diffusion. Also the

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W. LAUTERBORN et al.

bubble may become positional unstable in the sound field due to acoustic radiation

forces (Bjerknes forces). Moreover, when additionally surface oscillation of the bubble

are included [7] a restricted survival region results in the sound pressure amplitude –

equilibrium bubble radius plane. In Fig. 3 this stability region is shown shaded for bub-

bles with rest radii between 0 and 30 µm and for pressure amplitude between 0 and

300 kPa at a fixed frequency of the sound field of 20 kHz. Below the shaded region

the bubble is diffusion unstable, above the phenomenon of rectified diffusion prevents

the dissolution of the bubble. Above the shaded region the bubble gets unstable due to

surface oscillations that lead to a break-up of the bubble. There is a dotted line running

through the middle of the figure. This dotted line separates the bubbles being attracted

to the pressure antinode in the standing sound field (in a resonator or bubble trap) and

thus can be stably trapped and the bubbles above that are driven away from the pressure

antinode and are sentenced to move to and on a surface of lower pressure in the standing

sound field.

Fig. 3. Region of stable bubble oscillations in a standing sound field of ν = 20 kHz (shaded area).

This type of diagram can be calculated for other frequencies and for other ranges

of sound pressure and bubble radii. Figure 4 gives an example for ν = 40 kHz in

an extended bubble range normalized to the linear resonance radius Rlin

and for the acoustic pressure range 0 to 500 kPa. It can be noticed that the positionally

stable region (lightly shaded) is a rather small rugged stripe and located below the linear

resonance radius. Generally it can be stated from calculations for higher frequencies

that the survival space shrinks for higher frequencies to a small region in this parameter

space.

When the interior of the bubble is closer looked at, additional phenomena are ob-

served. The interior contents may not be homogeneously distributed, but pile up at the

bubble boundary or in the center, compression waves may run towards the center and

chemical reactions may take place. This problem has been addressed by molecular dy-

n

= 69 µm

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ACOUSTIC CAVITATION AND BUBBLE DYNAMICS

613

Fig. 4. Region of stable bubble oscillations in a standing sound field of ν = 40 kHz (shaded area).

namics calculations where the interior contents are modelled by a number of particles

that move around in the interior according to the laws of mechanics. They transfer im-

pulse and energy when they hit each other and the bubble wall. The interior dynamics is

calculated when the bubble is subject to a collapse in a sound field [10]. Figure 5 shows

the interior temperature distribution in a bubble for 800 ps around its strong collapse.

In this case with nonreacting particles (Argon) and reacting particles (water molecules)

maximum temperatures of about 18,000 K are reached in the center.

Fig. 5. The distribution of temperature inside a bubble versus time upon collapse. Rest radius of the bubble

Rn = 4.5 µm in a sound field of ν = 26.5 kHz at a pressure amplitude of 1.3 bar. Argon–vapour bubble

including chemical reactions.

3. Experiments

To study single bubble dynamics in a controlled way a method to prepare a bub-

ble and insert it in a sound field without negligible disturbance of the field must be

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W. LAUTERBORN et al.

conceived. One method is to produce a bubble by laser light at a pressure antinode of

a standing wave in an acoustic resonator [8]. Figure 6 shows an arrangement for this

purpose. The bubbles are generated by focusing a laser light pulse with a duration of

about 130 fs and an energy of a few µJ into a water filled cuvette that is excited by

a transducer to its (1,1,1)-mode at about 44 kHz. That way, a tiny bubble is placed at the

pressure antinode where it is kept by Bjerknes forces during its oscillations when being

able to reach its survival zone (compare Fig. 3). The bubble is photographed by a CCD

camera through a long distance microscope. Also the light emitted by the bubble can be

registered.

Fig. 6. Experimental arrangement for the investigation of laser generated bubbles in an ultrasonic field.

Figure 7 shows the evolution of a bubble inserted into the sound field versus the

number of acoustic cycles from one up to a million cycles. The conditions are set to

Fig. 7. Long-term evolution of the maximum bubble radius (left curve) and emitted light as captured by

an ICCD camera (right curve). The acoustic frequency is 43.68 kHz, and the driving pressure 1.55 bar

(Courtesy of T. Wilken).

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ACOUSTIC CAVITATION AND BUBBLE DYNAMICS

615

typical single bubble sonoluminescence conditions with an acoustic pressure of 1.55 bar

at 43.68 kHz and 200 mbar partial pressure of dissolved air. In this case the bubble sur-

vives for long time undergoing shrinking and growth presumably by diffusion processes

until it settles at a large maximum bubble radius and the maximum light emission during

its development. Before the onset of bright light output, but already at large maximum

bubble radius, the bubble experiences a jittering motion presumably by surface instabil-

ities leading to the shedding of microbubbles. After these adjustments of the bubble the

light emission grows strongly to a long-term stable state.

4. Acoustic cavitation bubble clouds

When many bubbles are present simultaneously as normally in acoustic cavitation,

the bubbles interact and form structures [16, 17, 19]. These highly dynamic structures

are not easily captured, but new technological developments in high speed videography

open up new possibilities. Figure 8 shows an arrangement where two high speed CCD

cameras are combined to capture a stereoscopic image of the bubble cloud to locate

the individual bubbles in three-dimensional space. Figure 9 gives an example of the

spatial distribution of bubbles as taken from 75 consecutive frames. The sound field

frequency is 22.8 kHz and the sound pressure amplitude is 132 kPa. It is seen that the

bubbles aggregate on a filamentary structure. This is due to attractive forces (besides

repulsive ones) among bubbles and acoustic radiation forces that drive small bubbles to

the pressure antinode.

