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Reaction-Diffusion Kinetics of a Single Sonoluminescing Bubble

Rüdiger Tögel

2002

Ph.D. thesisUniversity of Twente

Also available in print:http://www.tup.utwente.nl/catalogue/book/index.jsp?isbn=9036518407

T w e n t e U n i v e r s i t y P r e s s

http://www.tup.utwente.nl/catalogue/book/index.jsp?isbn=9036518407

REACTION–DIFFUSION KINETICS OF A SINGLESONOLUMINESCING BUBBLE

The research described in this thesis was funded by FOM (Fundamenteel Onderzoekder Materie). The research was carried out at the Physics of Fluids group of theUniversity of Twente.

Publisher:Twente University Press,P.O Box 217, 7500 AE Enschede, The Netherlandswww.tup.utwente.nl

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c© R.Togel, Enschede, The Netherlands 2002

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ISBN 90 365 18407

REACTION–DIFFUSION KINETICS OF A SINGLESONOLUMINESCING BUBBLE

PROEFSCHRIFT

ter verkrijging vande graad van doctor aan de Universiteit Twente,

op gezag van de rector magnificus,prof. dr. F.A. van Vught,

volgens besluit van het College voor Promotiesin het openbaar te verdedigen

op woensdag 11 december 2002 te 13.15 uur

door

Rudiger Togel

geboren op 10 mei 1971te Fulda, Duitsland.

Dit proefschrift is goedgekeurd door de promotor:

prof. dr. rer. nat. D. Lohse

Quam multa ... per secretum eunt, numquam humanis oculis orientia!Neque enim omnia Deus homini fecit.

Quota pars operis tanti nobis committitur... Multa venientis aevi populus ignota nobis sciet.

Multa saeculis tunc futuris cum memoria nostri exoleverit reservantur.... Rerum natura sacra sua non semel tradit.

Initiatos nos credimus, in vestibulo eius haeremus.Illa arcana non promiscue nec omnibus patent.

Reducta et in interiore sacrario clausa sunt,ex quibus aliud haec aetas, aliud quae post nos subibit aspiciet.

Seneca, naturales quaestionesVII,c.30,3ff

TABLE OF CONTENTS i

Table of Contents

1 Introduction 11.1 Single-bubble sonoluminescence – A brief overview . . . . . . . . . 11.2 A guide through the chapters . . . . . . . . . . . . . . . . . . . . . 4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Squeezing alcohols into SBSL bubbles 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 A phenomenological model . . . . . . . . . . . . . . . . . . . . . . 132.4 Theoretical estimate of the parameter Aalc . . . . . . . . . . . . . 17References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 A drunken bubble 233.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Does water vapor prevent upscaling SBSL 294.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Experimental setup and results . . . . . . . . . . . . . . . . . . . . 304.3 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.1 Bubble dynamics . . . . . . . . . . . . . . . . . . . . . . . 314.3.2 Mass diffusion . . . . . . . . . . . . . . . . . . . . . . . . 334.3.3 Heat diffusion . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 34References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 The effect of excluded volume 415.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.2 Thermodynamics of a reactive Van Der Waals gas . . . . . . . . . . 42

ii TABLE OF CONTENTS

5.3 A refined theoretical model . . . . . . . . . . . . . . . . . . . . . 445.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6 Phase diagrams for sonoluminescing bubbles 536.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2 The hydro/thermodynamical ODE model . . . . . . . . . . . . . . 56

6.2.1 Bubble dynamics . . . . . . . . . . . . . . . . . . . . . . . 566.2.2 Gas pressure . . . . . . . . . . . . . . . . . . . . . . . . . 576.2.3 Mass transport . . . . . . . . . . . . . . . . . . . . . . . . 576.2.4 Heat transport . . . . . . . . . . . . . . . . . . . . . . . . . 596.2.5 Transport parameters . . . . . . . . . . . . . . . . . . . . . 596.2.6 Chemical reactions . . . . . . . . . . . . . . . . . . . . . . 616.2.7 Energy balance . . . . . . . . . . . . . . . . . . . . . . . . 626.2.8 Summary of the ODE model . . . . . . . . . . . . . . . . . 66

6.3 Phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.3.1 Air bubbles at f = 20.6 kHz . . . . . . . . . . . . . . . . . 676.3.2 Argon-Nitrogen bubbles at f = 33.4 kHz . . . . . . . . . . 696.3.3 Nitrogen bubbles at f = 33.4 kHz . . . . . . . . . . . . . . 696.3.4 Air bubbles at f = 33.4 kHz . . . . . . . . . . . . . . . . . 696.3.5 Xenon-Nitrogen bubbles at f = 33.4 kHz . . . . . . . . . . 70

6.4 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.5 Robustness of the model . . . . . . . . . . . . . . . . . . . . . . . 74

6.5.1 Finite rate of condensation . . . . . . . . . . . . . . . . . . 756.5.2 The boundary layer thickness . . . . . . . . . . . . . . . . 77

6.6 Conclusions and summary . . . . . . . . . . . . . . . . . . . . . . 78References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7 Conclusions and outlook 89References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Summary 95

Samenvatting 97

Zusammenfassung 99

Acknowledgments 101

About the Author 103

CHAPTER 1. INTRODUCTION 1

Chapter 1

Introduction

1.1 Single-bubble sonoluminescence – A brief overview

Sonoluminescence composed from the latin sonus, the sound, lumen, the light andescere, to become, is, as the name already indicates, the rather unique process ofconversion of sound into light. The instrument of this conversion is a typically mi-crometer sized gas bubble which is excited to radial oscillations by means of an ex-ternal acoustical field. With suitable external parameters such a bubble exhibits arapid compression phase – a consequence of conservation of energy –, which resultsin an energy focusing of up to 9 orders of magnitude. Gas temperatures as high as104 K, and molecular energies on the order of electron volt, i.e., in the energy rangeof visible light,can be produced in this way.

Though Frenzel and Schultes [1] had discovered already 68 years ago that pho-tographic plates in the vicinity of a cloud of cavitating bubbles get blackened, thisphenomenon experimentally has become accessible in a controlled manner only 12years ago, when Felipe Gaitan, then a doctoral student with Larry Crum at the Uni-versity of Mississippi, conducted experiments on levitation of gas bubbles in a stand-ing acoustical wave [2]. With a moderate pressure amplitude of the acoustical field(Pa ≈ 1.2−1.4 bar) and the water degassed to around twenty percent of its saturationconcentration, he observed that

’as the pressure was increased the degassing action of the sound field was reduc-ing the number of bubbles, causing the cavitation streamers to become very thin untilonly a single bubble remained. The remaining bubble was approximately 20µm inradius ... and was remarkably stable in position and shape, remained constant in sizeand seemed to be pulsating in a purely radial mode. With the room lights dimmed,a greenish luminous spot the size of a pinpoint could be seen with the unaided eye,near the bubbles position in the liquid.’

It turned out that with the proper external conditions such a bubble could be

2 1.1. SINGLE-BUBBLE SONOLUMINESCENCE – A BRIEF OVERVIEW

FIGURE 1.1: Picture of a sonoluminescing bubble. One recognizes the water filledflask and one of the two piezo electric disks, which are placed opposite to each otheralong the equatorial circumference and provide the necessary acoustical excitation.The glowing bubble is witnessed by the bright spot in the center. Note that for bettervisibility the bubble is illuminated by a laser.

maintained stable for hours, regularly emitting a short flash of light once every cycle.It is this amazing stability of the phenomenon – which was given the name singlebubble sonoluminescence (SBSL) – that opened a whole new field of experimentaland theoretical research which has led to good understanding of the basic principles


involved. [3]. Being one of the simplest two-phase systems possible – a single pul-sating sphere without any translational motion – it is sometimes referred to as the“hydrogen atom” of cavitation physics.

The experimental setup is astoundingly simple. It consists of a few standardelectronic components, a container for the liquid and some purified water. Its cen-tral component is a water filled flask, which is mechanically excited at one of itseigenfrequencies, see Fig. 1.1 As a consequence, a standing wave field develops inthe antinode of which a bubble is automatically trapped and starts to undergo strongradial oscillations.

Fig. 1.2 (upper part) depicts the bubble dynamics for one acoustical cycle. Inthis case it has been monitored by Mie-scattering [4, 5], i.e., a laser is shone ontothe bubble and the scattered light is collected with a photomultiplier tube. As theintensity of the scattered light is roughly proportional to the surface area of the bub-ble, 1 it provides a convenient means to get information about its relative size. Thelower graph gives in addition the acoustic driving pressure amplitude recorded witha PVDF hydrophone at a distance of 1cm from the bubble.The response of the bubble to the external forcing can be subdivided into three dif-ferent stages:(i) During the phase of negative pressure amplitude, seen in the lower graph of Fig.1.2, the bubble expands from its ambient radius roughly tenfold for typically one halfof an acoustical cycle.(ii) As the pressure amplitude approaches zero the force exerted by the acoustical fieldvanishes. The pressure in the bubble has essentially dropped to zero at this point. Asa consequence atmospheric pressure and the inertia of the liquid rapidly compress thecavity to about one tenth of its ambient size 2, the collapse eventually being stoppedby the enormous pressure inside the bubble (≈ 1GPa) that build up during the col-lapse. The last phase of the compression furthermore takes place so fast that the gasin the bubble behaves nearly adiabatically and compressional heating to temperaturesof more than 104K occurs. A small amount of the bubble contents gets ionized at thisstage and the interaction of the electrons with the gas in the bubble is observed as ashort flash of light typically lasting 100−200 ps. The bubble has now lost about 90%of the energy it possessed at its maximum radius. Not by emission of light though(the energy loss by light emission is typically 0.01%), but by acoustical radiation, asat this stage the negative acceleration of the bubble wall has generated an outgoingshockwave in the liquid. It is reflected by the spike in the acoustical signal as the

1with a small aperture of the PMT one observes some additional fine structure due to interference,when the bubble radius equals a multiple of the wavelength of the employed light

2It should be emphasizes that the optical technique that is used here to monitor the bubble dynamicshas its limitations. It fails when the minimum bubble radius is of the same order as the wavelengthof the laser light. In addition, disturbances due to a highly compressed liquid layer around the bubblebecome important [5].

4 1.2. A GUIDE THROUGH THE CHAPTERS

0

5

10

R(t

)/R

0

0 0.2 0.4 0.6 0.8−1

0

1

time t / T

Pa (

norm

aliz

ed)

FIGURE 1.2: Plot of the radial dynamics of an SBSL bubble (upper graph) and thedriving pressure Pa (lower graph), respectively. The bubble radius is monitored byMie scattering techniques [4, 5], the driving pressure has been recorded with a PVDFhydrophone at a distance of 1cm from the bubble. The shock wave generated bythe bubble during its main collapse is nicely witnessed by the spike in the acousticalsignal at t/T ≈ 0.5.

shock passes the hydrophone, which in this example happens at t/T ≈ 0.5.(iii) The collapse is followed by a series of free, damped oscillations of the bubblewith essentially its eigenfrequency (Minnaert frequency [6, 7]) until the next cyclebegins and expansion starts anew.

1.2 A guide through the chapters

The remarkable energy focusing ability of SBSL naturally has raised the questionwhether the phenomenon can be even enhanced, and the gas in the bubble can be


pushed to an even more extreme state of matter. In other words, how can one upscaleSBSL?

Many different ideas have been proposed and tried in experiment, such as forexample, the application of a strong magnetic field or higher harmonics in the exter-nal driving, the use of other liquids or the addition of other substances to the water.Indeed, aliphatic alcohols were found to have an enormous effect on the light emis-sion, though in the wrong direction: A mole fraction of alcohol as small as 10−4

– equivalent to one drop of pure alcohol in a bucket of water – is found to entirelyextinguish the light emission [8].

Chapter 2 (see also Squeezing alcohols into sonoluminescing bubbles: The uni-versal role of surfactant [9]) presents experimental results for three different alcohols– ethanol, propanol and butanol – which show that the observed light quenching ef-fect is the stronger the longer the carbon chain of the employed alcohol. As theunderlying mechanism it is proposed that the alcohol molecules, which accumulateon the bubble surface due to their hydrophobic nature, are squeezed into the bubbleduring its main collapse, where they increase the heat capacity of the bubble contentsand consequently lower its adiabatic exponent. Accordingly, less heating occurs andless light is emitted. The light reduction predicted from this phenomenological modelis in quantitative agreement with the experimental data.

As a surface active substance alcohols also reduce the tension of the bubble-water interface and thereby change the bubble dynamics and the stability againstperturbation from the spherical form. The influence of the reduced surface tensionon Pa − R0 phase diagrams and in particular the transition from stable to unstableSBSL, which is observed in experiment, is investigated in chapter 3 (see also Sono-luminescence in alcohol contaminated water: a drunken bubble [10]).

Another promising candidate for upscaling SBSL has been the reduction of theacoustical driving frequency. Moreover, SBSL at low frequency was considered akey experiment to test the quality and range of applicability of the theory developedhitherto. Such experiments should in particular reveal the role of water vapor whichso far had been neglected in the theory. For frequencies of around 5 kHz two featureshave been expected: (i) The bubble should emit 100-1000 times as many photons ascompared to the usual frequency of 20 kHz or higher. (ii) In contrast to the standardfrequency the light pulse should be clearly longer in the red regime of the spectrumthan in the blue regime. Chapter 4 presents experimental results obtained at such lowfrequencies (around 7 kHz), which give evidence however, that neither of these ex-pectations is confirmed. The observed discrepancy is theoretically explained by thepresence of water vapor – more precisely by its large heat capacity – , which turnsout to dominate the scenario at low driving frequency and to have considerable influ-ence also at the standard frequencies, see also Does water vapor prevent upscalingsonoluminescence [11]).

6 1.2. A GUIDE THROUGH THE CHAPTERS

The presence of large quantities of vapor in the bubble has an immediate conse-quence: The vapor will dissociate when heated during the collapse and thereby con-sume a major part of the compressional work applied to the bubble, as was demon-strated theoretically by Szeri and Storey [12]. Their simulation suggests that thechemical reactions of the vapor provide an upper bound for the maximum tempera-ture at around 7000 K.

These temperatures however, are not compatible with the (commonly accepted)thermal bremsstrahlung model for light emission, since at 7000 K virtually no atomsor molecules in the bubble get excited or ionized and correspondingly no bremsstrahl-ung occurs.

Chapter 5 (see also Suppressing dissociation in sonoluminescing bubbles: Theeffect of excluded volume [13]) resolves this discrepancy. In an ab initio thermody-namical treatment of the multi component gas in the bubble, it is shown that for thehigh densities achieved at collapse the finite size of the gas molecules leads to a shiftof the equilibrium constants of the various chemical processes and thereby to a pro-nounced suppression of particle producing reactions. As most of these reactions areendothermic, their suppression then again leads to temperatures considerably higherthan the ones reported by Szeri and Storey – temperatures that in particular are com-patible with the occurrence of thermal bremsstrahlung.

One of the most spectacular discoveries in the field of SBSL was that undersonoluminescence conditions an initial air bubble equilibrates on a diffusive timescaleof seconds to an essentially pure argon bubble [14]. The argon originates from itssmall natural abundance in air. First suggested in 1997 by Lohse et al. it is nowadaysfrequently referred to as dissociation hypothesis.

Lohse et al. for the first time took into account the air chemistry that occursin the bubble and thereby could explain the overall topology of Pa − R0 phase dia-grams. In particular, the equilibration to an argon bubble in the SBSL regime appearsas a natural consequence in that model. A quantitative comparison however had notbeen attempted because (i) the theory was not fully developed those days (chemicalreactions and heat losses were treated in a simplified way only and water vapor wasnot taken into account at all) (ii) only little reliable experimental data were avail-able. This has changed meanwhile as among others Apfel and Ketterling [15, 16]extensively mapped out Pa −R0 phase diagrams for various different gas mixtures.

Chapter 6 (see also Phase diagrams for sonoluminescing bubbles: A compari-son between experiment and theory [17]) extends the model of the preceding sectionsto air bubbles and provides such a detailed comparison of experimental phase dia-grams and their theoretical counterpart. Striking quantitative agreement is found inall considered cases.

Chapter 7 forms the conclusion. It summarizes the limitations the bubble tem-perature is subjected to and gives some recommendations for its optimization. The


enhancement of the collapse through addition of harmonics to the driving signal [18]is one of the possibilities discussed in that context. The employment of completelydifferent pulse forms like shock waves seems to be another more extreme option. Theeffects of such waves on the bubble and its surroundings will briefly be consideredin that chapter [19, 20]. Finally, the problem of spatial control is addressed since ina general flow situation usually many different forces act on a bubble which can leadto surprising translational motion [21].

8 REFERENCES

References[1] H. Frenzel and H. Schultes, Z. Phys. Chem. 27B, 421 (1934).

[2] D. F. Gaitan, Ph.D. thesis, The University of Mississippi, 1990.

[3] M. P. Brenner, S. Hilgenfeldt, and D. Lohse, Rev. Mod. Phys. 74, 425 (2002).

[4] F. Gaitan, Physics World 12, 20 (1999).

[5] B. Gompf and R. Pecha, Phys. Rev. E 61, 5253 (2000).

[6] T. G. Leighton, The acoustic bubble (Academic Press, London, 1996).

[7] C. E. Brennen, Cavitation and Bubble Dynamics (Oxford University Press, Ox-ford, 1995).

[8] B. P. Barber, R. A. Hiller, R. Lofstedt, S. J. Putterman, and K. R. Weninger,Phys. Rep. 281, 65 (1997).

[9] R. Togel, S. Hilgenfeldt, and D. Lohse, Phys. Rev. Lett. 84, 2509 (2000).

[10] R. Toegel, S. Hilgenfeldt, and D. Lohse, in IUTAM Symposium on Free Sur-face Flows, edited by A. C. King and Y. D. Shikhmurzaev (Kluwer AcademicPublishers, Dordrecht, 2001), p. 297.

[11] R. Toegel, B.Gompf, R.Pecha, and D.Lohse, Phys. Rev. Lett. 85, 3165 (2000).

[12] B. Storey and A. Szeri, Proc.R. Soc. London, Ser. A 456, 1685 (2000).

[13] R. Toegel, S.Hilgenfeldt, and D.Lohse, Phys. Rev. Lett. 88, 034301 (2002).

[14] D. Lohse, M. P. Brenner, T. Dupont, S. Hilgenfeldt, and B. Johnston, Phys. Rev.Lett. 78, 1359 (1997).

[15] J. A. Ketterling and R. E. Apfel, Phys. Rev. Lett. 81, 4991 (1998).

[16] J. A. Ketterling and R. E. Apfel, Phys. Rev. E 61, 3832 (2000).

[17] R. Toegel and D. Lohse, J. Chem. Phys., in press .

[18] X. Lu, A. Prosperetti, R. Toegel, and D. Lohse, Harmonic enhancement of sin-gle bubble sonoluminescence, preprint.

REFERENCES 9

[19] T. J. Matula, P. R. Hilmo, B. Storey, and A. J. Szeri, Phys. Fluids 14, 913 (2002).

[20] M. C. Jullien, C. D. Ohl, R. Toegel, and D. Lohse, Dynamical Response of abubble submitted to two shock waves, preprint.

[21] J. Rensen, D. Bosman, J. Magnaudet, C. D. Ohl, A. Prosperetti, R. Togel, M.Versluis, and D. Lohse, Phys. Rev. Lett. 86, 4819 (2001).

10 REFERENCES

CHAPTER 2. SQUEEZING ALCOHOLS INTO SBSL BUBBLES 11

Chapter 2

Squeezing alcohols intosonoluminescing bubbles: Theuniversal role of surfactants †

We conduct an experimental study of the dependence of single bubble sonolumines-cence intensity on the concentration of various alcohols. The light intensity is re-duced by one half at a molar fraction of ≈ 2.5 · 10−5; butanol achieves the samereduction at a 10 times smaller concentration. We account for the results by a theo-retical model, in which the alcohols are assumed to be mechanically forced into thebubble at collapse, modifying the adiabatic exponent of the gas. The increasing hy-drophobicities of the alcohols lead to decreasing effective adiabatic exponents, andthus to less heating and therefore less light. Support for this model is obtained by re-plotting the experimental light intensity values vs the calculated exponents, yieldinga collapse of all data onto a universal curve.

2.1 Introduction

In 1990 Felipe Gaitan [1] discovered that a single air bubble trapped in an acousticalfield can emit bursts of light so strong as to be visible to the naked eye. During recentyears extensive theoretical and experimental research has been done in this field, andnow the basic mechanisms of this so-called single bubble sonoluminescence (SBSL)seem to be resolved [2–17]. The parameter regime of SBSL is set by the shapestability of the bubble [3–5], its diffusive stability [3, 18] and its chemical stability

†See also R. Togel, S. Hilgenfeldt and D. Lohse, Squeezing Alcohols into Sonoluminescing Bubbles:The Universal Role of Surfactants, Phys. Rev. Lett. 84, 2509-2512 (2000)

12 2.2. EXPERIMENT

[8, 12, 13]. The light seems to originate from thermal bremsstrahlung [10, 11, 16, 17],induced by (nearly) adiabatic heating of the bubble in the final stages of the radialcollapses induced by the acoustic driving.

2.2 Experiment

Inspired by the experimental finding that one drop of alcohol can extinguish SBSL[19, 20] and by the study on the effect of alcohols in multi bubble sonoluminescence(MBSL) by Grieser’s group [21, 22], we experimentally study the influence of variousalcohols on SBSL in the present paper.