Delay

LED

Delay

Trigger

Function

Generator

Amplifier

y

x

z

Spotlight

Synchronization

Cuvette

HiSIS2

HiSIS1

Pulnix

Fig. 8. Experimental arrangement for stereoscopic high speed videography of bubble clouds in a sound

field.

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W. LAUTERBORN et al.

z [mm]

z [mm]

x [mm]

y [mm]

-6

-4

-2

0

2

4

-4

-2

0

2

4

-4

-2

0

2

4

Fig. 9. Spatial distribution of bubbles from stereoscopic high speed videography at 2250 frames/s.

Acknowledgments

The authors thank the members of the Nonlinear Dynamics Group at the Third Phys-

ical Institute for numerous discussions and help, in particular U. Parlitz, R. Geisler and

D. Krefting.

References

[1] AKHATOV I., METTIN R., OHL C.-D., PARLITZ U., LAUTERBORN W., Bjerknes force threshold

for stable single bubble sonoluminescence, Phys. Rev., E55, 3747–3750 (1997).

[2] BRENNEN C.E., Cavitation and Bubble Dynamics, Oxford University Press, Oxford 1995.

[3] CRUM L.A., Rectifid diffusion, Ultrasonics, 22, 215–223 (1984).

[4] ELLER A.I., Growth of bubbles by rectified diffusion, J. Acoust. Soc. Am., 46, 1246–1250 (1969).

[5] GAITAN D.F., CRUM L.A., CHURCH C.C., ROY R.A., An experimental investigation of acoustic

cavitation and sonoluminescence from a single bubble, J. Acoust. Soc. Am., 91, 3166–3183 (1992).

[6] KELLER J.B., MIKSIS M.,Bubbleoscillationsoflargeamplitude,J.Acoust.Soc.Am.,68,628–633

(1980).

[7] KOCH P., Particle modelling of structure formation of acoustic cavitation bubbles interacting with

a sound field [in German], PhD Thesis, University of Göttingen, 2006.

[8] KURZ T., KRÖNINGER D., GEISLER R., LAUTERBORN W., Optic cavitation in an ultrasonic field,

Phys. Rev., E74, 066307-1–7 (2006).

[9] LAUTERBORN W., Cavitation by laser light [in German], Acustica, 31, 51–78 (1974).

Page 9

ACOUSTIC CAVITATION AND BUBBLE DYNAMICS

617

[10] LAUTERBORN W., KURZ T., METTEN B., GEISLER R., SCHANZ D., Molecular dynamics ap-

proach to sonoluminescent bubbles, [in:] Theoretical and Computational Acoustics 2003, A. TOL-

STOY, YU-CHIUNG TENG, E.C. SHANG [Eds.], pp. 233–243, World Scientific, New Jersey 2004.

[11] LAUTERBORN W., KURZ T., METTIN R., OHL C.-D., Experimental and theoretical bubble dy-

namics, Advances in Chemical Physics, 110, 295–380 (1999).

[12] LAUTERBORN W., PARLITZ U., Methods of chaos physics and their application to acoustics,

J. Acoust. Soc. Am., 84, 1975–1993 (1988).

[13] LEIGHTON T.G., The Acoustic Bubble, Academic Press, London 1994.

[14] MATULA T.J., CORDRY S.M., ROY R.A., CRUM L.A., Bjerknes force and bubble levitation under

single-bubble sonoluminescence conditions, J. Acoust. Soc. Am., 102, 1522–1527 (1997).

[15] METTEN B., Molecular dynamics simulations for sonoluminescence [in German], PhD Thesis, Uni-

versity of Göttingen, 2000.

[16] METTIN R., Bubble structures in acoustic cavitation, [in:] Bubble and particle dynamics in acoustic

fields: Modern trends and applications, A. DOINIKOV [Ed.], pp. 1–36, Research Signpost, Kerala,

India, 2005.

[17] METTIN R., LUTHER S., OHL C.-D., LAUTERBORN W., Acoustic cavitation structures and simu-

lations by a particle model, Ultrasonics Sonochemistry, 6, 25–29 (1999).

[18] PARLITZ U., ENGLISCH V., SCHEFFCZYK C.,LAUTERBORN W., Bifurcation structure of bubble

oscillators, J. Acoust. Soc. Am., 88, 1061–1077 (1990).

[19] PARLITZ U., METTIN R., LUTHER S., AKHATOV I., VOSS M., LAUTERBORN W., Spatio-

temporal dynamics of acoustic cavitation bubble clouds, Phil. Trans. R. Soc. Lond., A357, 313–334

(1999).

[20] SCHEFFCZYK C., PARLITZ U., KURZ T., KNOP W., LAUTERBORN W., Comparison of bifurcation

structures of driven dissipative nonlinear oscillators, Phys. Rev., A43, 6495–6502 (1991).

[21] VUONG V.Q., SZERI A.J., YOUNG D.A., Shock formation within sonoluminescing bubbles, Phys.

Fluids, 11, 10–17 (1999).