A SBSL bubble was created in a spherical flask containing purified water, drivenby two piezo transducers at a frequency of 35 kHz. Degassing of the dissolved air wasachieved through boiling. The oxygen concentration was directly monitored throughoxyometry from which the argon concentration that is the relevant one in SBSL [8]can easily be found. The light from the bubble was focused onto a high-sensitivityphoto diode. The resulting photocurrent signal was amplified, digitized, and read outby a computer.

Small amounts of different alcohols (ethanol C2H5OH, 2-propanol C3H7OH,and 1-butanol C4H9OH) were added. Subsequent stirring was necessary because ofthe long diffusive time scale of alcohol in water ( τdiff = R2

flask/D ≈ 29 days withthe flask radius Rflask = 5 cm and the typical diffusion constant D ≈ 10−5 cm2/s ofthe abovementioned alcohols in water). The driving frequency must be close to a res-onance frequency of the flask in order to obtain high driving pressures and thereforeSBSL. Adding a small liquid volume detunes the resonator, so that small adjustmentsof the driving frequency ( 10Hz) were necessary in order to keep the driving pres-sure constant (at the peak of the resonance). Possible effects of this adjustment ofresonance frequency were checked by adding an equal amount of water instead ofalcohol. No intensity changes were observed in this case.

The light intensity as a function of the bulk molar fraction of alcohol nb is seenin Fig. 2.1. Tiny amounts of alcohol strongly decrease the intensity in all cases.The effect is the more pronounced the longer the carbon chain of the alcohol is: forbutanol a 10 times smaller concentration is sufficient to halve the light intensity, seeTable 2.1.

Qualitatively, Ashokkumar et al. [21] observe the same trend in MBSL, butat considerably larger alcohol concentrations, with half-intensity concentrations ofnb ≈ 2 · 10−3 (ethanol) and nb ≈ 1 · 10−4 (butanol). They postulate [21] that the hy-drophobic alcohol molecules accumulate at the bubble surface, are then forced insideduring bubble collapse, and there quench the light emission. The surface accumula-tion will be stronger for longer carbon chains. Our work picks up these ideas for theSBSL case with some modifications, and proceeds to develop a quantitative model.


FIGURE 2.1: Logarithmic plot of the normalized SBSL light intensities vs bulk al-cohol concentration nb. The symbols refer to measurements with different alcoholspecies. The solid lines show fits according to Eq. (2.5). Note that the dependenceI(nb) is in general not exponential, but approaches exponential decay for very smallnb.

2.3 A phenomenological model

The calculation of the expected light intensity within our theory can be subdividedinto four steps: (i) Molar fraction ng(nb) in the gas bubble: The alcohol bulk con-centration nb determines the molar alcohol fraction ng inside the bubble. This depen-dence is, in general, nontrivial, with hydrophobicity, vapor pressure, surface tension,and even the bubble dynamics itself factoring in [23]. For low concentrations, how-ever, we can assume a linear dependence,

ng = Aalcnb , (2.1)

14 2.3. A PHENOMENOLOGICAL MODEL

with an alcohol dependent fit parameter Aalc . Later we will theoretically calculateAalc , finding good agreement with the fitted results.

(ii) Adiabatic exponent γ(ng) of the alcohol-argon mixture: After the alcoholhas been squeezed into the bubble, it basically contains a mixture of argon, whichaccumulates in sonoluminescing air bubbles [8], and the vapor of the employed al-cohol. The effective adiabatic exponent of the mixture can be derived from Dalton’slaw,

γ(ng) =ng(falc + 2) + (1− ng)(fAr + 2)

ngfalc + (1− ng)fAr. (2.2)

Here, fAr = 3 and falc indicate the number of degrees of freedom of argon andalcohol, respectively. As temperatures inside a collapsing SBSL bubble very likelycorrespond to energies of 1-2 eV [16], while typical energies for vibrational quantaare ≤ 0.1 eV (those of rotational quanta are even two orders of magnitude smaller),we can safely assume that all degrees of freedom are available. Thus, falc = 6+2×(3N − 6), where N is the number of atoms in the molecule. Note that γ in (2.2) willonly slightly change in the early stages of heating of the bubble, when the alcoholmolecules begin to dissociate, as γ only depends on the product ng(falc − 3), andfalc 1: E.g., dissociation in two products of about equal size roughly halves thenumber of degrees of freedom of each but doubles ng.

(iii) Maximum temperature Tmax (γ) in the bubble: SBSL light emission hasbeen explained satisfactorily as thermal bremsstrahlung of the (nearly) adiabaticallyheated gas inside the bubble [10, 11, 16, 17]. Calculations coupling bubble dynamicswith heat exchange have shown [24] that the gas inside the bubble maintains theambient temperature most of the time. Only at collapse it is heated adiabatically fora brief time. To keep the model simple, we will replace the actual change in thermalcoupling by an abrupt crossover from isothermal to adiabatic behavior at a certainbubble radius Rad . From Rad , the bubble is assumed to adiabatically collapse downto Rmin , which is very close to the van der Waals hard core radius h. From the vander Waals equation of state, we therefore have

Tmax (ng) ∝(R3

ad − h3

R3min − h3

)γ(ng)−1

. (2.3)

As a priori we do not know the length ratio Balc ≡ ((R3ad − h3)/(R3

min − h3))1/3, itis the second fit parameter of the model.

(iv) Light intensity I(Tmax ): Refs. [10, 11, 16, 17] show that the light intensity I(as a function of the temperature) is essentially obtained as the product of two factors:(a) blackbody radiation according to the actual temperature of the bubble (∝ T 4),and (b) the finite opacity of the bubble (dominated by a factor ∝ exp(−Eion/2kBT ),Eion being the ionization energy of argon and kB the Boltzmann constant). The


relative intensity reduction due to the addition of alcohol is thus approximately:

I(ng)I(0)

=(Tmax (ng)Tmax (0)

)4

exp

(−Eion

2kB

(1

Tmax (ng)− 1Tmax (0)

)). (2.4)

We now plug the four elements of the model together and obtain a well-defined func-tion of light intensity vs nb,

I(nb)I(0)

= F (nb, falc , Aalc , Balc) , (2.5)

with two fit parameters Aalc and Balc to which we have attributed physical meaningabove. The nonlinear fit (2.5) indeed describes our data sets very well, see Fig. 2.1.Furthermore, we find that the parameter Balc is virtually the same for all alcohols(it varies by about ±10% around Balc = 1.3). It can therefore be fixed independentof the alcohol species without affecting the quality of the fit. The physical reasonfor this is that the radius Rad is determined by the external parameters like argonconcentration (viaR0), driving pressure, etc., on which the addition of small amountsof alcohol has very little influence. Here we fixed it at Balc = 1.25. Best fits of theparameter Aalc for Tmax (nb = 0) = 15000K [16] are listed in Table 2.1 [25].

alcohol nb(I(0)/2) falc Aalc Aalc,th rdif∂σ∂nb/gs−2 pv/kPa

ethanol 2.6 · 10−5 48 475 564 14.9 −1213 7.87propanol 5.3 · 10−6 66 1568 1681 4.2 −3614 2.76butanol 2.7 · 10−6 84 1920 4190 1.3 −9007 0.86

TABLE 2.1: Characteristic parameters of the intensity data. nb(I(0)/2) is the mea-sured bulk alcohol concentration at which the SL intensity is halved, falc the numberof degrees of freedom. Aalc results from the fit Eq. (2.5) to the data and Aalc,th fromtheory. rdif is obtained from eq. (2.9). The change in surface tension with nb andthe vapor pressure of the pure substance (last two columns) are material parameterstaken from Refs. [26, 27] (for a temperature of 25oC).

The small value ofB shows that adiabaticity is achieved close toRmin , in agree-ment with Rayleigh-Plesset simulations [16].

The most striking result of the fit is the strong increase of Aalc with increas-ing carbon chain length. This should be expected as the longer alcohols are morehydrophobic and accumulate more strongly at the gas-water interface. For the sameconcentration inside the bubble, and consequently the same amount of light reduc-tion, very different alcohol bulk concentrations are needed, as observed. We also findthat the required alcohol concentration ng inside the gas bubble for halving the inten-sity is always very similar, ng ≈ 0.01 − 0.02, regardless of the type of alcohol. This

16 2.3. A PHENOMENOLOGICAL MODEL

resembles the result in [21] for MBSL, where the light intensity is roughly indepen-dent of the kind of alcohol added if plotted against the alcohol surface concentrationat the gas-water interface.

FIGURE 2.2: Relative light intensity data as in Fig. 2.1, now plotted vs the effec-tive adiabatic exponent γ of the argon-alcohol mixture. All data collapse onto oneuniversal curve.

According to the theory presented above, the resulting light intensities are afunction of both ng and falc , so that one would not expect completely universal con-centrations ng. We observe, however, that after fixing B, the relative intensity is aunique function of the adiabatic exponent γ only. We can thus test the theory byplotting I vs γ, where all the data for the different alcohols should collapse ontoone universal curve. Indeed, Fig. 2.2 reveals that the data, when plotted in this fash-ion, show a degree of universality even higher than the aforementioned dependenceof MBSL intensity on surface concentration of alcohol (figure 8 of Ref. [21]). Fig-ure 2.2 is also an a posteriori confirmation of our assumption that the light is of


essentially thermal origin and that it is the adiabatic heating of the argon-alcohol gasmixture which causes the emission.

2.4 Theoretical estimate of the parameter Aalc

We will now theoretically calculate the proportionality constants Aalc in Eq. (2.1)and thereby get further insight in the mechanism that squeezes the alcohols into thebubble. For dilute solutions the bulk concentration nb of surfactant is related to itssurface excess Γs and the change in surface tension σ via the Gibbs relation [28],

Γs = − nbkBT

(∂σ

∂nb

)T. (2.6)

The maximum achievable (critical) surface excess is given by close-packing of thesurfactant molecules on the surface, each of which occupies an area of about 0.25 nm2

[29], thus defining Γs,crit = (0.25nm2)−1. We will show below that the results of thetheory do not depend on the exact area per molecule. Γs,crit implies a critical num-ber of alcohol molecules sitting on the interface at bubble collapse, Ncrit(Rmin) =Γs,crit4πR2

min . This number must be compared with the number of alcohol moleculeswhich are “loaded” onto the surface at the bubble radius maximum, N(Rmax ) =Γs4πR2

max . We now assume that the alcohol molecules will not desorb or diffuseaway in the short time interval of bubble collapse, but that the total excess of alcoholat bubble minimum, ∆N = N(Rmax )−Ncrit(Rmin ), will enter the bubble, resultingin the alcohol concentration

ng = ∆N/(NAr +∆N) (2.7)

in the gas bubble. When the collapse is complete, this alcohol will be “burned” (justas the nitrogen or oxygen molecules sucked into the bubble [8]), and the reactionproducts will dissolve in the water.

For the forcing pressures (≈ 1.3 atm) and argon concentrations (≈ 0.2 − 0.4%of saturation) of our experiment, a typical value for the ambient radius is R0 = 5µm[3, 4], corresponding to NAr ≈ 1.67 · 1010. Minimum and maximum radii in thisregime are Rmin ≈ R0/10 and Rmax ≈ 10R0, respectively [7]. With ∂σ/∂nb fromTable 2.1 we get Ncrit(Rmin) ≈ 1.3 · 107, whereas in the experimental range ofvalues for nb we haveN(Rmax ) ≈ 1.3 ·109 for ethanol andN(Rmax ) ≈ 1.2 ·109 forbutanol. In all cases, NAr N(Rmax ) Ncrit(Rmin) so that (2.7) simplifies tong ≈ N(Rmax )/NAr. Thus, the exact value of Γs,crit does not matter, as nearly allalcohol molecules which accumulate at the bubble surface at maximum get squeezedinto the bubble at collapse.

From Eqs. (2.6), (2.7) we obtain an a posteriori justification of the assumed

linear relation between ng and nb for small nb < NAr

(4πR2

maxkBT

∣∣∣( ∂σ∂nb

)T

∣∣∣)−1. The

18 2.4. THEORETICAL ESTIMATE OF THE PARAMETERAALC

desired theoretical value of Aalc is then

Aalc,th =4πR2

max

NArkBT

∣∣∣∣(∂σ

∂nb

)T

∣∣∣∣ . (2.8)

We find good agreement with the fitted values, see Table 2.1. As Aalc,th depends onRmax , the experiments were conducted such that the external conditions like fillingheight of the flask, pressure amplitude and degree of degassing were roughly the samegiving roughly the same R0 and Rmax .

We stress that, as in MBSL [22], it is not the liquid vapor pressure which de-termines γ: The liquid vapor pressure is highest for ethanol and lowest for butanol(Table 2.1), which would suggest ethanol as the most efficient light quenching agent.

Two crucial assumptions were made in deriving (2.8), which have to be checked:(i) Eq. (2.6) is an equilibrium formula; therefore, there has to be enough time for thebubble to accumulate N(Rmax ) alcohol molecules within, say, half a driving period.(ii) The collapse must be fast enough to ensure that the excess alcohol does not diffuseback into the liquid but jumps into the bubble.

Condition (i) implies that Ndif , the number of alcohol molecules that can attachto the surface within half a cycle, be greater than N(Rmax ). We assume a diffusion-limited adsorption process and estimate Ndif ≈ 4πR2

max ldif nbNAρ/M , where NA

is Avogadro’s constant, ρ the density of water, M its molecular mass, and ldif =(D/2f)1/2 the diffusive length scale. Thus we demand

rdif ≡ Ndif

N(Rmax )=(D

2f

)1/2 GT

|(∂σ/∂nb)T |ρ

M> 1, (2.9)

where G is the universal gas constant. Table 2.1 shows that (2.9) is well fulfilled forethanol and propanol, and is just marginally valid for butanol. The latter may accountfor the relatively large deviation between Aalc and Aalc,th for butanol (see table I).As surface excesses become even larger for higher alkanols, we predict deviationsfrom this theory in experiments with e.g. pentanol or hexanol.

To assess condition (ii) we make a worst-case estimate, disregarding the surfaceaffinity of the alcohols, allowing them to diffuse freely during the collapse which laststypically ∆tcol ∼ 1 ns. A typical diffusion distance is then ∼ (D∆tcol )1/2 ∼ 1 nm,which is on the order of a molecule length, and implies that the alcohol cannot escapefrom the bubble surface during collapse.

For MBSL, Grieser and coworkers suggested that the alcohols and their reactionproducts accumulate in the bubble only over many cycles [22]. Such an accumulationmight also occur in SBSL [20]. However, we believe that SBSL bubbles becometoo hot to sustain and accumulate non-soluble molecules such as C2H2. A crucialexperiment to distinguish between the picture of refs. [20, 22] and the present onewould be to suddenly increase the forcing pressure of a non-SL argon bubble in the


alcohol-water mixture as done in [13] for air and argon bubbles in pure water: Ifaccumulative processes over many cycles play a role, the bubble should still glowbrightly for some period of time, in spite of the alcohol.

In summary, the observed SBSL light extinction effect due to small amountsof alcohols can be understood as follows: At the bubble radius maximum, alco-hol molecules accumulate at the gas-water interface, an effect that is the more pro-nounced the more hydrophobic the alcohol is. When the bubble is compressed, nearlyall the alcohol is forced into the bubble, leading to a reduction of the adiabatic ex-ponent γ and thus less heating of the gas inside the bubble, resulting in less light.Longer-chain alcohols are most effective in quenching the light as they are (i) morehydrophobic and enter the bubble in larger numbers, and (ii) result in a stronger re-duction of γ because of their larger number of degrees of freedom.

The experimentally observed relative light intensity displays universality as afunction of the effective adiabatic exponent, a finding that supports the suggestedmodel and provides another hint at the thermal origin of SBSL.

Acknowledgments

It is our pleasure to acknowledge stimulating discussions with F. Grieser, S. Gross-mann, T. Matula, A. Prosperetti, and K. Suslick. The work is part of the researchprogram of FOM, which is financially supported by NWO.

20 REFERENCES

References[1] D. F. Gaitan, Ph.D. thesis, The University of Mississippi, 1990.

[2] L. A. Crum, Physics Today 47, 22 (1994).

[3] S. Hilgenfeldt, D. Lohse, and M. P. Brenner, Phys. Fluids 8, 2808 (1996).

[4] G. Holt and F. Gaitan, Phys. Rev. Lett. 77, 3791 (1996).

[5] A. Prosperetti, Quart. Appl. Math. 34, 339 (1977); M. Brenner, D. Lohse, andT. Dupont, Phys. Rev. Lett. 75, 954 (1995); A. Prosperetti and Y. Hao, Phil.Trans. Roy. Soc. 357, 203 (1999).

[6] V. Q. Vuong and A. J. Szeri, Phys. Fluids 8, 2354 (1996).

[7] B. P. Barber et al., Phys. Rep. 281, 65 (1997).

[8] D. Lohse et al., Phys. Rev. Lett. 78, 1359 (1997); D. Lohse and S. Hilgenfeldt,J. Chem. Phys. 107, 6986 (1997).

[9] B. Gompf et al., Phys. Rev. Lett. 79, 1405 (1997); R. Pecha, B. Gompf, G. Nick,and W. Eisenmenger, Phys. Rev. Lett. 81, 717 (1998).

[10] W. Moss, D. Clarke, and D. Young, Science 276, 1398 (1997); W. C. Mosset al., Phys. Rev. E 59, 2986 (1999).

[11] L. Frommhold, Phys. Rev. E 58, 1899 (1998).


[13] T. J. Matula and L. A. Crum, Phys. Rev. Lett. 80, 865 (1998).

[14] R. Apfel, Nature 398, 378 (1999).

[15] F. Gaitan, Physics World 12, 20 (1999).

[16] S. Hilgenfeldt, S. Grossmann, and D. Lohse, Nature 398, 402 (1999); Phys.Fluids 11, 1318 (1999).

[17] K. Yasui, Phys. Rev. E 60, 1754 (1999).

[18] M. M. Fyrillas and A. J. Szeri, J. Fluid Mech. 277, 381 (1994).

REFERENCES 21

[19] K. R. Weninger et al., J. Phys. Chem. 99, 14195 (1995).

[20] M. Ashokkumar et al., Proceedings the 16th International Congress on Acous-tics and 135th Meeting Acoust. Soc. Am, Seattle, WA, Vol 3, 1543, 1998; T.Matula, talk given at the Int. Symposium on Nonlinear Acoustics, Gottingen,Sep. 1999.

[21] M. Ashokkumar, R. Hall, P. Mulvaney, and F. Grieser, J. Phys. Chem. 101,10845 (1997).

[22] M. Ashokkumar, P. Mulvaney, and F. Grieser, J. Am. Chem. Soc. 121, 7355(1999).

[23] M. M. Fyrillas and A. S. Szeri, J. Fluid Mech. 311, 361 (1996).

[24] M. Plesset and A. Prosperetti, Ann. Rev. Fluid Mech. 9, 145 (1977).

[25] In principle, Tmax (nb = 0) could be considered a (third) fit parameter. It is,however, known at least in principle from the theory without alcoholic surfac-tants [16]. Also, both Aalc and Balc show only a weak dependence on changesin Tmax (0): varying Tmax (0) between 15000K and 30000 K, Balc changes by14% and Aalc by only 1%.

[26] Handbook of Chemistry, edited by N. A. Lange (McGraw-Hill, New York,1961).

[27] D. R. Lide, Handbook of Chemistry and Physics (CRC Press, Boca Raton,1991).

[28] P. W. Atkins, Physical Chemistry (Oxford University Press, Oxford, 1995).

[29] G.L. Gaines Insoluble Monolayers at liquid/gas interfaces (Interscience: JohnWiley, New York, 1966).

22 REFERENCES

CHAPTER 3. A DRUNKEN BUBBLE 23

Chapter 3

Sonoluminescence in alcoholcontaminated water: A drunkenbubble †

The addition of one drop of alcohol to the water of a flask with a sonoluminescingbubble not only strongly quenches the observed light intensity but can also cause atransition from stable to unstable SBSL. We theoretically account for the effect byconsidering the surface active properties of alcohols. The reduction of the surfacetension significantly influences the diffusive equilibrium of the bubble and the shapestability and thereby induces a transition to unstable SBSL.

3.1 Introduction

In the past few years the strong quenching effect of small amounts of alcohols onsonoluminescence in water has been studied extensively (see e.g. [1–5]). Besidesthe drastic decrease of the light intensity observed already at tiny (bulk) concentra-tions (molar fraction α ≈ 10−4−10−5) another feature eventually occurs, which hasnot been addressed so far: Above a certain critical (bulk) concentration of alcoholthe originally stable SBSL turns unstable and the bubble starts to ”blink” on a slowdiffusive time scale τgas ∼ 1 s. Moreover, it starts to erratically ”dance”. Both fea-tures characterize unstable sonoluminescence ( [6, 7]). Further addition of alcoholincreases the frequency of the blinking and dancing. In this article we present exper-imental results (section 3.2) and a theoretical explanation (section 3.3) which relatesthe phenomenon to the surface active properties of alcohols, namely the reduction of

†See also R. Toegel, S. Hilgenfeldt, D. Lohse, Sonoluminescence in Alcohol Contaminated Water:A Drunken Bubble, IUTAM 2000 proceedings (Kluwer Academic Publishers, Dordrecht, 2001), p.297

24 3.2. EXPERIMENT

the interfacial tension σ of the bubble wall.

3.2 Experiment

SBSL was achieved by exciting a spherical water-filled flask (≈ 300ml) at one ofits resonance frequencies and trapping a bubble in the corresponding standing wavefield. We used the first spherically symmetric harmonic which was around 35.5 kHz.The gas concentration was monitored via the oxygen concentration. The light fromthe bubble was optically mapped onto a highly sensitive photodiode. The time de-pendence of the intensity is shown in Figure 3.1 for three different alcohol fractions:α = 0, α = 1.4 × 10−4 and α = 1.9 × 10−4. The relative argon saturation wasc∞,Ar/c0,Ar ≈ 2× 10−3 where c∞,Ar is the concentration far away from the bubbleand c0,Ar the saturation concentration. It is observed that above a molar alcohol frac-tion α ≈ 1.0 × 10−4 the light intensity exhibits roughly periodic behavior (unstableSBSL) whose mean period T is the shorter the larger the alcohol fraction. The sharpbreakdown of the light intensity that is usually observed in unstable SBSL (cf. [8]) issomewhat washed out in the experimental data (see Figure 3.1) as the photo diode’slarge capacitance allows for a time resolution of only ∼ 50ms.

The respective alcohol concentrations were achieved by successively addingsmall amounts of alcohol. Subsequent stirring was necessary because of the longdiffusive time scale of alcohol in water (τdiff = R2

flask/D ≈ 30 days). The proce-dure was checked by adding the same amount of water instead of alcohol followingthe same experimental protocol. No effect was observed then; the bubbles glowed ina stable way, the intensity varying by no more than ±5%.

3.3 Mechanism

The regime for stable SBSL in the Pa–R0 phase space is mainly determined by threecriteria ( [7]): (i) The condition of diffusive equilibrium; this line indicates pointsin the phase space of zero net mass change of the bubble over one cycle (ii) Thecondition of shape stability, which ensures that distortions from the spherical bubbleshape are damped out. (iii) The bubble collapse has to be violent enough so thatadiabatic heating of the bubble’s interior is possible.

As pointed out in [7], the bubble’s equilibrium radius R0 follows the stablediffusive equilibrium line when increasing the driving pressure Pa. The sonolumi-nescence is stable when for given Pa the corresponding value of R0 is smaller thana critical value R0,crit given by the shape stability line and unstable otherwise. Thekey towards a theoretical understanding of the above observed phenomenon is thatthe addition of alcohol causes a pronounced decrease of the interfacial tension σ ofthe bubble wall which turns out to considerably ”shift” the diffusive equilibrium and


0 1 2 3 4 5 6 7

1.4

1.6

time / s

1.4

1.6

Inte

nsity

(ar

bitr

ary

units

)

3.0

3.5

4.0

FIGURE 3.1: SBSL light intensity as a function of time for different concentrations ofethanol. From top to bottom: α = 0: constant intensity is observed, α = 1.4× 10−4:mean period T = 1.7 s, α = 1.9× 10−4: mean period T = 0.9 s.

the onset of shape instabilities in Pa–R0 phase space. In particular, an initially stableconfiguration can turn unstable. However, there is only little data on the surface ten-sion of an water-alcohol mixture at very low alcohol concentration available. There-fore, we restrict ourselves to a qualitative treatment and assume σ = 72.5×10−3 N/mwithout alcohol and σ = 50 × 10−3 N/m with alcohol. For a mathematical descrip-tion we combine the model for the bubble dynamics of [9], with the stability analysisof [7]. It should be noted that the latter one has been shown to slightly underestimateR0,crit ( [10]) but is nevertheless sufficient for the qualitative statement we are mak-ing here. Figure 3.2 shows the diffusive equilibrium lines (A and A’) and the shapestability threshold (B and B’) for the abovementioned values of the surface tension.

The probable scenario is the following (cf. Figure 3.2): Initially the bubble islocated at a stable point in the Pa-R0 phase space, i.e., along the curve A, belowthe stability line B. Note that we have tuned Pa to achieve the maximum possibleintensity from a stable bubble. In this case the stable point will in the vicinity ofthe stability threshold B – indicated by the dot– because large R0 results in brighter

26 3.3. MECHANISM

FIGURE 3.2: The diffusive equilibrium (curve A and A’) and the (parametric) shapestability line (curve B and B’) for σ = 72.5 × 10−3 N/m (dashed lines) and σ =50 × 10−3 N/m (solid lines). The driving frequency is f = 35 kHz and the argonconcentration c∞,Ar

c0,Ar= 2 × 10−3. The diffusively stable bubble indicated by the dot

starts to grow after the addition of alcohol and runs into the shape stability.

bubbles. Now alcohol is added and the surface tension drops. Correspondingly, (i)the stability line is lowered a little bit (curve B’) which was shown in [11], and at thesame time (ii) the diffusive equilibrium line is shifted towards smaller Pa (curve A’),which – at constant driving pressure – effects an increase ofR0. As a result the bubblemoves upwards in the phase diagram towards the unstable region above curve B’. Inother words, crossover to unstable sonoluminescence takes place. This increase ofthe bubble’s equilibrium size R0 under the influence of alcohols was also observedexperimentally by [5]. Upon further addition of alcohol the difference between thedesired equilibrium size on curve A’ and the maximum size achievable on curve B’is further enlarged which according to [12], increases the diffusive growth rate of thebubble

d

dτR0(τ) ∝

(c∞,Ar

c0,Ar− 〈p(t)〉4

P0

)

and hence the mean frequency of the unstable sonoluminescence. Here, τ character-


izes a slow time scale, i.e., the equation disregards mass exchange processes fasterthan 1/f . The expression 〈p(t)〉4 appearing in the above equation is a weighted meanof the bubble’s pressure over one cycle [12] and P0 is the ambient pressure. The in-crease of the diffusive growth rate is also clearly observed in experiment, see Figure3.1.

In summary, the transition from stable to unstable SBSL under the influenceof alcohols can be rationalized by taking the surface active properties of alcoholsinto account. Experimental evidence for the suggested mechanism is given by [5],who indeed observe a considerable increase of R0 under the influence of alcohols.In addition, it might be possible not only to extinguish SBSL through the additionof alcohol but also to “switch it on”: The size of bubbles especially at the onset ofSBSL, i.e., around the turnaround of curve A, will be effectively increased throughthe addition of alcohol which could results in a higher light intensity. This could beused as a further evidence for the above suggested mechanism.

Acknowledgement

The work is part of the research program of FOM, which is financially supported byNWO.

28 REFERENCES

References[1] K. R. Weninger, R. A. Hiller, B. P. Barber, D. Lacoste, and S. J. Putterman, J.

Phys. Chem. 99, 14195 (1995).

[2] M. Ashokkumar, R. Hall, P. Mulvaney, and F. Grieser, J. Phys. Chem. 101,10845 (1997).

[3] F. Grieser, M. Ashokkumar, and K. Barbour, J. Acoust. Soc. Am. 103, 2924(1998).

[4] R. Toegel, S. Hilgenfeldt, and D. Lohse, Phys. Rev. Lett. 84, 2509 (2000).

[5] M. Ashokkumar, L. A. Crum, C. A. Frensley, , F. Grieser, T. J. Matula, W. B.McNamara, and K. Suslick, J. Phys. Chem. 104, 8462 (2000).

[6] D. F. Gaitan, Ph.D. thesis, The University of Mississippi, 1990.


[8] B. P. Barber, K. Weninger, R. Lofstedt, and S. J. Putterman, Phys. Rev. Lett. 74,5276 (1995).


[10] A. Prosperetti and Y. Hao, Philos. Trans. R. Soc. London, Ser. A 357, 203(1999).

[11] S. Hilgenfeldt, M. P. Brenner, S. Grossmann, and D. Lohse, J. Fluid Mech. 365,171 (1998).


CHAPTER 4. DOES WATER VAPOR PREVENT UPSCALING SBSL 29

Chapter 4

Does water vapor prevent upscalingsonoluminescence? †

Experimental results for single bubble sonoluminescence of air bubbles at very lowfrequency f = 7.1 kHz are presented: In contrast to the predictions of a recent model[S. Hilgenfeldt and D. Lohse, Phys. Rev. Lett. 82, 1036 (1999)], the bubbles areonly as bright ( 104 − 105 photons per pulse ) and the pulses as long ( ≈ 150 ps ) asat f = 20 kHz. We can theoretically account for this effect by incorporating watervapor into the model: During the rapid bubble collapse a large amount of watervapor is trapped inside the bubble, resulting in an increased heat capacity and hencelower temperatures, i.e., hindering upscaling. At this low frequency water vapor alsodominates the light emission process.

4.1 Introduction

Upscaling single bubble sonoluminescence (SBSL) [1–4] is of prime importance bothfor possible application and for understanding the phenomenon. It has been suggested[5] that lowering the acoustical driving frequency from the standard f = 20−35 kHzto f ≈ 5 kHz should yield a 100 to 1000 times higher gain of light, due to the largerambient radius R0 and the prolonged expansion phase. These numbers are based onthe thermal bremsstrahlung model [6–9]; however, water vapor was not taken intoconsideration. Its relevance for SBSL has been revealed recently [10–13] and its roleis of increasing importance at lower frequencies: Water vapor diffuses into the bubbleduring the expansion and is trapped at the subsequent collapse [13]. The result is anincreased heat capacity due to the additional number of particles and correspondingly

†See also R. Toegel, B. Gompf, R. Pecha and D. Lohse, Does Water Vapor Prevent UpscalingSonoluminescence?, Phys. Rev. Lett. 85, 3165-3168 (2000)

30 4.2. EXPERIMENTAL SETUP AND RESULTS

less heating. As at low driving frequencies the bubble expands to a larger maximalradius and thus has lower internal pressure at maximum, more water vapor can becollected and its role becomes more pronounced, as already suggested in [14].

In this paper we present experimental results obtained at a frequency of 7.1 kHzwhich give rise to the idea that the above mechanism sets an upper limit for the max-imal temperature achievable in SBSL experiments. We then extend the theoreticalmodel of refs. [5, 7, 14, 15] by taking the water vapor into consideration. In this waythe flux of vapor and its influence on the temperature can be quantitatively described.

4.2 Experimental setup and results

SBSL is achieved in the usual way by trapping an air bubble in a spherical resonator.The volume of this resonator is approximately 6 l corresponding to a resonance fre-quency of 7.1 kHz. As predicted in [5] degassing down to about 5% of the naturalair saturation (monitored through the oxygen concentration) is necessary to achievestable sonoluminescence. The gas concentration is expressed in terms of the relativeargon saturation cAr/c0,Ar which is the relevant one for sonoluminescence [16, 17]and which originates from the 1% argon contained in air. The pulse width and thenumber of photons being emitted were measured by time correlated single photoncounting [18, 19] . Fig.4.1 displays the result for different gas concentrations andtemperatures:

(i) The total number of photons lies around 104 − 105 per pulse, i.e., roughlythe same as at 20 kHz [18, 19]. (ii) The pulse widths are found to be between 100 psfor the darkest bubbles and 180 ps for the brightest ones. Again, these pulse widthsare comparable with pulse widths at 20 kHz [18, 19]. (iii) Figure 4.2 displays theautocorrelation of the light flash once measured in the UV regime (300 − 400 nm)and once in the red regime (590−650 nm) for the same bubble (maximum normalizedto unity). There is no measurable difference, again as for SBSL bubbles at 20 kHz[18, 19] .

These results give rise to the idea that the maximum temperature of the bubble,which in the thermal bremsstrahlung model is closely related to the light emission,will also be roughly the same as at 20 kHz,i.e. around 15000K. In contrast, the modelof ref. [5], which neglected water vapor, predicts temperatures beyond 50000 K. Wetherefore set out to include the effect of water vapor in that model. Clearly, the mostaccurate treatment of sonoluminescing bubbles is through full numerical simulationsof the gas dynamical partial differential equations (PDE) as done e.g. in refs. [6,11, 13], where effects like vapor evaporation and condensation, mass segregationand even many chemical reactions are considered [20]. The price to pay for theselarge PDE simulations is that only singular points of the enormous parameter spaceof SBSL can be calculated. An approximate ordinary differential equation (ODE)


1x10 4 2x10 4 3x10 4 4x10 4100

120

140

160

180

puls

ewid

th (

FW

HM

) / p

s

total number of photons

FIGURE 4.1: The pulse width of the light flash for sonoluminescing bubbles at 7.1kHz as a function of the total number of photons. The relative argon saturation andthe liquid temperature are: 1.0×10−3 and 21.9 oC (squares), 6.3×10−4 and 19.6 oC(triangles), 6.2× 10−4 and 22.6 oC (diamonds), 7.0× 10−4 and 22.8 oC (stars). Typ-ical pulse widths are around 150ps, typical number of photons around 104 − 105 perpulse.

approach in the spirit of refs. [7, 10, 14, 15] as presented here is useful to easily scanthe parameter space. We therefore derive an ODE model, which consists of threecoupled differential equations, describing the radial motion of the bubble as well asthe heat loss and the mass change due to evaporation and condensation of water vapor.

4.3 Theoretical model

4.3.1 Bubble dynamics

The bubble motion is described by the Keller-Miksis equation [21, 22], which takessound loss and compressibility effects of the water into account,(

1− Rcl

)RR+

32R2

(1− R

3cl

)=

32 4.3. THEORETICAL MODEL

-500 -400 -300 -200 -100 0 100 200 300 400 500

0.0

0.2

0.4

0.6

0.8

1.0

au

toco

rrel

atio

n

time / ps

red filtered UV filtered

FIGURE 4.2: Direct comparison of the pulse width in the UV (300 − 400 nm) andin the red regime (590 − 650 nm) of the SL-spectrum (cAr/c0,Ar : 5.2 · 10−4, T0 =22 oC). No significant difference in the pulse width is observed.

(1 +

R

cl

)1ρl

(pg − Pa − P0) +R

ρlclpg − 4ν

R

R− 2σρR. (4.1)

Here P0 = 1 atm is the ambient pressure and ρ = 1000 kg/m3 the density of water.The viscosity ν , the speed of sound in water cl, and the surface tension σ are chosenaccording to the liquid temperature T0. For the gas pressure pg and its time derivativea van der Waals type equation of state is used where the total number of particlesNtot(t) is allowed to vary according to condensation and evaporation of water vapor,

pg(t) =Ntot(t)kT

4π3

((R(t))3 − (R0(t)/8.86)

3) . (4.2)

Correspondingly, the equilibrium radius R0(t) under ambient condition is now alsotime dependent and is another way to express the time dependence of the number ofparticles Ntot(t) in the bubble. Both are connected by

4π3

(P0 +

2σR0

)((R0)3 −

(R0

8.86

)3)

= NtotkT0. (4.3)


Taking a common hard core radius R0/8.86 is justified by the fact that the covolumeof argon and water vapor differs by only 5% [23] and therefore both species can betreated equally.

4.3.2 Mass diffusion

From ref. [24] it is known that the surface temperature of the bubble exceeds the watertemperature only for a very brief moment during collapse. Therefore, we divide thebubble into two parts, namely, a ’cold’ boundary layer being in thermal equilibriumwith the liquid (this implies condensation to be fast enough to maintain equilibrium)and an eventually hot, homogeneous core. An analysis of the instantaneous diffu-sive penetration depth ldiff =

√RD|R| shows that two cases occur. During expansion

and a major part of the afterbounces the penetration depth exceeds the bubble radius(ldiff ≥ R) implying the total volume to be in equilibrium with the liquid. Duringcollapse however, ldiff becomes as small as 0.01 ·R and the thickness of the bound-ary layer is negligible as compared to the total bubble volume. Hence in both casesthe bubble can be regarded as homogeneous and we can estimate the rate of particlechange to be

NH2O = 4πR2D∂rn|r=R ≈ 4πR2DnR − nldiff

. (4.4)

nR = nR(T0) corresponds to the equilibrium density at the wall and n is the actual

concentration, n = NH2O

V .So far we did not take into account that ldiff cannot exceed a length comparableto the bubble radius R, as it would be the case when R tends to zero. In thiscase the convection-diffusion equation for the water vapor inside the bubble be-comes a pure diffusion PDE, ∂t (rn) = D∂2

r (rn) , with the boundary conditionn (r = R) = nR. Expressing the solution in terms of a Fourier series, r (n− nR) =∑∞

k=1 ak (t) ·sin (kπr/R), and assuming that the system will be dominated by k = 1a characteristic length and hence the desired cutoff is found to be R

π , thus:

ldiff = min

(√RD

|R| ,R

π

). (4.5)

In order to close the equation we have to determine an effective binary diffusionconstant D of the mixture. We calculate this quantity – from the kinetic theory ofgases [25] – with the properties of the boundary layer, i.e., (i) the water temperatureT0 and (ii) the number density n = nR + Nar

V .

4.3.3 Heat diffusion

Because of D/χ ≈ 1 and the complete analogy between thermal and mass diffusionwe now apply the same approach as for the water vapor to the temperature field. An

34 4.4. NUMERICAL RESULTS

analogous equation for the mass balance (Eq.4.4) is obtained from the first law ofthermodynamics for an open system [26], dE = hdNH2O + dQ− dW or

E = hNH2O + Q− W . (4.6)

Here, h = h(p, V, T,N) denotes the enthalpy per water molecule, dQ is the heattransferred to the bubble and dW represents the work done by the bubble.Internal Energy: The internal energy is made up of (i) the translational energy of theargon atoms (ii) the translational and internal energy of the water molecules:

E =32NArkT +

(62+∑(

θi/T

eθi/T − 1

))NH2OkT,

θi are the characteristic vibrational temperatures [27].Enthalpy: The water molecules condense at the ’cold’ bubble surface. The enthalpyper water molecule is hence given by h ≈ 8

2kT0.Heat Transfer: Analogously to Eq. (4.4) the heat loss is estimated to be

Q = 4πR2λmixT0 − Tlth

, lth = min

(R

π,

√Rχ

|R|

).

In principle, the thermal diffusivity χ = λmixρmixcp,mix

is found from ρmix · cp,mix =82nRk+

52nArk.Unfortunately, there is no rigorous method to derive an effective ther-

mal conductivity λmix of a mixture of polyatomic gases. However, a semi-empiricalexpression is given e.g. in [25]. We strictly follow that approach using for the tem-perature the liquid temperature T0. For further details we refer to [25].Work: Because of the constant temperature condition at the wall (λH2O λgas)the latent heat of the water vapor does not contribute to the energy balance and inparticular the work done by the bubble reduces to the expansion work W = pV

Now plugging everything together the temperature change of the bubble finallyis

T =Q

Cv− pVCv

+(82T0 − 6

2T − T

∑(θi/T

eθi/T − 1

))Nk

Cv

Cv =32NArk +

(62+∑(

(θi/T )2eθi/T

(eθi/T − 1)2

))NH2Ok.

4.4 Numerical results

To verify the model we calculate R(t), R0(t), and NH2O(t) for the same parametersf = 26.5 kHz, RAr

0 = 4.5µm, and Pa = 1.2 bar as used in the full PDE approachof ref. [13]. Indeed, Fig 4.3 exactly resembles Figs.1,2 of ref. [13]. Moreover, table


0 5 10 15 20 25 30 350

10

20

30

R (

t) /

µm

0 5 10 15 20 25 30 354

6

8

10

R0 (

t) /

µm

0 5 10 15 20 25 30 3510

8

1010

time / µs

NH

2O

FIGURE 4.3: R(t), R0(t), and NH2O(t) for f = 26.5 kHz, RAr0 = 4.5µm, Pa =

1.2 bar and T0 = 300K within the presented ODE model.

1 compares the amount of vapor trapped in the bubble during collapse accordingto the above model to the full numerics of ref. [13]. We find good qualitative andquantitative agreement.

Fig.4.4 shows the maximum temperature of an argon bubble with RAr0 = 5µm,

Pa = 1.3 bar and T0 = 293.15K for different driving frequencies. The solid linegives the result according to the above model, the dotted line shows the predictionsof the approach made in [5]. Rather than a monotonic increase in temperature withdecreasing frequency we observe that the bubble’s peak temperature goes through amaximum around 16 kHz and 15000K and then decreases again. Only at very smallf it increases again. This can be reasoned as follows. When the driving frequencyis lowered the bubble expands to larger maximum radius. The subsequent collapseis hence more violent and one would observe an increase in temperature. On theother hand the expansion to a larger maximum radius is accompanied by a larger

36 4.4. NUMERICAL RESULTS

RAr0 Pa T0 f Szeri present

(µm) (bar) (K) (kHz) +Storey (%) model (%)4.5 1.2 300 26.5 14 14.56 1.4 293.15 20.6 33 334 1.32 293.15 20.6 27 23

2.1 1.29 293.15 20.6 22 14

TABLE 4.1: Comparison of the amount of vapor (mole fraction) trapped during col-lapse calculated from the presented model and from the full numerics of Szeri andStorey [13].

amount of vapor being trapped during collapse, i.e., an increased heat capacity andhence a lower temperature. At low frequencies (below 16 kHz) the latter mechanismdominates and thus the net effect is a decrease of the peak temperature. Note thatfor the parameters of Fig.4.4 and a frequency f = 7.1 kHz the mole fraction ofvapor is already found to be 67.5% at collapse. The increase at very low f againoriginates from the stronger collapse, but now the collapse of a basically pure watervapor bubble with its corresponding (temperature and time dependent) polytropicexponent. Inclusion of chemical reactions may change this very low f property.

Finally, we want to calculate the model-analog to our experimental results, i.e.,both pulse width and light intensity. The complication is that all ions and moleculesin the bubble (Ar, H2O and its eventual reaction products) with their different cross-sections and ionization potentials contribute [11, 28]. However, what can be saidis that even at relative large frequencies oxygen and/or H2O become relevant forthe light emitting process as they have the lowest ionization potential (12 eV for O,12.6 eV for H2O in contrast to 15.8 eV for Ar). For f = 7.1 kHz they clearly domi-nate the spectrum. The preliminary calculations give pulse widths around 150 ps andintensities of some 104 photons per pulse, both in agreement with figure 4.1. Alsothe trends seen in figure 4.1 can be reproduced. This more detailed approach takingthe different light emitting processes of the ions and also the chemical reactions intoaccount is work in progress.

In conclusion, we state that (i) at room temperatures water vapor prevents up-scaling SBSL through lowering the driving frequency, (ii) ODE models of Rayleigh-Plesset type can be extended to regard the role of water vapor [10, 12, 14] and (iii)the competition between upscaling through lower f or larger Pa and cooling throughmore water vapor manifests in non-monotonic behavior as seen in Fig.4. At lowerwater temperatures around 0 oC the water vapor is only 1/4 of that at room tempera-ture and the presented model suggests that intensities ≈ 100 times so high as here and


5 10 15 20 25 300

2

4

6

8

x 104

frequency / kHz

tem

pera

ture

/ K

FIGURE 4.4: Maximum temperature of an argon bubble with RAr0 = 5µm, Pa =

1.3 bar, T0 = 293.15 K (Solid line: present model, dotted line: approach made in [5].

gas temperatures in the bubble as high as ≈ 25000K can be achieved. Experimentalwork to confirm or falsify this prediction is in progress.

Acknowledgments

It is our pleasure to acknowledge stimulating discussions with S. Hilgenfeldt. Thework is part of the research program of FOM, which is financially supported byNWO.

38 REFERENCES



[3] S. Cordry, Ph.D. thesis, The University of Mississippi, 1995.

[4] B. P. Barber et al., Phys. Rep. 281, 65 (1997).

[5] S. Hilgenfeldt and D. Lohse, Phys. Rev. Lett. 82, 1036 (1999).

[6] W. Moss, D. Clarke, and D. Young, Science 276, 1398 (1997).

[7] S. Hilgenfeldt, S. Grossmann, and D. Lohse, Nature 398, 402 (1999).

[8] D. Hammer and L. Frommhold, Phys. Fluids 12, 472 (2000).

[9] K. Yasui, Phys. Rev. E 59, 1754 (1999).

[10] K. Yasui, Phys. Rev. E 56, 6750 (1997).

[11] W. C. Moss et al., Phys. Rev. E 59, 2986 (1999).

[12] A. J. Colussi and M. R. Hoffmann, J. Phys. Chem. A103, 11336 (1999).

[13] B. D. Storey and A. J. Szeri, Proc. Roy. Soc. London A 456, 1685 (2000).



[16] D. Lohse et al., Phys. Rev. Lett. 78, 1359 (1997).

[17] J. A. Ketterling and R. E. Apfel, Phys. Rev. Lett. 81, 4991 (1998)

[18] B. Gompf et al., Phys. Rev. Lett. 79, 1405 (1997).

[19] R. Pecha, B. Gompf, G. Nick, and W. Eisenmenger, Phys. Rev. Lett. 81, 717(1998).

[20] For most features to be discussed here the latter are found to be less importantas compared to the role of water vapor [13] and are therefore neglected for thetime being. We also have good reason to speculate that chemical reactions maybe suppressed, cf. upcoming work.

REFERENCES 39

[21] A. Prosperetti and A. Lezzi, J. Fluid Mech. 168, 457 (1986).

[22] C. E. Brennen, Cavitation and Bubble Dynamics (Oxford University Press, Ox-ford, 1995).

[23] CRC Handbook of Chemistry and Physics, edited by D. R. Lide (CRC Press,Boca Raton, 1995).

[24] V. Kamath, A. Prosperetti, and F. Egolfopoulos, J. Acoust. Soc. Am. 94, 248(1993).

[25] J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular theory of gases andliquids (Wiley, New York, 1954).

[26] G. N. Hatsopoulos and J. H. Keenan, Principles of General Thermodynam-ics,(Wiley, New York, 1965)

[27] J. A. Fay, Molecular Thermodynamics,(Addison-Wesley, Massachusetts, 1965)

[28] W. Moss and D. Young, J. Acoust. Soc. Am. 103, 3076 (1998).

40 REFERENCES

CHAPTER 5. THE EFFECT OF EXCLUDED VOLUME 41

Chapter 5

Suppressing dissociation insonoluminescing bubbles: the effectof excluded volume †

Recent theoretical work in single bubble sonoluminescence has suggested that watervapor in the collapsing bubble leads to energy-consuming chemical reactions, re-stricting the peak temperatures to values for which hardly any light emission couldoccur. Analyzing the reaction thermodynamics within the dense, collapsed bubble,we demonstrate that the excluded volume of the non-ideal gas results in pronouncedsuppression of the particle-producing endothermic reactions. Thus, sufficiently hightemperatures for considerable bremsstrahlung emission can be achieved.

5.1 Introduction

A violently collapsing gas bubble can emit short flashes of light so intense as to bevisible to the naked eye. Stable clock-like light emission from an isolated gas bubbleknown as single bubble sonoluminescence (SBSL) was first reported in 1990 [1] andhas been studied extensively since [2–4]. The probable origin of SBSL light emissionhas been identified as thermal bremsstrahlung and recombination radiation from theoptically thin bubble heated to a few 104 K peak temperature [5–8]. Recently, it hasbeen pointed out that water vapor may significantly reduce the heating at collapseof the bubble [9–12]. Water vapor invades and escapes the bubble as it expands andcollapses. However, the final phase of collapse is so fast that vapor cannot readily dif-fuse to the bubble wall to maintain the equilibrium composition. Thus, a considerable

†See also R. Toegel, S. Hilgenfeldt and D. Lohse, Suppressing Dissociation in SonoluminescingBubbles: the Effect of Excluded Volume, Phys. Rev. Lett. 88, 034301 (2002)

42 5.2. THERMODYNAMICS OF A REACTIVE VAN DER WAALS GAS

portion of vapor is trapped [10–12], which acts to reduce heating in two ways: (i) Asthe rest of the bubble consists largely of noble gas [13], the presence of water reducesthe effective adiabatic exponent of the mixture, restricting the maximum temperatureof the bubble to about 15000–20000 K [6, 8, 10, 12]. These temperatures are never-theless sufficiently high for thermal bremsstrahlung emission. (ii) Taking chemicalreactions of the water vapor into account [9, 14] drastically decreases the tempera-ture since most of the reactions are endothermic and hence consume a major part ofthe thermal energy of the bubble. At the residual temperature of 6000–8000 K [10]hardly any thermal bremsstrahlung or recombination radiation would occur. Thesetemperature calculations rely on chemical reaction rates and chemical equilibriumdata that were taken under conditions much less extreme than those in a collapsedSBSL bubble, where the gas reaches almost solid state density. There is thereforesome uncertainty about the quantitative results. In this paper we attempt a morefundamental approach to obtain reaction rates for high-density gases. We find signif-icant suppression of chemical reactions under typical sonoluminescence conditions,resulting in sufficiently high peak temperatures for light emission. Qualitatively, themechanism at work is Le Chatelier’s principle applied to a reactive van der Waalsgas: dissociated water vapor molecules in the bubble take up more space than undis-sociated molecules. When the bubble volume becomes comparable to the excludedvolume of the gas molecules, little free space remains, favoring the undissociatedstate.

5.2 Thermodynamics of a reactive Van Der Waals gas

Ansatz and crucial assumptions:1. Following the dissociation hypothesis [13, 15] the incondensable gas in the bubbleis assumed to be argon. In addition water vapor and its chemical reaction productsare taken into account.2. In a first step we consider as an example of a particle producing reaction thedissociation of water into H and OH radicals.

H2O+ 5.1eV ←→ OH+H . (5.1)

3. As in [12] we assume a hard-sphere potential for the molecular interaction with acommon excluded volume B ≈ 5.1× 10−29 m3 [16] for all species.4. We define the zero energy level as the energy of an OH radical at rest. In this frameof reference an H2O molecule, unlike the other species, has an additional potentialenergy Eb=5.17 eV due to the H–OH bond energy.5. The various internal degrees of freedom of the molecules, i.e., rotations and vibra-tions, do not couple.


6. Electronic excitation is negligible compared to the other degrees of freedom, sothat only the statistical weight of the electronic ground state needs to be taken intoaccount.7. The formalism described below leads to a van der Waals corrections to the re-action equilibrium constant of (5.1). Because of the common value of B, the samecorrection can then be applied to all particle-producing reactions in a more elaboratescheme taken from ref. [9, 14], consisting of 8 forward and backward reactions.

Partition function: The partition function of the bubble contents is [17, 18]

Z =ξNH2O

H2O

NH2O!ξNOHOH

NOH!ξNHH

NH!ξNArAr

NAr!

(1− NB

V

)N

. (5.2)

Here N = NAr +NH2O +NOH +NH is the total number of particles in the bubble,T its temperature and V its volume. The factors ξX are perfect gas partition functionsfor single particles of species X, namely [17, 18]

ξX = gXV/λ3X for X = Ar,H , (5.3)

ξOH = gOHV/λ3OH

T

γOHθr,OH(1− e−θv,OH/T ), (5.4)

ξH2O = gH2OV/λ3H2Oe

Eb/kT ×

×(

πT 3

γ2H2O

∏θr,H2O,i

)1/21∏(

1− e−θv,H2O,i/T) , (5.5)

where λX = h/(2πmXkT )1/2, with the Planck and Boltzmann constants h and k,and the masses mX of single X particles. The constants gX are statistical weights ofthe electronic ground state, γX characterize the molecular symmetry. θr,X and θv,Xare characteristic vibrational and rotational temperatures, respectively (see Table 5.1for numerical values).

species i θr,X,i/K θv,X,i/K gX γXH2O 1 13.37 2295 1 2

2 20.87 52553 40.00 5400

OH 1 27.18 5370 4 1

TABLE 5.1: Numerical values of the material constants in the partition function [18].Note also gAr = 1, gH = 2.

Chemical equilibrium: The chemical potentials of the various species in thebubble are readily calculated from µi = −kT ∂ lnZ

∂Ni. Applying the general condition

44 5.3. A REFINED THEORETICAL MODEL

for chemical equilibrium∑µjδNj = 0 and bearing in mind that in reaction (5.1),

δNAr=0, δNH=δNOH=−δNH2O, we find µOH + µH − µH2O = 0, or equivalently:

N2OH

NH2O= KeqV = K0TV

∏(1− e−θv,H2O,i/T

)(1− e−θv,OH/T

) e−Eb/kT

×(1− NB

V

)e

−NBV −NB , (5.6)

where we used NOH = NH. The constant K0 is given by

K0 =gOHgHγH2O

gH2OγOH

(mOHmH2πkmH2Oh2

)3/2(∏

θr,H2O,i

πθ2r,OH

)1/2

. (5.7)

Equation (5.6) is the law of mass action for a van der Waals gas. The last twoterms are specific to the excluded-volume gas we are considering, and lead to expo-nential suppression of dissociation under SBSL conditions when the particle densityN/V approaches the critical value 1/B. Note that the right hand side of this equation– the equilibrium constant Keq – is now density-dependent and (5.6) hence becomesimplicit. Only in the low density limit it reduces to the formula for a perfect gas.Equation (5.6) is known in chemistry as the extension of the Guldberg and Waagelaw of mass action to real gases [19]. It can also be derived by means of the partialfugacities of a van der Waals gas.

Fig. 5.1 plots the equilibrium constant Keq as a function of temperature fordifferent excluded volume fractions NB/V . With increasing density one observesa shift towards lower values, equivalent to suppression of dissociation. The circlesshow for comparison the values taken from [20].

5.3 A refined theoretical model

We now incorporate high density reaction thermodynamics into the model of Ref. [12]in order to evaluate the effects of excluded volume on temperature and compositionof a typical SBSL bubble. Due to the simplifications in the model, the results shouldbe understood as approximative, but they show a robust trend.

Bubble dynamics: The Keller-Miksis equation [21] is used as equation of motionfor the bubble wall radius R. The pressure pg inside the bubble is obtained from Zby pg = kT ∂ lnZ

∂V = NkTV−NB . The remaining task is to determine the state variables

N and T , as shown in the following paragraphs.Mass and heat transport: As the concentration of reaction products at the inter-

face is orders of magnitude less than the argon concentration [10] its contribution tothe total particle density and hence its influence on the transport parameters is small.


500 1000 1500 2000 2500 300010

−60

10−40

10−20

100

1020

T / K

Keq

/ m

−3

NB/V =0.1, 0.85, 0.95 (top to bottom)

data taken from Ref. [20]

FIGURE 5.1: The equilibrium constant Keq for water dissociation as a function oftemperature for different excluded volume fractionsNB/V . As the latter is increasedthe equilibrium constant shifts to lower values, i.e., it favors the undissociated state.For comparison, the circles show values from the extensive data base gri mech [20].

We can thus model the changes in particle number by mass diffusion Nd and thediffusive heat losses Q using the same boundary layer formalism as in [12],

NdH2O = 4πR2D

nH2O,0 − nH2O

ld, ld = min

(√RD

|R| ,R

π

),

Q = 4πR2κT0 − Tlth

, lth = min

(√Rχ

|R| ,R

π

).

Here, nH2O,0 and T0 are the equilibrium values of the water vapor number densityand bubble temperature, and κ, χ are the thermal conductivity and thermal diffusivityfor the gas mixture. We have only given Nd

H2O because diffusive transport for otherspecies is negligible. For further details we refer to Ref. [12].

46 5.4. RESULTS

Reaction scheme: We focus on the reactions no. 1–8 of [9, 14], involving thespecies Ar, H2O, OH, H, O, H2, and O2, as species like H2O2 and HO2 are reportedto appear only in trace amounts [10]. Since the forward and backward reaction rates(kf ,kb) are related through their equilibrium constant [22] all rate constants of reac-tions of the type A→ B + C are corrected for high density such that

Keq = kf/kb ∝(1− NB

V

)e

−NBV −NB , (5.8)

where N =∑

XNX is the total number of particles. Note that there is some am-biguity in how to distribute the correction factor among the forward and backwardreaction rates. The result however is found to be independent on how this is done.This is mainly because the bubble contents remain at thermochemical equilibriumduring collapse and hence only the equilibrium constant, i.e., the ratio kf /kb, matters.

Temperature: Using once more the partition function, the total energy E of thebubble and its change with respect to time are found to be:

E = kT 2∂ lnZ∂T

, (5.9)

E =∑X

∂E

∂NXNX +

∂E

∂TT +

∂E

∂VV . (5.10)

Note, that, since more species are involved now, ∂E/∂T and∑

X∂E∂NX

NX have beengeneralized accordingly. The latter expression is computed using

∑X

∂E

∂NXNX = −V

∑j

rj∆Ej , (5.11)

where rj is the net reaction rate of elementary reaction j (per unit volume) and ∆Ej isthe corresponding reaction energy (including the difference of degrees of freedom ofinitial and final state). The rj are obtained from kf,j , kb,j and the densities of particlespecies participating in reaction no. j. The binding energies and characteristic tem-peratures necessary to compute ∂E/∂T and the ∆Ej have been taken from [16, 18].

Comparing (5.10) to the first law of thermodynamics for an open system [23],E = Q− pgV + hwNd, and rearranging for T yields the differential equation for Tnecessary to close the model. Here hw ≈ 8

2kT0 is the molecular enthalpy of watervapor at the (cold) bubble wall.

5.4 Results

Figure 5.2 depicts the dynamics of the fraction of vapor + reaction products ζ =(N − NAr)/N and the temperature T for a time span of 6ns around collapse for


typical SBSL conditions. The high-density correction of the reaction rates leads toa suppression of vapor+reaction products. Accordingly, as the particle productionis energy-consuming, the peak temperature rises substantially, from ≈ 7000K to≈ 10000K (Fig. 5.2b).

At the very moment of collapse the density in the bubble becomes so high thatthe model predicts recombination of already dissociated molecules, witnessed by thesmall dip of the ζ curve in Fig. 5.2a. Whether this feature reflects reality remainsunclear as the recombination phase lasts for only ∼ 100 − 200 ps; the validity ofthermochemical equilibrium on this timescale is not obvious.

0.35

0.4

0.45

0.5

ζ

−3 −2 −1 0 1 2 30

5000

10000

t / ns

T /

K

FIGURE 5.2: (a) The combined fraction of vapor molecules + reaction products inan oscillating SBSL bubble as a function of time around collapse. (b) Bubble tem-perature for the same time interval. Dashed lines show calculations omitting ex-cluded volume effects, solid lines correspond to the corrected reaction rates. Theexcluded volume correction raises the maximum temperature. The equilibrium ra-dius is RAr

0 = 5µm, the driving pressure Pa = 1.4 bar, the liquid temperatureT0 = 293.15 K, and the driving frequency f = 20 kHz.

48 5.4. RESULTS

0

0.2

0.4

0.6

ζ*

1 1.1 1.2 1.3 1.4 1.50

5000

10000

15000

Pa / bar

T* /

K

FIGURE 5.3: Fraction of vapor + reaction products (a) and temperature (b) at thetime of collapse as a function of the driving pressure amplitude (T0, f , and R0 as inFig. 5.2). Dashed lines stem from the model without excluded volume effects, solidlines show corrected rates. In addition the plot shows the results obtained using theinertially corrected Rayleigh-Plesset equation of Ref. [24] (dotted lines).

Figure 5.3 shows ζ∗ and T ∗, the fraction of vapor + reaction products and thetemperature at the very moment of collapse, for different Pa. The argon content ofthe bubble was fixed (ambient radius RAr

0 = 5µm). In both models ζ∗ grows as thedriving pressure is increased; the absolute numbers, however, differ. For Pa =1.5 bar,e.g., the uncorrected model predicts ζ∗ = 0.58 whereas the corrected rates only yieldζ∗ = 0.52. This translates to 25% fewer particles in the bubble and, correspondingly,significantly enhanced temperatures. Neglecting the high density correction leads toa maximum temperature of T ∗ ≈ 7000K around 1.25–1.3 bar. As Pa is further in-creased T ∗ drops again due to the dominant role of chemical dissociation. If, on theother hand, the excluded volume effect on the equilibrium constant is taken into ac-count, the temperature is found to monotonically increase with Pa, until at Pa=1.5 bar


we find 10000 K rather than 7000 K.A more sophisticated model of bubble dynamics [24] takes inertial effects of the

gas in the bubble into account. The gas pressure at the bubble wall pg in this casebecomes

pg = pc − ρRR/2, (5.12)

where ρ is the mean mass density of the gas mixture and pc the center pressure,respectively. Interpreting pg as the mean gas pressure and using (5.12), strongercompression is achieved, and the excluded-volume effect described here is morepronounced. Temperatures of up to 13000 K are found then (see the dotted line inFig. 5.3).

In conclusion we propose a mechanism that, when combined with recent SBSLmodels [10, 11], predicts temperatures high enough for light emission, even whenthe endothermic reaction chemistry of water vapor is taken into account. Due to thevanishing free volume as the bubble approaches maximum compression, the particle-producing dissociation reactions are found to be suppressed by Le Chatelier’s princi-ple for the excluded volume gas, to a degree far beyond what could be expected in anideal gas.

Finally note that this result has bearing on high density reaction thermodynamicsbeyond SBSL. For instance, empirical chemical reaction rates, obtained from hydro-gen flame studies [25], should not be extrapolated via a modified Arrhenius law intoa regime where the finite size of the particles becomes important.

Acknowledgments

This work is part of the research program of FOM, which is financially supported byNWO.

50 REFERENCES




[4] D. Hammer and L.Frommhold, J. Mod. Optic. 48, 239 (2001).

[5] B. Gompf, R. Gunther, G. Nick, R. Pecha, and W. Eisenmenger, Phys. Rev. Lett.79, 1405 (1997).


[7] S. Hilgenfeldt, S. Grossmann, and D. Lohse, Nature 398, 402 (1999).

[8] W. C. Moss, D. A. Young, J. A. Harte, J. L. Levatin, B. Rozsnyai, G. B. Zim-merman, and I. H. Zimmerman, Phys. Rev. E 59, 2986 (1999).

[9] K. Yasui, Phys. Rev. E 56, 6750 (1997).






[15] T. J. Matula and L. A. Crum, Phys. Rev. Lett. 80, 865 (1998); J. A. Ketterlingand R. E. Apfel, Phys. Rev. Lett. 81, 4991 (1998).

[16] CRC Handbook of Chemistry and Physics, edited by D. R. Lide (CRC Press,Boca Raton, 1991).

[17] T. L. Hill, Introduction to Statistical Thermodynamics (Addison-Wesley, Read-ing, Massachusetts, 1960).

REFERENCES 51

[18] J. A. Fay, Molecular Thermodynamics (Addison-Wesley, Reading, Mas-sachusetts, 1965).

[19] I. Prigogine and R. Defay Chemical Thermodynamics (Longmans, Green andCo Ltd, London, 1962).

[20] Gregory P. Smith, David M. Golden, Michael Frenklach, Nigel W. MoriatyBoris Eiteneer, Mikhail Goldenberg, C. Thomas Bowman, Ronald K. Hanson,Soonho Song, William C. Gardiner Jr., Vitali V. Lissianski, and Zhiwei Qin,http://www.me.berkeley.edu/gri mech/

[21] J. B. Keller and M. Miksis, J. Acoust. Soc. Am. 68, 628 (1980); A. Prosperettiand A. Lezzi, J. Fluid Mech. 168, 457 (1986).

[22] W. C. Gardiner, Gas-Phase Combustion Chemistry (Springer Verlag, New York,2000).

[23] G. N. Hatsopoulos and J. Keenan, Principles of General Thermodynamics (Wi-ley, New York, 1965).

[24] H. Lin and B. Storey and A.Szeri, J. Fluid Mech. 452, 145 (2002).

[25] F. N. Egolfopoulos and C. K. Law, in Twenty-Third Symposium (International)on Combustion (The Combustion Institute, 1990), pp. 333–340.

52 REFERENCES

CHAPTER 6. PHASE DIAGRAMS FOR SONOLUMINESCING BUBBLES 53

Chapter 6

Phase diagrams for sonoluminescingbubbles:A comparison between experimentand theory †

Phase diagrams for single bubble sonoluminescence (SBSL) are calculated. The em-ployed model is based on a set of ordinary differential equations and accounts forthe bubble hydrodynamics, heat exchange, phase change of water vapor, chemicalreactions of the various gaseous species in the bubble (N2, O2 and H2O being themost important among these), and diffusion/dissolution of the reaction products inthe liquid. The results of the model are compared in detail to various phase diagramdata from recent experimental work, among which are air-water systems as well assystems with a xenon-nitrogen mixture as the saturated gas. Excellent quantitativeagreement is found for all considered cases. Moreover, we find that the onset of SBSLis hysteretic in the driving pressure Pa. When starting with air typical temperaturesbefore onset are 5500 K and 15000 K thereafter. In the light emitting regime the bub-bles are found to nearly entirely consist of argon.

6.1 Introduction

Single bubble sonoluminescence (SBSL) is the light emission of a micrometer sizedsingle gas bubble levitated in a standing acoustical field with pressure amplitudesPa ≈ 1.2 − 1.5 bar [1, 2]. The phenomenon is found to be amazingly stable. Thebubble can exist for hours emitting a short burst of light once every acoustical cycle.

†See also Journal of Chemical Physics, in press.

54 6.1. INTRODUCTION

The light emission is so intense that it can clearly be detected with the naked eye. Fora recent review of the phenomenon see [3].

Detailed experiments [4] showed that when starting with air bubbles, relative gasconcentrations of typically 20-40% are required for stable SBSL, whereas for argonbubbles one requires only 0.2-0.4%. Moreover, for air bubbles the pressure regimewhere light emission occurs is preceded by a regime where the ambient radius of thebubble R0 considerably shrinks with increasing acoustical pressure – up to a criticalpressure amplitude, where the bubble is hardly detectable anymore. Beyond thisthreshold a sudden growth in R0 takes place and light emission sets on [4–6].

Lohse et al. explained this behavior with the dissociation hypothesis [7, 8],which meanwhile is experimentally confirmed [6, 9–11]. The main idea is that themolecular ingredients of air dissociate at the high temperatures achieved at bubblecollapse and react with water vapor to soluble gases as NO, NH, which dissolve inwater.

The dissociation hypothesis [7, 8] predicts three different equilibrium branches“A”, “B”, and “C” in the Pa vs R0 phase diagram, see figure (6.1), which is repro-duced from reference [7].

(i) Equilibrium branch “A” for low driving pressure amplitudes (Pa =1.0 atm-1.1 atm), where no significant heating and hence no chemical reactions occur. Thebubble accordingly consists of air. This branch turns out to be unstable: As is in-dicated by the arrows, perturbations of the ambient radius R0 from its equilibriumvalue are amplified; the bubble either shrinks until dissolution or grows until it runsinto the shape instability (not shown in the figure), which sets an upper threshold forthe maximal ambient radius at which sphericity can be maintained [12, 13]. Bubblesbeyond this threshold will develop strong surface deformations and eventually breakapart. Branch A is therefore not observed in experiment.

(ii) Equilibrium branch “B”: For somewhat higher pressures (Pa =1.1 atm-1.3 atm) the bubble already reaches peak temperatures of typically 4000 K-6000 K,according to the model of [7]. Chemical activity sets on and the molecular compo-nents N2 and O2 together with the water vapor that is present in the bubble form avariety of reaction products most of which are highly soluble in water. Consequently,they are continuously removed from the bubble and thereby balance the influx of themolecular species N2 and O2 due to rectified diffusion from the surrounding water.A dynamical equilibrium, branch B, evolves which is stable and descending.

(iii) Equilibrium branch “C”: Once a critical pressure amplitude (Pa ≈ 1.3 atm)is exceeded, the bubble burns off the molecular components faster than they can diffu-sive into the bubble. It is more and more depleted from these species and eventuallyonly the chemically inert argon is left: A third equilibrium branch C has appearedwhich is dominated by argon originating from its small natural abundance in air.Also this branch turns out to be stable.


1.0 1.2 1.4Pa/atm

1

3

5

7R

0/µm

A B

C

9000K

FIGURE 6.1: Pa-R0 phase diagram taken from Ref. [7]. The parameters arec∞,air/c0 = 0.2, f = 26.5 kHz and T0 = 293K. Three different equilibria, denotedas A, B and C, can be distinguished. As is elaborated in that reference no significantheating occurs along branch A , the bubble essentially consists of air. The branch isunstable, indicated by the arrows. Upward arrows denote growth, downward arrowsshrinkage, respectively. Along branch B the diffusive influx of molecular gases, andthe ejection of their reaction products balance each other. Bubbles on branch C es-sentially contain pure argon as the molecular components are burned off faster thanthey can diffuse into the bubble. Only the chemically inert argon is left. Both branchB and branch C are stable as can be seen from the arrows. The additional thin linein between branch B and C shows for comparison the threshold, where the bubbletemperature exceeds 9000 K (in the model of Ref. [7]).

The dissociation hypothesis of Ref. [7] explains the overall topology of Pa−R0

phase diagrams and the underlying physics. However, no quantitative comparisonbetween the resulting phase diagram with experimental data had been attempted in

56 6.2. THE HYDRO/THERMODYNAMICAL ODE MODEL

that paper, because those days (i) hardly any high precision measurements of Pa−R0

phase spaces were available, and because (ii) also the theory was underdeveloped,treating heat losses and chemical reactions only in a simplified way and neglectingwater vapor.

It is the aim of this study to now provide a quantitative comparison betweenthe meanwhile available high-precision experimental phase diagrams with the phasediagrams resulting from a refined model for sonoluminescing bubbles. This modelwill be developed here and includes heat flux [13], mass transfer, evaporation andcondensation of liquid vapor [14, 15], and chemical reactions [16–18]. The modelis in the spirit of the hydrodynamical-chemical approach of our former publications[7, 8, 12, 19–21] or of references [15, 17, 18]. In particular, it is an ODE type model,which has the advantage that in contrast to full PDE models [14, 22–27] extensivescans of the SBSL parameter space can be performed at moderate computationaleffort.

The paper is organized as follows: The model is elaborated in section 6.2. Insection 6.3 various experimental phase diagrams and their theoretical counterpart arecompared, finding excellent quantitative agreement. Section 6.4 elaborates on thehysteresis that is typically observed upon transition from branch B to C and viceversa. Section 6.5 investigates the robustness of the presented results to variations inthe model. Section 6.6 finally contains conclusions, highlightening in particular theeffect of the water temperature [28, 29] and the importance of some ingredients tothe model.

6.2 The hydro/thermodynamical ODE model

6.2.1 Bubble dynamics

The radial motion of the bubble is described by the Keller-Miksis equation, a variantof the Rayleigh-Plesset equation which takes into account first order corrections forthe liquid compressibility [30]:

(1− R

c

)RR+

32R2

(1− R

3c

)=

(1 +

R

c

)1ρ(pg − Pa sinωt− P0) +

Rpgρc− 4νR

R− 2γρR

(6.1)

with ρ = 1000 kg/m3, c = 1484m/s, γ = 72 × 10−3 N/m, and ν = 10−6 m2/s thedensity, speed of sound, interfacial tension, and viscosity of the liquid. These valuescorrespond to a water temperature T0 = 293.15 K. They are adapted accordingly, incases for which the temperature is different. In addition, pg denotes the gas pres-


sure at the bubble surface, P0 the ambient pressure (typically 1bar) and Pa, ω theamplitude and frequency of the acoustic driving.

6.2.2 Gas pressure

Following Ref. [31] we approximate the gas pressure at the bubble wall as the sumof two terms, a uniform contribution – we take a van der Waals type equation of stateherefore – and an inertial correction, the relevance of which is is investigated in detailin [31],

pg =NtotkT

V −NtotB− 1

2ρBRR. (6.2)

Ntot denotes here the total number of particles, B = 5.1×10−29 m3 [32] the molec-ular covolume, which – for simplicity – is taken to be equal for all species and ρB themean mass density of the bubble.It should be mentioned that the inertial correction in Eq.(6.2) has been derived underthe assumption of constant total mass, which strictly speaking is not the case here,since the mass of the bubble changes due to diffusion. Yet, as the authors of Ref. [31]state themselves, it becomes relevant only in the very last phase of collapse, whenthe mass of the bubble is fairly constant and we shall therefore use it here. Notefurthermore, that Eq.(6.1) with Eq.(6.2) actually becomes third order as it involvesthe time derivative of pg.The equilibrium radius R0 of the bubble follows from

(P0 +

2σR0

)(4π3R3

0 −NtotB

)= NtotkT0. (6.3)

It is a measure for the mass of the bubble and is the quantity which is usually mon-itored in experiment. As in experiment to be accounted for [6], we calculate it fromthe total number of particles at the zero crossing of the driving pressure.

6.2.3 Mass transport

Depending on the solubility of the species we will use different approaches to modelthe diffusive transport across the bubble wall:(i) The moderately soluble species in the bubble, H2, O2, N2, Ar/Xe that are stableunder ambient conditions are treated in the long time scale, adiabatic approximationof Ref. [8, 12, 33]. Following these references the diffusive change per cycle is givenby:

∆Ni = 4πDi,liqτRmaxni,sat

(ni,∞ni,sat

− 〈p〉i,4P0

)(6.4)


with Di,liq the diffusion constant of species i in the surrounding liquid, τ the periodof the acoustical driving, Rmax the maximum radius of the bubble, ni,sat the satura-tion concentration of species i in the liquid (Henry’s constant), ni,∞ its actual liquidconcentration, and 〈p〉i,4 its weighted mean partial gas pressure [33]:

〈p〉i,4 =∫ T0 pi,gR

4dt∫ T0 R

4dt, pi,g =

NikT

V −NB . (6.5)

(ii) An extension of the boundary layer approach of Ref. [20, 21] is used for thehighly soluble and the unstable, radical species:

Ndi = 4πR2Di

ni,0 − nili,d

, (6.6)

li,d = min

(√RDi

|R| ,R

π

)(6.7)

where the various symbols denote the following: Ndi is the diffusive loss of particles

of species i (per unit time), ni its momentary concentration, and ni,0 the respective(equilibrium) concentration at the bubble wall. Di is the mass diffusion constant ofspecies i in the mixture of all other species – it will be considered in detail in section6.2.5 – and li,d finally is the diffusive boundary layer thickness of species i.The structure of li,d in Eq.(6.7) had already been suggested in [12] in the context ofdiffusive vorticity transport in the fluid around the bubble. It can be understood bymeans of the Peclet number

PeDi :=R|R|Di

, (6.8)

which compares the diffusive and dynamical time scales R2/Di andR/|R| with eachother. Large values of PeDi accordingly indicate that the dominant timescale is givenby the dynamics and li,d is found from dimensional analysis to be

li,d =

√RDi

|R| =R√PeDi

. (6.9)

This formula is obviously not applicable in the limit PeDi −→ 0 as li,d would divergethen. We introduce the bubble length scale R/π as an upper cutoff for li,d leading toEq.(6.7). The additional factor 1/π arises from spherical symmetry, see Ref. [20] forfurther details.In order to close Eq.(6.6), the equilibrium concentrations ni,0 must be set. For H2Oit is given by the number density corresponding to the equilibrium vapor pressure attemperature T0, since we assume isothermal boundary conditions at the bubble wall,


cf. subsection 6.2.5: nH2O,0 = Pv(T0)/kT0 ≈ 5.9× 1023 m−3 (for T0 = 293.15 K).Again, in cases where the liquid temperature T0 is different Pv is adapted accordingly.For the highly soluble species we naturally set ni,0 = 0. Note that the moderatelysoluble species, where ni,0 = ni,∞, are already treated under (i).

6.2.4 Heat transport

Analogously to Eq.(6.6) we approximate the conductive heat loss and the thermalboundary layer thickness to be

Q = 4πR2λmixT0 − Tlth

, (6.10)

lth = min

(√Rχ

|R| ,R

π

), (6.11)

T being the actual temperature in the bubble, T0 the (liquid) temperature at the bubblewall, λmix and χ the heat conductivity and thermal diffusivity of the gas mixturerespectively, and lth the thickness of the thermal boundary layer.

6.2.5 Transport parameters

The transport parameters, i.e., the conductivities λi, viscosities ηi and mass diffu-sivities Di of the various components play a key role as they control the heat andmass exchange of the bubble. In order to determine them, the local concentrationsand temperature are needed, an information that an ODE approach inherently cannotprovide. Consequently, one has to make reasonable assumptions about the tempera-ture and the composition of the gas in the region of interest, i.e., at the bubble wall.v.Kamath and Prosperetti [16, 34] have shown that – essentially due to the large heatcapacity of liquid water – the bubble wall is much colder than its interior, which waslater confirmed by the PDE simulation of Szeri and Storey [14]. Therefore, as in ourprevious work [20, 21], we assume that the temperature of the boundary layer willroughly equal the liquid temperature T0.When it comes to the composition, things are even more complicated as the strongcoupling between temperature and composition (via chemical reactions), inertial ef-fects, and different mass diffusivities all have influence on the concentration profiles.Only a full PDE simulation of the transport equations as performed in [14] can pro-vide a detailed answer on the spatial distribution of the various species. For thepurpose here, we will simply consider all concentrations to be spatially uniform withone exception: the water vapor in the boundary layer is assumed to be in thermalequilibrium with the liquid phase. As a matter of fact, the various chemical prod-ucts are found to have marginal influence on the heat and mass transport parameters


anyhow, as they appear in significant quantities only for a very short time around themain collapse of the bubble.With these assumptions, the transport parameters are now readily calculated from thekinetic theory of gases. We give here only the result and refer to [35] for a detailedderivation.(i) Diffusion coefficient. The diffusion coefficient Di of species i in the mixture of allother species can be constructed from the various binary diffusion coefficient Di,j:

Di,j =38

√πkT0/µi,j

ni,jπσ2i,jΩ

(1,1)∗i,j

(6.12)

The various symbols denote: µi,j = 2mimj/(mi + mj) is the reduced molecu-lar mass of the two species i, j, ni,j = ni + nj their joint concentration, σi,j =0.5 (σi + σj) their effective collision diameter and Ω(1,1)∗

i,j a dimensionless correc-tion of O(1), which describes the deviation of the collisional cross section from thehard sphere cross section. It is a function of the reduced temperature of the twospecies, T ∗i,j = kT0/

√εiεj , and is tabulated e.g. in [35]. εi, σi are the parameters of

the Lennard-Jones interaction potentials. They are provided in Table 6.2. An effec-tive mass diffusion constant, Di, is eventually obtained as the weighted average ofthe reciprocals of the binary diffusion constants [14, 35]:

1Di

=∑j =i

ξj(1− ξi)Di,j

(6.13)

with ξi the mole fraction of component i at the bubble wall(ii) Viscosity and conductivity: The viscosity of the pure substance i is given by

ηi =516

√πmikT0

πσ2iΩ

(2,2)∗i,i

, (6.14)

where Ω(2,2)∗i,i again is a correction to the collisional cross section. The conductivity

of component i is related to its viscosity by

λi =15k4mi

ηi

(4fi30

+35

), (6.15)

with the last factor the so called Eucken-correction which accounts for the contribu-tion of internal degrees of freedom to the heat conduction. fi is here the number ofdegrees of freedom of species i at the bubble wall. They are also provided in table6.2. Analogously to the mass diffusion constant an effective heat conductivity is now


constructed from the viscosities and conductivities of the pure substances [36]:

λmix =∑i

ξiλi∑j ξjΦi,j

, (6.16)

Φi,j =1√8

(1 +

mi

mj

)− 12

1 +

(ηiηj

) 12 (mj

mi

) 14

2

(iii) Thermal diffusivity: For an estimate of the thickness of thermal boundary layerthe thermal diffusivity will also be needed: Knowing the heat conductivity and theconcentrations (and thus the heat capacity, cp, per unit volume) at the bubble wall thethermal diffusivity is readily found to be:

χ =λmix

cp(6.17)

cp =∑i

fi + 22

kni

6.2.6 Chemical reactions

Altogether, we consider one inert gas, either argon or xenon, and 16 chemically re-active species which are listed in table 6.2 and characterized through the index i. Weallow for 45 chemical reactions which are listed in table 6.1 and labelled with theindex j. Each of these elementary reactions has a forward and backward rate perunit volume, rf,j , rb,j , which is described by means of modified Arrhenius laws [37].The shift of the equilibrium constant under high density (Le Chatelier effect) and thecorresponding suppression of particle producing reactions is included in our model.Following Ref. [21] it is accounted for by an additional term βtj in the forward re-action rate. Here the exponent tj indicates whether the reaction is particle producing(tj = 1 for one extra particle, tj = 2 for two extra particles,...), particle numberconserving (tj = 0), or reducing the number of particles (tj = −1 for one particleless,...). For a chemical process M+ A+ B←→M+ C the reaction rates thenbecomes:

rf,j = βtjkf,jntotnAnBTcf,j exp

(−Ef,j

kT

), (6.18)

rb,j = kb,jntotnCTcb,j exp

(−Eb,j

kT

), (6.19)

β =exp

(ntotB

(1−ntotB)

)1− ntotB (6.20)


with kf,j, kb,j the frequency factors (basically the time constant) of the forward andbackward rate, nA,...,C, ntot the concentration of the participating species and thecollider M (given by the total number concentration as every particle can act as acollider), T the gas temperature and Ef,j , Eb,j the activation energies.

The concentrations in reactions of the type A+ B←→ C+ D must be re-placed accordingly. As effectively no particles are produced in those reactions theytake the same form as in the low density limit, i.e.,

rf,j = kf,jnAnBT cf,j exp(−Ef,j

kT

), (6.21)

rb,j = kb,jnCnDT cb,j exp(−Eb,j

kT

). (6.22)

Table 6.1 lists the parameters we used.The net rate of reaction j (cf. table 6.1) is now given by the difference of forward andbackward rate

rj = rf,j − rb,j . (6.23)

and the chemical rate of change of species i by the sum over all elementary reactionrates with their proper stoichiometric weight αi,j

N ci = V ×

∑j

αi,jrj. (6.24)

To illustrate this, consider for example O radicals (i = O). In reaction j = 1 two Oradicals react (to an O2 molecule) and none arises. Therefore, we have αO,1 = −2.In reaction j = 2 it correspondingly is αO,1 = −1. For the hydroxyl radical i = OHwe have αOH,2 = 1 for the reaction j = 2 and αOH,1 = 0 for the reaction j = 1, inwhich no hydroxyl radical is involved.Finally, the total rate of change is the sum of the diffusive contribution Nd

i , Eq.(6.4)and Eq.(6.6), and the chemical contribution N c

i , Eq.(6.24)

Ni = N ci + N

di (6.25)

6.2.7 Energy balance

In order to obtain a differential equation for the temperature we start from the globalenergy balance of the bubble’s interior [39]:

E = Q− pgV +∑i

(hw,i + hform,i) Ndi (6.26)

E is the change of the total energy, Q is the conductive heat loss which had been givenin section 6.2.4, pgV is the work applied to the bubble and

∑i (hw,i + hform,i) Nd

i


TABLE 6.1: The 45 considered reactions of the 16 chemically active species of table6.2. Arrhenius parameters for the reaction rates given by Eq.(6.18) - (6.22). Thescheme is taken from [38]. tj is the power of the excluded volume term β, seeEq.(6.20). The frequency factors kf,j ,kb,j are given in cm3/mol/s for the two bodyreactions and in cm6/mol2/s for the three body reactions. The activation temperaturesEf,j/k and Eb,j/k are given in K and the reaction energies ∆Ej in kJ/mol. The ratesalso contain a power law of the temperature T, see Eqs.(6.18) - (6.22) cf,j and cb,j arethe corresponding powers.

j reaction tj kf,j cf,j Ef,j/k kb,j cb,j Eb,j/k ∆Ej

1 O+O+M←→ O2+M 1 1.2e17 -1 0 3.16e19 -1.3 59893 4982 O+H+M←→ OH+M 1 5e17 -1 0 3.54e17 -0.9 51217 4283 O+H2 ←→ H+OH 0 3.87e4 2.7 3150 1.79e4 2.7 2200 -84 H+O2 ←→ O+OH 0 2.65e16 -0.7 8576 9e13 -0.3 -83 -705 H+H+M←→ H2+M 1 1e18 -1 0 7.46e17 -0.8 52177 4366 H+OH+M←→ H2O+M 1 2.2e22 -2 0 3.67e23 -2 59980 4987 OH+H2 ←→ H+H2O 0 2.16e8 1.5 1726 5.2e9 1.3 9529 628 OH+OH←→ O+H2O 0 3.57e4 2.4 1062 1.74e6 2.2 7693 709 N+NO←→N2+O 0 2.7e13 0 179 5.32e13 0.1 37989 31410 N+O2←→NO+O 0 9.0e09 1 3271 8.33e08 1.1 19115 13311 N+OH←→NO+H 0 3.36e13 0 194 1.16e15 -0.3 24715 20312 N2O+O←→N2+O2 0 1.4e12 0 5440 2.05e08 1 44858 33113 N2O+O←→NO+NO 0 2.9e13 0 11651 2.05e08 1 29105 15014 N2O+H←→N2+OH 0 3.87e14 0 9502 1.55e08 1.4 40259 26115 N2O+M←→N2+O+M -1 5.59e12 -0.7 28480 6.48e06 0.5 8002 -16716 NO+O+M←→NO2+M 1 1.06e20 -1.4 0 4.98e24 -2.2 37005 30517 NO2+O←→NO+O2 0 3.9e12 0 -121 9.73e09 0.6 22763 19318 NO2+H←→NO+OH 0 1.32e14 0 181 1.85e07 0.9 14397 12319 NH+O←→NO+H 0 4.0e13 0 0 1.25e15 -0.2 35783 29220 NH+H←→N+H2 0 3.2e13 0 166 6.36e13 0.1 12388 9721 NH+OH←→HNO+H 0 2.0e13 0 0 6.52e17 -0.9 9192 6822 NH+OH←→N+H2O 0 2.0e09 1.2 0 8.93e10 1.1 20029 15923 NH+O2←→HNO+O 0 4.61e05 2 3271 5.41e07 1.1 8706 -1524 NH+O2←→NO+OH 0 1.28e06 1.5 50 1.08e05 1.7 27164 22225 NH+N←→N2+H 0 1.5e13 0 0 9.16e14 -0.1 73602 60626 NH+H2O←→HNO+H2 0 2.0e13 0 6970 2.92e16 -0.7 8361 -7027 NH+NO←→N2+OH 0 2.16e13 -0.2 0 3.1e13 0 49091 40328 NH+NO←→N2O+H 0 3.65e14 -0.5 0 1.13e21 -1.6 18337 14329 NH2+O←→OH+NH 0 3.0e12 0 0 2.95e15 -0.6 14365 4330 NH2+O←→H+HNO 0 3.9e13 0 0 3.82e16 -0.6 14360 9831 NH2+H←→NH+H2 0 4.0e13 0 1837 1.27e12 0.4 7970 5132 NH2+OH←→NH+H2O 0 9.0e07 1.5 -232 6.38e07 1.7 13698 11333 N2H←→N2+H -1 3.3e08 0 0 4.09e07 0.4 3748 3234 N2H+M←→N2+H+M -1 1.3e14 -0.1 2506 1.5e13 0.3 6256 3235 N2H+O←→OH+N2 0 2.5e13 0 0 1.07e12 0.6 54976 -48536 N2H+O←→NH+NO 0 7.0e13 0 0 1.67e12 0.4 5883 5637 N2H+H←→H2+N2 0 5.0e13 0 0 4.61e12 0.6 55923 46738 N2H+OH←→H2O+N2 0 2.0e13 0 0 4.14e13 0.4 63728 53139 H+NO+M←→HNO+M 1 4.48e19 -1.3 372 2.87e22 -1.9 25004 20740 HNO+O←→NO+OH 0 2.5e13 0 0 2.4e10 0.7 26593 22141 HNO+H←→H2+NO 0 9.0e11 0.7 332 1.05e09 1.5 27878 22942 HNO+OH←→NO+H2O 0 1.3e07 1.9 -478 2.91e05 2.5 34867 29143 NH3+H←→NH2+H2 0 5.4e05 2.4 4990 1.07e03 2.9 2257 -1344 NH3+OH←→NH2+H2O 0 5.0e07 1.6 481 4.56e06 1.8 5543 4945 NH3+O←→NH2+OH 0 9.4e06 1.9 3251 2.37e04 2.3 -435 -21


is the energy loss due to mass diffusion, with hw,i = (1 + fi/2)kT0 the molecularenthalpy of component i at the cold bubble wall, cf. table 6.2. We stress that, becausechemical reactions take place, the total energy of the bubble is not purely thermalanymore (i.e., translational, rotational and vibrational). Each species possesses anadditional potential energy – the enthalpy of formation hform,i – differences of whichdetermine the reaction energy of the chemical reactions. Hence we write:

E =∑i

(eth,i + hform,i)Ni, (6.27)

where the thermal energy per molecule eth,i is given by

eth,i =fi2kT +

∑l

kΘi,l

exp Θi,l

T − 1(6.28)

and fi, Θi,l, l = 1, . . . lmax, denotes the number of translational and rotational de-grees of freedom and the characteristic vibrational temperatures of species i. In gen-eral, lmax = 3m − 6 expcept for diatomic molecules where lmax = 3m − 5. Table6.2 gives an overview of the values that we used in the calculation. Taking the timederivative of Eq.(6.27) one finds:

E = T∑i

∂eth,i∂T

Ni +∑i

hform,iNci +

∑i

eth,iNci

+∑i

hform,iNdi +

∑i

eth,iNdi (6.29)

with Ndi , N c

i given by Eqs.(6.6) and (6.24). Note, that the diffusive change of themoderately soluble species during one cycle is negligible and therefore is not takeninto account here.The second term in the last equation is readily recognized as the net reaction energyper unit time ∑

i

hform,iNci = −V

∑j

rj∆Ej, (6.30)

∆Ej being the reaction energy of process j, cf. table 6.1. Replacing it and com-paring Eq.(6.29) to Eq.(6.26) we obtain a differential equation for the temperature T.Note that the second but last term in Eq.(6.29) cancels with a corresponding term ofEq.(6.26) and the enthalpies of formation hform,i thus only enter in the net reactionenergy – as one might expect. The final result becomes:

T∑i

∂eth,i∂T

Ni = Q− pgV +∑i

hw,iNdi

−∑i

eth,iNdi + V

∑j

rj∆Ej −∑i

eth,iNci (6.31)


TA

BL

E6.

2:C

hem

ical

lyre

activ

esp

ecie

si=

1...16

cons

ider

edin

our

mod

el.

Inad

ditio

n,ei

ther

argo

n(i=

17)

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non

(i=17

’)is

allo

wed

for.

Giv

enar

eth

enu

mbe

rfi

ofth

etra

nsla

tiona

l+ro

tatio

nald

egre

esof

free

dom

,the

char

acte

ristic

vibr

atio

nal

tem

pera

ture

sΘ

i,l,

and

the

Len

nard

-Jon

espa

ram

eter

sε i

andσi

whi

char

ere

quire

dto

calc

ulat

eth

ebi

nary

diff

usio

nco

nsta

nts,

see

the

text

belo

weq

uatio

n(6

.12)

.T

heva

lues

are

take

nfr

omre

fs.[

32,3

5,40

].T

hem

olec

ular

enth

alpy

follo

ws

from

the

num

ber

ofde

gree

sof

free

dom

ashw,i=

fi+

22kT

0.

For

som

eof

the

unst

able

radi

cals

peci

esno

liter

atur

eva

lues

ofth

eL

enna

rdJo

nes

para

met

ers

wer

eav

aila

ble.

Inth

ese

case

sw

eta

kety

pica

lval

uesσi=

3.5×

10−1

0m

andε i/k

=150

K.T

here

sults

are

foun

dto

bein

sens

itive

with

resp

ectt

oth

epr

ecis

eva

lue,

whi

chis

prob

ably

due

toth

efa

ctth

atth

ese

spec

ies

can

atm

ostc

ontri

bute

durin

ga

shor

tmom

enta

roun

dth

em

ain

colla

pse.

Fort

hest

able

,mod

erat

ely

solu

ble

spec

ies

we

furth

erm

ore

give

the

diff

usio

nco

nsta

ntD

i,liq

inth

eliq

uid

and

the

satu

ratio

nco

ncen

tratio

nni,sa

tat

one

barp

artia

lpre

ssur

ean

dT

0=

298.15

K[4

1].

isp

ecie

sf i

Θi,l

Θi,2

Θi,3

Θi,4

Θi,5

Θi,6

ε i/k

σi

Di,

liq

n i,s

at

KK

KK

KK

K10

−10m

10−

9m

2/s

mol

/m3

1H

25

6325

––

––

–33

2.97

5.1

0.78

2H

3–

––

––

–15

03.

5–

–3

O3

––

––

––

150

3.5

––

4O

25

2273

––

––

–11

33.

432.

41.

35

OH

553

70–

––

––

150

3.5

––

6H

2O

622

9552

5554

00–

––

380

2.65

––

7N

3–

––

––

–15

03.

5–

–8

NH

534

45–

––

––

150

3.5

––

9N

H2

646

2821

5247

46–

––

150

3.5

––

10N

H3

647

9813

6649

5149

5123

3923

3932

02.

6–

–11

N2H

546

5899

032

53–

––

150

3.5

––

12N

O5

2710

––

––

–11

93.

47–

–13

NO

26

1878

1064

2272

––

–15

03.

5–

–14

N2O

530

9882

218

19–

––

220

3.88

––

15H

NO

538

9521

5822

50–

––

150

3.5

––

16N

25

3350

––

––

–92

3.68

20.

617

Ar

3–

––

––

–12

43.

422.

51.

417

’X

e3

––

––

––

229

4.1

1.47

4.3

66 6.3. PHASE DIAGRAMS

6.2.8 Summary of the ODE model

The preceding subsections represent a closed set of 20 coupled first order ordinarydifferential equations for the 20 time dependent quantities, namely the number ofmolecules of 16 reactive species (i = 1− 16 in table 6.2), the bubble radius R(t), itsfirst and second derivatives R(t) and R(t), and the temperature in the bubble T (t).This set of coupled ODEs is solved numerically. We stress that the model does notcontain any free parameter. The summary of the physical ingredients is:

1. The bubble dynamics given by Eq.(6.1). As it is effectively third order owingto the inertially corrected gas pressure (Eq. (6.2)) it leads to three first orderequations for R, R and R.

2. The compositional change of the gas in the bubble due to

(a) mass diffusion, modeled by Eqs. (6.4) and (6.6)

(b) chemical reaction, described by Eq. (6.24).

The total rate of change of the various components is the sum of these twocontributions and yields 16 coupled first order equations for the 16 chemicallyactive species considered in the model.

3. The First Law of Thermodynamics which together with the total energy of thebubble yields an additional first order ODE for the gas temperature, Eq. (6.31).

Altogether, these are 20 first order ODEs.They are completed by the diffusive change of the employed noble gas according

to Eq.(6.6) [42] and by Eq.(6.2) for the gas pressure pg inside the bubble, which iscompletely determined by items 1–3, thus closing the set.

6.3 Phase diagrams

In this section we calculate phase diagrams in the Pa−R0 parameter plane for variousgas compositions, concentrations, frequencies, and water temperatures, and comparethem with those which have been mapped out experimentally [5, 6, 10] in the recentyears.

How to calculate the equilibrium points within the model, starting from someinitial bubble size and gas mixture? As already pointed out in the introduction, SBSLbubbles automatically adjust to a (dynamical) equilibrium composition and corre-sponding equilibrium radius R0 (which is characterized by the condition that the netnumber change over one cycle equals to zero for any involved species). So one couldstart with any size and composition, in particular with that corresponding to the dis-solved gas. As Eqs. (6.4), (6.6) and (6.24) completely determine the compositional


change of the bubble contents, in course of time the bubble automatically adjusts toan equilibrium composition and size (or dissolves). This equilibration can howevertake thousands of acoustical cycles as the diffusion of gas from and into the liquid israther slow – see the diffusion constants Di,liq in table 6.2 – and the computation thusis rather time consuming. Moreover, we want to find both equilibrium branches “B”and “C”. Therefore we developed a numerical method how to find both compositionalequilibria in a very efficient way. This method is given in Appendix A. We tested itagainst the “real” equilibration and found the results to be in excellent agreement.

6.3.1 Air bubbles at f = 20.6 kHz

First, we compare with the experimental phase diagram for air bubbles in watermeasured by Holt and Gaitan [5]. The external parameters were T0 = 21.7 oC,f = 20.6 kHz, and the relative saturation nair,∞/nair,sat = 14%. Fig. 6.2 depicts thetheoretical equilibrium radius R0 according to our model as a function of the drivingpressure Pa (solid lines). For comparison Holt and Gaitan’s [5] experimental data arealso given. Diamonds denote stable sonoluminescing bubbles and circles diffusivelystable bubbles that do not emit light. The overall agreement is excellent: The de-scending branch B along which the diffusion of the stable molecular species and theejection of their various reaction products compensate each other is reproduced notonly qualitatively but quantitatively. We reemphasize that the model does not containany free parameters.

Even the abrupt, discontinuous transition from branch C to B at Pa = 1.27 barcan be nicely observed in the simulation. It is witnessed by the steep decrease of R0

around Pa = 1.27 bar. Here we already mention that the transition from branch B tobranch C is hysteretic, i.e., the transition pressure amplitude is different on approachfrom above and below, respectively. The corresponding hysteresis is dealt with indetail in section 6.4. Here we only indicate this behavior by the tail of branch B thatcoexists with branch C.

In order to further highlight the properties of the equilibrium points, Fig. 6.3shows the corresponding composition in terms of particle numbers, the argon fractionξAr = NAr/Ntot and the maximum temperature of the bubble, again as a function ofthe driving pressure. The discontinuity is clearly seen in all three plots.

At the critical Pa the number of nitrogen molecules (dashed line) drops by ordersof magnitude whereas the amount of argon (solid line) increases considerably. Themiddle graph shows that the corresponding mole fraction of argon rises from 20% tomore than 90%, reflecting the dissociation hypothesis. Now the adiabatic exponentof argon is 5/3, much larger than that of the reactive molecular gases which domi-nate below the critical point. Therefore, the adiabatic heating at collapse gets moreefficient. Consequently, at the critical point the maximal bubble temperature sharplyrises. From figure 6.3 we see that beyond the critical point it is larger than 15000 K,


1.1 1.2 1.3 1.4

2

3

4

5

6

Pa / bar

R0 /

µm B C

FIGURE 6.2: Equilibrium radius R0 as a function of the driving pressure Pa, for thesame parameters as in Ref. [5]: T0 = 21.7 oC, f = 20.6 kHz, nair,∞/nair,sat = 0.14.The experimental data are plotted as well. Filled diamonds denote glowing bubbles,open circles diffusively stable bubbles without light emission. The error bars of theexperimental data are estimated to be ∆R0 = ±1µm, ∆Pa = ±0.025 bar [5].

and light emission sets in. The coincidence of argon rectification and light emissionwas already remarked by Gaitan and Holt [5]. and is also seen by other experimen-talists: All so called waterfall diagrams of Barber et al. [4] show light emission onlywhen the ambient radius is already increasing with Pa (i.e., when branch C has al-ready been reached), see figures 18, 21, 32, 33, 70, of ref. [4, 43]. The present modelcan now clearly relate the light emission to the composition and the correspondingchemical activity of the bubble. They are the limiting factors for the maximallyachievable temperature, and therefore for the degree of ionization and thus for the re-sulting light emission. Note that the temperatures of around 4000−6000 K at the endof branch B match those measured in multi bubble sonoluminescence [44–46]. This


is not a coincidence: In both cases the bubbles contain large quantities of moleculargases, the chemical reactions of which provide an upper limit for the temperature.

Interestingly, bubbles below the critical Pa are more or less completely depletedfrom O2, see the dashed-dotted line in Fig.6.3. Above the critical Pa they containroughly twice as much O2 asH2.

6.3.2 Argon-Nitrogen bubbles at f = 33.4 kHz

According to the dissociation hypothesis one does not necessarily have to take air asthe saturated gas. Any gas mixture whose reaction products dissolve in water shouldshow the same phenomena as long as it contains a suitable amount of a noble gas.Ketterling and Apfel [6] carried out experiments with such “artificial air” consistingof 99% N2 and 1% argon saturated to 10% to the water and indeed found the verysame behavior. Fig. 6.4 depicts their experimental result and the corresponding theo-retical counterpart. As above open circles denote stable bubbles that do not emit lightand filled diamonds light emitting bubbles, respectively. The solid line shows theresult of the calculation. It accurately follows the experimental data points. Glowingbubbles are found to consist of argon (not shown in the figure) non-glowing bubblescontain a considerable portion ofN2 and someH2/O2 from water vapor dissociation.

6.3.3 Nitrogen bubbles at f = 33.4 kHz

In the same paper Ketterling and Apfel also investigate bubbles in water that had beenpartially saturated with nitrogen only. In this case only the descending branch B couldbe observed. Fig. 6.5 shows the experimental data of that reference (circles) as wellas the corresponding result of the simulation (solid line). The saturation was 10%,the liquid temperature T0 293.15K. The descending trend is reproduced nicely, thequantitative agreement with the experiment is reasonable. Above a driving pressurePa ≈ 1.3 bar the bubbles are reported to be no longer stable and to dissolve. Thereason herefore presumably is the shrinking of the basin of attraction with increasingPa. It will be considered in detail in section 6.4.

6.3.4 Air bubbles at f = 33.4 kHz

Next we consider once more air as the saturated gas, now at f = 33.4 kHz andT0 = 293.15 K. Fig. 6.6 depicts the theoretical equilibrium radius R0 (solid line)as a function of the driving pressure Pa for bubbles in water saturated with air to10%, 20%, and 40%, respectively, in comparison with the corresponding experimen-tal data obtained by Ketterling and Apfel [47]. As above, diamonds denote exper-imental data points of stable sonoluminescing bubbles and circles those of stablebubbles that do not glow. Good agreement is achieved in all three cases.


106

109

1012

Ni

0

0.5

1

ξ Ar

1.15 1.2 1.25 1.3 1.35 1.40

1

2x 104

Pa / bar

Tm

ax /

K

FIGURE 6.3: Particle numbers (upper graph), argon fraction (middle graph) andmaximum temperature (lower graph) as a function of the driving pressure for thesame parameters as in Fig.6.2. The lines in the upper graph denote: Ar (solid),N2 (dashed), H2 (dotted), and O2 (dashed-dotted). The abrupt transition from thenon-sonoluminescing to the sonoluminescing regime is nicely reflected in all threegraphs.

6.3.5 Xenon-Nitrogen bubbles at f = 33.4 kHz

Finally, we want to compare our model results to the corresponding experimentaldata for gas mixtures containing a noble gas other than argon. As an experimental


1.1 1.2 1.3 1.4 1.5 1.61

2

3

4

5

6

Pa / bar

R0 /

µm

FIGURE 6.4: Comparison of the numerical results and the experimental data fora gas mixture of 99% N2 and 1% Ar saturated to 10% to the water. The drivingfrequency and the water temperature are f = 33.4 kHz and T0 = 293.15 K. Thesymbols denote the same as in Fig.6.2. The error bars of the experimental data arereported as ∆R0 = ±0.9µm, ∆Pa = ±0.07 bar.

reference we use again the data of Ketterling and Apfel for a gas mixture consistingof 99% nitrogen and 1% xenon saturated to 5%, 10% and 20% in water. We stressagain that our model does not contain any free parameter and though the transportparameters of xenon typically differ by 250% [48] from the ones of argon we findexcellent agreement. Fig. 6.7 plots the result in the familiar way of the precedingsections. The temperature (not shown in the figure) at the end of branch B in all threecases is again 6000K. On branch C it varies between 35000K for 5% saturation and18000K for 20% saturation.

72 6.4. HYSTERESIS

1 1.1 1.2 1.3 1.4 1.51

2

3

4

5

6

Pa / bar

R0 /

µm

FIGURE 6.5: Comparison of the numerical results and the experimental data from [6]for pure nitrogen saturated to 10%. The driving frequency is f = 33.4 kHz andthe water temperature T0 = 293.15 K. The error bars of the experimental data arereported as ∆R0 = ±0.9µm, ∆Pa = ±0.07 bar

6.4 Hysteresis

In SBSL experiments the bubble is found to be subject to hysteresis [49], i.e., a pres-sure regime is observed where the composition and the ambient radius of the bubbleare not unique: Depending on whether this regime is approached with ascending ordescending driving pressure, two different stable states – a light emitting one and anon-light emitting – are reported at the same Pa. Also this feature is nicely capturedin our model. We stress that such hysteretic behavior is an inherent property of SBSLbubbles, when starting with a gas mixture of molecular gases with inert gases. It ispresent in all cases considered in section 6.3 indicated by the tails of branch B whichcoexists with branch C at its onset.


1 1.25 1.51

2

3

4

5

6

Pa / bar

R0 /

µm

1 1.25 1.5

2

4

6

Pa / bar

1 1.25 1.5

2

4

6

Pa / bar

FIGURE 6.6: Comparison of the numerical results and the experimental data for airbubbles at 10%, 20% and 40% saturation (from left to right). The driving frequencyand the water temperature are f = 33.4 kHz and T0 = 293.15 K.

The mechanism is as follows, see figure 6.8: As is indicated by the arrows, withascending driving pressure the bubble follows branch B up to Ptrans 1.31 atm,i.e., beyond the critical pressure Pcrit 1.28 atm, where branch C appears. Sincebranch B is dominated by molecular gases the temperature hardly exceeds 6000Kand consequently, the bubble hardly emits any light – if at all. At Ptrans a transitionto branch C takes place. The temperature rises to 15000 K (seen in the lower graphof Fig. 6.8) since the bubble now essentially contains argon with a much higher adia-batic exponent, cf. also 6.3.1; Correspondingly, light emission sets in. At first glancethe transition to branch C seems to be contradictory as branch B is locally stable. Ascan be seen from Fig. 6.1 however, branch B is encapsulated between the unstablebranch A and the (negatively sloped) unstable part of branch C. Its region of attrac-tion correspondingly tends to zero with increasing driving pressure. At Ptrans it has

74 6.5. ROBUSTNESS OF THE MODEL

1 1.25 1.51

2

3

4

5

6

Pa / bar

R0 /

µm

1 1.25 1.5

2

4

6

Pa / bar

1 1.25 1.5

2

4

6

Pa / bar

FIGURE 6.7: Comparison of the numerical results and the experimental data for agas mixture of 99% nitrogen and 1% xenon saturated to 5%, 10% and 20% (from leftto right). The driving frequency and the water temperature are again f = 33.4 kHzand T0 = 293.15 K.

become so small that a small increase or perturbation of the driving pressure bringsthe bubble to a region in phase space (above the negatively sloped, unstable part ofbranch C) where it is attracted by the stable, upper part of branch C rather than bybranch B; transition takes place.

When the driving pressure is lowered again, the bubble now follows the stablepart of branch C down to its turnaround at Pcrit and then drops back to branch B.

6.5 Robustness of the model

The reader may worry on the robustness of the model. How do the results dependon the detailed model assumptions? In this section we slightly vary our model and


12345

R0 /

µm

1.2 1.25 1.3 1.35 1.40

1

2x 104

Pa / atm

Tm

ax /

K

Pcrit

Ptrans

B

C

FIGURE 6.8: Demonstration of hysteresis in SBSL. With increasing driving pressurethe crossover from branch B to C does not take place at Pcrit where the slope ofbranch C changes sign but only later at Ptrans. With decreasing Pa the bubble followsthe complete stable part of branch C.

investigate the consequences. It turns out that the numbers slightly change while thephysical statements remain robust.

6.5.1 Finite rate of condensation

The rate of condensation is subjected to another physical constraint which has notbeen accounted for so far: Even with vanishing li,d, i.e., with an infinitely thin bound-ary layer, it will not exceed a certain threshold due to the presence of a finite conden-sation time. Note that condensation of vapor and uptake of chemical reaction prod-ucts are limited by the same mechanism. In the following we will therefore refer toboth processes as “condensation”.

76 6.5. ROBUSTNESS OF THE MODEL

The existence of such an upper bound to the rate of mass change is evident: Itis set by the rate at which molecules strike the surface. A crude estimate is readilyobtained from the kinetic theory of gases [50]. A reliable quantification however isdifficult as it involves the so called “uptake coefficient” for molecules on the surface,which already for water vapor is reported to span two orders of magnitude [51].

A threshold to the rate of condensation has been employed earlier in modelsof the interior of an SBSL bubble [14, 15, 17, 18]. Following these references andrewriting the corresponding formulae in terms of the density rather than the partialpressure of component i one finds:

Nd,maxi = 4πR2ψi

√kT0

2πmi(ni,0 − ni) (6.32)

with ψi the uptake coefficient of species i and T0 the interface temperature, cf. sub-section 6.2.5. As mentioned above there is considerable uncertainty on the value ofψi. We follow once more refs. [14, 15, 18] and use ψi = 0.4, which is the value rec-ommended for water vapor [51]. Note that one minor simplification has been madein Eq.(6.32). It usually involves an additional correction factor (frequently denotedas Γ) which accounts for bulk motion of the gas. As it stays close to unity [50] it hasbeen omitted here.

Eq.(6.32) enables us to investigate the importance of the finite rate of uptake andcondensation with respect to the diffusive rate of mass change, modeled by Eq.(6.6).Indeed, there is a brief moment around the main collapse when both Eq.(6.32) and(6.6) are of the same order of magnitude, while for most of an acoustical cycle thediffusive rate of mass change stays more than an order of magnitude below the con-densation threshold. Limiting the rate of mass change by Eq.(6.32) [15, 18] conse-quently has small influence only.

Fig. 6.9 depicts the argon fraction (upper graph) and the maximum temperatureof an SBSL bubble (lower graph) with and without condensation threshold. The pa-rameters are chosen as in 6.3.1. The dashed lines correspond to a rate of mass changelimited by Eq.(6.32), the solid lines denote the unbound case. The findings are asfollows:(i) The cutoff hardly influences the temperature along branch B, which is plotted inthe lower graph. Consequently, since in this pressure regime the temperature con-trols the chemical output of the bubble, the equilibrium composition is virtually un-affected, as is in particular the argon fraction (upper graph).(ii) There is a visible reduction of the maximum temperature of the bubble (lowergraph) along branch C. This holds in particular for the crossover region to branch B.The reason herefore is that due to the condensation threshold especially small bub-bles loose vapor less easily which then again leads to a larger heat capacity, enhancedchemical activity, and accordingly weaker heating of those bubbles.


0

0.5

1

ξ Ar

1.15 1.2 1.25 1.3 1.35 1.40

1

2x 104

Pa/ bar

Tm

ax /

K

FIGURE 6.9: Demonstration of the effects of a maximal condensation rate. Thedashed lines corresponds to a bound, the solid lines to an unbound rate of condensa-tion. The parameters are chosen as in 6.3.1.

(iii) Though in the bound case bubbles entrap somewhat more vapor this does notaffect the argon fraction and the equilibrium composition (not shown) along branchC.

In summary, the bubble’s maximum temperature along branch C to a certainextent is affected by the employment of a condensation threshold; the phase diagramshowever are virtually unchanged.

6.5.2 The boundary layer thickness

The diffusive and thermal boundary layers li,d and lth, Eq.(6.9), (6.11) originate fromdimensional analysis and correspondingly possess an uncertainty of O(1). In order tounderline the robustness of the model we investigate in this subsection the sensitivity

78 6.6. CONCLUSIONS AND SUMMARY

of the presented results to a change of this quantities. Fig.6.10 depicts the equilibriumbranches B and C for the parameters of subsection 6.3.1 with modified boundary layerthickness ζ × li,d, ζ × lth. Dashed, solid and dotted lines correspond to ζ = 0.5, 1and 2, respectively.

One observes a slight change of the equilibrium branch B which however doesnot affect the overall quantitative agreement with the experimental results. It becomesclear from the fact that the exact position of branch B is determined by the onset ofnitrogen dissociation. Changing the boundary layer thickness has a twofold effect onthis onset. (i) The maximum radius and the amount of trapped water vapor slightlychange. Both slightly modify the maximum temperature and thereby the chemicalactivity. (ii) Water vapor has a catalytic effect on the nitrogen dissociation [18],which is also slightly changed.

As one might expect, the equilibrium branch C is essentially unaffected by themodified boundary layer thicknesses, since it is mainly determined by the chemicallyinert argon.

6.6 Conclusions and summary

To conclude, in Fig. 6.11 we directly compare the phase diagram of ref. [7] (dashedlines, which we have shown as Fig. 6.1 in the introductory section) to its counterpartwithin the present model (solid lines).

Despite the simplifying assumptions made in [7] the overall agreement withthe present model is not bad. However, three things are worth mentioning: First,branch B is much steeper in the present model. Second, the wiggling structure of theequilibrium branches, caused by the afterbounces, has completely disappeared. Thisis due to stronger thermal damping in the present model, which leads to fewer andsmaller afterbounces. Third, though the model of Ref. [7] does not account for vapor,which leads to a smaller maximum radius Rmax and via 〈p〉i,4 /P0 ∼ (R0/Rmax)

3

[4] in general to a smaller R0, branch C is matched quite accurately for large Pa. Thisseems to be a coincidence. We compared the bubble dynamics for the two models andfound that differences of the maximum radius are balanced by the enhanced presenceof afterbounces in the model of ref. [7], eventually leading to a similar 〈p〉i,4 andhence to a similar ambient radius R0.

Next, to further underline the importance of water vapor [14, 15, 17, 20, 52–54],we show in Fig.6.12 how equilibria are influenced by the liquid temperature andhence by the presence of vapor. The frequency and the saturation are the same asin Fig. 6.1. The solid line corresponds to a liquid temperature T0 = 293K and thedashed lines to T0 = 274K.

Along branch C the reduced vapor pressure in the cold liquid leads to a smallermaximum radius and consequently (because 〈p〉i,4 /P0 ∼ (R0/Rmax)

3 [4]) to a


1.1 1.2 1.3 1.4

2

3

4

5

6

Pa / bar

R0 /

µm B C

FIGURE 6.10: Equilibrium branches B and C for the parameters of section 6.3.1 butwith modified boundary layer thickness ζ × li,d, ζ × lth, see Eq.(6.9), (6.11). Thevalues are ζ = 0.5, 1, 2 for dashed, solid and dotted lines, respectively.

smaller R0.

Also the shift of branch B can be explained by the difference of Rmax. Forchemical activity to occur the temperature of the bubble must exceed a certain crit-ical value (≈ 4000 − 6000K). At a higher liquid temperature the bubble reachesa larger Rmax, it consequently collapses more violently which (in this parameterregime) leads to stronger heating and the necessary temperatures for chemical activ-ity are achieved already at a lower driving pressure Pa. Note that this is not the caseanymore along branch C. In that regime a considerable portion of vapor is trapped inthe bubble, the large heat capacity and chemical reactions of which result in a lowereffective adiabatic exponent and less heating occurs, see also Fig. 6.3.

In summary, we presented an ODE approach to sonoluminescing bubbles withthe main building blocks bubble dynamics, heat transfer, mass transfer, among which


1 1.1 1.2 1.3 1.4 1.51

2

3

4

5

6

7

Pa/ atm

R0 /

µm

B C

FIGURE 6.11: Comparison of the equilibrium branches for the present model (solidlines) and the model of Ref. [7]. The parameters are chosen as in Fig. 6.1.

non-equilibrium phase change of liquid vapor plays an important role, and chemicalreactions. The parameter-free model is able to accurately and quantitatively giveequilibrium points in the R0-Pa phase space, as comparison with experimental datafor a variety of different cases showed. In the SBSL regime temperatures around10000 − 15000K (needed for light emission from thermal bremsstrahlung) evolvefrom the model. These relatively large temperatures are a consequence of the argonaccumulation on branch “C”, as the adiabatic exponent of argon is larger than thatof reactive molecular gases and such efficient adiabatic heating is achieved at bubblecollapse. On branch B, where molecular gases dominate the bubble content, thetemperature remains in the range of 5500K, just as large as in MBSL [46], whereinert gases cannot accumulate due to the shape instability of the bubbles at theircollapse.

The developed model has predictive power. E.g., phase diagrams for different


1 1.1 1.2 1.3 1.4 1.51

2

3

4

5

6

7

Pa/ atm

R0 /

µm

B C

FIGURE 6.12: Influence of the water temperature and thus the vapor pressure oncompositional equilibria. The parameters are chosen as in Fig. 6.1. The liquid tem-peratures were T0 = 274K (dashed line), and T0 = 293K (solid line), respectively

gases, different compositions, different frequencies, or different water temperaturescan easily be calculated. The model also gives both the bubble temperature andthe chemical output of the bubble [55] and therefore provides a sound basis for thecomplete understanding and possibly optimization of a micro bubble chemical reactor[56].

Acknowledgments

We wish to thank Jeffrey Ketterling for making his experimental data available tous. It is furthermore our pleasure to acknowledge S. Hilgenfeldt, C. D. Ohl, andA. Prosperetti for critical comments and valuable discussions on the present work.The work is part of the research program of FOM, which is financially supported by


NWO.

Appendix A: Numerical calculation of compositional equilibria

As already pointed out, compositional equilibration can take thousands of acousticalcycles, which is unacceptable if one wants to map out R0–Pa phase diagrams. Wepresent here an alternative method to find equilibrium points based on the Newton–Raphson algorithm. For a given Pa it is found to reach a steady state typically after100-200 acoustical cycles, depending on how far from equilibrium the compositioninitially was.Starting point is the net change of species i over one cycle:

∆Ni = ∆Ndi +∆N c

i . (6.33)

The second term denotes the chemical change of component i per cycle,

∆N ci =

∫ T

0N c

i dt. (6.34)

The first term denotes the diffusive change of component i per cycle, which either is

∆Ndi = 4πDi,liqτRmaxni,sat

(ni,∞ni,sat

− 〈p〉i,4P0

)(6.35)

for the species H2, O2, N2, or Ar/Xe which do not react with water, or

∆Ndi =

∫ T

0Nd

i dt (6.36)

for all remaining species.The condition for compositional equilibrium reads:

∆Ni = 0 (6.37)

for all species i=1,...,17, i.e., the net number change over one cycle must be zero forany involved species.As a first approximation we consider in the following only the chemically stable(under ambient conditions), moderately soluble species, H2, O2, N2, and Ar/Xe.This is justified by the fact that apart from a brief moment during the main collapseall other species appear in trace amounts only and thus have negligible influence onthe equilibrium composition.

∆N := (∆Ni)i=H2,O2,N2,Ar/Xe (6.38)


can be regarded as a multidimensional function of the variables

N := (Ni)i=H2,O2,N2,Ar/Xe. (6.39)

The zero crossings ∆N(N) = 0 are the desired equilibrium points. The Newton-Raphson algorithm is conveniently used to calculate these zero crossings: We henceupdate the composition at the end of each cycle according to

Nnew = Nold − J−1∆N (6.40)

with

J =

(∂∆Ni

∂Nj

)i,j

(6.41)

the Jacobian matrix of ∆N. The indices i and j run over the species H2, O2, N2, andAr/Xe.

The remaining step is to determine J. We do that in an approximate analyticalway, which has the advantage that the final result allows for simultaneous calculationof the Jacobian and integration of the ODE - system. The fact that the number ofparticles of the species under consideration is only slightly changing during one cycleplays a key role hereby as in this case necessary differentiations and integrations canbe interchanged to a good approximation.

The Jacobian matrix J has two contributions:(i) diffusive contribution:The diffusive change of component i is determined by its weighted mean partial pres-sure, introduced in Eq.(6.5). Since Ni ≈ const., we extract it from the integral andwrite:

〈p〉i,4 Ni 〈p〉4 , 〈p〉4 :=

∫ T0

kTV−NBR

4dt∫ T0 R

4dt(6.42)

Note that strictly speaking also the dynamics and therefore all related quantities like〈p〉4 or Rmax weakly depend on the composition of the bubble (in fact, primarily onthe total number of particles). Neglecting these contribution though, one finds:

∂ 〈p〉i,4∂Nj

=∂Ni 〈p〉4∂Nj

δi,j 〈p〉4 (6.43)

with δi,j the Kronecker symbol. The diffusive part of the Jacobian is with theseassumptions readily found from differentiation of Eq. (6.35):

∂∆Ndi

∂Nj −δi,j4πDi,liqTRmaxni,sat

〈p〉4P0

(6.44)


(ii) chemical contribution:The chemical part of J is computed in a similar way. Since theNi can be regarded asroughly constant we interchange the differentiation of Eq. (6.34) with the integrationappearing herein.

∂∆N ci

∂Nj∫ T

0

∂N ci

∂Njdt. (6.45)

The N ci however are available analytically, see Eq.(6.18) - Eq.(6.24) and the integrand

can therefore be explicitly calculated by differentiating Eq.(6.24) with respect to Nj .The Jacobian is finally given by the sum of the diffusive and the chemical con-

tribution:∂∆Ni

∂Nj=∂∆N c

i

∂Nj+∂∆Nd

i

∂Nj(6.46)

REFERENCES 85

References[1] D. F. Gaitan, Ph.D. thesis, The University of Mississippi (1990).

[2] D. F. Gaitan, L. A. Crum, R. A. Roy, and C. C. Church, J. Acoust. Soc. Am. 91,3166 (1992).

[3] M. P. Brenner, S. Hilgenfeldt, and D. Lohse, Rev. Mod. Phys. 74, 425 (2002).


[5] G. Holt and F. Gaitan, Phys. Rev. Lett. 77, 3791 (1996).



[8] D. Lohse and S. Hilgenfeldt, J. Chem. Phys. 107, 6986 (1997).

[9] T. J. Matula and L. A. Crum, Phys. Rev. Lett. 80, 865 (1998).

[10] J. A. Ketterling and R. E. Apfel, Phys. Rev. E 61, 3832 (2000).

[11] Y. T. Didenko and K. S. Suslick, Nature 418, 394 (2002).


[13] A. Prosperetti and Y. Hao, Philos. Trans. R. Soc. London, Ser. A 357, 203(1999).




[17] K. Yasui, Phys. Rev. E 56, 6750 (1997).

[18] B. D. Storey and A. J. Szeri, Phys. Rev. Lett. 88, 074301 (2002).

[19] S. Hilgenfeldt, S. Grossmann, and D. Lohse, Phys. Fluids 11, 1318 (1999).

86 REFERENCES


[21] R. Toegel, S.Hilgenfeldt, and D.Lohse, Phys. Rev. Lett. 88, 034301 (2002).

[22] C. C. Wu and P. H. Roberts, Phys. Rev. Lett. 70, 3424 (1993).

[23] L. Kondic, J. I. Gersten, and C. Yuan, Phys. Rev. E 52, 4976 (1995).

[24] V. Q. Vuong and A. J. Szeri, Phys. Fluids 8, 2354 (1996).


[26] W. C. Moss, D. A. Young, J. A. Harte, J. L. Levatin, B. Rozsnyai, G. B. Zim-merman, and I. H. Zimmerman, Phys. Rev. E 59, 2986 (1999).

[27] L. Yuan, C. Y. Ho, M. C. Chu, and P. T. Leung, Phys. Rev. E 64, 016317 (2001).

[28] S. Hilgenfeldt, M. P. Brenner, S. Grossmann, and D. Lohse, J. Fluid Mech. 365,171 (1998).

[29] V. Q. Vuong, M. M. Fyrillas, and A. J. Szeri, J. Acoust. Soc. Am. 104, 2073(1998).

[30] J. B. Keller and M. Miksis, J. Acoust. Soc. Am. 68, 628 (1980).

[31] H. Lin, B. Storey, and A. Szeri, J. Fluid Mech. 452, 145 (2002).

[32] D. R. Lide, ed., CRC Handbook of Chemistry and Physics (CRC Press, BocaRaton, 1991).


[34] F. Lepoint-Mullie, T. Lepoint, and A. Henglein, in NATO-ASI on Sonolumines-cence and Sonochemistry, edited by L. Crum (Kluwer Academic Publishers,Dordrecht, 1998), p. 285.

[35] J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular theory of gases andliquids (Wiley, New York, 1954).

[36] E. U. Condon and H. Odishaw, Handbook of Physics (McGraw-Hill, New York,1958).

[37] W. C. Gardiner, Gas-Phase Combustion Chemistry (Springer Verlag, New York,2000).

[38] G. P. Smith, D. M. Golden, M. Frenklach, N. W. Moriaty, B. Eiteneer, M. Gold-enberg, C. T. Bowman, R. K. Hanson, S. Songand, W. C. Gardiner Jr., et al.,Gri-mech 3.0, http://www.me.berkeley.edu/gri mech/.

REFERENCES 87

[39] G. N. Hatsopoulos and J. Keenan, Principles of General Thermodynamics (Wi-ley, New York, 1965).

[40] J. A. Fay, Molecular Thermodynamics (Addison-Wesley, Reading, Mas-sachusetts, 1965).

[41] NIST, Chemistry Webbook (2001), http://webbook.nist.gov/chemistry/.

[42] Note that this is no extra ODE as the slow diffusive timescales are treated ac-cording to the timescale separation method by Fyrillas and Szeri [33].

[43] The only experimental exception we found is figure 38 of ref. [4], where verydim light emission sets in already slightly before the minimum in theRequi

0 (Pa)curve.

[44] E. B. Flint and K. S. Suslick, Science 253, 1325 (1991).

[45] K. S. Suslick, E. B. Flint, M. W. Grinstaff, and K. A. Kemper, J. Phys. Chem.97, 3098 (1993).

[46] W. B. McNamara III, Y. T. Didenko, and K. S. Suslick, Nature 401, 772 (1999).

[47] The data stem from the experimental investigation of J. A. Ketterling and R.E.Apfel. Part of it has been published in [6] and [10].

[48] An easy estimate is obtained from the fact that Di,χi ∝ 1σ2

i

√mi

, cf. Eq.6.12 and

table 6.2.

[49] T. Asai and Y. Watanabe, Jpn. J. Appl Phys 39, 2269 (2000).

[50] J. C. Collier, Convective Boiling and Condensation (McGraw-Hill, London,1972).

[51] I. Eames, N. Marr, and H. Sabir, Int. J. Heat Mass Transfer 40, 1997 (2963-2973).

[52] S. Putterman, P.G.Evans, G.Vazquez, and K. Weninger, Nature 409, 782 (2001).

[53] S.Hilgenfeldt, S. Grossmann, and D.Lohse, Nature 409, 783 (2001).

[54] K. Yasui, Phys. Rev. E 60, 1754 (1999).

[55] The latter in principle can be measured experimentally with the methods devel-oped by Didenko and Suslick [11].

[56] D.Lohse, Nature 418, 381 (2002).

88 REFERENCES

CHAPTER 7. CONCLUSIONS AND OUTLOOK 89

Chapter 7

Conclusions and outlook

What is thus the limiting factor for the maximum temperature of an SBSL bubble?The preceding chapters give a clear answer to that: It is the number of internal

degrees of freedom of the bubble contents.The numerous vibrational degrees of freedom which are generated through the

addition of small amounts of complex molecules, as elaborated in chapter 2, are onesuch example. The dominating presence of liquid vapor in the bubble at low fre-quency, see chapter 4, is another one. Chemical reactions with their large number ofdegrees of freedom, see chapter 5 and 6, complete this picture. The bubble tempera-ture as a function of the input energy is sketched in Fig. 7.1. It turns out to stronglydepend upon the composition of the bubble:

Air-vapor bubbles for example, which are found in multi-bubble sonolumines-cence or along the equilibrium branch B in SBSL, undergo a chemical phase transi-tion at around 5000K-6000K. At this point most of the thermal energy is spent on thedissociation of the molecular components in the bubble. Not enough input energyis available though to complete this phase transition 1 and therefore 5000K-6000Krepresents an upper temperature limit for such bubbles.

The noble gas-vapor bubbles, which are found along branch C in the Pa − R0

phase diagrams, show stronger heating – accompanied by significant light emission– because they possess a considerably smaller number of internal degrees of freedomthan air. Nonetheless, the chemical degrees of freedom of trapped liquid vapor alsolimit the temperature of such bubbles to 15000K-20000 K.

Let us go one step further: Suppose, one could somehow remove the liquidvapor from the bubble or prevent it from entering. Still ionization chemistry and

1An estimate for the available energy is given by the potential energy of the liquid-bubble systemat maximum expansion, P0 (4π/3) R3

max, which is only of the same order of magnitude as theenergy needed to dissociate all molecules in the bubble. The bubbles would have to collapse completelyadiabatically in order to provide enough energy for complete dissociation.

90

FIGURE 7.1: Sketch of the bubble temperature as a function of the input energy.Horizontal lines indicate phase transitions of the gas in the bubble. The SBSL regimeis shaded.

electronic excitation set in at temperatures of some 104 K – otherwise obviously nolight emission could be observed. The associated numerous electronic degrees offreedom do not yet play a role for a typical SBSL bubble, but they would for slightlyhigher temperatures and probably represent an unsurmountable obstacle for furtherheating.

Here is nevertheless an outline of what an SBSL system which is to give hotterbubbles would have to look like:

One needs not necessarily take water as the surrounding liquid. Improvementof the energy focusing abilities can be expected if the liquid fulfills the followingrequirements:

• For obvious reasons the vapor pressure of the liquid should be as small aspossible. Otherwise, the material properties, e.g. mass density, surface tensionor viscosity, should roughly equal those of water to ensure a similar radialdynamics and in particular the occurrence of the necessary inertial collapse.

CHAPTER 7. CONCLUSIONS AND OUTLOOK 91

• Even tiny amounts of insoluble reaction products will accumulate in the courseof time. Hence, in order to guarantee noble gas rectification, the reaction prod-ucts of the gas-vapor chemistry must all be soluble in the liquid.

The interior of the bubble:

• A noble gas is recommended because such gases possess a small number of in-ternal degrees of freedom – equivalent to a high adiabatic exponent – and hencegive rise to the strongest heating. Furthermore, they are the only species forwhich chemical stability (apart from ionization chemistry) is trivially fulfilled.

• Helium seems the perfect candidate among these as it possesses only two elec-trons and hence the smallest number of electronic degrees of freedom. 2

The external driving:

• Upscaling through lowering the driving frequency failed due to the presence ofwater vapor. It might however be an option if it is combined with the use of aliquid with low vapor pressure.

• The use of driving signals other than a sine – which necessarily involve har-monics – still seems to be one of the most promising candidates for upscalingSBSL. Preliminary theoretical results indicate that if one neglects ionizationchemistry for the time being, small bubbles (R0 = 2 − 3µm) can reach tem-peratures of some 105 K through suitable addition of harmonics to the drivingsignal [1], the bubble still being shape stable. However, it is not clear howmuch of this effect will remain when ionization and electronic excitation isaccounted for.

• Complications are likely to occur with harmonic driving: As compared to thesingle driving case the bubble reaches a considerably larger maximum radius(5 − 10 times as large as under the usual SBSL conditions). The collapse isdelayed accordingly – since the expansion phase simply takes longer. Once thedelay becomes too large the Bjerknes force of the first kind which keeps thebubble in place – i.e., at the antinode of the acoustical field – changes sign andbecomes repulsive. The bubble is pushed away from the desired high pressureregion, respectively.

Finally, it should be pointed out that cavitation physics has more to offer thanonly the pulsation of a single, stationary bubble. For example, cavitating bubblesare known to play an important role in the ultrasonic treatment of kidney stones

2Note that, if one is interested in optimizing the light output rather than the temperature, xenon is tobe preferred, since the light output is coupled to the degree of ionization, which is largest for xenon.

92

(lithotripsy) [2–4] and they can transfer mass in a yet unresolved way to living cellsin their vicinity [5–8]. Driven by the tensile part of a shock wave, such bubblesexpand to a size of about 1mm. Their subsequent collapse generates considerableforces on neighbouring boundaries – like kidney stones or cell membranes. The dy-namics of such bubbles turns out to be completely dominated by water vapor [9].Another research area within the field of cavitation physics is the control of bubblesin complex flow situations. As mentioned above, in SBSL there is essentially onlyone force acting on the bubble, namely the Bjerknes or pressure force, which keepsit in place. In general flow situations however a bubble is subject to a variety ofcompeting forces like drag, lift, added mass, etc. , the interaction of which leadsto often complex translational motion of the bubble [10]. The role and influence ofsome of these forces like the history force is not understood at all. And the situationbecomes even more complicated when more than only one bubble is involved as thenalso bubble-bubble interaction [11], i.e. Bjerknes forces of the 2nd kind have to betaken into account. A sound understanding of these forces and their interplay formsthe basis for any application involving bubbles.

In conclusion I dare say that a single, spherical bubble, pulsating but fixed inposition, in essence is understood. The challenge now is to go beyond one bubbleand to better understand bubble-bubble and bubble-wall interaction.

REFERENCES 93

References[1] X. Lu, A. Prosperetti, R. Toegel, and D. Lohse, Harmonic enhancement of sin-

gle bubble sonoluminescence, preprint.

[2] W. Sass, M. Braeunlich, H. P. Dreyer, E. Matura, W. Folberth, H. G. Priesmeyer,and J. Seifert, Ultrasound in Med. & Biol. 17, 239 (1991).

[3] M. Delius, Ultrasound in Med. & Biol. 23, 611 (1997).

[4] M. Delius, F. Ueberle, and W. Eisenmegner, Ultrasound in Med. & Biol. 24,1055 (1998).

[5] U. Lauer, E. Buergelt, K. Messmer, P. H. Hofschneider, M. Gregor, and M.Delius, Gene Therapy 4, 710 (1997).

[6] K. Tachibana, T. Uchida, N. Yamashita, and K. Tamura, Lancet 353, 1409(1999).

[7] M. Lokhandwalla and B. Sturtevant, Phys. Med. Biol. 46, 413 (2001).

[8] M. Lokhandwalla, J. A. McAteer, J. C. Williams, and B. Sturtevant, Phys. Med.Biol. 46, 413 (2001).

[9] M. C. Jullien, C. D. Ohl, R. Toegel, and D. Lohse, Dynamical Response of abubble submitted to two shock waves, preprint.

[10] J. Rensen, D. Bosman, J. Magnaudet, C. D. Ohl, A. Prosperetti, R. Togel, M.Versluis, and D.Lohse, Phys. Rev. Lett. 86, 4819 (2001).

[11] R. Mettin, I. Akhatov, U. Parlitz, C. D. Ohl, and W. Lauterborn, Phys. Rev. E56, 2924 (1997).

94 REFERENCES

SUMMARY 95

Summary

The mechanisms of single bubble sonoluminescence (SBSL) – in particular the originof the intense, visible radiation – were a widely disputed subject since its discoveryback in 1990. Meanwhile it is commonly accepted that the light originates fromthermal bremsstrahlung which is generated through quasi-adiabatic compression ofthe bubble. A brief overview is given in chapter 1.

Chapter 2 deals with the dependence of the SBSL light intensity on the additionof small amounts of alcohol to the surrounding water. For ethanol the light intensityis found to be reduced by one half at a molar fraction as tiny as 2.5 × 10−5; butanolachieves the same reduction even at a 10 times smaller concentration. The results areaccounted for by a phenomenological model, in which the alcohols are assumed to bemechanically forced into the bubble at collapse. Once in the bubble, their large num-ber of degrees of freedom reduces the adiabatic exponent of the gas, which leads toless heating and correspondingly to less light. The increasing hydrophobicities of thealcohols make this mechanism the more effective the longer the carbon chain of theemployed alcohol. Support for this model is obtained by replotting the experimentallight intensity values vs the calculated exponents, yielding a collapse of all data ontoa universal curve.

The addition of a small amount alcohol to the water not only strongly quenchesthe observed light intensity; it can also cause a transition from stable to unstableSBSL, as the experiments presented in chapter 3 show. The effect is theoreticallyaccounted for by considering the surface active properties of alcohols. The reductionof the surface tension turns out to significantly influence the diffusive equilibriumof the bubble and the shape stability and thereby induces the transition to unstableSBSL.

Chapter 4 presents experimental results for single bubble sonoluminescence ofair bubbles at very low frequency (f = 7.1 kHz) – where stronger bubble collapseoccur. The experiments reveal however that the bubbles are nevertheless only asbright ( 104 − 105 photons per pulse ) and the pulses only as long ( ≈ 150 ps ) asat f = 20 kHz. The reason for this is the presence of water vapor: During the rapidbubble collapse large amounts of water vapor are trapped inside the bubble, whichresults in an increased heat capacity and hence lower temperatures and less light.

96 SUMMARY

Next, recent theoretical work in single bubble sonoluminescence had suggestedthat the water vapor in the collapsing bubble undergoes energy-consuming chemicalreactions which restrict the peak temperatures to values for which hardly any lightemission could occur. Chapter 5 resolves this obvious paradox: Analyzing the re-action thermodynamics within the dense, collapsed bubble, it is demonstrated thatthe excluded volume of the non-ideal gas results in pronounced suppression of theparticle-producing endothermic reactions. Thus, sufficiently high temperatures forconsiderable bremsstrahlung emission can be achieved.

Finally, phase diagrams for single bubble sonoluminescence – i.e., equilibriumpoints of the bubble in parameter space – are presented in chapter 6. The employedmodel accounts for the bubble hydrodynamics, heat exchange, phase change of watervapor, chemical reactions of the various gaseous species in the bubble (N2, O2 andH2O being the most important among these), and diffusion/dissolution of the reac-tion products in the liquid. Altogether it consists of 20 coupled ordinary differentialequations. The results of the model are compared in detail to various phase diagramdata from recent experimental work, among which are air-water systems as well assystems with a xenon-nitrogen mixture as the saturated gas. Remarkable quantitativeagreement is found for all considered cases. In the light emitting regime the bubblesare found to nearly entirely consist of argon (dissociation hypothesis).

SAMENVATTING 97

Samenvatting

Sinds de ontdekking van sonoluminescentie van een enkele bel (SBSL) in 1990 vorm-den de mechanismen – in het bijzonder de oorsprong van de duidelijk zichtbaar stral-ing – een hevig betwist onderwerp. Inmiddels is het echter algemeen geaccepteerd,dat het daarbij zeer waarschijnlijk om thermische bremsstrahlung gaat, die door bi-jna adiabatische compressie van de bel wordt opgewekt. Hoofdstuk 1 geeft een kortoverzicht over het verschijnsel.

Hoofdstuk 2 houdt zich bezig met de invloed van kleine hoeveelheden verschil-lende alcoholen in het water op de intensiteit van het uitgestraald licht. Daarbij blijktdat al een ethanol aandeel van 2.5 × 10−5 voldoende is om de lichtintensiteit tehalveren. Bij butanol is zelfs een tiende hiervan toereikend. De resultaten kunnenworden verklaard met een fenomenologisch model, waarin aangenomen wordt dat dealcoholmoleculen tijdens de collaps mechanisch in de bel worden gedwongen. Hungrote aantal vrijheidsgraden reduceert de effectieve adiabatische exponent van hetgasmengsel, wat tot minder verhitting en in dezelfde mate tot minder licht leidt. Detoenemende hydrophobiciteit van de alcoholen maakt dit mechanisme des te effec-tiever naar mate de koolstofketting van het gebruikte alcohol langer is. Het modelwordt gesteund door het feit, dat alle meetwaarden op een universele lijn terechtkomen, wanneer ze tegen de effectieve adiabatische exponent worden uitgezet.

De toevoeging van kleine hoeveelheden alcohol aan het water verzwakt nietalleen de lichtuitstraling, ze kan ook een verandering van stabiele in instabiele sono-luminescentie veroorzaken. Dergelijke experimentele resultaten worden in hoofdstuk3 besproken. Het effect laat zich met de oppervlakteaktiviteit van de alcoholen the-oretisch verklaren. Daarbij blijkt dat de verlaging van de oppervlaktespanning groteinvloed heeft op het diffusieve evenwicht en de vormstabiliteit van de bel, waardoorde overgang naar instabiele sonoluminescentie veroorzaakt wordt.

Hoofdstuk 4 geeft experimentele resultaten voor sonoluminescentie van lucht-bellen bij een lage akoestische frequentie (7.1 kHz) – de collaps van de bel is duidelijksterker in dit geval. De experimenten onthullen echter dat de lichtflits desondanksdezelfde helderheid en dezelfde lengte heeft als bij gebruik van de standaardfre-quentie f = 20 kHz. De reden hiervoor is de aanwezigheid van waterdamp in debel; tijdens de snelle-compressie fase worden grote hoeveelheden daarvan in de bel

98 SAMENVATTING

gevangen, wat tot een vergrote warmtecapaciteit, lagere temperaturen en minder lichtleidt.

Recent theoretisch werk op het gebied van sonoluminescentie had bovendienaan het licht gebracht dat de waterdamp in de bel door zijn energie absorberendechemische reacties de temperaturen tot waardes beperkt, die lichtemissie praktischonmogelijk maken. Hoofdstuk 5 lost deze ogenschijnlijke tegenspraak op: Eenanalyse van de thermodynamica van de chemische reacties binnen de sterk gecom-primeerd bel laat duidelijk zien dat het covolume van het gas de energie vragende dis-sociatiereacties in hoge mate onderdrukt en dus voldoende hoge temperaturen voorthermische bremsstrahlung kunnen worden bereikt.

In hoofdstuk 6 worden tenslotte fasediagrammen voor sonoluminescentie, datbetekent evenwichtspunten van de bel in de parameter ruimte, gepresenteerd. Hetgebruikte model houdt naast de hydrodynamica ook rekening met warmtewisselling,verdamping en condensatie van waterdamp, chemische reacties van de talloze stof-fen (waaronder de belangrijksten N2,O2 en H2O zijn) en diffusie/oplossing van dechemische producten in de vloeistof. Bij elkaar bevat het model 20 gewone gekop-pelde differentiaalvergelijkingen. De theoretische resultaten worden in detail verge-leken met experimentele data voor fasediagramen, waaronder lucht-water systemenmaar ook systemen met een xenon-stikstof mengsel. In alle gevallen is er een op-merkelijke kwantitatieve overeenstemming. In het sonoluminescentiegebied blijkende bellen bovendien voornamelijk uit argon te bestaan (dissociatiehypothese).

ZUSAMMENFASSUNG 99

Zusammenfassung

Seit ihrer Entdeckung im Jahr 1990 waren die Mechanismen der Einzelblasensono-lumineszenz – insbesondere der Ursprung der deutlich sichtbaren Strahlung – einheftig umstrittener Gegenstand. Mittlerweile hat sich allgemein durchgesetzt, daß essich dabei wohl um thermische Bremsstrahlung handelt, die durch quasi-adiabatischeKompression der Blase erzeugt wird. Ein kurzer Uberblick wird in Kapitel 1 gegeben.

Kapitel 2 beschaftigt sich dann mit dem Einfluß kleiner Mengen verschiedenerAlkohole im umgebenden Wasser auf die Lichtintensitat. Dabei stellt sich heraus,daß schon ein Athanolanteil von 2.5 × 10−5 ausreicht, um die Lichintensitat zu hal-bieren; bei Buthanol ist sogar 1/10 dieser Menge ausreichend. Die Ergebnisse wer-den mithilfe eines phanomenologischen Modells erklart, worin angenommen wird,daß die Alkoholmolekule wahrend des Kollapses mechanisch in die Blase gezwun-gen werden. Ihre große Anzahl von Freiheitsgraden reduziert dort den effektivenAdiabatenexponenten des Gasgemisches, was zu schwacherer Erwarmung und ent-sprechend zu weniger Licht fuhrt. Die wachsende Wasserfeindlichkeit der Alkoholemacht diesen Mechanismus umso effektiver je langer die Kohlenstoffkette des ver-wendeten Alkohols ist. Das Modell wird gestutzt durch die Tatsache, daß alle Daten-punkte auf einer einzigen universellen Linie zusammenfallen, wenn sie gegen denentsprechenden effektiven Adiabatenexponenten aufgetragen werden.

Die Zugabe kleiner Alkoholmengen zum umgebenden Wasser schwacht nichtnur die Lichtemission, sie kann auch einen Wechsel von stabiler zu instabiler Sono-lumineszenz hervorrufen. Entsprechende experimentelle Ergebnisse werden in Kapi-tel 3 vorgestellt. Der Effekt laßt sich theoretisch mit den oberflachenaktiven Eigen-schaften der Alkohole erklaren. Wie sich herausstellt, hat die Herabsetzung der Ober-flachenspannung erheblichen Einfluß auf das diffusive Gleichgewicht und die Form-stabilitat der Blase, was den Ubergang zu instabiler Sonolumineszenz hervorruft.

Kapitel 4 stellt experimentelle Ergebnisse fur Einzelblasensonolumineszenz beiniedriger akustischer Frequenz (f = 7.1 kHz) vor – man beachte, daß der Blasenkol-laps deutlich “starker” ist in diesem Fall. Die Experimente zeigen jedoch, daß deremittierte Lichtblitz nichtsdestotrotz nur die gleiche Helligkeit und die gleiche Langewie bei Verwendung der Standardfrequenz f = 20 kHz hat. Der Grund hierfur istdas Vorhandensein von Wasserdampf in der Blase: Wahrend der schnellen Kompres-

100 ZUSAMMENFASSUNG

sionsphase werden große Mengen davon eingefangen. Die entsprechend vergroßerteWarmekapazitat fuhrt zu niedrigeren Temperaturen und weniger Licht.

Neuere theoretische Ergebnisse uber Einzelblasensonolumineszenz hatten daru-ber hinaus nahe gelegt, daß der Wasserdampf in der kollabierenden Blase durch seineEnergie verzehrenden chemischen Reaktionen die Temperaturen auf Werte beschrankt,die Lichtemission praktisch unmoglich machen. Kapitel 5 lost den augenschein-lichen Widerspruch auf: Eine Analyse der Thermodynamik der chemischen Reak-tionen in der stark komprimierten Blase zeigt, daß das Kovolumen des Gases dieenergetisch aufwendigen Dissoziationsreaktionen in erheblichem Maße unterdrucktund somit hinreichend hohe Temperaturen fur thermische Bremsstrahlung erreichtwerden konnen.

In Kapitel 6 schließlich werden Phasendiagramme fur Einzelblasensonolumi-neszenz, d.h., Gleichgewichtspunkte der Blase im Parameterraum, prasentiert. Daszugrunde liegende Modell berucksichtigt neben der Hydrodynamik auch Warmeaus-tausch, Verdampfung und Kondensation von Wasserdampf, chemische Reaktionender zahllosen Spezies in der Blase (N2, O2 und H2O sind hier als Wichtigste zu nen-nen) und Diffusion/Losung von Reaktionsprodukten in der umgebenden Flussigkeit.Insgesamt besteht das Modell aus 20 gewohnlichen, gekoppelten Differentialglei-chungen. Die theoretischen Ergebnisse werden im Detail mit entsprechenden ex-perimentellen Daten verglichen. Dabei werden unter anderem Systeme mit Luftals gelostem Gas aber auch Systeme mit anderen Gasgemischen, wie etwa Xenon-Stickstoff, betrachtet. In allen Fallen wird bemerkenswerte quantitative Ubereinstim-mung erzielt. Es stellt sich zudem heraus, daß Blasen im Sonolumineszenzregimenahezu vollstandig aus Argon bestehen (Dissoziationshypothese).

ACKNOWLEDGMENTS 101

Acknowledgments

First, I would like to express my gratitude to my promoter Detlef Lohse, for givingme the opportunity to earn my doctoral degree at the University of Twente. Withouthis support and open mind for new concepts, without the freedom to explore newideas, and eventually without the patience he has had with me, this work would nothave been as fruitful as it was.

I also would like to thank my family and my relatives. I found invaluable supportin times when I needed it.

I would like to thank Sascha Hilgenfeldt. He was strongly involved in a majorpart of this thesis and helped to rigorously shape its contents.

I would like to thank Andrea Prosperetti for motivating and stimulating discus-sions and for the opportunity to visit his group at the Johns Hopkins University inBaltimore.

I would like to thank Leen van Wijngaarden. With his knowledge and experiencehe gave good advice on numerous occasions.

I would like to thank the technical staff, Henni Scholten, Gert-Wim Bruggert,Bas Benschop and Joanita Leverink. They form the backbone of this group in techni-cal as well as in social respect. Special thanks goes to Marianne van der Linde. Shehelped me overcome many difficulties in the beginning.

I would like to thank Gerrit de Bruin. He helped me to understand what definesa good scientist.

Finally, I would like to thank all friends and colleagues who directly or indirectlycontributed to the completion of this work. This holds for the permanent staff Kovan der Weele, Devaraj van der Meer, and Michel Versluis, for the postdocs ClausDieter Ohl, Stefan Luther, Phillip Marmottant, and Marie-Caroline Jullien, for myfellow PhD students, Irene Mazzitelli, Judith Rensen, Anna von der Heydt, FlorineMeijer, Rene Mikkelsen, Mark Stijnman, and Manish Arora and of course also forthe undergraduate students.

Rudiger TogelEnschede, December 2002

102 ACKNOWLEDGMENTS

ABOUT THE AUTHOR 103

About the Author

Rudiger Togel was born on Mai 10, 1971 in Fulda, Germany. He grew up in Mar-bach, a small village near Fulda, and graduated in 1990 at the Wigbert GymnasiumHunfeld. In September 1990 he started his civil service at the Arbeiter-Samariter-Bund in Wiesbaden/Hessen, which lasted until December 1991. In March 1992 hebegan to study mathematics and physics at the Phillips University Marburg with theaim to become a teacher . After his Zwischenprufung (batchelor) in 1994 he took atutorial position for mathematics at the same university until 1996. In 1998 he got hismaster degree in mathematics and physics and accepted an offer from Prof. DetlefLohse to join him as a PhD-student at the University of Twente.

Documents

COnnecting REpositoriescore.ac.uk/download/pdf/11457159.pdf · The research described in this thesis was funded by FOM (Fundamenteel Onderzoek der Materie). The research was carried